- Email: [email protected]
Proton-antiproton scattering and annihilation into two mesons
Nuclear Physics @ North-Holland
A454 (1986) 429-452 Publishing Company
PROTON-ANTIPROTON SCA’ITERING AND ANNIHILATION INTO TWO MESONS*
M. KOHNO Institute of Theoretical Physics, University of Regensburg, D-8400 Regensburg,
W. Germany
w. WEISE’
Institute
Department of Physics, SUNY, Stony Brook, NY 11794, USA and of Theoretical Physics, University of Regensburg**, D-8400 Regensburg,
W. Germany
Received 12 December 1985 (Revised 31 January 1986) We present a combined semiphenomenological analysis of the low-energy pp annihilation into two mesons and of the pp scattering processes which enter into the description of the initial state before two-meson annihilation. Angular distributions for pp + rrtrT- and pp + K+K- are investigated with the aim of studying the relative importance of quark rearrangement and quark-antiquark annihilation mechanisms. Such simple quark models describe basic features of the pp+ TT+Y reaction, but fail to reproduce details of the pp -) K+K- angular distributions. Possible improvements are discussed.
Abstract:
1. Introduction Low energy proton-antiproton processes are a rich source of information on the dynamics of strong interactions. Now that the LEAR facility at CERN has come into operation i), both the quantity and the quality of pp data are about to reach an entirely new level. At the same time, these developments stimulate increasing theoretical interest in the low energy pp system ‘). The outstanding feature of pp physics is the annihilation into multimeson states ‘). By detailed selective analysis of such annihilation channels, one may hope to gain insight into the complex features of the low energy strong interaction dynamics of quarks and gluons. Recent theoretical analysis of pp annihilation into mesons has been carried out in models based on quark rearrangement 4-6), or quark-antiquark annihilation mechanisms 9), or combinations of both “,‘l ). In particular, such models have been used to study the absorptive part of the NN optical potential 5,7*83’2).As in ref. “), Work supported in part by Bundesministerium fiir Forschung und Technologie, grant MEP 33-REA. in part by USDOE, grant no. DE-AC 02-76 ER 13001, and by Deutsche Forschungsgemeinschaft. ‘* Permanent address. l
’ Supported
429 June
1986
430
our interest
M. Kohno, W. Weise / pp scattering
here will be primarily
in the annihilation pp+ MrMz
into two mesons
(I)
M, and MZ. In view of their kinematic
simplicity,
such processes
are particularly suitable to study details of the annihilation dynamics, such as the relative importance of rearrangement versus annihilation mechanisms at the quark level 9-11,13 ). At the same time, potential model calculations r4) are available for processes of type (1) and should be discussed in comparison with quark model descriptions. An investigation of the pp + rrr+Y and pp+ K+K- processes is of particular interest ‘O,“). The latter channel must necessarily involve quark-antiquark annihilation and strange quark pair creation via qq + sS, so that the ratio a(pp + K+K-)/a(pp + rr+C) sets constraints for the relative weight of quark rearrangement and qq annihilation processes. In ref. ‘l) we have found both these mechanisms to be about equally important at low energy. In the present paper we would like to further extend this investigation by a calculation of pp+ M,M, angular distributions. Our primary aim is to evaluate the capacity and also the limits of simple quark models to describe the relevant pp annihilation mechanisms. An essential part in the description of processes (1) is the proper treatment of pp initial state interactions prior to the annihilation into two mesons. This requires a detailed discussion of the NN optical potential describing pp elastic, inelastic and pp + nii charge exchange channels. Phenomenological NN potentials, with large imaginary parts representing annihilation, have been discussed repeatedly in the literature 2*15-17).We shall have to reanalyse the Nfi optical potential in our context, however, since the description of hadron sizes in the quark model approach to the pp+ M,M2 amplitude has to be matched consistently meter entering in the Nfi scattering potential. We present the results of our pp scattering analysis to follow. We will process, using the mentioned. Final tant, especially in
with the geometric in a separate
size para-
section,
the one
then proceed with the quark model description of the pp+ MlM2 pp wave functions obtained with the optical potential previously state rescatterings in the two-meson M,M, system may be importhe S-wave. We take into account the possible effect in a simple
way, based on the phase
shift analysis
of rrrr scattering
“).
2. Optical potential for NN scattering The wave function
of the initial
pp pair in the process
K'i%(r)@N@I;I,
(1) is of the form (2)
where QN and 0~ are the intrinsic wave functions of the nucleon and anti-nucleon which will be given in terms of valence quarks and antiquarks (see sect. 3). The relative motion of the nucleon and antinucleon at a center-of-mass energy E is
M. Kohnq
described energy,
by qE(r), (am
W. Weise / pp scattering
where r = RN -RR is the distance
satisfies
431
between
their centers.
At low
the equation h2V2 --+V,,+V,-E M 1
c&r)=O,
(3)
)
where M is the nucleon mass. Here VN~ is the Ns strong interaction potential and V,(r) is the Coulomb potential. The complex potential VNfi= U+iW,
optical
(4)
is constructed as follows. The long range properties of its real part U are assumed to be well described by the G-parity transform of a realistic one-boson exchange NN potential, Vo,,(NN)
=
V%E, C *=?r,p,w,...
(5)
as in refs. “-l’). In practice we use the static part of the G-transformed Ueda potential “) at distances r 3 R0 = 1 fm. This potential has P, p-, CT-and o-exchange as its dominant components, with additional small contributions from other weakly coupled mesons. The potential parameters are given in ref. I’). In the inner region (rc R,), we assign iitde significance to OBE and G-parity arguments. This is the domain in which the nucIeon-antinucleon density distributions overlap, and where the relation between the NN and Nfi potentials based on the OBE picture is presumably of little relevance. For r < R. we simply assume a smooth extrapolation by Woods-Saxon forms continuously matched at r = R. with the OBE potentials: U=Re
I&P;=
c
(-F
{
i= *,p.o,...
+
V%&)~(r- &I
2V&2&=Ro) 1 -f-exp (P(r-
(1_8(r_R,)) Rd)
I
,
where Gi is odd for i = T, w, . . . and even for i = a, p, . . . . The parameters p are uniquely determined by the smooth matching conditions at r = R,, in each spinisospin channel; we choose R. = 1.O fm*. The imaginary part W of V&N summarizes the effect of all annihilation channels. Assuming that annihilation takes place when the valence quark and antiquark cores of the nucleon and antinucleon overlap, one arrives at a geometric picture in which the imaginary NR potential is proportional to the convolution of the respective quark and antiquark distributions 12,‘9). For two gaussian distributions with r.m.s. radii 6 = (r2)“2, one obtains: W(r) = W( r = 0) x exp (-z( r/ b)*) ,
(7)
The potential used here for TZ R, is to be regarded as a typical representative of a class of realistic OBE potentials. We have also used the G-parity transformed Bonn potential 43f and found variations in the final results at the few percent level. l
432
M. Kohnq
where the c.m. degree of freedom
W. Weise / pp sca!tering
has been properly
subtracted.
We argue that the
r.m.s. radius b should not be identified with the proton charge radius, which results from a combination of quark core and meson cloud sizes in chiral quark models 20*21), but rather
with the size of the quark core alone,
for which the nucleon
axial form
factor gives an upper limit 2’). We therefore prefer radii b = OS-O.6 fm. For such radii, total, elastic and charge exchange pIs; cross sections are well described with an imaginary potential depth W( r = 0) = -1.5 GeV. As we shall see, excellent fits to pp total, elastic and charge exchange cross sections can be obtained with the slightly modified form
W(p)= - Wol{l +ew ((r- roValt, with a radius ro=0.55 fm closely corresponding to the gaussian radius b, and a surface parameter a = 0.2 fm. This form will be used in the actual calculations. The quality of the fits obtained with this potential are comparable to or better than the ones obtained with the potential of ref. 16). We prefer the potential (6-8) for our purpose since its characteristic range is consistent with the quark model picture to be developed in sect. 3.
2.1. TOTAL,
ELASTIC
AND
CHARGE
EXCHANGE
PI’, CROSS
SECTIONS
Using the N‘i;i optical potential U-tiW of eqs. (6) and (8) together with the Coulomb interaction, we have calculated the total pp cross section o;,,, the elastic cross section a,, = jdL?[d~(pp+pp)/dL?] and the charge exchange cross section ~(pp -+ nii). The results are shown in fig. 1 and compared with data in the energy region Elab < 250 MeV. The energy dependence of the total cross section is mainly determined by the geometric size of the imaginary potential W. We have performed systematic variations of the parameters r, and W. of this potential and found that the best fit over the entire energy region is obtained with r. = 0.55 fm and Wo= 1.2 GeV together with a = 0.2 fm. Increasing the annihilation radius r, can be compensated by decreasing the potential depth Wo. Very small radii (rO< 0.2 fm, as suggested e.g. in the baryon exchange model 14) would require Wo> 10 GeV and an explicit energy dependence of W to reproduce the data. These results are consistent
2.2. ELASTIC
with the earlier
AND
CHARGE
ones in refs. ‘s*‘6).
EXCHANGE
DIFFERENTIAL
CROSS
SECTIONS
Using the preferred parameter set W, = 1.2 GeV, r,= 0.55 fm, results for the differential cross sections da(pg + pp)/dL? and da(pp + mi)/dLI are shown in figs. 2-4 in comparison with selected sets of experimental data. For the peripheral partial waves with I > 5 at low energy and with I> 7 at higher energies, the analytic one-pion exchange Born amplitudes have been used. The Coufomb interaction has been incorporated throughout.
hf. Kohno, W. Weise / pp scattering
433
Fig. 1. Total, elastic and charge exchange pp cross sections as a function of 0 ia~orato~ momentum. The curves show resuhs of a calculation with the N@ pore&al VW% = U 9-W lsee eqs. f6>, (8)). The parameters of the real part U for ~a R, = 1.Q_fmare taken from ref. “1, The parameters of the imaginary. potendai (8) are W, = 1.2 CeV, r, = 0.55 fm and a = 0.2 Frn. J%e Coulomb interaction is included in the calculations; the proton and neutron masses are taken to be equal. The otot9,data are from ref. “). The elwic cross sections are taken from ref. “)+ The charge exchange cross sections o(pB-+ niij are from ref. “1.
Considering the simplicity of the potential l&p3with no explicit energy dependence and on@ a central imaginary potential W, the overaIl agreement with data is quite satisfactory. The results demonstrate that the gross features of elastic and charge exchange p$ scattering can be understood in terms of meson exchange at Iarge distance and a strong annihj~ati~n potential in the region of overlap of the proton and antjproton vaIence quark distributions (see also ref. “).
2.3. REAL
PART OF THE pii FORWARD SCATfERlNC
AMPLITUDE
Recent accurate measurements of pp differential cross sections at extreme forward angles permit to extract, by the interference of Coulomb and strong interaction ampIitudes, the ratio p(B)==ReF(E,B~Q)/Xm
F(E, 8=0),
(91
M. Kohno, W. Weise / pF scattering
434
lo2 ,N
09L cm OLOb
Fig. 2. Forward pp- pp elastic differential cross sections calculated with the NN potential VN~ plus Coulomb potential. The parameters are the same as in fig. 1. The data are taken from ref. 25).
of the real and imaginary parts of the spin-isospin averaged forward pfi scattering amplitude F( E, 8 = 0). Its imaginary part Im F( E, 0) is related to ottotarby the optical theorem. In terms of the singlet amplitude T,,(8) and the triplet amplitudes T,,(8) and T,,( 0) summed over isospins and defined in ref. *‘), we have
Re {T,,(O)+ ~dO)+2~~~(0)~
P=Im{T,,(0)+~oo(O)+2~,~(O)}’ The Coulomb
interference $’
analysis
involves
at the same time the parameter
IT,m+ loo -27xo)12+I~oo(wT,m12 ILw+ Mo)+27-,*(w ’
(10) *“)
(11)
for which a value has to be assumed in order to extract p from the experimental data. In ref. 25), two sets of analysis (with v2= 0 and ~~~0.1) have been given, whereas our potential yields T*= 0.06. In the comparison between our results for p(E) and the data, the small difference in 72 has been taken into account by shifting the data points for 72 = 0 by Ap = -0.05. (See fig. 5.) The standard parameterization of the annihilation potential with W, = 1.2 GeV and r. = 0.55 fm accounts well for the empirical data above plab b 300 MeV/c. However,
the calculated
p(E)
decreases
continuously
below plab s 200 MeV/c.
This
M. Kohno, W. Weise / p/i scattering
Fig. 3. Elastic
pp differential
cross sections.
The curves are obtained from ref. 26).
435
as in fig. 2. The data are taken
is in disagreement with the analysis of ref. 30) based on recent LEAR data which suggests values p = 0 at very low energy. The optical potential alone cannot account for this effect.
2.4. SUMMARY
We have presented a simple NR optical potential which reproduces the main features of pp elastic and charge exchange scattering. The potentials based on a combination of long range boson exchange and an energy independent imaginary (annihilation) potential W with a radius parameter r. = 0.55 fm and a depth W. = 1.2 GeV. The geometry reflects the size of the quark core in the proton or antiproton. The range of the imaginary potential is considerably larger than the range (2&f-’ characteristic of baryon exchange mechanisms 14). The simple N6I potential introduced here serves its purpose in the subsequent treatment of absorptive distortions in specific pp annihilation channels. Further improvements - not discussed here -
436
M. Kohno,
W. Weise / pp scaiiering 0
30"
60'
90"
IZP
cm lx)0 1
@p-En
1.0
05
0
-05
-1.0
cosB,, Fig. 4. Differential obtained
cross sections for the pfi -L nii charge exchange reaction. The calculated curves with the same potential parameters as in fig. 1. The data are from ref. z6),
are
700
500 P,,,lMeVicl
Fig. 5. Ratio of real to imaginary part of the forward pp scattering amplitude as a function of energy. The curve is obtained using the potential parameters as in fig. 1. The extracted data are from ref. “) (dots) corrected for 17’ = 0.06 instead of Q’ = 0, and from ref. 30) (squares).
M. Kohno,
can be made introducing
by adding additional
of development, might become the tensor
spin-orbit
and tensor
free parameters.
an explicit necessary
interaction
treatment
terms to W, but at the expense
It should
partial
of
also be noted that at a later stage
of coupled
in a more refined
in higher
437
W. Weise J pp scattering
NN c, A R, Nd and Ad channels
treatment
*), especially
for the role of
waves of the pp + M1M2 process.
3. Quark model for pp annihilation
into two mesons
Having discussed the ingredients relevant for the treatment of pp distortions before annihilation, we now turn to the description of the annihilation process pp + (meson), + (meson), itself. The conventional theory of the NN + 7r+F reaction based on the baryon exchange mechanism has been quite successful r4). However, the results depend very strongly on cutoffs associated with meson-baryon vertex factors. Moreover, the characterisic range AK’ -0.2 fm of the baryon exchange process appears to be unnatural, given the fact that baryon and antibaryon sizes are much larger than that. It is therefore desirable to have a description in terms of basic quark degrees of freedom, although it is clear that such a description can at present not be regarded as complete, given the complexities of non-perturbative QCD. In a quark model description of pp annihilation into two mesons, at least one quark-antiquark pair is annihilated. The two basic quark line diagrams are shown in fig. 6. We refer to them as the single pair annihilation plus rearrangement process (R-(rearrangement) process, fig. 6a) and annihilation plus pair creation process (A-(annihilation) process, fig. 6b). As mentioned before, there exists an extensive discussion in the literature as to the relative importance of the R- and A-processes. In the following, our basic framework will be the non-relativistic (constituent) quark model. This model permits a systematic separation of intrinsic and center-of-mass variables
in the description
of process
M,
L
;u
M,
n II fn Xl ------
I_..-
-
--_
-----
~
-
N
-
i
(a)
Fig. 6. Quark
M,
M,
----i-
(1).
-1
I-
----
L
--
-----
-
---
1
---___
-11 N
N
(b)
line diagrams for pp + two mesons: (a) single pair annihilation plus rearrangement (R-process) (b) annihilation plus creation process (A-process).
process
M. Kohno, W. Weise / pp scattering
438 3.1. CROSS
SECTION
AND
WAVE
FUNCTIONS
The cross section for pfi+M,M* c.m. frame as:
with two mesons M, and MZ, is given in the
(12)
(PI+P2)‘(E,+E,-Ei)I(~‘,IVIEFl)12,
Here v,, is the relative velocity of the proton and antiproton in the initial state and V is a transition potential, to be specified in the following, taken between the initial and final states I$i) and II&). The initial state wave function (see eq. (2)) is ~i(l, 2,, . . , 6) = (PC& - hd%(L
2,3)@i(4,5,6)
(13)
,
with the NN relative wave function Q as given in the previous section and RN= f( rl + r, + r3), RR = $( r, + r, + r6). The quarks in the nucleon are labelled by (1,2,3), the antiquarks in the antinucleon by (4,5,6). The final state wave function is: 9’x l’, 2’, 5’, 6’) = cpf(&, -Rl,,)~,,(r;-r;)~,,,(r;-rk).
(14)
Here pf describes the relative motion of the mesons M, and MZ, with
J&l,= and constituent wave state
m,r: + m5r;
RM,=
’
m,+m,
m2ri + m6rk
(15)
’
m2+m6
quark masses mi. The two mesons are assumed to be in a plane
rpf(&, -&J
= exp (& - &,
+ ip2 - 4~~) ,
(1%’
The additional effect of the M,M2 final state interaction is introduced by a factor &j eiSOin the rr S-wave. Throughout the paper we use fi = 0.35 and 6, = 50”. The intrinsic quark wave functions of the proton (c&), the antiproton (@s) and the mesons (#J& are assumed to be given by OSharmonic oscihator orbits. Their spin and flavour parts &t&, have the standard SU(6) structure. Combined with color, their explicit forms are as follows: @N(l, 2~3, = 4
Eiik14iqj9k>ol,,X~~~r(N)
x(fi&)-3’2exp
n=l
- i
(r,--RN)’
2b&
1 ,
@d4,5,6) =~E~kIQiiij~~k)colorXs~(N) / 1, x (v?nb$)-3’2
[
exp
-~.+-RN)~
I
2bk
(174 (17b)
and &&r) = ~~ijlqie)colorX~~~*~(M)(rbL)-3’4
exp (-r2/2bZ,)
.
(18)
M. Kohno,
In this parametrization
439
W. Weise / pp scattering
the proton
(antiproton)
and meson
and
= d$bM ,
r.m.s. radii are given
by (r&)“2 = bN
(r$1’2
(19)
respectively.
3.2. TRANSITION
POTENTIALS
As a model for the effective transition potential V in eq. (12) we consider the diagrams, figs. 7(a) and (b). We use the gluon exchange interaction as a guideline to develop the characteristic spin-flavour dependence of the transition potential, although with effective parameters not necessarily related to a perturbative picture, since the basic mechanisms must clearly be thought of as non-perturbative ones. The gluon exchange model adopted here is less restrictive than the ‘PO model in which quark and antiquark are annihilated into vacuum quantum numbers. In the 3P0 model, the transition matrix element is simply given by the overlap of the quark and antiquark wave functions, the qq pair being in a 3P,, state. Gluon exchange offers instead a different spin-space structure of (he transition potential. n
m
J
hH
1
2
m
(Cl1
Fig. 7. Generating interactions is followed by effective gluon
3
n
3
L
1
2
1
j
k
I
(b)
for the R-process (a) and the A-process (b) of fig. 6. The qq annihilation exchange (wavy line). The letters i, j, . refer to color indices [eq. (23)].
The interaction derived from figs. 7(a) and (b) is obtained by considering the leading terms in the non-relativistic expansion of the corresponding Feynman diagrams. These processes involve the effective gluon propagator [ qz - q2 - m,$ + i&l-’ once or twice. Here q. and q are the energy and momentum transfer carried by the gluon, and md is an effective gluon mass. This gluon mass reflects the fact that the gluon modes are localized within the confinement domain, so that their propagation length m&’ is restricted to short distances. The dominant low-lying gluon modes 31) have their energies in the range around 1 GeV, the value we adopt as a typical order of magnitude for mG. Now, the energy transfer q. is determined by the energy of the annihilating qq pair, i.e. q,,a 2m, where m, - 300 MeV is the (u- or d-) quark constituent mass. The typical momentum transfers 1q(deduced from
440
M. Kohno, W. W&e / pp scattering
the kinematics of pp+ MIMZ processes under consideration For mG- I GeV, it follows that ?&%+q9;:-q2.
are of the same order.
WV
We therefore replace the gluon propagators in figs. 7(a) and (b) approximately by -m;=. In the potential for the R-process, fig. 7(a), an additional factor $m, comes from the reduction of the quark spinor matrix element. The results of the reduction for R-process (fig. 7(a)) and A-process (fig* 7(b)) transition potentials are the foBowing*:
x{u(l)x(o(2)xo(34))io(2)x(u(1)~~r(34))}+~~~)~
(22)
I-Iere r&= ri -q and the coordinate numberings refer to notations in fig. 7**, The r;k and CA in eqs. (ZI), (22) are operators in color space given by their matrix element5 (jfC&[mfi))=
$h,q:h”,,,
trliiPt>icAIri~~ckt~=~~~ihfihn)nnrh”~,
W4
(23b)
where the color indices 8,j, . . . are again specified in fig. 7 and the h” are standard SU(3) matrices. For VA in eq. (22) only the dominant terms proportional to the momentum transfer between particles 3 and 4 (see fig. 7(b)) have been kept. The kinematics of the A-process justifies a static approximation for the intermediate quark propagator. This yields the V,,(exp (-~r~J/r~~) factor in eq. (22), where JA= m,, the strange constituent mass for a final KfK- state, and y = pnqfor non-strange two-meson finat states. The second term in V, of eq. (21) has a single spin operator acting between the quark and antiquark, Its structure is the same as the one of the ‘PO model. In fact this term contributes in initial pp odd states for the rr+7~- channel, whereas the first term in V, contributes in even states. The strengths of the transition potentials are determined by effective quark-gluon coupling constants (Ye and CY*.They need not necessarily be equal, but the final * These V, and VA differ from the ones in ref. I’) by the refined approa& to the eixective gh~on pmpagator and by thr: additional term (see also ref. 42)f ~r~~~~i~na~ to oj23f . V, in abe present work. ** In taking matrix elements of the spin operators w between Pat& spinors of qzsark and antiquark, remember that they are related by charge conjugation.
M. Kuh?q
w. W&e
1@
.mm?ring
441
results will show them to be of comparable magnitude. To obtain a rough estimate for artJm&, one can use the transition potential V, in the quark model description of the decays p + nn and A + n-N. From such an analysis one finds (YR/rr& = 0.4 fm2. Further evaluation of the matrix elements (ly,] V,,,l W;) leads to the results summarized in appendix A. Note that the intrinsic quark distributions of the proton, antiproton and mesons introduce form factors as functions of the squared meson c.m. momentum p’, which depends on the mass of the mesons involved. Light mesons carry a larger m~mentnm 1~~1,so that their matrix elements are reduced as compared to the ones for heavier mesons. This difference becomes signi~~nt already in the comparison of the pp+ rrt rr- and pp+ K’K- matrix elements of V.&. For pp at rest, the pion momentum in the former process is jpNIi= 4.7 fm-‘, while the kaon momentum in the latter one is IpNII= 4 fm-‘. As a consequence, for typical hadronic radii ( r2)“2 - f fm*, matrix elements (~T+T-I V,lpp) are suppressed by roughly a factor of two as compared to (K”K-IVA(pp). Furthermore, the form factors strongly favour matrix elements with two vector mesons in the final state, such as @“p-l tkjpfs) as compared to {~T+T-\V,Ipp). For example, the p meson c.m. momentum for pfi+pcpp at rest is refativety smalf, /p%$\=2.7 fm-‘. For hadron sizes as previously mentioned, the formfactor effect enhances the p+p- process by roughly a factor of five over the q+rr”‘ process. Thus hadron sizes evidently play an important role in the rearrangement and annihilation dynamics so that they wilf have to be discussed in some detail. We shall return to this question in sect. 4.1.
3.3. SELECTION
RULES
Parity, isospin and G-parity ~onse~a~~on impose selection rules on the R-process and A-process matrix elements. For two non-strange mesons in the final state, G-parity requires that St-f~L,+S,i-I,+S,+I,=even,
(24)
where S, I and L denote the spin, isospin and relative orbital angular momentum of the piij system; Sl,2 and I,,z are the spin and isospin of meson 1 or 2 in the final state. We note that, for the R-process, this setection rule is automat~caIly satisfied when one sums up all possible configurations of diagrams shown in fig. 8. Alternatively, the last two diagrams fig. gb are included if G-parity conservation, eq. (241, is imposed on the matrix elements of the first two diagrams, frg. 8a, together with an obvious factor of two. Consider now the R-process matrix elements (?~+yr-l VR/pp) with V, of eq. (21). The first term proportional to [o(l) x0(23)] *V,2S3(t12) contributes oniy in spin Note that the strong interaction radii should radii; see the discussion in sect. 4.1.
not be identified
naively with the corresponding
charge
442
M. Kohno,
W. Weise / pp scattering
(of
lttf
Fig. 8. Diagrams contributing to the R-process far p@+ M,M,.
and isospin triplet channels of the pp system; it acts in the 33S, 33D, _“. protonantiproton partial waves (which we denote in general by 21+‘*2s+‘Lj). On the other hand, the second term of V,, the one propo~ional to ~3(~,*)~(23) * V,, operates in the spin-triplet and isospin-singlet *3P0, 13P2,. . . channels only. The annihilation potential V, of eq. (22) has nonvanishing matrix elements only in spin and isospin triplet states. For pij+ ?T+T-, it therefore interferes with the first term of the rearrangement potential V, (eq. (21)) in the corresponding L-even partial waves, and in the ‘?S channel in particular. The pp-+ rr+C P-wave channels have no contributions from VA in this model. For pp+ K*K-, there are no C-parity restrictions. However, due to the specific structure of VA in our model, the isospin singlet channels are suppressed, so that contributions come only from the spin-isospin triplet channels 33S, 33P,. . . etc. 4. Results We concentrate here primarily on differential cross sections for pp+ r%- and pij-., K”‘K- at low energies (ptab<600 MeV/c). The corresponding total cross sections have been investigated already in ref. If) using the same modet, but several refinements and details will be discussed in the following, Furthermore, we shall comment on channels with one or two vector mesons in the final state, in particular the pp -+ r-p- channel for which detailed experimental information is now becoming available.
4.1. INPUT PARAMETERS
Apart from the pp optical potential state, the following parameters enter (a) the effective coupling strength %, eq. (21); (b) the effective coupting strength Vn, eq. (22);
describing the distortion effects in the initial in the pp- M,M2 process: cyR/m& of the R-process transition potential aY,jrn& of the A-process transition potential
hf. Kohno, W. W&se / pj7 scattering
443
the size parameters bN(ff~) and bM related to the intrinsic quark dist~butions of proton (antiproton) and outgoing mesons, respectively (see eqs. (17-19)), and the constituent quark masses, @t”,dand m,. The size parameter of proton and antiproton is taken to be (c)
& = bi;i=:(ri)“* = 0.55 fm .
(25)
This should be interpreted as the quark core radius* for which an analysis of the nucleon axial form factor *O)gives (0,5-4.6) fm as an upper limit, It is smaller than the proton charge radius which is influenced more strongly by the charged pion cloud surrounding the core ‘9X20).Note that the same size as in eq. (25) has been used consistently in the imaginary part of the pp opticat potential, eqs. (7) and (8). The quark core radii of the mesons are uncertain. For the pion, a large part of its empirical charge radius of about 0.7 fm can be understood simply in terms of the p-meson dominance picture, in which the form factor is (1 - q2/ mt)-’ with an otherwise pointlike pion. The intrinsic pion radius is therefore expected to be considerably smalier than 0.7 fm. Detailed models of the pion structure 32*33)give an upper limit of about 0.4 fm to its intrinsic size, the value which we adopt:
The empirical kaon charge radius ““) is estimated to be around 0.5 fm. This is an upper limit for the intrinsic kaon size. We use b,= b, with eq. (26). We use constituent quark masses m”.d= 330 MeV and m, = 530 MeV. The effective strength (LY,Jm&) of the A-process potential is determined by the pp -, K’K- cross section since the R-process does not contribute to strangeness-producing reactions. As in ref. rl> we obtain: ~~~/rn&==O.l5frn~.
(27)
(For exampte, with an effective gtuon mass m, = 800 MeV, this corresponds to iTA= 2.5). The pp + ?rf-i7- reaction is described as a ~ombjnation of the A- and R-processes. Given LY,J~& one finds that values around curJm&=0.25 fm’,
Gw
for the effective coupling strength of the rearrangement process yield a cross section a(pp+ 7rfn-) as shown in fig. 9*. The absolute values of p@+ M,M2 cross sections depend on the proton (antiproton) size parameter b,. Smaller proton and antiproton core radii tend to increase * The new KEK data 35f*aiso shown in fig. 9, are syst~rnat~~~lyhigher than the older ones used for the determination of ctR and LY~.A fit to the KEK data would require aA/& = 0.18 fm” and a,/& = 0.35 fm2.
200
a
I
200
^X^
LW
600
4ao
PLob [MN/c
1
1
ml
600 PLab [MeVlcl
1
(b)
la1
Fig. 9. Total cross sections a(ppZT’Y) and v(pp+ K’K-) as a function of the antiproton lab momentum. The data are taken from the compilation in ref. 23)” The open squares refer to the recent data at KEK 35). The curves are the results of calculations using parameters as specified in eqs. (25), (26) and eqs. (27), (28).
the calculated two meson production cross sections and would require a corresponding reduction of the coupling strengths, eqs. (27,28). The cross sections are considerably less sensitive to the meson size parameters bM as long as these are varied within *20% of their values, eq. (26). Large pion r.m.s. radii enhance the rearrangement process relative to the annihilation process. Very small pion radii around (rtjl/‘= 0.1 fm tend to suppress the R-process with respect to the A-process by more than a factor of three. However, such small values of (& of tr(pp+ r7r) by a l’arger factor which would unreasonably large coupling strength EYJ m&.
4.2. DISTCIRTION
EFFECTS
IN THE
also reduce the overall magnitude have to be compensated by an
pP-CHANNEL
The results just mentioned are obtained with the pp optical potential derived in chapter 2. Distortion effects are large and depend strongly on the pp partial wave involved. A characteristic comparison with plane wave Born approximation results for pp+ GT~T-” and pp + K*K- at Sa,, = 500 MeV/c are shown in table 1. One finds that the S-wave cantribution to the cross sections at this energy are reduced by a factor around 20. The reduction of the P-wave contributions is considerably less as seen in table 1. It is due to a combination of the pi-, initial state F-wave distortion and the strong S-wave interaction in the ~T+?T- final state.
M. Kohno, W. Weise / pp scattering
445
TABLE 1 Comparison of cross sections (in mb) for pp + ~+rand pp + K+Kplane wave. (PWBA) and distored wave (DWBA) Born approximation chapter 2
o(pp+
a(pp + r+?r-)/mb channel
R-process
at p,.,, = 500 MeV/c calculated with the pp optical potential
A-process
in of
K+K-)/mb (A-process.
R+A
channel
0.12 0.23 0.06
‘9, =P UP0z
0.11 0.00 0.04
2.03 1.52 0.05
9, =P 0 33P2
2.85 0.01 0.07
only)
DWBA 0.52 0.23 0.06
0.26 0.0 0.0
13.63 1.52 0.05
5.14 0.0 0.0
PWBA
The cross sections are given for the leading partial waves. Also shown are the separate contributions from the R- and A-processes for pp+ r+Y. The input parameters are specified in sect. 4.1.
Effects of the attractive Coulomb interaction in the pp channel are small in our cross section with and without region of interest. Calculations of the pp+ ntY the Coulomb potential Vc show in fact that the difference is only at the percent level.
4.3. DIFFERENTIAL
CROSS
SECTIONS
FOR
pp+~~+sr-
Results of calculated differential cross sections da(pp+ r+n-)/dfl at several low energies are compared with experimental data 35) in fig. 10. The angle OoM is defined as the angle between the incoming proton and outgoing 7~+ momentum. The low energy data show a pronounced
peak at t?cM = 0. In our model, this structure
arises
from the combined contributions of S- and P-wave amplitudes. In the S-wave, the first term of V, (eq. (21)) interferes destructively with the annihilation potential VA (eq. (22)). The second term of V, contributes in the P-wave only. It is seen from fig. 10 that the combination of V, and VA, with parameters given in eqs. (25,26) and eqs. (27), (28), accounts reasonably well for gross features of the observed pp+ r*Y angular distribution, although a stronger forward peaking would still be required in the model. In any case, it is clear from the previous discussion that both R- and A-processes contribute significantly to da(pp + rr+Y)/dfl.
4.4. DIFFERENTIAL
CROSS
The pp+ K+K- process The calculated differential
SECTIONS
FOR
pP+K+K-
is determined by the A-process alone, as stated earlier. cross section at low energies arises from a dominant
S-wave amplitude with a smaller P-wave contribution (see also table 1). The rise of the angular distribution at forward angles is reproduced (see fig. 11). However, the
h4. Kohnff. W Wefse / pij sentfeting
0
0
-10
-05 co* o,,
-10 -0.5 cosecn
lb1
(al
I
OS
L
0
(cl
-05 cos
e,,
J -10
Fig. SO. ~iffereutiai cross sections for pp+ T+T- at low energies, compared to the data measured at KEK35). The results obtained using the input parameters specified in eqs. (25), (26) and eqs. (27), (28) are normalized to the experiment 35)by a factor of 1.3 (390 MeV/cf, 2.0 (490 MeV/c) and 2.0 (590 MeV/c) to compensate for the difference of the KEK data from the old ones shown in fig. 9.
model fails to account for the steep backward peaking which is a characteristic feature of the pp + K’K-‘ reaction even at low energy 35). Similar observations are made at higher energies 36). The failure to reproduce the backward peaking indicates that an important piece is still missing in the description of the annihilation mechanism via the potential VA, and that this piece is specific to strangeness-producing processes. One possible source might be a two-step mechanism via pij + AA + K’K- 37).The pp + Ax interac-
t pjF--K+K590 M&/c
IO
0
05
-05
-10 cos o,,
10
05
0
-0.5 -10 CO5O~rn
(al (bl Fig. 11. Differential cross sections for pp + K+K- at low energies, compared to the data measured at KEK 35). The curves are obtained using the input parameters specified in eqs. (251, (27) together with (rjy = 0.4 fm.
tion has a strong tensor force component through kaon exchange with a corresponding mixing of higher partial waves. 4.5. THE RATIO aipp-tK*K-)/of~ji-,~~rr-)
IN S- AND P-WAVES
relative yield of K+K- and r+rr- following pp annihilation at rest in S- and P-waves has been an interesting point of recent discussion. The older bubble chamber data correspond primarily to S-wave annihilation and give 3*38): The
T(pp+K+K-) T(PP+
with 38):
T(pp-,
K+K-)
rtpp+
T+r-)
~0.5
I S-wave
_$ I liquid
r+r-)
and
'
T(pp+ K+K-) f&q+
p+d
= 0.01 . ! P-wave
New LEAR data taken with a gas target 39) are supposedly representative annihilation with the result: f (pl? + K+K-) T(PP + r+r-)
=0.15*0.03. pss
(291
of P-wave
(30)
It is instructive to evaluate corresponding numbers in our model. We do this by choosing a very low energy (plab = IO0 MeV/c) for comparison. While it should be kept in mind that the treatment of pfi distortions becomes less reliable at these energies, the magnitude of p = Re F/Im F being too large (see fig. 5), one expects that uncertainties in the overall distortion factors cancel out to a large extent when taking the ratios, eqs. (29), (30).
M. Kohno, W. Weise / pp scattering
448
Our results are summarized in table 2: our values for the S- and P-wave ratios are 0.63 and 0.01, respectively. Also shown in table 2 is the cross section for pp+ nor0 at plab= 100 MeV/ c. This is a direct measure of the P-wave R-type process since the S-wave is forbidden by selection rules. Our result for the ratio u(pp + rr’rr’)/~(pp?r+7~--)is 0.27 at PI=,,= 100 MeV/c and 0.36 at pi&,= 500 MeV/c. The rather large P-wave ratio is due to the cancellation between R- and A-processes in the S-wave.
4.6. COMMENTS
ON
OTHER
TWO-MESON
ANNIHILATION
CHANNELS
In ref. 11)we have reported results obtained within the present model for branching ratios of pp+M,M, channels with one or two vector mesons in the final state. Generally, the rearrangement mechanism (R-process) favours vector mesons in the final state. This follows from the discussion of transition form factors in sect. 3.2; vector mesons are generated with low momenta, so the form factors (see also appendix A) do not cut down their amplitudes. Of course, these statements depend on the unknown sizes of vector mesons and are therefore subject to uncertainties. Here we would like to point out a problem related to the pp-, 7rcp- channel which is faced by any model based on a combination of quark-antiquark annihilation and rearrangement mechanisms, including the present one. Empi~cally, the isospin f = 1 contribution to pp + rip- was found long ago to be negligibly small in S-waves as compared to the I = 0 contribution 40). Recently, the analysis of LEAR data 39) has shown that I = 1 P-wave pp+ -rr+.p- annihilation is also suppressed. Our results for pp+ rr+p- and pp+ rope cross sections at low energy are shown in table 3. In S-waves the I= 0 component dominates the I = 1 component by a
TABLE
2
Cross sections at piab= 100 MeV/ e for pp + TUTand pp --f K+K- in mb for s- and p-wave. channels, given separately for R- and A-processes and with the combined amplitude (R+A) R-process
Channel pp+
7r+n-
9, 13 PO 13Pz
pp --*K+K-
9 33PA 3)P,
pp + $ro*a
335, 13Pa ‘3Pz
8.03 2.30 0.15
1.14 0.07
A-process
R+A
3.16 0.0 0.0
2.06 2.30 0.15
1.28 0.00 0.03
1.28 0.00 0.03
0.0 0.0
1.14 0.07
The input parameters are specilied in eqs. (ZS), (26) and differences for no, n* are taken into account.
(27),
(28). The mass
M. Kohno, U? Weise / pp scattering TABLE
Contributions
of various
pp partial
cr(pp+ Channel 3’S I$ 1 ‘IP, =P Xjp’z ‘)D,
R-process 4.81 48.06 0.05 1.40 0.01 0.27
waves
3
to the pp+ rrn+p- and pp+ 100 MeV/ c
rr+p-)/mb
A-process 2.58 24.85 0.05 0.10 0.07 0.14
449
rrOpo cross
a(ppR+A 0.70 5.99 0.08 1.25 0.13 0.03
R-process
sections
at plab=
r’p’)/rnb A-process
R+A
47.92 0.05
24.85 0.05
5.95 0.08
0.07
0.14
0.03
Other partial waves not shown (e.g. “So) are forbidden by selection those specified in eqs. (25), (26) with b, = b, and eqs. (27), (28).
rules. The input parameters
are
large factor in both R- and A-type processes, compatible with the experimental observation and consistent with ref. 41). In fact, the A-process suppresses all but the 13S1 and 3’S,, partial waves, the 3S, state being dominant over the ‘S,, state by about an order of magnitude. However, the R-process, via the second term of V, (eq. (21)), the one proportional to ~(23) * VI, contributes a relatively large P-wave amplitude in the 33P1 channel, which is very sensitive to the S-wave np final state interaction. Now, one could think of mechanisms which suppress this term dynamically. However, we recall that this part of V, is necessary to obtain the forward peaking of the pp+ ~T+T- differential cross section: without it, the T+~T- angular distribution would be flat. Hence there are still important ingredients missing in the attempt to achieve a consistent description of all pp+ M,M2 channels.
5. Summary and conclusions In the first part of this paper, we have introduced a minimal NN optical potential capable of reproducing pp elastic and charge exchange differential cross sections together with total cross sections in the low energy region up to about pIa,,= 600 MeV/c. The basic potential parameters are the proton and antiproton sizes and the strength of the imaginary potential, which is proportional to the overlap of the quark and antiquark distributions in the proton and antiproton. Quark core rms radii of 0.55-0.6 fm are found to be optimal, consistent with chiral quark model descriptions of various other hadronic properties. In the second part, we have developed a quark model description of the low energy proton-antiproton annihilation into two mesons. The pp+ 7r+C and pp+ K+K- channels are treated with special emphasis. The model is based on a combination of quark rearrangement (R-process) and quark-antiquark annihilation (Aprocess) mechanisms. In deriving the spin structure of the corresponding effective
450
hf. K~MZO, W Weis
/ pp scatferirtg
interactions, we have been guided by gluon exchange, but with effective coupling strengths and propagators to simulate non-perturbative features. We find both Rand A-processes to be about equally important in the description of pp + 7rh7r- and pj5+ K’K- cross sections and angular distributions. In these caicuiations, the pp optical potential discussed in the first part proves useful in the calculation of important distortion effects in the pIj initial state. Final state interactions in rrrr S-waves have also been found to be important. Apart from rhe improved treatment of distortion effects, this paper differs from our previous investigation I’) by the additional velocity dependent term in eq, (21). This term has also been included in recent work by Henfey et al. 42). As far as the relative importance of this term is concerned, we are in qualitative agreement with their results. The quark model approach with combined R- and A-type mechanisms adequately accounts for the gross behaviour of the pp+ 7r+7~- and pp-+ K+K- cross sections and the T?V- angular distributions, as well as for the K”K^- and 7ret7f’-yield ratios close to threshold in both S- and P-wave channels. At the same time, there are ~ha~~te~sti~ diffi~u~t~es~however, which indicate the need for further refinements. First, the strong backward peaking observed in the pp+ K’X- angular distributions at tow energy cannot be matched by the present model. This backward peaking requires an admixture of partial waves higher than S- and P-waves, but selectively so for strangeness-producing reactions. We have rnent~o~ed the explicit in~arporation of the pp + Ail * K’K- channet as a possibility to mix in D-waves via the kaon exchanges tensor force in the pp-, Ax interaction. Secondly, the empirical data on pp+ rr’p”” indicate a strong suppression of the I = 1 S- and P-wave amplitudes. While this feature can be well understood in S-waves within the present model, a part of the R-process leads to a large pp -+ ~?p- amplitude in the ‘P, channel, contrary to observation. On the other hand, this particular part of the R-type interaction cannot be easily abandon~d~ since it is necessary to reproduce the pp + r+rr- angular distributions. We would Iike to thank Prof. K. Nakamura for correspondence concerning the KEK data. We are grateful to Carl Dover and Rudolph Tegen for many helpful discussions. Appendix A We present here explicit expressions for the matrix elements f lkCjVI ?ui) with the transition potentials V, (eq. (21)) and V, (eq. (22)), which incorporate form factors from the intrinsic quark distributions of the proton, antiprolon and mesons. The relevant wave functions are defined by eqs, (13) to (1X) in the text;
451
M. Kohno, W. Weise / p@ scattering x[(e,-az)
* (3ir/bL-2pM)-a2.
(6ir/bh+5pM/2)1
3(2b2,+ b2,)
’ crdV.d+i)=I d3rdr) [b’(36’9+2b:)a]3’2i Id’r(a,.r) NN -‘2(8bZ,+5bfJPM’
r
’ (3r-z)2 -$-F-S N
- iph2* W-+(1
(A.1)
3r2 N
M
-P)z) ,
(A.21
1
where bN = & is assumed, pM denotes the momentum of the final meson in the c.m. frame, and (p, II) = (4, mu,J for non-strange quark pair creation or (/3, CL)= (m”,dl(m”,d f m,), m,) for strange quark pair creation. The vectors al, a2 and aj involve spin, flavor and color matrix elements: 41=-
z,”f$36(~tIC~t(-i)(u(2) X~(34))I~J 9
q
G
a2=- ~~36(y,ICttu(34)I’I’J 4 a,=4a
9
2 36(‘P&(tr(34)
x (~(25) ~~(5’6’))
+u(~S)X(U(~~)XU(~‘~‘))}[~J
y
(A3)
where one of the possible configurations of diagrams shown in fig. 8 is given as an example. Using eqs. (17), (18) and (23), the color matrix elements are obtained as ( FrI CR1?lii)color = -$ and ( TfI CA TJcolor = hi. References 1) R. KJapisch, Nucl. Phys. A434 (1985) 207c, and references therein 2) A.M. Green and J.A. Niskanen, Int. Rev. of Nucl. Phys. 1 (1984) 569 (World Scientific, Singapore) and references therein and B. French, in High energy physics IV, ed. E.H.S. Burhop 3) R. Armenteros 1969) and H. Stern, Phys. Lett. 21 (1966) 447; 4) H. Rubinstein J. Harte, R. Socoiow and J. Vandermeuien, Nuovo Cim. 49 (1967) 555 and T. Ueda, Nuct. Phys. A364 (1981) 297 5) M. Maruyama and T. Ueda, Phys. Lett. 124B (1983) 121; 6) M. Maruyama M. Maruyama, Prog. Theor. Phys. 69 (1983) 937
(Academic
Press, NY,
452
M. X&no, W. W&se / pp sc~~~~~jng
7) A.M. Green and J.A. Niskanen, Nuci. Phys. A412 (1984) 448 8) S. Furui and A. Faessier, Nucl. Phys. A424 (1984) 525 9) H. Genz, Phys. Rev. D28 (1983) 1094; Phys. Rev. 031 (1985) 1136 10) C.B. Dover and P.M. Fishbane, Nucl. Phys. B244 (1984) 349 11) M. Kohno and W. Weise, Phys. Lett. 152B (1985) 303 12) R.A. Freedman, W.Y.P. Hwang and L. Wiiets, Phys. Rev. D23 (1981) 1103; M.A. Alberg et al., Phys. Rev. D27 (1983) 536 13) U. Hartmann, E. Kiempt and J.G. Kiirnet, Phys. Lett. 1556 (1985) 163 14) B. Moussa~~am, Nucl. Phys. Ad07 (1983) 413; Nucl. Phys. A429 (1984) 429 15) R.A. Bryan and R.J.N. Phillips, Nucl. Phys. BS (1968) 201 16) C.B. Dover and J.M. Richard, Phys. Rev. C21 (1980) 1466 17) T. Ueda, Prog. Theor, Phys. 62 (1979) 1670; 63 (1980) I9S 18) B. Hyams er at, Nucf. Phys. 864 (1973) 134 19) F. Myhrer and R. Tegen, Univ. of South Carolina preprint (1985) 20) A.W. Thomas, Adv. Nucl. Phys. 13 (1984) 1; G.A. Millet, Int. Rev. of Nucl. Phys. 1 (1984) 189 (World Scientific, Singapore) 21) W. Weise, Int. Rev. of Nucl. Phys. 1 (1984) 57 (World Scientific, Singapore); E. Oset, R. Tegen and W. Weise, Nucl. Phys. Ad26 (1984) 456; R. Tegen and W. Weise, Z. Phys. A314 (1983) 357 22j A.S. Clough et nt, Phys. Lett. 146B (1984) 299 23) V. Flaminio, W.G. Moorhead, D.R.O. Morrison and N. Rivoire, Compilation of cross sections: p and p induced reactions, CERN-HERA 84-01 (1984) 24) R.P. Hamilton ef a& Phys. Rev. Lett. 44 (1980) 1182 25) V. Ashford er al., Phys. Rev. Lett. 54 (1985) 518 26) K. Nakamura et ni, Phys. Rev. Lett. 53 (1984) 885 and private communication 27) T. Mizutani, F.‘Myhrer and R. Tegen, Phys. Rev. D32 (1985) 1663 281 M.L. Goldberger and K.M. Watson, Collision theory (J. Wiley, 1964) 29) C. Bourrely er al, Nucl. Phys. B77 (1974f 386; M. Lacombe et al., Phys. Lett. 124B (1983) 443 30) W. Bruckner et al., Phys. Lett. 158B (1985) IS0 31) R.L. Jaffe and K. Johnsan, Phys. I&t. SOB (1976) 201; C.E. Carbon er nL, Phys. Rev. D27 (1983) 1556 32) S.J. Brodsky, Springer tracts in Modern Physics 100 (1982) 81 33) V. Bernard, R. Brockmann, M. Schaden, W. Weise and E. Werner, Nucl. Phys. A412 (1984) 349; V. Bernard, R. Brockmann and W. Weise, Nucf. Phys. A440 (1985) 605; W. We&e, Nucl. Phys. A434 (1985) 68% 34) E.B. Dally er nf., Phys. Rev. Lett. 45 (1980) 232 35) T. Tanimori, Ph.D. thesis (University of Tokyo), UT-HE-85/03 (1985); T. Tanimori ef ai, Phys. Rev. Lett. 55 (1985) 183.5 36) E. Eisenhandler er al, Nucf. Phys. B96 (1975) 109 37) J.A. Niskanen, Phys. Lett. 1548 (1985) 351 38) S. Devons et af., Phys. Rev. Lett. 27 (1971) 1614; R Bizzarri et nL, Nucl. Phys. 369 (1974) 307 39) ASTERIX Collaboration, preprint CERN-EP/M-I 16 40) M. Foster et at, Nucl. Phys. B6 (1968) 107 41) J.A. Niskanen and F. Myhrer, Phys. Len. i57B (1985) 247 42) E.M. Henley, M. Oka and J. Vergados, Phys. Lett. 166B (1986) 274 43) K. Erkelenz, Phys. Reports 13 (1974) 191; K. Ho&de and R. Machleidt, Nuci. Phys. A256 (1976) 479
A454 (1986) 429-452 Publishing Company
PROTON-ANTIPROTON SCA’ITERING AND ANNIHILATION INTO TWO MESONS*
M. KOHNO Institute of Theoretical Physics, University of Regensburg, D-8400 Regensburg,
W. Germany
w. WEISE’
Institute
Department of Physics, SUNY, Stony Brook, NY 11794, USA and of Theoretical Physics, University of Regensburg**, D-8400 Regensburg,
W. Germany
Received 12 December 1985 (Revised 31 January 1986) We present a combined semiphenomenological analysis of the low-energy pp annihilation into two mesons and of the pp scattering processes which enter into the description of the initial state before two-meson annihilation. Angular distributions for pp + rrtrT- and pp + K+K- are investigated with the aim of studying the relative importance of quark rearrangement and quark-antiquark annihilation mechanisms. Such simple quark models describe basic features of the pp+ TT+Y reaction, but fail to reproduce details of the pp -) K+K- angular distributions. Possible improvements are discussed.
Abstract:
1. Introduction Low energy proton-antiproton processes are a rich source of information on the dynamics of strong interactions. Now that the LEAR facility at CERN has come into operation i), both the quantity and the quality of pp data are about to reach an entirely new level. At the same time, these developments stimulate increasing theoretical interest in the low energy pp system ‘). The outstanding feature of pp physics is the annihilation into multimeson states ‘). By detailed selective analysis of such annihilation channels, one may hope to gain insight into the complex features of the low energy strong interaction dynamics of quarks and gluons. Recent theoretical analysis of pp annihilation into mesons has been carried out in models based on quark rearrangement 4-6), or quark-antiquark annihilation mechanisms 9), or combinations of both “,‘l ). In particular, such models have been used to study the absorptive part of the NN optical potential 5,7*83’2).As in ref. “), Work supported in part by Bundesministerium fiir Forschung und Technologie, grant MEP 33-REA. in part by USDOE, grant no. DE-AC 02-76 ER 13001, and by Deutsche Forschungsgemeinschaft. ‘* Permanent address. l
’ Supported
429 June
1986
430
our interest
M. Kohno, W. Weise / pp scattering
here will be primarily
in the annihilation pp+ MrMz
into two mesons
(I)
M, and MZ. In view of their kinematic
simplicity,
such processes
are particularly suitable to study details of the annihilation dynamics, such as the relative importance of rearrangement versus annihilation mechanisms at the quark level 9-11,13 ). At the same time, potential model calculations r4) are available for processes of type (1) and should be discussed in comparison with quark model descriptions. An investigation of the pp + rrr+Y and pp+ K+K- processes is of particular interest ‘O,“). The latter channel must necessarily involve quark-antiquark annihilation and strange quark pair creation via qq + sS, so that the ratio a(pp + K+K-)/a(pp + rr+C) sets constraints for the relative weight of quark rearrangement and qq annihilation processes. In ref. ‘l) we have found both these mechanisms to be about equally important at low energy. In the present paper we would like to further extend this investigation by a calculation of pp+ M,M, angular distributions. Our primary aim is to evaluate the capacity and also the limits of simple quark models to describe the relevant pp annihilation mechanisms. An essential part in the description of processes (1) is the proper treatment of pp initial state interactions prior to the annihilation into two mesons. This requires a detailed discussion of the NN optical potential describing pp elastic, inelastic and pp + nii charge exchange channels. Phenomenological NN potentials, with large imaginary parts representing annihilation, have been discussed repeatedly in the literature 2*15-17).We shall have to reanalyse the Nfi optical potential in our context, however, since the description of hadron sizes in the quark model approach to the pp+ M,M2 amplitude has to be matched consistently meter entering in the Nfi scattering potential. We present the results of our pp scattering analysis to follow. We will process, using the mentioned. Final tant, especially in
with the geometric in a separate
size para-
section,
the one
then proceed with the quark model description of the pp+ MlM2 pp wave functions obtained with the optical potential previously state rescatterings in the two-meson M,M, system may be importhe S-wave. We take into account the possible effect in a simple
way, based on the phase
shift analysis
of rrrr scattering
“).
2. Optical potential for NN scattering The wave function
of the initial
pp pair in the process
K'i%(r)@N@I;I,
(1) is of the form (2)
where QN and 0~ are the intrinsic wave functions of the nucleon and anti-nucleon which will be given in terms of valence quarks and antiquarks (see sect. 3). The relative motion of the nucleon and antinucleon at a center-of-mass energy E is
M. Kohnq
described energy,
by qE(r), (am
W. Weise / pp scattering
where r = RN -RR is the distance
satisfies
431
between
their centers.
At low
the equation h2V2 --+V,,+V,-E M 1
c&r)=O,
(3)
)
where M is the nucleon mass. Here VN~ is the Ns strong interaction potential and V,(r) is the Coulomb potential. The complex potential VNfi= U+iW,
optical
(4)
is constructed as follows. The long range properties of its real part U are assumed to be well described by the G-parity transform of a realistic one-boson exchange NN potential, Vo,,(NN)
=
V%E, C *=?r,p,w,...
(5)
as in refs. “-l’). In practice we use the static part of the G-transformed Ueda potential “) at distances r 3 R0 = 1 fm. This potential has P, p-, CT-and o-exchange as its dominant components, with additional small contributions from other weakly coupled mesons. The potential parameters are given in ref. I’). In the inner region (rc R,), we assign iitde significance to OBE and G-parity arguments. This is the domain in which the nucIeon-antinucleon density distributions overlap, and where the relation between the NN and Nfi potentials based on the OBE picture is presumably of little relevance. For r < R. we simply assume a smooth extrapolation by Woods-Saxon forms continuously matched at r = R. with the OBE potentials: U=Re
I&P;=
c
(-F
{
i= *,p.o,...
+
V%&)~(r- &I
2V&2&=Ro) 1 -f-exp (P(r-
(1_8(r_R,)) Rd)
I
,
where Gi is odd for i = T, w, . . . and even for i = a, p, . . . . The parameters p are uniquely determined by the smooth matching conditions at r = R,, in each spinisospin channel; we choose R. = 1.O fm*. The imaginary part W of V&N summarizes the effect of all annihilation channels. Assuming that annihilation takes place when the valence quark and antiquark cores of the nucleon and antinucleon overlap, one arrives at a geometric picture in which the imaginary NR potential is proportional to the convolution of the respective quark and antiquark distributions 12,‘9). For two gaussian distributions with r.m.s. radii 6 = (r2)“2, one obtains: W(r) = W( r = 0) x exp (-z( r/ b)*) ,
(7)
The potential used here for TZ R, is to be regarded as a typical representative of a class of realistic OBE potentials. We have also used the G-parity transformed Bonn potential 43f and found variations in the final results at the few percent level. l
432
M. Kohnq
where the c.m. degree of freedom
W. Weise / pp sca!tering
has been properly
subtracted.
We argue that the
r.m.s. radius b should not be identified with the proton charge radius, which results from a combination of quark core and meson cloud sizes in chiral quark models 20*21), but rather
with the size of the quark core alone,
for which the nucleon
axial form
factor gives an upper limit 2’). We therefore prefer radii b = OS-O.6 fm. For such radii, total, elastic and charge exchange pIs; cross sections are well described with an imaginary potential depth W( r = 0) = -1.5 GeV. As we shall see, excellent fits to pp total, elastic and charge exchange cross sections can be obtained with the slightly modified form
W(p)= - Wol{l +ew ((r- roValt, with a radius ro=0.55 fm closely corresponding to the gaussian radius b, and a surface parameter a = 0.2 fm. This form will be used in the actual calculations. The quality of the fits obtained with this potential are comparable to or better than the ones obtained with the potential of ref. 16). We prefer the potential (6-8) for our purpose since its characteristic range is consistent with the quark model picture to be developed in sect. 3.
2.1. TOTAL,
ELASTIC
AND
CHARGE
EXCHANGE
PI’, CROSS
SECTIONS
Using the N‘i;i optical potential U-tiW of eqs. (6) and (8) together with the Coulomb interaction, we have calculated the total pp cross section o;,,, the elastic cross section a,, = jdL?[d~(pp+pp)/dL?] and the charge exchange cross section ~(pp -+ nii). The results are shown in fig. 1 and compared with data in the energy region Elab < 250 MeV. The energy dependence of the total cross section is mainly determined by the geometric size of the imaginary potential W. We have performed systematic variations of the parameters r, and W. of this potential and found that the best fit over the entire energy region is obtained with r. = 0.55 fm and Wo= 1.2 GeV together with a = 0.2 fm. Increasing the annihilation radius r, can be compensated by decreasing the potential depth Wo. Very small radii (rO< 0.2 fm, as suggested e.g. in the baryon exchange model 14) would require Wo> 10 GeV and an explicit energy dependence of W to reproduce the data. These results are consistent
2.2. ELASTIC
with the earlier
AND
CHARGE
ones in refs. ‘s*‘6).
EXCHANGE
DIFFERENTIAL
CROSS
SECTIONS
Using the preferred parameter set W, = 1.2 GeV, r,= 0.55 fm, results for the differential cross sections da(pg + pp)/dL? and da(pp + mi)/dLI are shown in figs. 2-4 in comparison with selected sets of experimental data. For the peripheral partial waves with I > 5 at low energy and with I> 7 at higher energies, the analytic one-pion exchange Born amplitudes have been used. The Coufomb interaction has been incorporated throughout.
hf. Kohno, W. Weise / pp scattering
433
Fig. 1. Total, elastic and charge exchange pp cross sections as a function of 0 ia~orato~ momentum. The curves show resuhs of a calculation with the N@ pore&al VW% = U 9-W lsee eqs. f6>, (8)). The parameters of the real part U for ~a R, = 1.Q_fmare taken from ref. “1, The parameters of the imaginary. potendai (8) are W, = 1.2 CeV, r, = 0.55 fm and a = 0.2 Frn. J%e Coulomb interaction is included in the calculations; the proton and neutron masses are taken to be equal. The otot9,data are from ref. “). The elwic cross sections are taken from ref. “)+ The charge exchange cross sections o(pB-+ niij are from ref. “1.
Considering the simplicity of the potential l&p3with no explicit energy dependence and on@ a central imaginary potential W, the overaIl agreement with data is quite satisfactory. The results demonstrate that the gross features of elastic and charge exchange p$ scattering can be understood in terms of meson exchange at Iarge distance and a strong annihj~ati~n potential in the region of overlap of the proton and antjproton vaIence quark distributions (see also ref. “).
2.3. REAL
PART OF THE pii FORWARD SCATfERlNC
AMPLITUDE
Recent accurate measurements of pp differential cross sections at extreme forward angles permit to extract, by the interference of Coulomb and strong interaction ampIitudes, the ratio p(B)==ReF(E,B~Q)/Xm
F(E, 8=0),
(91
M. Kohno, W. Weise / pF scattering
434
lo2 ,N
09L cm OLOb
Fig. 2. Forward pp- pp elastic differential cross sections calculated with the NN potential VN~ plus Coulomb potential. The parameters are the same as in fig. 1. The data are taken from ref. 25).
of the real and imaginary parts of the spin-isospin averaged forward pfi scattering amplitude F( E, 8 = 0). Its imaginary part Im F( E, 0) is related to ottotarby the optical theorem. In terms of the singlet amplitude T,,(8) and the triplet amplitudes T,,(8) and T,,( 0) summed over isospins and defined in ref. *‘), we have
Re {T,,(O)+ ~dO)+2~~~(0)~
P=Im{T,,(0)+~oo(O)+2~,~(O)}’ The Coulomb
interference $’
analysis
involves
at the same time the parameter
IT,m+ loo -27xo)12+I~oo(wT,m12 ILw+ Mo)+27-,*(w ’
(10) *“)
(11)
for which a value has to be assumed in order to extract p from the experimental data. In ref. 25), two sets of analysis (with v2= 0 and ~~~0.1) have been given, whereas our potential yields T*= 0.06. In the comparison between our results for p(E) and the data, the small difference in 72 has been taken into account by shifting the data points for 72 = 0 by Ap = -0.05. (See fig. 5.) The standard parameterization of the annihilation potential with W, = 1.2 GeV and r. = 0.55 fm accounts well for the empirical data above plab b 300 MeV/c. However,
the calculated
p(E)
decreases
continuously
below plab s 200 MeV/c.
This
M. Kohno, W. Weise / p/i scattering
Fig. 3. Elastic
pp differential
cross sections.
The curves are obtained from ref. 26).
435
as in fig. 2. The data are taken
is in disagreement with the analysis of ref. 30) based on recent LEAR data which suggests values p = 0 at very low energy. The optical potential alone cannot account for this effect.
2.4. SUMMARY
We have presented a simple NR optical potential which reproduces the main features of pp elastic and charge exchange scattering. The potentials based on a combination of long range boson exchange and an energy independent imaginary (annihilation) potential W with a radius parameter r. = 0.55 fm and a depth W. = 1.2 GeV. The geometry reflects the size of the quark core in the proton or antiproton. The range of the imaginary potential is considerably larger than the range (2&f-’ characteristic of baryon exchange mechanisms 14). The simple N6I potential introduced here serves its purpose in the subsequent treatment of absorptive distortions in specific pp annihilation channels. Further improvements - not discussed here -
436
M. Kohno,
W. Weise / pp scaiiering 0
30"
60'
90"
IZP
cm lx)0 1
@p-En
1.0
05
0
-05
-1.0
cosB,, Fig. 4. Differential obtained
cross sections for the pfi -L nii charge exchange reaction. The calculated curves with the same potential parameters as in fig. 1. The data are from ref. z6),
are
700
500 P,,,lMeVicl
Fig. 5. Ratio of real to imaginary part of the forward pp scattering amplitude as a function of energy. The curve is obtained using the potential parameters as in fig. 1. The extracted data are from ref. “) (dots) corrected for 17’ = 0.06 instead of Q’ = 0, and from ref. 30) (squares).
M. Kohno,
can be made introducing
by adding additional
of development, might become the tensor
spin-orbit
and tensor
free parameters.
an explicit necessary
interaction
treatment
terms to W, but at the expense
It should
partial
of
also be noted that at a later stage
of coupled
in a more refined
in higher
437
W. Weise J pp scattering
NN c, A R, Nd and Ad channels
treatment
*), especially
for the role of
waves of the pp + M1M2 process.
3. Quark model for pp annihilation
into two mesons
Having discussed the ingredients relevant for the treatment of pp distortions before annihilation, we now turn to the description of the annihilation process pp + (meson), + (meson), itself. The conventional theory of the NN + 7r+F reaction based on the baryon exchange mechanism has been quite successful r4). However, the results depend very strongly on cutoffs associated with meson-baryon vertex factors. Moreover, the characterisic range AK’ -0.2 fm of the baryon exchange process appears to be unnatural, given the fact that baryon and antibaryon sizes are much larger than that. It is therefore desirable to have a description in terms of basic quark degrees of freedom, although it is clear that such a description can at present not be regarded as complete, given the complexities of non-perturbative QCD. In a quark model description of pp annihilation into two mesons, at least one quark-antiquark pair is annihilated. The two basic quark line diagrams are shown in fig. 6. We refer to them as the single pair annihilation plus rearrangement process (R-(rearrangement) process, fig. 6a) and annihilation plus pair creation process (A-(annihilation) process, fig. 6b). As mentioned before, there exists an extensive discussion in the literature as to the relative importance of the R- and A-processes. In the following, our basic framework will be the non-relativistic (constituent) quark model. This model permits a systematic separation of intrinsic and center-of-mass variables
in the description
of process
M,
L
;u
M,
n II fn Xl ------
I_..-
-
--_
-----
~
-
N
-
i
(a)
Fig. 6. Quark
M,
M,
----i-
(1).
-1
I-
----
L
--
-----
-
---
1
---___
-11 N
N
(b)
line diagrams for pp + two mesons: (a) single pair annihilation plus rearrangement (R-process) (b) annihilation plus creation process (A-process).
process
M. Kohno, W. Weise / pp scattering
438 3.1. CROSS
SECTION
AND
WAVE
FUNCTIONS
The cross section for pfi+M,M* c.m. frame as:
with two mesons M, and MZ, is given in the
(12)
(PI+P2)‘(E,+E,-Ei)I(~‘,IVIEFl)12,
Here v,, is the relative velocity of the proton and antiproton in the initial state and V is a transition potential, to be specified in the following, taken between the initial and final states I$i) and II&). The initial state wave function (see eq. (2)) is ~i(l, 2,, . . , 6) = (PC& - hd%(L
2,3)@i(4,5,6)
(13)
,
with the NN relative wave function Q as given in the previous section and RN= f( rl + r, + r3), RR = $( r, + r, + r6). The quarks in the nucleon are labelled by (1,2,3), the antiquarks in the antinucleon by (4,5,6). The final state wave function is: 9’x l’, 2’, 5’, 6’) = cpf(&, -Rl,,)~,,(r;-r;)~,,,(r;-rk).
(14)
Here pf describes the relative motion of the mesons M, and MZ, with
J&l,= and constituent wave state
m,r: + m5r;
RM,=
’
m,+m,
m2ri + m6rk
(15)
’
m2+m6
quark masses mi. The two mesons are assumed to be in a plane
rpf(&, -&J
= exp (& - &,
+ ip2 - 4~~) ,
(1%’
The additional effect of the M,M2 final state interaction is introduced by a factor &j eiSOin the rr S-wave. Throughout the paper we use fi = 0.35 and 6, = 50”. The intrinsic quark wave functions of the proton (c&), the antiproton (@s) and the mesons (#J& are assumed to be given by OSharmonic oscihator orbits. Their spin and flavour parts &t&, have the standard SU(6) structure. Combined with color, their explicit forms are as follows: @N(l, 2~3, = 4
Eiik14iqj9k>ol,,X~~~r(N)
x(fi&)-3’2exp
n=l
- i
(r,--RN)’
2b&
1 ,
@d4,5,6) =~E~kIQiiij~~k)colorXs~(N) / 1, x (v?nb$)-3’2
[
exp
-~.+-RN)~
I
2bk
(174 (17b)
and &&r) = ~~ijlqie)colorX~~~*~(M)(rbL)-3’4
exp (-r2/2bZ,)
.
(18)
M. Kohno,
In this parametrization
439
W. Weise / pp scattering
the proton
(antiproton)
and meson
and
= d$bM ,
r.m.s. radii are given
by (r&)“2 = bN
(r$1’2
(19)
respectively.
3.2. TRANSITION
POTENTIALS
As a model for the effective transition potential V in eq. (12) we consider the diagrams, figs. 7(a) and (b). We use the gluon exchange interaction as a guideline to develop the characteristic spin-flavour dependence of the transition potential, although with effective parameters not necessarily related to a perturbative picture, since the basic mechanisms must clearly be thought of as non-perturbative ones. The gluon exchange model adopted here is less restrictive than the ‘PO model in which quark and antiquark are annihilated into vacuum quantum numbers. In the 3P0 model, the transition matrix element is simply given by the overlap of the quark and antiquark wave functions, the qq pair being in a 3P,, state. Gluon exchange offers instead a different spin-space structure of (he transition potential. n
m
J
hH
1
2
m
(Cl1
Fig. 7. Generating interactions is followed by effective gluon
3
n
3
L
1
2
1
j
k
I
(b)
for the R-process (a) and the A-process (b) of fig. 6. The qq annihilation exchange (wavy line). The letters i, j, . refer to color indices [eq. (23)].
The interaction derived from figs. 7(a) and (b) is obtained by considering the leading terms in the non-relativistic expansion of the corresponding Feynman diagrams. These processes involve the effective gluon propagator [ qz - q2 - m,$ + i&l-’ once or twice. Here q. and q are the energy and momentum transfer carried by the gluon, and md is an effective gluon mass. This gluon mass reflects the fact that the gluon modes are localized within the confinement domain, so that their propagation length m&’ is restricted to short distances. The dominant low-lying gluon modes 31) have their energies in the range around 1 GeV, the value we adopt as a typical order of magnitude for mG. Now, the energy transfer q. is determined by the energy of the annihilating qq pair, i.e. q,,a 2m, where m, - 300 MeV is the (u- or d-) quark constituent mass. The typical momentum transfers 1q(deduced from
440
M. Kohno, W. W&e / pp scattering
the kinematics of pp+ MIMZ processes under consideration For mG- I GeV, it follows that ?&%+q9;:-q2.
are of the same order.
WV
We therefore replace the gluon propagators in figs. 7(a) and (b) approximately by -m;=. In the potential for the R-process, fig. 7(a), an additional factor $m, comes from the reduction of the quark spinor matrix element. The results of the reduction for R-process (fig. 7(a)) and A-process (fig* 7(b)) transition potentials are the foBowing*:
x{u(l)x(o(2)xo(34))io(2)x(u(1)~~r(34))}+~~~)~
(22)
I-Iere r&= ri -q and the coordinate numberings refer to notations in fig. 7**, The r;k and CA in eqs. (ZI), (22) are operators in color space given by their matrix element5 (jfC&[mfi))=
$h,q:h”,,,
trliiPt>icAIri~~ckt~=~~~ihfihn)nnrh”~,
W4
(23b)
where the color indices 8,j, . . . are again specified in fig. 7 and the h” are standard SU(3) matrices. For VA in eq. (22) only the dominant terms proportional to the momentum transfer between particles 3 and 4 (see fig. 7(b)) have been kept. The kinematics of the A-process justifies a static approximation for the intermediate quark propagator. This yields the V,,(exp (-~r~J/r~~) factor in eq. (22), where JA= m,, the strange constituent mass for a final KfK- state, and y = pnqfor non-strange two-meson finat states. The second term in V, of eq. (21) has a single spin operator acting between the quark and antiquark, Its structure is the same as the one of the ‘PO model. In fact this term contributes in initial pp odd states for the rr+7~- channel, whereas the first term in V, contributes in even states. The strengths of the transition potentials are determined by effective quark-gluon coupling constants (Ye and CY*.They need not necessarily be equal, but the final * These V, and VA differ from the ones in ref. I’) by the refined approa& to the eixective gh~on pmpagator and by thr: additional term (see also ref. 42)f ~r~~~~i~na~ to oj23f . V, in abe present work. ** In taking matrix elements of the spin operators w between Pat& spinors of qzsark and antiquark, remember that they are related by charge conjugation.
M. Kuh?q
w. W&e
1@
.mm?ring
441
results will show them to be of comparable magnitude. To obtain a rough estimate for artJm&, one can use the transition potential V, in the quark model description of the decays p + nn and A + n-N. From such an analysis one finds (YR/rr& = 0.4 fm2. Further evaluation of the matrix elements (ly,] V,,,l W;) leads to the results summarized in appendix A. Note that the intrinsic quark distributions of the proton, antiproton and mesons introduce form factors as functions of the squared meson c.m. momentum p’, which depends on the mass of the mesons involved. Light mesons carry a larger m~mentnm 1~~1,so that their matrix elements are reduced as compared to the ones for heavier mesons. This difference becomes signi~~nt already in the comparison of the pp+ rrt rr- and pp+ K’K- matrix elements of V.&. For pp at rest, the pion momentum in the former process is jpNIi= 4.7 fm-‘, while the kaon momentum in the latter one is IpNII= 4 fm-‘. As a consequence, for typical hadronic radii ( r2)“2 - f fm*, matrix elements (~T+T-I V,lpp) are suppressed by roughly a factor of two as compared to (K”K-IVA(pp). Furthermore, the form factors strongly favour matrix elements with two vector mesons in the final state, such as @“p-l tkjpfs) as compared to {~T+T-\V,Ipp). For example, the p meson c.m. momentum for pfi+pcpp at rest is refativety smalf, /p%$\=2.7 fm-‘. For hadron sizes as previously mentioned, the formfactor effect enhances the p+p- process by roughly a factor of five over the q+rr”‘ process. Thus hadron sizes evidently play an important role in the rearrangement and annihilation dynamics so that they wilf have to be discussed in some detail. We shall return to this question in sect. 4.1.
3.3. SELECTION
RULES
Parity, isospin and G-parity ~onse~a~~on impose selection rules on the R-process and A-process matrix elements. For two non-strange mesons in the final state, G-parity requires that St-f~L,+S,i-I,+S,+I,=even,
(24)
where S, I and L denote the spin, isospin and relative orbital angular momentum of the piij system; Sl,2 and I,,z are the spin and isospin of meson 1 or 2 in the final state. We note that, for the R-process, this setection rule is automat~caIly satisfied when one sums up all possible configurations of diagrams shown in fig. 8. Alternatively, the last two diagrams fig. gb are included if G-parity conservation, eq. (241, is imposed on the matrix elements of the first two diagrams, frg. 8a, together with an obvious factor of two. Consider now the R-process matrix elements (?~+yr-l VR/pp) with V, of eq. (21). The first term proportional to [o(l) x0(23)] *V,2S3(t12) contributes oniy in spin Note that the strong interaction radii should radii; see the discussion in sect. 4.1.
not be identified
naively with the corresponding
charge
442
M. Kohno,
W. Weise / pp scattering
(of
lttf
Fig. 8. Diagrams contributing to the R-process far p@+ M,M,.
and isospin triplet channels of the pp system; it acts in the 33S, 33D, _“. protonantiproton partial waves (which we denote in general by 21+‘*2s+‘Lj). On the other hand, the second term of V,, the one propo~ional to ~3(~,*)~(23) * V,, operates in the spin-triplet and isospin-singlet *3P0, 13P2,. . . channels only. The annihilation potential V, of eq. (22) has nonvanishing matrix elements only in spin and isospin triplet states. For pij+ ?T+T-, it therefore interferes with the first term of the rearrangement potential V, (eq. (21)) in the corresponding L-even partial waves, and in the ‘?S channel in particular. The pp-+ rr+C P-wave channels have no contributions from VA in this model. For pp+ K*K-, there are no C-parity restrictions. However, due to the specific structure of VA in our model, the isospin singlet channels are suppressed, so that contributions come only from the spin-isospin triplet channels 33S, 33P,. . . etc. 4. Results We concentrate here primarily on differential cross sections for pp+ r%- and pij-., K”‘K- at low energies (ptab<600 MeV/c). The corresponding total cross sections have been investigated already in ref. If) using the same modet, but several refinements and details will be discussed in the following, Furthermore, we shall comment on channels with one or two vector mesons in the final state, in particular the pp -+ r-p- channel for which detailed experimental information is now becoming available.
4.1. INPUT PARAMETERS
Apart from the pp optical potential state, the following parameters enter (a) the effective coupling strength %, eq. (21); (b) the effective coupting strength Vn, eq. (22);
describing the distortion effects in the initial in the pp- M,M2 process: cyR/m& of the R-process transition potential aY,jrn& of the A-process transition potential
hf. Kohno, W. W&se / pj7 scattering
443
the size parameters bN(ff~) and bM related to the intrinsic quark dist~butions of proton (antiproton) and outgoing mesons, respectively (see eqs. (17-19)), and the constituent quark masses, @t”,dand m,. The size parameter of proton and antiproton is taken to be (c)
& = bi;i=:(ri)“* = 0.55 fm .
(25)
This should be interpreted as the quark core radius* for which an analysis of the nucleon axial form factor *O)gives (0,5-4.6) fm as an upper limit, It is smaller than the proton charge radius which is influenced more strongly by the charged pion cloud surrounding the core ‘9X20).Note that the same size as in eq. (25) has been used consistently in the imaginary part of the pp opticat potential, eqs. (7) and (8). The quark core radii of the mesons are uncertain. For the pion, a large part of its empirical charge radius of about 0.7 fm can be understood simply in terms of the p-meson dominance picture, in which the form factor is (1 - q2/ mt)-’ with an otherwise pointlike pion. The intrinsic pion radius is therefore expected to be considerably smalier than 0.7 fm. Detailed models of the pion structure 32*33)give an upper limit of about 0.4 fm to its intrinsic size, the value which we adopt:
The empirical kaon charge radius ““) is estimated to be around 0.5 fm. This is an upper limit for the intrinsic kaon size. We use b,= b, with eq. (26). We use constituent quark masses m”.d= 330 MeV and m, = 530 MeV. The effective strength (LY,Jm&) of the A-process potential is determined by the pp -, K’K- cross section since the R-process does not contribute to strangeness-producing reactions. As in ref. rl> we obtain: ~~~/rn&==O.l5frn~.
(27)
(For exampte, with an effective gtuon mass m, = 800 MeV, this corresponds to iTA= 2.5). The pp + ?rf-i7- reaction is described as a ~ombjnation of the A- and R-processes. Given LY,J~& one finds that values around curJm&=0.25 fm’,
Gw
for the effective coupling strength of the rearrangement process yield a cross section a(pp+ 7rfn-) as shown in fig. 9*. The absolute values of p@+ M,M2 cross sections depend on the proton (antiproton) size parameter b,. Smaller proton and antiproton core radii tend to increase * The new KEK data 35f*aiso shown in fig. 9, are syst~rnat~~~lyhigher than the older ones used for the determination of ctR and LY~.A fit to the KEK data would require aA/& = 0.18 fm” and a,/& = 0.35 fm2.
200
a
I
200
^X^
LW
600
4ao
PLob [MN/c
1
1
ml
600 PLab [MeVlcl
1
(b)
la1
Fig. 9. Total cross sections a(ppZT’Y) and v(pp+ K’K-) as a function of the antiproton lab momentum. The data are taken from the compilation in ref. 23)” The open squares refer to the recent data at KEK 35). The curves are the results of calculations using parameters as specified in eqs. (25), (26) and eqs. (27), (28).
the calculated two meson production cross sections and would require a corresponding reduction of the coupling strengths, eqs. (27,28). The cross sections are considerably less sensitive to the meson size parameters bM as long as these are varied within *20% of their values, eq. (26). Large pion r.m.s. radii enhance the rearrangement process relative to the annihilation process. Very small pion radii around (rtjl/‘= 0.1 fm tend to suppress the R-process with respect to the A-process by more than a factor of three. However, such small values of (& of tr(pp+ r7r) by a l’arger factor which would unreasonably large coupling strength EYJ m&.
4.2. DISTCIRTION
EFFECTS
IN THE
also reduce the overall magnitude have to be compensated by an
pP-CHANNEL
The results just mentioned are obtained with the pp optical potential derived in chapter 2. Distortion effects are large and depend strongly on the pp partial wave involved. A characteristic comparison with plane wave Born approximation results for pp+ GT~T-” and pp + K*K- at Sa,, = 500 MeV/c are shown in table 1. One finds that the S-wave cantribution to the cross sections at this energy are reduced by a factor around 20. The reduction of the P-wave contributions is considerably less as seen in table 1. It is due to a combination of the pi-, initial state F-wave distortion and the strong S-wave interaction in the ~T+?T- final state.
M. Kohno, W. Weise / pp scattering
445
TABLE 1 Comparison of cross sections (in mb) for pp + ~+rand pp + K+Kplane wave. (PWBA) and distored wave (DWBA) Born approximation chapter 2
o(pp+
a(pp + r+?r-)/mb channel
R-process
at p,.,, = 500 MeV/c calculated with the pp optical potential
A-process
in of
K+K-)/mb (A-process.
R+A
channel
0.12 0.23 0.06
‘9, =P UP0z
0.11 0.00 0.04
2.03 1.52 0.05
9, =P 0 33P2
2.85 0.01 0.07
only)
DWBA 0.52 0.23 0.06
0.26 0.0 0.0
13.63 1.52 0.05
5.14 0.0 0.0
PWBA
The cross sections are given for the leading partial waves. Also shown are the separate contributions from the R- and A-processes for pp+ r+Y. The input parameters are specified in sect. 4.1.
Effects of the attractive Coulomb interaction in the pp channel are small in our cross section with and without region of interest. Calculations of the pp+ ntY the Coulomb potential Vc show in fact that the difference is only at the percent level.
4.3. DIFFERENTIAL
CROSS
SECTIONS
FOR
pp+~~+sr-
Results of calculated differential cross sections da(pp+ r+n-)/dfl at several low energies are compared with experimental data 35) in fig. 10. The angle OoM is defined as the angle between the incoming proton and outgoing 7~+ momentum. The low energy data show a pronounced
peak at t?cM = 0. In our model, this structure
arises
from the combined contributions of S- and P-wave amplitudes. In the S-wave, the first term of V, (eq. (21)) interferes destructively with the annihilation potential VA (eq. (22)). The second term of V, contributes in the P-wave only. It is seen from fig. 10 that the combination of V, and VA, with parameters given in eqs. (25,26) and eqs. (27), (28), accounts reasonably well for gross features of the observed pp+ r*Y angular distribution, although a stronger forward peaking would still be required in the model. In any case, it is clear from the previous discussion that both R- and A-processes contribute significantly to da(pp + rr+Y)/dfl.
4.4. DIFFERENTIAL
CROSS
The pp+ K+K- process The calculated differential
SECTIONS
FOR
pP+K+K-
is determined by the A-process alone, as stated earlier. cross section at low energies arises from a dominant
S-wave amplitude with a smaller P-wave contribution (see also table 1). The rise of the angular distribution at forward angles is reproduced (see fig. 11). However, the
h4. Kohnff. W Wefse / pij sentfeting
0
0
-10
-05 co* o,,
-10 -0.5 cosecn
lb1
(al
I
OS
L
0
(cl
-05 cos
e,,
J -10
Fig. SO. ~iffereutiai cross sections for pp+ T+T- at low energies, compared to the data measured at KEK35). The results obtained using the input parameters specified in eqs. (25), (26) and eqs. (27), (28) are normalized to the experiment 35)by a factor of 1.3 (390 MeV/cf, 2.0 (490 MeV/c) and 2.0 (590 MeV/c) to compensate for the difference of the KEK data from the old ones shown in fig. 9.
model fails to account for the steep backward peaking which is a characteristic feature of the pp + K’K-‘ reaction even at low energy 35). Similar observations are made at higher energies 36). The failure to reproduce the backward peaking indicates that an important piece is still missing in the description of the annihilation mechanism via the potential VA, and that this piece is specific to strangeness-producing processes. One possible source might be a two-step mechanism via pij + AA + K’K- 37).The pp + Ax interac-
t pjF--K+K590 M&/c
IO
0
05
-05
-10 cos o,,
10
05
0
-0.5 -10 CO5O~rn
(al (bl Fig. 11. Differential cross sections for pp + K+K- at low energies, compared to the data measured at KEK 35). The curves are obtained using the input parameters specified in eqs. (251, (27) together with (rjy = 0.4 fm.
tion has a strong tensor force component through kaon exchange with a corresponding mixing of higher partial waves. 4.5. THE RATIO aipp-tK*K-)/of~ji-,~~rr-)
IN S- AND P-WAVES
relative yield of K+K- and r+rr- following pp annihilation at rest in S- and P-waves has been an interesting point of recent discussion. The older bubble chamber data correspond primarily to S-wave annihilation and give 3*38): The
T(pp+K+K-) T(PP+
with 38):
T(pp-,
K+K-)
rtpp+
T+r-)
~0.5
I S-wave
_$ I liquid
r+r-)
and
'
T(pp+ K+K-) f&q+
p+d
= 0.01 . ! P-wave
New LEAR data taken with a gas target 39) are supposedly representative annihilation with the result: f (pl? + K+K-) T(PP + r+r-)
=0.15*0.03. pss
(291
of P-wave
(30)
It is instructive to evaluate corresponding numbers in our model. We do this by choosing a very low energy (plab = IO0 MeV/c) for comparison. While it should be kept in mind that the treatment of pfi distortions becomes less reliable at these energies, the magnitude of p = Re F/Im F being too large (see fig. 5), one expects that uncertainties in the overall distortion factors cancel out to a large extent when taking the ratios, eqs. (29), (30).
M. Kohno, W. Weise / pp scattering
448
Our results are summarized in table 2: our values for the S- and P-wave ratios are 0.63 and 0.01, respectively. Also shown in table 2 is the cross section for pp+ nor0 at plab= 100 MeV/ c. This is a direct measure of the P-wave R-type process since the S-wave is forbidden by selection rules. Our result for the ratio u(pp + rr’rr’)/~(pp?r+7~--)is 0.27 at PI=,,= 100 MeV/c and 0.36 at pi&,= 500 MeV/c. The rather large P-wave ratio is due to the cancellation between R- and A-processes in the S-wave.
4.6. COMMENTS
ON
OTHER
TWO-MESON
ANNIHILATION
CHANNELS
In ref. 11)we have reported results obtained within the present model for branching ratios of pp+M,M, channels with one or two vector mesons in the final state. Generally, the rearrangement mechanism (R-process) favours vector mesons in the final state. This follows from the discussion of transition form factors in sect. 3.2; vector mesons are generated with low momenta, so the form factors (see also appendix A) do not cut down their amplitudes. Of course, these statements depend on the unknown sizes of vector mesons and are therefore subject to uncertainties. Here we would like to point out a problem related to the pp-, 7rcp- channel which is faced by any model based on a combination of quark-antiquark annihilation and rearrangement mechanisms, including the present one. Empi~cally, the isospin f = 1 contribution to pp + rip- was found long ago to be negligibly small in S-waves as compared to the I = 0 contribution 40). Recently, the analysis of LEAR data 39) has shown that I = 1 P-wave pp+ -rr+.p- annihilation is also suppressed. Our results for pp+ rr+p- and pp+ rope cross sections at low energy are shown in table 3. In S-waves the I= 0 component dominates the I = 1 component by a
TABLE
2
Cross sections at piab= 100 MeV/ e for pp + TUTand pp --f K+K- in mb for s- and p-wave. channels, given separately for R- and A-processes and with the combined amplitude (R+A) R-process
Channel pp+
7r+n-
9, 13 PO 13Pz
pp --*K+K-
9 33PA 3)P,
pp + $ro*a
335, 13Pa ‘3Pz
8.03 2.30 0.15
1.14 0.07
A-process
R+A
3.16 0.0 0.0
2.06 2.30 0.15
1.28 0.00 0.03
1.28 0.00 0.03
0.0 0.0
1.14 0.07
The input parameters are specilied in eqs. (ZS), (26) and differences for no, n* are taken into account.
(27),
(28). The mass
M. Kohno, U? Weise / pp scattering TABLE
Contributions
of various
pp partial
cr(pp+ Channel 3’S I$ 1 ‘IP, =P Xjp’z ‘)D,
R-process 4.81 48.06 0.05 1.40 0.01 0.27
waves
3
to the pp+ rrn+p- and pp+ 100 MeV/ c
rr+p-)/mb
A-process 2.58 24.85 0.05 0.10 0.07 0.14
449
rrOpo cross
a(ppR+A 0.70 5.99 0.08 1.25 0.13 0.03
R-process
sections
at plab=
r’p’)/rnb A-process
R+A
47.92 0.05
24.85 0.05
5.95 0.08
0.07
0.14
0.03
Other partial waves not shown (e.g. “So) are forbidden by selection those specified in eqs. (25), (26) with b, = b, and eqs. (27), (28).
rules. The input parameters
are
large factor in both R- and A-type processes, compatible with the experimental observation and consistent with ref. 41). In fact, the A-process suppresses all but the 13S1 and 3’S,, partial waves, the 3S, state being dominant over the ‘S,, state by about an order of magnitude. However, the R-process, via the second term of V, (eq. (21)), the one proportional to ~(23) * VI, contributes a relatively large P-wave amplitude in the 33P1 channel, which is very sensitive to the S-wave np final state interaction. Now, one could think of mechanisms which suppress this term dynamically. However, we recall that this part of V, is necessary to obtain the forward peaking of the pp+ ~T+T- differential cross section: without it, the T+~T- angular distribution would be flat. Hence there are still important ingredients missing in the attempt to achieve a consistent description of all pp+ M,M2 channels.
5. Summary and conclusions In the first part of this paper, we have introduced a minimal NN optical potential capable of reproducing pp elastic and charge exchange differential cross sections together with total cross sections in the low energy region up to about pIa,,= 600 MeV/c. The basic potential parameters are the proton and antiproton sizes and the strength of the imaginary potential, which is proportional to the overlap of the quark and antiquark distributions in the proton and antiproton. Quark core rms radii of 0.55-0.6 fm are found to be optimal, consistent with chiral quark model descriptions of various other hadronic properties. In the second part, we have developed a quark model description of the low energy proton-antiproton annihilation into two mesons. The pp+ 7r+C and pp+ K+K- channels are treated with special emphasis. The model is based on a combination of quark rearrangement (R-process) and quark-antiquark annihilation (Aprocess) mechanisms. In deriving the spin structure of the corresponding effective
450
hf. K~MZO, W Weis
/ pp scatferirtg
interactions, we have been guided by gluon exchange, but with effective coupling strengths and propagators to simulate non-perturbative features. We find both Rand A-processes to be about equally important in the description of pp + 7rh7r- and pj5+ K’K- cross sections and angular distributions. In these caicuiations, the pp optical potential discussed in the first part proves useful in the calculation of important distortion effects in the pIj initial state. Final state interactions in rrrr S-waves have also been found to be important. Apart from rhe improved treatment of distortion effects, this paper differs from our previous investigation I’) by the additional velocity dependent term in eq, (21). This term has also been included in recent work by Henfey et al. 42). As far as the relative importance of this term is concerned, we are in qualitative agreement with their results. The quark model approach with combined R- and A-type mechanisms adequately accounts for the gross behaviour of the pp+ 7r+7~- and pp-+ K+K- cross sections and the T?V- angular distributions, as well as for the K”K^- and 7ret7f’-yield ratios close to threshold in both S- and P-wave channels. At the same time, there are ~ha~~te~sti~ diffi~u~t~es~however, which indicate the need for further refinements. First, the strong backward peaking observed in the pp+ K’X- angular distributions at tow energy cannot be matched by the present model. This backward peaking requires an admixture of partial waves higher than S- and P-waves, but selectively so for strangeness-producing reactions. We have rnent~o~ed the explicit in~arporation of the pp + Ail * K’K- channet as a possibility to mix in D-waves via the kaon exchanges tensor force in the pp-, Ax interaction. Secondly, the empirical data on pp+ rr’p”” indicate a strong suppression of the I = 1 S- and P-wave amplitudes. While this feature can be well understood in S-waves within the present model, a part of the R-process leads to a large pp -+ ~?p- amplitude in the ‘P, channel, contrary to observation. On the other hand, this particular part of the R-type interaction cannot be easily abandon~d~ since it is necessary to reproduce the pp + r+rr- angular distributions. We would Iike to thank Prof. K. Nakamura for correspondence concerning the KEK data. We are grateful to Carl Dover and Rudolph Tegen for many helpful discussions. Appendix A We present here explicit expressions for the matrix elements f lkCjVI ?ui) with the transition potentials V, (eq. (21)) and V, (eq. (22)), which incorporate form factors from the intrinsic quark distributions of the proton, antiprolon and mesons. The relevant wave functions are defined by eqs, (13) to (1X) in the text;
451
M. Kohno, W. Weise / p@ scattering x[(e,-az)
* (3ir/bL-2pM)-a2.
(6ir/bh+5pM/2)1
3(2b2,+ b2,)
’ crdV.d+i)=I d3rdr) [b’(36’9+2b:)a]3’2i Id’r(a,.r) NN -‘2(8bZ,+5bfJPM’
r
’ (3r-z)2 -$-F-S N
- iph2* W-+(1
(A.1)
3r2 N
M
-P)z) ,
(A.21
1
where bN = & is assumed, pM denotes the momentum of the final meson in the c.m. frame, and (p, II) = (4, mu,J for non-strange quark pair creation or (/3, CL)= (m”,dl(m”,d f m,), m,) for strange quark pair creation. The vectors al, a2 and aj involve spin, flavor and color matrix elements: 41=-
z,”f$36(~tIC~t(-i)(u(2) X~(34))I~J 9
q
G
a2=- ~~36(y,ICttu(34)I’I’J 4 a,=4a
9
2 36(‘P&(tr(34)
x (~(25) ~~(5’6’))
+u(~S)X(U(~~)XU(~‘~‘))}[~J
y
(A3)
where one of the possible configurations of diagrams shown in fig. 8 is given as an example. Using eqs. (17), (18) and (23), the color matrix elements are obtained as ( FrI CR1?lii)color = -$ and ( TfI CA TJcolor = hi. References 1) R. KJapisch, Nucl. Phys. A434 (1985) 207c, and references therein 2) A.M. Green and J.A. Niskanen, Int. Rev. of Nucl. Phys. 1 (1984) 569 (World Scientific, Singapore) and references therein and B. French, in High energy physics IV, ed. E.H.S. Burhop 3) R. Armenteros 1969) and H. Stern, Phys. Lett. 21 (1966) 447; 4) H. Rubinstein J. Harte, R. Socoiow and J. Vandermeuien, Nuovo Cim. 49 (1967) 555 and T. Ueda, Nuct. Phys. A364 (1981) 297 5) M. Maruyama and T. Ueda, Phys. Lett. 124B (1983) 121; 6) M. Maruyama M. Maruyama, Prog. Theor. Phys. 69 (1983) 937
(Academic
Press, NY,
452
M. X&no, W. W&se / pp sc~~~~~jng
7) A.M. Green and J.A. Niskanen, Nuci. Phys. A412 (1984) 448 8) S. Furui and A. Faessier, Nucl. Phys. A424 (1984) 525 9) H. Genz, Phys. Rev. D28 (1983) 1094; Phys. Rev. 031 (1985) 1136 10) C.B. Dover and P.M. Fishbane, Nucl. Phys. B244 (1984) 349 11) M. Kohno and W. Weise, Phys. Lett. 152B (1985) 303 12) R.A. Freedman, W.Y.P. Hwang and L. Wiiets, Phys. Rev. D23 (1981) 1103; M.A. Alberg et al., Phys. Rev. D27 (1983) 536 13) U. Hartmann, E. Kiempt and J.G. Kiirnet, Phys. Lett. 1556 (1985) 163 14) B. Moussa~~am, Nucl. Phys. Ad07 (1983) 413; Nucl. Phys. A429 (1984) 429 15) R.A. Bryan and R.J.N. Phillips, Nucl. Phys. BS (1968) 201 16) C.B. Dover and J.M. Richard, Phys. Rev. C21 (1980) 1466 17) T. Ueda, Prog. Theor, Phys. 62 (1979) 1670; 63 (1980) I9S 18) B. Hyams er at, Nucf. Phys. 864 (1973) 134 19) F. Myhrer and R. Tegen, Univ. of South Carolina preprint (1985) 20) A.W. Thomas, Adv. Nucl. Phys. 13 (1984) 1; G.A. Millet, Int. Rev. of Nucl. Phys. 1 (1984) 189 (World Scientific, Singapore) 21) W. Weise, Int. Rev. of Nucl. Phys. 1 (1984) 57 (World Scientific, Singapore); E. Oset, R. Tegen and W. Weise, Nucl. Phys. Ad26 (1984) 456; R. Tegen and W. Weise, Z. Phys. A314 (1983) 357 22j A.S. Clough et nt, Phys. Lett. 146B (1984) 299 23) V. Flaminio, W.G. Moorhead, D.R.O. Morrison and N. Rivoire, Compilation of cross sections: p and p induced reactions, CERN-HERA 84-01 (1984) 24) R.P. Hamilton ef a& Phys. Rev. Lett. 44 (1980) 1182 25) V. Ashford er al., Phys. Rev. Lett. 54 (1985) 518 26) K. Nakamura et ni, Phys. Rev. Lett. 53 (1984) 885 and private communication 27) T. Mizutani, F.‘Myhrer and R. Tegen, Phys. Rev. D32 (1985) 1663 281 M.L. Goldberger and K.M. Watson, Collision theory (J. Wiley, 1964) 29) C. Bourrely er al, Nucl. Phys. B77 (1974f 386; M. Lacombe et al., Phys. Lett. 124B (1983) 443 30) W. Bruckner et al., Phys. Lett. 158B (1985) IS0 31) R.L. Jaffe and K. Johnsan, Phys. I&t. SOB (1976) 201; C.E. Carbon er nL, Phys. Rev. D27 (1983) 1556 32) S.J. Brodsky, Springer tracts in Modern Physics 100 (1982) 81 33) V. Bernard, R. Brockmann, M. Schaden, W. Weise and E. Werner, Nucl. Phys. A412 (1984) 349; V. Bernard, R. Brockmann and W. Weise, Nucf. Phys. A440 (1985) 605; W. We&e, Nucl. Phys. A434 (1985) 68% 34) E.B. Dally er nf., Phys. Rev. Lett. 45 (1980) 232 35) T. Tanimori, Ph.D. thesis (University of Tokyo), UT-HE-85/03 (1985); T. Tanimori ef ai, Phys. Rev. Lett. 55 (1985) 183.5 36) E. Eisenhandler er al, Nucf. Phys. B96 (1975) 109 37) J.A. Niskanen, Phys. Lett. 1548 (1985) 351 38) S. Devons et af., Phys. Rev. Lett. 27 (1971) 1614; R Bizzarri et nL, Nucl. Phys. 369 (1974) 307 39) ASTERIX Collaboration, preprint CERN-EP/M-I 16 40) M. Foster et at, Nucl. Phys. B6 (1968) 107 41) J.A. Niskanen and F. Myhrer, Phys. Len. i57B (1985) 247 42) E.M. Henley, M. Oka and J. Vergados, Phys. Lett. 166B (1986) 274 43) K. Erkelenz, Phys. Reports 13 (1974) 191; K. Ho&de and R. Machleidt, Nuci. Phys. A256 (1976) 479