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Total cross sections and elastic scattering at high energies
TOTAL CROSS SECTIONS AND ELASTIC SCATTERING AT HIGH ENERGIES
G.GIACOMELLI University of Bologna, Bologna, Italy, INFN, Sezione di Bologna, Fermilab, Batavia, Illinois
q~c NORTH-HOLLAND PUBLISHING COMPANY
—
AMSTERDAM
PHYSICS REPORTS (Section C of Physics Letters) 23, No. 2 (1976) 123—235. NORTH-HOLLAND PUBLISHING COMPANY
TOTAL CROSS SECTIONS AND ELASTIC SCATTERING AT HIGH ENERGIES G. GIACOMELLI University of Bologna, Italy, JNFN, Sezione di Bologna, Fermilab, Batavia, Illinois Received 18 December 1974
Single orders for this issue PHYSICS REPORTS (Section C of PHYSICS LETTERS) 23, No. 2(1976)123—235. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 36.00, postage included.
G. Giacomelli, Total cross sections and elastic scattering at high energies
125
Contents: 1. Introduction 126 2. Notations and general theorems 128 2.1. Variables of particle i 129 2.2. Relativistic invariants 129 2.3. Scattering amplitudes and cross sections 130 2.4. Lorentz transformations 131 2.5. Optical theorem 132 2.6. High energy theorems 132 2.6.1. Total cross sections. Froissart bound 132 2.6.2. Elastic cross sections. Consequences of the Froissart bound 133 2.6.3. Diffraction scattering 133 2.6.4. Eden and Kinoshita bound 134 2.6.5. Pomeranchuk theorems 134 2.6.6. Pomeranchuk-like theorems for elastic scattering 134 2.6.7. Relation between total cross section differences and charge exchange scattering 135 2.6.8. Lower bounds for total cross sections 135 2.7. Structure of the total hadron—hadron cross sections in terms of Regge poles 135 2.8. MacDowell—Martin bound 136 3. Experimental 136 3.1. The Serpukhov 76 GeV proton synchrotron 137 3.2. The separated-function proton synchrotron at Fermilab 138 3.3. The CERN—ISR 140 3.4. Secondary beams of particles 143 3.5. Charged particle identification 145 3.6. Neutral particle identification 146 3.7. Main features of high energy hadron—hadron interactions 146 4. Total cross sections 147 4.1. Introduction 147 4.2. Experimental methods 148 4.2.1. The transmission method 148 4.2.2. Direct measurement of the pp total cross section at the ISR 149 4.2.3. Bubble chamber measurements 150 4.2.4. Indirect methods for charged particles 150 4.2.5. Other methods 152 4.3. The shadow correction in deuterium 153 4.4. Energy dependence of the total cross sections 155 4.4.1. “Weak, electromagnetic and strong” cross sections 157 4.4.2. Energy dependence of the cross sections of the long-lived hadrons 159
4.4.3. Energy dependence of other hadron—hadron cross-section 165 4.5. Relations among hadron —hadron cross sections 165 4.5.1. Total cross-section differences 165 4.5.2. Structure of the total cross-section differences in terms of Regge poles 167 4.5.3. Relations from the quark model 169 4.5.4. The Johnson—Treiman relations 170 4.5.5. p and w-universality 171 4.6. The energy dependence at asymptotic energies 172 5. Elastic scattering 173 5.1. Introduction 173 5.2. The Coulomb-Nuclear Interference Region 2] 173 [0.001 < Iti <0.01(GeV/c) 5.3. Elastic scattering in the diffraction region [0.01 < ti <0.5(GeV/c)2j 179 5.3.1. pp—* pp 181 5.3.2. Other types of elastic scattering 183 5.3.3. The cross-over phenomenon 183 5.4. Scattering at large angles 186 5.4.1. pp * pp 187 5.4.2. Other types of elastic scattering 189 5.5. Backward scattering 193 5.6. Total elastic cross sections 196 6. Charge exchange scattering and K~regeneration 199 6.1. The np —~ ir°nreaction 200 6.2. Other charge exchange scattering processes 203 6.3. K~regeneration 204 7. Polarization 205 7.1. Polarization in i~~p elastic scattering 205 7.2. Spin effects in pp scattering 207 7.2.1. ‘tot 207 7.2.2. The P parameter in pp —+ pp 208 7.2.3. pp elastic scattering in pure spin states 208 7.3. Conclusion 209 8. Dispersion relations for the forward scattering amplitude 210 9. Impact parameter expansion 212 10. Theoretical models 217 10.1. Amplitude analyses for irN -+ irN scattering 217 10.2. Thermodynamic or statistical models 219 10.3. Regge-pole models 219 10.4. Constituent models 221 10.5. Shadow scattering models 221 10.6. Optical models 223 10.7. Mixed models 226 11. Conclusions 227 12. Acknowledgements 227 References 227
126
G. Giacomelli, Total cross sections and elastic scattering at high energies
1. Introduction Inthis review we shall summarize the present experimental and phenomenological situation of high energy total cross sections and elastic scattering measurements. Charge exchange scattering and polarization phenomena will be mentioned in the context of their relations to total cross sections and elastic scattering. We shall also make a few remarks on photon—hadron, electron—positron and neutrino—hadron total cross sections. Only high-energy scattering will be considered, where we define high energy as greater than 10 GeV in the laboratory system. This number will sometimes be brought down to 5 GeV when lack of data or continuity reasons will require us to do so. In any case this means that we shall be considering experiments from only a few accelerators. Two new high-energy accelerators became operational three and two years ago respectively. The first is the CERN Intersecting Storage Rings and the second is the Fermilab Separated Function Proton—synchrotron. During this time these accelerators have provided a wealth of new information in a new energy region. It is thus appropriate to review elastic scattering and total cross-section measurements at this time. The measurements of total cross section and of elastic scattering processes are among the first measurements to be performed when a new accelerator opens up a new energy region; both types of measurements are repeated, as time goes by, with improved accuracy, finer spacing and, for elastic scattering, over a larger angular range. From an experimental point of view the measurements of total cross sections and of elastic scattering in the diffraction region are relatively easy to make; the difficulties arise when one tries to achieve the highest precision or when one wants to measure the differential cross sections at large angles. From a phenomenological point of view, both the total and elastic cross sections are simple processes with a relatively straightforward analysis. Even crude measurements give an immediate idea of the main features of the type of interaction under study in the energy region considered. Most of the systematic total cross-section measurements for ir~p,KIN, pIN, and nN have been performed with the transmission method, which is capable of high precisions. The recent measurements at IHEP (Serpukhov) and at Fermilab achieved point-to-point precisions of ±0.2%, with a systematic scale error of about ±0.4%. The recent construction of high-quality high-energy beams of tagged photons, of neutrinos, of hyperons, etc., have allowed an extension of the total crosssection measurements to these particles. Total cross sections depend on one kinematic variable, the energy, and on the type of colliding particles. Thus, we shall be interested in (i) the energy dependence of the total cross sections, and (ii) the relations among various cross sections. The differential cross section for the elastic scattering of unpolarized particles from unpolarized targets depends on two variables: the energy and an angular variable which is usually chosen to be the square of the four-momentum transfer, t. One may divide the angular distributions into five regions: (i) The Coulomb region (for Iti ~ 0.00 l(GeV/c)2). (ii) The Coulomb-nuclear interference region for 0.001 ~ It < 0.01. Measurements in this region are used to obtain information on the ratio of the real to imaginary part of the forward scattering amplitude. (iii) The diffraction region proper (for 0.01 ~ ti ~ 0.5). The most important parameter here is the slope or the slopes of the diffraction pattern.
G. Giacomelli, Total cross sections and elastic scattering at high energies
127
(iv) The large-angle region (for ItI> 0.5) is characterized by the smallness of the cross section and by the presence of a dip-bump structure which resembles that coming from diffraction from an opaque disc. Point-like constituents within the nucleon should radically affect the large-angle cross section at high energies. (v) The backward region for scattering close to 180°(not for pp). Here the cross sections are very small and all have a bump at u = 0. The experimental results from IHEP, TSR, and Fermilab have brought about a considerable change in the phenomenological picture of strong interactions at high energies. The change is summarized in fig. 1.1. The three sketches at the top illustrate what most of us would have expected for the total cross sections, for the ratios of the real-to-imaginary part of the forward scattering amplitudes (p), and for the elastic scattering differential cross sections at asymptotic energies. The expectations, based on the available experimental data and on the then prevailing theoretical models, were the following: (i) The total cross sections of particle on protons and of antiparticles on protons [atot(xp) and a505(~p)]should have decreased toward a common, constant, non-zero value as the energy increased. (ii) The ratios p should have increased toward zero from below. (iii) The differential cross section for proton—proton elastic scattering in the diffraction region should have continued having an almost exponential behaviour in t, with a slow increase of the slope as the energy increased. (iv) The ratios between various total cross sections should be governed by some internal symmetry. The total cross-section measurements performed in Serpukhov (1969—1972) showed first that some cross sections were not decreasing as fast as expected and later that the K~ptotal cross section was increasing with energy. In 1973, measurements at the CERN ISR showed that the pp total cross section was also rising with energy. In reality previous cosmic-ray measurements and do df
TOT!
PP—PP
1968 0.05
BLAB
0TOT
,,
-~
~.
o~ ,,~LAB
~~—-—~
BLAB
dt ~
1975
0.15
i~~T
Fig. 1.1. Illustration of the expectations of high-energy behaviours in 1968 and 1975.
1 28
G. Giacomelli, Total cross sections and elastic scattering at high energies
calculations had indicated rising pp total cross sections, but this was not taken very seriously. In 1972 measurements of the differential cross section for pp elastic scattering at the CERN ISR showed that the diffraction pattern had a break at ti 0.13 (GeV/c)2, a minimum at about 1.5 (GeV/c)2 and continued shrinkage. In 1972—1973, measurements at the ISR and at Fermilab of the ratio p for pp scattering indicated that p was becoming positive. The measurements of the total cross sections and of differential elastic cross sections at Fermilab completed the change in the overall picture as sketched in fig. 1 1. Present experimental results and the predictions of optical model calculations lead us to expect the following high-energy behaviour: (i) All total hadron—hadron cross sections will rise with an approximately (1n2 s)-dependence, while the differences a~ 0~(~p)u~0~(xp) decrease toward zero. (ii) The ratios p of the real-to-imaginary part of the forward scattering amplitude should become positive, reach a positive maximum, and then decrease toward zero. (iii) The diffraction pattern for pp elastic scattering will continue to shrink;Eat theabout diffraction 2 and a minimum pattern have whose a change of slope at it ~ in 0.13 iti 1.5should (GeV/c)2] location decreases iti (GeV/c) as the energy increases. No anticipation is however possible about the very large angle cross section, which is expected to be sensitive to hadron substructures. It may be interesting to note that our energy scale also changed. For instance, what is often called the beginning of the high-energy region (or of the asymptotic region), in the early 1960’s used to be at a few GeV; in the late 1960’s it increased to several tens of GeV; now it is more of the order of hundreds of GeV if one assumes that the asymptotic region starts after all total cross sections have started rising. It may be of the order of thousands of GeV if one assumes that the asymptotic region begins when the ratios of real-to-imaginary parts of the forward scattering amplitude have become positive and start decreasing towards zero. As far as the theoretical interpretations of high-energy scattering, there is no theory in the real sense of the word, i.e., there is no mathematical formalism that can give numerical predictions of the experimental quantities via well-defined and consistent procedures. Most of the theoretical work is done with specific dynamic models, like Regge pole models and optical models. We shall discuss some of their implications together with those of high-energy theorems. The notations used as well as the implications of some high-energy theorems and of some models are described in section 2. In section 3 are discussed the main features of the three highest energy accelerators, from which come most of the experimental results relevant to this review. The main discussions on total cross sections and elastic scatterings are in sections 4 and 5 respectively, while some connections with charge exchange scattering are discussed in section 6. Spin effects are briefly considered in section 7. The phenomenological discussion continues in sections 8 and 9, where the implications of dispersion relations and of optical models are considered. Section 10 gives an overall view of the interpretation of high energy total cross sections and elastic-scattering data. .
—
2. Notations and general theorems This section summarizes the notations and some general relations used in this review. The notation followed will be that of the Landolt—Börnstein data compilation [73C5] and concern the
G. Giacomelli, Total cross sections and elastic scattering at high energies
1 29
general two-body reaction l+2-÷3+4.
(2.1)
In elastic scattering one has: particle I = particle 3 and particle 2 particle 4. In the laboratory system the target 2 is at rest; particle 1 will be referred to as the incident particle, 3 as the scattered particle, and 4 as the recoil particle. Quantities defined in the center-of-mass system (c.m.) will be denoted by an asterisk, whereas quantities in the laboratory system are asterisk free. 2. 1. Variables of particle i (i = 1, 2, 3, 4) The notations used are listed in table 2.1. The c =
=
h/2ir = 1 system of units is used.
Table 2.1 Variables of particle i (i = 1, 2, 3,4) Center of mass system momentum (3-dimensional) kinetic energy total energy
~$
4-momentum scattering angle
P7 = (E7, 07
rest mass solid angle velocity/velocity of light transverse momentum
m1
Laboratory system
T$ E7
T. E 1
~$)
P1 = (E1,
~)
mf
n~’ 137
The following relations hold (in c.m. or lab.):
Pt 2 — p2 = E T=E m.
=
2.2. Relativistic invariants Convenient kinematical variables for the scattering processes are the s, t, u Lorentz-invariant quantities (Mandeistam variables). We shall define these quantities and express them in terms of lab or c.m. quantities: Total c.m. energy squared s
=
(P
2
=
(Er ÷ Er)2
1÷ P2)
=
m~÷ m~+ 2E
2 1m2
=
+
(m1 ÷ m2)
2T 1m2.
(2.2)
Four-momentum transfer squared between particles 1 and 3 2 = m~÷ m~ 2ErEr ÷2p~p~cosO~ = m~ + m~ 2E t = (P1 P3) 1E3 + 2p1p3cosO3 2 = m~+ m~ 2ErE~+ 2p~p~’cosO~’ = (m 2 2m = (P2 P4) 2 rn4) 2(E4 m4). —
—
—
—
—
(2.3)
—
—
Four-momentum transfer squared between particles I and 4
—
(2.4)
130
G. Giacomelli, Total cross sections and elastic scattering at high energies
u
=
2
(P1
—
=
m~+ m~ 2E?E —
2 = m~+ rn~ 2ErEr —
=
(P2
—
+
2p7p~cosO~ = m~ ÷ m~ 2E
+
2p~p~cosO~’ = (m
—
P4) P3)
2 2
—
rn3)
—
1E4 + 2p1p4cosO4 2m 2(E3 m3). —
(2.5) (2.6)
These three invariants satisfy the relation s+t+u~m~.
(2~7)
For elastic scattering
Pi*=
*=
P2
*—
*=
*
P3P4—P,
Er=E~,
Er=E~,
(2.9)
and the expressions for t and u may be simplified: 2 sin2.~O~c = —2m t = —4p” 2(E4 m4) = —2m2(E1 2(I + cosOr) + (m~ m~)2/s. u = _2p* At high energies —
—
E3)
—
(2.10) (2.11)
(2.12)
2.3. Scattering amplitudes and cross sections The elastic scattering amplitude for irN scattering will be written in the formt f(E*,O*)g(E*,O*)+i~ñh(E*,O*),
(2.13)
where g and Ii are the non-spin-flip and spin-flip amplitudes, respectively,a represents the Pauli-spin matrices and ,i is a unit vector perpendicular to the scattering plane. The differential cross-section for elastic scattering is given by
=
if(E*,
O*)12
=
Ig(E*, O*)12
+
ih(E*, O*)i2.
(2.14)
Using relativistic invariants one may write (c~)
dt
=
el
1 2 iF(s, t)i2’ 641r5P*
= ~.
1 2 F(s, t)i2 l6irs
,
(2.15)
where F(s, t)
=
81
/,~f(E*,0*),
(2.16)
tFor nucleon—nucleon scattering one has to add to add a spin—spin term. At high energy this term has not been measured. It is presumed to be small and is often neglected (see section 7).
G. Giacomelli, Total cross sections and elastic scattering at high energies
(~fl
=—~---(---~----~
‘~dt/et p*2\dfZ*J
1 31
(2.17)
.
The polarization parameters P, R and A are defined as 2Im(gh*) Igi2+ hi2
(2.18)
R~
I 12—1h12 Ig12+ hi2
(2.19)
A
2 Re(gh*) 1gi2+ hi2
(2.20)
2.4. Lorentz transformations The Lorentz transformations between c.m. and lab frames of reference for the momentum and energy of particle i may be written as: p 5p,sin01p~sin07
(2.21)
p, cos 0~= 7cm(Pt COS Or ÷f3cmE7)
(2.22)
E1 =
7cm(E7 +
I3cmPr cos Or),
where ~cm
=
7cm
=
p1/(E1 + m2)
Ifs/I
—
1~~m = (E1 +
(2.23) m2)/s/i
(2.24)
The following relation is often useful =
p~= p~m2/sJ~
(2.25)
In the case of elastic scattering the relation between the angles O~and 07 with i = 3, 4, is given by tg01tg~07.
(2.26)
m
The elastic differential cross sections da/dt, da/d~2*and do/d~2are related by da 7r 2 dt, p7
da ir 1/~/m, \2 d~27 p~ iil—u----—sin0.i Y \rn 2 ‘I
da d~2,
.—.
(2.27)
G. Giacomelli, Total cross sections and elastic scattering at high energies
1 32
2.5. Optical theorem Neglecting spin—spin effects, the imaginary part of the non-spin-flip forward scattering amplitude may be related to the total cross-section [h(E, 0) = 0]: p* Img(E*, 0) =—a~O~.
(2.28)
4ir
Hence, Idu\
=
da
diti (—)
p
~
iRe g(E*,
—~
p*2 a~
0)12+
1
(2.29)
2+__. I6ir
(2.30)
iReg(E*,0)i
At high energies, where the ratio p good approximation
=
Re g(E*, 0)/Tm g(E*, 0) is presumably small, one has to a
(~)
2cJ~ 10m~eV~21
5.095 X l0
2.
(2.31)
0~(mb)
2.6. High energy theorems On the basis of the general properties of unitarity, analyticity and crossing, a number of highenergy theorems have been derived for the limit s oo~We shall briefly discuss here their implications for total cross sections and elastic scattering at very high energies. In this context the asymptotic region may be defined as that region where the number of open channels is very large and the cross section for each channel is very small. In practice, if one assumes that the asymptotic region starts when all total cross sections have started to rise, then it cannot start below a laboratory momentum of 300 GeV/c. In terms of the ratio p of the real-to-imaginary part of the forward scattering amplitude, the asymptotic region starts when all ratios have become positive and have started decreasing toward zero. This presumably happens at laboratory momenta larger than a few thousands of GeV. It is, therefore, not clear if the analyses of the present experimental results in terms of high-energy theorems are really meaningful. In the following, the high-energy theorems will be stated without any proof. For a more thorough analysis, the reader is referred to theoretical reviews of the subject [72R11. -*
2.6.1. Total cross sections. Froissart bound From positivity, analyticity, and unitarity, it is possible to derive the following upper limit [61F11: ~
(2.32)
G. Giacomelli, Total cross sections and elastic scattering at high energies
133
where ir/m~~ 60 mb and the unknown constant ~ fixes the energy scale (it may be the energy value above which the cross section rises).* 2.6.2. Elastic cross sections. Consequences of the Froissart bound It is possible to show that (i) if Gtot ln~s, then Gei 1n2 s and thus tlei/Utot const; (ii) if ~ ln s, then Uel/Otot ~ anything. For energies smaller than S~it would seem that case (ii) applies, while for s ~ s 0, one should have case (i). For energies around s~or slightly larger, the situation is more complex. 2.6.3. Diffraction scattering One may define the width of the diffraction peak F as 0•i
(2.33)
F(s, 0)/s 2s then F ~ 1n2s and If ln F(s,t) = fit/ ln2 — s\ lim = ftr), s—3.=F(s,0) \
(2.34)
50/
that is the limit is a function of r = t 1n2(s/s 0) only. This and other considerations lead to the expectations that the differential cross section cannot be of the form 2). (2.35) du/dt = A exp(Bt + Ct An example of an acceptable form is the one proposed by Cheng and Wu E71C2]: F(s, t)
=
a,’.
isJ
2.
(2.36)
0(V’~ilns)(lns)
General ideas on analyticity in the laboratory energy E and the crossing property allow one to write for the even (F4) and odd (F) signature amplitudes at high energies: i 4 E(lg E — ~ iir)13~ 7 F(E)~a‘y’~E(lgEf ~i3’y(lgE)’3~. F4(E)
(2.37)
—
If
ln~sone has ~ Re F(s, 0) ImF(s,0)
=
2; moreover, if F ~ F4 one has
Re F’~(s0) ImF4(s,0)
iT
(2.38)
lgE
that is at very high energies, the ratio of the real-to-imaginary part of the forward scattering amplitude is positive and decreases towards zero. *The presence of the pion mass in the Froissart bound may be thought to arise from the tail in the Yukawa distribution (~1/2mg) which becomes more and more important as the energy increases.
G. Giacomelli, Total cross sections and elastic scattering at high energies
134
2.6.4. Eden and Kinoshita bound Idcr\
Is \
2~_) < (const)u~ t=o s—~= 1ln so
(2.39)
iT
I—i
< —1n41—I. a—4’.’. #2o \S /
~ t—+O
(2.40)
-~
0
2.6.5. Pomeranchuk theorems First Pomeranchuk theorem (Okun—Pomeranchuk rule). In the limit of high energies, the total cross section on the same target particle of particles belonging to the same isospin multiplet become equal: Otot(lrp)
=
a~05(?p)= a~0~(ir°p)
a50~(pp)= cr~05(np) 4p) = u 4n)= o~ a~0~(K 50~(K 0~(K°p) = a~05(K°n).
(2.41)
Second Pomeranchuk theorem. At high energies if the difference between particle and antiparticle cross section has a limit and if [(F P)/(s ln s/s0)] -÷ 0 ass oo, the total cross sections on the same target particle of particle and antiparticle become equal —
-+
a~0~(pp) = Utot(PP) a105(K~p)= a~01(Kp),etc.
(2.42)
One can also guess how the “Pomeranchuk” region is approached, that is one may guess the energy dependence of the difference in the total cross sections. From formulae (2.37) one has: a10~(xp)
a~
—
Gtot(XP)
13
+ /2
< const(ln E)
—
01(xp)+ a501(xp) 2 const/lns. (2.43) and if atot (lns) The available data on total cross section differences agree better with a power dependence of s (see sections 2.6.7 and 4.5). 2.6.6. Pomeranchu k-like theorems for elastic scattering They may be expressed by the following limits as the energy goes to infinity: rdu(xp)/dt~ lim i i= 1 Lda(xp)/dtJ
(2.44)
lim F
(2.45)
=
0
a—’
lim [F(xp)/F(i~p)]
=
1
(2.46)
G. Giacomelli, Total cross sections and elastic scattering at high energies
lim [Ue1(XP)/Gei(itP)]= 1.
135
(2.47)
These relations express the general statement that the differential cross sections of particle and antiparticle scatterings on protons tend to become equal as the energy increases. 2.6.7.
Relation between total cross-section differences and charge exchange scattering 4p)i <—\/a(irp-# iT°n) ln[ vT~
-
a~0~(ir ~)
—
m
s-’
a~01(iT
2
___
(2.48)
-
s0a(1rp~~* 7T0n)
5.
4n) a —
a~05(K
105(Kn)i~
Ju(K~p-~ K~p)ln[2(K
K~p)]~
(2.49)
The usual parametrization =
Ap~ = Bs~
(2.50)
for total cross-section differences at high energies would eventually violate the above bounds. 2.6.8. Lower bounds for total cross sections Jim—Martin
~
> ~-+~
Cornille—Martin
a~~(s)~
6
s (lns/s0)
(2.51)
2
const
(2.52)
2.7. Structure of the total hadron—hadron cross sections in terms of Regge poles In the framework of a simple Regge pole model, the total hadron—hadron cross sections may be expressed as a sum of contributions from a few basic trajectories: the vacuum trajectories P (Pomeron) and P’, and the trajectories with the quantum numbers of the p, w, A2 and f2 mesons. For the irN, KN, pN, and pN total cross sections we have: fit ot(1T~P)= a~(7TN)T a~(irN)+ (J~i~(1rN)
a~0~(K~p) = u~(KN)~ a~(KN)~ a~’.,(KN)+ cA2(KN)
f(KN)
(2.53)
a~~~(KN)
(2 54
+ api
~tot(P~P) = a~(NN);u~(NN);a~(NN)+ aA2(NN) + a~0~(K~n) = a~(KN)±c7~(KN);a~(KN) aA2(KN)
+
atot(p~n)= u~(NN)±a~(NN)~ a~(NN)—aA2(NN) + UP? ~(NN). The P, P’, A2 and f2 contributions to particle and antiparticle cross sections are the same. Each term of (2.53, 2.54) may be written in the Regge pole form u.(ab) = B~bp°~’’
(2.55) (2.56)
G. Giacomelli, Total cross sections and elastic scattering at high energies
1 36
where p is the laboratory momentum, i= P, P’, p, w, A2 and f; a, bare the incident (irk, K~,~ and target particles (p, n). The form (2.56) implies factorization of the residues. 2.8. Mac Dowell—Martin bound The slope of the forward elastic peak at t
=
0 may be defined as
Ed ln(da/dt)1 B0 = [ dt i~=0 The Mac Dowell—Martin bound [74B14] states that 1
2
U~0~
B0 > — —i--
-
-_
-~
1 ~ot —i--.
(2.57)
(2.58)
3. Experimental The progress in the field of subatomic phenomena in general and of elementary particles in particular has always been connected with the development of new accelerators of ever increasing energies. They allowed the production of secondary particle beams of higher energies and of beams of altogether new particles. The experimental results which will be reported in this review come from a limited number of accelerators: the 12 GeV Zero Gradient Synchrotron (ZGS) at Argonne National Laboratory (ANL), the 28 GeV Proton Synchrotron (PS) of the European Organization for Nuclear Research (CERN), the 33 GeV Alternating Gradient Synchrotron (AGS) of Brookhaven National Laboratory (BNL), the 40 GeV Stanford Linear Accelerator (SLAC), the 76 GeV Proton Synchrotron of the Institute for High Energy Physics (IHEP), the 200—500 GeV Fermi National Accelerator Laboratory Proton Synchrotron, and the CERN Intersecting Storage Rings (ISR). These will soon be joined by the CERN 400 Proton Synchrotron. Table 3. 1 summarizes the main features of these accelerators. The table gives the main parameters of each accelerator, like the maximum energy, the energy at which the accelerator is usually operated, the average intensity per pulse, and the repetition rate. In addition it gives two parameters connected with the operation of the accelerator for physics: these are taken as the total number of “Stations” at which experiments may be performed and the number of “Stations” which can operate at the same time E 74S I]. The experiments that can be performed at an accelerator depend critically on the quality and quantity of the available secondary beams. In a conventional fixed-target accelerator the beams may originate either from an internal target placed in the circulating beam or may come from a target placed in an extracted beam. While the first system was used extensively in the past, the trend for the future will be towards targets in external beams. The main reason for this trend is connected with problems of radiation damage, which become serious when internal beam intensities exceed 1013 particles per pulse. We shall briefly recall some of the features of the three highest energy accelerators, at IHEP Fermilab, and CERN, from the point of view of an experimenter, neglecting all technical details.
G. Giacomelli, Total cross sections and elastic scattering at high energies
137
Table 3.1 Some features of the high-energy accelerators with E> 12 GeV. The average intensity, the number ofbeams, and the statistics on the nuinber of experiments refer to the situation in the Summer of 1974 and are only approximative.* Laboratory
ANL
CERN
BNL
Accelerator Maximum energy (0eV) Normal energy (GeV) Average intensity (10’ 2 particles/pulse) Repetition rate (pulses/sec) Burst length (sec) Total number of stations Stations served at same time
ZGS 12.5 12 3
PS 28 23
AGS 33 28.5 5
5
2
IHEP
Fermilab
CERN
SLAC
—
—
ISR
—
76 70
500 300 8
5
1/2
1/2
1/7
0.7 17
0.5 17 + ISR
1 10
10
13
13
ISR
6
5
+
*For the CERN—ISR we quote the luminosity L
L
1/7
1—1.5
in cm2
2062 500—1500
0.8 20
=
22.1 20 40 ~zA
3’
i0
360
Continuous 8
10
8
1.67 X 106 7 6
CERN
SPS 400 200—400 —
—
—
—
sect and the number of intersecting regions (8).
3.1. The Serpukhov 76 Ge V proton synchrotron The 76 GeV proton synchrotron of the Institute for High-Energy Physics (IHEP) at Serpukhov, USSR, is an alternating-gradient, strong-focusing accelerator with a 200 MeV linear accelerator as an injector. The accelerator usually runs at 70 GeV, with a repetition rate of one burst every 7 sec. The
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~
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8Con,
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p—Fast
28 Beam 2 25 2A
0iot
Beam 14
‘~
P(,r±p~~~_,r±p) ~r-p---AAn Fig. 3.1. Beam layouts at the IHEP accelerator.
Target Area
138
G. Giacomelli, Total cross sections and elastic scattering at high energies
circulating proton beam averages about 5 X 1012 protons. The present beam layout is shown in fig. 3. 1. There are seven beams of pions, kaons, and nucleons, fed by separate targets which intercept the internal proton beam. Of these secondary beams, four are negative beams for counter experiments, one neutral, and a test beam. Moreover there are an external slow-extracted proton beam, from which originate two secondary-charged beams, and two fast-extracted proton beams from which originate the pions which eventually decay into neutrinos and an RF separated beam for bubble chambers. There are also internal-target facilities. 3.2. The separated-function proton synchrotron at Fermilab The separated-function proton synchrotron of the Fermi National Accelerator Laboratory in Batavia, Illinois (Fermilab), has four acceleration stages [74W1]: (i) a 750 keV Cockcroft—Walton electrostatic accelerator; (ii) a 200 MeV linear accelerator; (iii) a 10 GeV booster fast-cycling proton synchrotron; (iv) the main ring proper, 1 km in radius. This is the first accelerator where deflection and focusing are done by separate magnetic elements, i.e., Particle Beams and Facilities at Fermilab S,ngIo-Arn, Spectrom,te,
——
_..~M38M4
Meson Areg_.—~~M2
Booster
b~L~~oc~
Area ~
‘.MI~rom.torC=flter
~
Area
Eo,t
‘.
P1
P2
Prima, S
5 Beam Secondary Boom PrimoryTorgetLocotion
Meson Area Ml General- Purpose Charged Particle Beam M2 Diffracted Proton Beam (Sometimes Pions) M3 Neutral Beom (Neutrons) M4 Neutral Beam (Koons and Neutrons) M6 Charged Particle Beam
Neutrino Area NO Ni N3 N5
Neutrino Beam Muon Beam (Sometimes Pions) Charged Beam to 30’ Bubble Chamber Charged Beam to 15’ Bubble Chamber
Proton Area Pt
Neutral Beam (Photons or Neutrons)
P2 Tagged Photon Beam (Sometimes Electrons) P3 Charged Particle Beam
Fig. 3.2. Diagram of the experimental beams available at Fermilab in the summer of 1974 [75G1].
G. Giacomelli, Total cross sections and elastic scattering at high energies
139
Flat Top
:
/
Le3lon
Time After Last Cycle tn Seconds
~
1.0
Proton Beam Use d. Internally
: Injection
-
2”,
Beam Extracted
—_.~
/
B
\
I
~-Slowly
\
/
/
I
~
.2
for Counter
Experiments
/ /
.
l’—.’
0
1
2 3 4 Time After Last Cycle in Seconds
Fast Extraction a— for Bubble Chamber Exp.j t I 5 6
Fig. 3.3. Energy and intensity cycles at the FNAL accelerator [74S 1].
bending magnets and quadrupoles, respectively. At lower energy accelerators, both functions are performed by bending magnets with pole pieces of appropriate shape. Fig. 3.2 shows the general layout of the accelerator. The fast-cycling booster synchrotron is capable of accelerating protons to 10 GeV, 15 times per sec; 13 of these pulses are injected into the main ring from t = 0. 1 to t = 0.8 sec of the main cycle time, as shown schematically in fig. 3.3. After injection the magnetic field rises, the protons are accelerated by a RF cavity at the rate of about 100 GeV per 0.8 sec. During the acceleration cycle a small fraction of the protons may be intercepted by a gas jet, or thin targets of various shapes, for experiments using directly the proton beam of the accelerator. At the end of the accelerating cycle, the magnetic field stops rising and is kept at a constant value for about one second, during which time the internal beam is extracted partly in a short time and partly during 0.8 sec. The fast-extracted beam is used by bubble chambers and for neutrino experiments, while the slow-extracted beam is used by counter experiments. There is only one extraction point from the main ring, from which protons come out into one beam line. The external proton beam is then split into two, at the first splitting station, by a combination of electric and magnetic fields. After subsequent splits one has the possibility of operating six primary proton beams in addition to the internal beam (fig. 3.2). This allows four experimental areas to be run simultaneously: (i) The Internal Target Area utilizes directly the internal proton beam during its acceleration cycle. Here one can perform proton—proton and proton—light nuclei collisions at all energies from 20 to 500 GeV.
G. Giacomelli, Total cross sections and elastic scattering at high energies
140
(ii) The Meson Area is fed by a single 300 GeV proton beam which hits a beryllium target from which originate five secondary beams, three charged and two neutrals, which are all used simultaneously. Here are performed counter experiments with secondary beams of pions, kaons, nucleons, and hyperons. (iii) The Proton Area is in reality a complex of three areas, each with an independent primary proton beam. Here are performed counter experiments with primary proton beams; there are also secondary photon-electron facilities. (iv) The Neutrino Area has a more complex situation designed to satisfy the requirements of bubble-chamber, neutrino, and muon experiments. The area may be subdivided into at least two areas: the muon area, where a muon beam and a dichromatic neutrino beam are utilized, and a bubble-chamber area where one can have secondary hadron beams for the two bubble chambers and neutrino beams of different momentum bands for use with counters and bubble-chamber experiments. Table 3.3 gives a list of the major experiments performed at Fermilab which yielded results relevant to this review. 3.3. The CERN—ISR Experiments at a storage-ring machine are closely connected with the machine itself, much more so than for conventional accelerators [74G3]. The Intersecting Storage Rings (ISR) of the European Organization for Nuclear Research (CERN) consist of two concentric and slightly distorted rings, each 300 m in diameter, fig. 3.4 [73J 1]. The two rings intersect horizontally eight times, with a crossing angle a = 14.8°.They are filled in succession by pulses of about 1012 protons ejected from the CERN 28 GeV proton synchrotron. In each ring every pulse is “stacked” at a slightly different horizontal location from the previous one. After filling, each circulating beam has the form of a ribbon with approximate transverse dimensions of 3—5 cm wide, horizontally, and 0.4—0.7 cm high, vertically (see fig. 3.5). Currents of up to 30 A have been stacked. The momentum spread of each beam is approximately ±1%. Owing to the small intersecting angle, the interaction region has a “diamond” shape, approximately 50 cm long, 5 cm wide, and 0.4 cm high. When operated in the “Terwilliger mode” the transverse dimensions of the beams, and thus of the diamond, become considerably smaller, about 20 cm long, 3 cm wide, and 0.3 cm high. Table 3.2 gives the main parameters of the ISR. The ISR is presently operated at five energies, which cover the range between 23 and 63 GeV in the center of mass. The top energy of 63 GeV, which requires acceleration of the protons in the ISR itself, corresponds to the energy of a 2062 GeV conventional accelerator. At present this is the highest accelerator energy available. At each intersection region the total collision rate N is given by N—Lcj~0~, (3.1) 0tot is the proton—proton total cross section, and L is the luminosity. L is connected to the where parameters of the machine by the following equation 1112
e2ch~~~tg(a/2)
(3.2)
G. Giacomelli, Total cross sections and elastic scattering at high energies
T12
TTI
Ii
/ 113
//
~
ISR
H
~4.
Fig. 3.4. Scheme of CERN PS and ISR.
BEAM-BEAM DIAMOND
Fig. 3.5. Horizontal distribution of 65 000 reconstructed events. Also shown are the single beam profiles [71 Hi I.
141
G. Giacomelli, Total cross sections and elastic scattering at high energies
142
Table 3.2. Some parameters of the CERN ISR and present operation for physics Beam momenta Crossing angle Beam losses Average pressure Pressure in intersections Design luminosity Transverse beam Jhorizontal
11—31 GeV/c, s~sp/p 2% 14.8° ‘~- 0.1—1% per hour < 10-1 1 torr ~ i0-’ 2 torr 2 sec~ 4— X3—5 1030 cmcnc
dimensions
—
[vertical
Beam momenta (0eV/c)
cm energy (0eV)
s = F*2
11.8/11.8 15.3/15.3 22.4/22.4
23.4 30.4 44.4 52.6 62.3
26.5/26.5 31.4/31.4
0.4—0.7 cm
(0eV2)
Equivalent lab. momentum (0eV/c)
Present average luminosity (10~° cm~seC1)
548 924 1971 2767 3881
291 491 1053 1474 2062
2 4 10 15 3
In addition there is a low-fl section in which one may achieve luminosities of (3—4) X i0~’cm2 sect
where I~and ‘2 are the proton currents in the two rings, c is the velocity of light, e is the electron charge, a = 14.8°is the crossing angle, and heff is the effective height of the overlap of the two beams. heff is defined in terms of integrals over the direction (vertical) perpendicular to the plane of the two beams: 1
=
fp 1(z)p2(z)dz
heff
(3 3)
fp1(z)dzfp2(z)dz
Here Pi and P2 are the beam densities as a function of z, the vertical coordinate. The measurement of the luminosity is one of the most crucial at the ISR, since on it depends the knowledge of the absolute values of the measured cross sections. Inasmuch as all parameters in eq. (3.2) except heff are known or measured to better than 0.1%, the determination of heff becomes the most delicate task in measuring L. heff may be measured by a number of methods. The direct measurement according to formula (3.3) requires the measurement with spark or proportional chambers, first of the densities of one beam (Pi and P2, by recording secondary particles originated from the collisions of one beam against the residual gas), and then of events coming from beam—beam collisions (Pi P2)- At present this method does not yield high accuracies, because of the small vertical dimensions of the beams. A method which is more widely used is the Van der Meer method [68Vl} of recording the event rates in some monitors, while displacing the two beams vertically from one another. In this way, one measures directly2 the overlap height heff. sec’effective has been obtained; since a~At the c.m. energy of 53 GeV a luminosity L 2 X 1031 cm 0~(pp) 40 mb, one has 9 X I 0~interactions per second in each intersection region. At Ecm = 53 GeV the average multiplicity of the produced particles is 18 [74R 11; thus about 1.5 X 1 0~particles per second are produced at each intersection.
G. Giacomelli, Total cross sections and elastic scattering at high energies
143
Table 3.3 List of ISR and Fermilab experiments which provided data relevant to this review*. For the results from other accelerators see the Landolt—Bornstein compilation [73C5] and the proceedings of the latest conference [74D6l, [75W1 I. Experiment number
Collaboration
Process studied
c.m. energy (GeV)
i-range 2 (GeV/c)
R401 R601
CERN—Hamburg—Orsay—Vienna CERN—Rome
pp -~ pp
0.8—3 0.001—0.014
R602 R801
Aachen—CERN—Genova—Harvard—Torino
23, 63 23,31 23,31,45,53 23,31,45,53 23,31,45,53,63 lab, momenta 20—200 30—280 50200
1A 4 7 21A 36 96 104 111
Pisa—SUNY Fermilab—Harvard—Pennsylvania—Wisconsin Berkeley—Michigan ANL—Fermilab—Indiana—Michigan Caltech—Fermilab Fermilab—Dubna—Rochester—Rockefeller ANL—Bari—Brown— CERN—Cornell—FL—etc. Brookhaven—Fermilab—Rockefeller Caltech—Berkeley
pp—opp °tot(PP) pp—’ pp °tot(PP)
o~ot(vN) otot(np) (iT~p,K~p,P~P)eI at 0t(vN) pp -+ pp (iT~p,K~p,P~P)el atot(1r~,K~p,pip) 1rp—olr°n
40—100 50—400 50—150 35 —280 21—101
—
0.05—3.0 —
— —
0.13 —
0.001—0.09 0.1—1 —
0—1.3
*Not listed in the table are the bubble-chamber experiments. Furthermore, it is expected that experiments R604, R805 at the ISR and 8, 12, 25A, 69, 82, 97, 177A, 198A, 248, 313 at Fermilab will soon yield further results relevant to the subjects of this review.
The decrease in the circulating beam intensities is between 0.1% and 1% per hour; it is then possible to keep useful beams for physics experiments up to 40 hours without “restacking”. Such conditions were achieved mainly because of the extraordinary vacuum in the beam pipes, which averages about 10_li mm Hg in the whole accelerator and is kept below 10_12 mm Hg in the intersecting regions. Finally, it should be noted that since the two beams collide at a small angle (14.8°)the ISR laboratory system is, to a good approximation, equal to the center-of-mass system. Table 3.3 gives a list of the experiments performed at the ISR which yielded results pertinent to this review. 3.4. Secondary beams of particles
A fixed target accelerator allows the possibility of producing secondary beams of many different types of particles. Thus one can study a large variety of interactions. At a proton-synchrotron one can produce beams of neutrons, K~and photons, using a sequence of collimator-sweeping magnet repeated a number of times. One may even achieve a partial separation of these particles by preferential absorption in a low-Z material (to remove neutrons) or in a high-Z material (to remove photons). A neutrino beam is more complex: it requires optimization of particle production, of acceptances, of a decay region, of p-absorption, etc. A charged particle beam requires a large number of bending magnets and of quadrupole focusing elements. We shall only discuss further this last case. High energy secondary beams of charged particles tend to be very long physically because of the large magnetic rigidity of the particles. It is relatively easy to make beams with good optical properties for which the laws of geometrical optics are completely valid. Fig. 3.6 shows as an
144
G. Giacomelli, Total cross sections and elastic scattering at high energies
Meson Produclion Target Septum Magnet f-Stop Collimntor
Quadrupole Focusing Doublet
Main Bend Magnets
First Focus. ~P Collimator
Bend
~
Second Focus.llalo Collimator
Recombination Bends
Third Focus. Experimenters Target
Fig. 3.6. Equivalent optical diagram of the M6 beam line of the Meson Laboratory at Fermilab.
example the M6 beam of the Meson Area at Fermilab [74T 1]. The beam is produced at an angle of 2.5 mrad; it can be tuned in the momentum range of 20—180 GeV/c and is designed to have a momentum resolution of better than ±0.1%.It is a classical beam, as one may have in geometrical optics, with three stages and three foci. The first set of quadrupoles make a parallel beam (they make a point-to-parallel transformation) which is then analyzed in momentum; the second set of quadrupoles focuses the beam horizontally and vertically at F1. At this focus the beam has a horizontal dispersion of about 6 cm/(% .i~p/p);the movable aperture of a momentum collimator defines the momentum acceptance of the beam. A similar optical arrangement, point-to-parallel and parallel-to-point is repeated through two more sections with shorter focal lengths. Vertical and horizontal clean-up slits are located at the second focus F2. In the final stage of the beam
G. Giacomelli, Total cross sections and elastic scattering at high energies
145
there is momentum recombination in both positions and angle at the third focus’F3. If the produc3 and if the momentum bite of the beam is fixed at tion target is a rod 0.5 X at150 = ±0.1%, the 0.5 finalX image F mm 2 in size with a horizontal 3 isone approximately X 0.5 divergence of ±1 mrad and a vertical of ±0.5mrad.0.5The totalcmlength of the beam is 440 m. 3.5. Charged particle identification The fraction of the incident charged particles to be used for a scattering experiment are usually defined geometrically with scintillation counters and identified in velocity, and thus in mass, with a combination of differential and threshold ~erenkov counters. High-energy beam lines have several ~erenkov counters, for a total length of up to 100 m. The counters differ considerably from the simplest threshold counter, with a simple mirror and one photomultiplier, to the most sophisticated differential counter, with a precision mirror, a refined optics for coma and chromatic corrections and many photomultipliers. Fig. 3.7 shows the mass spectrum obtained with a gas differential ~erenkov counter [7 SC 1]. Small counter hodoscopes or proportional wire chambers are often placed in the beam in order: (i) to improve the momentum resolution; for this purpose a hodoscope is usually placed at a dispersive focus in order to locate exactly each particle and thus its momentum; (ii) to select only those particles which have specific directions; for this purpose the outputs of two sets of proportional wire chambers are fed into a fast logic, a coincidence matrix, which allows the selection of the appropriate trajectories. In this case the proportional wire chambers are used as a sort of “electronic collimator” [74C 1].
pn300Geht l0’
P a~$c~ Ge\6t 9n 35 n~ad
P ia
10 io~
II
irK
(2
13
14
5
6
rT
PRESSU~(PSIA OF HEIJUM) Fig. 3.7. Mass spectrum of the particles present in a 280 0eV/c positive beam produced at 3.5 mrad by 300 GeV protons impinging on a beryllium target. The curve was obtained with a differential ~erenkov counter filled with helium [75C1 1.
G. Giacomell4 Total cross sections and elastic scattering at high energies
146
3.6. Neutral particle identification Neutral particles can only be identified via their interaction and the subsequent observation of the charged products. Thus charged-particle detectors can only be placed at the end of the beams, after the secondary target. Neutral-particle detectors are in practice designed for specific experiments and are more part of the experiment than of the beam line. Calorimeters play a major role in neutral-particle detection. ‘y-rays are detected with total absorption shower detectors, most of which are lead-glass ~erenkov detectors or lead-scintillator sandwiches. Many neutron calorimeters are made of iron-scintillator sandwiches. The same type of detector, but much larger, is used for i’-detection. K°’sare detected through their decay products in magnetic V°spectrometers. 3.7. Main features of high-energy hadron—hadron interactions In planning an experiment one has to keep well in mind the main features of strong interactions at high energies, in particular: (i) Most of the interactions are inelastic, Uel/Gtot being 15—25% (see fig. 5.31). 1111111
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S [co.v?] Fig. 3.8. The average multiplicity of sr~,~r, K~,K, p, ~, and for all charged particles plotted versus s. The dashed lines represent the results of the fits according to the formula (n1> = A + B in s + C ~i /2 [74R1].
G. Giacomelli, Total cross sections and elastic scattering at high energies
5
~ 10
(GeV) 20 I
50
500 —
I—
w 400
T~(MeV)
180 170
—
D
~
-O’~
P
t~o
._
_._.
K:
ISO 140
—
a~
—
• _~•~
—
—.
—
_._
—
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150 11.0 130
•
120? I 150
•~.•__—o—-o charged
Lii
•— —:~1it 300
~
140
I~~II
10 LABORATORY
I
102 MOMENTUM
3 io (GeVIc)
io’
Fig. 3.9. Average transverse momentum for the production of pions, kaons, antiprotons, and all charged particles plotted versus laboratory momentum. On the right are shown “temperature” scales for different particles computed in the framework of a thermodynamic model [74R1 1.
(ii) For each interaction the average multiplicity grows approximately as ln s and becomes large at high energy. At p = 1500 GeV, corresponding to = 53 GeV, one has:
~
=
~ch>
+
4.7 ir~+4,5
~.O
+
~ 12 ÷6
4.~3ir~+ 0.43 K~+ 0.31 K ÷l.6p+ 0.lSj5+
...
(see fig. 3.8) [74Rl].
(iii) The average transverse momentum is small and becomes essentially energy-independent. It ranges between 350 to 450 MeV/c, depending on the type of produced particle (fig. 3.9). Thus large p~(t) events are rare. A specific process has to be “fished out” of the many possible channels. This is easy for elastic scattering at small angles; it becomes difficult at large angles.
4. Total cross sections 4.1. Introduction As already stated, we shall be concerned with high-energy hadron—hadron total cross sections, and shall only make a few remarks about photon—hadron, ~lectron—positron and neutrino—hadron cross sections. The magnitude and the energy dependence of a high-energy total cross section gives an intuitive idea of the type of interaction and of the size of the particles involved. The neutrino—nucleon total cross section is very small and increases linearly with laboratory energy. This is typical of the collision between point-like objects. Any deviation from linearity
G. Giacomelli, Total cross sections and elastic scattering at high energies
148
may yield informations on the non-locality of the interaction. In contrast, hadron—hadron total cross sections show a wealth of structures in the Resonance Region, that is, for laboratory momen~abelow 5 GeV/c, while in the High Energy Region, above 5 GeV/c, the cross-sections first decrease slowly, reach a minimum, and then slowly rise. The cross-section values of about 20—40 mb are typical of objects of about 1 fermi radius. The photon—hadron cross sections are about two orders of magnitude smaller, but may have an energy dependence similar to that of hadron—hadron ones. Neglecting the recently discovered narrow resonances, the e~e -~ hadrons cross sections are small (2 x 10~° cm2 at E~= 4GeV) and decrease slowly with energy. We shall first describe some experimental methods and then discuss the experimental results. For a more thorough discussion of the experimental details the reader is referred to other reviews [69G2, 70G2]. 4.2. Experimental methods The transmission method I) Charged particles. The standard transmission method is illustrated by the experimental set-up of a Fermilab experiment, fig. 4.1 (74C1). Schematically the equipment consists of: (i) a system of counters to count the incoming particles, (ii) a number of targets where the incoming beam is absorbed, and (iii) a series of cylindrical transmission counters, each subtending a different solid angle. The measurement consists in determining the fraction of transmitted particles with and without the target. This leads to the determination of the partial cross section a~for each of the transmission counters i. ~tot is then obtained by an extrapolation of the partial cross sections to zero solid angle. Several improvements in the technique and in data handling were made in the last few years. They made it possible to measure total cross sections with point-to-point precisions of ±0.1% and with systematic errors of about ±0.4%. 4.2.1.
—
—
Aj
A2 B1
Q
/~ii
82 L~i
~
02
I Scintillation Counters
E LI
~erenkov Counters PWC
0
Quadrupoles
B3 j C3
P1
Targets
T 1-T12
D2 E
Cei,~
~
_____________
~2 ~ H2
~
p(feet)
Fig. 4.1. Layout of the charged-particle cross section measurement performed at Fermilab [74C1 —21. The incoming beam is counted as B = B12 3A1 2C,, where C1 is one or a combination of the three gas ~erenkov counters. PWC1 —3 are proportional wire chambers. D2, VAC and H2 are targets. T1 —T12 are transmission counters. E1 and E2 are small counters used for beam tuning and for the measurement of the efficiencies of the transmission counters. Ce is a lead glass ~erenkov counter; i.u is a combination of two large scintillation counters.
G. Giacomelli, Total cross sections and elastic scattering at high energies
149
A few technical details will now be briefly discussed. The beam intensity is typically (1—2) X 1 0~particles per accelerator cycle, with a momentum resolution of ±0.5%. In the Fermilab experiment, two proportional wire chambers (PWC) in the beam were used to eliminate particles with wrong directions. Thus the PWC acted as sort of “electronic collimators”. Particle identification is performed with gas differential ~erenkov counters, capable of resolutions up to i~43 10~(see fig. 3.7). The targets (hydrogen, deuterium and dummy), typically 1—3 m long, have a double jacket design, which provides long-term density stability to ±0.04% and a knowledge of the absolute density to about ±0.1%. The transmission counters usually cover the range 0.01 < ti < 0.07 (GeV/c)2, the t-range being kept fixed, by changing the distance between the hydrogen target and the transmission counters. Some experiments used counter hodoscopes [69F 1, 74A6] or proportional wire chambers [74C 11 in order to have a complete knowledge of the beam shape and of the form of the extrapolation to very small angles. The electronics is standard, the main emphasis being on reliability and stability. A computer on-line is needed if hodoscopes or proportional wire chambers are used; the computer is also useful to “keep an eye” on the stability of the apparatus and on the time structure of the beam. Beam impurities of electrons and muons are measured directly by means of total absorption lead-glass counters and with a muon filter made up of a few meters of iron, respectively. In order to obtain nuclear cross sections, the results have to be corrected for single Coulomb scattering and Coulomb-nuclear interference. Even after these corrections it is not completely clear if one has obtained nuclear cross sections or if other contributions are present [71B3]. II) Neutral particles. The method for measuring the total cross sections of neutral particles is ‘illustrated in fig. 4.2 [74L2] The Fermilab neutron beam, indirectly monitored, is absorbed in a target placed before a neutral particle detector. Here, neutrons are converted with about 20% efficiency into charged particles in a slab of iron, 2 cm thick. The forward going charged particles produced are counted in one of the transmission counters and the neutron energy is measured in the hadron calorimeter. This method becomes particularly useful and yields precise results at high energies because one can make better neutral particle detectors. .
Direct measurement of the pp total cross section at the ISR At intersecting storage rings machines it is not possible to use the traditional method of deducing c~ 0~ by measuring the transmission of a beam through a known target. Instead a~0~ may be measured via the direct application of the luminosity formula N = u~0~L, by performing sep4.2.2.
Neutron Beam,
i~ ~
A ,°
TeleSCOPeS
I lull I
A,F.
i carg.~
D~ - D7 2OO~f
,,~
[~J
Total Abwpticn Calorimeter (C)
Fig. 4.2. Layout of the neutron total cross section measurement performed at Fermilab [74L2].
G. Giacomelli, Total cross sections and elastic scattering at high energies
150
~
Left
Right
0
1
2
3
4
5m
Fig. 4.3. Layout of the proton—proton total cross-section measurement performed at the ISR. A system of counter hodoscopes (H, L) surrounding the intersection region, counts almost all the interactions and allows an extrapolation to 4ir coverage. The H 1 hodoscopes are binned in the ~ direction; the H10 hodoscopes are split into 0-bins; the L-hodoscopes are four planes of x- and ycounters interspersed with lead plates; TB are triggering counters [73A4J.
arate measurements of N and L. The total number of interactions N per unit time is measured with large scintillation counters; extrapolations have to be made to take into account the missing number of interactions both at small and large angles. The absolute value of the cross section is then obtained via a luminosity measurement as discussed in section 3.3. Fig. 4.3 shows the apparatus used at the ISR to measure the proton—proton total cross section. About 500 scintillation counters, subdivided into left and right hodoscopes and into large-angle counters, were used to detect the total number of interactions [73A4]. Systematic uncertainties of about ±2%were achieved. Bubble-chamber measurements I) Charged particles. Bubble-chamber pictures are scanned for all types of interactions inside a well-defined fiducial volume; the total cross section is then computed on the basis of the number of incident tracks. Corrections have to be applied for beam contamination, for scanning losses, and for the loss of elastic scattering events at small angles. Measurements of this type were for instance performed with the 30 in. Fermilab hydrogen bubble chamber exposed to incident protons of 100, 200, 300, and 400 GeV. Other exposures using particles tagged with a ~erenkov counter have also been performed (100, 150, 200 GeV/c ir, 100 GeV/c ?, etc.) [73W1, 75G 11. These measurements are accurate (with statistical and systematic accuracies of about ±2%), but cannot compete with systematic counter measurements. II) Neutral particles. In this case one still measures all the interactions inside a well-defined fiducial volume. The energy spectrum and the absolute flux of the incident neutral particles have to be determined by indirect measurements. The incident energy may be determined for some individual events if they lead to a kinematically overconstrained situation or if the total incident energy is deposited in the bubble chamber. Neutrino cross sections, production by photons of strongly interacting particles, K~,n, and IT total cross sections were measured using this technique. Many exposures of large bubble chambers to broad band and dichromatic neutrino beams are planned for the future. 4.2.3.
Indirect methods for charged particles Two methods of determining the pp total cross section from a measurement of the differential elastic cross sections were used extensively at high energies. (j) In the first method the pp elastic differential cross section was measured in the diffraction 4.2.4.
G. Giacomelli, Total cross sections and elastic scattering at high energies
region, for 0.01 < ti < 0.12 (GeV/c)2. The data were then extrapolated to t formula da/dt
151 =
0 using the
(da/dt)~, 0èxp(bt).
(4.1)
Then, using the optical theorem, one has: l6ir(da/dt) =
to
-
(4.2)
t~O
where p is the ratio of the real-to-imaginary part of the forward elastic scattering amplitude. p is either computed with dispersion relations or set equal to zero, in which case eq. (4.2) yields an upper limit. The computation of the cross section via formula (4.2) requires a knowledge of the luminosity since da/dt (dN/dt)/L where dN/dt is the measured number of events. (ii) The second method is based on measurements of the pp elastic scattering differential cross 2]. Using the sectionstheorem in the Coulomb-Nuclear region 0.001 ~ ti 0.01 (GeV/c) optical and assuming theinterference t-dependence of eq.[for (4.1) for nuclear scattering, the pp -~ pp differential cross section may be written as: cia
—
a~t I
I
+
(1
+ ii2)
Coulomb
-j-~-exP(bt) 2(p
2a +
Nuclear
aq5)
CTtt
ri.G2 ~j~= exp(bt/2)
(4.3)
C—N interference
where C is the Coulomb amplitude: C = -~~-G2(t)exp(ia~t)
(4.4)
and G(t) =
=
proton form factorns (1
+
t/0.7l)2,
phase of Coulomb amplitude
a = fine structure constant
a(ln 0.08/iti — 0.577)0.025,
(4.5)
1/137.
Eq. (4.3) is valid on the hypothesis that p does not depend on t and that spin effects are negligible. It contains three parameters, p, b and ~ Several types of fits may be done, fixing b and either leaving both p and a~ 0~ free or only one. When using eq. (4.3) one has an absolute normalization based on the small-angle Coulomb cross section, which is theoretically known. Thus for the ISR data one does not need a separate determination of the luminosity. The above methods applied at IHEP, Fermilab, and at the ISR yielded values of the total cross sections with overall errors between ±0.5and ±2mb [73A1, 73A~,73B3, 73B4]. (iii) A third method, used at the ISR, is based on the simultaneous measurements of the total collision rate (see section 4.2.2) and of the elastic-scattering differential cross section in the diffraction region [method (i) above]. Dividing (3.1) by (4.2) one has t1tot
=
1 611’(dN/dt)~..0 N(l + 2)
(4.6)
G. Giacomelli, Total cross sections and elastic scattering at high energies
152
In this case the measurement of a~0~does not depend on the luminosity, thus removing one of the largest uncertainties. Other methods Among the large variety of other methods used to measure total cross sections, we shall mention the following: (i) Hadron production by photons. These experiments have been performed with tagged photon beams, using a system of hadron counters for charged and neutral hadrons, and a system of anti-coincidences for the forward-going electrons and photons (see fig. 4.4). Experimental results are available up to 30 GeV. They will soon be followed by measurements up to 200 GeV. (ii) Neutrino total cross sections. At high energies it is possible to perform relatively good measurements with neutrinos because (a) the neutrino cross section increases with energy, (b) better neutrino beams of high intensity in a narrow forward cone can be made, and (c) large detectors are easier to use. Intense beams of high-energy neutrinos are produced from the decays of pions and kaons. If the pions and kaons are momentum analyzed, before the decay region, one obtains a dichromatic neutrino beam, where the higher energy component comes from K-decay and the lower energy one from ir-decay. If pions and kaons are focused and not momentum analyzed one obtains a wide band neutrino beam. Because of the smallness of the neutrino cross section one needs very large detectors with tens of tons (and very soon hundreds Of tons) of material. Fig. 4.5 shows the layout of one of the two counter experiments performed at Fermilab using the dichromatic neutrino beam. There are also many results obtained with large bubble chambers. (iii) Hyperon total cross sections. At high energies the Lorentz dilatation factor becomes important; thus beams of short-lived particles (like the hyperons) can be made and with these one can perform experiments similar to those for the other hadrons, but clearly not with the same accuracies. In particular, the beams and the detecting equipment have to be very short physically, in order to minimize decays. The CERN—Heidelberg Collaboration measured at the CERN PS the u~0~(A°p) and the u000(A°n) in the 6 to 21 GeV/c momentum range using a wire chamber set-up, where the A°above the median plane of the equipment interacted in carbon, while those below the median plane interacted in a CH2 target. Thus they obtained the A°p cross sections by what may be called a selfmonitoring difference [72G11. 4.2.5.
~—
r
I
PHOTON BEAM
r- SHOWER
\ COUNTER
/
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S2a A
S2b
C,
Fig. 4.4. Layout of the tagged photon beam and of the apparatus used for the measurement of the total photonucleon cross section at SLAC [69C1J.
G. Giacomelli, Total cross sections and elastic scattering at high energies TARGET-CALORIMETER 10cm iron
BEAM
TOROIDAL IRON CORE MAGNET
:~
~
153
~
Steel target
0 Spark chamber Calonmeter
0
Im
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Fig. 4.5. Layout of one of the two counter experiments which study uM(l3~)interactions at Fermilab [74B1]. The neutrino beam, arriving from the left, interacts in the target calorimeter; forward going muons are momentum-analyzed in the muon spectrometer. This apparatus is operated with the dichromatic neutrino beam, while the other [74A8, 74Bl I operates with both dichromatic and wide-band neutrino beams.
(iv) Cosmic-ray measurements. At even higher energies there are some cosmic-ray measurements of pp inelastic cross sections [70A 11 and calculations of total cross sections obtained comparing the fluxes of cosmic rays at the top of the atmosphere and at various atmospheric depths; from these calculations the mean-free paths in air are obtained; then, using the Glauber model, lower limits for the free nucleon cross sections are computed [72Y1, 74B8]. 4.3. The shadow correction in deuterium Direct measurements of neutron cross sections may be performed only for the np case. In all other cases the free neutron cross sections ~ are derived from deuterium (~~d)and hydrogen measurements using the formula ~Xd
=
+
a~
—
8g.
(4.7)
The cross section defect 6ci arises from the shadow of one nucleon on the other. 6o is substantial only in the case of hadron—hadron collisions; for neutrinos and for photons the nucleon is transparent. 6cj is usually computed at very high energies with the following simple formula 2) (r ~ -~--—a~u~(l—p~p~) (4.8) where p~and p~,are the ratios of the real-to-imaginary parts of the forward scattering amplitudes. (r2) is the average inverse square separation of the proton and neutron inside the deuteron. To first approximation, should be a constant, independent of energy. On the basis of the nuclear physics of the deuteron one finds (r2> = 0.031 mb1 using the Gartenhaus wave function for the deuteron. 6a and thus (r2) may be measured directly at high energies by measuring the total cross sections of ir~and 7( mesons in hydrogen and deuterium. With these four measurements at each energy, the pion—nucleon system is overdetermined, thus allowing the computation of 6cr:
G. Giacomelli, Total cross sections and elastic scattering at high energies
154
6cr
=
airp
+
a irp -
—
~-(a . 2 ,rd
+
a,rd -
(4.9)
)
Because of charge symmetry the cross sections ~ and ~ should be equal. It is more advantageous to use in eq. (4.9) their arithmetic mean because this decreases the influence of systematic errors and of statistical fluctuations. For the 1 5—200 GeV/c interval the agreement between 7rd and ir~dcross sections is good [73D2,74C2, 75Cl], 01rd/0~d
=
1.003
(4.10)
±0.002
while at lower energies there seems to be a difference of about 2.5% [69G1, 73C5]. Similarly one may obtain the shadow correction 6cr from a measurement of pd, pp, and np total cross sections: (411) bao +a —1(a +a ) PP
flP
2
pd
nd
The results for 6cr from pion data are shown in fig. 4.6; they indicate that 6cr is either constant or increases slowly with momentum. The same comment applies to the energy dependence of (r2). In the Glauber—Wilkin model the shadow effect arises from the multiple scatterings of the incident particle by the nucleons inside the deuteron [5 5G 1, 66W 1]. At low momenta such scatterings are predominantly elastic and (r2) is a constant determined by the internal structure of the deuteron and independent of the incident particle energy. At high energies inelastic processes can make a significant contribution to the shadow correction and the quantity (r2) must at present be regarded not as the neutron—proton distance inside the deuteron, but as an energy-dependent 0.06
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I
I
I
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2
~
ii I
10
I
II
III
I
20 50 100 200 LABORATORY MOMENTUM (0eV/c)
Fig. 4.6. The energy dependence of the shadow correction determined from 1r±pand ~±d total cross sections: (a) the shadow correction &a for pions; (b) the (r2) parameter.
G. Giacomelli, Total cross sections and elastic scattering at high energies
155
phenomenological parameter to be determined experimentally [72A1, 73Q1]. In the recent experiments (r2> was taken as 0.039 mb for K~scattering and 0.03 5 for p and ~ scattering. Intuitively one expects that the percentage of screening increases with the increase of the elementary cross sections off free proton and free neutron targets. Thus the percentage screening should be larger for p than for K~and it should increase with momentum if the cross section increases with momentum. 4.4. Energy dependence of the total cross sections Figs. 4.7—4.14 show compilations of various total cross sections plotted versus laboratory momentum. We shall first discuss the accuracy of the data, then analyze the gross features of the cross sections, those which differentiate between strong, weak and electromagnetic interactions, and then proceed to discuss more thoroughly the hadron—hadron interaction. In general the authors of total cross-section measurements quote two types of errors: (a) a point-to-point error, which includes the statistical error and those uncertainties which are energy dependent; (b) a systematic (or scale) error which is energy independent. The typical uncertainties quoted for high-energy total cross sections are approximately those given in the following table. Table 4.1
Uncertainties in total cross section measurements Cross section
Point-to-point errors
Scale error
~
±0.2%
±0.4%
±0.4% ±0.6%
±0.4% ±0.7%
±0.4%
±1.5%
±3% ±10,±20%
±2% ±20%
~
K~p,pp K~n pn,np ~yp
Clearly the precisions of neutrino total cross sections are not comparable to those of the longlived hadrons. The table does not show the precision of 7r~d,Kid, ~d, pd and nd total cross sections: their percentage statistical errors are about the same as for hydrogen, while their systematic error is somewhat larger (about 0.5%) because of greater uncertainties in the deuterium density, in the deuterium contamination and in the extrapolation. As discussed in section 4.3, the extraction of the free neutron cross sections presents several uncertainties. Therefore the typical errors for neutron cross sections are considerably greater than for hydrogen and deuterium measurements. The error bars shown in figs. 4.7—4. 14 represent point-to-point errors only. A scale error should be added when comparing data from different experiments. Thus the agreement among the different experiments may be considered reasonable and the energy dependence of the data well established. There are some minor differences around 60 GeV/c for i(p total cross sections and around 200 GeV/c for pp. The nd and pd total cross sections are equal, within errors; similarly, and to a higher degree of accuracy, the ird and ir~dcross sections are equal. The np and pn cross sections scatter considerably (in particular around 50 GeV/c), though not outside their relatively large errors.
G. Giacomelli, Total cross sections and elastic scattering at high energies
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4.4.1. “Weak, electromagnetic and strong” cross sections Figs. 4.7—4.10, which show the total cross sections for vN, ‘yN, e~eand charged hadrons respectively, give an immediate idea of the drastic differences of the interactions involved. The neutrino cross sections for muon production rise linearly with laboratory energy. There is no visible deviation from linearity over the energy range of few MeV to 200 GeV, that is over a factor of 100 000 in laboratory energy. Instead, the hadron—hadron cross sections change only very slowly for energies above 5 GeV and the energy dependence of the photon—nucleon cross section resembles that of hadron—hadron collisions. The e~e-÷ hadrons cross section also changes slowly with momentum, apart from the recently discovered spikes. In absolute values the cross sections differ considerably. At ~ 10 GeV the neutrino cross section is about 4 X l0” cm2, while the (eke -+ hadrons) is probably 10_32 cm2, the photo-absorption cross section is about 10-28 cm2 and the typical hadron—hadron cross section is about 3 X 10~~ cm2. Their respective ratios are thus approximately 1.3 X l0”: 0.4 X 106: 1/300:1 at~J~ 10GeV. The (i) linear rise of ~N and 17N cross sections, the (ii) relation —‘
G. Giacomelli, Total cross sections and elastic scattering at high energies
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~nN(
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GeV/c.
(4.12)
and (iii) the ratio cJ~N/cJVN 0.36 are consistent with the hypothesis of a local interaction, from point-like objects. In particular, these facts are consistent with a parton model of the nucleon in which the partons are relativistic and predominantly spin ~ fermions, not anti-fermions. In contrast the hadron—hadron cross sections are typical of objects of about 1 fm radius. The photon—nucleon cross sections are qualitatively consistent with the vector-dominance model: the photons couple directly to vector mesons, which then interact with the nucleon. The consistency is qualitative since quantitative tests of the vector-dominance model may have some difficulties. It is interesting to look at the cross section for the process e~e-÷ hadrons, fig. 4.9. The most remarkable feature is the presence of the very narrow structures at ~ = 3~1 and 3.7 GeV [74A9, 74A10, 74Al1] and the broader structure at 4.15 GeV. This brings in the question: could one have missed narrow, large-mass resonances in the other total cross section measurements? This is —‘
G. Giacomelli, Total cross sections and elastic scattering at high energies
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certainly possible, because the spacing of the data points is relatively large, because one would have tended to disregard a point deviating from the others, and because the overall experimental resolution was not too good. 4.4.2. Energy dependence of the cross sections of the long-lived hadrons Turning our attention to the long-lived hadron—hadron cross sections, one may “expand the vertical scale” and take a more detailed look at what is happening. It is apparent from fig. 4. 10 that the total cross sections of ira, K~,p, and 15 on protons first decrease, reach a minimum, and then increase with increasing energy. The location of the minimum (see table 4.2) changes with the type of incoming hadron, being lowest for Kp and increasing, in order, for ?p, pp, Kp, ~rp, and ~p. The location of these minima reflect via dispersion relations in the location at which the ratio of real-to-imaginary part of the forward scattering amplitude crosses zero. The energy dependence of ir~d,Kid, pd, and pd cross sections is very similar to that in hydrogen. This is also so for the data on neutrons (fig. 4.12), within their larger errors. It may be interesting to single out the main features of hadron—hadron cross sections by averages of particles and antiparticles and also by averaging the cross sections for particles which
G. Giacomelli, Total cross sections and elastic scattering at high energies
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are members of the same isomultiplet; this means considering the following averages: (cr(irN)>
=
~
[a~0~(ir~p)+ Utot(7(p)]
(4.13)
(a(KN))
=
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(4.14)
(a(NN))
=
~[a~0~(pp) + u~0~(pp) + u~0~(pn) + a~0~(~n)].
(4.15)
In the Regge pole language (see section 2.7) these average cross sections have contributions from the exchange of the Pomeron P, of the second vacuum trajectory P’, whose contribution decreases with energy, and from possible cuts, like the Pomeron cuts. In formulae: Table 4.2 Approximate location of the laboratory momentum at which the total cross section has a minimum (pa) and at which the ratio of real to imaginary part crosses zero (p2). Pi (GeV/c) K~p Kp lIp pp pp
P2 (GeV/c)
15 70 50
100 15 140
90 70 250
70 270 80
G. Giacomelli, Total cross sections and elastic scattering at high energies
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20 40 60 80 100 200 400 600 1000 LABORATORY MOMENTUM (GeV!c) Fig. 4.14. Cross sections in pure isospin states.
(ci(xN))
=
a~(xN)+ a~~(xN) + cr~(xN).
(4.16)
The values of the averaged cross sections are given in fig. 4.13. They all have a minimum in the region experimentally covered. Many analyses are performed on pure isospin cross sections. Only the ?p, K~pand pp are pure isospin states. The other experimentally accessible cross sections contain a mixture of two isostates. Therefore the determination of the pure isostates requires the measurements of two cross sections. The pure isospin cross sections shown in fig. 4.14 have been computed from the measured values via the formulae: a112(7rN) = ~-[3a~0~(i(p)
—
u~0~(ir~p)]
(4.17)
ci0(KN) = 2a~0~(Kp) — a~0~(ICn)
(4.18)
cr0(pN) = 2o~0~(K~n)a~0~(K~p)
(4.19)
—
G. Giacomelli, Total cross sections and elastic scattering at high energies
162
2a~
u
0~(!5p) a~0~(~n).
0(pN) =
(4.20)
—
The errors on these cross sections are larger than those quoted in table 4. 1, since they are obtained from differences which involve cross sections from neutrons. It may be worthwhile to point out that the shape of the hadron—hadron cross sections versus energy resembles that of the energy loss of particles in matter. One may explain the differences among the various cross sections in the frame of Regge poles. For instance, the fall-off of the ir~ptotal cross section is attributed to the p and f trajectories, while the absence of a fall-off in the K~pcross section comes from the exchange degeneracy of the f and w, p, A2 trajectories [see the total cross-section structures in terms of Regge poles, eqs. (2.53—2.54)]. In reality the simple Regge picture breaks down, because one has difficulties in explaining the low-energy fall-off of the pp cross section: it may result from a breaking of the exchange degeneracy, a lower lying singularity or some other mechanism. We next face the problem of parametrizing the energy dependence of the total cross sections, in particular of their rise after the minimum. We may assume the type of dependences suggested by Regge-pole theory, which need a diffractive contribution, due to Pomeron exchange, which rises with energy, and non-diffractive contributions, which decrease with energy. The non-diffractive parts behave as p°, with n-values which may be computed from the analysis of total crosssection differences. The diffractive component is expected to have a constant term plus a term which increases logarithmically with energy. The specific type of logarithmic behaviour will have to be found experimentally. Thus in the Regge pole frame the general energy dependence will be of the types: a~0~(xp)D+ND =
(4.21)
a~+ b,(ln p/c,)°’+ B1p”
(4.22)
p +m’ a1 ln( b. :)+Bp~
(4.23)
where p is the laboratory momentum. Hendrick et al. [74H 1], using a parametrization of the type (4.23), found that the bulk of the total cross-section data is explained by the following dependences: 49 •a(K~p) =
a(~r~p) = 3.27 ln
a(pp)
(4.24)
3.27 ln p+l
4.91 ~
p+206 0.33
16.8 +
2.62 —
p°567 p0.426
(4.25)
(4.26)
These parametrizations should be considered a reasonable interpolation of the data, without any further significance. If it would hold true at higher energies, it would imply that the cross sections rise with a dependence ln (p) and that ~K~p and UlI+p will approach each other, and that both approach asymptotically the value of 2/3 predicted by the naive quark model.
G. Giacomelli, Total cross sections and elastic scattering at high energies
163
Table 4.3 Fits of the hadron—proton cross sections above 8 GeVIc to formulae (4.27) [74D6l. Only the point-to-point errors 2 have been used for computing the x ~r~p 13.5
±1.6
C 02
2.4
02
0.37 22.8 0.37
±0.2 ±0.03 ±0.7 ±0.06
x2 DOF
K~p
p~p
14.2 ±1.1 0.010 ±0.002 8.9 ±0.6
25.7
±2.5
25.8
±1.6
0.53 ±0.02 14.7 ±2.3 0.52 ±0.16 225 204 — 16 = 188
0.58 43.7 0.35
±0.02 ±1.3 ±0.05
Many other types of energy dependences may be found in the literature. For instance Bartel and Diddens [74D6] tried the following parametrization: —
+
a~ 02 = 2a2E a~ 0~(xp) 0~(xp) = 2(a1E°’ + a,,) (4.27)
The results of the fits to the ir~p,K~pand p~pcross sections above 8 GeV/c are given in table2s4.3. Other formulae, basedbecome on optical models, yield dependences areInslower than ln at present energies and of the ln2s type at energy asymptotic energies which [73C 1]. some of these models all total cross sections become equal at infinite energies. So we did not really solve the problem of the energy dependence of the cross sections, but we found some semi-empirical relations which are useful for practical considerations. If there is any hope of finding out the true energy dependence this would have to be based on the measurement of the K~por possibly of the pp total cross sections, because the first starts rising at very low energy and the second has been measured at the higher energies provided by the ISR. Fig. 4.15 shows the K~ptotal cross section plotted versus (In ~)3; the behaviour seems to be consistent with
21
-
-
~2O-
-
—
I.) Id
U
18-
•‘
— -
100
200 p (in
300
s) 3
Fig. 4.15. The K~ptotal cross section plotted versus In3 s.
G. Giacomelli, Total cross sections and elastic scattering at high energies
164
~
o(p*p~++
+
+
+ 10
I
I
I
:
10
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
~
-
I
0
I
I
I
I
I
5
I
10
I
I
I
15
I
I
20
GeWc
Fig. 4.16. The A°pand A°ntotal cross sections versus lab. momentum. Also shown are two points for atot(Ep) and °tot(~~) [72G1 I.
G. Giacomelli, Total cross sections and elastic scattering at high energies
165
a constant value at low energies and then with a straight line rise at higher energies. It would thus seem that the energy dependence of the rise is approximately of the ln3s type. This would not violate the Froissart bound, since the coefficient of a possible ln2s term would be about 100 times smaller than that of the Froissart bound. This is so also for pp [see eq. (4.27)], thus indicating that the rise of the measured cross sections in the 20 to 2000 GeV/c range has nothing to do with the asymptotic Froissart bound. 4.4.3. Energy dependence of other hadron—hadron cross sections Fig. 4.16 shows the A°pand A°ntotal cross sections versus laboratory momentum. For comparison a d~ 0~(~N) point at 18.7 GeV/c is also shown. The A°pand A°ncross sections are constant in the 6 to 21 GeV/c region and have the average values quoted in table 4.4. The results of table 4.4 indicate that all the hyperon—nucleon total cross sections in the 10—20 GeV/c range are approximately equal within their relatively large errors. 4.5. Relations among hadron—hadron cross sections A large number of high-energy relations among total cross sections and total cross section differences were derived on the basis of asymptotic theorems, SU(3) symmetry, absence of exotic states, additive quark model, etc. In the following we shall discuss the applications of some of these relations to the available data. 4.5.1. Total cross-section differences The differences of the total cross sections for particles and antiparticles on protons ~a(x~p)
=
a~05(xp)— a~0~(x~p)
(4.28)
are shown in fig. 4. 17. The errors on the differences are smaller than those based on the total cross sections themselves, since some systematic errors cancel out and the final errors are mainly of a statistical nature. A similar statement is valid for i~a(x~d) (fig. 4.18). Instead, the differences ~a(x~n) for scatterings on neutrons have considerably larger errors. The differences of particle and antiparticle cross sections on the same target must eventually become zero in order to satisfy the Pomeranchuk theorem. We want now to discuss the problem Table 4.4 Average values of the hyperon—nucleon total cross sections and comparison with the predictions of various high-energy sum rules. (Interpolations of available total cross sections were used for the calculations.) [72G1 I Total cross section
Ap An
Laboratory momentum (GeV/c) 6—21 6—21 18.7 18.7
Ap An
6—21 6—21
Measured value (mb)
Predicted value (mb)
Equation no.
34.6 34.0
±0.4 ±0.8
35.2
±0.6
(4.41)
34.0 30.0
±1.1 ±1.2
35.0 34.4
±0.9 ±0.3
(4.42) (4.43)
56 46
±11 ±20
I
I
I
1111
I
20
E 10
I
V GALBRAITH
£ FOLEY
65 67
• DENISOV
73
—
i
L
2
~
74
~
5p)
~~(K C-’
0.6 0.4 AdH1t~p) 0.2 10
I
I
I
20
111111
40
60
I
90 100
LABORATORY
I
I
200
MOMENTUM
I
600
(0eV/c)
Fig. 4.17. The differences of total cross sections for 11±, K±,p and ~ interactions with protons. The solid lines represent fits of the data to the power dependence (4.29). I
20
~
I
I
liii
I
I
~
~
I
I
GAIBRAIT 65 DEN(SOV H 73 74
~(p*d)
:
4)
1’
~ U, o.e . Ill o
,I,J,
~d~CKtfl)
0.64
-
O
Q4.
0.2
10
-
~(Ktdl ~aip5n) I
- I~
I
I
I
I
I
I
I I
I
20 60 60 80 100 LABORATORY MOMENTUM
Fig. 4.18. Differences of total cross sections of
11±,K~,p
I
200
I
I
400
I
(0eV/c)
and ~ interactions with deuteróns. The lines are fits of the data to the power dependence of eq. (4.29).
G. Giacomelli, Total cross sections and elastic scattering at high energies I
I
I
11111
I
I
I
167
II
-
.0
E ~
2
0.4
—‘0.2 4 0
n-
p
I-
I
10
20
I
I
I
I
~
I
I
40 60 80 100 200 400 LABORATORY MOMENTUM (0eV/c)
I
I
I
600
Fig. 4.19. Differences of total cross sections of K±,p and ~ on protons and neutron targets.
of what is the energy dependence of these differences in order to learn some information on how the asymptotic region is approached. The straight line behaviour of the total cross-section differences in the log--log plots of figs. 4.17—4.18 indicate that their energy dependence may be represented by a power function of the laboratory momentum =
Ap~.
(4.29)
The results of fitting the total cross section differences to this expression are given in table 4.5. Though the fits have reasonable x2-values, one cannot consider that the power dependence parametrization (4.29) is unique. In fact, fits with a dependence ~aA+B/lnp
(4.30)
may represent the data at higher energies. The differences of total cross sections for particles belonging to the same isospin multiplet interacting with the same target are shown in fig. 4. 19. The discussion of these differences follows the same line as the previous discussion on particle—antiparticle differences. The behaviour with energy of the total cross-section differences is consistent with their approaching a value of zero, as required by the Pomeranchuk theorems. On the other hand, the energy dependence is relatively weak, thus indicating that the asymptotic region is far away. Structure of the total-cross section differences in terms of Regge poles In terms of Regge poles the total cross section differences have a much simpler structure than the total cross sections themselves (see section 2.7). For instance,
4.5.2.
G. Giacomelli, Total cross sections and elastic scattering at high energies
168
Table 4.5 Fits of total cross-section differences to formula (4.29).
~o(K~p) ~a(p~p)
~o(K~n) &s(p±n) ~o(K~d) &s(p~d)
p-range (GeV/c)
A (mb)
44—200 3— 60 3— 60 35—240 3-- 60 35—200 3— 60 3— 60 3— 60 35—240 3— 60 35—200
5.24 4.0 18.1 21.7 63 58 13.0 49 29.0 32 106 95
n ±0.10
0.43 0.32 0.54 0.58 0.64 0.59 0.67 0.61 0.58 0.58 0.64 0.54
±0.30 ±0.3 ±3.9 ±2 ±4 ±0.4 ±7 ±0.5 ±5 ±8 ±7
±0.01 ±0.02 ±0.02 ±0.04 ±0.02 ±0.02 ±0.02 ±0.05 ±0M2 ±0.04 ±0.04 ±0.02
2a~(-irp) B~p0P1 &i(p~p)= ~a(K~p) = 2a~(Kp)— 2a~(Kp) B~p0P~_B~p0~l. ~a(ir~p)
(4.31)
=
(4.32)
The equality between ~a(p~p) and z~u(K~p) is clearly not verified experimentally (see fig. 4. 1 7). Thus another parameter has to be added, for instance a sort of threshold effect (writing p — Po), or some other explanation. From eq. (4.31) one can obtain the values of B~and ct~directly, as given in table 4.6. The computation of the same parameters for the K~pand p~pcases is not as straight forward. One may use the total cross section differences in hydrogen and the predictions of -÷
p universality
—~
=
(4.33)
=
o universality = (4.34) to obtain the parameters listed in table 4.6. Another approach is to consider also the cross sections on neutrons and make combinations of cross section differences for which only one trajectory is important: Table 4.6 Fists of total cross-section differences in terms of Regge poles 2 formulae (4.31—4.36) [74H1 Quantity
Value
Op
0.57 0.43 =
~
=
= =
am,
3~
Comment
±0.01 ±0.01
p~ 1,
=
I.
(1.31 (1.14 (23.9
±0.03) mb
1”2
±0.01) (mb) ±0.4) mb
determined from ~~(ir~p) l3~pdetermined from ~(K~p)
(4.88 ±0.04) (mb)112 (16.8 ±0.8) mb (3.44 ±0.18) (mb)’12 0.42 ±0.05
Assumes a~= Assumes ~‘!,= Effective trajectory for
(11.1
Effective residue for
±0.3)
mb
G. Giacomelli, Total cross sections and elastic scattering at high energies
169
L~a(K~p) ~a(K~n)
=
4a~(KN)
(4.35)
~a(K~p)
=
4a~,,(KN).
(4.36)
—
+
~o(K~n)
Some results of different fits are given in table 4.6. 4.5.3. Relations from the quark model From the simplest quark model in which baryons are made of three quarks and bosons of one quark and one antiquark, assuming (i) that a sort of Pomeranchuk theorem holds for quark collisions, and (ii) that these collisions are independent of one another, then for pions and nucleons the average cross sections should become proportional to the number of constituent quarks. Thus one should have (a(7rN)) = ~(a(NN)). It is better to study the energy dependence of the ratio
(4.37)
~
(a(KN)> (a(irN))
(4.38)
===~‘l. ~—~oo
For the averaged cross sections one may also propose the following relation ~
2(a(irN)),
(4.39)
==~1. a(KN) s—~
(4.40)
=
thus R3=
2(a(lrN)) ~(a(NN))
+
I
I
II~
I
1.4
-
12
ic -
06 -
10
A
~
:
• •...~
~
•
I
11111
•S •~• •••••
20 50 100 LABORATORY MOMENTUM
1,uv-iui 2LN 2/3NN+KN
I
I
200 1GeV/c)
I
I
500
Fig. 4.20. Test of the additive quark model relations (4.37—4.40).
G. Giacomelli, Total cross sections and elastic scattering at high energies
170
Fig. 4.20 shows the ratios R1, R2 and R3 plotted versus laboratory momentum. The ratios (4.37) and (4.38) are slowly decreasing towards 1, while ratio (4.40) is very close to 1 already at 20 GeV/c. For hyperon—nucleon total cross sections the simple additive quark model yields the relations a~0~(Ap) = a~0~(pp) + a~0~(Kn) a~0~(7r~p)
(4.41)
—
=
a~0~(pp) + a10~(Kp)— a~0~(7(p) + 2{a~0~(K~n) a~0~(K~p)}
(4.42)
=
a~0~(pp) + a~0~(Kp) a~0~(7(p).
(4.43)
—
—
Table 4.4 shows that there is a reasonably good agreement between measured and computed values according to eqs. (4.41)—(4.43), with the possible exception of a~0~(~n). 4.5.4. The Johnson—Treiman relations The relations ~a(K~p) — 2&j(K~n)
(4.44)
&r(ir~p)~~r z~a(K~n)
(4.45)
were first derived in the framework of SU(6) symmetry [65J1]. They may also be obtained in the context of SU(3) symmetry, assuming universality for the p and w couplings; in the framework of the quark model they are obtained from the condition ~a(q~q~) = z~a(q~q~). Relation (4.44) connects the o- and p-contributions to the total cross section for KN interactions a~(KN) —
3a~(KN).
(4.46)
Relation (4.45) corresponds to the condition a~(KN) a~(KN)~ a~(7rN). —
(4.47)
Instead of (4.44) and (4.45) one may use &i(K~p) 2.Aa(ir~p) or R4=
z~a(K~p) ==~l, 2~a(ir~p) ~
(4.48)
and &r(K~p)+ ~a(K~n)
=
3z~u(ir~p)
or R5=
3&i(ir~p) &~(K~p) + ~a(K~n)
===~l.
(4.49)
As may be seen in fig. 4.21, the relation R4 is approximately verified, within large errors, at energies above 20 GeV.
G. Giacomelli, Total cross sections and elastic scattering at high energies
I
1.5
I
11111)
-
+
1.2
~
If ~
j
1.20.5-
-
2~
I
I
s—=
-
UNIVERSALITY
s
+
+
-
“
-
-
-
111.1
50
1-c
0.8
5p) 2~ai1t /~aiKtp)
R~ I
10
H
_________(b)
/
w—UNIVER$ALITY
I
-
1
(~)
171
I
100
LABORATORY MOMENTUM
200 1GeV/c)
Fig. 4.21. Test of the Johnson—Treiman relation.
I
10
I
20 LABORATORY
I
I
11111
50
100
I
200
MOMENTUM 1GeV/c)
Fig. 4.22. Test of(a) the w-universality relation (4.52) and (b) of the p—a’ universality relation (4.53).
4.5.5. p- and o-universality As examples of many other relations we quote a number of formulae for total cross-section differences, all written in a form so as to give a value of 1 at infinite energies. (i) p-universality or additivity of quark interactions leads to &r(p~p) &r(ptn) —
R 6=
~l;
(4.50)
(ii) p-universality plus isospin invariance ~o(K~p) &i(K~n) ==~1; (~±~) —
R7=
(4.51)
(iii) co-universality or equality of cross sections for q~,,and ~ interacting with R8
=
3[&~(K~p)+ &i(K~n)]1 &i(p~p)+ &i(p~n)
(4.52)
s—~
(iv) p- and co-universality 3i~a(K~p) ~a(ir~p) ~1. (p~~) —
R9=
(4.53)
The experimental results on the ratios (4.52) and (4.53) are shown in fig. 4.22. Within relatively large errors they seem to be consistent with 1 at energies above 20 GeV. In conclusion all asymptotic formulae for total cross-section differences are either verified at present energies or have the proper tendency to be satisfied at larger energies. This conclusion is not unique, since the deviations from the predictions of asymptotic theorems could be of a systematic nature [75L1].
G. Giacomelli, Total cross sections and elastic scattering at high energies
172
4.6. The energy dependence at asymptotic energies The discovery of rising hadron—proton total cross sections has generated many speculations about their asymptotic behaviour. A wide range of theoretical speculations give very different asymptotic extrapolations. This is illustrated in fig. 4.23 for the case of pp scattering. The various curves correspond to: (i) An empirical ln2s dependence: = 38.4+ 0.5 1n2(s/137). (4.54) (ii) The prediction of the impact picture based on studies of massive quantum electrodynamic prediction eventually leads to an asymptotic ln2s dependence, but is between a In s and ln2s dependence in the energy range covered by the figure. (iii) An empirical ln s dependence from a geometrical scaling model with a logarithmicallygrowing radius [73C 1]. This
u~ 01(pp)= 28.2 (1
+
0.068 ln s).
(4.55)
(iv) A dependence arising from a threshold type behaviour due to particle production a~0~(pp) = 73
—
325[ln(s/0.l
1)]’.
(4.56)
The predictions of these formulae differ considerably at energies higher than those of the tSR. One could also add other types of behaviour like: (v) Asymptotic constant behaviour from Regge-cut models. (vi) Oscillating behaviour from complex Regge poles associated with dynamical thresholds. (vii) Asymptotically growing cross sections from s- and t-channel unitarity arguments. Until a new generation of accelerators is constructed, the only hope of resolving the question of asymptotic behaviour lies with cosmic-ray measurements of the proton—nucleon cross sections. The existing data on extensive air-showers have been recently analyzed with the purpose of obtaining the absorption-cross section of protons on air nuclei. From these one tries to obtain the free pp total cross section using the Glauber multiple scattering theory. The present data and the 15C
“-I,,
.g
/ 125
125
lao
~
/‘
100
~j\-’+o (‘~“
75 50
Q
50
25
~ -
---.—.,~——-.
-
;:;~32~nc5b01
----~
LABORATORY MOI~IENTUM (GeVk)
Fig. 4.23. Typical high-energy model extrapolations of the proton—proton total cross section [74B8J.
G. Giacomelli, Total cross sections and elastic scattering at high energies
173
method of extracting the pp total cross sections are not adequate to reach any conclusion on the energy dependence in the laboratory momentum region between l0~to l0~GeV/c [74B8]. In fact one cannot distinguish between a constant cross section at 42 mb and a ln2s dependence. This is because (a) the absorption cross sections are poorly known; (b) the elementary pp total cross section is large and therefore the air nucleus is like a black disc to protons and the absorption cross section is independent of a~ 01(NN);(c)the conversion from cabs to cr~01(NN),according to the Glauber theory, depends on the assumptions on the impact parameter profile of the proton and is therefore model-dependent. One may wonder and ask a sort of philosophical question about what happens to the size of the elementary particles when the incoming energy becomes infinite: does the proton become as large as a billiard ball? Well, at most one can have the whole energy of the universe concentrated in the incoming proton. In this case if a formula like (4.54) holds one gets Utot(pp) 66 b, which corresponds to an interaction radius of 46 fm, to be compared with 1 .1 fm at present energies! The fact is that a logarithmic dependence is a very weak one indeed. 5. Elastic scattering 5.1. Introduction The main features of all high energy differential distributions for hadron—hadron elastic scattering processes are: (i) a large and narrow peak at small angles, probably with two slopes; (ii) some wiggles at intermediate angles; (iii) a very small cross section at large angles, and (iv) a backward peak. With increasing energy the slope of the forward peak becomes steeper (shrinking the forward peak), while the large angle and the backward cross sections decrease very fast with increasing energy. It is customary to classify the high energy angular distributions in five regions (see Introduction). We shall follows this classification, neglecting the very small-t region, the Coulomb region, which is theoretically well understood. 2) 5.2. The Coulomb-Nuclear Interference Region (0.001 < ti < 0.01 (GeV/c) Experiments in this angular region study the interference between Coulomb and nuclear scatter-
ing, which yields information on the real part of the forward non-spin-flip part of the nuclear scattering amplitude (the spin-flip part contains a sinO term and therefore does not contribute at small angles). The absence of spin—spin terms makes the irN and KN systems more amenable to a phenomenological analysis in the interference region than the NN system. In practice for the NN system one often neglects spin—spin effects, hoping that they are small. Three different experimental methods have been used to measure the differential elastic scattering cross section in this angular range. In the first method one measures angle and range of the recoil proton, which has a very low energy and recoils at angles close to 90°.The method makes use of the fact that the laboratory kinetic energy, T~,of the recoil proton is directly related to the four-momentum transfer through
G. Giacomelli, Total cross sections and elastic scattering at high energies
174
He Gas OUfL :Liquid He Supply
~Vl
-
V2
-
Hydrogen Supply
-
Frozen---~~ Hydrogen
Liquid He
—
i!~
Gate Valve
—
Beam
r
L.~..L.
,.J
.
I
-i
Beam Chamber
L-.~ II
I
Fig. 5.1. Diagram of the gas-jet target used in experiments with the internal proton beam at the Fermilab accelerator. (See text for a description of the operation [73B41.)
the relation t—2m~T~.
(5.1)
In order to reach very small values of t, the target has to be very thin. In practice, experiments of this type were performed with the internal proton beams of the IHEP and Fermilab accelerators, striking a gas jet of hydrogen moving at supersonic speed. The gas-jet target was designed to produce a 200 msec burst of hydrogen gas into the circulating proton beam, usually three times for each accelerator cycle. The principle on which it operates is sketched in fig. 5.1: at a given time, valve V1 opens and lets through about 50 cm3 of hydrogen gas under pressure of 1 to 4 atm; this gas was originally confined between the valves V 1 and V2. A single pulse of value hydrogen 3 to the jet intolerable of 2 gas X 1 would 0~ raise the pressure of the whole accelerator volume of 50 m torr. Thus one has to trap the hydrogen. This is done by cryopumping and precooling the hydrogen and then freezing it in the helium-cooled cup enclosure. Additional diffusion and ion pumps restore between accelerator cycles the vacuum of the accelerator in this section to 10-6_b 7 torr. The gas density is about 5 x 1 0~g/cm3 and the full width at half height of the jet is about 12 mm. In the Fermilab experiment, when the protons were accelerated at the rate of 100 GeV/sec, the beam had a diameter of ~ 5 mm, cirôulating with a period of 20 ~.isec.The beam traversed the gas jet about 1 0~times during the jet pulse. Thus with 2 X 1012 circulating protons per machine pulse and a 6 sec repetition period, the effective luminosity was l0~~ cm2 sec’,
G. Giacomelli, Total cross sections and elastic scattering at high energies
175
Jet Target
Deto~\~
-
Fig. 5.2. Layout of the small angle pp elastic scattering experiment in the internal target area at Fermilab [73B4]. The internal proton beam is intercepted by the gas jet target. Recoil protons are detected at large angles by means of solid-state detectors.
500 I
I I
I
I
I I
I
I
I
E:400GeV
400
300
BEST FIT FOR cr~4Omb pr+0.040 2/~67.4/42di. X
~200 .0
E
b
10080
a I
60
U
50 II,,,I.II’I”
.005
.010 .015
.020 .025 .030 .035 ti (GeV/c)2—~
Fig. 5.3. The differential cross section for pp elastic scattering at 400 GeV measured with the layout of fig. 5.2 [73B4I curve represents the best fit to the data, which are normalized as discussed in the text.
-
The solid
G. Giacomelli, Total cross sections and elastic scattering at high energies
176
~4Ai
2
TOP VIEW
SIDE VIEW -
__—
_--_---~_~
~—
-
SIDE
VIEW
6 mmd
VIEW ALONG BEAM
Fig. 5.4. Sketch of the layout used at the ISR by the CERN—Rome Collaboration to measure the elastic scattering in the C—N region [73A1 I. The two protons scattered in the vertical plane were detected by tiny scintillation counter hodoscopes, which could be lowered or raised close to the circulating beams.
11.8
+
15.4 + 15.4 GeVk
11.8 0eV/c
3O0~
~
:~
:~
7
)
NUCLEAR ‘~‘~—cOu1CMB
50
NUCLEAR ‘~—CO(JL0MB
_______________________________________________
O
1
2
3
4
5
6
7
8
I
01
2
ti
3 4 (0eV’)
5
I
6
78
9
101112
13x10’
Fig. 5.5. pp elastic scattering angular distributions measured with the layout of fig. 5.4 at equivalent laboratory momenta of 290 and 500 GeV/c [73A1 I - The solid line interpolating the points is a fit to the data. The separate contributions of’ Coulomb and nuclear scattering are indicated by dashed lines.
G. Giacomelli, Total cross sections and elastic scattering at high energies
177
corresponding to 1 08_ 1 0~interactions per second. The target is so thin that even very slow recoils, lower than 0.2 MeV, could be recorded. For elastic scattering at small angles, the angle of the recoil proton is practically independent of beam energy. Thus one may use the same apparatus for measurements at various primary energies as the accelerator cycle proceeds. Fig. 5.2 shows the layout of the experiment performed by a USSR—USA Collaboration. The angular region of the detectors was 1.2°to 6.3°from 90°when measurements in the Coulomb-nuclear interference region were performed; for scattering at larger angles the detectors were moved to cover the 1 .7°to 12° angular range. There are also measurements of pd elastic scattering. From the measurements of pd and pp elastic scattering at Serpukhov, Beznogikh et al. [73B2] computed pn elastic scattering using the Glauber theory of multiple scattering. Fig. 5.3 shows the experimental angular distribution at 400 GeV. From the angular distributions it is difficult to see the Coulomb-nuclear interference. Moreover the gas-jet target does not allow an easy normalization of the data. Thus, from formulae (4.7)—(4.lO), the best determination of p was obtained assuming exact values of ~ At the ISR an experiment employed small scintillation counters placed a few millimeters from the ISR beams. An elastic event was defined as a coincidence between a small counter above the beam on the left and the collinear counter below the beam on the right (see fig. 5.4). Angles of 10 mrad were involved and were measured to a precision of better than 0.1 mrad. Fig. 5.5 shows two angular distributions obtained at equivalent laboratory momenta of 270 and 500 GeV/c. The previous two methods are quite specialized to match the available accelerators and may be used only for pp collisions. For measuring the elastic scattering of other particles in the Coulombnuclear interference region one resorts to the measurement of angle and momentum of the forward-going particle only. In this case, very small scattering angles are involved and problems of multiple scattering may be severe. Fig. 5.6 shows a sketch of the set-up used at the Brookhaven AGS for the measurement of elastic scattering at very small angles [69F 11 The experiment used hodoscopes of scintillation counters; historically it was the first experiment to make an extensive use of a computer on line. A similar layout was used at Serpukhov [74A6], while experiments of the same type presently being performed at Fermilab employ proportional wire chambers of very high spatial resolution [75A1]. Some typical differential cross sections for irt p elastic scattering in the Coulomb-nuclear region are shown in fig. 5.7. As already stated, the elastic-scattering data in the 0.001 ~ ti ~ 0.01 (GeV/c)2 region are fitted to a formula like (4.7), which contains the three parameters p, b, 0tot~ Assuming values for 0tot and for b, one may obtain p, the ratio of the real-to-imaginary part of —
.
iT ABSORBER LIQJ-12 TGT BEAM
Al ~
CTI SI
42
~43
/
~
COUNTER
M~NETS
CT2 1-101 S2 H02 HI HI H2 (XX) (XX) (X)(Y)(X)
H4(x,Y)
Fig. 5.6. Experimental layout of the equipment used by a Brookhaven group to measure elastic scattering in the Coulomb-nuclear interference region [69F1 I. CT1 and CT2 are beam gas ëerenkov counters. The H symbol denotes hodoscopes of scintillation counters.
178
G. Giacomelli, Total cross sections and elastic scattering at high energies
(~)
IOO\
40 ~
20O~
b.— ~
100 80 60 40
7T’p~7T’p 1796 GeV/c
.‘.
+
20
0
003 2 -t (GeV/c)
001
-
—-..I-..~-- -
I
~5O~
,~ _._. —~ ~
-I.— i—
•~ •~ -•—
~
-
0~~~ - •— - —
— - — -
4 I
I
I
0.01
—t
I
0.02
—
I
0.03
(0eV/c)’
Fig. 5.7. Typical elastic scattering cross sections in the Coulomb-nuclear interference region measured at the Brookhaven AGS [69F1I and at IHEP [74A6l. (a) ,rp at 26.23 GeV/c, ir~pat 17.96 GeV/c and (b) 33.5 GeV/c. The solid lines are fits to the data using eq. (4.7), while the dashed lines represent fits with p = 0. The bottom graph in (b) gives the differential cross section after subtraction of single Coulomb scattering.
the forward-scattering amplitude. At high energies, these values of p have relatively large systematic errors because p is small and thus the effects on the angular distribution are small, as can be seen in fig. 5.7. Moreover, the parameters p, b, u~ 0~ are strongly correlated and one needs very good measurements of b and a~0~ to obtain reliable values of p. The measured p-values are plotted in fig. 8. 1, where they are compared with the predictions of dispersion relations. Many of the measured values scatter considerably outside the quoted statistical errors. They do not always agree with the predictions of dispersion relations (see section 8).
G. Giacomelli, Total cross sections and elastic scattering at high energies
179
5.3. Elastic scattering in the diffraction region [0.01 < ti < 0.5(GeV/c)2] Experiments in this angular range yield information on the over-all spatial structure of the particles. Phenomenologically one would like to determine: (i) the shape of the angular distribution, in particular, of its slope or slopes; (ii) the energy dependence of the slope, and (iii) the relation among the slopes for different elastic-scattering processes. Over a limited t-range the differential cross section may be well represented with an exponential function in t: du/dt = a exp (bt).
(5.2)
Over a larger t-region the cross section deviates from an exponential form and it may be represented with formulae of the type du/dt = A exp (Br + Ct2) da/dt
=
(5.3)
A [exp(b 1t)
+
a exp(b2t)].
(5.4)
Eq. (5.3) eventually leads to nonsense at large angles. Instead of eq. (5.4) one often quotes the “local” slope from a fit of a small part of the angular distribution to eq. (5.2). In order to study the energy dependence of the slope parameter, one has to define precisely the
g.
I
I
I
32GeV
-
b=—Ii.59±.06 4.
-
—3-
-
0)
I—
2 . . i
>-
<
~ 85~.
“~
—6—...-
5. 4 -
S.
...
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-
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-
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. 5.
-
.
0.0
I
I
.04
.08
t
I
I
.12
.16
(GeV/c)U 2 for pp elastic scattering at
incident energy 132 GeY [73B4j. Fig. 5.8. The differential cross section da/dt an in arbitrary units inofthe interval 0.005
G. Giacomelli, Total cross sections and elastic scattering at high energies
180
~&iJ
~.2.
~UUMCHAMBER W1NOCM’ THPI inTn WWCW/ Wi Si W2 S2 W3
PROION 2
MINIMUM DETECTED ~
~
~T~T_
____
~imrrJlH—H\
i~J,
S~~L
-~1_ \I~
~
~3irn~ ~
4,,ttm~
6.5mR
~
urn
4,
Sm -
—,tim,, ~ l85m
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~7TAWAYTM~~~E
SFIARK CHAMBER WI
Fig. 5.9. Apparatus used by the Aachen—CERN—Genoa—Harward—Torino Collaboration (ACGHT) to measure pp elastic scattering 2 at the ISR. The scattering takes place in the vertical plane and the scattered protons are detected in the 0.05 < ItI <0.4 (GeV/c) by two sets of spark chambers W 1—W3,
W~—W~ [72B3I.
t-range over which the slope is determined. The slope parameters b, B, b1, or b2 are connected to the radius of interaction (rb, rB. rbl or rb2). In the optical model with constant opacity the relation between r. and b. is: r.
=
~
(5.5)
Thus, the steeper the slope, the larger the radius of interaction. Some of the experimental equipment employed in this angular range is similar to that used in experiments at smaller t-values. 2 [73B4], (see fig. In the Fermilab experiment, covering the range 0.005 ~ ti < 0.09 (GeV/c) 5.2), the thickness of the solid-state detectors was not adequate to stop the recoil protons at all scattering angles. Thus some copper degraders were placed in front of the detectors on the high-t side of the apparatus, so that recoil protons would always stop in the detectors. Fig. 5.8 shows an angular distribution measured at Fermilab. At the ISR, an experiment was performed in the range 0.05 < ti < 0.4 (GeV/c)2, using two telescopes of six spark chambers located on either side of the interaction region, above and below the vacuum chamber respectively (see fig. 5.9) [72B3]. An elastic event was defined as such if it
G. Giacomelli, Total cross sections and elastic scattering at high energies
181
gave rise to two collinear tracks and nothing else. Fig. 5.10 shows the pp differential cross section at ..,/~= 53 GeV. One clearly observes a change in slope at —t 0.13 (GeV/c)2, the local slope changing from 12.8 to 10.8 (GeV/c12. In a typical set-up used in more conventional experiments one measures in angle and momentum the forward-going particle, as in the scattering at lower t-values; but now, in addition, one measures also the recoil proton angle with a side detector (for instance a spark chamber). This extra requirement becomes a necessity at larger values of it i, since the elastic cross section becomes small, and the contamination from inelastic events becomes progressively worse. We shall in fact see in the next section that at larger scattering angles one is forced to measure also the momentum of the recoil proton. 5.3.1. pp-÷pp We shall first discuss the pp elastic scattering data, since these are most abundant and are available up to the highest energies. Many models have been proposed to explain the change in shape of the angular distribution at —t ~ 0.13 (GeV/c)2, fig. 5.10. In this section we shall mainly try to get a feeling of the physical meaning of the change in shape, without going into detailed analyses of the models. One may write two possible rough representations of the data in terms of the sum of two exponentials:
10’
-
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EcMr267+26.7GeV 0.98- ~6 EVENTS
\\\ \ \\
~ \—
_________________________
()
{(Gev/c)~]
‘
I
I
I
I
10’-
\
C.)
0
e~~8t
i
~
t
~
3 ~ BARBIELLINI ET AL 4 + BEZNOGIKH ET AL 5 )< CHERNEV ET AL
8
1
6 0 BELLETTINI ET AL
S—(GeV)2
-
I 1o2~L
0
I
005 OX)
I
0.15 -t
I
I
020 025 (0eV’)
I
030
I
035
Fig. 5.10. t-distribution for pp elastic scattering at a c.m.
•0 AMALDI THIS EXPERIMENT ET AL
I 2
/
10
20
50
100
200
500
000
5000
Fig.
5.11. The slope of the diffraction peak, b(s), for ItI ~ 0.12 (GeV/c)2, as a function of the square of the c.m. energy. The solid line is a fit of b (s) = + 2~In (xis
2 energy of 53 GeV [72B3J.
2000
to the data points
fori ~
100 0eV
0) only (73B4I.
-
G. Giacornelli, Total cross sections and elastic scattering at high energies
182
do/dt
=
A1[exp(— 11 in)
0.37 exp(—341t1)]
(incoherent sum)
0.27 exp(—l8iti)J 2
(coherent sum).
+
(5.6)
do/dt
A2[exp(—Siti)
+
Recalling the optical model relation (5.5), eq. (5.6) implies that the “feel” 2] two andinteracting 2.3 fm [b ~protons 35 (GeV/cY2I. twocourse regions, radii of approximately 1.3 fm 11 (GeV/c) Of thiswith is too simple an interpretation, but[bin fact in many models the scattering amplitude is written as the sum of at least two contributions with different t-dependences, i.e. which differ in their space behaviour. In other models the change in slope is due to effects of the multiple scattering type inside the nucleons. Fig. 5. 11 shows a compilation of the energy dependence of the b coefficient for data in 0.02< iti < 0.12 (GeV/c)2. The energy dependence for b determined in the 0.15< in < 0.3 (GeV/c)2 range is shown in fig. 5.12. In both cases the increase is fast up to s ~ 100 GeV’; then the energy dependence becomes weaker and is almost linear in a semilog plot. There are some systematic differences among the different data (one has to remember that the slopes are affected by systematic errors of the order of 0.2—0.3 units). A fit of the data with 5> 100 GeV2 to the formula, suggested by Regge-pole theory, b(s, t) = b 0(t)
(5.7) 0.44 for the region 0.02< ti < 0.12 yields b0 =and 8.23 (GeV/c)’ b ± 0.27 and a = 0.278 ± 0.024 with X 2 region 0 = 9.21 ± 0.942). and a = if 0.100 0.062 the 0.15
2a(t) ln (s/s0)
2/DOF
Compilation of Elastic Slopes at<—t= ~O.2(GeV/c)2
I ~
I
I
I
IlIlIl
I
I
1111111
I
I
I
I
I
I
111111
I
I
I
I
I
I
I
I
1111111
I
I
I
I
I
~
~~:
11111111
~
111111
~ U
I
20
I
I
111111
50
100
I
200
1111111
I
500 1000 2000
s(GeV)2 Fig. 5.12. Compilation of elastic slopes b of eq. (5.7) at an average value (—t)
0.2 (GeV/c)2.
G. Giacomelli, Total cross sections and elastic scattering at high energies
183
tuitive explanation of the energy dependence for the slope parameter b is that the size of the interaction region grows with increasing energy. 5.3.2. Other types of elastic scattering Fig. 5.13 shows 7(p elastic scattering data in the diffraction region at seven laboratory momenta between 5 and 200 GeV/c. Since the data above 100 GeV/c [74A3, 74A7], are still preliminary, they have been normalized to the optical point (with p = 0) at t = 0. The figure is illustrative of the quality of the data as well as of the t- and s-dependences. The n-dependence is of the quadratic exponential type, eq. (5.3). There is no clear indication of a break, like that observed in pp elastic scattering at 0.13 (GeV/c)2. In fact, the data are not good enough to prove or disprove the existence of such a break. Fig. 5.13 shows that the energy dependence of the angular distribution is weak, if the$ is any. The differetices and similarities among ~ K~p,and p~pelastic scattering are illustrated in fig. 5. 14, which displays lines interpolated through the data at 100 GeV/c. With increasing energy the differences between particle and antiparticle cross sections at t = 0 and t 0 tend to disappear. The differences between irp and Kp scattering also become smaller. Only a small amount of data exist for the elastic scattering of short-lived hadrons. As an example, fig. 5. 1 5 shows the differential cross section for ~p -~ ~p at 23 GeV/c. Within the limited statistics it would seem that the scattering is similar to that of the other long-lived hadrons [74M2, 75B2]. In order to make more quantitative statements one has to perform fits of the data, for instance with equations of type (5.3). The fits yield values of the dimensionless quantity C/B2 of the order of 5%. From the fits one may compute the local slope at t = —0.2 (Ge V/c)2
mi
b~_ 0.2= (B + 2Ct)~,_2= B
—
0.4G.
(5.8)
The results are shown in fig. 5.12 for ir~p,K~p,and p~pelastic scattering. The points scatter con-
siderably. Part of this is due to the fact that systematic errors are not included and also because the t-range should have been chosen to be rigorously equal at all energies. In any case the graphs allow the following conclusions to be drawn: (i) The b-coefficients for K~pand pp elastic scattering have a considerable energy dependence (shrinking of the diffraction pattern). The energy dependence decreases with increasing energy. Within errors the slopes of pp and pn scattering are the same. (ii) The b-coefficients for ir~pand Kp elastic scattering increase slowly with energy. (iii) The antiproton—proton is still expanding its diffraction pattern, that is ~ decreases with increasing energy (antishrinking), but its slope is approaching that of pp. (iv) It seems that there is a tendency at high energy to have ~ bKsP bK-P ~ The shrinking of the diffraction pattern is very likely a universal phenomenon at high energy, but the rate of shrinking differs with energy and with the types of particles involved. 5.3.3. The cross-over phenomenon The elastic differential cross section differences ~=~(~p)
—~(xp)
(5.9)
are positive at small values of t and become negative at larger t. This comes about because of the
G. Giacomelli, Total cross sections and elastic scattering at high energies
184
7
lryg-lr-p
ANL’72-
5 GeV/c
IHEP’73-25
IO*.
Q IOa:
IO*:
IO -
IO-
+125 GeV/c
12
7 IO -0
1 .
.
lip7
. .
PP ‘,
_\
le
.
-P
‘i
IO
.
l
.
.
% s
?
IO0 GeV/c
Qg
IO-
‘I
Ii I4OGeVk
I
r-p ZP b, K-P ,& K*p \;:
1
?? #f 200
0.1
GeV/c
I-
0.1 :
If
0l.Olo
I
I,
Q2
I
I
0.4
I
0.6
I
I
0.6
f
,I
f 1.0
-t(GeV/c)’
Fig: 5.13. Compilation of w-p elastic scattering angular distributions for laboratory momenta from 5 to 200 GeV/c [75Gl].
02
04 -t
0.6 08 (GeV/c12
3
1.0
Fig. 5.14. Illustration of n*p, K*p and p’p elastic scattering angular distributions at 100 GeV/c.
G. Giacomelli, Total cross sections and elastic scattering at high energies
~7T-p
+ ~
~I0.
185
-
8 6
o
.05
.10
.15
-t (GeV/c)2
.20
.25
Fig. 5.15. i(p and ~p differential cross sections at 23 GeV/c [74M2].
difference in the total cross sections (and thus of the t = 0 points) and of the difference in the slopes of the diffraction patterns. Thus the differential cross sections for particle and antiparticle on protons cross each other. The cross-over point, t~,defined as that value of four-momentum transfer at which particle and antiparticle have equal cross sections, is around 0.1—0.2 (GeV/c)2. The cross-over point is well localized at laboratory momenta around 5 GeV/c; it is much more difficult to localize it at 100 GeV/c, because the differential cross sections of particle and antiparticle have become quite similar. Table 5.1 gives values of t~at different momenta. At energies below 20 GeV/c, t~was determined from linear fits of the data near the cross-over point. At ‘higher energies, approximate values of t~were deduced from experiments, not specifically designed to measure t~,using the formula =
rI ln L
a~ 2(i~p)-1 / 2 I / [B(~p) 1 + p2 a~05(~p) 0~(xp) 1 + p (xp)
—
B(xp)]
(5.10)
.~‘
where only the slopes B were obtained from elastic scattering data; a~0~ was taken from tabulations of total cross section measurements and p from dispersion-relation calculations. Present experimental data are not adequate to establish if t~is always at a constant location, independent of momentuni, or if it moves to larger or smaller absolute values. The average values of t~are slightly different for the different particles. Differences between particle and antiparticle cross sections isolate interference terms between amplitudes of opposite C-parity in the t-channel. To a good approximation the difference is proportional to the imaginary part of the helicity non-flip vector exchange amplitude. There are many models which try to predict the energy dependence of t~.For instance, in a geometrical model ImF~J0(R\/7).
(5.11)
Thus equating the first zero of J0(R~/~i) with the cross-over point, one may calculate the effective radius for the source of the C = —1 amplitude according to the relation
G. Giacomelli, Total cross sections and elastic scattering at high energies
1 86
Table 5.1 Cross-over points, —ta, at different laboratory momenta (74A1, 75B1, 74A3]. Laboratory momentum (GeV/c)
t~(7rp) (GeV/c)Z
t~(Kp) (GeV/c)2
t~(pp) 2 (GeV/c)
3 3.65 5 6 10.4 25 40—43 100
0.094 0.166 0.158 0.125 0.231
0.186 0.189 0.189 0.202 0.211
0.169 0.157 0.159 0.172 0.146 0.15 0.3 0.17
±0.040 ±0.023 ±0.022 ±0.016 ±0.007
—
—
—
—
0.25
±0.05
0.15
±0.006 ±0.006 ±0.007 ±0.010 ±0.004
±0.05
±0.005 ±0.004 ±0.006 ±0.010 ±0.002 ±0.03 ±0_i ±0.08
R~ 1
0.475/~/~7~.
(5.12) 2. One obtains Rc=_
R~_1is in fermi if t~is in (GeV/c) 1 0.8—1.2 fm. In geometrical models t~ is expected to move to smaller values as the energy increases, while in factorizable Regge pole t~ would be energy independent and in other models t~could move to larger values. The available data are probably not adequate to discriminate among these possibilities. In particular, they are not adequate to establish if the sources of the C = + 1 and C = 1 amplitudes are correlated or not and if the source of the C I is the periphery of the strongly absorbing region. —
—
5.4. Scattering at large angles The large-angle elastic-scattering differential cross section decreases very fast with increasing energy. Thus its measurement presents severe problems of counting rates and of background. One is forced to use beams of high intensity, detectors which cover large solid angles and measure both secondary particles in. angle and momentum. Fig. 5.16 shows one of the layouts employed in a Brookhaven experiment [73C6]. The beam
PROTON
_______
:W34W5~
~IIIJ
H2 PION
Fig. 5.16. Layout of a Brookhaven experiment on large-angle elastic scattering. S5—S10, Vj and V2 are scintillation counters. Wi—W12 are wire spark chambers and PC is a proportional wire chamber [73C6].
G. Giacomelli, Total cross sections and elastic scattering at high energies
187
Magnet 2 beo.~
intersection
Magnet I WI -W6 magnetostrictive wire- chamber I
I
I
0
1
2
I
S1-S4,M1-M4,AI,A2 scintillation counters
3m
Fig. 5.17. Layout of the large angle pp elastic scattering experiment at the CERN—ISR. The scattering takes place in the vertical plane l74Bl0].
impinged on a 60 cm long liquid hydrogen target. Both scattered and recoil particles were momentum-analyzed using counter hodoscopes and magnetostrictive read-out wire spark chambers. Fig. 5. 17 shows the apparatus used at the ISR to measure pp elastic scattering at large angles [74B 10]. Also this apparatus allowed the measurement of both angles and momenta of the secondary particles, thus reducing the inelastic background to negligible values. 5.4.1. pp—~pp We shall again start discussing the pp system for which there is considerable data. Fig. 5. 1 8 shows two angular distributions measured at the ISR. The main features of these distributions are: (i) the fall-off of the small-angle cross section by six orders of magnitude when t goes from zero to 1, (ii) the appearance of a sharp minimum, followed by (iii) a secondary maximum and then by (iv) a slower fall-off. The angular distributions of fig. 5.18 are clearly reminiscent of diffraction patterns. The CHOV group [75K 1] fitted the t-distributions with the formula (5.13) dcr/dt ~ Al exp(Bt/2) ÷ \/~7Aexp(Dt/2) + iipl2. —
One of these fits is shown in fig. 5. 18 (b). From the fits one can derive the location of the dip and the height of the secondary peak, which are given in table 5.3 and plotted in fig. 5.19. The location of the minimum moves inward with increasing energy: it is at 1.45 (GeV/c)2 at ‘/~= 23 GeV and at 1.3 (GeV/c)2 at \/~= 62 GeV. The height of the secondary peak is, however, increasing with energy. The inward moving of the dip and the increasing of the secondary maximum may come about as the result of an interference effect of a shrinking small-t amplitude with a large-t amplitude which is energy independent (this is a justification for using eq. (5.13)). We shall discuss the implications of the data of fig. 5. I 8 extensively in section 9. Fig. 5.20 shows a compilation of pp elastic scattering data, over a broad energy range; the figure illustrates the gradual development of an inflection point at ItI = 1.4 (GeV/c)2 into a deep minimum as the energy increases. The new results from Fermilab [75A2] agree with this picture. For comparison it is also shown 4(t), 0G where G(t) is the dipole expression of the electromagnetic form factor du/dt =
(do/dt)~
(5.14)
G. Giacomelli, Total cross sections and elastic scattering at high energies
188
(a) V~z53.2GeV
10.1~
o Borbiettini et at. (1972) I This experiment — OpticaL model
2_
1O.
io~-
10
-
do/dt
[pb/Gev2]
\
1.
PP~PP ~Fs.23GeV
(b)
-
~ b 0
1~ -÷ 1 is a Fig. 5.18. da/dt prediction. (b) /~= (pp 23 pp). GeV(a)l74N1]. -t ~j~” (0eV2) 53.2 TheGeV; solidthe curve data are four normalized parametertofit1 at as texplained = 0ieve [74B10]. in theThe text. solid Theline large is.t[Gev2] an angle optical data model have a scale error of about ±20%.
G(t)
(1
÷t/0.7l)2.
(5.15)
If one assumes that the matter distribution inside the proton is equal to its charge distribution, then the differential cross section should be, in first order, proportional to G4(t). This is not born out by the data. Fig. 5.21 illustrates the energy dependence from a different point of view, i.e. by considering the s-dependence of the differential cross section at various values of t. It is evident that at very low energies and high four-momentum transfers the s-dependence is much stronger than at higher energies and lower momentum transfers. This suggests that part of the energy dependence may be due to a sort of kinematical effect, since large t-values are only allowed at higher energies. The shrinking of the diffraction pattern nevertheless continues at high energies, as was illustrated in the discussion of the b-parameter in the previous section.
G. Giacomelli, Total cross sections and elastic scattering at high energies
do/dt (2 nd max.) 0.15
.
189
S CHOV 0 ACHGT
(a)
[tiblGev2]
::: I
1.6
-
I
I
tdlp [Gw2}
(b) -
6~
~[GevJ Fig. 5.19. (a) Height of the second maximum in pp elastic scattering; (b) Position of the minimum near t
2 [74N1].
1.4 (GeV/c)
5.4.2. Other types of elastic scattering The general behaviour of the ~ K~pand p~pelastic differential cross sections at 5 GeV/c is shown in fig. 5.22 over the whole kinematical range. Data at 10 GeV/c are shown in fig. 5.23 for ~ and K~pelastic scattering. np data for np np at different energies are shown in fig. 5.24. The angular distributions exhibit a rather complex structure and a strong s-dependence. The following specific comments can be made: (i) The ~r~pelastic scattering cross sections have dips at —t ~ 0.8, 2.8, 5(GeV/c)2 and u = —0. 1 5 (GeV/c)2 (see table 5.4). These dips are clearly observed at 5 GeVIc and have become -~
Table 5.3 Position of the diffraction minimum in pp elastic scattering at various energies [74B10, 74N1]. E*
minimum
(GeV)
(GeV/c)2
23.0 30.7 40.9 53.0 62.0
1.45 1.45 1.38 1.37 1.30
±0.02 ±0.10 ±0.04 ±0.04 ±0.05
1 90
G. Giacomelli, Total cross sections and elastic scattering at high energies
102 I
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I
~
I
6
ItI
8
(6eV2)
pp differential cross sections at different energies.
Table 5.4 Position of the dips (or of the shoulders) in low-energy elastic scattering Scattering
Laboratory
ifp
momentum (GeV/c) 5 10 5
t t2
t3
u
(GeV/c)2
(GeV/c)2
(GeV/c)2
(GeV/c) 0.8 0.8 0.8
2.8 2.8 2.8
5
0.8
2.8
—
—
1
Kp K~p np pp
10 11 11 11 13—21 3—20
2
2 0.5 1.3 1.3
2.2
— —
5
—
— —
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G. Giacomelli, Total cross sections and elastic scattering at high energies INCIDENT BEAM MOMENTUM
10i26
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.
(1968,1972)
\~~~Ietat
•Jctyde (19661 Ar,kerbirsttetal (1968) t ~ Atlabyetat (1966,1968)
08
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10 BEHAVI01I~OF ELASTIC SCATTERING AT 5.0 GeV/c 10
1~
I
e
\\NQ ~t
•‘~“
\.
/1
~pp
E
TTp
K’p 0
io~.
I. S
I
0
20
~
I
I
I
4.0
60
8.0
—t (Ge~)2 Fig. 5.22. The general behaviour of the elastic differential cross sections at 5 GeV/c for lr±p,K~p,pp and ~p when smooth curves are drawn through the original data points [73E3].
G. Giacomelli, Total cross sections and elastic scattering at high energies
192
100
• •
2
•
10
1
I
(a)
I
I
I
• • I
101 100
(b) KpKp 10 GeV
101..
-
\
ii p—-rrp 10GeV/c .1r+ orr
~
• •
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~ ~
10~ 3
I
~l0
M
+
~t0~
: +
tt
~.
~
-t (6eV/cl2
l~
l~6
5
—t (0eV/c)2 10
~5
Fig. 5.23. The complete elastic scattering angular distributions at 10 GeV/c. (a) ir~p[73B7]. (b) K~p[73B8].
only shoulders at 10 GeV/c. Thus the structures seem to diminish as the energy is increased. This is in contrast with the dip in pp scattering at 1.4 (GeV/c)2, which instead becomes more prominent as the energy increases. The structure in pp is interpreted as a diffraction-like structure; this is probably not the case for the low energy structures in ir~pscattering, which may be connected to low energy s-channel effects. (ii) At 10 GeV/c the ir~pcross section flattens out in the region around 0cm = 90°(from 7< tI < 11 (GeV/c)2) reaching the value of 1 nb/GeV2. It has decreased by a factor of about 100 while going from 5 to 10 GeV/c. The energy dependence may be expressed with the form (da/dt) 0=90~ A s_fl
A
~—(7±1)
(5.16)
which is expected if the scattering process proceeds via the interactions between point-like constituents (see section 10.4). (iii) Similar statements, concerning dips and energy dependences, can be made about K~pand ~p scatterings.0cm = 90° all the cross sections for ir~p,K~pand pp seem to be approximately equal; on At hand the pp cross section is about a factor of 100 larger. the(iv) other Note added in proof. The recent measurements of ir~pelastic scattering angular distributions at 100 and 200 GeV/c [75A2
I suggest the presence of two breaks, at —t~0.5and 1.5 (GeV/c)2, which develop as the momentum increases. Thus they would seem to be of the same type of the break at —t — 1.4 (GeV/c)2 in pp elastic scattering.
G. Giacomelli, Total cross sections and elastic scattering at high energies
ir
193
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-t (GeV/c)2 Fig. 5.24. Neutron—proton elastic scattering differential cross sections da/dt versus t for five different incident momenta.
5.5. Backward scattering The experimental study of backward elastic scattering presents all the problems previously mentioned for large angle scattering. Fig. 5.25 shows the layout used at the CERN—PS to study backward scattering in the 5 to 10 GeV/c region. The system was triggered requiring that (i) there was a forward positive particle not anticoincidized by the gas threshold ~erenkov counter C 3, (ii) there was a backward going charged particle and (iii) no counts in the anticoincidence counters. Only the forward proton was momentum analyzed using a magnet and sets of wire spark chambers [68Bl]. We have already seen from the scattering data at 5 GeV/c shown in fig. 5.21 that all elastic processes at intermediate energies have a backward bump2.(except pp naturally). The magnitude of theMoreover backwardthe peaks ~differs distribution exhibits also a dip at u ~ —0.15 (Ge V/c) considerably: the 180°ir~pcross section is almost 10 times that of 7Cp, while that for K~p is about the same as for ?p and about 100 times larger than that for either Kp or~~p scattering. Fig. 5.26 shows the energy behaviour of the differential cross section for ?p backward scattering. Above 10 GeV/c the only data which exist concern ir~pelastic scattering only. The energy dependence of the u = 0 cross sections is shown in fig. 5.27. The dependence is of a power type
G. Giacomelli, Total cross sections and elastic scattering at high energies
194
4~J~
—4-I-I
w4
Rt.—..R~
Fig. 5.25. Layout of an experiment at the CERN—PS on backward elastic scattering [68B1J. L110 are Counter hodoscopes used for triggering purposes. W1—W4 are wire spark chambers; A are anticoincidence counters and C3 is a threshold gas ~erenkov counter.
0.01
I
f
I (a) 17.07
—
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BACKWARD WIZKID MODEL ir,p
0.0
(b)
0.0
lr-p BACKWARD ELASTIC GORE MODEL (~ only)
0.0
625
0.01
—
0.01
—
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—
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~
I
-0.1
I
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I
-0.7
I
-1.0
-1.3
0.2
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2 -0.7
-1.0
—1.3
(G.V/c)° u(GeVk) Fig. 5.26. Backward scattering of (a) lrp and (b) irp [71B4]. The solid lines represent fits with the GORE model (see text).
G. Giacomelli, Total cross sections and elastic scattering at high energies
195
30
‘‘‘I
~
‘‘
ltp
— 01
.0
~
1—
-
ltp
~
t~N T
e
::~ ~
-
‘~
-
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-
~‘io-
-
-
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‘~,,
—
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5~_~-
-
I 6 8 10 LABORATORY
,
-~
~
~I 8 10 LABORATORY
-
I
20 40 60 80 MOMENTUM (6eV/c)
6
Fig. 5.27. The energy dependence of backward ir~pelastic scattering at u = 0. The lines represent possible eyeball fits to the data l74B14l.
20 60 MOMENTUM
60 (6ev/c)
Fig. 5.28. The slope of the backward peak in ir~pelastic scattering plotted versus energy. The lines are only meant to guide the eye 174B14].
with roughly ~ There may be a break at 10 GeV/c in the energy dependence of ~ scattering. The backward peak for ir~pis narrower than the forward one. Both ir~pdistributions may be fitted to exponential functions in t or u: (dG/dt)backward
=
(da/dt)~,~ exp{—bB(tmax
—
t)}
.
(5.17)
Fig. 5.28 shows the behaviour of the backward slope, bB, for 7rtp: it is apparent that the backward peak shrinks with increasing energy. Below 5 GeV/c the backward ?p cross section at fixed u plotted versus energy exhibits a number of structures which are usually interpreted as due to s-channel resonances. Above 5 GeV/c there are many interpretations in the framework of Regge models, parton models and optical models. Most theoretical papers try to interpret backward scattering in the frame of Regge pole models. In these models backward scattering arises from the exchange of baryon trajectories in the t-channel, namely of the £‘11236) in ?p scattering and of the z~°(1236), N~(l5l5)in ~rp scattering. Several specific Regge models have been used to analyze quantitatively the backward scattering data with: (i) Only Regge poles without cuts (in the notation of [71B4]: GORE = Good Old Regge); (ii) with poles plus “weak” cuts (WIZKID). The cuts, needed to remedy problems of unitarity in the s-channel, are generated with “absorption prescription”. (iii) poles plus “strong” cuts (SCRAM); (iv) poles plus cuts plus a prescription to remove effects from baryon parity doublets (HIPPIE = Hide Parity Partners). These four models allow fits of comparable quality. They differ considerably in their predictions at higher energies, where hopefully they will be subjected to more stringent tests, using differential
196
G. Giacomelli, Total cross sections and elastic scattering at high energies
Fig. 5.29. Illustration of t-channel baryon exchange for ir~pbackward scattering.
distributions of various types of scatterings and studying the energy dependence at fixed u-values. For instance the energy dependence of the differential cross section at u = 0 could in principle give indications of the presence of Regge cuts. On the other hand one could interpret backward scattering in terms of parton models; in this case the presence of a break in the established power dependence on the energy could give indications of the existence of parton—parton scattering, which could dominate backward scattering at high energies. In terms of optical models the explanations of backward scattering are reminiscent of those given for the forward peak. Clearly we are not close to a simple or unique explanation of backward scattering. 5.6. Total elastic cross sections The total elastic cross sections are obtained by integration of differential cross sections, which are usually not available for the whole angular range, in particular for very small and very large scattering angles. Therefore, in order to obtain the total elastic cross section extrapolations have to be made. Since the differential cross section is peaked at small angles it is usually the extrapolation here which introduces the largest errors. The errors from the extrapolations and the contributions from other sources yield a large total systematic uncertainty for the total elastic cross section. For this reason a plot of the data shows a considerable scatter of the points outside their statistical errors. Thus fits to the data yield large x2-values, if systematic uncertainties are not considered. A compilation of the measured total elastic cross sections is shown in fig. 5.30, while fig. 5.31 shows the ratios ~~eII~~tor In the momentum range 5—200 GeV/c all elastic cross sections decrease slowly with energy, the rate of change decreasing as the energy increases. Above 200 GeV/c at present there are only data for pp elastic scattering: they indicate that the elastic cross section also increases with energy. It does not take too much imagination to expect that all elastic cross sections may eventually increase with energy. The comparison of particle and antiparticle cross sections shows some interesting features: at low energies the negative particle cross sections are larger than the cross sections of the corresponding positive antiparticles; then, as the energy increases, the two cross sections tend to become equal
G. Giacomelli, Total cross sections and elastic scattering at high energies
11111
I
I
I
11111 6 8 tO
I 20
I
I 60
LABORATORY
11111
I
I
I
111111
111111 60 80 100
MOMENTUM
111111
I
I
I
I
197
I
I
200
400
(0eV/c)
I
I
I
111111
15—
—
11111 8 10
6
I
20
I
11(1111
40
60 80 100
LABORATORY
MOMENTUM
I
200
II
11111
400 600
1GeV/c)
Fig. 5.30. Compilation of integrated elastic cross sections.
1000
20OD
198
G. Giacomelli, Total cross sections and elastic scattering at high energies
11111
I
I
I
11111
I
I
I
111111
I
I
I
I
I
I
I
111T
~
c5
(0
111111 50 100
II 500
000
Laboratory Momentum 1GeV/c)
Fig. 5.31. Interpolations of the Oul/Otot ratios versus laboratory momentum.
at relatively low energies (compared to what happens to the total cross sections). It may even be that there is a kind of crossover between particle and antiparticle cross sections, but the data are not really adequate to establish this. For the ratios (1eI/0~o( the energy dependence is also relatively fast up to 40 GeV/c, and then slows down considerably. Table 5.5 gives the average values of UellUtot at about 100 GeV/c. For protons there is almost a plateau from 80 to 1 500 GeV/c, with indications that the ratio reaches a minimum and then increases slowly. The average value of Uel/Utot in this plateau or broad minimum is about 0.18, a value which is rather close to the Van Hove limit of 0.185 (see section 10). Information on the asymptotic region may only come from the pp system: the indication is that Uel/Gtot increases with energy in the ISR energy range. The U~)/tj~O~ratios for irp and ?p are within errors equal for laboratory momenta above 50 GeV/c. The same applies for K~pand at higher energies for pip. There are differences in Uel/tJtot for the different particles, indicating that the opacity of the proton to the various particles is different. Table 5.5 Average values of °el’°totand Oe)/atot at about 100 GeV/c laboratory momentum. The opacity may be defined as 0 2aei/otot.
=
irp Kp Kp pp
°eI
0tot
3.1
±0.2
24.09
±0.07
3.1 2.3 2.2
±0.2
±0.07
7.2
±0.5
7.0
±0.3
23.25 20.45 18.87 42.10 38.50
±0.2 ±0.2
Gel/Otot 0.130 ±0.009 ±0.009
±0.08
0.132 0.110 0.117
±
0.15
0.171
± 0.015
±0.06
0.181
±0.008
±0.07
±0.010 ±0.011
G. Giacomelli, Total cross sections and elastic scattering at high energies
199
6. Charge exchange scattering and K~ regeneration In this section the charge exchange reactions will be briefly discussed, mainly from the point of view of further understanding their relations to total cross section differences and to elastic scattering. Some of these relations arise from isospin conservation and from the optical theorem. Further relations may be obtained in the framework of some specific models, like the Regge-pole model, where charge exchange scattering plays a dominant role since it can be explained with a small number of exchanged trajectories. It may be worthwhile to recall that while total cross section measurements provide informations on the imaginary part of the forward scattering amplitude, charge exchange processes and K~regeneration have the added feature of measuring also the real part. The best studied charge exchange reaction is irp ir°nwhich has been measured up to FNAL energies. We shall briefly consider the following reactions (the main exchanged trajectory is shown in parenthesis): -~
7rp-~7r°n
(p)
(6.1)
Kp irp
-~
K°n
(p~A2)
(6.2)
-÷
r~°n
(A2)
(6.3)
(7r)
(6.4)
~ip—* ñn
1000
I
I
I
I
I
V CERN ~ CERN • IHEP 0 Fermilab S Fermilab 0 Fermilab
100 .
68 ‘65 ‘74 74 74 74
I
5GeV/c 10GeV/c 21GeV/c 40.6GeV/c 10$ GeV/c — 200GeV/c
1000
100
10
0.1
BORIGHT ET. AL. SONDEREGGERET AL
~
~
1.0
0.01
01
0.00~ ~
2 —t 1GeV/c)
Fig. 6.1. Compilation of angular distributions for i(p -4 ir°n scattering from 5 to 101 GeV/c in the 0 < ti < 1.3 (GeV/c)2 region [75G1]. The lines are only meant to guide the eye.
010
10
10
1f~ll~fff~ —t (GeV/c)C
~
0.l
lb
.01
Fig. 6.2. The complete angular distribution for irp -4 1T°n scattering at 5.9 GeV/c [74B6 J. Error bars represent counting statistics only; there is an overall normalization uncertainty of ~20%.
G. Giacomelli, Total cross sections and elastic scattering at high energies
200
np-f pn
(7r)
K~p-÷K~p
(w,p),
2-*K~C12
K~,C’
(w).
(6.5) (6.6)
A large variety of experimental methods have been used to study the various charge exchange reactions at high energy. All methods have to be able to detect a small two-body cross section, with at least one neutral particle in the final state, among a large sea of multi-body final states, in particular with ir°’s.For example the typical trigger for a charge exchange measurement (6. l)—(6.4) is a “neutral trigger” which requires that an incoming charged particle interacted in the target without producing any charged particle; forward going neutral particles eventually convert or decay into charged particles. 6.1. The i(p
-~
ir°nreaction
Fig. 6. 1 shows a compilation of the differential cross sections at laboratory momenta between Sand 100 GeV/c [74B4, 75Gl]. Data have been taken up to tI 1.5 (GeV/c)2 the complete angular distribution has been measured only at lower energies and a sample is shown in fig. 6.2. There are some normalization differences between the CERN, IHEP and Fermilab data. The main features of the i(p ~r°ndifferential cross sections at high energies are the following: i) All the distributions have two dips: at t = 0 and at —t = 0.6 (GeV/c)2 respectively. The dip at t = 0 is due to a large spin-flip amplitude, which has a sinO dependence and therefore vanishes at t = 0. The dip at t = —0.6 has a logical explanation in the Regge-pole model: it is the location at which the p-trajectory crosses zero. ii) The 5.9 GeV/c angular distribution shows the presence of another dip at —t = 10 (GeV/c)2, which corresponds to u = 0.11 (GeV/c)2. Changing energy the location of the dip remains constant in u. iii) In the 0.1 < tI < 0.5 (GeV/c)2 interval the differential cross section is approximately exponential. The slope of the exponential grows slowly with energy, thus indicating a shrinking of the forward peak. In order to analyze the energy dependence of the data over a larger t-interval, it is more instructive to look at the differential cross section at fixed t as a function of laboratory momentum. Fig. 6.3 clearly shows that at higher t-values the cross sections fall off faster with energy than for smaller t-values, thus yielding a shrinking of the forward peak. The angular distributions may be fitted to a single Regge-pole form -+
4~?
~9
(6.7)
52OIp(t)—2
dt
p*
Fits of the experimental data to eq. (6.7) yielded the effective p-trajectory shown in fig. 6.4, whose equation is approximately a~ 0.50 0.75 t. The solid curves shown in fig. 6.3 are the results of such a fit. iv) The integrated charge exchange cross section, shown together with other cross sections in fig. 6.5, has the power law dependence of eq. (2.50) with the coefficients quoted in table 6.1. v) The differential cross sections at t = 0 may be obtained from an extrapolation of the angular distributions. The optical theorem and charge independence yield the following relation between the total cross section difference ~a(ir~p) and the forward charge exchange cross section —
G. Giaco,nelli, Total cross sections and elastic scattering at high energies
I
I
111111
-t~O.36
0.5
\\
0.2 0.1
201
-t~0.45
-
0.05
0.02 -t ~0.55 I
11111111
2
5
10
p
I
111.1111
20
50
I
100
200
I
III
500
(GeV/c)
Fig. 6.3. Plots of do/dt versus laboratory momentum for the reaction irp -4 lr°n at a number of different t-values. The solid curves are the results of a fit to expression (6.7) [74B9]
1
dcr
ce(t
iT
0)=~-~--[u~0~(7rp) ~ 2 .
(6.8) (6.9)
25.5(1 + P~e)(L~U) Re ~ is the real part of the non-spin-flip scattering amplitude and Pce = [Re g~~/Im gee] to From eq. (6.9) one can compute z~u(7r~p) if a reasonable estimate of Pce (t = 0) can be made. One possible form is =
P~
(6.10) It may be worthwhile to point out that equations (6.8—6.10) are obtained from (i) isospin conservation, (ii) the optical theorem and (iii) assumptions on the p-Regge trajectory. The agreement between measurements of ~a and the predictions from charge exchange is reasonable (see fig. 6.6), within errors. 0~
tg(~ira~(0)).
202
G. Giacomelli, Total cross sections and elastic scattering at high energies
600
rp,°n
400
• MANNEU.I 65 I GIJISAN 65 65
‘~
2(30
— ST1RUNG V WAHLIG ~VERT
~
A
~
S
100 80 60
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..
o 0.4 0.2-
•
~cl -0.2
6,50
6eV/c 6.100 6eV/c
+
\\
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I
(0
~
8
0)
6
0
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-
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~
74
0
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-
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BOLOTOV ~p—nI,
68
o BOLOTOV VLEE 0 AST~~ 74 66
\
0
0.6 ________________________________ I I
66
0
z
68 74 74
ASTBIJRY VB 0
‘~
40 20\
o w
BARNES
• eowro~
0.8
i
0.6
-
iP~~fl~\
04 O4~
-o.e
-
t~l -1.2
I -08
t
I
0.2
~o
-04 2
Fig. 6.4. Effective p-trajectory for iip
-4
6
(0
20
40
60
100 200
LABORATORY MOMENTUM (GeV/c)
(6eV/c)
ir°n [74B9].
Fig. 6.5. Plot of integrated charge-exchange cross sections versus laboratory momentum. The solid curves are eyeball fits of the data.
I
I
I
I
r
Z2.Ob
~
El
(.0
0.5 ______________________________ I
I,,,.I
5
(0
I
I
20
50
(00 200
LABORATORY MOMENTUM (GeV/c) Fig. 6.6. The difference &s(sr±p)of the srp and ir~ptotal cross sections, plotted versus laboratory momentum. The solid line represents the prediction from forward charge exchange scattering, according to the relation (6.9), using Pce = 1.08 and o~(t= 0) = 0.50 [74B9l.
G. Giacomelli, Total cross sections and elastic scattering at high energies
203
6.2. Other charge exchange scattering processes Fig. 6.5 also shows the integrated cross sections for the following reactions: jip -÷ñn, Kp -÷K°nand irp -÷~°n (i~°-÷2’y). It is apparent from the figure that all the integrated cross sections have an energy dependence of type (2.50). The amount of data on these reactions is still rather scarce, while a considerable amount of information exists on the np pn differential cross sections (see fig. 6.7). The data are characterized by a sharp peak (with a slope of about 1 /m~up to tI < 0.02 (GeV/c)2) followed by a less steep dependence. The t-distribution does not seem to change in shape with increasing energy, while the absolute value decreases approximately as as one would have expected on the basis of the one-pion-exchange mechanism. In [74K 11 the authors parametrize the t- and s-dependence of the differential cross section with a two exponent formula: -*
da/dt~ A p’~81~°~25{exp[(5l ±8)t] +(0.8 ±0.l)exp[(4.50 ±0.15)t]
}.
(6.11)
L
IHfllfIl 1fff
21.0
~
6W/c .1 ~
I
1$
6eV/c
I
.~
II f
H~111~~ ~ ~
I
25.3f
.~,
.0I
lII
‘~
I
a .c~.d4 .d6 .~ a
.i
.~
.1
~
I
27~4 6eV/c
If
.~ .~ .i’
.~
.1 4.0
lul(6W/c? Fig. 6.7. Differential cross sections for neutron—proton charge-exchange scattering [72D2] - The low.t region is shown with an expanded scale.
G. Giacomelli, Total cross sections and elastic scattering at high energies
204
Table 6.1 Results of fitting integrated charge exchange cross sections to the formula a Reaction
Laboratory momentum range (GeV/c)
A ( 1.zb)
6—101
7rp—~?r°fl
679
20—200 20—200 6—40 6—50
irp—~X°n L. ‘y~y irp -+ f°n irp-~ w°n
=A
p_fl.
n
±
719 ± 292 ± 3.2 ± 25 ±
47
1.15
±0.07
20 20 0.8 4
1.17 1.47 1.1 2.3
± 0.04 ± 0.02
18 1680
±
Kp -÷K°n
6—50 4—35
4 220
2.7 1.5
~o(K°n)
4—50
7300
± 3000
0.53
±
±0.1 ±0.1 ±0.1 ±0.1
±0.12
6.3. K~regeneration In reaction (6.6) the interference between the iT~ir decay of the regenerated K~with the CP violating ir~Ir decay of K~in the transmitted beam depends on two weak-interaction parameters (i~÷ and ~m, the K~—K~ mass difference) and two strong interaction parameters. At high energies one may assume the known values of the weak-interaction parameters and compute the two stronginteraction parameters, that is the regeneration phase at forward angles =
arg[iA(K~p -÷K~p)0~
+
=
(6.12)
iT
and the imaginary part of the forward scattering amplitude, which may be expressed via the optical theorem in terms of a total cross-section difference: ~u(K0p)atot(k0~i)
T-at0t(K°p)
—~Im [ftO) —j~0)].
10
~
I
‘~.‘
I
(6.13)
111111
2 I b
_~o.(p).(7.3±3)xp0~OI2 FREITAG St al
~
4
I
~ANDEN&JRGet oP ~ FiRESTONE et dl
~
~ DARRIULAT et al I ALBRECI-IT 61 01 74 ~ ARO4JSON et al I
2
46
K)
I
20
I
I
1,,
4060100
LABORATORY MOMENTUM (6eV/c) Fig. 6.8. The measured ~at0t(K°p)
values obtained from K~regeneration experiments in hydrogen. The dashed line is the fit through the total cross section differences ~a(K±n).
G. Giacomelli, Total cross sections and elastic scattering at high energies
205
ftO) and ftO) are the forward scattering amplitudes for K°pand ~Op elastic scattering. Because of isospin conservation the difference ~c(K°p) is equal to i~cr(K~n): ~a(K°p)
= i~a(K~n) = a~ 0~(Kn) a~0~(K~n). —
(6.14)
Thus, for our purposes, K~-regenerationexperiments are useful to determine z~u(K~n) at high energy. In fact the experiments are rather sensitive to the difference and even a rough experiment is capable of yielding information on ~a(K~n) with precisions comparable to those achieved only in the most precise total cross section measurements. Fig. 6.8 shows the &r~0~(K°p) differences computed from regeneration experiments in hydrogen. The dashed line is an interpolation through the ~u(K~n) data from total cross-section measurements. The direct measurements and the values obtained from regeneration experiments agree rather well.
7. Polarization The present general picture, which emerges from the available data on polarization at high energies, is that polarization effects decrease with increasing energy. Thus we shall limit the discussion to some qualitative aspects and try to estimate what are the possible errors which one makes when one neglects spin—spin and spin—orbit effects. In the last few years a number of technological advances have made it possible to measure polarization effects using refined polarized targets and, more recently, very high intensity polarized proton beams. 1~target>’ of The more recent experiments use butanol targets, which have average polarizations, ‘~ about 50% and which contain about 10% free hydrogen. The other recent achievement was the acceleration of polarized protons in the Zero Gradient Synchrotron (ZGS) at Argonne. Using special precautions at some “depolarizing resonances” the polarized protons were accelerated to 8.5 GeV without any measurable loss of polarization. This is possible because of the special magnetic field configuration of the ZGS. The extracted proton beam has (73 ±8)% transverse polarization, with intensities ranging from 1 0~to 108 protons per pulse. By scattering this high-intensity highly polarized beam from a polarized target, and eventually measuring the recoil proton polarization, one has the capability of measuring all possible polarization parameters in pp elastic scattering [74F1, 74K2]. 7.
1. Polarization in 1r~pelastic scattering
The irp system is relatively simple to analyze [see formulae (2. 1 8)—(2.40)] and many experimental data on polarization parameters are available. Fig. 7. 1 shows the angular distribution of the P polarization parameter for i(p elastic scattering at 14 and 40 GeV/c [75G21. The shape of the t-distribution at the two energies is similar: P is negative and it has a zero at t = —0.6 (GeV/c)2, besides those at t = 0 and at t —1.9 (GeV/c)2. This behaviour is consistent with the hypothesis that the P polarization parameter behaves like the logarithmic derivative of the differential elastic distribution. The zero at t = 0 is due to the vanishing of the spin-flip term. In so far as the energy dependence is concerned, the maximum polarization at —t 0.15 (GeV/c)2 has decreased from about 9% at 14 GeV/c to 5% at 40 GeV/c. The energy dependence is
G. Giacomelli, Total cross sections and elastic scattering at high energies
206
(b)
.5 p
1.
15
I I
I
PL~ts45 0eV/c 05
pp
Ia)
I
‘If
-
-0.05-
-0.10
.2
40
-
-
+0.05-~-
~
+
.05
0
-
(
~I~I
-0.15
-
14GeV/a
f-
~II~V~
0.2
I
~
I
I
cc
~-*
PLV
-
I
..2
-0.05-
-0.10
I
~
40GV/
2
-
—t I
(Gev/C) 0.6
2
I
•c
~
1.0
.~
Fig. 7.1. The P polarization parameter (a) for i(p —~ irp at 14 and 40 GeV/c [72G1] around 40 GeV/c [74D6] , [75G2].
, and (b) for pp,
1.
_t@ev/c~ 1.5
pp and Kp elastic scattering
of the power type, eq. (2.59), with n 0.53 (fig. 7.2). Thus the P parameter decreases rather fast with energy, but polarization effects may really be neglected only at the highest energies, The P parameter for ir~pelastic scattering is essentially positive and seems to be the mirror image of that for i(p scattering. Thus the above conclusions about the energy dependence are valid also for ir~pscattering (and more generally for K~pscattering also, fig. 7.2). The main features of the P parameter, that is its t- and s-dependence and the relation P(ir~p) —P(i(p) may be explained in the framework of Regge-pole models assuming that the polarization arises from the interference of P and P’ with the p-trajectory: P~Im[(G~+G~c)H*].
(7.1)
The first measurements of the polarization parameter R in i(p and ?p elastic scattering at 6 and 16 GeV/c are shown in fig. 7.3. The average values are R(irp) —0.2 and R(?p) ~ —0.1. From the knowledge of P and R one may compute the A parameter from the relation (7.2) A2-l-P2+R2= 1. The values of A obtained from relation
(7.2)
and from direct measurements yield A(irp)
1.
G. Giacomelli, Total cross sections and elastic scattering at high energies
1.OC
I
1111
I
I
I
I
I
0.60
-n
I
Kp
0.59
Vp TLp
0.60 0.53 0.65 -0.90 -0.30
pp Kp ~p
0.40
III
-
~
LU
0.20
4
K~p
~
I
I
-
0.02
~‘N
I
$4~+
ci
~~1NN
I
~
R(Jp),641
~
~~
I
~R(irp),6Gey~
~0.04.
207
I
~0.5j ~p),I6
G~
~ R(7rp),4OGe~
I
1111
6
8
I
I
I
20
10
LABORATORY
I
40
I
I
I
ir
60 80
MOMENTUM (GeVic)
Fig. 7.2. Compilation of the energy dependence of the P 2polarization for sr~p,parameter K~p,pp and averaged ~p elastic overscattering. 0.1 < (t( Note <0.3 that (GeV/c) the sign of P(i(p) is negative. 7.2.
-a5L L
I
0.1
02
I
0.3 2 0.4
-t(GeVk)
I
05
and 16 GeV/c (72G1]. Fig. 7.3. The R polarization parameter for ir~p..+ ir~pat 6
Spin effects in pp scattering
The study of polarization effects in the pp system is complicated because both particles have spin. Thus one has to worry about spin—spin and spin—orbit terms. In total there are nine measurable quantities. 7.2.1. 0tot A measurement of the pp total cross section in pure isospin states was performed at 3.5 GeV/c using the Argonne polarized proton beam and a polarized target [73P 1]. The measurement was a standard geometry transmission experiment, which measured the difference
R,(CC)
= a~~(tt) a~
1
—
0~(t4~) = R~(N)PBPTN’
(7.3)
G. Giacomelli, Total cross sections and elastic scattering at high energies
208
where R~(tt)and R.(t4.) are normalized counting rates with the beam and target spins parallel and antiparallel respectively; ~B and ~T are the polarizations of the beam and of the target; N is the number of hydrogen nuclei in the target. In reality the measurement was performed with the four combinations offered by the incident beam and target spins: t t, 4.4., t 4., 4. t. But rotational invariance of space requires the equalities o~0~(tt) = ~
cr~0~(t4.) = o~~(4.t). (7.4) values of(7.3) are averages. The recent results of the measurement indicate that
0tot
Thusdifference the two the =
u~(tt)
—
o~ 0~(C4.)
(7.5)
is about 8 mb at 2 GeV/c and decreases to about 0.5 mb at 6 GeV. Thus at 6 GeV/c the two cross sections differ by about 4%. This may be thought to imply that at high energy the proton size will be unique for different spin states. The Pparameterin pp-÷pp The behaviour of the average P polarization parameter versus energy shown in fig. 7.2, suggests that also for pp elastic scattering the spin—orbit effects decrease with energy and may become essentially negligible for energies above 100 GeV. 7.2.2.
7.2.3. pp elastic scattering in pure spin states At 6 GeV/c there are now measurements of all the polarization parameters in pp pp scattering measured with the ZGS polarized beam impinging on a polarized target and measuring the polarization of the final state protons with scattering on carbon targets [74F1, 74K2] We shall restrict the discussion to the simplest case. Fig. 7.4 shows the differential cross sections for elastic scattering in pure initial-spin states, without looking at the polarization of the final protons. The cross sections have been normalized to the spin-averaged cross sections. The spins are measured perpendicular to the scattering plane. 2 exhibit the folThe data at the laboratory momentum of 6 GeV/c and for 0.5 < p~< 2 (GeV/c) lowing features: -~
.
(i)
do
(tt)>
do
(t4.)
—
=
do do —~-(4.t)>—(44),
(7.6)
the difference between the first and the last being about a factor of two. do/d~(t4.)is equal to do/d~2(4.t)because of space rotational invariance. do
(ii) ~~~-(44)
do
(44)>
do
(11)
in the 0.5
(N)
forp~>0.8 (GeV/c)2
—~~-
do
that is there is a change in the angular distribution of do/d~Z(14) at about —t 0.8 (GeV/c)2. This feature and the other features previously mentioned are better observed in fig. 7.5, which shows the differential distributions for the three spin states. They have been interpreted as arising
G. Giacomelli, Total cross sections and elastic scattering at high energies
I -
~
•
T
_______
.
~—0ai -
I
ido\
I.
0 0~P~IV4~80 \~. ‘0
-
~‘
‘~
NORMALIZATION PRELIMINARY
I
-
. ___
•~~
~
6 GeV/c
~
I
-
0
209
~
-
pep—pep
~I
o-~ ~
-
.01
—~
preliminary
~
2
1 P$(GVc) Fig. 7.4. The ratio of the differential elastic pp cross section for each spin state (da~ 1/dt)to the spin averaged cross section (da/dt) plotted versus p~at 6 GeV/c [74F1, 74K2]. The spins are measured perpendicular to the scattering plane. The if subscripts refer to the cases where only the initial spins are measured, while indicate cross sections for the elastic process when both initial and final spins are observed. The frame of reference is such that the scattering of the forward particle is observed to the left.
~2
(GeV/c)2
2
1 Fig. 7.5. The differential elastic proton—proton cross section for pure initial spin states plotted versus p~at 6 (GeV/c) [74K21.
from different spatial regions in the proton, each with different spin states. 2 where the spins of both the initial In fig. 7.4 measurements are also shown at p~ = 0.5 (GeV/c) and final protons are observed. The interesting feature is that the double-flip cross sections are typically ten times smaller than the non-flip cross sections, more specifically do
ldu
do
-~
44)
—~(t4.-~ 4.t)rai -~ -~
ldo
(N. -~ N)
1 dci
-÷ tt)~
j-~-~ (tt
-+
N.).
(7.7)
The Wolfenstein A and C,~parameters have also been measured at 6 GeV/c: for ItI < I both average the value of around 0.1. 7.3. Conclusion In conclusion, in the 5—20 GeV/c laboratory momentum range there are sizeable spin—spin and spin—orbit effects. The latter decrease considerably with energy and may probably be neglected for laboratory momenta above 50—100 GeV/c; there is still no information on the energy dependence of spin—spin effects in pp collisions, though they are expected to decrease with energy too.
G. Giacomelli, Total cross sections and elastic scattering at high energies
210
8. Dispersion relations for the forward scattering amplitude Dispersion relations give the real part of an amplitude in terms of an integral over its imaginary part. Many calculations are actually concerned with forward (t = 0) dispersion relations, where the imaginary part may be obtained from the total cross section via the optical theorem. Our discussion will therefore be centered on the computation of the real part of the forward scattering amplitude from the experimental knowledge of the total cross section. The results of the computations will be compared with the direct measurements of the ratio between the real part and the imaginary part of the elastic forward amplitude using elastic scattering data in the Coulomb—nuclear interference region (see section 5.2). Such a comparison serves practical and theoretical purposes. From the first point of view, one can check the consistency of the total cross section data with the elastic scattering data at forward angles. From a theoretical point of view, the comparison allows a check of the validity of dispersion relations and of the assumption of spin-independence for forward and near-forward scattering. The real part is obtained as an integral over 0tot~ Thus the connection between the two is nonlocal, the real part depending upon the behaviour of 0tot at all energies, including the low-energy region and the high-energy region where no measurements exist. At high energy the relation between the real and the imaginary part becomes almost local, since the effects of the far away parts of the integral become small. One may estimate the energy at which the contributions of the unphysical region, of the subtraction constants and of pole terms become small by computing the real part with the exact formulae and with approximate formulae. One finds that above 50 GeV/c for irp and Kp and above 1 50 GeV/c for pp the difference between the two methods is smaller than the experimental uncertainties. The estimate of the uncertainty arising from the neglect of the high-energy part of the integral is more difficult to estimate. Nevertheless, we shall consider that for energies higher than 50—150 GeV the relation between the real and the imaginary part is almost local. It is thus pedagogically interesting to write down the approximate local relation. Bronzan et al. [74B3] write: .-1 1
[ReF(xp)] +
5.0
...~
iT
(+)
(—)
~
~~~°tot — fl
————-—+
[tg~-~--)O~
+~s° ~
/7Ta
d
/0tot
\1 (8.1)
where o~ = o~0~(xp) ±o50~(x~p). Expression (8.1) maybe even simplified further. For instance, for pp collisions one may also write: Ref (E, t0)
ira ——-——Imf (E, t0). ~ 2 alns
(8.2)
In (8.1) the o-parameters are found empirically to be o 0.68 for irN, ~ 0.44 for KN and 0.39 for NN. Several authors have computed p using various approximations for the dispersion relations [73Bll, 74B3, 74H1, 74H2] ; there are also several calculations performed in the framework of various theoretical models [73C2, etc.] The results of these calculations are quite similar, apart from some numerical details. Fig. 8. 1 shows a compilation of forward real parts computed via complete dispersion relations [74Hl I. The shaded bands represent the results of these calculations up to the energies where measurements of °tot exist. The vertical widths of the bands represent estimated uncertainties arising from the uncertainties in the total cross section data, in the theoret.
G. Gi.acomelli, Total cross sections and elastic scattering at high energies
~2
~
-0.3
211
___
~
-0.3
-0.4
‘°irp
03
)Il~
~
O.2~ ~
~
0.+l
I
III)
I
III
I
-0.4 Ill
I
III)
I
1111
I
1111
I—I—I—)
o~ 0.2
0
_//__
-0.2 -0.2
¶~
-0.3 -0.4
PK~P I
o
I—
III
1111 I
II))
I
III
I
III)
-0.4
I
-0.6 1111
I
1111
0.4
-0.2
I —I—I—I
0.6
.1
10
I
00
I.
1000
.1
10
•
tOO
I
-0.4
1000
LABORATORY MOMENTUM 1GeV/c) Fig. 8.1. Compilation of the ratios p versus energy [74H1] . The bands are calculations from dispersion relations in the energy region where there are data on °tot’ while the solid lines are the predictions at higher energies where no measurements have been performed. The points are the computation from measurements of elastic scattering in the Coulomb-Nuclear interference region.
Note added in proof: There are now
data from Exp.# 69 at Fermilab between 70 and 150
GeV/c [75A1I ,which agree with the
dispersion relation predictions.
ical approximations in the treatment of the unphysical region and in the uncertainties in the coupling constants. The solid lines extending beyond 200 GeV/c are predictions of the real parts based on the parametrization (4.25)—(4.27) of the unmeasured high-energy cross sections above 200 (280) GeV/c. Fig. 8.1 shows also the real parts directly obtained from measurements of elastic scattering in the Coulomb-nuclear interference region. The agreement between dispersion relation calculations and direct measurements is not too good, with the exception of the pp system, where on the other hand most measurements exist. As stressed in section 5.2, the disagreements are very probably due to the measurements and to the methods of analyses of the elastic scattering data at small t, since the effects of the real parts on the differential cross sections are very small and the computation of p depends considerably also on the absolute values. Thus the comparisons between dispersion relation calculations and direct measurements are in
G. Giacomelli, Total cross sections and elastic scattering at high energies
212
general not really meaningful, with the exception of the pp data, for which the agreement may be considered to be good over the entire energy range. This agreement is consistent with the validity of dispersion relations and with the assumption of spin-independence of the scattering amplitude at small angles. The dispersion relation results of fig. 8. 1 clearly show that all the ratios p are negative in the energy range 5—50 GeV and become positive at higher energies. As already stated in the Introduction and in section 3, one expects that p reaches a positive maximum and that it eventually decreases towards zero.
9. Impact parameter expansion Analyses of high-energy elastic scattering data in terms of impact parameter expansions on ISR data have yielded a simple picture of very high energy scattering {73A3, 73C1, 73C4, 73Hl, 73H3, 74M1]. Assuming that (i) the real part of the scattering amplitude is negligible at all angles, and that (ii) there are no spin effects, we may write: IF(s, t)12= IReF(s, t)12+ IImF(s, t)I2~IImF(s,
t)12.
(9.1)
The neglect of Re F(s, t) is only an approximation since we know that some real parts exist. Performing a Fourier transform to impact parameter space, we have Tm F(s, b) =
exp(—iq~b)Im F(s, q) dq,
(2ir)
(9.2)
where q = ~ and b is the impact parameter. The opacity &2 may be defined so that the following relation is valid: ImF(s, b)
=
~
—
exp[—&2(s, b)]
}.
(9.3)
This form of Im F(s, b) satisfies automatically the unitarity condition. Eqs. (9. 1 )—(9.3) yield the required informations on Tm F(s, b) and ~2(s,b) [73A3]. In terms of the invariant quantities defined in section 2.3, we may define the helicity-non-flip elastic amplitude transformed in impact space as [73H3]: G(s, b)
=
f G(s, t)J0(b~t) 2p~~
(9.4)
Neglecting spin—spin and spin-flip effects we may write do dt — ~jei=
______
2
(9.5)
641rp*s IG(s, t)1
f
~dt~ _dt
flG(5,b)I2~~-~P
8ir
(9.6)
G. Giacomelli, Total cross sections and elastic scattering at high energies
=J[
1m(
213
~
Fig. 9.1. Illustration of the s-channel unitarity equation.
ImG(s, t = 0) 2p*~/~
=
tot
JIm G(s, b)bdb.
(9.7)
Unitarity in the s-channel implies: Im G(s, b)
2 + I(s, b) 8ir IG(s, b)1
= —~---
=
E(s, b) + I(s, b),
(9.8)
where E(s, b) and I(s, b) are the overlap functions due to elastic and non-elastic s-channel intermediate states. Eq. (9.8) is illustrated in fig. 9.1. In this context, diffraction scattering is interpreted as the shadow of absorption due to the existence of elastic and inelastic channels. By combining eqs. (9.6) and (9.8) we have
0in
=
0tot
—
0el
= /I(s, b)bdb
=
f I(s, b)bdbJ0(b~~)~0.
(9.9)
Finally the eikonal form is given by G(s, b)
=
4iri[ 1
—
exp{—~(s,b)/4iri}I.
(9.10)
From the unitarity relation we have the following limits: 0< Im G(s, b) ~ 4ir,
0 ~ 1(s, b) ~ 2ir,
Im ~7~(s,b) ~ 0.
(9.11)
A calculation was performed using the equation
Tm G(s, b)
=
~f dtJ0(b~f~i)[l6iT{(~)
—
~~real)
hh1~~2.
(9.12)
An estimate of the real part of the scattering amplitude at very high energies may be obtained using the derivative analyticity relations of Bronzan [74B3]. Assuming that above 2 the differential elastic cross sections et areal. dominated by the imaginary part of the shelicity ~ 1000non-flip GeV amplitude, they neglect spin effects and obtain the approximate relation: Re F(s, t) ~
—~-—
2dlns
Im F(s, t)
~
-~ —~----
2dlns
dt
(9.13)
that is one may compute the real part of the scattering amplitude at a certain t-value via a logarithmic derivative in s of the square root of the differential cross section. The results of a numerical calculation using pp elastic data at ~ 30.8 and 53 GeV, indicate that the real part is always considerably smaller than the imaginary part, with the exception of the region around the dip at —t ~ 1.4 (GeV/c)2 (see fig. 9.2).
G. Giacomelli, Total cross sections and elastic scattering at high energies
214
__________________________________________
2
__
0
.2
~3~eI1\II~~
.4
6
.8
1.0
21,2
-t (0ev/c)
14
15
0
1620
—1.0
t —2.0 1GeV)2 —3.0
—4.0
Fig. 9.2. The real part of the pp elastic scattering amplitude at -..fl= 53 GeV computed via formula (9.13) [73F13I.
The computations according to (9.2) and (9.12) yield very similar values, independent of the choice of the real part. This is so because Im G(s, b) is computed with an integral, whose largest contribution is for b 0, where the real part is very small (say 1% of the imaginary part). Thus even if at larger angles the real part becomes larger this has little effect in the integration. Fig. 9.3 shows Im G(s, b) computed via eq. (9.12) from the ISR data on pp elastic scattering at ~ = 53 GeV. Also shown are I(s, b) and &~2(s,b). The following important features may be pointed out: (i) At small values of b, Im G(s, b) is very nearly Gaussian. This is more vividly shown in fig. 9.4. The Gaussian form of Im G(s, b) is no surprise, since an exponential form in t transforms into a Gaussian in b. (ii) There is a large tail at large b. Above 3 fm the curves are not very reliable because of difficulties caused by small discontinuities in the experimental data. Henyey et al. [73H31 emphasize that the tail is related to the break observed in do/dt at —t ~ 0.13 (GeV/c)2. For practical purposes the amplitude may be parametrized as the sum of Gaussians [73H31 Im G(s, b)
=
8.9 exp(—b2/21) + 0.5 exp(—b2/62)
—
0.24
exp(—b2/0.92).
(9.14)
The first Gaussian dominates at small b out to about 2 fm, beyond which the second term takes over. The third term yields a small contribution, which is important only around the dip at —t 1 .4 (GeV/c)2. The first term represents central collisions and has a mean radius of 0.8 fm; instead the second term provides a peripheral contribution and has a mean radius of 1 .4 fm. In four-momentum transfer, eq. (9.14) becomes Im G(s, ~ 2p
~
exp(5.3t) + 15.6 exp(15.6t) —0.11 exp(0.23t).
(9.15)
*,V./~
It is surprising that the t = 1 .4 (GeV/c)2 dip does not show up at all in the graphs of figs. 9.3 and 9.4. The dip corresponds to a small flattening of Im G(s, b) near b = 0, a flattening which is too small to be seen in the figure. Thus when looked in b-space the diffraction minimum does not look impressive at all. It rather appears as an accident. —
G. Giacomelli, Total cross sections and elastic scattering at high energies
I
2It
I
I
21
-
\&——-—
I
‘/i~c53 0eV
5
pp-
2
-
(sb)
—
- —
-—
1
(s • b)
I
~
-
I(s,b)
E (s~b)
-
1
‘\
E~-
‘N
0.2 01
scattering
Im G(s.b)
Im (sb)
I
215
c
\
BLACK DISK LIMIT
2it
\
°
‘Na 005
0 2 (fermi)2
b
Fig. 9.3. Impact structure of proton—proton scattering at ~/~= 53 GeV. The imaginary part of the non-spin-flip elastic amplitude, Im G(s, b), the inelastic overlap function, J(s, b), and the eikonal or opacity, n(s, b), computed from the proton—proton elastic scattering data at .,/i~ 53 GeV [73H3I.
b (fermi)
Fig. 9.4. Im G(s, b),I(s, b), and E(s, b) at ~/~= 53 GeV. The horizontal line denoted “black disk limit” indicates the maximum value of the inelastic overlap function allowed by unitarity (100% absorption) [74M1I.
It is interesting to remark, from the graphs of fig. 9.4, that at b = 0 one has I(s, b) 94% of the maximum value allowed by unitarity, represented in the figure by the “black disk limit” line. This means that in a head-on collision two protons have a probability of 6% to pass straight through each other without any absorption at all. On the other hand E(s, b) is only 50% of its black disk limit. The difference between the b = 0 values of E and I raises the question as to whether one has to add to elastic processes also the contribution from diffractive inelastic processes. The graphs of figs. 9.3 and 9.4 show that there is really no major advantage or simplicity in discussing E or We have now to investigate the energy dependence and, in particular, try to answer the question: “from which impact parameters does the rise of ~ and of 0tot come from?” Fig. 9.5 shows I(s, b) at four ISR energies. The difference in I(s, b) between the lowest and highest energy is shown in fig. 9.5b. This graph has some uncertainties at small b, but even so the picture is quite clear. The increment in 0in -comes from an average b of 1 fm and is more peripheral then I itself. Thus with increasing energy the proton radius becomes larger (by about 5% over the ISR range), but the values of the opacity at b = 0 does not change: the proton becomes bigger, but not blacker.
G. Giacomelli, Total cross sections and elastic scattering at high energies
216
I
1.C -
I
a) 53GeV
0.5
\\\
-
0
b)
1~~~2O
b (fermi)
Fig. 9.5. (a) Inelastic overlap function, I(s, b), computed from the ~ = 21, 31, 44 and 53 GeV ISR pp data [74M1 I. (b) Difference of the ~ = 53 and 31 GeV inelastic overlap functions, i~J(s,b) It shows that the cross section increase comes from a region around 1 fm.
It may be worthwhile to point out that in an optical model with constant opacity one has: 0e1
7~2a2,
00o0 =
2irR2a,
Oei/OtOt
(9.16)
~
where R is the radius up to which there is a constant opacity (or greyness). a = 1 for a black disc and a = 0 for a transparent one. Instead in an optical model with Gaussian opacity [0 a exp(—5b2/2R2)I: 0e1
~7~20I2,
°tot =~7rR2a,
OeI/Otot
~
(9.17)
where a is the opacity at b = 0. Thus in this case ciel/0000 cannot exceed 0.185. The experimental data on Gel/cOot (see fig. 5.31) are not conclusive as far as optical model analyses since the ratio may increase with energy. The impact parameter analyses of the data at the present energies suggest a picture in agreement with (9.17).
G. Giacomelli, Total cross sections and elastic scattering at high energies
217
10. Theoretical models A large amount of theoretical and phenomenological work has been done to understand high energy hadron—hadron elastic scattering and total cross sections. The following will only be a brief list of the work in progress, without attempting a comprehensive survey. The emphasis will be on trying to understand the main lines of approach and to discuss the phenomenological background needed to make comparisons with the experimental data. We refer the reader to the many theoretical reviews which have been published recently [71C3, 71 Z 1, 73K 1, 74B 13, 74J 1, etc.]. Table 10.1 illustrates the predictions of various models for the high energy behaviour of 0tot’ Gei/Otot, b/a 000 and of the cross section for the secondary maximum for the pp system [74B13]. 10. 1. Amplitude analyses for irN
-~
irN scattering
Amplitude analyses seem to be the best way to analyse the elastic scattering data in the intermediate energy region independently of any specific model. These amplitudes are becoming the meeting point between experiments and theory following the same trend one had with phaseshift analyses at lower energies. On the other hand their f’ble at higher energies is modest. These analyses need much experimental information, particularly on some polarization parameters which are difficult to measure. Thus, in practice, quantitative amplitude analyses could be performed only for irN -÷ irN scattering. The new measurements with the polarized beam at the ZGS are yielding much data on all polarization parameters which may be useful for amplitude analyses. Our discussion will be limited to the irN case. The measurement at each energy and at each angle of the eight quantities, (do/do) (irp), (da/d~2)(?p), (da/d~)(ir°n),P(irp), P(ir~p),P(ir°n),R(i(p), R(ir~p)and of the sign of A(i(p) allow the complete determination of the irN elastic scattering amplitudes in the s-, t-, or u-channels. In each of these channels the amplitudes are determined by seven numbers (four moduli and three phases, with an undefined overall phase, which could be determined, for example, by Coulomb-nuclear interference). In the last year several amplitude analyses of the irN system were performed at 6 and 16 GeV/c, where the spin rotation parameters A and R have been measured (in irN scattering there are 12 measurable quantities, 8 of which are independent). The analyses of the experimental data into s-channel helicity amplitudes employ formulae of the type: do —=
R
2÷ IF~j2 IF~I —2 Im(F~~F.~’)
(10.1) (10.2)
—(IF~~I2 F÷J2)cos0~ —
—
2
Re(F~~Fjsin09,
(10.3)
where ~ and F~_are s-channel helicity-non-flip and flip amplitudes, and is the recoil proton angle in the laboratory system. Other authors divide the ~ and F~_amplitudes by p*/V’~,where p* is the c.m. momentum.
G. Giacomelli, Total cross sections and elastic scattering at high energies
218
Table 10.1 Illustration of the predictions of various models for the energy behaviour of “tot’ °eI/°tot’b/a
000 and of the height of the secondary maximum for pp scattering [74B13J (b is the slope of the diffraction pattern). “tot 2(s)
“e1/~~tot
b/at00
constant
constant
(da/dt)secondary (do/dt)p0 constant
Geometrical scaling Cheng—Walker—Wu Asymptotic black disc model
~
R
—~
increases (by 7%)
decreases (by 7%)
increases (by factor >2) -+l.7X102
Regge poles and cuts [with np(O) “l I
rises to a constant
decreases 0
increases
—*
—~
decreases or constant
Factorizable eikonal
arbitrary
increases (by 7%)
decreases
In2 s
(b)
‘Im 0.10
ISOSPIN 0
,
increases (by factor >2)
F°
ISOSPIN 1
IFLI
1O005~1~00
~5~40~009
Fig. 10.1. Non—nucleon scattering amplitude analysis at 6 GeV/c [72C3]. a) Moduli of s-channel helicity amplitudes. The heavy line is the result of the fit, while the thin lines indicate the error limit. b) Complex diagrams of ratio of s-channel helicity amplitudes.
G. Giacomelli, Total cross sections and elastic scattering at high energies
219
The t-channel I = 0 and I = 1 isospin decomposition is given by: f~~(?p-÷ ?p)F.~~
f~~(irp-+ir°n)=
~
—‘~,/2F.~~
(10.4) (10.5)
and similarly for the helicity-flip amplitudes F~,F~. The results of the analyses at 6 (fig. 10.1) and 16 GeV/c show that F÷° + is the dominant amplitude and that the energy dependence of all amplitudes is consistent with Regge-pole pictures. 10.2. Thermodynamic or statistical models The fact that the high-energy, large-angle differential cross sections are characterized by a slower t-dependence and a faster s-dependence than at smaller angles, had led a number of authors to hypothesize a statistical (or thermodynamic) mechanism for large-angle scattering. Following the compound-model picture for nuclear reactions, one may visualize the statistical model with the following naive interpretation: in central, high-energy collisions the two colliding particles form a compound system. The total available energy is concentrated in a small volume, of about 1 fm radius, in which a kind of thermal equilibrium is reached. A short time later, the volume starts to expand, the equilibrium ceases and the compound nucleus decays into the many channels open, one of which is the initial state. Thus if the large-angle scattering is due to a mechanism of this sort, its cross section will be proportional to the probability of formation of the compound state and to the branching ratio for the decay of the compound state into the elastic channel. The analogy with nuclear physics does not really go very far, because of the very short lifetimes involved in particle physics, which means that a statistical equilibrium cannot be reached. The thermodynamic model of Hagedorn et al. [65H2] gave the qualitative prediction that the energy dependence of the 90° cross section should fall as an exponential in transverse momentum, with a slope about equal to that found experimentally. The model gave no information about the angular distribution. However, other statistical considerations by Ericson [66E 11 in analogy with low-energy nuclear reactions, lead to the prediction that the angular distribution at fixed energy’ should show a characteristic structure in the form of fluctuations, arising from the random nature of the partial wave scattering amplitude in a statistical model. Such fluctuations have been observed in nuclear reactions at 10 MeV, but failed to materialize in pp elastic scattering at 16.9 GeV/c incident momentum in the 67°< 0~I~ < 90° range [66A1, 73A6]. Frautschi [72F 11 rediscussed the Ericson fluctuations suggesting that a necessary condition for their generation is the presence of direct channel resonances. Since the pp system is exotic and either does not have resonances or has non-dominant ones, the experiments performed do not settle the problem of statistical fluctuations, which have to be looked for in other scattering processes. It may be that some of the wiggles observed in large angle irp elastic scattering have this origin. 10.3. Regge-pole models
It was hoped that Regge-pole theories would provide a dynamical basis for strong interactions and, in particular, for hadron—hadron elastic scattering. Such a goal has not been reached, but a considerable amount of useful phenomenology based on Regge poles has emerged. One of the
220
G. Giacomelli~Total cross sections and elastic scattering at high energies
irp -. EXTRAPOLATION OF 5 POLE
2t
REGGE EXCHANGE MODEL
I0
~L*B6
GeV
02
~2
U
IO2[~~3~ 02
t Fig. 10.2. Extrapolations of Barger—Phillips five-pole model compared with IHEP and FNAL irp elastic data [74B131.
main advantages of the Regge-pole approach is that of accommodating easily the general principles of analyticity and symmetry. Practical calculations of total cross sections and of elastic scattering in the diffraction region require the use of two vacuum trajectories, P, P’, and of some other trajectories like p, w, A 2, f (see section 2.7). Thus the number of independent adjustable parameters becomes large and one can easily obtain good fits to the data. Some particularly simple situations, from the point of view of Regge theories, are offered by some total cross section differences, charge exchange scattering and backward elastic scattering, where only a small number of trajectories are involved. Regge-poie models have had their ups and downs. From a phenomenological point of view one can certainly use Regge models to fit, correlate and explain many features of the experimental data as we have done several times in this review (see sections 4.4.2, 4.5.5, 4.6, 5.3, 5.5, 6.1, 7.1 and, for instance, the impressive fit of irp elastic scattering data with a five-pole model, fig. 10.2). From a theoretical point of view there are several open problems which range from the question of what is the Pomeron to which Regge cuts and of what type exist [71B4, 73T 11.
G. Giacomelli. Total cross sections and elastic scattering at high energies
221
10.4. Constituent models It is particularly interesting to investigate the consequences on elastic scattering arising from the hypothesis that hadrons are made of a small or large number of substructures (constituents) which can be identified with quarks or partons. Since the constituents are usually assumed to be point-like, their presence, revealed through constituent—constituent collisions, gives rise to small cross sections, which may be visible in hard collisions, for instance in high energy large-angle scattering, that is, scattering with large momentum transfers. The predictions from constituent models are often computed using extreme simplifications; for instance it is often assumed that very large Iti collisions between two hadrons may be considered as arising from the collisions between two constituents, which take place in a way which is independent of the presence of the other constituents (additive quark model). At somewhat smaller It I-values multiple scattering models are used, where one assumes that there are many constituents which shadow one another and thus a single hadron—hadron collision may be the result of a number of collisions inside them. This picture leads to a multiple scattering formalism of the Glauber type and to a connection with optical models. For instance, the higher order diffraction minima of the optical models are generated from the interference between different orders of multiple scatterings, that is from single and double scattering, double and triple scattering, etc. Higher order scattering processes become more important than single scatterings at medium and large momentum transfers. Thus, in this picture one could say that there is no true high momentum transfer elastic scattering, that is, they are not generated from a one-step process, but originate from a series of steps. The additive quark model leads to simple rules, which correlate the energy dependence of two-body cross sections at highs and fixed c.m. scattering angle 0* (for 0* large) with the number n of constituent quarks of the four hadrons participating in the reaction: ~
5(2-E~1n~)J~0*)
1meson
= 2,
~baryon =
(larges,
0*)
(10.6)
3; thus
where ~8f(~*)
for meson—baryon elastic scattering
(10.7)
~loft~*)
for baryon—baryon elastic scattering.
(10.8)
Comparison of the energy dependence predictions at fixed 0 * yield reasonable agreement with pp and ir~pelastic scattering data, fig. 10.3. Attempts to account simultaneously for energy and angular dependences by constituent models have not been as successful until now. 10.5. Shadow scattering models At high energies the elastic-scattering amplitude is mainly imaginary; then elastic scattering arises mainly from the absorption of the incident waves by the many open inelastic channels. Thus elastic scattering is the shadow of inelastic processes and is completely determined by the knowledge of the inelastic collisions. These statements have been formalized at high energies by
222
G. Giacomelli, Total cross sections and elastic scattering at high energies I0~ iv —3
ir~p— -irp
~
0
istL 4~tI ~I
10-4 —5
Ik t4i~~~*.~,4t
~5
~+ ~ I,
IO_6 .‘
7
ir~p-. ,r~p
ii
b
106
-,
4’ •f (~l~t 4.
_3 10
Iv
~
7L~t.I
~ii
,.,—4
~
~.
t•1
~ IO~’1:~
~
cose
- ~
pp-+pp
2
I0_
—-
•
3
-~
cose
pp—pp 1014
~
~4••..~1~4•,
•
~ :
59.
5. 8.
4
iO~ 108 .4-i
~LCB 4.
IS+It •~
~LAB
GeV/c
= 8— 9
0
0
—
-3
2
“.
‘
0
•
~.
:.
• •
2
I..t—I(
~
I
I
-
14.2
•
•
24
~
I
•
•
• • • •
•
.
•
~
‘
0
‘~.
~
~O
~
•
-6 0
0.75
0.50
cose
0.25
• •
24.
‘
‘
.
0_B .00
0
.9.25 10.8 • 12.15
19.2
0
8.IGeV/c
•
15.1
-...
0°
2 (1)
~I0
19—21
.~
4
—
. ..
•
,,,,
.00
•
•
4.25 ~6.9
:
I
0.75
0.50
0.25
0
cose
Fig. 10.3. Comparison of the quark model rule (10.6) for the energy dependence of large-angle elastic scattering. The three graphs on the right show the raw data for ir~pand pp elastic scattering, while the three graphs on the left show the same data multiplied by 58 for ir~pand ~ for pp. One obtains a kind of universal curve for pp, thus proving that eq. (10.8) is well satisfied at most scattering angles. The situation is not as clear for ir~pscattering [74B13]
Van Hove [64V 1] through the overlap function, which describes the contribution of all inelastic final states to the unitary conditions (9.8). There are many specific models, each of which assumes specific forms for the overlap function, either for empirical reasons or for general dynamic reasons. For instance, the multiperipheral models make specific assumptions about the inelastic processes. In a simpler form Krisch [67K21 related elastic scattering to inelastic processes and was able to obtain a very good fit to the largeangle pp elastic scattering data (over 1 2 orders of magnitude, see fig. 10.4) with the formula ~
~c1exp(_~2p~/g~),
(10.9)
where c, and g, are six parameters, j3 is the c.m. proton velocity and Pt is the transverse momentum. 2p~was interpreted The fact that three regions appeared which were exponential in the variable f3
G. Giacomelli, Total cross sections and elastic scattering at high energies 10
-25
223
_______________________________________________________
I
10~26
I
I
I
I
I
I
I
I
I
I
-
2
• ISR (This experiment) 5=2950 GeV o BNL Fbleyetal. GeWc Berkeley Clyde eta!. 24.6 5—7GeWc ~ Argonne Akerlofeta). 5~.i3.40eV/c . CERN Allabyet aL ‘ BNL Cocconi et al.
-
10
-
8—21 0eV/c 10—32 GeY/c
:~ 10~29~~
~
-
~~,,~e_3~45p2Pi2
10~°
‘~E1031 1k4!
L~J
—.
2
-
S io-35~-
-~
V
-
1O36~ -
I
I
I
I
I
I
I
I
I
012345678910111213
~2~2 (0eV/c)2
Fig. 10.4. Plot of da~/dtversus
for high-energy pp elastic scattering data [75Wl 1.
as being the result of diffraction scattering associated with three different types of inelastic processes, namely pion production, kaon production and anti-proton production, which occupy three different spatial regions of decreasing size inside the proton. More recent data on both elastic and inelastic scattering spoiled this simple and attractive picture [75W 1]. 10.6. Optical models Most optical models used in high-energy scattering start with the formula of the impact parameter expansion of the scattering amplitude and assume different expressions for the matter density distribution inside the hadrons. If the interaction is spin independent and the real part of the scattering is small, the impact parameter expansion may be written as [70G 1]
224
G. Giacomelli, Total cross sections and elastic scattering at high energies
g(E*, 0*)
ip*f b{ 1
—
Re a(E*, b)}Jo(p*bO*)db (10.10)
h(E*,
0*)
—
The various optical models differ in the form of a(E*, b). The above formula is usually written for one type of elastic scattering at one energy. In this case the cross section depends on the functional form of a(E*, b) and on two parameters, the opacity (connected to the ratio Gei/Otot) and the radius of interaction. If one wants to describe the energy dependence and to connect the elastic scattering of two hadrons with that of other hadrons, the number of parameters increases considerably. According to the geometrical scaling hypothesis, the energy dependence of the elastic amplitude is contained in R only and the amplitude scales in impact parameter as: g(E,0)zg(b/R).
(10.11)
The grey disc model and the Gaussian one have this property. From geometric scaling it follows that: R2,
0tot
do/dt
0e1
R2,
~
R2,
B
R2
2 (10.12) 51(R~JT7)I [see also eqs. (9.15) and (9.16)1. If one allows the proton radius to grow with energy, then geometric scaling accounts for the growth of 000t’ the essentially constant value of(u 61/a000) and the shrinkage of the forward peak. The combined ISR data are compatible with a In s growth of 2: R R2R~+R2lns (10.13) =
R4IF
R 0
=
0.84 fm,
R1-
0.22
fm.
The expectation of geometrical scaling that [(1/o~00)(dcr/dt)J should be independent of energy is compatible with the tSR data (see fig. 10.5). This implies the inward moving of the diffraction minimum and the increase of the secondary maximum as the energy increases. As stated in section 9, the “matter” density should have essentially a Gaussian distribution. The next step is to find out the reason for this distribution. Wu and Yang [65W11 made the suggestion that the proton matter distribution would be identical to the electromagnetic charge distribution as determined by electron—proton scattering. This means that the differential cross section for pp elastic scattering should be proportional to the fourth power of the proton electromagnetic form factor: ~(pp~pp)~F~
.
(10.14)
Fig. 5.20 shows that this approximation is not adequate to represent the ISR data. Cheng, Walker and Wu [7lC2, 73Cl, 73C2, 74C31,analysing the implications of relativistic quantum field theory for scattering at very high energies, arrived at an optical model description,
G. Giacomelli, Total cross sections and elastic scattering at high energies
225
I0~ 0
-
GEOMETRIC SCALING
2
0 l0~
a. .&
•
496 GeV
•
073 GeV 501 GeV
‘~
l0~
-
S
0
-50
-100
-150
—200
mb(GeV)2
~
I,
-250
Fig. 10.5. Geometrical scaling ofpp elastic scattering data. [(da/dt)/o~
001 versus [at0tItll is predicted to have a universal shape independent of energy. The dip location is predicted to be proportional to 1/a~ and the secondary maximum is predicted to rise like °~ot [74B131.
which they called the “impact picture” of high-energy scattering. The model pictures the nucleon as made of a partially transparent inner core whose size grows with energy and of a “skin”, about 0.2 fm thick, which stays constant. The practical formulation of the model contains a large number of parameters (seventeen) and thus its predictive power is not totally clear. On the other hand, starting from the knowledge of the pp elastic scattering distribution at small angle at the ISR, of the total cross sections measured at Serpukhov (which contained the rising K~ptotal cross sections), they were able to predict correctly the rising of all the other hadron—hadron cross sections, the change in sign of the real parts2.of the forward scattering amplitudes and the dip in pp elastic scattering at tI ~ behaviour 1.4 (GeV/c)of the Cheng—Walker—Wu model corresponds to a black disc picThe asymptotic ture of the proton leading to the following asymptotic predictions (table 10. 1): Oel/Otot
—~ 4,
2s,
b/a -÷ 0.1,
~
(dO/dt)secondary maximum ~ 1.7 ~
10-2.
000
-~
ln
a 000(1rp)/a~0~(pp) -÷1,
(10.15) (10.16)
(da/dt)5..0 The ISR data are still far away from these limits; in fact this model suggests that the asymptotic region starts at much higher energies (fig. 10.6).
226
G. Giacomelli, Total cross sections and elastic scattering at high energies
FACTORIZABLE EIKONAL: CHENG-WALKER -WU
>
13.
-
12.
-ff~, ~
II.
-
to. 0.40
-
DATA ItI0.i 3
102
tO
5000
--i
0.35 > 0.30
-O o~
~
~
~
~ -
0.25 0.20
I
102
5000
0.2C
I
~E~ 0.19
0.15 0.14
102
i03
5000
SLAB (GeVI Fig. 10.6. Predicted energy dependences of B, act/at, and a 0/B for pp scattering for the Cheng—Walker—Wu model.
10.7. Mixed models The previous discussion indicates that it is difficult for a model to give a complete and satisfactory description of all high-energy data. Thus in some models it is assumed that the scattering process arises from the combination of two mechanisms, one of which dominates in a certain energy (or in a certain angular region) and the other in a different one. Several attempts have been made along these lines, with improvements in the fits to the data, but whose meaning is difficult to assess. The situation is somewhat reminiscent of that of nuclear physics: strong interactions are very complicated and each model sees some aspects of the truth, but not the whole truth.
G. Giacomelli, Total cross sections and elastic scattering at high energies
227
11. Conclusions
The basic conclusions concerning total cross sections and elastic scattering at high energies have already been sketched in fig. 1. 1 and discussed in the Introduction. One may ask how believable is the picture which emerges from the existing data and from our theoretical prejudices, and also if it applies to all hadron—hadron collisions. The answers to these questions require experiments at even higher energies and precise experiments with short-lived hadrons. Assuming that the conclusions of fig. 1 lb are correct, one may ask the following questions: where does the rise of 000t come from (both experimentally and theoretically)? What are its deep theoretical consequences? What are the implications (or the sources) of the optical-type picture which emerges from the study of elastic scattering? Many more features or problems are not summarized in the graphs of fig. 1 1. Some of these may be classified as details, though they could turn out to be fundamental. For instance, one should answer experimentally the question of whether spin effects are really negligible above, say, 100 GeV. The same is true for the real parts at t = 0. Other problems are usually classified as more fundamental. For instance, very little is known about the high energy large-angle elastic cross sections. It is here that one would think of observing effects arising from possible substructures of the hadrons. Present experimental information does not necessarily require substructures, but as already stated, the information available is not really adequate to reach any final conclusion. In summary: the recently commissioned high energy accelerators have opened up a new energy region. A wealth of new experimental information has become available very quickly, forcing a change in our ideas about the asymptotic region. It would not be surprising if new experimental results would lead to even further modifications. .
.
12. Acknowledgements I would like to gratefully acknowledge the fruitful discussions I had with many colleagues, too numerous to mention. They provided me with clarifications, advice and data before publication. Among those who contributed more directly I would like to mention U. Amaldi, F. Bonaudi, A. Costa, A. Diddens, M. Einhorn, M. Jacob, A. Greene, B. Lee, J. Sanford, G. Venturi and J. Walker. I would like to thank many people on the staff of the University of Bologna, of the tSR at CERN, at Fermilab and all the colleagues of experiment No. 104 at Fermilab and of the first CERN-IHEP Collaboration.
References The list includes references with total cross-section and differential cross-section data above 10 GeV/c. References before 1970 are not complete; in general the list includes articles and only those letters with tables of data. For a more complete list of references see the Landolt—Bärnstein compilation [73C5]. [55G11 R.J. Glauber, Cross sections in deuterium at high energies, Phys. Rev. 100 (1955) 242. [6lFl]
M. Froissart, Asymptotic behaviour and subtractions in the Mandeistam representation, Phys. Rev. 123 (1961) 1053.
228
G. Giacomelli, Total cross sections and elastic scattering at high energies
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[72B5] [72B6] [72C1]
[72C2]
[72C3] [72D1] [72D2] [72D3]
[72Fl] [72G!] [72G2]
(7211!] [72L1] [72M1] l72Rl] [72Y1] [73A!]
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[75B3] T. Buran, A. Eide, P. Helgaker, P. Lehmann, A. Lundby, A. Navanro-Savoy, L. Staunset, 0. Strum, C. Baglin, P. Briandet, P. Fleury, G. de Rosny, J.M. Ihenard, P.J. Carlson, K.E. Johansson, B. D’Almagne, F. Richard, D. Treille and V. Gracco, Antiproton—proton elastic scattering at 6.2 GeV/c, Nucl. Phys. B97 (1975) 11. [75C!] A.S. Carroll, 1.-H. Chiang, T.F. Kycia, K.K. Li, P.O. Mazur, P. Mockett, D.C. Rahm, R. Rubinstein, W.F. Baker, D.P. Eartly, G. Giacomelli, P.F.M. Koehler, K.P. Pretzl, A.A. Wehmann, R.L. Cool and 0. Facklen, Total cross sections of sri, K~,p and ~ion protons and deutenons between 35 and 280 GeV/c, 1975 APS meeting at Anaheim and Phys. Letters, to be published. [75D!] B. Drevillon, S. Borenstein, B. Chauraud, J.M. Gago, R.A. Salmeron, R. Barloutaud, A. Borg, M. Spiro, C. WohI, K. Paler, C. Comber, S.N. Tovey, T.P. Shah, Kp elastic scattering at 14.3 GeV/c, LPNHE/X/75 (1975). [75D2] R. Diebold, D.S. Ayres, S.L. Kramer, A.J. Pawlicki and A.B. Wicklund, Measurement of the proton—neutron elastic scattering polarization from 2 to 6 GeV/c, Phys. Rev. Letters 35 (1975) 632. [75G1] G. Giacomelli, A.F. Greene and J.R. Sanford, A survey of the Fermilab research program, Phys. Reports 19C (1975) 169. [75G2] A. Gaidot, C. Bruneton, 1. Bystricky, G. Cozzika, J. Deregel, A. Derevschikov, Y. Ducross, Yu.M. Kazarinov, M.Yu. Kazarinov, V.P. Kanavets, F. Khantine-Langlois, B. Khatchaturov, F. Lehan, A. Dr Lesquen, Yu. Matulenko, J.P. Menlo, A.P. Meschanin, S. Miyashita, J. Movchet, S.B. Nunushev, J. Piernard, I. Potashnikova, J.C. Raoul, L. Van Rossum, A. Saraykin, V. Siksin, E. Smirnov, V. Solovyanov and J. Strakhota, Polarization measurements in sip, Kp and ~p elastic scattering at 40 GeV/c, Phys. Letters 57B (1975) 389. [75K!] N. Kwak, E. Lohrmann, E. Nagy, M. Regler, W. Schmidt-Parzefall, K.R. Schubert, F. Winter, A. Brandt, H. Dibon, G. Flugge, F. Niebergall, P.E. Schumacher, J.J. Aubert, C. Broll, G. Coignet, J. Favien, L. Massonet, M. Vivargent, W. Bartl, H. Eichinger, Ch. Gottfried and G. Neuhofen, Experimental results on large angle elastic pp scattering at the CERN-ISR, Phys. Letters 58B (1975) 233. [75Ll] H.J. Lipkin, New systematics in hadron total cross sections, Phys. Rev. Dl! (1975) 1827. [75M1] W. Michael, G.5. Abrams, M. Alston-Garnjost, A. Barbaro-Galtieri, B.Y. Daugeras, G. Gidal, A.D. Johnson, G.R. Lynch, F.I. Solmitz and F.C. Winkelman, Backward elastic sr~pscattering at 3.7 and 7.! GeV/c, P~ys.Rev. Letters 35 (1975) 193. [75P1] 1. Pierrard, C. Bruneton, J. Bystricky, G. Cozzika, J. Denegal, A. Denevschikov, Y. Ducros, A. Gaidot, Yu.M. Kazaninov, M.Yu. Kazarinov, V.P. Kanavets, F. Khantine-Langlois, B. Khatchaturov, F. Lehar, A. De Lesquen, Yu. Matulenko, J.P. Menlo, A. Meschanin, S. Miyashita, 3. Movchet, S.B. Nunushev, I. Potashnikova, J.C. Raoul, L. Van Rossum, A. Saraykin, V. Siksin, E. Smirnov, V. Solovyanov and 3. Strakhota, Measurement of the spin rotation parameter R in sip and Kp elastic scattering at 40 GeV/c, Phys. Letters 578 (1975) 393. [7SRl] A.M. Rossi, G. Vannini, A. Bussiere, F. Albini, D. D’Alessandro and G. Giacomelli, Experimental study of the energy dependence in proton—proton inclusive reactions, Noel. Phys. 884 (1975) 269. [75W!] A.M. Wethenell, Elastic and total cross sections and implications, Proc. 1975 EPS Int. Conf. on High Energy Physics, Palermo.
G.GIACOMELLI University of Bologna, Bologna, Italy, INFN, Sezione di Bologna, Fermilab, Batavia, Illinois
q~c NORTH-HOLLAND PUBLISHING COMPANY
—
AMSTERDAM
PHYSICS REPORTS (Section C of Physics Letters) 23, No. 2 (1976) 123—235. NORTH-HOLLAND PUBLISHING COMPANY
TOTAL CROSS SECTIONS AND ELASTIC SCATTERING AT HIGH ENERGIES G. GIACOMELLI University of Bologna, Italy, JNFN, Sezione di Bologna, Fermilab, Batavia, Illinois Received 18 December 1974
Single orders for this issue PHYSICS REPORTS (Section C of PHYSICS LETTERS) 23, No. 2(1976)123—235. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 36.00, postage included.
G. Giacomelli, Total cross sections and elastic scattering at high energies
125
Contents: 1. Introduction 126 2. Notations and general theorems 128 2.1. Variables of particle i 129 2.2. Relativistic invariants 129 2.3. Scattering amplitudes and cross sections 130 2.4. Lorentz transformations 131 2.5. Optical theorem 132 2.6. High energy theorems 132 2.6.1. Total cross sections. Froissart bound 132 2.6.2. Elastic cross sections. Consequences of the Froissart bound 133 2.6.3. Diffraction scattering 133 2.6.4. Eden and Kinoshita bound 134 2.6.5. Pomeranchuk theorems 134 2.6.6. Pomeranchuk-like theorems for elastic scattering 134 2.6.7. Relation between total cross section differences and charge exchange scattering 135 2.6.8. Lower bounds for total cross sections 135 2.7. Structure of the total hadron—hadron cross sections in terms of Regge poles 135 2.8. MacDowell—Martin bound 136 3. Experimental 136 3.1. The Serpukhov 76 GeV proton synchrotron 137 3.2. The separated-function proton synchrotron at Fermilab 138 3.3. The CERN—ISR 140 3.4. Secondary beams of particles 143 3.5. Charged particle identification 145 3.6. Neutral particle identification 146 3.7. Main features of high energy hadron—hadron interactions 146 4. Total cross sections 147 4.1. Introduction 147 4.2. Experimental methods 148 4.2.1. The transmission method 148 4.2.2. Direct measurement of the pp total cross section at the ISR 149 4.2.3. Bubble chamber measurements 150 4.2.4. Indirect methods for charged particles 150 4.2.5. Other methods 152 4.3. The shadow correction in deuterium 153 4.4. Energy dependence of the total cross sections 155 4.4.1. “Weak, electromagnetic and strong” cross sections 157 4.4.2. Energy dependence of the cross sections of the long-lived hadrons 159
4.4.3. Energy dependence of other hadron—hadron cross-section 165 4.5. Relations among hadron —hadron cross sections 165 4.5.1. Total cross-section differences 165 4.5.2. Structure of the total cross-section differences in terms of Regge poles 167 4.5.3. Relations from the quark model 169 4.5.4. The Johnson—Treiman relations 170 4.5.5. p and w-universality 171 4.6. The energy dependence at asymptotic energies 172 5. Elastic scattering 173 5.1. Introduction 173 5.2. The Coulomb-Nuclear Interference Region 2] 173 [0.001 < Iti <0.01(GeV/c) 5.3. Elastic scattering in the diffraction region [0.01 < ti <0.5(GeV/c)2j 179 5.3.1. pp—* pp 181 5.3.2. Other types of elastic scattering 183 5.3.3. The cross-over phenomenon 183 5.4. Scattering at large angles 186 5.4.1. pp * pp 187 5.4.2. Other types of elastic scattering 189 5.5. Backward scattering 193 5.6. Total elastic cross sections 196 6. Charge exchange scattering and K~regeneration 199 6.1. The np —~ ir°nreaction 200 6.2. Other charge exchange scattering processes 203 6.3. K~regeneration 204 7. Polarization 205 7.1. Polarization in i~~p elastic scattering 205 7.2. Spin effects in pp scattering 207 7.2.1. ‘tot 207 7.2.2. The P parameter in pp —+ pp 208 7.2.3. pp elastic scattering in pure spin states 208 7.3. Conclusion 209 8. Dispersion relations for the forward scattering amplitude 210 9. Impact parameter expansion 212 10. Theoretical models 217 10.1. Amplitude analyses for irN -+ irN scattering 217 10.2. Thermodynamic or statistical models 219 10.3. Regge-pole models 219 10.4. Constituent models 221 10.5. Shadow scattering models 221 10.6. Optical models 223 10.7. Mixed models 226 11. Conclusions 227 12. Acknowledgements 227 References 227
126
G. Giacomelli, Total cross sections and elastic scattering at high energies
1. Introduction Inthis review we shall summarize the present experimental and phenomenological situation of high energy total cross sections and elastic scattering measurements. Charge exchange scattering and polarization phenomena will be mentioned in the context of their relations to total cross sections and elastic scattering. We shall also make a few remarks on photon—hadron, electron—positron and neutrino—hadron total cross sections. Only high-energy scattering will be considered, where we define high energy as greater than 10 GeV in the laboratory system. This number will sometimes be brought down to 5 GeV when lack of data or continuity reasons will require us to do so. In any case this means that we shall be considering experiments from only a few accelerators. Two new high-energy accelerators became operational three and two years ago respectively. The first is the CERN Intersecting Storage Rings and the second is the Fermilab Separated Function Proton—synchrotron. During this time these accelerators have provided a wealth of new information in a new energy region. It is thus appropriate to review elastic scattering and total cross-section measurements at this time. The measurements of total cross section and of elastic scattering processes are among the first measurements to be performed when a new accelerator opens up a new energy region; both types of measurements are repeated, as time goes by, with improved accuracy, finer spacing and, for elastic scattering, over a larger angular range. From an experimental point of view the measurements of total cross sections and of elastic scattering in the diffraction region are relatively easy to make; the difficulties arise when one tries to achieve the highest precision or when one wants to measure the differential cross sections at large angles. From a phenomenological point of view, both the total and elastic cross sections are simple processes with a relatively straightforward analysis. Even crude measurements give an immediate idea of the main features of the type of interaction under study in the energy region considered. Most of the systematic total cross-section measurements for ir~p,KIN, pIN, and nN have been performed with the transmission method, which is capable of high precisions. The recent measurements at IHEP (Serpukhov) and at Fermilab achieved point-to-point precisions of ±0.2%, with a systematic scale error of about ±0.4%. The recent construction of high-quality high-energy beams of tagged photons, of neutrinos, of hyperons, etc., have allowed an extension of the total crosssection measurements to these particles. Total cross sections depend on one kinematic variable, the energy, and on the type of colliding particles. Thus, we shall be interested in (i) the energy dependence of the total cross sections, and (ii) the relations among various cross sections. The differential cross section for the elastic scattering of unpolarized particles from unpolarized targets depends on two variables: the energy and an angular variable which is usually chosen to be the square of the four-momentum transfer, t. One may divide the angular distributions into five regions: (i) The Coulomb region (for Iti ~ 0.00 l(GeV/c)2). (ii) The Coulomb-nuclear interference region for 0.001 ~ It < 0.01. Measurements in this region are used to obtain information on the ratio of the real to imaginary part of the forward scattering amplitude. (iii) The diffraction region proper (for 0.01 ~ ti ~ 0.5). The most important parameter here is the slope or the slopes of the diffraction pattern.
G. Giacomelli, Total cross sections and elastic scattering at high energies
127
(iv) The large-angle region (for ItI> 0.5) is characterized by the smallness of the cross section and by the presence of a dip-bump structure which resembles that coming from diffraction from an opaque disc. Point-like constituents within the nucleon should radically affect the large-angle cross section at high energies. (v) The backward region for scattering close to 180°(not for pp). Here the cross sections are very small and all have a bump at u = 0. The experimental results from IHEP, TSR, and Fermilab have brought about a considerable change in the phenomenological picture of strong interactions at high energies. The change is summarized in fig. 1.1. The three sketches at the top illustrate what most of us would have expected for the total cross sections, for the ratios of the real-to-imaginary part of the forward scattering amplitudes (p), and for the elastic scattering differential cross sections at asymptotic energies. The expectations, based on the available experimental data and on the then prevailing theoretical models, were the following: (i) The total cross sections of particle on protons and of antiparticles on protons [atot(xp) and a505(~p)]should have decreased toward a common, constant, non-zero value as the energy increased. (ii) The ratios p should have increased toward zero from below. (iii) The differential cross section for proton—proton elastic scattering in the diffraction region should have continued having an almost exponential behaviour in t, with a slow increase of the slope as the energy increased. (iv) The ratios between various total cross sections should be governed by some internal symmetry. The total cross-section measurements performed in Serpukhov (1969—1972) showed first that some cross sections were not decreasing as fast as expected and later that the K~ptotal cross section was increasing with energy. In 1973, measurements at the CERN ISR showed that the pp total cross section was also rising with energy. In reality previous cosmic-ray measurements and do df
TOT!
PP—PP
1968 0.05
BLAB
0TOT
,,
-~
~.
o~ ,,~LAB
~~—-—~
BLAB
dt ~
1975
0.15
i~~T
Fig. 1.1. Illustration of the expectations of high-energy behaviours in 1968 and 1975.
1 28
G. Giacomelli, Total cross sections and elastic scattering at high energies
calculations had indicated rising pp total cross sections, but this was not taken very seriously. In 1972 measurements of the differential cross section for pp elastic scattering at the CERN ISR showed that the diffraction pattern had a break at ti 0.13 (GeV/c)2, a minimum at about 1.5 (GeV/c)2 and continued shrinkage. In 1972—1973, measurements at the ISR and at Fermilab of the ratio p for pp scattering indicated that p was becoming positive. The measurements of the total cross sections and of differential elastic cross sections at Fermilab completed the change in the overall picture as sketched in fig. 1 1. Present experimental results and the predictions of optical model calculations lead us to expect the following high-energy behaviour: (i) All total hadron—hadron cross sections will rise with an approximately (1n2 s)-dependence, while the differences a~ 0~(~p)u~0~(xp) decrease toward zero. (ii) The ratios p of the real-to-imaginary part of the forward scattering amplitude should become positive, reach a positive maximum, and then decrease toward zero. (iii) The diffraction pattern for pp elastic scattering will continue to shrink;Eat theabout diffraction 2 and a minimum pattern have whose a change of slope at it ~ in 0.13 iti 1.5should (GeV/c)2] location decreases iti (GeV/c) as the energy increases. No anticipation is however possible about the very large angle cross section, which is expected to be sensitive to hadron substructures. It may be interesting to note that our energy scale also changed. For instance, what is often called the beginning of the high-energy region (or of the asymptotic region), in the early 1960’s used to be at a few GeV; in the late 1960’s it increased to several tens of GeV; now it is more of the order of hundreds of GeV if one assumes that the asymptotic region starts after all total cross sections have started rising. It may be of the order of thousands of GeV if one assumes that the asymptotic region begins when the ratios of real-to-imaginary parts of the forward scattering amplitude have become positive and start decreasing towards zero. As far as the theoretical interpretations of high-energy scattering, there is no theory in the real sense of the word, i.e., there is no mathematical formalism that can give numerical predictions of the experimental quantities via well-defined and consistent procedures. Most of the theoretical work is done with specific dynamic models, like Regge pole models and optical models. We shall discuss some of their implications together with those of high-energy theorems. The notations used as well as the implications of some high-energy theorems and of some models are described in section 2. In section 3 are discussed the main features of the three highest energy accelerators, from which come most of the experimental results relevant to this review. The main discussions on total cross sections and elastic scatterings are in sections 4 and 5 respectively, while some connections with charge exchange scattering are discussed in section 6. Spin effects are briefly considered in section 7. The phenomenological discussion continues in sections 8 and 9, where the implications of dispersion relations and of optical models are considered. Section 10 gives an overall view of the interpretation of high energy total cross sections and elastic-scattering data. .
—
2. Notations and general theorems This section summarizes the notations and some general relations used in this review. The notation followed will be that of the Landolt—Börnstein data compilation [73C5] and concern the
G. Giacomelli, Total cross sections and elastic scattering at high energies
1 29
general two-body reaction l+2-÷3+4.
(2.1)
In elastic scattering one has: particle I = particle 3 and particle 2 particle 4. In the laboratory system the target 2 is at rest; particle 1 will be referred to as the incident particle, 3 as the scattered particle, and 4 as the recoil particle. Quantities defined in the center-of-mass system (c.m.) will be denoted by an asterisk, whereas quantities in the laboratory system are asterisk free. 2. 1. Variables of particle i (i = 1, 2, 3, 4) The notations used are listed in table 2.1. The c =
=
h/2ir = 1 system of units is used.
Table 2.1 Variables of particle i (i = 1, 2, 3,4) Center of mass system momentum (3-dimensional) kinetic energy total energy
~$
4-momentum scattering angle
P7 = (E7, 07
rest mass solid angle velocity/velocity of light transverse momentum
m1
Laboratory system
T$ E7
T. E 1
~$)
P1 = (E1,
~)
mf
n~’ 137
The following relations hold (in c.m. or lab.):
Pt 2 — p2 = E T=E m.
=
2.2. Relativistic invariants Convenient kinematical variables for the scattering processes are the s, t, u Lorentz-invariant quantities (Mandeistam variables). We shall define these quantities and express them in terms of lab or c.m. quantities: Total c.m. energy squared s
=
(P
2
=
(Er ÷ Er)2
1÷ P2)
=
m~÷ m~+ 2E
2 1m2
=
+
(m1 ÷ m2)
2T 1m2.
(2.2)
Four-momentum transfer squared between particles 1 and 3 2 = m~÷ m~ 2ErEr ÷2p~p~cosO~ = m~ + m~ 2E t = (P1 P3) 1E3 + 2p1p3cosO3 2 = m~+ m~ 2ErE~+ 2p~p~’cosO~’ = (m 2 2m = (P2 P4) 2 rn4) 2(E4 m4). —
—
—
—
—
(2.3)
—
—
Four-momentum transfer squared between particles I and 4
—
(2.4)
130
G. Giacomelli, Total cross sections and elastic scattering at high energies
u
=
2
(P1
—
=
m~+ m~ 2E?E —
2 = m~+ rn~ 2ErEr —
=
(P2
—
+
2p7p~cosO~ = m~ ÷ m~ 2E
+
2p~p~cosO~’ = (m
—
P4) P3)
2 2
—
rn3)
—
1E4 + 2p1p4cosO4 2m 2(E3 m3). —
(2.5) (2.6)
These three invariants satisfy the relation s+t+u~m~.
(2~7)
For elastic scattering
Pi*=
*=
P2
*—
*=
*
P3P4—P,
Er=E~,
Er=E~,
(2.9)
and the expressions for t and u may be simplified: 2 sin2.~O~c = —2m t = —4p” 2(E4 m4) = —2m2(E1 2(I + cosOr) + (m~ m~)2/s. u = _2p* At high energies —
—
E3)
—
(2.10) (2.11)
(2.12)
2.3. Scattering amplitudes and cross sections The elastic scattering amplitude for irN scattering will be written in the formt f(E*,O*)g(E*,O*)+i~ñh(E*,O*),
(2.13)
where g and Ii are the non-spin-flip and spin-flip amplitudes, respectively,a represents the Pauli-spin matrices and ,i is a unit vector perpendicular to the scattering plane. The differential cross-section for elastic scattering is given by
=
if(E*,
O*)12
=
Ig(E*, O*)12
+
ih(E*, O*)i2.
(2.14)
Using relativistic invariants one may write (c~)
dt
=
el
1 2 iF(s, t)i2’ 641r5P*
= ~.
1 2 F(s, t)i2 l6irs
,
(2.15)
where F(s, t)
=
81
/,~f(E*,0*),
(2.16)
tFor nucleon—nucleon scattering one has to add to add a spin—spin term. At high energy this term has not been measured. It is presumed to be small and is often neglected (see section 7).
G. Giacomelli, Total cross sections and elastic scattering at high energies
(~fl
=—~---(---~----~
‘~dt/et p*2\dfZ*J
1 31
(2.17)
.
The polarization parameters P, R and A are defined as 2Im(gh*) Igi2+ hi2
(2.18)
R~
I 12—1h12 Ig12+ hi2
(2.19)
A
2 Re(gh*) 1gi2+ hi2
(2.20)
2.4. Lorentz transformations The Lorentz transformations between c.m. and lab frames of reference for the momentum and energy of particle i may be written as: p 5p,sin01p~sin07
(2.21)
p, cos 0~= 7cm(Pt COS Or ÷f3cmE7)
(2.22)
E1 =
7cm(E7 +
I3cmPr cos Or),
where ~cm
=
7cm
=
p1/(E1 + m2)
Ifs/I
—
1~~m = (E1 +
(2.23) m2)/s/i
(2.24)
The following relation is often useful =
p~= p~m2/sJ~
(2.25)
In the case of elastic scattering the relation between the angles O~and 07 with i = 3, 4, is given by tg01tg~07.
(2.26)
m
The elastic differential cross sections da/dt, da/d~2*and do/d~2are related by da 7r 2 dt, p7
da ir 1/~/m, \2 d~27 p~ iil—u----—sin0.i Y \rn 2 ‘I
da d~2,
.—.
(2.27)
G. Giacomelli, Total cross sections and elastic scattering at high energies
1 32
2.5. Optical theorem Neglecting spin—spin effects, the imaginary part of the non-spin-flip forward scattering amplitude may be related to the total cross-section [h(E, 0) = 0]: p* Img(E*, 0) =—a~O~.
(2.28)
4ir
Hence, Idu\
=
da
diti (—)
p
~
iRe g(E*,
—~
p*2 a~
0)12+
1
(2.29)
2+__. I6ir
(2.30)
iReg(E*,0)i
At high energies, where the ratio p good approximation
=
Re g(E*, 0)/Tm g(E*, 0) is presumably small, one has to a
(~)
2cJ~ 10m~eV~21
5.095 X l0
2.
(2.31)
0~(mb)
2.6. High energy theorems On the basis of the general properties of unitarity, analyticity and crossing, a number of highenergy theorems have been derived for the limit s oo~We shall briefly discuss here their implications for total cross sections and elastic scattering at very high energies. In this context the asymptotic region may be defined as that region where the number of open channels is very large and the cross section for each channel is very small. In practice, if one assumes that the asymptotic region starts when all total cross sections have started to rise, then it cannot start below a laboratory momentum of 300 GeV/c. In terms of the ratio p of the real-to-imaginary part of the forward scattering amplitude, the asymptotic region starts when all ratios have become positive and have started decreasing toward zero. This presumably happens at laboratory momenta larger than a few thousands of GeV. It is, therefore, not clear if the analyses of the present experimental results in terms of high-energy theorems are really meaningful. In the following, the high-energy theorems will be stated without any proof. For a more thorough analysis, the reader is referred to theoretical reviews of the subject [72R11. -*
2.6.1. Total cross sections. Froissart bound From positivity, analyticity, and unitarity, it is possible to derive the following upper limit [61F11: ~
(2.32)
G. Giacomelli, Total cross sections and elastic scattering at high energies
133
where ir/m~~ 60 mb and the unknown constant ~ fixes the energy scale (it may be the energy value above which the cross section rises).* 2.6.2. Elastic cross sections. Consequences of the Froissart bound It is possible to show that (i) if Gtot ln~s, then Gei 1n2 s and thus tlei/Utot const; (ii) if ~ ln s, then Uel/Otot ~ anything. For energies smaller than S~it would seem that case (ii) applies, while for s ~ s 0, one should have case (i). For energies around s~or slightly larger, the situation is more complex. 2.6.3. Diffraction scattering One may define the width of the diffraction peak F as 0•i
(2.33)
F(s, 0)/s 2s then F ~ 1n2s and If ln F(s,t) = fit/ ln2 — s\ lim = ftr), s—3.=F(s,0) \
(2.34)
50/
that is the limit is a function of r = t 1n2(s/s 0) only. This and other considerations lead to the expectations that the differential cross section cannot be of the form 2). (2.35) du/dt = A exp(Bt + Ct An example of an acceptable form is the one proposed by Cheng and Wu E71C2]: F(s, t)
=
a,’.
isJ
2.
(2.36)
0(V’~ilns)(lns)
General ideas on analyticity in the laboratory energy E and the crossing property allow one to write for the even (F4) and odd (F) signature amplitudes at high energies: i 4 E(lg E — ~ iir)13~ 7 F(E)~a‘y’~E(lgEf ~i3’y(lgE)’3~. F4(E)
(2.37)
—
If
ln~sone has ~ Re F(s, 0) ImF(s,0)
=
2; moreover, if F ~ F4 one has
Re F’~(s0) ImF4(s,0)
iT
(2.38)
lgE
that is at very high energies, the ratio of the real-to-imaginary part of the forward scattering amplitude is positive and decreases towards zero. *The presence of the pion mass in the Froissart bound may be thought to arise from the tail in the Yukawa distribution (~1/2mg) which becomes more and more important as the energy increases.
G. Giacomelli, Total cross sections and elastic scattering at high energies
134
2.6.4. Eden and Kinoshita bound Idcr\
Is \
2~_) < (const)u~ t=o s—~= 1ln so
(2.39)
iT
I—i
< —1n41—I. a—4’.’. #2o \S /
~ t—+O
(2.40)
-~
0
2.6.5. Pomeranchuk theorems First Pomeranchuk theorem (Okun—Pomeranchuk rule). In the limit of high energies, the total cross section on the same target particle of particles belonging to the same isospin multiplet become equal: Otot(lrp)
=
a~05(?p)= a~0~(ir°p)
a50~(pp)= cr~05(np) 4p) = u 4n)= o~ a~0~(K 50~(K 0~(K°p) = a~05(K°n).
(2.41)
Second Pomeranchuk theorem. At high energies if the difference between particle and antiparticle cross section has a limit and if [(F P)/(s ln s/s0)] -÷ 0 ass oo, the total cross sections on the same target particle of particle and antiparticle become equal —
-+
a~0~(pp) = Utot(PP) a105(K~p)= a~01(Kp),etc.
(2.42)
One can also guess how the “Pomeranchuk” region is approached, that is one may guess the energy dependence of the difference in the total cross sections. From formulae (2.37) one has: a10~(xp)
a~
—
Gtot(XP)
13
+ /2
< const(ln E)
—
01(xp)+ a501(xp) 2 const/lns. (2.43) and if atot (lns) The available data on total cross section differences agree better with a power dependence of s (see sections 2.6.7 and 4.5). 2.6.6. Pomeranchu k-like theorems for elastic scattering They may be expressed by the following limits as the energy goes to infinity: rdu(xp)/dt~ lim i i= 1 Lda(xp)/dtJ
(2.44)
lim F
(2.45)
=
0
a—’
lim [F(xp)/F(i~p)]
=
1
(2.46)
G. Giacomelli, Total cross sections and elastic scattering at high energies
lim [Ue1(XP)/Gei(itP)]= 1.
135
(2.47)
These relations express the general statement that the differential cross sections of particle and antiparticle scatterings on protons tend to become equal as the energy increases. 2.6.7.
Relation between total cross-section differences and charge exchange scattering 4p)i <—\/a(irp-# iT°n) ln[ vT~
-
a~0~(ir ~)
—
m
s-’
a~01(iT
2
___
(2.48)
-
s0a(1rp~~* 7T0n)
5.
4n) a —
a~05(K
105(Kn)i~
Ju(K~p-~ K~p)ln[2(K
K~p)]~
(2.49)
The usual parametrization =
Ap~ = Bs~
(2.50)
for total cross-section differences at high energies would eventually violate the above bounds. 2.6.8. Lower bounds for total cross sections Jim—Martin
~
> ~-+~
Cornille—Martin
a~~(s)~
6
s (lns/s0)
(2.51)
2
const
(2.52)
2.7. Structure of the total hadron—hadron cross sections in terms of Regge poles In the framework of a simple Regge pole model, the total hadron—hadron cross sections may be expressed as a sum of contributions from a few basic trajectories: the vacuum trajectories P (Pomeron) and P’, and the trajectories with the quantum numbers of the p, w, A2 and f2 mesons. For the irN, KN, pN, and pN total cross sections we have: fit ot(1T~P)= a~(7TN)T a~(irN)+ (J~i~(1rN)
a~0~(K~p) = u~(KN)~ a~(KN)~ a~’.,(KN)+ cA2(KN)
f(KN)
(2.53)
a~~~(KN)
(2 54
+ api
~tot(P~P) = a~(NN);u~(NN);a~(NN)+ aA2(NN) + a~0~(K~n) = a~(KN)±c7~(KN);a~(KN) aA2(KN)
+
atot(p~n)= u~(NN)±a~(NN)~ a~(NN)—aA2(NN) + UP? ~(NN). The P, P’, A2 and f2 contributions to particle and antiparticle cross sections are the same. Each term of (2.53, 2.54) may be written in the Regge pole form u.(ab) = B~bp°~’’
(2.55) (2.56)
G. Giacomelli, Total cross sections and elastic scattering at high energies
1 36
where p is the laboratory momentum, i= P, P’, p, w, A2 and f; a, bare the incident (irk, K~,~ and target particles (p, n). The form (2.56) implies factorization of the residues. 2.8. Mac Dowell—Martin bound The slope of the forward elastic peak at t
=
0 may be defined as
Ed ln(da/dt)1 B0 = [ dt i~=0 The Mac Dowell—Martin bound [74B14] states that 1
2
U~0~
B0 > — —i--
-
-_
-~
1 ~ot —i--.
(2.57)
(2.58)
3. Experimental The progress in the field of subatomic phenomena in general and of elementary particles in particular has always been connected with the development of new accelerators of ever increasing energies. They allowed the production of secondary particle beams of higher energies and of beams of altogether new particles. The experimental results which will be reported in this review come from a limited number of accelerators: the 12 GeV Zero Gradient Synchrotron (ZGS) at Argonne National Laboratory (ANL), the 28 GeV Proton Synchrotron (PS) of the European Organization for Nuclear Research (CERN), the 33 GeV Alternating Gradient Synchrotron (AGS) of Brookhaven National Laboratory (BNL), the 40 GeV Stanford Linear Accelerator (SLAC), the 76 GeV Proton Synchrotron of the Institute for High Energy Physics (IHEP), the 200—500 GeV Fermi National Accelerator Laboratory Proton Synchrotron, and the CERN Intersecting Storage Rings (ISR). These will soon be joined by the CERN 400 Proton Synchrotron. Table 3. 1 summarizes the main features of these accelerators. The table gives the main parameters of each accelerator, like the maximum energy, the energy at which the accelerator is usually operated, the average intensity per pulse, and the repetition rate. In addition it gives two parameters connected with the operation of the accelerator for physics: these are taken as the total number of “Stations” at which experiments may be performed and the number of “Stations” which can operate at the same time E 74S I]. The experiments that can be performed at an accelerator depend critically on the quality and quantity of the available secondary beams. In a conventional fixed-target accelerator the beams may originate either from an internal target placed in the circulating beam or may come from a target placed in an extracted beam. While the first system was used extensively in the past, the trend for the future will be towards targets in external beams. The main reason for this trend is connected with problems of radiation damage, which become serious when internal beam intensities exceed 1013 particles per pulse. We shall briefly recall some of the features of the three highest energy accelerators, at IHEP Fermilab, and CERN, from the point of view of an experimenter, neglecting all technical details.
G. Giacomelli, Total cross sections and elastic scattering at high energies
137
Table 3.1 Some features of the high-energy accelerators with E> 12 GeV. The average intensity, the number ofbeams, and the statistics on the nuinber of experiments refer to the situation in the Summer of 1974 and are only approximative.* Laboratory
ANL
CERN
BNL
Accelerator Maximum energy (0eV) Normal energy (GeV) Average intensity (10’ 2 particles/pulse) Repetition rate (pulses/sec) Burst length (sec) Total number of stations Stations served at same time
ZGS 12.5 12 3
PS 28 23
AGS 33 28.5 5
5
2
IHEP
Fermilab
CERN
SLAC
—
—
ISR
—
76 70
500 300 8
5
1/2
1/2
1/7
0.7 17
0.5 17 + ISR
1 10
10
13
13
ISR
6
5
+
*For the CERN—ISR we quote the luminosity L
L
1/7
1—1.5
in cm2
2062 500—1500
0.8 20
=
22.1 20 40 ~zA
3’
i0
360
Continuous 8
10
8
1.67 X 106 7 6
CERN
SPS 400 200—400 —
—
—
—
sect and the number of intersecting regions (8).
3.1. The Serpukhov 76 Ge V proton synchrotron The 76 GeV proton synchrotron of the Institute for High-Energy Physics (IHEP) at Serpukhov, USSR, is an alternating-gradient, strong-focusing accelerator with a 200 MeV linear accelerator as an injector. The accelerator usually runs at 70 GeV, with a repetition rate of one burst every 7 sec. The
IHEP( 1975) 76GeV —25x IO12ppp
Kt p—.K
I/7sec —(7 sec Spill
P
np—pn n
Be—.DDX
+
&OQfl)q
1.~—1
Beam 18 fr-p-—trn
~
~K’~n—.K°p
I1~4?oQ,,”)1/
ftj~J
9
°~ Beom6 (Test)
Beom5..,,
iMirabellei
RF
,T-p——1r-p
~
7rp—p+MM n.—p—.~~°n
45 6 4A
K A—A+K,,-,~ 1r~r
Internal
8Con,
Beam7
p—Fast
28 Beam 2 25 2A
0iot
Beam 14
‘~
P(,r±p~~~_,r±p) ~r-p---AAn Fig. 3.1. Beam layouts at the IHEP accelerator.
Target Area
138
G. Giacomelli, Total cross sections and elastic scattering at high energies
circulating proton beam averages about 5 X 1012 protons. The present beam layout is shown in fig. 3. 1. There are seven beams of pions, kaons, and nucleons, fed by separate targets which intercept the internal proton beam. Of these secondary beams, four are negative beams for counter experiments, one neutral, and a test beam. Moreover there are an external slow-extracted proton beam, from which originate two secondary-charged beams, and two fast-extracted proton beams from which originate the pions which eventually decay into neutrinos and an RF separated beam for bubble chambers. There are also internal-target facilities. 3.2. The separated-function proton synchrotron at Fermilab The separated-function proton synchrotron of the Fermi National Accelerator Laboratory in Batavia, Illinois (Fermilab), has four acceleration stages [74W1]: (i) a 750 keV Cockcroft—Walton electrostatic accelerator; (ii) a 200 MeV linear accelerator; (iii) a 10 GeV booster fast-cycling proton synchrotron; (iv) the main ring proper, 1 km in radius. This is the first accelerator where deflection and focusing are done by separate magnetic elements, i.e., Particle Beams and Facilities at Fermilab S,ngIo-Arn, Spectrom,te,
——
_..~M38M4
Meson Areg_.—~~M2
Booster
b~L~~oc~
Area ~
‘.MI~rom.torC=flter
~
Area
Eo,t
‘.
P1
P2
Prima, S
5 Beam Secondary Boom PrimoryTorgetLocotion
Meson Area Ml General- Purpose Charged Particle Beam M2 Diffracted Proton Beam (Sometimes Pions) M3 Neutral Beom (Neutrons) M4 Neutral Beam (Koons and Neutrons) M6 Charged Particle Beam
Neutrino Area NO Ni N3 N5
Neutrino Beam Muon Beam (Sometimes Pions) Charged Beam to 30’ Bubble Chamber Charged Beam to 15’ Bubble Chamber
Proton Area Pt
Neutral Beam (Photons or Neutrons)
P2 Tagged Photon Beam (Sometimes Electrons) P3 Charged Particle Beam
Fig. 3.2. Diagram of the experimental beams available at Fermilab in the summer of 1974 [75G1].
G. Giacomelli, Total cross sections and elastic scattering at high energies
139
Flat Top
:
/
Le3lon
Time After Last Cycle tn Seconds
~
1.0
Proton Beam Use d. Internally
: Injection
-
2”,
Beam Extracted
—_.~
/
B
\
I
~-Slowly
\
/
/
I
~
.2
for Counter
Experiments
/ /
.
l’—.’
0
1
2 3 4 Time After Last Cycle in Seconds
Fast Extraction a— for Bubble Chamber Exp.j t I 5 6
Fig. 3.3. Energy and intensity cycles at the FNAL accelerator [74S 1].
bending magnets and quadrupoles, respectively. At lower energy accelerators, both functions are performed by bending magnets with pole pieces of appropriate shape. Fig. 3.2 shows the general layout of the accelerator. The fast-cycling booster synchrotron is capable of accelerating protons to 10 GeV, 15 times per sec; 13 of these pulses are injected into the main ring from t = 0. 1 to t = 0.8 sec of the main cycle time, as shown schematically in fig. 3.3. After injection the magnetic field rises, the protons are accelerated by a RF cavity at the rate of about 100 GeV per 0.8 sec. During the acceleration cycle a small fraction of the protons may be intercepted by a gas jet, or thin targets of various shapes, for experiments using directly the proton beam of the accelerator. At the end of the accelerating cycle, the magnetic field stops rising and is kept at a constant value for about one second, during which time the internal beam is extracted partly in a short time and partly during 0.8 sec. The fast-extracted beam is used by bubble chambers and for neutrino experiments, while the slow-extracted beam is used by counter experiments. There is only one extraction point from the main ring, from which protons come out into one beam line. The external proton beam is then split into two, at the first splitting station, by a combination of electric and magnetic fields. After subsequent splits one has the possibility of operating six primary proton beams in addition to the internal beam (fig. 3.2). This allows four experimental areas to be run simultaneously: (i) The Internal Target Area utilizes directly the internal proton beam during its acceleration cycle. Here one can perform proton—proton and proton—light nuclei collisions at all energies from 20 to 500 GeV.
G. Giacomelli, Total cross sections and elastic scattering at high energies
140
(ii) The Meson Area is fed by a single 300 GeV proton beam which hits a beryllium target from which originate five secondary beams, three charged and two neutrals, which are all used simultaneously. Here are performed counter experiments with secondary beams of pions, kaons, nucleons, and hyperons. (iii) The Proton Area is in reality a complex of three areas, each with an independent primary proton beam. Here are performed counter experiments with primary proton beams; there are also secondary photon-electron facilities. (iv) The Neutrino Area has a more complex situation designed to satisfy the requirements of bubble-chamber, neutrino, and muon experiments. The area may be subdivided into at least two areas: the muon area, where a muon beam and a dichromatic neutrino beam are utilized, and a bubble-chamber area where one can have secondary hadron beams for the two bubble chambers and neutrino beams of different momentum bands for use with counters and bubble-chamber experiments. Table 3.3 gives a list of the major experiments performed at Fermilab which yielded results relevant to this review. 3.3. The CERN—ISR Experiments at a storage-ring machine are closely connected with the machine itself, much more so than for conventional accelerators [74G3]. The Intersecting Storage Rings (ISR) of the European Organization for Nuclear Research (CERN) consist of two concentric and slightly distorted rings, each 300 m in diameter, fig. 3.4 [73J 1]. The two rings intersect horizontally eight times, with a crossing angle a = 14.8°.They are filled in succession by pulses of about 1012 protons ejected from the CERN 28 GeV proton synchrotron. In each ring every pulse is “stacked” at a slightly different horizontal location from the previous one. After filling, each circulating beam has the form of a ribbon with approximate transverse dimensions of 3—5 cm wide, horizontally, and 0.4—0.7 cm high, vertically (see fig. 3.5). Currents of up to 30 A have been stacked. The momentum spread of each beam is approximately ±1%. Owing to the small intersecting angle, the interaction region has a “diamond” shape, approximately 50 cm long, 5 cm wide, and 0.4 cm high. When operated in the “Terwilliger mode” the transverse dimensions of the beams, and thus of the diamond, become considerably smaller, about 20 cm long, 3 cm wide, and 0.3 cm high. Table 3.2 gives the main parameters of the ISR. The ISR is presently operated at five energies, which cover the range between 23 and 63 GeV in the center of mass. The top energy of 63 GeV, which requires acceleration of the protons in the ISR itself, corresponds to the energy of a 2062 GeV conventional accelerator. At present this is the highest accelerator energy available. At each intersection region the total collision rate N is given by N—Lcj~0~, (3.1) 0tot is the proton—proton total cross section, and L is the luminosity. L is connected to the where parameters of the machine by the following equation 1112
e2ch~~~tg(a/2)
(3.2)
G. Giacomelli, Total cross sections and elastic scattering at high energies
T12
TTI
Ii
/ 113
//
~
ISR
H
~4.
Fig. 3.4. Scheme of CERN PS and ISR.
BEAM-BEAM DIAMOND
Fig. 3.5. Horizontal distribution of 65 000 reconstructed events. Also shown are the single beam profiles [71 Hi I.
141
G. Giacomelli, Total cross sections and elastic scattering at high energies
142
Table 3.2. Some parameters of the CERN ISR and present operation for physics Beam momenta Crossing angle Beam losses Average pressure Pressure in intersections Design luminosity Transverse beam Jhorizontal
11—31 GeV/c, s~sp/p 2% 14.8° ‘~- 0.1—1% per hour < 10-1 1 torr ~ i0-’ 2 torr 2 sec~ 4— X3—5 1030 cmcnc
dimensions
—
[vertical
Beam momenta (0eV/c)
cm energy (0eV)
s = F*2
11.8/11.8 15.3/15.3 22.4/22.4
23.4 30.4 44.4 52.6 62.3
26.5/26.5 31.4/31.4
0.4—0.7 cm
(0eV2)
Equivalent lab. momentum (0eV/c)
Present average luminosity (10~° cm~seC1)
548 924 1971 2767 3881
291 491 1053 1474 2062
2 4 10 15 3
In addition there is a low-fl section in which one may achieve luminosities of (3—4) X i0~’cm2 sect
where I~and ‘2 are the proton currents in the two rings, c is the velocity of light, e is the electron charge, a = 14.8°is the crossing angle, and heff is the effective height of the overlap of the two beams. heff is defined in terms of integrals over the direction (vertical) perpendicular to the plane of the two beams: 1
=
fp 1(z)p2(z)dz
heff
(3 3)
fp1(z)dzfp2(z)dz
Here Pi and P2 are the beam densities as a function of z, the vertical coordinate. The measurement of the luminosity is one of the most crucial at the ISR, since on it depends the knowledge of the absolute values of the measured cross sections. Inasmuch as all parameters in eq. (3.2) except heff are known or measured to better than 0.1%, the determination of heff becomes the most delicate task in measuring L. heff may be measured by a number of methods. The direct measurement according to formula (3.3) requires the measurement with spark or proportional chambers, first of the densities of one beam (Pi and P2, by recording secondary particles originated from the collisions of one beam against the residual gas), and then of events coming from beam—beam collisions (Pi P2)- At present this method does not yield high accuracies, because of the small vertical dimensions of the beams. A method which is more widely used is the Van der Meer method [68Vl} of recording the event rates in some monitors, while displacing the two beams vertically from one another. In this way, one measures directly2 the overlap height heff. sec’effective has been obtained; since a~At the c.m. energy of 53 GeV a luminosity L 2 X 1031 cm 0~(pp) 40 mb, one has 9 X I 0~interactions per second in each intersection region. At Ecm = 53 GeV the average multiplicity of the produced particles is 18 [74R 11; thus about 1.5 X 1 0~particles per second are produced at each intersection.
G. Giacomelli, Total cross sections and elastic scattering at high energies
143
Table 3.3 List of ISR and Fermilab experiments which provided data relevant to this review*. For the results from other accelerators see the Landolt—Bornstein compilation [73C5] and the proceedings of the latest conference [74D6l, [75W1 I. Experiment number
Collaboration
Process studied
c.m. energy (GeV)
i-range 2 (GeV/c)
R401 R601
CERN—Hamburg—Orsay—Vienna CERN—Rome
pp -~ pp
0.8—3 0.001—0.014
R602 R801
Aachen—CERN—Genova—Harvard—Torino
23, 63 23,31 23,31,45,53 23,31,45,53 23,31,45,53,63 lab, momenta 20—200 30—280 50200
1A 4 7 21A 36 96 104 111
Pisa—SUNY Fermilab—Harvard—Pennsylvania—Wisconsin Berkeley—Michigan ANL—Fermilab—Indiana—Michigan Caltech—Fermilab Fermilab—Dubna—Rochester—Rockefeller ANL—Bari—Brown— CERN—Cornell—FL—etc. Brookhaven—Fermilab—Rockefeller Caltech—Berkeley
pp—opp °tot(PP) pp—’ pp °tot(PP)
o~ot(vN) otot(np) (iT~p,K~p,P~P)eI at 0t(vN) pp -+ pp (iT~p,K~p,P~P)el atot(1r~,K~p,pip) 1rp—olr°n
40—100 50—400 50—150 35 —280 21—101
—
0.05—3.0 —
— —
0.13 —
0.001—0.09 0.1—1 —
0—1.3
*Not listed in the table are the bubble-chamber experiments. Furthermore, it is expected that experiments R604, R805 at the ISR and 8, 12, 25A, 69, 82, 97, 177A, 198A, 248, 313 at Fermilab will soon yield further results relevant to the subjects of this review.
The decrease in the circulating beam intensities is between 0.1% and 1% per hour; it is then possible to keep useful beams for physics experiments up to 40 hours without “restacking”. Such conditions were achieved mainly because of the extraordinary vacuum in the beam pipes, which averages about 10_li mm Hg in the whole accelerator and is kept below 10_12 mm Hg in the intersecting regions. Finally, it should be noted that since the two beams collide at a small angle (14.8°)the ISR laboratory system is, to a good approximation, equal to the center-of-mass system. Table 3.3 gives a list of the experiments performed at the ISR which yielded results pertinent to this review. 3.4. Secondary beams of particles
A fixed target accelerator allows the possibility of producing secondary beams of many different types of particles. Thus one can study a large variety of interactions. At a proton-synchrotron one can produce beams of neutrons, K~and photons, using a sequence of collimator-sweeping magnet repeated a number of times. One may even achieve a partial separation of these particles by preferential absorption in a low-Z material (to remove neutrons) or in a high-Z material (to remove photons). A neutrino beam is more complex: it requires optimization of particle production, of acceptances, of a decay region, of p-absorption, etc. A charged particle beam requires a large number of bending magnets and of quadrupole focusing elements. We shall only discuss further this last case. High energy secondary beams of charged particles tend to be very long physically because of the large magnetic rigidity of the particles. It is relatively easy to make beams with good optical properties for which the laws of geometrical optics are completely valid. Fig. 3.6 shows as an
144
G. Giacomelli, Total cross sections and elastic scattering at high energies
Meson Produclion Target Septum Magnet f-Stop Collimntor
Quadrupole Focusing Doublet
Main Bend Magnets
First Focus. ~P Collimator
Bend
~
Second Focus.llalo Collimator
Recombination Bends
Third Focus. Experimenters Target
Fig. 3.6. Equivalent optical diagram of the M6 beam line of the Meson Laboratory at Fermilab.
example the M6 beam of the Meson Area at Fermilab [74T 1]. The beam is produced at an angle of 2.5 mrad; it can be tuned in the momentum range of 20—180 GeV/c and is designed to have a momentum resolution of better than ±0.1%.It is a classical beam, as one may have in geometrical optics, with three stages and three foci. The first set of quadrupoles make a parallel beam (they make a point-to-parallel transformation) which is then analyzed in momentum; the second set of quadrupoles focuses the beam horizontally and vertically at F1. At this focus the beam has a horizontal dispersion of about 6 cm/(% .i~p/p);the movable aperture of a momentum collimator defines the momentum acceptance of the beam. A similar optical arrangement, point-to-parallel and parallel-to-point is repeated through two more sections with shorter focal lengths. Vertical and horizontal clean-up slits are located at the second focus F2. In the final stage of the beam
G. Giacomelli, Total cross sections and elastic scattering at high energies
145
there is momentum recombination in both positions and angle at the third focus’F3. If the produc3 and if the momentum bite of the beam is fixed at tion target is a rod 0.5 X at150 = ±0.1%, the 0.5 finalX image F mm 2 in size with a horizontal 3 isone approximately X 0.5 divergence of ±1 mrad and a vertical of ±0.5mrad.0.5The totalcmlength of the beam is 440 m. 3.5. Charged particle identification The fraction of the incident charged particles to be used for a scattering experiment are usually defined geometrically with scintillation counters and identified in velocity, and thus in mass, with a combination of differential and threshold ~erenkov counters. High-energy beam lines have several ~erenkov counters, for a total length of up to 100 m. The counters differ considerably from the simplest threshold counter, with a simple mirror and one photomultiplier, to the most sophisticated differential counter, with a precision mirror, a refined optics for coma and chromatic corrections and many photomultipliers. Fig. 3.7 shows the mass spectrum obtained with a gas differential ~erenkov counter [7 SC 1]. Small counter hodoscopes or proportional wire chambers are often placed in the beam in order: (i) to improve the momentum resolution; for this purpose a hodoscope is usually placed at a dispersive focus in order to locate exactly each particle and thus its momentum; (ii) to select only those particles which have specific directions; for this purpose the outputs of two sets of proportional wire chambers are fed into a fast logic, a coincidence matrix, which allows the selection of the appropriate trajectories. In this case the proportional wire chambers are used as a sort of “electronic collimator” [74C 1].
pn300Geht l0’
P a~$c~ Ge\6t 9n 35 n~ad
P ia
10 io~
II
irK
(2
13
14
5
6
rT
PRESSU~(PSIA OF HEIJUM) Fig. 3.7. Mass spectrum of the particles present in a 280 0eV/c positive beam produced at 3.5 mrad by 300 GeV protons impinging on a beryllium target. The curve was obtained with a differential ~erenkov counter filled with helium [75C1 1.
G. Giacomell4 Total cross sections and elastic scattering at high energies
146
3.6. Neutral particle identification Neutral particles can only be identified via their interaction and the subsequent observation of the charged products. Thus charged-particle detectors can only be placed at the end of the beams, after the secondary target. Neutral-particle detectors are in practice designed for specific experiments and are more part of the experiment than of the beam line. Calorimeters play a major role in neutral-particle detection. ‘y-rays are detected with total absorption shower detectors, most of which are lead-glass ~erenkov detectors or lead-scintillator sandwiches. Many neutron calorimeters are made of iron-scintillator sandwiches. The same type of detector, but much larger, is used for i’-detection. K°’sare detected through their decay products in magnetic V°spectrometers. 3.7. Main features of high-energy hadron—hadron interactions In planning an experiment one has to keep well in mind the main features of strong interactions at high energies, in particular: (i) Most of the interactions are inelastic, Uel/Gtot being 15—25% (see fig. 5.31). 1111111
I
iiiirij
11111111
IIIiIr~
—
~
lo~
— -
-‘--
4
-
o ‘aJ o
1—
ii.
o
•
‘a •~ .5_•
‘
/
/
-
-
UI
-
•U
> —
/
/
— ..••
.~
/
o.0i—
/ /
/
/
—
/
0 /0
/
_3-___
-~---
/ /
o .—
—
.7
~
o_o—
.__-__
/
-
S
,~e
0.1—
I
U
— —
4
-
-
I
—
:
-~
—4_
~
———-a---
.
~
—4— —
III
—
-
o
~—~--~
I
/ /
/
/
/
/0
/
/0
/°
°
/
0/
/ •
0
1111111
10
1111111 102
1111111 10’
11111111 10’
I
Ii
S [co.v?] Fig. 3.8. The average multiplicity of sr~,~r, K~,K, p, ~, and for all charged particles plotted versus s. The dashed lines represent the results of the fits according to the formula (n1> = A + B in s + C ~i /2 [74R1].
G. Giacomelli, Total cross sections and elastic scattering at high energies
5
~ 10
(GeV) 20 I
50
500 —
I—
w 400
T~(MeV)
180 170
—
D
~
-O’~
P
t~o
._
_._.
K:
ISO 140
—
a~
—
• _~•~
—
—.
—
_._
—
.~-. ~
147
_o—o K
150 11.0 130
•
120? I 150
•~.•__—o—-o charged
Lii
•— —:~1it 300
~
140
I~~II
10 LABORATORY
I
102 MOMENTUM
3 io (GeVIc)
io’
Fig. 3.9. Average transverse momentum for the production of pions, kaons, antiprotons, and all charged particles plotted versus laboratory momentum. On the right are shown “temperature” scales for different particles computed in the framework of a thermodynamic model [74R1 1.
(ii) For each interaction the average multiplicity grows approximately as ln s and becomes large at high energy. At p = 1500 GeV, corresponding to = 53 GeV, one has:
~
=
~ch>
+
4.7 ir~+4,5
~.O
+
~ 12 ÷6
4.~3ir~+ 0.43 K~+ 0.31 K ÷l.6p+ 0.lSj5+
...
(see fig. 3.8) [74Rl].
(iii) The average transverse momentum is small and becomes essentially energy-independent. It ranges between 350 to 450 MeV/c, depending on the type of produced particle (fig. 3.9). Thus large p~(t) events are rare. A specific process has to be “fished out” of the many possible channels. This is easy for elastic scattering at small angles; it becomes difficult at large angles.
4. Total cross sections 4.1. Introduction As already stated, we shall be concerned with high-energy hadron—hadron total cross sections, and shall only make a few remarks about photon—hadron, ~lectron—positron and neutrino—hadron cross sections. The magnitude and the energy dependence of a high-energy total cross section gives an intuitive idea of the type of interaction and of the size of the particles involved. The neutrino—nucleon total cross section is very small and increases linearly with laboratory energy. This is typical of the collision between point-like objects. Any deviation from linearity
G. Giacomelli, Total cross sections and elastic scattering at high energies
148
may yield informations on the non-locality of the interaction. In contrast, hadron—hadron total cross sections show a wealth of structures in the Resonance Region, that is, for laboratory momen~abelow 5 GeV/c, while in the High Energy Region, above 5 GeV/c, the cross-sections first decrease slowly, reach a minimum, and then slowly rise. The cross-section values of about 20—40 mb are typical of objects of about 1 fermi radius. The photon—hadron cross sections are about two orders of magnitude smaller, but may have an energy dependence similar to that of hadron—hadron ones. Neglecting the recently discovered narrow resonances, the e~e -~ hadrons cross sections are small (2 x 10~° cm2 at E~= 4GeV) and decrease slowly with energy. We shall first describe some experimental methods and then discuss the experimental results. For a more thorough discussion of the experimental details the reader is referred to other reviews [69G2, 70G2]. 4.2. Experimental methods The transmission method I) Charged particles. The standard transmission method is illustrated by the experimental set-up of a Fermilab experiment, fig. 4.1 (74C1). Schematically the equipment consists of: (i) a system of counters to count the incoming particles, (ii) a number of targets where the incoming beam is absorbed, and (iii) a series of cylindrical transmission counters, each subtending a different solid angle. The measurement consists in determining the fraction of transmitted particles with and without the target. This leads to the determination of the partial cross section a~for each of the transmission counters i. ~tot is then obtained by an extrapolation of the partial cross sections to zero solid angle. Several improvements in the technique and in data handling were made in the last few years. They made it possible to measure total cross sections with point-to-point precisions of ±0.1% and with systematic errors of about ±0.4%. 4.2.1.
—
—
Aj
A2 B1
Q
/~ii
82 L~i
~
02
I Scintillation Counters
E LI
~erenkov Counters PWC
0
Quadrupoles
B3 j C3
P1
Targets
T 1-T12
D2 E
Cei,~
~
_____________
~2 ~ H2
~
p(feet)
Fig. 4.1. Layout of the charged-particle cross section measurement performed at Fermilab [74C1 —21. The incoming beam is counted as B = B12 3A1 2C,, where C1 is one or a combination of the three gas ~erenkov counters. PWC1 —3 are proportional wire chambers. D2, VAC and H2 are targets. T1 —T12 are transmission counters. E1 and E2 are small counters used for beam tuning and for the measurement of the efficiencies of the transmission counters. Ce is a lead glass ~erenkov counter; i.u is a combination of two large scintillation counters.
G. Giacomelli, Total cross sections and elastic scattering at high energies
149
A few technical details will now be briefly discussed. The beam intensity is typically (1—2) X 1 0~particles per accelerator cycle, with a momentum resolution of ±0.5%. In the Fermilab experiment, two proportional wire chambers (PWC) in the beam were used to eliminate particles with wrong directions. Thus the PWC acted as sort of “electronic collimators”. Particle identification is performed with gas differential ~erenkov counters, capable of resolutions up to i~43 10~(see fig. 3.7). The targets (hydrogen, deuterium and dummy), typically 1—3 m long, have a double jacket design, which provides long-term density stability to ±0.04% and a knowledge of the absolute density to about ±0.1%. The transmission counters usually cover the range 0.01 < ti < 0.07 (GeV/c)2, the t-range being kept fixed, by changing the distance between the hydrogen target and the transmission counters. Some experiments used counter hodoscopes [69F 1, 74A6] or proportional wire chambers [74C 11 in order to have a complete knowledge of the beam shape and of the form of the extrapolation to very small angles. The electronics is standard, the main emphasis being on reliability and stability. A computer on-line is needed if hodoscopes or proportional wire chambers are used; the computer is also useful to “keep an eye” on the stability of the apparatus and on the time structure of the beam. Beam impurities of electrons and muons are measured directly by means of total absorption lead-glass counters and with a muon filter made up of a few meters of iron, respectively. In order to obtain nuclear cross sections, the results have to be corrected for single Coulomb scattering and Coulomb-nuclear interference. Even after these corrections it is not completely clear if one has obtained nuclear cross sections or if other contributions are present [71B3]. II) Neutral particles. The method for measuring the total cross sections of neutral particles is ‘illustrated in fig. 4.2 [74L2] The Fermilab neutron beam, indirectly monitored, is absorbed in a target placed before a neutral particle detector. Here, neutrons are converted with about 20% efficiency into charged particles in a slab of iron, 2 cm thick. The forward going charged particles produced are counted in one of the transmission counters and the neutron energy is measured in the hadron calorimeter. This method becomes particularly useful and yields precise results at high energies because one can make better neutral particle detectors. .
Direct measurement of the pp total cross section at the ISR At intersecting storage rings machines it is not possible to use the traditional method of deducing c~ 0~ by measuring the transmission of a beam through a known target. Instead a~0~ may be measured via the direct application of the luminosity formula N = u~0~L, by performing sep4.2.2.
Neutron Beam,
i~ ~
A ,°
TeleSCOPeS
I lull I
A,F.
i carg.~
D~ - D7 2OO~f
,,~
[~J
Total Abwpticn Calorimeter (C)
Fig. 4.2. Layout of the neutron total cross section measurement performed at Fermilab [74L2].
G. Giacomelli, Total cross sections and elastic scattering at high energies
150
~
Left
Right
0
1
2
3
4
5m
Fig. 4.3. Layout of the proton—proton total cross-section measurement performed at the ISR. A system of counter hodoscopes (H, L) surrounding the intersection region, counts almost all the interactions and allows an extrapolation to 4ir coverage. The H 1 hodoscopes are binned in the ~ direction; the H10 hodoscopes are split into 0-bins; the L-hodoscopes are four planes of x- and ycounters interspersed with lead plates; TB are triggering counters [73A4J.
arate measurements of N and L. The total number of interactions N per unit time is measured with large scintillation counters; extrapolations have to be made to take into account the missing number of interactions both at small and large angles. The absolute value of the cross section is then obtained via a luminosity measurement as discussed in section 3.3. Fig. 4.3 shows the apparatus used at the ISR to measure the proton—proton total cross section. About 500 scintillation counters, subdivided into left and right hodoscopes and into large-angle counters, were used to detect the total number of interactions [73A4]. Systematic uncertainties of about ±2%were achieved. Bubble-chamber measurements I) Charged particles. Bubble-chamber pictures are scanned for all types of interactions inside a well-defined fiducial volume; the total cross section is then computed on the basis of the number of incident tracks. Corrections have to be applied for beam contamination, for scanning losses, and for the loss of elastic scattering events at small angles. Measurements of this type were for instance performed with the 30 in. Fermilab hydrogen bubble chamber exposed to incident protons of 100, 200, 300, and 400 GeV. Other exposures using particles tagged with a ~erenkov counter have also been performed (100, 150, 200 GeV/c ir, 100 GeV/c ?, etc.) [73W1, 75G 11. These measurements are accurate (with statistical and systematic accuracies of about ±2%), but cannot compete with systematic counter measurements. II) Neutral particles. In this case one still measures all the interactions inside a well-defined fiducial volume. The energy spectrum and the absolute flux of the incident neutral particles have to be determined by indirect measurements. The incident energy may be determined for some individual events if they lead to a kinematically overconstrained situation or if the total incident energy is deposited in the bubble chamber. Neutrino cross sections, production by photons of strongly interacting particles, K~,n, and IT total cross sections were measured using this technique. Many exposures of large bubble chambers to broad band and dichromatic neutrino beams are planned for the future. 4.2.3.
Indirect methods for charged particles Two methods of determining the pp total cross section from a measurement of the differential elastic cross sections were used extensively at high energies. (j) In the first method the pp elastic differential cross section was measured in the diffraction 4.2.4.
G. Giacomelli, Total cross sections and elastic scattering at high energies
region, for 0.01 < ti < 0.12 (GeV/c)2. The data were then extrapolated to t formula da/dt
151 =
0 using the
(da/dt)~, 0èxp(bt).
(4.1)
Then, using the optical theorem, one has: l6ir(da/dt) =
to
-
(4.2)
t~O
where p is the ratio of the real-to-imaginary part of the forward elastic scattering amplitude. p is either computed with dispersion relations or set equal to zero, in which case eq. (4.2) yields an upper limit. The computation of the cross section via formula (4.2) requires a knowledge of the luminosity since da/dt (dN/dt)/L where dN/dt is the measured number of events. (ii) The second method is based on measurements of the pp elastic scattering differential cross 2]. Using the sectionstheorem in the Coulomb-Nuclear region 0.001 ~ ti 0.01 (GeV/c) optical and assuming theinterference t-dependence of eq.[for (4.1) for nuclear scattering, the pp -~ pp differential cross section may be written as: cia
—
a~t I
I
+
(1
+ ii2)
Coulomb
-j-~-exP(bt) 2(p
2a +
Nuclear
aq5)
CTtt
ri.G2 ~j~= exp(bt/2)
(4.3)
C—N interference
where C is the Coulomb amplitude: C = -~~-G2(t)exp(ia~t)
(4.4)
and G(t) =
=
proton form factorns (1
+
t/0.7l)2,
phase of Coulomb amplitude
a = fine structure constant
a(ln 0.08/iti — 0.577)0.025,
(4.5)
1/137.
Eq. (4.3) is valid on the hypothesis that p does not depend on t and that spin effects are negligible. It contains three parameters, p, b and ~ Several types of fits may be done, fixing b and either leaving both p and a~ 0~ free or only one. When using eq. (4.3) one has an absolute normalization based on the small-angle Coulomb cross section, which is theoretically known. Thus for the ISR data one does not need a separate determination of the luminosity. The above methods applied at IHEP, Fermilab, and at the ISR yielded values of the total cross sections with overall errors between ±0.5and ±2mb [73A1, 73A~,73B3, 73B4]. (iii) A third method, used at the ISR, is based on the simultaneous measurements of the total collision rate (see section 4.2.2) and of the elastic-scattering differential cross section in the diffraction region [method (i) above]. Dividing (3.1) by (4.2) one has t1tot
=
1 611’(dN/dt)~..0 N(l + 2)
(4.6)
G. Giacomelli, Total cross sections and elastic scattering at high energies
152
In this case the measurement of a~0~does not depend on the luminosity, thus removing one of the largest uncertainties. Other methods Among the large variety of other methods used to measure total cross sections, we shall mention the following: (i) Hadron production by photons. These experiments have been performed with tagged photon beams, using a system of hadron counters for charged and neutral hadrons, and a system of anti-coincidences for the forward-going electrons and photons (see fig. 4.4). Experimental results are available up to 30 GeV. They will soon be followed by measurements up to 200 GeV. (ii) Neutrino total cross sections. At high energies it is possible to perform relatively good measurements with neutrinos because (a) the neutrino cross section increases with energy, (b) better neutrino beams of high intensity in a narrow forward cone can be made, and (c) large detectors are easier to use. Intense beams of high-energy neutrinos are produced from the decays of pions and kaons. If the pions and kaons are momentum analyzed, before the decay region, one obtains a dichromatic neutrino beam, where the higher energy component comes from K-decay and the lower energy one from ir-decay. If pions and kaons are focused and not momentum analyzed one obtains a wide band neutrino beam. Because of the smallness of the neutrino cross section one needs very large detectors with tens of tons (and very soon hundreds Of tons) of material. Fig. 4.5 shows the layout of one of the two counter experiments performed at Fermilab using the dichromatic neutrino beam. There are also many results obtained with large bubble chambers. (iii) Hyperon total cross sections. At high energies the Lorentz dilatation factor becomes important; thus beams of short-lived particles (like the hyperons) can be made and with these one can perform experiments similar to those for the other hadrons, but clearly not with the same accuracies. In particular, the beams and the detecting equipment have to be very short physically, in order to minimize decays. The CERN—Heidelberg Collaboration measured at the CERN PS the u~0~(A°p) and the u000(A°n) in the 6 to 21 GeV/c momentum range using a wire chamber set-up, where the A°above the median plane of the equipment interacted in carbon, while those below the median plane interacted in a CH2 target. Thus they obtained the A°p cross sections by what may be called a selfmonitoring difference [72G11. 4.2.5.
~—
r
I
PHOTON BEAM
r- SHOWER
\ COUNTER
/
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S2a A
S2b
C,
Fig. 4.4. Layout of the tagged photon beam and of the apparatus used for the measurement of the total photonucleon cross section at SLAC [69C1J.
G. Giacomelli, Total cross sections and elastic scattering at high energies TARGET-CALORIMETER 10cm iron
BEAM
TOROIDAL IRON CORE MAGNET
:~
~
153
~
Steel target
0 Spark chamber Calonmeter
0
Im
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Fig. 4.5. Layout of one of the two counter experiments which study uM(l3~)interactions at Fermilab [74B1]. The neutrino beam, arriving from the left, interacts in the target calorimeter; forward going muons are momentum-analyzed in the muon spectrometer. This apparatus is operated with the dichromatic neutrino beam, while the other [74A8, 74Bl I operates with both dichromatic and wide-band neutrino beams.
(iv) Cosmic-ray measurements. At even higher energies there are some cosmic-ray measurements of pp inelastic cross sections [70A 11 and calculations of total cross sections obtained comparing the fluxes of cosmic rays at the top of the atmosphere and at various atmospheric depths; from these calculations the mean-free paths in air are obtained; then, using the Glauber model, lower limits for the free nucleon cross sections are computed [72Y1, 74B8]. 4.3. The shadow correction in deuterium Direct measurements of neutron cross sections may be performed only for the np case. In all other cases the free neutron cross sections ~ are derived from deuterium (~~d)and hydrogen measurements using the formula ~Xd
=
+
a~
—
8g.
(4.7)
The cross section defect 6ci arises from the shadow of one nucleon on the other. 6o is substantial only in the case of hadron—hadron collisions; for neutrinos and for photons the nucleon is transparent. 6cj is usually computed at very high energies with the following simple formula 2) (r ~ -~--—a~u~(l—p~p~) (4.8) where p~and p~,are the ratios of the real-to-imaginary parts of the forward scattering amplitudes. (r2) is the average inverse square separation of the proton and neutron inside the deuteron. To first approximation,
G. Giacomelli, Total cross sections and elastic scattering at high energies
154
6cr
=
airp
+
a irp -
—
~-(a . 2 ,rd
+
a,rd -
(4.9)
)
Because of charge symmetry the cross sections ~ and ~ should be equal. It is more advantageous to use in eq. (4.9) their arithmetic mean because this decreases the influence of systematic errors and of statistical fluctuations. For the 1 5—200 GeV/c interval the agreement between 7rd and ir~dcross sections is good [73D2,74C2, 75Cl], 01rd/0~d
=
1.003
(4.10)
±0.002
while at lower energies there seems to be a difference of about 2.5% [69G1, 73C5]. Similarly one may obtain the shadow correction 6cr from a measurement of pd, pp, and np total cross sections: (411) bao +a —1(a +a ) PP
flP
2
pd
nd
The results for 6cr from pion data are shown in fig. 4.6; they indicate that 6cr is either constant or increases slowly with momentum. The same comment applies to the energy dependence of (r2). In the Glauber—Wilkin model the shadow effect arises from the multiple scatterings of the incident particle by the nucleons inside the deuteron [5 5G 1, 66W 1]. At low momenta such scatterings are predominantly elastic and (r2) is a constant determined by the internal structure of the deuteron and independent of the incident particle energy. At high energies inelastic processes can make a significant contribution to the shadow correction and the quantity (r2) must at present be regarded not as the neutron—proton distance inside the deuteron, but as an energy-dependent 0.06
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I
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I
I
I
I
E 0.02 V
2
~
ii I
10
I
II
III
I
20 50 100 200 LABORATORY MOMENTUM (0eV/c)
Fig. 4.6. The energy dependence of the shadow correction determined from 1r±pand ~±d total cross sections: (a) the shadow correction &a for pions; (b) the (r2) parameter.
G. Giacomelli, Total cross sections and elastic scattering at high energies
155
phenomenological parameter to be determined experimentally [72A1, 73Q1]. In the recent experiments (r2> was taken as 0.039 mb for K~scattering and 0.03 5 for p and ~ scattering. Intuitively one expects that the percentage of screening increases with the increase of the elementary cross sections off free proton and free neutron targets. Thus the percentage screening should be larger for p than for K~and it should increase with momentum if the cross section increases with momentum. 4.4. Energy dependence of the total cross sections Figs. 4.7—4.14 show compilations of various total cross sections plotted versus laboratory momentum. We shall first discuss the accuracy of the data, then analyze the gross features of the cross sections, those which differentiate between strong, weak and electromagnetic interactions, and then proceed to discuss more thoroughly the hadron—hadron interaction. In general the authors of total cross-section measurements quote two types of errors: (a) a point-to-point error, which includes the statistical error and those uncertainties which are energy dependent; (b) a systematic (or scale) error which is energy independent. The typical uncertainties quoted for high-energy total cross sections are approximately those given in the following table. Table 4.1
Uncertainties in total cross section measurements Cross section
Point-to-point errors
Scale error
~
±0.2%
±0.4%
±0.4% ±0.6%
±0.4% ±0.7%
±0.4%
±1.5%
±3% ±10,±20%
±2% ±20%
~
K~p,pp K~n pn,np ~yp
Clearly the precisions of neutrino total cross sections are not comparable to those of the longlived hadrons. The table does not show the precision of 7r~d,Kid, ~d, pd and nd total cross sections: their percentage statistical errors are about the same as for hydrogen, while their systematic error is somewhat larger (about 0.5%) because of greater uncertainties in the deuterium density, in the deuterium contamination and in the extrapolation. As discussed in section 4.3, the extraction of the free neutron cross sections presents several uncertainties. Therefore the typical errors for neutron cross sections are considerably greater than for hydrogen and deuterium measurements. The error bars shown in figs. 4.7—4. 14 represent point-to-point errors only. A scale error should be added when comparing data from different experiments. Thus the agreement among the different experiments may be considered reasonable and the energy dependence of the data well established. There are some minor differences around 60 GeV/c for i(p total cross sections and around 200 GeV/c for pp. The nd and pd total cross sections are equal, within errors; similarly, and to a higher degree of accuracy, the ird and ir~dcross sections are equal. The np and pn cross sections scatter considerably (in particular around 50 GeV/c), though not outside their relatively large errors.
G. Giacomelli, Total cross sections and elastic scattering at high energies
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4.4.1. “Weak, electromagnetic and strong” cross sections Figs. 4.7—4.10, which show the total cross sections for vN, ‘yN, e~eand charged hadrons respectively, give an immediate idea of the drastic differences of the interactions involved. The neutrino cross sections for muon production rise linearly with laboratory energy. There is no visible deviation from linearity over the energy range of few MeV to 200 GeV, that is over a factor of 100 000 in laboratory energy. Instead, the hadron—hadron cross sections change only very slowly for energies above 5 GeV and the energy dependence of the photon—nucleon cross section resembles that of hadron—hadron collisions. The e~e-÷ hadrons cross section also changes slowly with momentum, apart from the recently discovered spikes. In absolute values the cross sections differ considerably. At ~ 10 GeV the neutrino cross section is about 4 X l0” cm2, while the (eke -+ hadrons) is probably 10_32 cm2, the photo-absorption cross section is about 10-28 cm2 and the typical hadron—hadron cross section is about 3 X 10~~ cm2. Their respective ratios are thus approximately 1.3 X l0”: 0.4 X 106: 1/300:1 at~J~ 10GeV. The (i) linear rise of ~N and 17N cross sections, the (ii) relation —‘
G. Giacomelli, Total cross sections and elastic scattering at high energies
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and (iii) the ratio cJ~N/cJVN 0.36 are consistent with the hypothesis of a local interaction, from point-like objects. In particular, these facts are consistent with a parton model of the nucleon in which the partons are relativistic and predominantly spin ~ fermions, not anti-fermions. In contrast the hadron—hadron cross sections are typical of objects of about 1 fm radius. The photon—nucleon cross sections are qualitatively consistent with the vector-dominance model: the photons couple directly to vector mesons, which then interact with the nucleon. The consistency is qualitative since quantitative tests of the vector-dominance model may have some difficulties. It is interesting to look at the cross section for the process e~e-÷ hadrons, fig. 4.9. The most remarkable feature is the presence of the very narrow structures at ~ = 3~1 and 3.7 GeV [74A9, 74A10, 74Al1] and the broader structure at 4.15 GeV. This brings in the question: could one have missed narrow, large-mass resonances in the other total cross section measurements? This is —‘
G. Giacomelli, Total cross sections and elastic scattering at high energies
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certainly possible, because the spacing of the data points is relatively large, because one would have tended to disregard a point deviating from the others, and because the overall experimental resolution was not too good. 4.4.2. Energy dependence of the cross sections of the long-lived hadrons Turning our attention to the long-lived hadron—hadron cross sections, one may “expand the vertical scale” and take a more detailed look at what is happening. It is apparent from fig. 4. 10 that the total cross sections of ira, K~,p, and 15 on protons first decrease, reach a minimum, and then increase with increasing energy. The location of the minimum (see table 4.2) changes with the type of incoming hadron, being lowest for Kp and increasing, in order, for ?p, pp, Kp, ~rp, and ~p. The location of these minima reflect via dispersion relations in the location at which the ratio of real-to-imaginary part of the forward scattering amplitude crosses zero. The energy dependence of ir~d,Kid, pd, and pd cross sections is very similar to that in hydrogen. This is also so for the data on neutrons (fig. 4.12), within their larger errors. It may be interesting to single out the main features of hadron—hadron cross sections by averages of particles and antiparticles and also by averaging the cross sections for particles which
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20 40 60 80 100 200 400 LABORATORY MOMENTUM 1GeV/c)
600
Fig. 4.13. Energy dependence of averaged cross sections.
are members of the same isomultiplet; this means considering the following averages: (cr(irN)>
=
~
[a~0~(ir~p)+ Utot(7(p)]
(4.13)
(a(KN))
=
~[u~0~(K~p)+ cj~0~(Kp) + u~0~(K~n) ÷u~0~(Kn)]
(4.14)
(a(NN))
=
~[a~0~(pp) + u~0~(pp) + u~0~(pn) + a~0~(~n)].
(4.15)
In the Regge pole language (see section 2.7) these average cross sections have contributions from the exchange of the Pomeron P, of the second vacuum trajectory P’, whose contribution decreases with energy, and from possible cuts, like the Pomeron cuts. In formulae: Table 4.2 Approximate location of the laboratory momentum at which the total cross section has a minimum (pa) and at which the ratio of real to imaginary part crosses zero (p2). Pi (GeV/c) K~p Kp lIp pp pp
P2 (GeV/c)
15 70 50
100 15 140
90 70 250
70 270 80
G. Giacomelli, Total cross sections and elastic scattering at high energies
161
U)
U)
o 26.
iSli2}~:
:: ~ 11111
6
8 10
~ I
I
I
111111
I
I
I
1111,
20 40 60 80 100 200 400 600 1000 LABORATORY MOMENTUM (GeV!c) Fig. 4.14. Cross sections in pure isospin states.
(ci(xN))
=
a~(xN)+ a~~(xN) + cr~(xN).
(4.16)
The values of the averaged cross sections are given in fig. 4.13. They all have a minimum in the region experimentally covered. Many analyses are performed on pure isospin cross sections. Only the ?p, K~pand pp are pure isospin states. The other experimentally accessible cross sections contain a mixture of two isostates. Therefore the determination of the pure isostates requires the measurements of two cross sections. The pure isospin cross sections shown in fig. 4.14 have been computed from the measured values via the formulae: a112(7rN) = ~-[3a~0~(i(p)
—
u~0~(ir~p)]
(4.17)
ci0(KN) = 2a~0~(Kp) — a~0~(ICn)
(4.18)
cr0(pN) = 2o~0~(K~n)a~0~(K~p)
(4.19)
—
G. Giacomelli, Total cross sections and elastic scattering at high energies
162
2a~
u
0~(!5p) a~0~(~n).
0(pN) =
(4.20)
—
The errors on these cross sections are larger than those quoted in table 4. 1, since they are obtained from differences which involve cross sections from neutrons. It may be worthwhile to point out that the shape of the hadron—hadron cross sections versus energy resembles that of the energy loss of particles in matter. One may explain the differences among the various cross sections in the frame of Regge poles. For instance, the fall-off of the ir~ptotal cross section is attributed to the p and f trajectories, while the absence of a fall-off in the K~pcross section comes from the exchange degeneracy of the f and w, p, A2 trajectories [see the total cross-section structures in terms of Regge poles, eqs. (2.53—2.54)]. In reality the simple Regge picture breaks down, because one has difficulties in explaining the low-energy fall-off of the pp cross section: it may result from a breaking of the exchange degeneracy, a lower lying singularity or some other mechanism. We next face the problem of parametrizing the energy dependence of the total cross sections, in particular of their rise after the minimum. We may assume the type of dependences suggested by Regge-pole theory, which need a diffractive contribution, due to Pomeron exchange, which rises with energy, and non-diffractive contributions, which decrease with energy. The non-diffractive parts behave as p°, with n-values which may be computed from the analysis of total crosssection differences. The diffractive component is expected to have a constant term plus a term which increases logarithmically with energy. The specific type of logarithmic behaviour will have to be found experimentally. Thus in the Regge pole frame the general energy dependence will be of the types: a~0~(xp)D+ND =
(4.21)
a~+ b,(ln p/c,)°’+ B1p”
(4.22)
p +m’ a1 ln( b. :)+Bp~
(4.23)
where p is the laboratory momentum. Hendrick et al. [74H 1], using a parametrization of the type (4.23), found that the bulk of the total cross-section data is explained by the following dependences: 49 •a(K~p) =
a(~r~p) = 3.27 ln
a(pp)
(4.24)
3.27 ln p+l
4.91 ~
p+206 0.33
16.8 +
2.62 —
p°567 p0.426
(4.25)
(4.26)
These parametrizations should be considered a reasonable interpolation of the data, without any further significance. If it would hold true at higher energies, it would imply that the cross sections rise with a dependence ln (p) and that ~K~p and UlI+p will approach each other, and that both approach asymptotically the value of 2/3 predicted by the naive quark model.
G. Giacomelli, Total cross sections and elastic scattering at high energies
163
Table 4.3 Fits of the hadron—proton cross sections above 8 GeVIc to formulae (4.27) [74D6l. Only the point-to-point errors 2 have been used for computing the x ~r~p 13.5
±1.6
C 02
2.4
02
0.37 22.8 0.37
±0.2 ±0.03 ±0.7 ±0.06
x2 DOF
K~p
p~p
14.2 ±1.1 0.010 ±0.002 8.9 ±0.6
25.7
±2.5
25.8
±1.6
0.53 ±0.02 14.7 ±2.3 0.52 ±0.16 225 204 — 16 = 188
0.58 43.7 0.35
±0.02 ±1.3 ±0.05
Many other types of energy dependences may be found in the literature. For instance Bartel and Diddens [74D6] tried the following parametrization: —
+
a~ 02 = 2a2E a~ 0~(xp) 0~(xp) = 2(a1E°’ + a,,) (4.27)
The results of the fits to the ir~p,K~pand p~pcross sections above 8 GeV/c are given in table2s4.3. Other formulae, basedbecome on optical models, yield dependences areInslower than ln at present energies and of the ln2s type at energy asymptotic energies which [73C 1]. some of these models all total cross sections become equal at infinite energies. So we did not really solve the problem of the energy dependence of the cross sections, but we found some semi-empirical relations which are useful for practical considerations. If there is any hope of finding out the true energy dependence this would have to be based on the measurement of the K~por possibly of the pp total cross sections, because the first starts rising at very low energy and the second has been measured at the higher energies provided by the ISR. Fig. 4.15 shows the K~ptotal cross section plotted versus (In ~)3; the behaviour seems to be consistent with
21
-
-
~2O-
-
—
I.) Id
U
18-
•‘
— -
100
200 p (in
300
s) 3
Fig. 4.15. The K~ptotal cross section plotted versus In3 s.
G. Giacomelli, Total cross sections and elastic scattering at high energies
164
~
o(p*p~++
+
+
+ 10
I
I
I
:
10
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
~
-
I
0
I
I
I
I
I
5
I
10
I
I
I
15
I
I
20
GeWc
Fig. 4.16. The A°pand A°ntotal cross sections versus lab. momentum. Also shown are two points for atot(Ep) and °tot(~~) [72G1 I.
G. Giacomelli, Total cross sections and elastic scattering at high energies
165
a constant value at low energies and then with a straight line rise at higher energies. It would thus seem that the energy dependence of the rise is approximately of the ln3s type. This would not violate the Froissart bound, since the coefficient of a possible ln2s term would be about 100 times smaller than that of the Froissart bound. This is so also for pp [see eq. (4.27)], thus indicating that the rise of the measured cross sections in the 20 to 2000 GeV/c range has nothing to do with the asymptotic Froissart bound. 4.4.3. Energy dependence of other hadron—hadron cross sections Fig. 4.16 shows the A°pand A°ntotal cross sections versus laboratory momentum. For comparison a d~ 0~(~N) point at 18.7 GeV/c is also shown. The A°pand A°ncross sections are constant in the 6 to 21 GeV/c region and have the average values quoted in table 4.4. The results of table 4.4 indicate that all the hyperon—nucleon total cross sections in the 10—20 GeV/c range are approximately equal within their relatively large errors. 4.5. Relations among hadron—hadron cross sections A large number of high-energy relations among total cross sections and total cross section differences were derived on the basis of asymptotic theorems, SU(3) symmetry, absence of exotic states, additive quark model, etc. In the following we shall discuss the applications of some of these relations to the available data. 4.5.1. Total cross-section differences The differences of the total cross sections for particles and antiparticles on protons ~a(x~p)
=
a~05(xp)— a~0~(x~p)
(4.28)
are shown in fig. 4. 17. The errors on the differences are smaller than those based on the total cross sections themselves, since some systematic errors cancel out and the final errors are mainly of a statistical nature. A similar statement is valid for i~a(x~d) (fig. 4.18). Instead, the differences ~a(x~n) for scatterings on neutrons have considerably larger errors. The differences of particle and antiparticle cross sections on the same target must eventually become zero in order to satisfy the Pomeranchuk theorem. We want now to discuss the problem Table 4.4 Average values of the hyperon—nucleon total cross sections and comparison with the predictions of various high-energy sum rules. (Interpolations of available total cross sections were used for the calculations.) [72G1 I Total cross section
Ap An
Laboratory momentum (GeV/c) 6—21 6—21 18.7 18.7
Ap An
6—21 6—21
Measured value (mb)
Predicted value (mb)
Equation no.
34.6 34.0
±0.4 ±0.8
35.2
±0.6
(4.41)
34.0 30.0
±1.1 ±1.2
35.0 34.4
±0.9 ±0.3
(4.42) (4.43)
56 46
±11 ±20
I
I
I
1111
I
20
E 10
I
V GALBRAITH
£ FOLEY
65 67
• DENISOV
73
—
i
L
2
~
74
~
5p)
~~(K C-’
0.6 0.4 AdH1t~p) 0.2 10
I
I
I
20
111111
40
60
I
90 100
LABORATORY
I
I
200
MOMENTUM
I
600
(0eV/c)
Fig. 4.17. The differences of total cross sections for 11±, K±,p and ~ interactions with protons. The solid lines represent fits of the data to the power dependence (4.29). I
20
~
I
I
liii
I
I
~
~
I
I
GAIBRAIT 65 DEN(SOV H 73 74
~(p*d)
:
4)
1’
~ U, o.e . Ill o
,I,J,
~d~CKtfl)
0.64
-
O
Q4.
0.2
10
-
~(Ktdl ~aip5n) I
- I~
I
I
I
I
I
I
I I
I
20 60 60 80 100 LABORATORY MOMENTUM
Fig. 4.18. Differences of total cross sections of
11±,K~,p
I
200
I
I
400
I
(0eV/c)
and ~ interactions with deuteróns. The lines are fits of the data to the power dependence of eq. (4.29).
G. Giacomelli, Total cross sections and elastic scattering at high energies I
I
I
11111
I
I
I
167
II
-
.0
E ~
2
0.4
—‘0.2 4 0
n-
p
I-
I
10
20
I
I
I
I
~
I
I
40 60 80 100 200 400 LABORATORY MOMENTUM (0eV/c)
I
I
I
600
Fig. 4.19. Differences of total cross sections of K±,p and ~ on protons and neutron targets.
of what is the energy dependence of these differences in order to learn some information on how the asymptotic region is approached. The straight line behaviour of the total cross-section differences in the log--log plots of figs. 4.17—4.18 indicate that their energy dependence may be represented by a power function of the laboratory momentum =
Ap~.
(4.29)
The results of fitting the total cross section differences to this expression are given in table 4.5. Though the fits have reasonable x2-values, one cannot consider that the power dependence parametrization (4.29) is unique. In fact, fits with a dependence ~aA+B/lnp
(4.30)
may represent the data at higher energies. The differences of total cross sections for particles belonging to the same isospin multiplet interacting with the same target are shown in fig. 4. 19. The discussion of these differences follows the same line as the previous discussion on particle—antiparticle differences. The behaviour with energy of the total cross-section differences is consistent with their approaching a value of zero, as required by the Pomeranchuk theorems. On the other hand, the energy dependence is relatively weak, thus indicating that the asymptotic region is far away. Structure of the total-cross section differences in terms of Regge poles In terms of Regge poles the total cross section differences have a much simpler structure than the total cross sections themselves (see section 2.7). For instance,
4.5.2.
G. Giacomelli, Total cross sections and elastic scattering at high energies
168
Table 4.5 Fits of total cross-section differences to formula (4.29).
~o(K~p) ~a(p~p)
~o(K~n) &s(p±n) ~o(K~d) &s(p~d)
p-range (GeV/c)
A (mb)
44—200 3— 60 3— 60 35—240 3-- 60 35—200 3— 60 3— 60 3— 60 35—240 3— 60 35—200
5.24 4.0 18.1 21.7 63 58 13.0 49 29.0 32 106 95
n ±0.10
0.43 0.32 0.54 0.58 0.64 0.59 0.67 0.61 0.58 0.58 0.64 0.54
±0.30 ±0.3 ±3.9 ±2 ±4 ±0.4 ±7 ±0.5 ±5 ±8 ±7
±0.01 ±0.02 ±0.02 ±0.04 ±0.02 ±0.02 ±0.02 ±0.05 ±0M2 ±0.04 ±0.04 ±0.02
2a~(-irp) B~p0P1 &i(p~p)= ~a(K~p) = 2a~(Kp)— 2a~(Kp) B~p0P~_B~p0~l. ~a(ir~p)
(4.31)
=
(4.32)
The equality between ~a(p~p) and z~u(K~p) is clearly not verified experimentally (see fig. 4. 1 7). Thus another parameter has to be added, for instance a sort of threshold effect (writing p — Po), or some other explanation. From eq. (4.31) one can obtain the values of B~and ct~directly, as given in table 4.6. The computation of the same parameters for the K~pand p~pcases is not as straight forward. One may use the total cross section differences in hydrogen and the predictions of -÷
p universality
—~
=
(4.33)
=
o universality = (4.34) to obtain the parameters listed in table 4.6. Another approach is to consider also the cross sections on neutrons and make combinations of cross section differences for which only one trajectory is important: Table 4.6 Fists of total cross-section differences in terms of Regge poles 2 formulae (4.31—4.36) [74H1 Quantity
Value
Op
0.57 0.43 =
~
=
= =
am,
3~
Comment
±0.01 ±0.01
p~ 1,
=
I.
(1.31 (1.14 (23.9
±0.03) mb
1”2
±0.01) (mb) ±0.4) mb
determined from ~~(ir~p) l3~pdetermined from ~(K~p)
(4.88 ±0.04) (mb)112 (16.8 ±0.8) mb (3.44 ±0.18) (mb)’12 0.42 ±0.05
Assumes a~= Assumes ~‘!,= Effective trajectory for
(11.1
Effective residue for
±0.3)
mb
G. Giacomelli, Total cross sections and elastic scattering at high energies
169
L~a(K~p) ~a(K~n)
=
4a~(KN)
(4.35)
~a(K~p)
=
4a~,,(KN).
(4.36)
—
+
~o(K~n)
Some results of different fits are given in table 4.6. 4.5.3. Relations from the quark model From the simplest quark model in which baryons are made of three quarks and bosons of one quark and one antiquark, assuming (i) that a sort of Pomeranchuk theorem holds for quark collisions, and (ii) that these collisions are independent of one another, then for pions and nucleons the average cross sections should become proportional to the number of constituent quarks. Thus one should have (a(7rN)) = ~(a(NN)). It is better to study the energy dependence of the ratio
(4.37)
~
(a(KN)> (a(irN))
(4.38)
===~‘l. ~—~oo
For the averaged cross sections one may also propose the following relation ~
2(a(irN)),
(4.39)
==~1. a(KN) s—~
(4.40)
=
thus R3=
2(a(lrN)) ~(a(NN))
+
I
I
II~
I
1.4
-
12
ic -
06 -
10
A
~
:
• •...~
~
•
I
11111
•S •~• •••••
20 50 100 LABORATORY MOMENTUM
1,uv-iui 2LN 2/3NN+KN
I
I
200 1GeV/c)
I
I
500
Fig. 4.20. Test of the additive quark model relations (4.37—4.40).
G. Giacomelli, Total cross sections and elastic scattering at high energies
170
Fig. 4.20 shows the ratios R1, R2 and R3 plotted versus laboratory momentum. The ratios (4.37) and (4.38) are slowly decreasing towards 1, while ratio (4.40) is very close to 1 already at 20 GeV/c. For hyperon—nucleon total cross sections the simple additive quark model yields the relations a~0~(Ap) = a~0~(pp) + a~0~(Kn) a~0~(7r~p)
(4.41)
—
=
a~0~(pp) + a10~(Kp)— a~0~(7(p) + 2{a~0~(K~n) a~0~(K~p)}
(4.42)
=
a~0~(pp) + a~0~(Kp) a~0~(7(p).
(4.43)
—
—
Table 4.4 shows that there is a reasonably good agreement between measured and computed values according to eqs. (4.41)—(4.43), with the possible exception of a~0~(~n). 4.5.4. The Johnson—Treiman relations The relations ~a(K~p) — 2&j(K~n)
(4.44)
&r(ir~p)~~r z~a(K~n)
(4.45)
were first derived in the framework of SU(6) symmetry [65J1]. They may also be obtained in the context of SU(3) symmetry, assuming universality for the p and w couplings; in the framework of the quark model they are obtained from the condition ~a(q~q~) = z~a(q~q~). Relation (4.44) connects the o- and p-contributions to the total cross section for KN interactions a~(KN) —
3a~(KN).
(4.46)
Relation (4.45) corresponds to the condition a~(KN) a~(KN)~ a~(7rN). —
(4.47)
Instead of (4.44) and (4.45) one may use &i(K~p) 2.Aa(ir~p) or R4=
z~a(K~p) ==~l, 2~a(ir~p) ~
(4.48)
and &r(K~p)+ ~a(K~n)
=
3z~u(ir~p)
or R5=
3&i(ir~p) &~(K~p) + ~a(K~n)
===~l.
(4.49)
As may be seen in fig. 4.21, the relation R4 is approximately verified, within large errors, at energies above 20 GeV.
G. Giacomelli, Total cross sections and elastic scattering at high energies
I
1.5
I
11111)
-
+
1.2
~
If ~
j
1.20.5-
-
2~
I
I
s—=
-
UNIVERSALITY
s
+
+
-
“
-
-
-
111.1
50
1-c
0.8
5p) 2~ai1t /~aiKtp)
R~ I
10
H
_________(b)
/
w—UNIVER$ALITY
I
-
1
(~)
171
I
100
LABORATORY MOMENTUM
200 1GeV/c)
Fig. 4.21. Test of the Johnson—Treiman relation.
I
10
I
20 LABORATORY
I
I
11111
50
100
I
200
MOMENTUM 1GeV/c)
Fig. 4.22. Test of(a) the w-universality relation (4.52) and (b) of the p—a’ universality relation (4.53).
4.5.5. p- and o-universality As examples of many other relations we quote a number of formulae for total cross-section differences, all written in a form so as to give a value of 1 at infinite energies. (i) p-universality or additivity of quark interactions leads to &r(p~p) &r(ptn) —
R 6=
~l;
(4.50)
(ii) p-universality plus isospin invariance ~o(K~p) &i(K~n) ==~1; (~±~) —
R7=
(4.51)
(iii) co-universality or equality of cross sections for q~,,and ~ interacting with R8
=
3[&~(K~p)+ &i(K~n)]1 &i(p~p)+ &i(p~n)
(4.52)
s—~
(iv) p- and co-universality 3i~a(K~p) ~a(ir~p) ~1. (p~~) —
R9=
(4.53)
The experimental results on the ratios (4.52) and (4.53) are shown in fig. 4.22. Within relatively large errors they seem to be consistent with 1 at energies above 20 GeV. In conclusion all asymptotic formulae for total cross-section differences are either verified at present energies or have the proper tendency to be satisfied at larger energies. This conclusion is not unique, since the deviations from the predictions of asymptotic theorems could be of a systematic nature [75L1].
G. Giacomelli, Total cross sections and elastic scattering at high energies
172
4.6. The energy dependence at asymptotic energies The discovery of rising hadron—proton total cross sections has generated many speculations about their asymptotic behaviour. A wide range of theoretical speculations give very different asymptotic extrapolations. This is illustrated in fig. 4.23 for the case of pp scattering. The various curves correspond to: (i) An empirical ln2s dependence: = 38.4+ 0.5 1n2(s/137). (4.54) (ii) The prediction of the impact picture based on studies of massive quantum electrodynamic prediction eventually leads to an asymptotic ln2s dependence, but is between a In s and ln2s dependence in the energy range covered by the figure. (iii) An empirical ln s dependence from a geometrical scaling model with a logarithmicallygrowing radius [73C 1]. This
u~ 01(pp)= 28.2 (1
+
0.068 ln s).
(4.55)
(iv) A dependence arising from a threshold type behaviour due to particle production a~0~(pp) = 73
—
325[ln(s/0.l
1)]’.
(4.56)
The predictions of these formulae differ considerably at energies higher than those of the tSR. One could also add other types of behaviour like: (v) Asymptotic constant behaviour from Regge-cut models. (vi) Oscillating behaviour from complex Regge poles associated with dynamical thresholds. (vii) Asymptotically growing cross sections from s- and t-channel unitarity arguments. Until a new generation of accelerators is constructed, the only hope of resolving the question of asymptotic behaviour lies with cosmic-ray measurements of the proton—nucleon cross sections. The existing data on extensive air-showers have been recently analyzed with the purpose of obtaining the absorption-cross section of protons on air nuclei. From these one tries to obtain the free pp total cross section using the Glauber multiple scattering theory. The present data and the 15C
“-I,,
.g
/ 125
125
lao
~
/‘
100
~j\-’+o (‘~“
75 50
Q
50
25
~ -
---.—.,~——-.
-
;:;~32~nc5b01
----~
LABORATORY MOI~IENTUM (GeVk)
Fig. 4.23. Typical high-energy model extrapolations of the proton—proton total cross section [74B8J.
G. Giacomelli, Total cross sections and elastic scattering at high energies
173
method of extracting the pp total cross sections are not adequate to reach any conclusion on the energy dependence in the laboratory momentum region between l0~to l0~GeV/c [74B8]. In fact one cannot distinguish between a constant cross section at 42 mb and a ln2s dependence. This is because (a) the absorption cross sections are poorly known; (b) the elementary pp total cross section is large and therefore the air nucleus is like a black disc to protons and the absorption cross section is independent of a~ 01(NN);(c)the conversion from cabs to cr~01(NN),according to the Glauber theory, depends on the assumptions on the impact parameter profile of the proton and is therefore model-dependent. One may wonder and ask a sort of philosophical question about what happens to the size of the elementary particles when the incoming energy becomes infinite: does the proton become as large as a billiard ball? Well, at most one can have the whole energy of the universe concentrated in the incoming proton. In this case if a formula like (4.54) holds one gets Utot(pp) 66 b, which corresponds to an interaction radius of 46 fm, to be compared with 1 .1 fm at present energies! The fact is that a logarithmic dependence is a very weak one indeed. 5. Elastic scattering 5.1. Introduction The main features of all high energy differential distributions for hadron—hadron elastic scattering processes are: (i) a large and narrow peak at small angles, probably with two slopes; (ii) some wiggles at intermediate angles; (iii) a very small cross section at large angles, and (iv) a backward peak. With increasing energy the slope of the forward peak becomes steeper (shrinking the forward peak), while the large angle and the backward cross sections decrease very fast with increasing energy. It is customary to classify the high energy angular distributions in five regions (see Introduction). We shall follows this classification, neglecting the very small-t region, the Coulomb region, which is theoretically well understood. 2) 5.2. The Coulomb-Nuclear Interference Region (0.001 < ti < 0.01 (GeV/c) Experiments in this angular region study the interference between Coulomb and nuclear scatter-
ing, which yields information on the real part of the forward non-spin-flip part of the nuclear scattering amplitude (the spin-flip part contains a sinO term and therefore does not contribute at small angles). The absence of spin—spin terms makes the irN and KN systems more amenable to a phenomenological analysis in the interference region than the NN system. In practice for the NN system one often neglects spin—spin effects, hoping that they are small. Three different experimental methods have been used to measure the differential elastic scattering cross section in this angular range. In the first method one measures angle and range of the recoil proton, which has a very low energy and recoils at angles close to 90°.The method makes use of the fact that the laboratory kinetic energy, T~,of the recoil proton is directly related to the four-momentum transfer through
G. Giacomelli, Total cross sections and elastic scattering at high energies
174
He Gas OUfL :Liquid He Supply
~Vl
-
V2
-
Hydrogen Supply
-
Frozen---~~ Hydrogen
Liquid He
—
i!~
Gate Valve
—
Beam
r
L.~..L.
,.J
.
I
-i
Beam Chamber
L-.~ II
I
Fig. 5.1. Diagram of the gas-jet target used in experiments with the internal proton beam at the Fermilab accelerator. (See text for a description of the operation [73B41.)
the relation t—2m~T~.
(5.1)
In order to reach very small values of t, the target has to be very thin. In practice, experiments of this type were performed with the internal proton beams of the IHEP and Fermilab accelerators, striking a gas jet of hydrogen moving at supersonic speed. The gas-jet target was designed to produce a 200 msec burst of hydrogen gas into the circulating proton beam, usually three times for each accelerator cycle. The principle on which it operates is sketched in fig. 5.1: at a given time, valve V1 opens and lets through about 50 cm3 of hydrogen gas under pressure of 1 to 4 atm; this gas was originally confined between the valves V 1 and V2. A single pulse of value hydrogen 3 to the jet intolerable of 2 gas X 1 would 0~ raise the pressure of the whole accelerator volume of 50 m torr. Thus one has to trap the hydrogen. This is done by cryopumping and precooling the hydrogen and then freezing it in the helium-cooled cup enclosure. Additional diffusion and ion pumps restore between accelerator cycles the vacuum of the accelerator in this section to 10-6_b 7 torr. The gas density is about 5 x 1 0~g/cm3 and the full width at half height of the jet is about 12 mm. In the Fermilab experiment, when the protons were accelerated at the rate of 100 GeV/sec, the beam had a diameter of ~ 5 mm, cirôulating with a period of 20 ~.isec.The beam traversed the gas jet about 1 0~times during the jet pulse. Thus with 2 X 1012 circulating protons per machine pulse and a 6 sec repetition period, the effective luminosity was l0~~ cm2 sec’,
G. Giacomelli, Total cross sections and elastic scattering at high energies
175
Jet Target
Deto~\~
-
Fig. 5.2. Layout of the small angle pp elastic scattering experiment in the internal target area at Fermilab [73B4]. The internal proton beam is intercepted by the gas jet target. Recoil protons are detected at large angles by means of solid-state detectors.
500 I
I I
I
I
I I
I
I
I
E:400GeV
400
300
BEST FIT FOR cr~4Omb pr+0.040 2/~67.4/42di. X
~200 .0
E
b
10080
a I
60
U
50 II,,,I.II’I”
.005
.010 .015
.020 .025 .030 .035 ti (GeV/c)2—~
Fig. 5.3. The differential cross section for pp elastic scattering at 400 GeV measured with the layout of fig. 5.2 [73B4I curve represents the best fit to the data, which are normalized as discussed in the text.
-
The solid
G. Giacomelli, Total cross sections and elastic scattering at high energies
176
~4Ai
2
TOP VIEW
SIDE VIEW -
__—
_--_---~_~
~—
-
SIDE
VIEW
6 mmd
VIEW ALONG BEAM
Fig. 5.4. Sketch of the layout used at the ISR by the CERN—Rome Collaboration to measure the elastic scattering in the C—N region [73A1 I. The two protons scattered in the vertical plane were detected by tiny scintillation counter hodoscopes, which could be lowered or raised close to the circulating beams.
11.8
+
15.4 + 15.4 GeVk
11.8 0eV/c
3O0~
~
:~
:~
7
)
NUCLEAR ‘~‘~—cOu1CMB
50
NUCLEAR ‘~—CO(JL0MB
_______________________________________________
O
1
2
3
4
5
6
7
8
I
01
2
ti
3 4 (0eV’)
5
I
6
78
9
101112
13x10’
Fig. 5.5. pp elastic scattering angular distributions measured with the layout of fig. 5.4 at equivalent laboratory momenta of 290 and 500 GeV/c [73A1 I - The solid line interpolating the points is a fit to the data. The separate contributions of’ Coulomb and nuclear scattering are indicated by dashed lines.
G. Giacomelli, Total cross sections and elastic scattering at high energies
177
corresponding to 1 08_ 1 0~interactions per second. The target is so thin that even very slow recoils, lower than 0.2 MeV, could be recorded. For elastic scattering at small angles, the angle of the recoil proton is practically independent of beam energy. Thus one may use the same apparatus for measurements at various primary energies as the accelerator cycle proceeds. Fig. 5.2 shows the layout of the experiment performed by a USSR—USA Collaboration. The angular region of the detectors was 1.2°to 6.3°from 90°when measurements in the Coulomb-nuclear interference region were performed; for scattering at larger angles the detectors were moved to cover the 1 .7°to 12° angular range. There are also measurements of pd elastic scattering. From the measurements of pd and pp elastic scattering at Serpukhov, Beznogikh et al. [73B2] computed pn elastic scattering using the Glauber theory of multiple scattering. Fig. 5.3 shows the experimental angular distribution at 400 GeV. From the angular distributions it is difficult to see the Coulomb-nuclear interference. Moreover the gas-jet target does not allow an easy normalization of the data. Thus, from formulae (4.7)—(4.lO), the best determination of p was obtained assuming exact values of ~ At the ISR an experiment employed small scintillation counters placed a few millimeters from the ISR beams. An elastic event was defined as a coincidence between a small counter above the beam on the left and the collinear counter below the beam on the right (see fig. 5.4). Angles of 10 mrad were involved and were measured to a precision of better than 0.1 mrad. Fig. 5.5 shows two angular distributions obtained at equivalent laboratory momenta of 270 and 500 GeV/c. The previous two methods are quite specialized to match the available accelerators and may be used only for pp collisions. For measuring the elastic scattering of other particles in the Coulombnuclear interference region one resorts to the measurement of angle and momentum of the forward-going particle only. In this case, very small scattering angles are involved and problems of multiple scattering may be severe. Fig. 5.6 shows a sketch of the set-up used at the Brookhaven AGS for the measurement of elastic scattering at very small angles [69F 11 The experiment used hodoscopes of scintillation counters; historically it was the first experiment to make an extensive use of a computer on line. A similar layout was used at Serpukhov [74A6], while experiments of the same type presently being performed at Fermilab employ proportional wire chambers of very high spatial resolution [75A1]. Some typical differential cross sections for irt p elastic scattering in the Coulomb-nuclear region are shown in fig. 5.7. As already stated, the elastic-scattering data in the 0.001 ~ ti ~ 0.01 (GeV/c)2 region are fitted to a formula like (4.7), which contains the three parameters p, b, 0tot~ Assuming values for 0tot and for b, one may obtain p, the ratio of the real-to-imaginary part of —
.
iT ABSORBER LIQJ-12 TGT BEAM
Al ~
CTI SI
42
~43
/
~
COUNTER
M~NETS
CT2 1-101 S2 H02 HI HI H2 (XX) (XX) (X)(Y)(X)
H4(x,Y)
Fig. 5.6. Experimental layout of the equipment used by a Brookhaven group to measure elastic scattering in the Coulomb-nuclear interference region [69F1 I. CT1 and CT2 are beam gas ëerenkov counters. The H symbol denotes hodoscopes of scintillation counters.
178
G. Giacomelli, Total cross sections and elastic scattering at high energies
(~)
IOO\
40 ~
20O~
b.— ~
100 80 60 40
7T’p~7T’p 1796 GeV/c
.‘.
+
20
0
003 2 -t (GeV/c)
001
-
—-..I-..~-- -
I
~5O~
,~ _._. —~ ~
-I.— i—
•~ •~ -•—
~
-
0~~~ - •— - —
— - — -
4 I
I
I
0.01
—t
I
0.02
—
I
0.03
(0eV/c)’
Fig. 5.7. Typical elastic scattering cross sections in the Coulomb-nuclear interference region measured at the Brookhaven AGS [69F1I and at IHEP [74A6l. (a) ,rp at 26.23 GeV/c, ir~pat 17.96 GeV/c and (b) 33.5 GeV/c. The solid lines are fits to the data using eq. (4.7), while the dashed lines represent fits with p = 0. The bottom graph in (b) gives the differential cross section after subtraction of single Coulomb scattering.
the forward-scattering amplitude. At high energies, these values of p have relatively large systematic errors because p is small and thus the effects on the angular distribution are small, as can be seen in fig. 5.7. Moreover, the parameters p, b, u~ 0~ are strongly correlated and one needs very good measurements of b and a~0~ to obtain reliable values of p. The measured p-values are plotted in fig. 8. 1, where they are compared with the predictions of dispersion relations. Many of the measured values scatter considerably outside the quoted statistical errors. They do not always agree with the predictions of dispersion relations (see section 8).
G. Giacomelli, Total cross sections and elastic scattering at high energies
179
5.3. Elastic scattering in the diffraction region [0.01 < ti < 0.5(GeV/c)2] Experiments in this angular range yield information on the over-all spatial structure of the particles. Phenomenologically one would like to determine: (i) the shape of the angular distribution, in particular, of its slope or slopes; (ii) the energy dependence of the slope, and (iii) the relation among the slopes for different elastic-scattering processes. Over a limited t-range the differential cross section may be well represented with an exponential function in t: du/dt = a exp (bt).
(5.2)
Over a larger t-region the cross section deviates from an exponential form and it may be represented with formulae of the type du/dt = A exp (Br + Ct2) da/dt
=
(5.3)
A [exp(b 1t)
+
a exp(b2t)].
(5.4)
Eq. (5.3) eventually leads to nonsense at large angles. Instead of eq. (5.4) one often quotes the “local” slope from a fit of a small part of the angular distribution to eq. (5.2). In order to study the energy dependence of the slope parameter, one has to define precisely the
g.
I
I
I
32GeV
-
b=—Ii.59±.06 4.
-
—3-
-
0)
I—
2 . . i
>-
<
~ 85~.
“~
—6—...-
5. 4 -
S.
...
•S.
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-
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-
•... U..
. 5.
-
.
0.0
I
I
.04
.08
t
I
I
.12
.16
(GeV/c)U 2 for pp elastic scattering at
incident energy 132 GeY [73B4j. Fig. 5.8. The differential cross section da/dt an in arbitrary units inofthe interval 0.005
G. Giacomelli, Total cross sections and elastic scattering at high energies
180
~&iJ
~.2.
~UUMCHAMBER W1NOCM’ THPI inTn WWCW/ Wi Si W2 S2 W3
PROION 2
MINIMUM DETECTED ~
~
~T~T_
____
~imrrJlH—H\
i~J,
S~~L
-~1_ \I~
~
~3irn~ ~
4,,ttm~
6.5mR
~
urn
4,
Sm -
—,tim,, ~ l85m
SENSITIVE AREA 162 ~
~7TAWAYTM~~~E
SFIARK CHAMBER WI
Fig. 5.9. Apparatus used by the Aachen—CERN—Genoa—Harward—Torino Collaboration (ACGHT) to measure pp elastic scattering 2 at the ISR. The scattering takes place in the vertical plane and the scattered protons are detected in the 0.05 < ItI <0.4 (GeV/c) by two sets of spark chambers W 1—W3,
W~—W~ [72B3I.
t-range over which the slope is determined. The slope parameters b, B, b1, or b2 are connected to the radius of interaction (rb, rB. rbl or rb2). In the optical model with constant opacity the relation between r. and b. is: r.
=
~
(5.5)
Thus, the steeper the slope, the larger the radius of interaction. Some of the experimental equipment employed in this angular range is similar to that used in experiments at smaller t-values. 2 [73B4], (see fig. In the Fermilab experiment, covering the range 0.005 ~ ti < 0.09 (GeV/c) 5.2), the thickness of the solid-state detectors was not adequate to stop the recoil protons at all scattering angles. Thus some copper degraders were placed in front of the detectors on the high-t side of the apparatus, so that recoil protons would always stop in the detectors. Fig. 5.8 shows an angular distribution measured at Fermilab. At the ISR, an experiment was performed in the range 0.05 < ti < 0.4 (GeV/c)2, using two telescopes of six spark chambers located on either side of the interaction region, above and below the vacuum chamber respectively (see fig. 5.9) [72B3]. An elastic event was defined as such if it
G. Giacomelli, Total cross sections and elastic scattering at high energies
181
gave rise to two collinear tracks and nothing else. Fig. 5.10 shows the pp differential cross section at ..,/~= 53 GeV. One clearly observes a change in slope at —t 0.13 (GeV/c)2, the local slope changing from 12.8 to 10.8 (GeV/c12. In a typical set-up used in more conventional experiments one measures in angle and momentum the forward-going particle, as in the scattering at lower t-values; but now, in addition, one measures also the recoil proton angle with a side detector (for instance a spark chamber). This extra requirement becomes a necessity at larger values of it i, since the elastic cross section becomes small, and the contamination from inelastic events becomes progressively worse. We shall in fact see in the next section that at larger scattering angles one is forced to measure also the momentum of the recoil proton. 5.3.1. pp-÷pp We shall first discuss the pp elastic scattering data, since these are most abundant and are available up to the highest energies. Many models have been proposed to explain the change in shape of the angular distribution at —t ~ 0.13 (GeV/c)2, fig. 5.10. In this section we shall mainly try to get a feeling of the physical meaning of the change in shape, without going into detailed analyses of the models. One may write two possible rough representations of the data in terms of the sum of two exponentials:
10’
-
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EcMr267+26.7GeV 0.98- ~6 EVENTS
\\\ \ \\
~ \—
_________________________
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{(Gev/c)~]
‘
I
I
I
I
10’-
\
C.)
0
e~~8t
i
~
t
~
3 ~ BARBIELLINI ET AL 4 + BEZNOGIKH ET AL 5 )< CHERNEV ET AL
8
1
6 0 BELLETTINI ET AL
S—(GeV)2
-
I 1o2~L
0
I
005 OX)
I
0.15 -t
I
I
020 025 (0eV’)
I
030
I
035
Fig. 5.10. t-distribution for pp elastic scattering at a c.m.
•0 AMALDI THIS EXPERIMENT ET AL
I 2
/
10
20
50
100
200
500
000
5000
Fig.
5.11. The slope of the diffraction peak, b(s), for ItI ~ 0.12 (GeV/c)2, as a function of the square of the c.m. energy. The solid line is a fit of b (s) = + 2~In (xis
2 energy of 53 GeV [72B3J.
2000
to the data points
fori ~
100 0eV
0) only (73B4I.
-
G. Giacornelli, Total cross sections and elastic scattering at high energies
182
do/dt
=
A1[exp(— 11 in)
0.37 exp(—341t1)]
(incoherent sum)
0.27 exp(—l8iti)J 2
(coherent sum).
+
(5.6)
do/dt
A2[exp(—Siti)
+
Recalling the optical model relation (5.5), eq. (5.6) implies that the “feel” 2] two andinteracting 2.3 fm [b ~protons 35 (GeV/cY2I. twocourse regions, radii of approximately 1.3 fm 11 (GeV/c) Of thiswith is too simple an interpretation, but[bin fact in many models the scattering amplitude is written as the sum of at least two contributions with different t-dependences, i.e. which differ in their space behaviour. In other models the change in slope is due to effects of the multiple scattering type inside the nucleons. Fig. 5. 11 shows a compilation of the energy dependence of the b coefficient for data in 0.02< iti < 0.12 (GeV/c)2. The energy dependence for b determined in the 0.15< in < 0.3 (GeV/c)2 range is shown in fig. 5.12. In both cases the increase is fast up to s ~ 100 GeV’; then the energy dependence becomes weaker and is almost linear in a semilog plot. There are some systematic differences among the different data (one has to remember that the slopes are affected by systematic errors of the order of 0.2—0.3 units). A fit of the data with 5> 100 GeV2 to the formula, suggested by Regge-pole theory, b(s, t) = b 0(t)
(5.7) 0.44 for the region 0.02< ti < 0.12 yields b0 =and 8.23 (GeV/c)’ b ± 0.27 and a = 0.278 ± 0.024 with X 2 region 0 = 9.21 ± 0.942). and a = if 0.100 0.062 the 0.15
2a(t) ln (s/s0)
2/DOF
Compilation of Elastic Slopes at<—t= ~O.2(GeV/c)2
I ~
I
I
I
IlIlIl
I
I
1111111
I
I
I
I
I
I
111111
I
I
I
I
I
I
I
I
1111111
I
I
I
I
I
~
~~:
11111111
~
111111
~ U
I
20
I
I
111111
50
100
I
200
1111111
I
500 1000 2000
s(GeV)2 Fig. 5.12. Compilation of elastic slopes b of eq. (5.7) at an average value (—t)
0.2 (GeV/c)2.
G. Giacomelli, Total cross sections and elastic scattering at high energies
183
tuitive explanation of the energy dependence for the slope parameter b is that the size of the interaction region grows with increasing energy. 5.3.2. Other types of elastic scattering Fig. 5.13 shows 7(p elastic scattering data in the diffraction region at seven laboratory momenta between 5 and 200 GeV/c. Since the data above 100 GeV/c [74A3, 74A7], are still preliminary, they have been normalized to the optical point (with p = 0) at t = 0. The figure is illustrative of the quality of the data as well as of the t- and s-dependences. The n-dependence is of the quadratic exponential type, eq. (5.3). There is no clear indication of a break, like that observed in pp elastic scattering at 0.13 (GeV/c)2. In fact, the data are not good enough to prove or disprove the existence of such a break. Fig. 5.13 shows that the energy dependence of the angular distribution is weak, if the$ is any. The differetices and similarities among ~ K~p,and p~pelastic scattering are illustrated in fig. 5. 14, which displays lines interpolated through the data at 100 GeV/c. With increasing energy the differences between particle and antiparticle cross sections at t = 0 and t 0 tend to disappear. The differences between irp and Kp scattering also become smaller. Only a small amount of data exist for the elastic scattering of short-lived hadrons. As an example, fig. 5. 1 5 shows the differential cross section for ~p -~ ~p at 23 GeV/c. Within the limited statistics it would seem that the scattering is similar to that of the other long-lived hadrons [74M2, 75B2]. In order to make more quantitative statements one has to perform fits of the data, for instance with equations of type (5.3). The fits yield values of the dimensionless quantity C/B2 of the order of 5%. From the fits one may compute the local slope at t = —0.2 (Ge V/c)2
mi
b~_ 0.2= (B + 2Ct)~,_2= B
—
0.4G.
(5.8)
The results are shown in fig. 5.12 for ir~p,K~p,and p~pelastic scattering. The points scatter con-
siderably. Part of this is due to the fact that systematic errors are not included and also because the t-range should have been chosen to be rigorously equal at all energies. In any case the graphs allow the following conclusions to be drawn: (i) The b-coefficients for K~pand pp elastic scattering have a considerable energy dependence (shrinking of the diffraction pattern). The energy dependence decreases with increasing energy. Within errors the slopes of pp and pn scattering are the same. (ii) The b-coefficients for ir~pand Kp elastic scattering increase slowly with energy. (iii) The antiproton—proton is still expanding its diffraction pattern, that is ~ decreases with increasing energy (antishrinking), but its slope is approaching that of pp. (iv) It seems that there is a tendency at high energy to have ~ bKsP bK-P ~ The shrinking of the diffraction pattern is very likely a universal phenomenon at high energy, but the rate of shrinking differs with energy and with the types of particles involved. 5.3.3. The cross-over phenomenon The elastic differential cross section differences ~=~(~p)
—~(xp)
(5.9)
are positive at small values of t and become negative at larger t. This comes about because of the
G. Giacomelli, Total cross sections and elastic scattering at high energies
184
7
lryg-lr-p
ANL’72-
5 GeV/c
IHEP’73-25
IO*.
Q IOa:
IO*:
IO -
IO-
+125 GeV/c
12
7 IO -0
1 .
.
lip7
. .
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_\
le
.
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‘i
IO
.
l
.
.
% s
?
IO0 GeV/c
Qg
IO-
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r-p ZP b, K-P ,& K*p \;:
1
?? #f 200
0.1
GeV/c
I-
0.1 :
If
0l.Olo
I
I,
Q2
I
I
0.4
I
0.6
I
I
0.6
f
,I
f 1.0
-t(GeV/c)’
Fig: 5.13. Compilation of w-p elastic scattering angular distributions for laboratory momenta from 5 to 200 GeV/c [75Gl].
02
04 -t
0.6 08 (GeV/c12
3
1.0
Fig. 5.14. Illustration of n*p, K*p and p’p elastic scattering angular distributions at 100 GeV/c.
G. Giacomelli, Total cross sections and elastic scattering at high energies
~7T-p
+ ~
~I0.
185
-
8 6
o
.05
.10
.15
-t (GeV/c)2
.20
.25
Fig. 5.15. i(p and ~p differential cross sections at 23 GeV/c [74M2].
difference in the total cross sections (and thus of the t = 0 points) and of the difference in the slopes of the diffraction patterns. Thus the differential cross sections for particle and antiparticle on protons cross each other. The cross-over point, t~,defined as that value of four-momentum transfer at which particle and antiparticle have equal cross sections, is around 0.1—0.2 (GeV/c)2. The cross-over point is well localized at laboratory momenta around 5 GeV/c; it is much more difficult to localize it at 100 GeV/c, because the differential cross sections of particle and antiparticle have become quite similar. Table 5.1 gives values of t~at different momenta. At energies below 20 GeV/c, t~was determined from linear fits of the data near the cross-over point. At ‘higher energies, approximate values of t~were deduced from experiments, not specifically designed to measure t~,using the formula =
rI ln L
a~ 2(i~p)-1 / 2 I / [B(~p) 1 + p2 a~05(~p) 0~(xp) 1 + p (xp)
—
B(xp)]
(5.10)
.~‘
where only the slopes B were obtained from elastic scattering data; a~0~ was taken from tabulations of total cross section measurements and p from dispersion-relation calculations. Present experimental data are not adequate to establish if t~is always at a constant location, independent of momentuni, or if it moves to larger or smaller absolute values. The average values of t~are slightly different for the different particles. Differences between particle and antiparticle cross sections isolate interference terms between amplitudes of opposite C-parity in the t-channel. To a good approximation the difference is proportional to the imaginary part of the helicity non-flip vector exchange amplitude. There are many models which try to predict the energy dependence of t~.For instance, in a geometrical model ImF~J0(R\/7).
(5.11)
Thus equating the first zero of J0(R~/~i) with the cross-over point, one may calculate the effective radius for the source of the C = —1 amplitude according to the relation
G. Giacomelli, Total cross sections and elastic scattering at high energies
1 86
Table 5.1 Cross-over points, —ta, at different laboratory momenta (74A1, 75B1, 74A3]. Laboratory momentum (GeV/c)
t~(7rp) (GeV/c)Z
t~(Kp) (GeV/c)2
t~(pp) 2 (GeV/c)
3 3.65 5 6 10.4 25 40—43 100
0.094 0.166 0.158 0.125 0.231
0.186 0.189 0.189 0.202 0.211
0.169 0.157 0.159 0.172 0.146 0.15 0.3 0.17
±0.040 ±0.023 ±0.022 ±0.016 ±0.007
—
—
—
—
0.25
±0.05
0.15
±0.006 ±0.006 ±0.007 ±0.010 ±0.004
±0.05
±0.005 ±0.004 ±0.006 ±0.010 ±0.002 ±0.03 ±0_i ±0.08
R~ 1
0.475/~/~7~.
(5.12) 2. One obtains Rc=_
R~_1is in fermi if t~is in (GeV/c) 1 0.8—1.2 fm. In geometrical models t~ is expected to move to smaller values as the energy increases, while in factorizable Regge pole t~ would be energy independent and in other models t~could move to larger values. The available data are probably not adequate to discriminate among these possibilities. In particular, they are not adequate to establish if the sources of the C = + 1 and C = 1 amplitudes are correlated or not and if the source of the C I is the periphery of the strongly absorbing region. —
—
5.4. Scattering at large angles The large-angle elastic-scattering differential cross section decreases very fast with increasing energy. Thus its measurement presents severe problems of counting rates and of background. One is forced to use beams of high intensity, detectors which cover large solid angles and measure both secondary particles in. angle and momentum. Fig. 5.16 shows one of the layouts employed in a Brookhaven experiment [73C6]. The beam
PROTON
_______
:W34W5~
~IIIJ
H2 PION
Fig. 5.16. Layout of a Brookhaven experiment on large-angle elastic scattering. S5—S10, Vj and V2 are scintillation counters. Wi—W12 are wire spark chambers and PC is a proportional wire chamber [73C6].
G. Giacomelli, Total cross sections and elastic scattering at high energies
187
Magnet 2 beo.~
intersection
Magnet I WI -W6 magnetostrictive wire- chamber I
I
I
0
1
2
I
S1-S4,M1-M4,AI,A2 scintillation counters
3m
Fig. 5.17. Layout of the large angle pp elastic scattering experiment at the CERN—ISR. The scattering takes place in the vertical plane l74Bl0].
impinged on a 60 cm long liquid hydrogen target. Both scattered and recoil particles were momentum-analyzed using counter hodoscopes and magnetostrictive read-out wire spark chambers. Fig. 5. 17 shows the apparatus used at the ISR to measure pp elastic scattering at large angles [74B 10]. Also this apparatus allowed the measurement of both angles and momenta of the secondary particles, thus reducing the inelastic background to negligible values. 5.4.1. pp—~pp We shall again start discussing the pp system for which there is considerable data. Fig. 5. 1 8 shows two angular distributions measured at the ISR. The main features of these distributions are: (i) the fall-off of the small-angle cross section by six orders of magnitude when t goes from zero to 1, (ii) the appearance of a sharp minimum, followed by (iii) a secondary maximum and then by (iv) a slower fall-off. The angular distributions of fig. 5.18 are clearly reminiscent of diffraction patterns. The CHOV group [75K 1] fitted the t-distributions with the formula (5.13) dcr/dt ~ Al exp(Bt/2) ÷ \/~7Aexp(Dt/2) + iipl2. —
One of these fits is shown in fig. 5. 18 (b). From the fits one can derive the location of the dip and the height of the secondary peak, which are given in table 5.3 and plotted in fig. 5.19. The location of the minimum moves inward with increasing energy: it is at 1.45 (GeV/c)2 at ‘/~= 23 GeV and at 1.3 (GeV/c)2 at \/~= 62 GeV. The height of the secondary peak is, however, increasing with energy. The inward moving of the dip and the increasing of the secondary maximum may come about as the result of an interference effect of a shrinking small-t amplitude with a large-t amplitude which is energy independent (this is a justification for using eq. (5.13)). We shall discuss the implications of the data of fig. 5. I 8 extensively in section 9. Fig. 5.20 shows a compilation of pp elastic scattering data, over a broad energy range; the figure illustrates the gradual development of an inflection point at ItI = 1.4 (GeV/c)2 into a deep minimum as the energy increases. The new results from Fermilab [75A2] agree with this picture. For comparison it is also shown 4(t), 0G where G(t) is the dipole expression of the electromagnetic form factor du/dt =
(do/dt)~
(5.14)
G. Giacomelli, Total cross sections and elastic scattering at high energies
188
(a) V~z53.2GeV
10.1~
o Borbiettini et at. (1972) I This experiment — OpticaL model
2_
1O.
io~-
10
-
do/dt
[pb/Gev2]
\
1.
PP~PP ~Fs.23GeV
(b)
-
~ b 0
1~ -÷ 1 is a Fig. 5.18. da/dt prediction. (b) /~= (pp 23 pp). GeV(a)l74N1]. -t ~j~” (0eV2) 53.2 TheGeV; solidthe curve data are four normalized parametertofit1 at as texplained = 0ieve [74B10]. in theThe text. solid Theline large is.t[Gev2] an angle optical data model have a scale error of about ±20%.
G(t)
(1
÷t/0.7l)2.
(5.15)
If one assumes that the matter distribution inside the proton is equal to its charge distribution, then the differential cross section should be, in first order, proportional to G4(t). This is not born out by the data. Fig. 5.21 illustrates the energy dependence from a different point of view, i.e. by considering the s-dependence of the differential cross section at various values of t. It is evident that at very low energies and high four-momentum transfers the s-dependence is much stronger than at higher energies and lower momentum transfers. This suggests that part of the energy dependence may be due to a sort of kinematical effect, since large t-values are only allowed at higher energies. The shrinking of the diffraction pattern nevertheless continues at high energies, as was illustrated in the discussion of the b-parameter in the previous section.
G. Giacomelli, Total cross sections and elastic scattering at high energies
do/dt (2 nd max.) 0.15
.
189
S CHOV 0 ACHGT
(a)
[tiblGev2]
::: I
1.6
-
I
I
tdlp [Gw2}
(b) -
6~
~[GevJ Fig. 5.19. (a) Height of the second maximum in pp elastic scattering; (b) Position of the minimum near t
2 [74N1].
1.4 (GeV/c)
5.4.2. Other types of elastic scattering The general behaviour of the ~ K~pand p~pelastic differential cross sections at 5 GeV/c is shown in fig. 5.22 over the whole kinematical range. Data at 10 GeV/c are shown in fig. 5.23 for ~ and K~pelastic scattering. np data for np np at different energies are shown in fig. 5.24. The angular distributions exhibit a rather complex structure and a strong s-dependence. The following specific comments can be made: (i) The ~r~pelastic scattering cross sections have dips at —t ~ 0.8, 2.8, 5(GeV/c)2 and u = —0. 1 5 (GeV/c)2 (see table 5.4). These dips are clearly observed at 5 GeVIc and have become -~
Table 5.3 Position of the diffraction minimum in pp elastic scattering at various energies [74B10, 74N1]. E*
minimum
(GeV)
(GeV/c)2
23.0 30.7 40.9 53.0 62.0
1.45 1.45 1.38 1.37 1.30
±0.02 ±0.10 ±0.04 ±0.04 ±0.05
1 90
G. Giacomelli, Total cross sections and elastic scattering at high energies
102 I
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I
~
I
6
ItI
8
(6eV2)
pp differential cross sections at different energies.
Table 5.4 Position of the dips (or of the shoulders) in low-energy elastic scattering Scattering
Laboratory
ifp
momentum (GeV/c) 5 10 5
t t2
t3
u
(GeV/c)2
(GeV/c)2
(GeV/c)2
(GeV/c) 0.8 0.8 0.8
2.8 2.8 2.8
5
0.8
2.8
—
—
1
Kp K~p np pp
10 11 11 11 13—21 3—20
2
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5
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G. Giacomelli, Total cross sections and elastic scattering at high energies INCIDENT BEAM MOMENTUM
10i26
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10.0 150
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(1968,1972)
\~~~Ietat
•Jctyde (19661 Ar,kerbirsttetal (1968) t ~ Atlabyetat (1966,1968)
08
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10 BEHAVI01I~OF ELASTIC SCATTERING AT 5.0 GeV/c 10
1~
I
e
\\NQ ~t
•‘~“
\.
/1
~pp
E
TTp
K’p 0
io~.
I. S
I
0
20
~
I
I
I
4.0
60
8.0
—t (Ge~)2 Fig. 5.22. The general behaviour of the elastic differential cross sections at 5 GeV/c for lr±p,K~p,pp and ~p when smooth curves are drawn through the original data points [73E3].
G. Giacomelli, Total cross sections and elastic scattering at high energies
192
100
• •
2
•
10
1
I
(a)
I
I
I
• • I
101 100
(b) KpKp 10 GeV
101..
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ii p—-rrp 10GeV/c .1r+ orr
~
• •
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10~ 3
I
~l0
M
+
~t0~
: +
tt
~.
~
-t (6eV/cl2
l~
l~6
5
—t (0eV/c)2 10
~5
Fig. 5.23. The complete elastic scattering angular distributions at 10 GeV/c. (a) ir~p[73B7]. (b) K~p[73B8].
only shoulders at 10 GeV/c. Thus the structures seem to diminish as the energy is increased. This is in contrast with the dip in pp scattering at 1.4 (GeV/c)2, which instead becomes more prominent as the energy increases. The structure in pp is interpreted as a diffraction-like structure; this is probably not the case for the low energy structures in ir~pscattering, which may be connected to low energy s-channel effects. (ii) At 10 GeV/c the ir~pcross section flattens out in the region around 0cm = 90°(from 7< tI < 11 (GeV/c)2) reaching the value of 1 nb/GeV2. It has decreased by a factor of about 100 while going from 5 to 10 GeV/c. The energy dependence may be expressed with the form (da/dt) 0=90~ A s_fl
A
~—(7±1)
(5.16)
which is expected if the scattering process proceeds via the interactions between point-like constituents (see section 10.4). (iii) Similar statements, concerning dips and energy dependences, can be made about K~pand ~p scatterings.0cm = 90° all the cross sections for ir~p,K~pand pp seem to be approximately equal; on At hand the pp cross section is about a factor of 100 larger. the(iv) other Note added in proof. The recent measurements of ir~pelastic scattering angular distributions at 100 and 200 GeV/c [75A2
I suggest the presence of two breaks, at —t~0.5and 1.5 (GeV/c)2, which develop as the momentum increases. Thus they would seem to be of the same type of the break at —t — 1.4 (GeV/c)2 in pp elastic scattering.
G. Giacomelli, Total cross sections and elastic scattering at high energies
ir
193
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-t (GeV/c)2 Fig. 5.24. Neutron—proton elastic scattering differential cross sections da/dt versus t for five different incident momenta.
5.5. Backward scattering The experimental study of backward elastic scattering presents all the problems previously mentioned for large angle scattering. Fig. 5.25 shows the layout used at the CERN—PS to study backward scattering in the 5 to 10 GeV/c region. The system was triggered requiring that (i) there was a forward positive particle not anticoincidized by the gas threshold ~erenkov counter C 3, (ii) there was a backward going charged particle and (iii) no counts in the anticoincidence counters. Only the forward proton was momentum analyzed using a magnet and sets of wire spark chambers [68Bl]. We have already seen from the scattering data at 5 GeV/c shown in fig. 5.21 that all elastic processes at intermediate energies have a backward bump2.(except pp naturally). The magnitude of theMoreover backwardthe peaks ~differs distribution exhibits also a dip at u ~ —0.15 (Ge V/c) considerably: the 180°ir~pcross section is almost 10 times that of 7Cp, while that for K~p is about the same as for ?p and about 100 times larger than that for either Kp or~~p scattering. Fig. 5.26 shows the energy behaviour of the differential cross section for ?p backward scattering. Above 10 GeV/c the only data which exist concern ir~pelastic scattering only. The energy dependence of the u = 0 cross sections is shown in fig. 5.27. The dependence is of a power type
G. Giacomelli, Total cross sections and elastic scattering at high energies
194
4~J~
—4-I-I
w4
Rt.—..R~
Fig. 5.25. Layout of an experiment at the CERN—PS on backward elastic scattering [68B1J. L110 are Counter hodoscopes used for triggering purposes. W1—W4 are wire spark chambers; A are anticoincidence counters and C3 is a threshold gas ~erenkov counter.
0.01
I
f
I (a) 17.07
—
I
BACKWARD WIZKID MODEL ir,p
0.0
(b)
0.0
lr-p BACKWARD ELASTIC GORE MODEL (~ only)
0.0
625
0.01
—
0.01
—
—. 0.01
—
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—
0.01
~
I
-0.1
I
-0.4
I
-0.7
I
-1.0
-1.3
0.2
-0.1
-0.4
2 -0.7
-1.0
—1.3
(G.V/c)° u(GeVk) Fig. 5.26. Backward scattering of (a) lrp and (b) irp [71B4]. The solid lines represent fits with the GORE model (see text).
G. Giacomelli, Total cross sections and elastic scattering at high energies
195
30
‘‘‘I
~
‘‘
ltp
— 01
.0
~
1—
-
ltp
~
t~N T
e
::~ ~
-
‘~
-
N~
15~~’
-
~‘io-
-
-
Thp
‘~,,
—
0.1-
5~_~-
-
I 6 8 10 LABORATORY
,
-~
~
~I 8 10 LABORATORY
-
I
20 40 60 80 MOMENTUM (6eV/c)
6
Fig. 5.27. The energy dependence of backward ir~pelastic scattering at u = 0. The lines represent possible eyeball fits to the data l74B14l.
20 60 MOMENTUM
60 (6ev/c)
Fig. 5.28. The slope of the backward peak in ir~pelastic scattering plotted versus energy. The lines are only meant to guide the eye 174B14].
with roughly ~ There may be a break at 10 GeV/c in the energy dependence of ~ scattering. The backward peak for ir~pis narrower than the forward one. Both ir~pdistributions may be fitted to exponential functions in t or u: (dG/dt)backward
=
(da/dt)~,~ exp{—bB(tmax
—
t)}
.
(5.17)
Fig. 5.28 shows the behaviour of the backward slope, bB, for 7rtp: it is apparent that the backward peak shrinks with increasing energy. Below 5 GeV/c the backward ?p cross section at fixed u plotted versus energy exhibits a number of structures which are usually interpreted as due to s-channel resonances. Above 5 GeV/c there are many interpretations in the framework of Regge models, parton models and optical models. Most theoretical papers try to interpret backward scattering in the frame of Regge pole models. In these models backward scattering arises from the exchange of baryon trajectories in the t-channel, namely of the £‘11236) in ?p scattering and of the z~°(1236), N~(l5l5)in ~rp scattering. Several specific Regge models have been used to analyze quantitatively the backward scattering data with: (i) Only Regge poles without cuts (in the notation of [71B4]: GORE = Good Old Regge); (ii) with poles plus “weak” cuts (WIZKID). The cuts, needed to remedy problems of unitarity in the s-channel, are generated with “absorption prescription”. (iii) poles plus “strong” cuts (SCRAM); (iv) poles plus cuts plus a prescription to remove effects from baryon parity doublets (HIPPIE = Hide Parity Partners). These four models allow fits of comparable quality. They differ considerably in their predictions at higher energies, where hopefully they will be subjected to more stringent tests, using differential
196
G. Giacomelli, Total cross sections and elastic scattering at high energies
Fig. 5.29. Illustration of t-channel baryon exchange for ir~pbackward scattering.
distributions of various types of scatterings and studying the energy dependence at fixed u-values. For instance the energy dependence of the differential cross section at u = 0 could in principle give indications of the presence of Regge cuts. On the other hand one could interpret backward scattering in terms of parton models; in this case the presence of a break in the established power dependence on the energy could give indications of the existence of parton—parton scattering, which could dominate backward scattering at high energies. In terms of optical models the explanations of backward scattering are reminiscent of those given for the forward peak. Clearly we are not close to a simple or unique explanation of backward scattering. 5.6. Total elastic cross sections The total elastic cross sections are obtained by integration of differential cross sections, which are usually not available for the whole angular range, in particular for very small and very large scattering angles. Therefore, in order to obtain the total elastic cross section extrapolations have to be made. Since the differential cross section is peaked at small angles it is usually the extrapolation here which introduces the largest errors. The errors from the extrapolations and the contributions from other sources yield a large total systematic uncertainty for the total elastic cross section. For this reason a plot of the data shows a considerable scatter of the points outside their statistical errors. Thus fits to the data yield large x2-values, if systematic uncertainties are not considered. A compilation of the measured total elastic cross sections is shown in fig. 5.30, while fig. 5.31 shows the ratios ~~eII~~tor In the momentum range 5—200 GeV/c all elastic cross sections decrease slowly with energy, the rate of change decreasing as the energy increases. Above 200 GeV/c at present there are only data for pp elastic scattering: they indicate that the elastic cross section also increases with energy. It does not take too much imagination to expect that all elastic cross sections may eventually increase with energy. The comparison of particle and antiparticle cross sections shows some interesting features: at low energies the negative particle cross sections are larger than the cross sections of the corresponding positive antiparticles; then, as the energy increases, the two cross sections tend to become equal
G. Giacomelli, Total cross sections and elastic scattering at high energies
11111
I
I
I
11111 6 8 tO
I 20
I
I 60
LABORATORY
11111
I
I
I
111111
111111 60 80 100
MOMENTUM
111111
I
I
I
I
197
I
I
200
400
(0eV/c)
I
I
I
111111
15—
—
11111 8 10
6
I
20
I
11(1111
40
60 80 100
LABORATORY
MOMENTUM
I
200
II
11111
400 600
1GeV/c)
Fig. 5.30. Compilation of integrated elastic cross sections.
1000
20OD
198
G. Giacomelli, Total cross sections and elastic scattering at high energies
11111
I
I
I
11111
I
I
I
111111
I
I
I
I
I
I
I
111T
~
c5
(0
111111 50 100
II 500
000
Laboratory Momentum 1GeV/c)
Fig. 5.31. Interpolations of the Oul/Otot ratios versus laboratory momentum.
at relatively low energies (compared to what happens to the total cross sections). It may even be that there is a kind of crossover between particle and antiparticle cross sections, but the data are not really adequate to establish this. For the ratios (1eI/0~o( the energy dependence is also relatively fast up to 40 GeV/c, and then slows down considerably. Table 5.5 gives the average values of UellUtot at about 100 GeV/c. For protons there is almost a plateau from 80 to 1 500 GeV/c, with indications that the ratio reaches a minimum and then increases slowly. The average value of Uel/Utot in this plateau or broad minimum is about 0.18, a value which is rather close to the Van Hove limit of 0.185 (see section 10). Information on the asymptotic region may only come from the pp system: the indication is that Uel/Gtot increases with energy in the ISR energy range. The U~)/tj~O~ratios for irp and ?p are within errors equal for laboratory momenta above 50 GeV/c. The same applies for K~pand at higher energies for pip. There are differences in Uel/tJtot for the different particles, indicating that the opacity of the proton to the various particles is different. Table 5.5 Average values of °el’°totand Oe)/atot at about 100 GeV/c laboratory momentum. The opacity may be defined as 0 2aei/otot.
=
irp Kp Kp pp
°eI
0tot
3.1
±0.2
24.09
±0.07
3.1 2.3 2.2
±0.2
±0.07
7.2
±0.5
7.0
±0.3
23.25 20.45 18.87 42.10 38.50
±0.2 ±0.2
Gel/Otot 0.130 ±0.009 ±0.009
±0.08
0.132 0.110 0.117
±
0.15
0.171
± 0.015
±0.06
0.181
±0.008
±0.07
±0.010 ±0.011
G. Giacomelli, Total cross sections and elastic scattering at high energies
199
6. Charge exchange scattering and K~ regeneration In this section the charge exchange reactions will be briefly discussed, mainly from the point of view of further understanding their relations to total cross section differences and to elastic scattering. Some of these relations arise from isospin conservation and from the optical theorem. Further relations may be obtained in the framework of some specific models, like the Regge-pole model, where charge exchange scattering plays a dominant role since it can be explained with a small number of exchanged trajectories. It may be worthwhile to recall that while total cross section measurements provide informations on the imaginary part of the forward scattering amplitude, charge exchange processes and K~regeneration have the added feature of measuring also the real part. The best studied charge exchange reaction is irp ir°nwhich has been measured up to FNAL energies. We shall briefly consider the following reactions (the main exchanged trajectory is shown in parenthesis): -~
7rp-~7r°n
(p)
(6.1)
Kp irp
-~
K°n
(p~A2)
(6.2)
-÷
r~°n
(A2)
(6.3)
(7r)
(6.4)
~ip—* ñn
1000
I
I
I
I
I
V CERN ~ CERN • IHEP 0 Fermilab S Fermilab 0 Fermilab
100 .
68 ‘65 ‘74 74 74 74
I
5GeV/c 10GeV/c 21GeV/c 40.6GeV/c 10$ GeV/c — 200GeV/c
1000
100
10
0.1
BORIGHT ET. AL. SONDEREGGERET AL
~
~
1.0
0.01
01
0.00~ ~
2 —t 1GeV/c)
Fig. 6.1. Compilation of angular distributions for i(p -4 ir°n scattering from 5 to 101 GeV/c in the 0 < ti < 1.3 (GeV/c)2 region [75G1]. The lines are only meant to guide the eye.
010
10
10
1f~ll~fff~ —t (GeV/c)C
~
0.l
lb
.01
Fig. 6.2. The complete angular distribution for irp -4 1T°n scattering at 5.9 GeV/c [74B6 J. Error bars represent counting statistics only; there is an overall normalization uncertainty of ~20%.
G. Giacomelli, Total cross sections and elastic scattering at high energies
200
np-f pn
(7r)
K~p-÷K~p
(w,p),
2-*K~C12
K~,C’
(w).
(6.5) (6.6)
A large variety of experimental methods have been used to study the various charge exchange reactions at high energy. All methods have to be able to detect a small two-body cross section, with at least one neutral particle in the final state, among a large sea of multi-body final states, in particular with ir°’s.For example the typical trigger for a charge exchange measurement (6. l)—(6.4) is a “neutral trigger” which requires that an incoming charged particle interacted in the target without producing any charged particle; forward going neutral particles eventually convert or decay into charged particles. 6.1. The i(p
-~
ir°nreaction
Fig. 6. 1 shows a compilation of the differential cross sections at laboratory momenta between Sand 100 GeV/c [74B4, 75Gl]. Data have been taken up to tI 1.5 (GeV/c)2 the complete angular distribution has been measured only at lower energies and a sample is shown in fig. 6.2. There are some normalization differences between the CERN, IHEP and Fermilab data. The main features of the i(p ~r°ndifferential cross sections at high energies are the following: i) All the distributions have two dips: at t = 0 and at —t = 0.6 (GeV/c)2 respectively. The dip at t = 0 is due to a large spin-flip amplitude, which has a sinO dependence and therefore vanishes at t = 0. The dip at t = —0.6 has a logical explanation in the Regge-pole model: it is the location at which the p-trajectory crosses zero. ii) The 5.9 GeV/c angular distribution shows the presence of another dip at —t = 10 (GeV/c)2, which corresponds to u = 0.11 (GeV/c)2. Changing energy the location of the dip remains constant in u. iii) In the 0.1 < tI < 0.5 (GeV/c)2 interval the differential cross section is approximately exponential. The slope of the exponential grows slowly with energy, thus indicating a shrinking of the forward peak. In order to analyze the energy dependence of the data over a larger t-interval, it is more instructive to look at the differential cross section at fixed t as a function of laboratory momentum. Fig. 6.3 clearly shows that at higher t-values the cross sections fall off faster with energy than for smaller t-values, thus yielding a shrinking of the forward peak. The angular distributions may be fitted to a single Regge-pole form -+
4~?
~9
(6.7)
52OIp(t)—2
dt
p*
Fits of the experimental data to eq. (6.7) yielded the effective p-trajectory shown in fig. 6.4, whose equation is approximately a~ 0.50 0.75 t. The solid curves shown in fig. 6.3 are the results of such a fit. iv) The integrated charge exchange cross section, shown together with other cross sections in fig. 6.5, has the power law dependence of eq. (2.50) with the coefficients quoted in table 6.1. v) The differential cross sections at t = 0 may be obtained from an extrapolation of the angular distributions. The optical theorem and charge independence yield the following relation between the total cross section difference ~a(ir~p) and the forward charge exchange cross section —
G. Giaco,nelli, Total cross sections and elastic scattering at high energies
I
I
111111
-t~O.36
0.5
\\
0.2 0.1
201
-t~0.45
-
0.05
0.02 -t ~0.55 I
11111111
2
5
10
p
I
111.1111
20
50
I
100
200
I
III
500
(GeV/c)
Fig. 6.3. Plots of do/dt versus laboratory momentum for the reaction irp -4 lr°n at a number of different t-values. The solid curves are the results of a fit to expression (6.7) [74B9]
1
dcr
ce(t
iT
0)=~-~--[u~0~(7rp) ~ 2 .
(6.8) (6.9)
25.5(1 + P~e)(L~U) Re ~ is the real part of the non-spin-flip scattering amplitude and Pce = [Re g~~/Im gee] to From eq. (6.9) one can compute z~u(7r~p) if a reasonable estimate of Pce (t = 0) can be made. One possible form is =
P~
(6.10) It may be worthwhile to point out that equations (6.8—6.10) are obtained from (i) isospin conservation, (ii) the optical theorem and (iii) assumptions on the p-Regge trajectory. The agreement between measurements of ~a and the predictions from charge exchange is reasonable (see fig. 6.6), within errors. 0~
tg(~ira~(0)).
202
G. Giacomelli, Total cross sections and elastic scattering at high energies
600
rp,°n
400
• MANNEU.I 65 I GIJISAN 65 65
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2(30
— ST1RUNG V WAHLIG ~VERT
~
A
~
S
100 80 60
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..
o 0.4 0.2-
•
~cl -0.2
6,50
6eV/c 6.100 6eV/c
+
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~
8
0)
6
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~
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68
o BOLOTOV VLEE 0 AST~~ 74 66
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0
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0
z
68 74 74
ASTBIJRY VB 0
‘~
40 20\
o w
BARNES
• eowro~
0.8
i
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-
iP~~fl~\
04 O4~
-o.e
-
t~l -1.2
I -08
t
I
0.2
~o
-04 2
Fig. 6.4. Effective p-trajectory for iip
-4
6
(0
20
40
60
100 200
LABORATORY MOMENTUM (GeV/c)
(6eV/c)
ir°n [74B9].
Fig. 6.5. Plot of integrated charge-exchange cross sections versus laboratory momentum. The solid curves are eyeball fits of the data.
I
I
I
I
r
Z2.Ob
~
El
(.0
0.5 ______________________________ I
I,,,.I
5
(0
I
I
20
50
(00 200
LABORATORY MOMENTUM (GeV/c) Fig. 6.6. The difference &s(sr±p)of the srp and ir~ptotal cross sections, plotted versus laboratory momentum. The solid line represents the prediction from forward charge exchange scattering, according to the relation (6.9), using Pce = 1.08 and o~(t= 0) = 0.50 [74B9l.
G. Giacomelli, Total cross sections and elastic scattering at high energies
203
6.2. Other charge exchange scattering processes Fig. 6.5 also shows the integrated cross sections for the following reactions: jip -÷ñn, Kp -÷K°nand irp -÷~°n (i~°-÷2’y). It is apparent from the figure that all the integrated cross sections have an energy dependence of type (2.50). The amount of data on these reactions is still rather scarce, while a considerable amount of information exists on the np pn differential cross sections (see fig. 6.7). The data are characterized by a sharp peak (with a slope of about 1 /m~up to tI < 0.02 (GeV/c)2) followed by a less steep dependence. The t-distribution does not seem to change in shape with increasing energy, while the absolute value decreases approximately as as one would have expected on the basis of the one-pion-exchange mechanism. In [74K 11 the authors parametrize the t- and s-dependence of the differential cross section with a two exponent formula: -*
da/dt~ A p’~81~°~25{exp[(5l ±8)t] +(0.8 ±0.l)exp[(4.50 ±0.15)t]
}.
(6.11)
L
IHfllfIl 1fff
21.0
~
6W/c .1 ~
I
1$
6eV/c
I
.~
II f
H~111~~ ~ ~
I
25.3f
.~,
.0I
lII
‘~
I
a .c~.d4 .d6 .~ a
.i
.~
.1
~
I
27~4 6eV/c
If
.~ .~ .i’
.~
.1 4.0
lul(6W/c? Fig. 6.7. Differential cross sections for neutron—proton charge-exchange scattering [72D2] - The low.t region is shown with an expanded scale.
G. Giacomelli, Total cross sections and elastic scattering at high energies
204
Table 6.1 Results of fitting integrated charge exchange cross sections to the formula a Reaction
Laboratory momentum range (GeV/c)
A ( 1.zb)
6—101
7rp—~?r°fl
679
20—200 20—200 6—40 6—50
irp—~X°n L. ‘y~y irp -+ f°n irp-~ w°n
=A
p_fl.
n
±
719 ± 292 ± 3.2 ± 25 ±
47
1.15
±0.07
20 20 0.8 4
1.17 1.47 1.1 2.3
± 0.04 ± 0.02
18 1680
±
Kp -÷K°n
6—50 4—35
4 220
2.7 1.5
~o(K°n)
4—50
7300
± 3000
0.53
±
±0.1 ±0.1 ±0.1 ±0.1
±0.12
6.3. K~regeneration In reaction (6.6) the interference between the iT~ir decay of the regenerated K~with the CP violating ir~Ir decay of K~in the transmitted beam depends on two weak-interaction parameters (i~÷ and ~m, the K~—K~ mass difference) and two strong interaction parameters. At high energies one may assume the known values of the weak-interaction parameters and compute the two stronginteraction parameters, that is the regeneration phase at forward angles =
arg[iA(K~p -÷K~p)0~
+
=
(6.12)
iT
and the imaginary part of the forward scattering amplitude, which may be expressed via the optical theorem in terms of a total cross-section difference: ~u(K0p)atot(k0~i)
T-at0t(K°p)
—~Im [ftO) —j~0)].
10
~
I
‘~.‘
I
(6.13)
111111
2 I b
_~o.(p).(7.3±3)xp0~OI2 FREITAG St al
~
4
I
~ANDEN&JRGet oP ~ FiRESTONE et dl
~
~ DARRIULAT et al I ALBRECI-IT 61 01 74 ~ ARO4JSON et al I
2
46
K)
I
20
I
I
1,,
4060100
LABORATORY MOMENTUM (6eV/c) Fig. 6.8. The measured ~at0t(K°p)
values obtained from K~regeneration experiments in hydrogen. The dashed line is the fit through the total cross section differences ~a(K±n).
G. Giacomelli, Total cross sections and elastic scattering at high energies
205
ftO) and ftO) are the forward scattering amplitudes for K°pand ~Op elastic scattering. Because of isospin conservation the difference ~c(K°p) is equal to i~cr(K~n): ~a(K°p)
= i~a(K~n) = a~ 0~(Kn) a~0~(K~n). —
(6.14)
Thus, for our purposes, K~-regenerationexperiments are useful to determine z~u(K~n) at high energy. In fact the experiments are rather sensitive to the difference and even a rough experiment is capable of yielding information on ~a(K~n) with precisions comparable to those achieved only in the most precise total cross section measurements. Fig. 6.8 shows the &r~0~(K°p) differences computed from regeneration experiments in hydrogen. The dashed line is an interpolation through the ~u(K~n) data from total cross-section measurements. The direct measurements and the values obtained from regeneration experiments agree rather well.
7. Polarization The present general picture, which emerges from the available data on polarization at high energies, is that polarization effects decrease with increasing energy. Thus we shall limit the discussion to some qualitative aspects and try to estimate what are the possible errors which one makes when one neglects spin—spin and spin—orbit effects. In the last few years a number of technological advances have made it possible to measure polarization effects using refined polarized targets and, more recently, very high intensity polarized proton beams. 1~target>’ of The more recent experiments use butanol targets, which have average polarizations, ‘~ about 50% and which contain about 10% free hydrogen. The other recent achievement was the acceleration of polarized protons in the Zero Gradient Synchrotron (ZGS) at Argonne. Using special precautions at some “depolarizing resonances” the polarized protons were accelerated to 8.5 GeV without any measurable loss of polarization. This is possible because of the special magnetic field configuration of the ZGS. The extracted proton beam has (73 ±8)% transverse polarization, with intensities ranging from 1 0~to 108 protons per pulse. By scattering this high-intensity highly polarized beam from a polarized target, and eventually measuring the recoil proton polarization, one has the capability of measuring all possible polarization parameters in pp elastic scattering [74F1, 74K2]. 7.
1. Polarization in 1r~pelastic scattering
The irp system is relatively simple to analyze [see formulae (2. 1 8)—(2.40)] and many experimental data on polarization parameters are available. Fig. 7. 1 shows the angular distribution of the P polarization parameter for i(p elastic scattering at 14 and 40 GeV/c [75G21. The shape of the t-distribution at the two energies is similar: P is negative and it has a zero at t = —0.6 (GeV/c)2, besides those at t = 0 and at t —1.9 (GeV/c)2. This behaviour is consistent with the hypothesis that the P polarization parameter behaves like the logarithmic derivative of the differential elastic distribution. The zero at t = 0 is due to the vanishing of the spin-flip term. In so far as the energy dependence is concerned, the maximum polarization at —t 0.15 (GeV/c)2 has decreased from about 9% at 14 GeV/c to 5% at 40 GeV/c. The energy dependence is
G. Giacomelli, Total cross sections and elastic scattering at high energies
206
(b)
.5 p
1.
15
I I
I
PL~ts45 0eV/c 05
pp
Ia)
I
‘If
-
-0.05-
-0.10
.2
40
-
-
+0.05-~-
~
+
.05
0
-
(
~I~I
-0.15
-
14GeV/a
f-
~II~V~
0.2
I
~
I
I
cc
~-*
PLV
-
I
..2
-0.05-
-0.10
I
~
40GV/
2
-
—t I
(Gev/C) 0.6
2
I
•c
~
1.0
.~
Fig. 7.1. The P polarization parameter (a) for i(p —~ irp at 14 and 40 GeV/c [72G1] around 40 GeV/c [74D6] , [75G2].
, and (b) for pp,
1.
_t@ev/c~ 1.5
pp and Kp elastic scattering
of the power type, eq. (2.59), with n 0.53 (fig. 7.2). Thus the P parameter decreases rather fast with energy, but polarization effects may really be neglected only at the highest energies, The P parameter for ir~pelastic scattering is essentially positive and seems to be the mirror image of that for i(p scattering. Thus the above conclusions about the energy dependence are valid also for ir~pscattering (and more generally for K~pscattering also, fig. 7.2). The main features of the P parameter, that is its t- and s-dependence and the relation P(ir~p) —P(i(p) may be explained in the framework of Regge-pole models assuming that the polarization arises from the interference of P and P’ with the p-trajectory: P~Im[(G~+G~c)H*].
(7.1)
The first measurements of the polarization parameter R in i(p and ?p elastic scattering at 6 and 16 GeV/c are shown in fig. 7.3. The average values are R(irp) —0.2 and R(?p) ~ —0.1. From the knowledge of P and R one may compute the A parameter from the relation (7.2) A2-l-P2+R2= 1. The values of A obtained from relation
(7.2)
and from direct measurements yield A(irp)
1.
G. Giacomelli, Total cross sections and elastic scattering at high energies
1.OC
I
1111
I
I
I
I
I
0.60
-n
I
Kp
0.59
Vp TLp
0.60 0.53 0.65 -0.90 -0.30
pp Kp ~p
0.40
III
-
~
LU
0.20
4
K~p
~
I
I
-
0.02
~‘N
I
$4~+
ci
~~1NN
I
~
R(Jp),641
~
~~
I
~R(irp),6Gey~
~0.04.
207
I
~0.5j ~p),I6
G~
~ R(7rp),4OGe~
I
1111
6
8
I
I
I
20
10
LABORATORY
I
40
I
I
I
ir
60 80
MOMENTUM (GeVic)
Fig. 7.2. Compilation of the energy dependence of the P 2polarization for sr~p,parameter K~p,pp and averaged ~p elastic overscattering. 0.1 < (t( Note <0.3 that (GeV/c) the sign of P(i(p) is negative. 7.2.
-a5L L
I
0.1
02
I
0.3 2 0.4
-t(GeVk)
I
05
and 16 GeV/c (72G1]. Fig. 7.3. The R polarization parameter for ir~p..+ ir~pat 6
Spin effects in pp scattering
The study of polarization effects in the pp system is complicated because both particles have spin. Thus one has to worry about spin—spin and spin—orbit terms. In total there are nine measurable quantities. 7.2.1. 0tot A measurement of the pp total cross section in pure isospin states was performed at 3.5 GeV/c using the Argonne polarized proton beam and a polarized target [73P 1]. The measurement was a standard geometry transmission experiment, which measured the difference
R,(CC)
= a~~(tt) a~
1
—
0~(t4~) = R~(N)PBPTN’
(7.3)
G. Giacomelli, Total cross sections and elastic scattering at high energies
208
where R~(tt)and R.(t4.) are normalized counting rates with the beam and target spins parallel and antiparallel respectively; ~B and ~T are the polarizations of the beam and of the target; N is the number of hydrogen nuclei in the target. In reality the measurement was performed with the four combinations offered by the incident beam and target spins: t t, 4.4., t 4., 4. t. But rotational invariance of space requires the equalities o~0~(tt) = ~
cr~0~(t4.) = o~~(4.t). (7.4) values of(7.3) are averages. The recent results of the measurement indicate that
0tot
Thusdifference the two the =
u~(tt)
—
o~ 0~(C4.)
(7.5)
is about 8 mb at 2 GeV/c and decreases to about 0.5 mb at 6 GeV. Thus at 6 GeV/c the two cross sections differ by about 4%. This may be thought to imply that at high energy the proton size will be unique for different spin states. The Pparameterin pp-÷pp The behaviour of the average P polarization parameter versus energy shown in fig. 7.2, suggests that also for pp elastic scattering the spin—orbit effects decrease with energy and may become essentially negligible for energies above 100 GeV. 7.2.2.
7.2.3. pp elastic scattering in pure spin states At 6 GeV/c there are now measurements of all the polarization parameters in pp pp scattering measured with the ZGS polarized beam impinging on a polarized target and measuring the polarization of the final state protons with scattering on carbon targets [74F1, 74K2] We shall restrict the discussion to the simplest case. Fig. 7.4 shows the differential cross sections for elastic scattering in pure initial-spin states, without looking at the polarization of the final protons. The cross sections have been normalized to the spin-averaged cross sections. The spins are measured perpendicular to the scattering plane. 2 exhibit the folThe data at the laboratory momentum of 6 GeV/c and for 0.5 < p~< 2 (GeV/c) lowing features: -~
.
(i)
do
(tt)>
do
(t4.)
—
=
do do —~-(4.t)>—(44),
(7.6)
the difference between the first and the last being about a factor of two. do/d~(t4.)is equal to do/d~2(4.t)because of space rotational invariance. do
(ii) ~~~-(44)
do
(44)>
do
(11)
in the 0.5
(N)
forp~>0.8 (GeV/c)2
—~~-
do
that is there is a change in the angular distribution of do/d~Z(14) at about —t 0.8 (GeV/c)2. This feature and the other features previously mentioned are better observed in fig. 7.5, which shows the differential distributions for the three spin states. They have been interpreted as arising
G. Giacomelli, Total cross sections and elastic scattering at high energies
I -
~
•
T
_______
.
~—0ai -
I
ido\
I.
0 0~P~IV4~80 \~. ‘0
-
~‘
‘~
NORMALIZATION PRELIMINARY
I
-
. ___
•~~
~
6 GeV/c
~
I
-
0
209
~
-
pep—pep
~I
o-~ ~
-
.01
—~
preliminary
~
2
1 P$(GVc) Fig. 7.4. The ratio of the differential elastic pp cross section for each spin state (da~ 1/dt)to the spin averaged cross section (da/dt) plotted versus p~at 6 GeV/c [74F1, 74K2]. The spins are measured perpendicular to the scattering plane. The if subscripts refer to the cases where only the initial spins are measured, while indicate cross sections for the elastic process when both initial and final spins are observed. The frame of reference is such that the scattering of the forward particle is observed to the left.
~2
(GeV/c)2
2
1 Fig. 7.5. The differential elastic proton—proton cross section for pure initial spin states plotted versus p~at 6 (GeV/c) [74K21.
from different spatial regions in the proton, each with different spin states. 2 where the spins of both the initial In fig. 7.4 measurements are also shown at p~ = 0.5 (GeV/c) and final protons are observed. The interesting feature is that the double-flip cross sections are typically ten times smaller than the non-flip cross sections, more specifically do
ldu
do
-~
44)
—~(t4.-~ 4.t)rai -~ -~
ldo
(N. -~ N)
1 dci
-÷ tt)~
j-~-~ (tt
-+
N.).
(7.7)
The Wolfenstein A and C,~parameters have also been measured at 6 GeV/c: for ItI < I both average the value of around 0.1. 7.3. Conclusion In conclusion, in the 5—20 GeV/c laboratory momentum range there are sizeable spin—spin and spin—orbit effects. The latter decrease considerably with energy and may probably be neglected for laboratory momenta above 50—100 GeV/c; there is still no information on the energy dependence of spin—spin effects in pp collisions, though they are expected to decrease with energy too.
G. Giacomelli, Total cross sections and elastic scattering at high energies
210
8. Dispersion relations for the forward scattering amplitude Dispersion relations give the real part of an amplitude in terms of an integral over its imaginary part. Many calculations are actually concerned with forward (t = 0) dispersion relations, where the imaginary part may be obtained from the total cross section via the optical theorem. Our discussion will therefore be centered on the computation of the real part of the forward scattering amplitude from the experimental knowledge of the total cross section. The results of the computations will be compared with the direct measurements of the ratio between the real part and the imaginary part of the elastic forward amplitude using elastic scattering data in the Coulomb—nuclear interference region (see section 5.2). Such a comparison serves practical and theoretical purposes. From the first point of view, one can check the consistency of the total cross section data with the elastic scattering data at forward angles. From a theoretical point of view, the comparison allows a check of the validity of dispersion relations and of the assumption of spin-independence for forward and near-forward scattering. The real part is obtained as an integral over 0tot~ Thus the connection between the two is nonlocal, the real part depending upon the behaviour of 0tot at all energies, including the low-energy region and the high-energy region where no measurements exist. At high energy the relation between the real and the imaginary part becomes almost local, since the effects of the far away parts of the integral become small. One may estimate the energy at which the contributions of the unphysical region, of the subtraction constants and of pole terms become small by computing the real part with the exact formulae and with approximate formulae. One finds that above 50 GeV/c for irp and Kp and above 1 50 GeV/c for pp the difference between the two methods is smaller than the experimental uncertainties. The estimate of the uncertainty arising from the neglect of the high-energy part of the integral is more difficult to estimate. Nevertheless, we shall consider that for energies higher than 50—150 GeV the relation between the real and the imaginary part is almost local. It is thus pedagogically interesting to write down the approximate local relation. Bronzan et al. [74B3] write: .-1 1
[ReF(xp)] +
5.0
...~
iT
(+)
(—)
~
~~~°tot — fl
————-—+
[tg~-~--)O~
+~s° ~
/7Ta
d
/0tot
\1 (8.1)
where o~ = o~0~(xp) ±o50~(x~p). Expression (8.1) maybe even simplified further. For instance, for pp collisions one may also write: Ref (E, t0)
ira ——-——Imf (E, t0). ~ 2 alns
(8.2)
In (8.1) the o-parameters are found empirically to be o 0.68 for irN, ~ 0.44 for KN and 0.39 for NN. Several authors have computed p using various approximations for the dispersion relations [73Bll, 74B3, 74H1, 74H2] ; there are also several calculations performed in the framework of various theoretical models [73C2, etc.] The results of these calculations are quite similar, apart from some numerical details. Fig. 8. 1 shows a compilation of forward real parts computed via complete dispersion relations [74Hl I. The shaded bands represent the results of these calculations up to the energies where measurements of °tot exist. The vertical widths of the bands represent estimated uncertainties arising from the uncertainties in the total cross section data, in the theoret.
G. Gi.acomelli, Total cross sections and elastic scattering at high energies
~2
~
-0.3
211
___
~
-0.3
-0.4
‘°irp
03
)Il~
~
O.2~ ~
~
0.+l
I
III)
I
III
I
-0.4 Ill
I
III)
I
1111
I
1111
I—I—I—)
o~ 0.2
0
_//__
-0.2 -0.2
¶~
-0.3 -0.4
PK~P I
o
I—
III
1111 I
II))
I
III
I
III)
-0.4
I
-0.6 1111
I
1111
0.4
-0.2
I —I—I—I
0.6
.1
10
I
00
I.
1000
.1
10
•
tOO
I
-0.4
1000
LABORATORY MOMENTUM 1GeV/c) Fig. 8.1. Compilation of the ratios p versus energy [74H1] . The bands are calculations from dispersion relations in the energy region where there are data on °tot’ while the solid lines are the predictions at higher energies where no measurements have been performed. The points are the computation from measurements of elastic scattering in the Coulomb-Nuclear interference region.
Note added in proof: There are now
data from Exp.# 69 at Fermilab between 70 and 150
GeV/c [75A1I ,which agree with the
dispersion relation predictions.
ical approximations in the treatment of the unphysical region and in the uncertainties in the coupling constants. The solid lines extending beyond 200 GeV/c are predictions of the real parts based on the parametrization (4.25)—(4.27) of the unmeasured high-energy cross sections above 200 (280) GeV/c. Fig. 8.1 shows also the real parts directly obtained from measurements of elastic scattering in the Coulomb-nuclear interference region. The agreement between dispersion relation calculations and direct measurements is not too good, with the exception of the pp system, where on the other hand most measurements exist. As stressed in section 5.2, the disagreements are very probably due to the measurements and to the methods of analyses of the elastic scattering data at small t, since the effects of the real parts on the differential cross sections are very small and the computation of p depends considerably also on the absolute values. Thus the comparisons between dispersion relation calculations and direct measurements are in
G. Giacomelli, Total cross sections and elastic scattering at high energies
212
general not really meaningful, with the exception of the pp data, for which the agreement may be considered to be good over the entire energy range. This agreement is consistent with the validity of dispersion relations and with the assumption of spin-independence of the scattering amplitude at small angles. The dispersion relation results of fig. 8. 1 clearly show that all the ratios p are negative in the energy range 5—50 GeV and become positive at higher energies. As already stated in the Introduction and in section 3, one expects that p reaches a positive maximum and that it eventually decreases towards zero.
9. Impact parameter expansion Analyses of high-energy elastic scattering data in terms of impact parameter expansions on ISR data have yielded a simple picture of very high energy scattering {73A3, 73C1, 73C4, 73Hl, 73H3, 74M1]. Assuming that (i) the real part of the scattering amplitude is negligible at all angles, and that (ii) there are no spin effects, we may write: IF(s, t)12= IReF(s, t)12+ IImF(s, t)I2~IImF(s,
t)12.
(9.1)
The neglect of Re F(s, t) is only an approximation since we know that some real parts exist. Performing a Fourier transform to impact parameter space, we have Tm F(s, b) =
exp(—iq~b)Im F(s, q) dq,
(2ir)
(9.2)
where q = ~ and b is the impact parameter. The opacity &2 may be defined so that the following relation is valid: ImF(s, b)
=
~
—
exp[—&2(s, b)]
}.
(9.3)
This form of Im F(s, b) satisfies automatically the unitarity condition. Eqs. (9. 1 )—(9.3) yield the required informations on Tm F(s, b) and ~2(s,b) [73A3]. In terms of the invariant quantities defined in section 2.3, we may define the helicity-non-flip elastic amplitude transformed in impact space as [73H3]: G(s, b)
=
f G(s, t)J0(b~t) 2p~~
(9.4)
Neglecting spin—spin and spin-flip effects we may write do dt — ~jei=
______
2
(9.5)
641rp*s IG(s, t)1
f
~dt~ _dt
flG(5,b)I2~~-~P
8ir
(9.6)
G. Giacomelli, Total cross sections and elastic scattering at high energies
=J[
1m(
213
~
Fig. 9.1. Illustration of the s-channel unitarity equation.
ImG(s, t = 0) 2p*~/~
=
tot
JIm G(s, b)bdb.
(9.7)
Unitarity in the s-channel implies: Im G(s, b)
2 + I(s, b) 8ir IG(s, b)1
= —~---
=
E(s, b) + I(s, b),
(9.8)
where E(s, b) and I(s, b) are the overlap functions due to elastic and non-elastic s-channel intermediate states. Eq. (9.8) is illustrated in fig. 9.1. In this context, diffraction scattering is interpreted as the shadow of absorption due to the existence of elastic and inelastic channels. By combining eqs. (9.6) and (9.8) we have
0in
=
0tot
—
0el
= /I(s, b)bdb
=
f I(s, b)bdbJ0(b~~)~0.
(9.9)
Finally the eikonal form is given by G(s, b)
=
4iri[ 1
—
exp{—~(s,b)/4iri}I.
(9.10)
From the unitarity relation we have the following limits: 0< Im G(s, b) ~ 4ir,
0 ~ 1(s, b) ~ 2ir,
Im ~7~(s,b) ~ 0.
(9.11)
A calculation was performed using the equation
Tm G(s, b)
=
~f dtJ0(b~f~i)[l6iT{(~)
—
~~real)
hh1~~2.
(9.12)
An estimate of the real part of the scattering amplitude at very high energies may be obtained using the derivative analyticity relations of Bronzan [74B3]. Assuming that above 2 the differential elastic cross sections et areal. dominated by the imaginary part of the shelicity ~ 1000non-flip GeV amplitude, they neglect spin effects and obtain the approximate relation: Re F(s, t) ~
—~-—
2dlns
Im F(s, t)
~
-~ —~----
2dlns
dt
(9.13)
that is one may compute the real part of the scattering amplitude at a certain t-value via a logarithmic derivative in s of the square root of the differential cross section. The results of a numerical calculation using pp elastic data at ~ 30.8 and 53 GeV, indicate that the real part is always considerably smaller than the imaginary part, with the exception of the region around the dip at —t ~ 1.4 (GeV/c)2 (see fig. 9.2).
G. Giacomelli, Total cross sections and elastic scattering at high energies
214
__________________________________________
2
__
0
.2
~3~eI1\II~~
.4
6
.8
1.0
21,2
-t (0ev/c)
14
15
0
1620
—1.0
t —2.0 1GeV)2 —3.0
—4.0
Fig. 9.2. The real part of the pp elastic scattering amplitude at -..fl= 53 GeV computed via formula (9.13) [73F13I.
The computations according to (9.2) and (9.12) yield very similar values, independent of the choice of the real part. This is so because Im G(s, b) is computed with an integral, whose largest contribution is for b 0, where the real part is very small (say 1% of the imaginary part). Thus even if at larger angles the real part becomes larger this has little effect in the integration. Fig. 9.3 shows Im G(s, b) computed via eq. (9.12) from the ISR data on pp elastic scattering at ~ = 53 GeV. Also shown are I(s, b) and &~2(s,b). The following important features may be pointed out: (i) At small values of b, Im G(s, b) is very nearly Gaussian. This is more vividly shown in fig. 9.4. The Gaussian form of Im G(s, b) is no surprise, since an exponential form in t transforms into a Gaussian in b. (ii) There is a large tail at large b. Above 3 fm the curves are not very reliable because of difficulties caused by small discontinuities in the experimental data. Henyey et al. [73H31 emphasize that the tail is related to the break observed in do/dt at —t ~ 0.13 (GeV/c)2. For practical purposes the amplitude may be parametrized as the sum of Gaussians [73H31 Im G(s, b)
=
8.9 exp(—b2/21) + 0.5 exp(—b2/62)
—
0.24
exp(—b2/0.92).
(9.14)
The first Gaussian dominates at small b out to about 2 fm, beyond which the second term takes over. The third term yields a small contribution, which is important only around the dip at —t 1 .4 (GeV/c)2. The first term represents central collisions and has a mean radius of 0.8 fm; instead the second term provides a peripheral contribution and has a mean radius of 1 .4 fm. In four-momentum transfer, eq. (9.14) becomes Im G(s, ~ 2p
~
exp(5.3t) + 15.6 exp(15.6t) —0.11 exp(0.23t).
(9.15)
*,V./~
It is surprising that the t = 1 .4 (GeV/c)2 dip does not show up at all in the graphs of figs. 9.3 and 9.4. The dip corresponds to a small flattening of Im G(s, b) near b = 0, a flattening which is too small to be seen in the figure. Thus when looked in b-space the diffraction minimum does not look impressive at all. It rather appears as an accident. —
G. Giacomelli, Total cross sections and elastic scattering at high energies
I
2It
I
I
21
-
\&——-—
I
‘/i~c53 0eV
5
pp-
2
-
(sb)
—
- —
-—
1
(s • b)
I
~
-
I(s,b)
E (s~b)
-
1
‘\
E~-
‘N
0.2 01
scattering
Im G(s.b)
Im (sb)
I
215
c
\
BLACK DISK LIMIT
2it
\
°
‘Na 005
0 2 (fermi)2
b
Fig. 9.3. Impact structure of proton—proton scattering at ~/~= 53 GeV. The imaginary part of the non-spin-flip elastic amplitude, Im G(s, b), the inelastic overlap function, J(s, b), and the eikonal or opacity, n(s, b), computed from the proton—proton elastic scattering data at .,/i~ 53 GeV [73H3I.
b (fermi)
Fig. 9.4. Im G(s, b),I(s, b), and E(s, b) at ~/~= 53 GeV. The horizontal line denoted “black disk limit” indicates the maximum value of the inelastic overlap function allowed by unitarity (100% absorption) [74M1I.
It is interesting to remark, from the graphs of fig. 9.4, that at b = 0 one has I(s, b) 94% of the maximum value allowed by unitarity, represented in the figure by the “black disk limit” line. This means that in a head-on collision two protons have a probability of 6% to pass straight through each other without any absorption at all. On the other hand E(s, b) is only 50% of its black disk limit. The difference between the b = 0 values of E and I raises the question as to whether one has to add to elastic processes also the contribution from diffractive inelastic processes. The graphs of figs. 9.3 and 9.4 show that there is really no major advantage or simplicity in discussing E or We have now to investigate the energy dependence and, in particular, try to answer the question: “from which impact parameters does the rise of ~ and of 0tot come from?” Fig. 9.5 shows I(s, b) at four ISR energies. The difference in I(s, b) between the lowest and highest energy is shown in fig. 9.5b. This graph has some uncertainties at small b, but even so the picture is quite clear. The increment in 0in -comes from an average b of 1 fm and is more peripheral then I itself. Thus with increasing energy the proton radius becomes larger (by about 5% over the ISR range), but the values of the opacity at b = 0 does not change: the proton becomes bigger, but not blacker.
G. Giacomelli, Total cross sections and elastic scattering at high energies
216
I
1.C -
I
a) 53GeV
0.5
\\\
-
0
b)
1~~~2O
b (fermi)
Fig. 9.5. (a) Inelastic overlap function, I(s, b), computed from the ~ = 21, 31, 44 and 53 GeV ISR pp data [74M1 I. (b) Difference of the ~ = 53 and 31 GeV inelastic overlap functions, i~J(s,b) It shows that the cross section increase comes from a region around 1 fm.
It may be worthwhile to point out that in an optical model with constant opacity one has: 0e1
7~2a2,
00o0 =
2irR2a,
Oei/OtOt
(9.16)
~
where R is the radius up to which there is a constant opacity (or greyness). a = 1 for a black disc and a = 0 for a transparent one. Instead in an optical model with Gaussian opacity [0 a exp(—5b2/2R2)I: 0e1
~7~20I2,
°tot =~7rR2a,
OeI/Otot
~
(9.17)
where a is the opacity at b = 0. Thus in this case ciel/0000 cannot exceed 0.185. The experimental data on Gel/cOot (see fig. 5.31) are not conclusive as far as optical model analyses since the ratio may increase with energy. The impact parameter analyses of the data at the present energies suggest a picture in agreement with (9.17).
G. Giacomelli, Total cross sections and elastic scattering at high energies
217
10. Theoretical models A large amount of theoretical and phenomenological work has been done to understand high energy hadron—hadron elastic scattering and total cross sections. The following will only be a brief list of the work in progress, without attempting a comprehensive survey. The emphasis will be on trying to understand the main lines of approach and to discuss the phenomenological background needed to make comparisons with the experimental data. We refer the reader to the many theoretical reviews which have been published recently [71C3, 71 Z 1, 73K 1, 74B 13, 74J 1, etc.]. Table 10.1 illustrates the predictions of various models for the high energy behaviour of 0tot’ Gei/Otot, b/a 000 and of the cross section for the secondary maximum for the pp system [74B13]. 10. 1. Amplitude analyses for irN
-~
irN scattering
Amplitude analyses seem to be the best way to analyse the elastic scattering data in the intermediate energy region independently of any specific model. These amplitudes are becoming the meeting point between experiments and theory following the same trend one had with phaseshift analyses at lower energies. On the other hand their f’ble at higher energies is modest. These analyses need much experimental information, particularly on some polarization parameters which are difficult to measure. Thus, in practice, quantitative amplitude analyses could be performed only for irN -÷ irN scattering. The new measurements with the polarized beam at the ZGS are yielding much data on all polarization parameters which may be useful for amplitude analyses. Our discussion will be limited to the irN case. The measurement at each energy and at each angle of the eight quantities, (do/do) (irp), (da/d~2)(?p), (da/d~)(ir°n),P(irp), P(ir~p),P(ir°n),R(i(p), R(ir~p)and of the sign of A(i(p) allow the complete determination of the irN elastic scattering amplitudes in the s-, t-, or u-channels. In each of these channels the amplitudes are determined by seven numbers (four moduli and three phases, with an undefined overall phase, which could be determined, for example, by Coulomb-nuclear interference). In the last year several amplitude analyses of the irN system were performed at 6 and 16 GeV/c, where the spin rotation parameters A and R have been measured (in irN scattering there are 12 measurable quantities, 8 of which are independent). The analyses of the experimental data into s-channel helicity amplitudes employ formulae of the type: do —=
R
2÷ IF~j2 IF~I —2 Im(F~~F.~’)
(10.1) (10.2)
—(IF~~I2 F÷J2)cos0~ —
—
2
Re(F~~Fjsin09,
(10.3)
where ~ and F~_are s-channel helicity-non-flip and flip amplitudes, and is the recoil proton angle in the laboratory system. Other authors divide the ~ and F~_amplitudes by p*/V’~,where p* is the c.m. momentum.
G. Giacomelli, Total cross sections and elastic scattering at high energies
218
Table 10.1 Illustration of the predictions of various models for the energy behaviour of “tot’ °eI/°tot’b/a
000 and of the height of the secondary maximum for pp scattering [74B13J (b is the slope of the diffraction pattern). “tot 2(s)
“e1/~~tot
b/at00
constant
constant
(da/dt)secondary (do/dt)p0 constant
Geometrical scaling Cheng—Walker—Wu Asymptotic black disc model
~
R
—~
increases (by 7%)
decreases (by 7%)
increases (by factor >2) -+l.7X102
Regge poles and cuts [with np(O) “l I
rises to a constant
decreases 0
increases
—*
—~
decreases or constant
Factorizable eikonal
arbitrary
increases (by 7%)
decreases
In2 s
(b)
‘Im 0.10
ISOSPIN 0
,
increases (by factor >2)
F°
ISOSPIN 1
IFLI
1O005~1~00
~5~40~009
Fig. 10.1. Non—nucleon scattering amplitude analysis at 6 GeV/c [72C3]. a) Moduli of s-channel helicity amplitudes. The heavy line is the result of the fit, while the thin lines indicate the error limit. b) Complex diagrams of ratio of s-channel helicity amplitudes.
G. Giacomelli, Total cross sections and elastic scattering at high energies
219
The t-channel I = 0 and I = 1 isospin decomposition is given by: f~~(?p-÷ ?p)F.~~
f~~(irp-+ir°n)=
~
—‘~,/2F.~~
(10.4) (10.5)
and similarly for the helicity-flip amplitudes F~,F~. The results of the analyses at 6 (fig. 10.1) and 16 GeV/c show that F÷° + is the dominant amplitude and that the energy dependence of all amplitudes is consistent with Regge-pole pictures. 10.2. Thermodynamic or statistical models The fact that the high-energy, large-angle differential cross sections are characterized by a slower t-dependence and a faster s-dependence than at smaller angles, had led a number of authors to hypothesize a statistical (or thermodynamic) mechanism for large-angle scattering. Following the compound-model picture for nuclear reactions, one may visualize the statistical model with the following naive interpretation: in central, high-energy collisions the two colliding particles form a compound system. The total available energy is concentrated in a small volume, of about 1 fm radius, in which a kind of thermal equilibrium is reached. A short time later, the volume starts to expand, the equilibrium ceases and the compound nucleus decays into the many channels open, one of which is the initial state. Thus if the large-angle scattering is due to a mechanism of this sort, its cross section will be proportional to the probability of formation of the compound state and to the branching ratio for the decay of the compound state into the elastic channel. The analogy with nuclear physics does not really go very far, because of the very short lifetimes involved in particle physics, which means that a statistical equilibrium cannot be reached. The thermodynamic model of Hagedorn et al. [65H2] gave the qualitative prediction that the energy dependence of the 90° cross section should fall as an exponential in transverse momentum, with a slope about equal to that found experimentally. The model gave no information about the angular distribution. However, other statistical considerations by Ericson [66E 11 in analogy with low-energy nuclear reactions, lead to the prediction that the angular distribution at fixed energy’ should show a characteristic structure in the form of fluctuations, arising from the random nature of the partial wave scattering amplitude in a statistical model. Such fluctuations have been observed in nuclear reactions at 10 MeV, but failed to materialize in pp elastic scattering at 16.9 GeV/c incident momentum in the 67°< 0~I~ < 90° range [66A1, 73A6]. Frautschi [72F 11 rediscussed the Ericson fluctuations suggesting that a necessary condition for their generation is the presence of direct channel resonances. Since the pp system is exotic and either does not have resonances or has non-dominant ones, the experiments performed do not settle the problem of statistical fluctuations, which have to be looked for in other scattering processes. It may be that some of the wiggles observed in large angle irp elastic scattering have this origin. 10.3. Regge-pole models
It was hoped that Regge-pole theories would provide a dynamical basis for strong interactions and, in particular, for hadron—hadron elastic scattering. Such a goal has not been reached, but a considerable amount of useful phenomenology based on Regge poles has emerged. One of the
220
G. Giacomelli~Total cross sections and elastic scattering at high energies
irp -. EXTRAPOLATION OF 5 POLE
2t
REGGE EXCHANGE MODEL
I0
~L*B6
GeV
02
~2
U
IO2[~~3~ 02
t Fig. 10.2. Extrapolations of Barger—Phillips five-pole model compared with IHEP and FNAL irp elastic data [74B131.
main advantages of the Regge-pole approach is that of accommodating easily the general principles of analyticity and symmetry. Practical calculations of total cross sections and of elastic scattering in the diffraction region require the use of two vacuum trajectories, P, P’, and of some other trajectories like p, w, A 2, f (see section 2.7). Thus the number of independent adjustable parameters becomes large and one can easily obtain good fits to the data. Some particularly simple situations, from the point of view of Regge theories, are offered by some total cross section differences, charge exchange scattering and backward elastic scattering, where only a small number of trajectories are involved. Regge-poie models have had their ups and downs. From a phenomenological point of view one can certainly use Regge models to fit, correlate and explain many features of the experimental data as we have done several times in this review (see sections 4.4.2, 4.5.5, 4.6, 5.3, 5.5, 6.1, 7.1 and, for instance, the impressive fit of irp elastic scattering data with a five-pole model, fig. 10.2). From a theoretical point of view there are several open problems which range from the question of what is the Pomeron to which Regge cuts and of what type exist [71B4, 73T 11.
G. Giacomelli. Total cross sections and elastic scattering at high energies
221
10.4. Constituent models It is particularly interesting to investigate the consequences on elastic scattering arising from the hypothesis that hadrons are made of a small or large number of substructures (constituents) which can be identified with quarks or partons. Since the constituents are usually assumed to be point-like, their presence, revealed through constituent—constituent collisions, gives rise to small cross sections, which may be visible in hard collisions, for instance in high energy large-angle scattering, that is, scattering with large momentum transfers. The predictions from constituent models are often computed using extreme simplifications; for instance it is often assumed that very large Iti collisions between two hadrons may be considered as arising from the collisions between two constituents, which take place in a way which is independent of the presence of the other constituents (additive quark model). At somewhat smaller It I-values multiple scattering models are used, where one assumes that there are many constituents which shadow one another and thus a single hadron—hadron collision may be the result of a number of collisions inside them. This picture leads to a multiple scattering formalism of the Glauber type and to a connection with optical models. For instance, the higher order diffraction minima of the optical models are generated from the interference between different orders of multiple scatterings, that is from single and double scattering, double and triple scattering, etc. Higher order scattering processes become more important than single scatterings at medium and large momentum transfers. Thus, in this picture one could say that there is no true high momentum transfer elastic scattering, that is, they are not generated from a one-step process, but originate from a series of steps. The additive quark model leads to simple rules, which correlate the energy dependence of two-body cross sections at highs and fixed c.m. scattering angle 0* (for 0* large) with the number n of constituent quarks of the four hadrons participating in the reaction: ~
5(2-E~1n~)J~0*)
1meson
= 2,
~baryon =
(larges,
0*)
(10.6)
3; thus
where ~8f(~*)
for meson—baryon elastic scattering
(10.7)
~loft~*)
for baryon—baryon elastic scattering.
(10.8)
Comparison of the energy dependence predictions at fixed 0 * yield reasonable agreement with pp and ir~pelastic scattering data, fig. 10.3. Attempts to account simultaneously for energy and angular dependences by constituent models have not been as successful until now. 10.5. Shadow scattering models At high energies the elastic-scattering amplitude is mainly imaginary; then elastic scattering arises mainly from the absorption of the incident waves by the many open inelastic channels. Thus elastic scattering is the shadow of inelastic processes and is completely determined by the knowledge of the inelastic collisions. These statements have been formalized at high energies by
222
G. Giacomelli, Total cross sections and elastic scattering at high energies I0~ iv —3
ir~p— -irp
~
0
istL 4~tI ~I
10-4 —5
Ik t4i~~~*.~,4t
~5
~+ ~ I,
IO_6 .‘
7
ir~p-. ,r~p
ii
b
106
-,
4’ •f (~l~t 4.
_3 10
Iv
~
7L~t.I
~ii
,.,—4
~
~.
t•1
~ IO~’1:~
~
cose
- ~
pp-+pp
2
I0_
—-
•
3
-~
cose
pp—pp 1014
~
~4••..~1~4•,
•
~ :
59.
5. 8.
4
iO~ 108 .4-i
~LCB 4.
IS+It •~
~LAB
GeV/c
= 8— 9
0
0
—
-3
2
“.
‘
0
•
~.
:.
• •
2
I..t—I(
~
I
I
-
14.2
•
•
24
~
I
•
•
• • • •
•
.
•
~
‘
0
‘~.
~
~O
~
•
-6 0
0.75
0.50
cose
0.25
• •
24.
‘
‘
.
0_B .00
0
.9.25 10.8 • 12.15
19.2
0
8.IGeV/c
•
15.1
-...
0°
2 (1)
~I0
19—21
.~
4
—
. ..
•
,,,,
.00
•
•
4.25 ~6.9
:
I
0.75
0.50
0.25
0
cose
Fig. 10.3. Comparison of the quark model rule (10.6) for the energy dependence of large-angle elastic scattering. The three graphs on the right show the raw data for ir~pand pp elastic scattering, while the three graphs on the left show the same data multiplied by 58 for ir~pand ~ for pp. One obtains a kind of universal curve for pp, thus proving that eq. (10.8) is well satisfied at most scattering angles. The situation is not as clear for ir~pscattering [74B13]
Van Hove [64V 1] through the overlap function, which describes the contribution of all inelastic final states to the unitary conditions (9.8). There are many specific models, each of which assumes specific forms for the overlap function, either for empirical reasons or for general dynamic reasons. For instance, the multiperipheral models make specific assumptions about the inelastic processes. In a simpler form Krisch [67K21 related elastic scattering to inelastic processes and was able to obtain a very good fit to the largeangle pp elastic scattering data (over 1 2 orders of magnitude, see fig. 10.4) with the formula ~
~c1exp(_~2p~/g~),
(10.9)
where c, and g, are six parameters, j3 is the c.m. proton velocity and Pt is the transverse momentum. 2p~was interpreted The fact that three regions appeared which were exponential in the variable f3
G. Giacomelli, Total cross sections and elastic scattering at high energies 10
-25
223
_______________________________________________________
I
10~26
I
I
I
I
I
I
I
I
I
I
-
2
• ISR (This experiment) 5=2950 GeV o BNL Fbleyetal. GeWc Berkeley Clyde eta!. 24.6 5—7GeWc ~ Argonne Akerlofeta). 5~.i3.40eV/c . CERN Allabyet aL ‘ BNL Cocconi et al.
-
10
-
8—21 0eV/c 10—32 GeY/c
:~ 10~29~~
~
-
~~,,~e_3~45p2Pi2
10~°
‘~E1031 1k4!
L~J
—.
2
-
S io-35~-
-~
V
-
1O36~ -
I
I
I
I
I
I
I
I
I
012345678910111213
~2~2 (0eV/c)2
Fig. 10.4. Plot of da~/dtversus
for high-energy pp elastic scattering data [75Wl 1.
as being the result of diffraction scattering associated with three different types of inelastic processes, namely pion production, kaon production and anti-proton production, which occupy three different spatial regions of decreasing size inside the proton. More recent data on both elastic and inelastic scattering spoiled this simple and attractive picture [75W 1]. 10.6. Optical models Most optical models used in high-energy scattering start with the formula of the impact parameter expansion of the scattering amplitude and assume different expressions for the matter density distribution inside the hadrons. If the interaction is spin independent and the real part of the scattering is small, the impact parameter expansion may be written as [70G 1]
224
G. Giacomelli, Total cross sections and elastic scattering at high energies
g(E*, 0*)
ip*f b{ 1
—
Re a(E*, b)}Jo(p*bO*)db (10.10)
h(E*,
0*)
—
The various optical models differ in the form of a(E*, b). The above formula is usually written for one type of elastic scattering at one energy. In this case the cross section depends on the functional form of a(E*, b) and on two parameters, the opacity (connected to the ratio Gei/Otot) and the radius of interaction. If one wants to describe the energy dependence and to connect the elastic scattering of two hadrons with that of other hadrons, the number of parameters increases considerably. According to the geometrical scaling hypothesis, the energy dependence of the elastic amplitude is contained in R only and the amplitude scales in impact parameter as: g(E,0)zg(b/R).
(10.11)
The grey disc model and the Gaussian one have this property. From geometric scaling it follows that: R2,
0tot
do/dt
0e1
R2,
~
R2,
B
R2
2 (10.12) 51(R~JT7)I [see also eqs. (9.15) and (9.16)1. If one allows the proton radius to grow with energy, then geometric scaling accounts for the growth of 000t’ the essentially constant value of(u 61/a000) and the shrinkage of the forward peak. The combined ISR data are compatible with a In s growth of 2: R R2R~+R2lns (10.13) =
R4IF
R 0
=
0.84 fm,
R1-
0.22
fm.
The expectation of geometrical scaling that [(1/o~00)(dcr/dt)J should be independent of energy is compatible with the tSR data (see fig. 10.5). This implies the inward moving of the diffraction minimum and the increase of the secondary maximum as the energy increases. As stated in section 9, the “matter” density should have essentially a Gaussian distribution. The next step is to find out the reason for this distribution. Wu and Yang [65W11 made the suggestion that the proton matter distribution would be identical to the electromagnetic charge distribution as determined by electron—proton scattering. This means that the differential cross section for pp elastic scattering should be proportional to the fourth power of the proton electromagnetic form factor: ~(pp~pp)~F~
.
(10.14)
Fig. 5.20 shows that this approximation is not adequate to represent the ISR data. Cheng, Walker and Wu [7lC2, 73Cl, 73C2, 74C31,analysing the implications of relativistic quantum field theory for scattering at very high energies, arrived at an optical model description,
G. Giacomelli, Total cross sections and elastic scattering at high energies
225
I0~ 0
-
GEOMETRIC SCALING
2
0 l0~
a. .&
•
496 GeV
•
073 GeV 501 GeV
‘~
l0~
-
S
0
-50
-100
-150
—200
mb(GeV)2
~
I,
-250
Fig. 10.5. Geometrical scaling ofpp elastic scattering data. [(da/dt)/o~
001 versus [at0tItll is predicted to have a universal shape independent of energy. The dip location is predicted to be proportional to 1/a~ and the secondary maximum is predicted to rise like °~ot [74B131.
which they called the “impact picture” of high-energy scattering. The model pictures the nucleon as made of a partially transparent inner core whose size grows with energy and of a “skin”, about 0.2 fm thick, which stays constant. The practical formulation of the model contains a large number of parameters (seventeen) and thus its predictive power is not totally clear. On the other hand, starting from the knowledge of the pp elastic scattering distribution at small angle at the ISR, of the total cross sections measured at Serpukhov (which contained the rising K~ptotal cross sections), they were able to predict correctly the rising of all the other hadron—hadron cross sections, the change in sign of the real parts2.of the forward scattering amplitudes and the dip in pp elastic scattering at tI ~ behaviour 1.4 (GeV/c)of the Cheng—Walker—Wu model corresponds to a black disc picThe asymptotic ture of the proton leading to the following asymptotic predictions (table 10. 1): Oel/Otot
—~ 4,
2s,
b/a -÷ 0.1,
~
(dO/dt)secondary maximum ~ 1.7 ~
10-2.
000
-~
ln
a 000(1rp)/a~0~(pp) -÷1,
(10.15) (10.16)
(da/dt)5..0 The ISR data are still far away from these limits; in fact this model suggests that the asymptotic region starts at much higher energies (fig. 10.6).
226
G. Giacomelli, Total cross sections and elastic scattering at high energies
FACTORIZABLE EIKONAL: CHENG-WALKER -WU
>
13.
-
12.
-ff~, ~
II.
-
to. 0.40
-
DATA ItI0.i 3
102
tO
5000
--i
0.35 > 0.30
-O o~
~
~
~
~ -
0.25 0.20
I
102
5000
0.2C
I
~E~ 0.19
0.15 0.14
102
i03
5000
SLAB (GeVI Fig. 10.6. Predicted energy dependences of B, act/at, and a 0/B for pp scattering for the Cheng—Walker—Wu model.
10.7. Mixed models The previous discussion indicates that it is difficult for a model to give a complete and satisfactory description of all high-energy data. Thus in some models it is assumed that the scattering process arises from the combination of two mechanisms, one of which dominates in a certain energy (or in a certain angular region) and the other in a different one. Several attempts have been made along these lines, with improvements in the fits to the data, but whose meaning is difficult to assess. The situation is somewhat reminiscent of that of nuclear physics: strong interactions are very complicated and each model sees some aspects of the truth, but not the whole truth.
G. Giacomelli, Total cross sections and elastic scattering at high energies
227
11. Conclusions
The basic conclusions concerning total cross sections and elastic scattering at high energies have already been sketched in fig. 1. 1 and discussed in the Introduction. One may ask how believable is the picture which emerges from the existing data and from our theoretical prejudices, and also if it applies to all hadron—hadron collisions. The answers to these questions require experiments at even higher energies and precise experiments with short-lived hadrons. Assuming that the conclusions of fig. 1 lb are correct, one may ask the following questions: where does the rise of 000t come from (both experimentally and theoretically)? What are its deep theoretical consequences? What are the implications (or the sources) of the optical-type picture which emerges from the study of elastic scattering? Many more features or problems are not summarized in the graphs of fig. 1 1. Some of these may be classified as details, though they could turn out to be fundamental. For instance, one should answer experimentally the question of whether spin effects are really negligible above, say, 100 GeV. The same is true for the real parts at t = 0. Other problems are usually classified as more fundamental. For instance, very little is known about the high energy large-angle elastic cross sections. It is here that one would think of observing effects arising from possible substructures of the hadrons. Present experimental information does not necessarily require substructures, but as already stated, the information available is not really adequate to reach any final conclusion. In summary: the recently commissioned high energy accelerators have opened up a new energy region. A wealth of new experimental information has become available very quickly, forcing a change in our ideas about the asymptotic region. It would not be surprising if new experimental results would lead to even further modifications. .
.
12. Acknowledgements I would like to gratefully acknowledge the fruitful discussions I had with many colleagues, too numerous to mention. They provided me with clarifications, advice and data before publication. Among those who contributed more directly I would like to mention U. Amaldi, F. Bonaudi, A. Costa, A. Diddens, M. Einhorn, M. Jacob, A. Greene, B. Lee, J. Sanford, G. Venturi and J. Walker. I would like to thank many people on the staff of the University of Bologna, of the tSR at CERN, at Fermilab and all the colleagues of experiment No. 104 at Fermilab and of the first CERN-IHEP Collaboration.
References The list includes references with total cross-section and differential cross-section data above 10 GeV/c. References before 1970 are not complete; in general the list includes articles and only those letters with tables of data. For a more complete list of references see the Landolt—Bärnstein compilation [73C5]. [55G11 R.J. Glauber, Cross sections in deuterium at high energies, Phys. Rev. 100 (1955) 242. [6lFl]
M. Froissart, Asymptotic behaviour and subtractions in the Mandeistam representation, Phys. Rev. 123 (1961) 1053.
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G. Giacomelli, Total cross sections and elastic scattering at high energies
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[72B5] [72B6] [72C1]
[72C2]
[72C3] [72D1] [72D2] [72D3]
[72Fl] [72G!] [72G2]
(7211!] [72L1] [72M1] l72Rl] [72Y1] [73A!]
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