Physics (and technique) of gas jet experiments

PHYSICS (AND TECHNIQUE) OF GAS JET EXPERIMENTS

A.C. MELISSINOS and S.L. OLSEN Department of Physics and Astronomy, University of Rochester, Rochester, N. Y. 14627, USA

(~E NORTH-HOLLAND PUBLISHING COMPANY

—

AMSTERDAM

PFIYSICS REPORTS (Section C of Physics letters) 17. no. 3 (1975) 77

32. NORTH-HOLLAND PUBLISHING COMPANY

PHYSICS (AND TECHNIQUE) OF GAS JET EXPERIMENTS * AC. MELISSINOS and S.L. OLSEN Department of P/tt’sic~and

.4strooonn. Unii’ersitt’

of Rochester, Rochester, V. Y. /462

LSA

Received 22 October 1974 C sottenis

Scope and method .1. Introduction I .2. The experimental metltod I .3. The gas jet 1.4. The detection apparatus 1 .5. Background and its subtraction 2. Elastic scattering and the structure of the proton 2.t . Elastic differential cross-sections 2.2. Phenomenological analysis 2.3. Models for the slope parameter 6 2.4. Bounds from local field theory

79

79 81) ~4 87 9) 93 93 99 I 02 I 08

3. The real part of the forward nuclear amplitude 3.1. Determination of the real part 3.2. Interpretation of the results 4. Inelastic scattering 4.1. Introduction 4.2. Experimental details 4.3. I.ow mass results 5. Current and future program 6. Acknowledgments References and notes

110 110 1)6 120 20 1 23 125 129 130 I 31)

~lhstrac1 Recently gas jet targets have been used for the study of small momentum transfer p p and p- d elastic and inelastic scattering. In these experiments, which were performed at Serpukhov in the USSR and at the Fermilab in the USA, the gas jet is introduced in the main ring of the accelerator. We review the techniques associated with these measurements and summarize the published 2 and incident energies 8 400 GeV. Elastic scatterresults which cover 4-momentum transfers 0.001 t 0.12 (GeV/c1 ing data have yielded precise values for the slope of the nuclear scattering amplitude and for the ratio of the real to imaginary part as~~t ftii~ctionof energy. The implications of these data on the understanding of high energy hadronic interactions are discussed in some detail. We also present and discuss information on the diffractive dissociation of the proton to low mass states. obtained from inelastic scattering using the gas jet target.

Single orders fOr this issue PHYSICS REPORTS ISection C of PHYSICS LETTERS) 17, No. 3 (1975) 77-132. (‘opies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Singe issue price Dtl. 20.-

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postage included.

~This work was supported in part by the U.S. Atomic Energy (‘ommission. A.P. Sloan Foundation Fellow.

.4. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

79

1. Scope and method 1.1. Introduction The field of high energy physics is both rich and restrictive in the type of experimental observations that can be made. By rich we have in mind the many states that are found in the spectrum of the elementary particles and the interesting property that some of these particles do not partake in all the known interactions. By restrictive we imply that most of these particles are unstable and one is concerned either with scattering experiments, elastic or inelastic, or with the observation of the decay of the unstable particles. The investigations which we review here deal with proton—proton interactions. Indeed, because of their stability, protons are the traditional target in high energy physics; on the other hand many other particles are used as projectiles. Nevertheless, the p—p interaction is most extensively studied both because of its fundamental importance and because of the relative ease with which protons can be accelerated. The measurements were performed at the Fermi National Accelerator Laboratory by the U.S.A. —U.S.S.R. collaboration. A gaseous hydrogen jet was used as a target and placed in the path of the internal beam of the accelerator. Only the slow recoil proton was detected as described in more detail in the following section. Since the energy of the internal beam is continuously increasing during the acceleration cycle, simultaneous measurements are obtained over the entire energy range of the accelerator, namely from 8 to 400 GeV. Elastic scattering for spinless particles is fully characterized by two invariants: the square of the total c.m. energy s, and the 4-momentum transfer t, which we define as s>0 s(p1+p2)2, (1) t—(p 2, t<0 1—p’1) as shown in fig. la. The metric is such that s> 0 and t < 0 in a scattering experiment. Inelastic experiments are more complex, but we consider a special class of reactions, in which one of the colliding particles in only slightly perturbed in the scattering process. One must then specify, in addition to s and t, the invariant mass (missing mass), M, of all the other particles of the final state in order to characterize the inclusive process shown in fig. lb. As usual —

—

(1’) In this report we discuss elastic and inelastic scattering in which the 4-momentum transfer is very small when compared to the mass of the proton m~,

(a)

(b)

Fig. 1. Kinematic definitions (a) for elastic scattering, (b) for diffractive dissociation.

80

AC. Melissinos, S.L. Olsen, Physics (and technique) ofgas jet experiments 1<

0.2 (GeV/c)2.

On the other hand, the incident energies are in the range from 50 to 400 GeV which, on the same scale, are “high” energies. Scattering at high energy and small momentum transfer is often referred to as diffractive scattering because of its many similarities with the familiar phenomena of the diffraction of light. From this optical analogy, and more specifically from the uncertainty principle, it follows that collisions with small momentum transfer probe large spatial distances: one speaks of a peripheral interaction. While such interactions are not expected to reveal directly the internal structure of the colliding particles, they do measure their overall size and shape. Furthermore, because of the optical theorem, elastic scattering in the forward direction is related to the total absorption probability, namely elastic and inelastic scattering at all momentum transfers. In the case of small t (diffractive) inelastic scattering, one probes the excitation of the incident particle to states of higher mass, while preventing its fragmentation. Finally, from both geometrical considerations and the experimental data one can argue that diffractive phenomena will constitute an ever larger fraction of the total interaction cross section as the incident energy increases. The report is divided into four main parts. In the first part we discuss the experimental method, its advantages and limitations and also review the necessary kinematic relations. The second part deals with elastic scattering at small-i’ and the available information on the structure of the proton. In the third part we present elastic scattering data in the t-region where the nuclear and electromagnetic scattering interfere. From these data one obtains the phase of the nuclear amplitude which can be related to the total cross section by dispersion relations. Finally, in the last part we present data on the diffractive dissociation of the proton to states of higher mass. 1,2. The experimental method The scattering of elementary particles was first studied by Lord Rutherford [1] who measured the angle of deflection of ct-particles incident on a gold foil. If, in addition to the angle of deflection, the energy of the scattered particle is measured, one has available one constraint with which to establish the elasticity of the scattering. Alternatively, one can measure the angle and energy of the recoil particle. In this case the kinetic energy of the recoil, Tr, is a direct measure of the 4-momentum transfer since (in the laboratory frame) it holds It~21hirTr

(2)

It follows that in our [-range the recoiling protons are of very low energy and therefore one must utilize fairly thin targets. Scattering measurements based solely on the detection of the recoil particle were in use for some time and are generally referred to as missing mass experiments. V. Nikitin and his co-workers at the JINR in Dubna, USSR, realized the advantages of this technique for the study of small angle elastic scattering. In their earliest experiments they used nuclear emulsions [2] and later solid state detectors [3] to register slow recoil protons from an internal polyethylene foil at the Dubna Synchrophasotron. In the late 1960’s together with Y. Pilipenko they devised a supersonic hydrogen jet of adequate density for use as a target in an accelerator and used solid state detectors for the detection of the recoil protons. The first experiment was performed at the Serpukhov accelerator [41 and subsequently at the FERMILAB [5,6,7].

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas let experiments

~

~. b~.

81

~

STATIO~ C-O, tGAS JET

( ~

__

BEAM

-

‘

-

-~

‘

\

VALVE

43.~ 2.5m

5.L18. -

I

D~f~

\I ~

DETECTORS

Fig. 2 (a). Schematic layout of the detection apparatus.

A typical arrangement for such an experiment is shown in fig. 2. The active area of the solid state detectors is of the order of 1 cm2 and the transverse dimensions of the hydrogen jet are 1 cm. Thus with detectors placed at a distance of 2.5 m one obtains an angular resolution of

—

_

±0.75/250 = ±3

miliradians.

For an accurate determination of the recoil energy it is desirable that the protons stop in the detector. Given the practical limitations on the thickness of the detectors (5 mm) this corresponds to protons of energies Tr ~ 30 MeV, or It I ~ 0.05 6 (GeV/c)2 while this energy range can be extended by the use of absorbers in front of the detectors the method is limited to t ~ 0.2 (GeV/c)2. For elastic scattering the recoil energy is uniquely determined by the recoil angle. To see this express eq. (1’) in the laboratory frame, 2

2

r

.

M _mi2P1[P2s1n~_

m 2+E1 ~

T2j1

(2)

where ~ is the recoil angle as measured from 90°(see 2) and andmaking p1, m1 use is the 2 fig. = m~ ofincident eq. (2’), particle we obtain whereas p2, m2 is the recoil. For elastic scatteringM (E+m 2 )2 sin (3) 4m~+ItI (E2—m~) —

_____

-

~—

with E

E 1. For high energies E ~ m1, m2 and for small momentum transfer

tI

‘~

4m~the

82

.lle/jssjnos,.S / . ( )!o

I. t

0

Ph

~i

,id I

ItHh/IO I ‘,f

~

/

Vpe 0

n/cots

4

L

ic.

2/hi. \ en

ot

ic

sn guide and e.~let in the main ring I the I ermilab n celer I’r.

approximate relation sinØ

~tI/2ni.

=

~~71/2ni2

(3’)

is quite adequate. Since each detector is located at a fixed angle, the energy spectrum of the recoils registered by the detector shows a peak. corresponding to elastic scattering, at an energy determined by eq. (3’). An example of such an elastic peak is shown in fig. 3a. The background from which the peak must be separated is both instrumental and physical. When the elastic differential crosssection is large, as in fig. 3. the elastic signal can be measured with great accuracy: this is true in our range oft ~ 0.2 but fails for large t-values. A detailed discussion of background problems is given in section 1.5. An important advantage of the recoil technique is evident from eq. (3). We note that the recoil angle depends on It~and only weakly on the incident energy F. Furthermore, the 4-momentum

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

83

o 0 0

o

(a)

Co. I-. (‘I > Ui U. 0

a z

(b)

.

~.

.5 ...

0

50 CHANNEL NO.

00

50

100

Fig. 3. Typical pulse-height spectrum from a 2-mm thick solid-state detector at 50 GeV incident energy; the proton energy at the elastic peak is approximately 10 MeY. (a) Original spectrum, (b) spectrum after correction for “room” background.

transfer is directly measured independent of an exact knowledge of the energy and direction of the incident beam. Thus the technique is ideal for a study of the s-dependence of elastic scattering. Since the energy of the internal beam of the accelerator varies continuously during the acceleration cycle any desired energy can be selected by pulsing the jet at the appropriate time in the acceleration cycle. The high intensity of the internal beam and the multiple traversals through the target compensate for the small density of the hydrogen jet and result in luminosities of the order of 1 O~cm2 sec~ as discussed in section 1.3. Apart from the restricted t-range the technique has two other limitations. One is the inability to measure absolute cross sections in the sense that so far it has not been possible to obtain an independent measure of the beam-jet luminosity. The other limitation is the discrete nature of the measurements in the sense that each detector is located at a fixed tvalue. This is compensated in part by placing the detectors on a moveable carriage so as to be able to adjust their position at will. On the other hand these limitations may be overcome by the use of more sophisticated detection systems, studies of which are now in progress [8]. Each detector subtends a fixed solid angle d~2,and therefore measures da/d~2.To convert to the invariant cross-section dcr/dt we need to differentiate eq. (3) and obtain dada di’

2d[cos(90—~)]ir12m do df~ dt 2vj~ d~7

(4)

in the limit of t ~ 4m~and E>> see fromt eq. that the counting rate da/d~2de1111 m1,m2. when weWe approach -* 0.(4) This tendency is compensated in part creases relative to dcs/dt as N by an increasing do/di’ = A exp(—bItI), yielding a maximum counting rate for ItI = 1/2b 0.05 (GeV/c)2. A relevant parameter of the system is the resolution in missing mass that can be achieved. From eq. (2’) we note that the factors that influence the mass resolution are the uncertainty in the recoil energy and the recoil angle. The uncertainty in the incident momentum p 1 is not important since it acts as an overall scale factor. Using the small-t approximation and in the vicinity of the elastic peak one obtains from eq. (2’)

84

AC Melissinos, S.L. Olsen, Phj:sics (and technique) of gas jet experiments

(5) ~M2~2p

1’fT~J~.

(5’)

Since the typical energy resolution of the solid state detectors is z~Ts~ 50 keV = 5 X l0~GeV, the contribution of eq. (5) to the missing mass resolution is negligible even for p1 1 o~GeV. On the other hand given ~Øp ±3 X iO~radians eq. (5’) can2become significant.orFor instance, 2 and = 0.60 (GeV/c2)2 i~M ±0.30 at (GeV/c2) = 0.06 (GeV/c) 1 = 400 GeV/c, we .~M which is well past the inelastic threshold. As obtain expected, the missing mass resolution deteriorates rapidly with increasing incident energy. Finally, for completeness we give the optical theorem which relates the total cross section to the elastic differential cross section in the forward direction. It holds UT(S)—

~/!~ dt (s,t—:~ )

-~

--

-

where p is the ratio of the real to the imaginary part of the forward nuclear amplitude. Since the differential cross sections measured by this technique are not absolute, eq. (6) is used to provide the normalization, given a knowledge of UT. 1.3. The gas jet [9] The target consists of a jet of precooled gas which is injected across the beam of the accelerator. There are three basic problems that must be overcome: (a) The target must have adequate density and typically it operates at p 3 X 107g/cm3, (b) The vacuum must be maintained in the accelerator, where P 10—6 torr, (c) The jet must be confined in the direction transverse to the beam. These objectives are to some extent, contradictory in that high density implies large quantities of gas which therefore are more difficult to pump. Thus one seeks a design where the flow velocity is small; but small velocities make the collimation of the jet more difficult. The answer to these requirements lies in the use of cryogenic techniques. In a typical mode of operation, hydrogen gas is precooled to 40°K and emerges from the nozzle with velocities of the order of 1000 rn/sec. This requires a flow of 60 cm3 (stp) of hydrogen for a 200 msec pulse, to achieve p = 3 X 1 0~g/cm3. To pump such quantities of gas in a short time a liquid cryopump was used. This consists of a copper vessel in contact with a liquid helium bath; the jet is directed into the vessel and the hydrogen condenses on its surface. As the vessel is gradually coated with solid hydrogen the pumping efficiency is reduced at which point the whole assembly is removed from the accelerator beam pipe, isolated by means of a gate valve, and the hydrogen is allowed to sublimate. Once the sublimation is complete the assembly is re-introduced in the accelerator pipe and the target operation resumed. Typically, the jet can operate for 4—8 hours before sublimation, depending on the amount of gas injected, and has to be interrupted for approximately 1 hour. A schematic drawing of the target is shown in fig. 4. One notes the short distance from the nozzle to the opening of the cryopump, approximately 4.7 cm. Provisions must also be made to prevent freezing of the hydrogen in the nozzle and in the supply lines. Indeed, the jet density is determined by a combination of the pressure in the hydrogen supply line and the nozzle tempera-

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

—

85

BEAM

Fig. 4. Schematic cross section of the gas jet target in its retracted position. Legends: (1,2) hydrogen (or deuterium) supply valves; (3) heat exchanger; (4) nozzle and collimator system; (5) upper cryogenic pump; (6) lower cryogenic pump; (7) shield cooled by helium vapor; (8) target isolation chamber; (9) pressure gauge; (10) heat-exchanger control valve; (11) accelerator chamber; (12) vacuum gate; and (13) valve for evacuation of sublimated hydrogen. Table 1 Typical operating characteristics of gas jet Width of hydrogenjet (FWHM) Nozzle to beam center distance Pressure of injected H 2 Precooling temperature for H2 Density ofjet Ambient vacuum in proximity of jet Duration of pulsed H 2 jet

12 mm 15 mm 1.5

4.0 atm. 7g/cm3 40°K 3iO~ X 10— l0~ tori —

200 msec

ture. A typical set of operating parameters is given in table 1 and a detailed description can be found in ref. [9]. The jet target was constructed at the J.I.N.R., Dubna, USSR. The jet was operated in a pulsed mode typically 200 msec long twice during the 3 sec acceleration ramp. This is necessary ,in order (a) to keep the vacuum at the desired level of 1 0~torr, and (b) for an acceptable consumption of liquid helium; it also happens to match the data acquisition rate of the detection system. As a matter of fact, when the internal beam reached 1013 protons it became necessary to inject the hydrogen at sub-atmospheric pressure in order to keep the data rate at a tolerable level. Fig. 5 shows a typical pulsing sequence and the pressure variations as measured in the region of the jet. The jet has also been operated with deuterium gas, in this case three pulses can be used in one cycle due to the higher freezing point of deuterium. Deuterium has the further advantage of lower thermal gas velocities at the same temperature and thus higher density for the same quantity of injected gas. —.

86

-1. C. Melissinos. S. I.. (I/sen. Pht’sics (and technique) of gas jet experi/nents

H

2JET TARGET OPERATION

nj.—4~

—~

-•-•~

~-

—.—-—-—10~torr

~—

10

—

_____

-

.4J~’’~p~_~ -

•.

011~4~I ~

-

•---

--

torI

VACUUiMIN ACCELERATOR CHAtv1BEI

COUNTING RATE

BEAM POSITION CONTROL SIGNAL RADIAL BEAM POSITION

isec

Fig. 5. Oscilloscope tracings as a function of time of: (a) main ring vacuum in the vicinity of the gas jet: (b) counting rate in scintillation monitors; (c) beam position control signal (bump); (d) radial position of the beam.

Under the above operating conditions, the liquid helium consumption of the system is of the order of 25 liters/hour. This was supplied from a 500 liter dewar which was located in the accelerator tunnel next to the jet. The main supply of liquid helium was located outside the tunnel and transferred through a 40 m line to the dewar. Thus the jet could operate continuously without assess into the tunnel. The evaporated helium gas was returned to a 25 liter/hour CTI-1400 liquefier which maintained the main supply, so that in principle 3theorsystem closed. 6 X 1016was protons/cm3, a 1 cm To estimate the luminosity we can assume p = l0~g/cm wide jet and a beam of 1013 protons. The period of rotation at the Fermilab is 20 ,usec (50 000 traversals/sec), so that the instantaneous luminosity is L

=

3 X 10~~ cm2 sec’.

Given the two 200 msec pulses and the 8 sec acceleration cycle, the average luminosity is 1/20 of its instantaneous value quoted above. In practice, the interaction rate depends not only on the jet but also on the beam size and beam orbit which should properly intercept the jet. Thus, some radial steering of the beam through the control of the radio frequency was used. The main characteristics of the beam are given in table 2. From the point of view of the experiment it is important that the jet target appear as a point source as closely as possible. The best knowledge of the profile of the jet can be obtained from the width of the elastic peaks and these indicate a full width (at half maximum) of 12mm. This is the main restriction in the angular resolution that can be achieved. In addition, a certain amount of gas diffuses in the beam pipe, its density being of the order of 10~ of peak density at a distance of 10 cm from the jet. Interactions of the proton beam with the residual gas in the upstream beam pipe were the major source of background radiation incident on the detectors.

A.C Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

87

Table 2 Typical parameters of the Fermilab’s internal proton beam Intensity Vertical size of beam at highest energies Angular divergence in target region Rate of energy increase Energy span Radial bumps of beam during jet pulsing Beam losses over acceleration cycle

1012 1013 ~ —2—3 mm ~ 0.5 mrad 100 GeV/sec 50—400 GeV 10 mm ~ 5%

As mentioned before, the gas jet was operated with hydrogen or deuterium gas. The possibility of using helium is now under exploration [10] and presumably other heavier gases can also be injected. Before the installation of the gas jet polyethylene foils were used as targets. Either thin foils or filaments (6 ~cmin diameter) were mounted on a wheel rotating at 60 Hz so that the centrifugal force keeps the foil or filaments stretched. The carbon content of the CH2 introduces a large amount of background under the elastic peaks. Furthermore, with a circulating beam intensity of 1013 protons, the polyethylene targets deteriorate very rapidly making this mode of operation extremely tedious. 1.4. The detection apparatus The apparatus is shown in fig. 2 and the solid state detectors are contained in a vacuum box directly connected to the accelerator vacuum. As many as 12 detectors were placed on an arc of circle of radius 2.50 m from the center of the target. The detectors were mounted on a moveable carriage so that the entire assembly could be positioned at any angle from 5°< ~ < 11° where ç~iis measured from 90°.The spacing between detectors (which determines the density of t-values at which da/dz’ is measured) was typically 42 mm (17 mrad) for the nuclear elastic scattering measurements and 20 mm (8 mrad) for the measurements in the Coulomb interference region. It was also possible to rapidly move the whole detector assembly by 64 mrad towards smaller t values; this is called the “background” position. In this position the elastic peak is drastically shifted on the energy spectrum, or completely absent, so that one obtains a physical measure of the background under the peak. This is valid provided that the background is not strongly dependent on angle as is expected for room background. Indeed, during data taking the carriage was in normal position for 10 pulses and in the background position for 5 pulses. The detectors were silicon, both of the lithium drifted and surface barrier variety and connected to their own preamplifier. The signals were amplified and sent to individual ADC’s (amplitude to digital converters). A 2 MHz commutator would interrogate the 12 ADC’s in sequence and if a signal was found it would be transferred to a 2 word buffer memory and then to a PDP- 11 computer. The incident proton energy (i.e., the time in the magnet ramp) as well as some information on jet performance were also transmitted to the computer at regular intervals. The computer transferred all information to magnetic tape and generated displays of the data for monitoring purposes. When an ADC was presented with a signal, it reverted to a “busy” status until the digital information was read out. If a second signal was presented during the “busy” time an “event overflow” counter was incremented. The contents of this counter, one for each ADC, were also transferred —

88

A. C’. Melissinos, S. L. Olsen, Physics (arid technique) of gas jet experiments

to tape and used to determine the dead time correction. This correction did not exceed 3Ye. A discriminator threshold could be set so as to eliminate low level pulses such as produced by detector or preamplifier noise. The detectors were calibrated by using a polonium 210 o-source which was mounted inside the vacuum tank; this gave a quasi-monochromatic line at 5.8 MeV. Furthermore, the linearity and offset of each channel’s electronics was checked by feeding a pulse of known amplitude to the detector and recording the response through the system. While the o-source did not penetrate the detectors to any significant depth, background particles from the polyethylene foil could be used to find the overall sensitive depth of the detectors. In general all calibration procedures were performed with the equipment in its data taking configuration. The solid state detectors either stop the recoil protons or register the transversal of the proton. In the first case the total energy, E, is recorded while in the latter case dE/dx (
2 3 4 5 6 7

8 9 10 11 12

4 (mrad)

ti protons (GeV/c)2

7elastic protons (MeV)

Thickness Si (mm)

20 37 55 72 90 107 125 144 160 177 194 212

0.0014 0.0048 0.011 0.018 0.028 0.040 0.055 0.073 0.089 0.109 0.131 0.156

0.8 2.6 5.7 9.7 15.2 21.4 29.2 38.6 47.6 58.1 69~7 83.1

1.0 1.0 1.0 1.0 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5

Absorber (‘u (mm)

(~min).Emax For stopping protons tMeVt

-

--

0.45 0.70 1.16 2.50 3.90 4.82

(14.). (19), (25). (39). (49). (57t,

12 12 12 12 29 29 34 36 40 54 60 66

The solid angle subtended by each detector was defined by collimators. The areas of the collimators was measured both by conventional means and also by using an intense U-source which could be inserted at the position of the target. A further check of the acceptance is obtained by moving the carriage so that one detector overlaps the position previously occupied by another detector. The consistency of the elastic count obtained in these two different detectors is a measure of the correct evaluation of the acceptance, nuclear corrections and efficiency. The efficiency of all detectors was assumed to be 100% as also evidenced by the data. Nevertheless in the high radiation environment of the accelerator tunnel, the detectors deteriorate in time (in particular

A.C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

89

Fig. 6;(a). Scatter plot ofE

1 versus £2 for a stack of two solid state detectors. Front detector 1 mm thick, rear detector 5 mm thick. A carbon filament was used as a target and clear bands of protons, deuterons and tritons can be distinguished.

the lithium drifted ones) as evidenced by large leakage currents and poor alpha lines. Such detectors had to be replaced. A more sophisticated method of particle detection consists of the use of two detectors, a thin one for dE/dx followed by a thicker detector where the particle is stopped. Knowledge of dE/dx and E permits particle identification as shown in fig. 6a. Fig. 6b is a scatter plot of E1 versus F2 for a 200 pm (E1) followed by a 2000 pm (E2) detector when a deuterium target was used. Clear bands corresponding to deuterons and to protons can be seen. Once the energy of the deuterons reaches the point where they traverse the second detector as well, the separation becomes difficult and soon it is beyond the resolution of the system. While this approach was not necessary for p—p elastic scattering it is useful for the inelastic measurements since the background cannot be distinguished from the data (there is no “inelastic peak”). The price one has to pay is that the energy, E, of useful recoils is limited jgmax

.—

~‘

~.-

J’max 1+2

where EraX and E~X correspond to the maximum range in the first detector and the sum of first and second detector. In principle, the t-range can be extended by using thicker detectors such as Nal crystals. This introduces large corrections for interactions in the detector material, efficiency etc. The normalization of the data presents a difficult problem. As discussed in the previous section it is not possible to calculate to satisfactory accuracy, the luminosity of the system from the beam and jet parameters. Observation of p—e elastic scattering or of the optical radiation emitted

90

AC’. Melissinos, SI.. Olsen, Physics (and technique)

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ill

5

Ia

E 2

Ii

15

(MeV)—’

Fig. 6(b). Scatter plot of Ei versus E2 for a stack of two solid state detectors. I~rontdetector 0.5 mm thick rear detector 2 nun thick using a deuterium gas jet target. The stack was positioned at such an angle that recoil deuterons from p d elastic scattering stop in the rear detector; points with more than 35 counts are indicated by stars.

in the interaction region, have been considered as possible monitors of the luminosity but have so far proved to be unsuccessful. On the other hand, it is possible to obtain a relative measure of the luminosity by the use of a scintillation telescope placed at some angle (typically 60° from the forward direction) or by placing a solid state detector at a fixed angle in the vacuum tank. The scintillator telescope is useful only for relative normalization at a particular energy since production of secondaries at some fixed angle is a strong function of the incident energy. The fixed detector can be placed in the region of the elastic peak and thus its rate is independent of energy as discussed previously. One need only take into account the energy variation of the differential cross-section as given by eq. (6), which can be inferred from a knowledge of o.~(s). In particular, since for each energy, data were taken at several positions of the carriage, these sets of data could be normalized with respect to one another using the yield of the fixed detector. In practice the accuracy of such a procedure was not better than 1 2)2% it was to to and obtain the preferable best normaliuse the relative consistency of the data sets (minimize the overall x zation. However, for the measurement of inelastic cross sections the fixed detector yield was used as the primary monitor, the results being subject to the above quoted uncertainty. -

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

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1.5. Background and its subtraction

As already mentioned and as can be seen from fig. 3, the elastic peaks are superimposed on a small continuum which is attributed to background. This background is associated with the target and, broadly speaking, can be assigned to three categories. (a) Fast particles originating in the target that dE/dx in the detector. Some of these particles have been emitted in different directions but reach the detector through rescattering. (b) Particles produced in the diffused gas outside the central target region. From the upstream gas one obtains mainly fast particles but which may traverse the detector at large angle. From the downstream gas one obtains elastic protons of lower energy; this is the most serious background for measurements in the inelastic continuum. (c) Recoils from inelastic interactions which, because of the finite mass resolution, contaminate the elastic peak. These recoils are concentrated in the lower energy part of the elastic peak. Detector noise is not a serious problem but particles which partially traverse the detector collimators do contribute to the background. From a point of view of hardware, additional pumps upstream and downstream of the target are used to reduce the background of type (b) while the insertion of appropriate baffles in the vacuum tank reduces background of type (a). Further reduction of background can be achieved by the use of a dE/dx and E detector system as discussed in the previous section. This separates the fast from the slow particles and greatly reduces background of type (a). To a large extent the background is independent of the angular position of the detector and has typically a hE dependence where E is the energy recorded in the detector. Clearly, the background problem becomes more serious as the cross section decreases, that is for large It I. While the statistical accuracy of each data point is only a small fraction of a percent, the uncertainty in the background subtraction alone increases this uncertainty to the order of 1% depending on the particular detector. As a first step, the angle independence of the background can be used to measure it experimentally. To this effect the carriage on which the detector assembly is mounted was moved by 64 mrad towards lower ti. This eliminates the elastic peaks from the energy spectrum (or at least moves them drastically towards lower energies) so that the observed spectrum is primarily that of the angle-independent background. In fig. 7 is shown an oscilloscope display of the energy spectra of 11 detectors where both the normal and background spectra have been simultaneously recorded. By appropriately normalizing on a fixed detector (or other monitor) the background can be subtracted to yield a corrected spectrum. This is the procedure used to obtain the spectrum of fig. 3b from the original raw spectrum of fig. 3a. One can use alternate methods, such as fitting the background to a smooth curve, or fitting both background and peak with appropriate functions. This approach, however, is made difficult by the lack of knowledge of the exact shape of the elastic peak which depends strongly on the operating characteristic3 of the detector. In particular, a marginally defective detector will give a correct overall count but will distort the spectrum. Furthermore, due to the angular resolution the elastic peaks are wide and thus cover a large fraction of the available dynamic range of the detector. After the subtraction of the experimentally measured background one is still left with a contribution from inelastic recoils (type (c)). These contributions can be calculated, to sufficient

92

AC’. Melissinos. S.L. Olsen, Physics (and technique) of gas jet experiments

o.ooi

0.5 1

1.5 2 0.5 1 2 3 1 3 RECOIL PROTON ENERGY

5

Fig. 7. Oscilloscope display of energy spectra of recoil protons from p~pelastic scattering for eleven detectors positioned at different angles. The corresponding background spectra are also indicated.

accuracy, if the corresponding excitation cross sections are known. To that effect we used the data obtained [111 at 24 GeV assuming no energy dependence. That the energy dependence is indeed very small or absent is supported by the data presented in part 4. Furthermore, one can measure the inelastic contributions through the following approach: From eq. (2’) it can be seen that for fixed angle (~)and recoil energy (p2, T2) the missing mass squared is directly proportional to the incident momentum p1. Thus, at low incident momentum a given recoil energy corresponds to masses below the inelastic threshold while at high momentum the same recoil energy corresponds to the excitation of some state of mass M~.By comparing the spectra at p1 = 50 GeV/c with those at higher momenta one can then isolate the inelastic contributions. This approach was used in the determination of the excitation of the low lying mass states as well as to confirm the inelastic background subtraction. In addition to the statistical and other random errors, the data is subject to certain overall systematic errors. These systematics affect differently the particular physical variable which is extracted from the data and will be discussed later in the appropriate context. The most serious systematic effect arises from the background subtraction and the isolation of the elastic signal. In addition the following effects are present: (1) an overall angle uncertainty which depends on the central position of the jet; this has been observed to move by as much as ±2 mm; (2) a possible overall t-dependence of the detection efficiency due to the decrease of signal/noise for large t~, and the additional corrections due to the use of absorbers and/or thicker detectors; (3) an exact knowledge of ti is limited by the uncertainty in individual detector calibration. In general. the systematic effects are expected to affect the t-distribution but have a very small s-dependence, as was also evidenced by comparing the results obtained by different methods of analysis and different data sets.

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

93

2. Elastic scattering and the structure of the proton 2. 1. Elastic differential cross-sections In this part we discuss elastic p—p scattering in the range 0.0035 < I tI < 0.16 (GeV/c)2. The lower limit of t is chosen so that the effects on the electromagnetic (Coulomb) scattering are not dominant, while the upper limit is set by the capabilities of the apparatus. We call this the small-t region, in which the differential cross section can be represented by dcr/dtA ~

(7)

The value of the constant A can be obtained through eq. (6) and the slope parameter b ranges from 10—12 (GeV/cY2. Thus the small-t region contains approximately 70% of the total elastic cross section. In fig. 8 we show a typical differential cross section at 132 GeV incident energy. The statistical error on each point is given and the scatter of the points is an indication of the systematic uncertainty. The main information that can be extracted from these data is the parameter b in the exponent of eq. (7) and its s-dependence, b(s). The exponential t-dependence of eq. (7) seems to be typical of the small t behavior of many hadronic reactions at high energies and in particular of elastic scattering. Nevertheless one wishes

E 132 GeV b~II.36±0.06(GeV/cY 100

X2~96.60

-

‘I

> 11)

E

“...

b 0

10

-

0.0

-

0.02

0.04

0.06

0.08

itt

0.10

0.12

0.14

0.16

0.18

0.20

(GeV/c)2

Fig. 8. Typical differential cross section da/dt for p—p elastic scattering in the region 0.005< tI < 0.015. The incident energy is 132 GeV and a fit to the form do/dt =A exp(—bitl)yields b = 11.36 ±0.06 (GeV/cY2 with x2 = 96.60 for 64 degrees of freedom.

A. C. Melissions, S.L. Olsen, I-’ht’sics (and technique) of gas jet experiments

94

to inquire whether other representations are more adequate, or whether eq. (7) must be modified as a function of energy. For instance, data from the ISR have demonstrated the presence of a change in slope [12] between the small and medium t regions; this break appearing at It I 0. 15 (GeV/c)2. For a cm. energy ./~= 53.0 GeV the authors of ref. [12] find that b depends upon r in the following way: 2

A

dcr/dt with

1 exp(-- b1 ti) A 2 exp(—b2lti)

12.40 ±0.30,

(7

2 for tI> 0.15 (GeV/c) 2

b 2

both b

for it] < 0.15 (GeV/c)

=

10.80 ±0.20 (GeV/cY

1 and b2 being functions of energy. The data cannot distinguish between a continuous or a

discrete 2change in curvature. as shown in fig. 9. The “break” in the slope of dcs/dt appears in the vicinity of t 0. 1 5 (GeV/c) Beyond the break, the high energy du/dt appears to continue its exponential drop for over six decades [13]. Eventually, the cross section changes its behavior and a diffraction dip appears at t 1.3 (for\/s= 53.0 GeV). This behavior is shown in fig. 10 where we have also included data at lower c.m. energies. One notes that the exponential behavior ceases in the region of ItI (GeV/c)2 and that the steepness of the diffractive peak increases with energy. In view of these facts one wishes to know whether eq. (7) is the correct description of elastic scattering in the small t region or whether a polynomial or other dependence is more appropriate. 101

~r53GeV

~ io1

\\\OIO8I

1021_,~ 0 005

010

015 -t

020 025 (0eV2)

030

035

Fig. 9. Differential cross-section da/dt for p—p elastic scattering at a cm. energy squared s [12]. Note that the data exhibit two different slopes in different t-regions.

=

2800 (GeV)2 as obtained at the tSR

A. C. Melissinos, S. L. Olsen, Physics (and technique) of gas jet experiments

2

95

___________________________________________________________________

10

F

to’

I

Nl02

I

I

0

ANKENBRANDT et aI,(I968) CLYDE et at. ALLABY et 01. (1968)

• a

ALLABY et at. (1971) ACHGT et 01. (1974)

v

10°

I

0

E

10-3

—

b ~I0

_~

0 -

12

24 IO~ I I i’Y~ I I 0 2 4 6 82) FOUR-MOMENTUM TRANSFER SQUARED, I (6eV Fig. 10. Differential cross-section da/dt for p—p elastic scattering for large-t for different incident energies. Note the existence of a dip in the data from the ISR obtained at an equivalent lab energy of 1500 GeV (s = 2800 GeV2) [13]. The other data are from C.M. Akenbrandt et at., Phys. Rev. 170 (1968) 1223; AR. Qyde, UCRL-16275 (1966); J.V. Allaby et at., Nuci. Phys. B52 (1973) 316.

The data from ref. [6] averaged over all energies do not show a significant departure from a pure exponential. For instance if this departure is parametrized by da/dt—Aexp(—bItI)(l+6ItI2) one finds I 6/b2 I < 0.10. Unfortunately the suitability of these data to answer this question is limited because, inspite of the intrinsic accuracy of each data point these points have been acquired in a discrete fashion. Furthermore, the t-range is restricted so as to preclude a confrontation with the break observed at higher energies [12]. Another possibility considered by some authors [14] is that the differential cross section may be oscillatory in t with periods typically 0.1 (GeV/c)2. No evidence for such behavior is found in the data of ref. [6] but we remind the reader of the reservations expressed previously. To obtain the slope parameter b, the data were first corrected for the effects of the Coulomb amplitude which interferes with the nuclear amplitude. Since the real part of the nuclear amplitude has been accurately measured (see part 3) this procedure is rigorous. Secondly, the t-region was restricted by the application of appropriate t-cuts. At the lower end the cut was chosen at

AC’. Meliss,nos, Si.. Olsen, Physics (and technique) of gas jet experiments

96

ItI> 0.0035 in order to avoid a large Coulomb contribution while at the upper end the cut was ti < 0.16 (GeV/c)2. This was done to avoid the influence of the breal in the determination of b. Furthermore, the t-cuts were varied so as to check the stability of the values of b obtained, and to estimate the systematic uncertainty. The data points so selected were then fit to eq. (7) by a standard x2 minimization. Typically there were 70 data points at each energy and using only statistical errors the fit gave x2/d.f. 1.5 indicating the presence of some systematic effects. From the minimization procedure h and its error ~b are obtained: the error ~b was incremented in each case by \/3(2/d.f. The final values of L\b are given in table 4 for 16 different incident momentum bins between 40 and 400 GeV. ‘Fable 4 Energy dependence of h(s)

(GeV/c)

b(s) [(GeV/cY0

9 12 50 58 78 102 128 150

8.72 ±0.38 9.03±0.30 10.70± 0.18 10.83 ±0.07 10.84±0.20 11.24 ±0.13 11.30 ±0.20 11.57 ±0.23

~ab

-

I

;b (GeV/c)

[(GeV/cY2

175 199 239 270 312 348 371 396

11.52 ±0.11 11.56±0.12 11.61 ±0.19 11.69 ±0.10 11.90±0.28 11.96 ±0.15 11.87 ±0.15 11.77 ±0.10

I

The momentum bins are -~ 20GeV/c wide centered at the given Plab except for the 9-and 12-GeV/c points which have widths of — I and 4 GeV/c, respectively. ~b(syst) = ±0.2, independent of s. The errors in b(s) reflect both the statistical and background subtraction uncertainty.

Aside from the systematic instrumental errors enumerated in section 1.5, the final value is also influenced by some of the input parameters introduced into the fit. In particular the value of b depends on the values used for the: (1) Real to imaginary part of the nuclear amplitude, p. (2) Total cross-section, UT. (3) Overall angular position of detector assembly, Ø~. On the average, the corresponding correlation and the estimate of a possible error are as follows: 3.0 ~b/~uT

0.025/mb --

0.20/mrad

±(~P)max

=

0.02

±(~~T)max=

0.5 mb

±(z~4o)max =

0.5 mrad.

Note, however, that these systematics have only a small effect on the s-dependence of the data but tend to increase or decrease b by a uniform amount over our entire energy range. Therefore an overall systematic uncertainty of L~b= 0.2 (GeV/c)2 is assigned to the data. The data are shown in fig. 11 where we have included values of b(s) obtained from other experiments as well. Inspection of the data points of fig. 11 suggests that the slope parameter b grows linearly with ln s for s above 100 (GeV)2. Such behavior was one of the first predictions of the simple Regge pole model of strong interactions which we discuss in section 2.3 below. A linear

A.C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

b(s) [(Gev/c)2]

I

I

I

‘~

++

j

9 -8

1

,

J

1

I

~I

t

97

I

~

I

•

2

0 AMALDI ET AL.

3 4

~ BARBIELLINI ET AL. + BEZNOGIKH ET AL.

5

X CI-IERNEV ET AL.

6

~‘

REF.[6]

BELLETTINI

ET AL.

7

S—(GeV)2 10

20

50

100

200

-

500

000

2000

5000

Fig. 11. Slope of the diffraction peak b(s) for It I < 0.2 (GeV/c)2 as a function of the square of the c.m. energy s. The solid line is a fit of the form b(s) = b 2. The other data are from refs. [34,12,3]; K. Chernev et at., 36B et at., Lett. 14 (1965) 164. 0 +Phys. 2a’ lnLett. (5/5~) only(1971) for the266; dataand of G. ref.Bellettini [6] and for s >Phys. 100 (GeV)

fit to the Fermilab data yields the following values [15] b(s)=b 0+2a’lns with

(8) (8’)

2 a’ = 0.278 ±0.024 (GeV/c)2. (GeV/c) This fit, which is included in the figure, passes through the Serpukhov data points [3] even though if those data are treated independently one obtains b 2, 0 order 6.8 ± 0.3than andlinear a’ = 0.47 ±0.09 (GeV/c) namely a higher value for a’. This suggest that a higher fit is necessary, and such behavior can be accommodated by the data. More about this later. Here we would like to make a general comment about the increase of b with energy, what is called the “shrinkage” of the elastic peak. (1) The effect is small: The energy must increase by one decade for roughly a 10% change in b. (ii) In itself b is a derivative of a differential cross section and we are here seeking the second derivative of a physical process. As defined by eq. (8) = 8.23 ±0.27

sdrd

/da

(iii) In recoil experiments, such as described here, one can obtain the s-dependence in one measurement by pulsing the jet at two well separated energies. Analysis of the data in this fashion yields results which agree within errors with those of eq. (8’). (iv) The error on a’ given by eq. (8’) is mainly statistical and should be treated with care. It is estimated that inclusion of systematic effects raises the value of L~a’to ±0.04 (GeV/c)2. Once the value of b has been established one can obtain a reliable estimate of the total elastic cross section. One needs the total cross section a 7 [16], in order to normalize the differential cross sections through eq. (6). If one ~thenassumes the validity of eq. (7), it follows that

A.C Melissinos. S. I.. Olsen, Physics (and technique) of gas jet experiments

98

Table 5 Ratio of elastic to total cross-section Plah (GeV)

8.5 11.0 50.0 58.0 102.0 150.0 200.0 270.0 312.0

17.8 22.5 95.6 110.6 193.2 283.2 377.0 508.4 587.2

1

AU~

or (Inb)

2

1GeV/c)

40.00 39.70 38.46 38.44 38.40 38.70 39.00 39.10 40.40

±0.20 ±0.12 ± 0.12 ± 0.12 ± 0.10 ± 0.10 ± 0.10 ± 0.40 ± 0.28

°el 1mb)

°inel (rnb)

01/aT

10.80 ±0.32 9.98 ±0.24 7.62 ±0.12 7.49 ±0.08 7.08 ±0.09 6.94 ±0.11 6.99 ±0.08 6.91 ±0.15 7.27 ±0.17

29.20 0.38 29.72 ±0.26 30.84 ±0.17 30.95 ±0.15 3l.32 0.14 31.76 ±0.15 32.01 ±0.13 32.19 ±0.43 33.13 ±0.33

(.1.270 0.251 0.198 0.195 0.184 0.179 0.179 0.177 0.180

± 0.008 ± 0.006 ± 0.003 ±0.002 ± 0.002 ±0.003 ± 0.002 ±0.002 ±0.003

2

and the ratio of elastic to total cross sections is ~l

0T

0T

1

1 2)

167r —(l+p b

~—(l+p2)

U 1

20.5 b

where 0T is expressed in millibarns and b in (GeV/c)2. The values of 0T and 0e1 and 0jn and Gel/UT are given in table 5, and results [17] for Gel/UT are shown in fig. 12 together with results from other experiments. We note that the ratio Gel/UT seems to tend to a constant in spite of the rising 0T’ and we will discuss this observation in detail in the following section. Finally, for completeness we show in fig. 13 the total p—p cross section including the points recently measured at the Fermilab [16].

.4

.3

II[

~1~t I

-

•

VAN HOVE LIMIT

•

.1

,••~

L

• REF. ]~7~

~___..t

0

I._._j___1___1. .l.j_1__~_~~__

~

I00 ~LA8

~

LiL~

I

1000

0eV/c

Fig. 12. The ratio of the elastic to the total cross-section OCI/OT for p—p interactions as a function of the incident momentum. The dotted line is the “Van-Hove” limit [18]. The data are trom refs. [17, 20].

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

II~I

31’ 0

I 20 I

I

III~

It 50

II

100

99

GALBRAITHe16S

I 200 I

11111

I

500

1000

I

I

Iii 5000 I

PLAB (0eV/c) 100

2

I 1000

I I 10000

s (0eV) Fig. 13. The total cross section 07 for p—p and ~—pinteractions as a function of the incident momentum (from ref. [16]). The solid lines are phenomenological fits to show the trend of the data (see ref. [39]).

2.2. Phenomeno logical analysis

The present interpretation of elastic p—p scattering data is mostly phenomenological in nature, and this approach has yielded considerable insight into the absorption profile of the proton. Since we are in the short-wavelength domain, the optical (or eikonal) model for the scattering is widely used. Note for instance that for 40 GeV/c momentum, A 27r(h/p) 0.03 f, as compared to an interaction radius R 1 f. Furthermore the elastic scattering amplitude appears to be almost purely imaginary (see part 3) so that forward elastic scattering results as the shadow of the absorption of the incident wave. We develop the eikonal formalism by starting from the familiar partial wave expansion in which the elastic scattering amplitude is ‘-~

a(O)—~-I~(2l+l) 2 1=0 2 k with th~normalization da/dt =

iT I

)P 1(cosO)

a(O)I2

and k is the c.m. wave number. S~is the S-matrix element in the lth partial wave 1—5, = 1— exp(i~,) In terms of a real phase shift 6, and an absorbtion coefficient i7~we write

(9)

(9’)

AC’. Melissinos, SI,. Olsen, Physics (and technique) of gas jet experiments

100

I -—S,

1 --~1exp(2i6,)

with ö1, m real and 0< m < 1: complete absorption corresponds toTh of eq. (9’) leads to

=

0, or

-~

ioo.

Integration

2 Gel

(10)

k2 ~ 1=0 ~(2l+1)I1 ---S11

=

and through unitarity to the additional relations =

~ (21+ l)( I

-~

1S 2) 1I

---

l=0

10’)

—~~(2l+l)(1~Re[S

0T

2

1=0

1]). (10”) In the high energy (eikonal) approximation the trajectory of the incident particle is not deflected through large angles so that each partial wave, 1, can be considered to correspond to scattering at a well defined impact parameter x through k

kxL

1.

=

We can then transform expressions (9) and (10) into impact parameter space by noting that for large k and small It I the following hold cosO

=

1— iti/2k2

limP and

2/212 —*0

2/212)=J

1(1 —y -*

1=0

for .v

0(y)

f~

1.

0

Thus a(t) =

if

(1

S(s,x1))J0(x1~x1 ~

(11)

where S1 S(s,x1) is now a continuous function of the impact parameterx1. The energy dependence is contained in S(s, x1) and for the moment we shall ignore it. Because of the familiar representation of J0, eq. (11) can be written also as -*

a(t)

~f(l 0

~S(x1))

f

2x, exp(ix1~~cosØ)dØx1dx1 = ~—f(l

0

---

S(x1))exp(iKx1)d

(11’)

where K and x 2 = -t and x~= x~.We see from the repre2-dimensional sentation of eq.1 are (11’) that [1 S(xvectors such that K 1)] is the Fourier transform of the elastic scattering amplitude, a(t). Thus S(x1) can be found from the data or alternatively constructed from various models. -

—

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

101

In the limit where the elastic scattering is purely absorptive, S is real and bounded by unitarity. For this case, a useful representation of S has been given by Van Hove [18] in terms of an “overlap function” G(x1) which measures the absorption of the incident wave at the impact

parameter x1. We write S(x1) = ~/1 G(x1) and it follows from eq. (11’) by definition, that the distribution of elastic scattering in impact parameter space is given by 2Xi = (1 1 —G(x 2 (11”) dGei/d 1)) and because of eqs. (10) we obtain the analogous expressions —

—‘~/

da~~/d2x

2xj 1G(x1),

2(1 —V’l —G(x

daT/d

1)).

Unitarity implies, as it must, that 2xl = dGei/d2Xi + dGjn/d2Xj.. daT/d We can then speak of elastic, inelastic and total overlap functions respectively. In the present context it suffices to note that G(x 1) represents the absorption profile, or opacity, of the proton as a function ofx1 and is limited between 0 ~ G(x1) ~ 1; for complete absorption G = 1, whereas G = 0 means no interaction. As we noted, G(x1) can be obtained from the elastic da/dt by inverting eq. (11’). Such calculations have been performed by many authors [19,201 and in fig. 14 we show G(x1), the inelastic overlap function as obtained in ref. [191, for \/~= 7.5 and ~ = 53 GeV. The relevant features of fig. 14 are that: (i) The absorption profile of the proton is very close to a Gaussian. (ii) At its center the proton appears to be almost black (at x1 0 the inelastic overlap function approaches its unitarity limit G(x1) -÷ 1); however, this is the region which is probed by -*

2.0

1

I

Total 2(i-./C~(x1))

-

30 GeV/c 1500 GeV/c

1.0 I~~ic~Gx~\

0.0

—

~~‘I.O R30

~Unitarity

Limit of G(x5)

2.0

R1500

Fig. 14. The elastic, inelastic and total overlap functions for p—p interactions at incident energies of 30 and 1500 GeV. Note that the inelastic overlap function is a measure of the absorption profile of the proton [19].

102

.4. C Mthssinos, S. L. Olsen, Physics (and technique) of gas jet experiments

large-t elastic scattering and the experimental information at high energies is not very extensive. (iii) The mean interaction radius is of the order of 1 f and grows with increasing energy. (iv) With increasing energy, the opacity of the proton at large radii appears to increase whereas it is relatively constant at small radii. It is possible that the absorption profile of the proton “scales” with energy but at present one cannot make a conclusive statement on this point. In any event the recently observed rise in the total cross section can be interpreted as being due mainly to the growth of the absorbtion at large radii, namely to an increase in the peripheral interactions. To reproduce the diffraction dip of the experimental cross section of fig. 10 at ~ = 53 the Gaussian G(x1) must be slightly flattened close to the origin and to account for the break shown in fig. 9 the opacity at x1 I f must slightly exceed the Gaussian form [20]. The above conclusions can also be reached without recourse to the complete determination of G(x1). One argues that in an optical model a uniformly absorbing disk of radius R gives rise to a diffraction pattern of the form [J1(R\/Ti{)/R\/TTi ] 2 which has secondary maxima of the order of 1/100 of peak amplitude. On the other hand the data are almost purely exponential in t (i.e. Gaussian in ‘~/17)for six order of magnitude which implies that the opacity distribution must be very close to a Gaussian, rather than that of a black disk. To estimate the mean radius of the opacity profile of the proton we note that for most distributions (including the Gaussian). the mean radius is related to the exponential slope b, through KR>~—~ Since b has been observed to increase, one concludes that the interaction radius grows with energy. Numerically, KR> = 0.395 ~/7J 2 and KR> in (fermi) which is comparable in magnitude to the values indicated with in (GeV/c)— in fig.b 14. Finally, a strong clue to the absorptive properties of the proton can be obtained from the ratio of the elastic to the total cross section. We know that for a black disk the ratio Gel/UT = ~ and this represents the unitarity limit. From the data of fig. 12 we note that we are far from this limit, and that the proton must be at least partially gray. The reason for the observed values of Ge 1/U7 can be traced to the Gaussian distribution of the opacity shown in fig. 14. Indeed, it was proved by Van Hove [18] that if the opacity is Gaussian in impact parameter space and G(x1) 1 as x1 -÷ 0 (i.e. G(x1) reaches the unitarity limit for a Gaussian), then Gel/UT = 1 (4--- 4ln 2)—’ = 0.185. This so called “Van Hove” limit is shown in fig. 12 and it is remarkable that the data given in table 5 are very close to this value, and that Gel/UT remains constant in spite of the rising UT. According to ref. [18] completely uncorrelated inelastic absorptive processes give rise to a Gaussian overlap function. —

2.3. Models for the slope parameter b The slope parameter for p—p elastic scattering has been measured with considerable accuracy because of the availability of intense proton beams combined with the sizeable cross-section for forward scattering. Thus b and its variation with energy b(s) are known to reasonable accuracy from treshold up to the highest c.m. energies achieved so far, notably at the ISR where ~ = 61 GeV.

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

103

It is therefore not surprising that the subject has received considerable theoretical attention and has even been treated as a test case in the early days of Regge theory. On the other hand, any model developed in order to predict b(s) for p—p scattering must account for all the features of the p—p data and also describe the elastic scattering of the other hadrons, ~p, ir~p,K~p.All such attempts lead to complications which destroy the simplicity of the original model and necessitate the introduction of an overabundance of free parameters. The large number of models that have appeared in the literature can be divided into three broad classes as follows: (i) Models based on Regge theory, (ii) special models developed to predict b, and (iii) general models of elastic scattering. In all cases the models are of phenomenological nature, even though they have to a varying extent some underlying dynamic basis. From our point of view the more satisfactory models, in spite of their asymptotic nature, are contained in class (iii) because of their generality. Nevertheless we feel that none of these models is compelling, especially when the possible errors in the published determinations of b(s) are taken into account in a realistic fashion. The general features of b(s) are shown in fig. 1 5a where the data are plotted from threshold up through ISR energies. We show b(s) as determined in the smallest t-region, pertinent to the gas jet measurements; namely we can characterize the data as referring to b ~ We note that b(s) rises rapidly and eventually reaches a linear behavior in ln s or a variation slightly slower than ins. However, we cannot exclude that a (lns)2 behavior sets in at s 200 (GeV)2 and continues from then on. Thus we are faced with the unfortunate situation that we do not know the asymptotic behavior of b(s), which is the deciding factor amongst the different models. (i) Regge models: The Regge model gives a scattering amplitude of the form 1

±exp{.i

ira.(t)}

,~

A(t,s) = ~ ‘y~(t) sin 71~ (:s:;;-) (12) with the normalization da/dt = IA(t, 5)12 and where the sum is over all contributing Regge poles and cuts. y 1(t) is related to the residue and a1(t) is the angular momentum of the trajectory at that value of t. The phase 0 is related to a1(t) as shown in (12). At high energies we expect only the leading trajectory to contribute, which for elastic scattering will be the Pomeron for which we write ap(t) =

1 +a~t=

1

—

and by convention Op(t da/dt

a’ItI = 0) =

ir/2. Thus the differential cross section becomes

F(t)exp {—2a’iti ln(s/s0)}

(12’)

which results in the linear dependence of b(s) as given by eq. (8), b(s) = b0 + 2a’ ln s with the scale factors0 absorbed in b0. A fit to the data linear in ln(s) is shown in2. fig. and it is clear that if eq. is assumed This15a is in contradiction with the(12’) presumed successes valid, it must be which restricted s 100 (GeV)in a much lower s-region. A more satisfactory predicof Regge theory havetobeen observed tion can be obtained by including the contributions of other Regge trajectories in addition to the Pomeron. Furthermore, one can simultaneously accommodate the ~p data. An example of this approach has been given by Barger, Phillips and Geer [21] who use the P, f and w trajectories to fit the jip and pp data. It holds A(pp)Ap+Af—AW,

A(j5p)A~+A~+A~

104

AC’. Melissinos, SI,. Olsen, Physics (and technique) of gas jet experiments

4

L

~

I

I

I

I

ilT

i,-. ~

‘3 I2H

~‘tPP~

AMALDI

-

9

,1 ~

—

ET AL

-

*

BARBIELLINI ET AL BEZNOGIKH ET AL

-~

FOLEY

8— 7

~I_I

I

I_~~ilj

.

I

ET AL

00

.JIIIIJ.!

I

0000

I 000

2

s (0eV) (II)

14

~r

‘‘‘‘I’’’’’’

-

/

//

L

I

I

•

‘‘‘‘‘I

REF.L6~

BARBIELLINI DAMALDI A

4

:

I

o

-,

BEZNOGIKH LASINSKI

I 10

• BELLETTINI

100

I

000

10000

s (GeV)2 (b)

Fig. 15. The energy dependence of the slope of the diffraction peak h(s) for p--p elastic scattering for Ft

<-. 0.1 (GeV/c)2. (a) The solid line is a Regge fit including only the Pomeron; the dashed line is a Regge fit including the Pomeron, f and w trajectories [21, 22). A few ~p points from K.J. Foley et at., Phys. Rev. Letters 11(1963) 503 are also shown. Note however that the slope for ~p is obtained using a much wider t-interval and thus cannot be compared directly to the p—p data. (b) The solid line is the fit of the geometrical model [251.The dashed line is the fit of an eikonal model with Reggeized energy dependence [31]. In addition to the references cited in fig. lithe data of Lasinski et at., Phys. Rev. 179 (1969) 1426 are also included.

and the energy dependence ofAf, A~ is of the form (s/s 6.Thus by an appropriate choice of 0~° the t-dependence of the residues ‘y 1(t) one can fit UT, du/dt as well as the antishrinkage of jip. This is shown by the dashed curve in fig. 15a [22].

A. C. Melissinos, S.L. Olsen, Physics (and technique) ofgas jet experiments

105

In general, lower lying trajectories or cuts yield an energy dependence of the form (s/s0 Y°~6, so that the asymptotic behavior of b(s) is still controlled by the Pomeron. If one accepts a sloping E trajectory, shrinkage will continue to infinity and should be manifested by all hadrons. On the other hand one can assume that the Pomeron is a simple fixed pole at ap = 1 (and include appropriate cuts) so that asymptotically b(s) reaches a finite limit. In this case the shrinkage observed at present energies is due to the decreasing influence of destructive cuts. A discussion of these possibilities has been given by Barger et al. [23] and an example of a calculation using cuts can be found in Fujisaki and Akiba [24]. As is evident from the work of refs. [21—24] the inclusion of lower lying trajectories and/or cuts allows the adequate fitting of b(s) and other elastic data. However, these models contain a large number of free parameters and/or assumptions so that they lack predictive power and must be considered, at best, as phenomenological. Furthermore, the exact behavior of the Pomeron still remains obscured because of the necessary recourse to auxiliary trajectories. It is an old saying in Regge language that data at higher energies are needed to provide definitive tests between competing models. However, with the observed ln s dependence of high energy phenomena this may take a long time. (ii) Special models: As examples of special models we will mention those of Krisch [251 and of Leader and Pennington [261. They are based on geometrical considerations and have the characteristic property that b(s) tends to a limiting value b~furthermore b(s) reaches 99% of its asymptotic value at s = 400 (GeV)2 which is within the range of the available data. Krisch proposes that the interaction density is a pure 3-dimensional Gaussian, which becomes Lorentz contracted along the direction of incidence. Thus Pei(~’)=B

expf—i 2 + (yx 2]} 2[(x1/R) 11/R) The square of the Fourier transform f Pe do/dtAexp{—R2p~(1—l/y2)}

3x gives the differential cross section

1Cv) exp (ik ~x)d

(13)

where a term exp(—R2~32m~) has been absorbed in the constant A. Identifyingp~= ItI, eq. (13) becomes analogous to eq. (7) with

(13’)

b=b,,,(l—l/y2)

where ~yis evaluated in the p—p c.m. system, namely, ~2 = s/4m2. The prediction of eq. (13’) is shown in fig. 15b by the solid curve and while it describes the low energy behavior adequately it predicts that b is almost flat for s 400 GeV. A fit to the data yields b 0,, = 11.61 132 but misses the ISR points. One can introduce a rising b(s) in this model by postulating an increasing interaction radius as an explanation for the rising UT. Equation (13’) is then modified to read b = b’,~(uT/o~)(l l/y2) where a~is a suitably chosen scale factor. Leader and Pennigton base their model on a group theoretical approach, and find that 2+m2)/s+(M2—m2)2/s2] (14) b(s)b[l—2(M where b 0~is a free parameter different for each pair of interacting hadrons. For the p—p case (M = m) their result is identical with that of Krisch and thus not in agreement with the data. Similar ideas have been developed by Barshay and Chao [27]. (iii) General models of elastic scattering: Several authors have proposed models which are directed at a global, if not detailed, description of hadronic scattering. The most familiar among —

AC. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

106

these models are that of Chou, Yang and Wu [281and of Cheng, Wu and Walker [29]. Both models are asymptotic in the sense that they become fully valid ass even though several improvements and variants have been developed to obtain reasonable fits to the existing data at presently available energies. Wu and Yang proposed in 1965 that at high energies the absorption of the incident wave is given by the folding of the matter distribution of the two interacting particles; this distribution is presumed to be specified by the (Fourier transform of the) charge form factor of the particles. Namely. the S-matrix element in eq. (11’) is given by -*

SAB(XI) =

~

exp~--~AB(xl)~

x~)DB(xi)d2x’T where D(x1) are 2-dimensional opaqueness distributions. For p—p scattering and using I S(x1) ~(x1) one obtains the Wu----Yang limit to elastic p—p scattering 4(t) du/dt pp elastic = constant G where G(t) can be represented by the dipole expression G(t) = (1 +ItI/0.71Y2 with tin (GeV/cY2. This prediction appears to approximate the correct t-dependence, but as shown by the dot ----dash curve in fig. 16 it fails quantitatively; the model is equivalent to the Regge expression of eq. (12) assuming a simple fixed pole for the Pomeron and the square of the form factor for the residue. Chou and Yang showed that one must take into account the higher order terms in exp{ ~AB(xI)~ in which case much better agreement with the data are obtained (see ref. [28]). In the framework of the Regge model this corresponds to the introduction of cuts or multiple pomeron exchanges. Further improvements can be made on the model as shown by Durand and Lipes [30] where now diffraction minima are predicted at high energies. Their result is given by the dashed curve in fig. 16, in fair agreement with the ISR data at \/~= 53 GeV. The Chou—Yang—Wu model has no energy dependence, in contradiction to the observed rise in UT and the present behavior of b(s). One can, however, introduce, a posteriori, an energy dependence into the model as shown by Carreras and White [31]. By fitting the then existing data these authors find values for b(s), which however, depend on tI. Their results for I [I 0.008 are shown by the dashed curve in fig. 1 Sb. A somewhat different approach to the scattering of hadrons was pioneered by Cheng and Wu who use an analogy to electrodynamic processes. The exchanged quantities are assumed to be massive photons and in the high energy limit it is possible to sum the contribution from complicated tower graphs. From these considerations they deduce an optical model where the absorption strength is a function of the incident energy. The model contains a large number of free parameters (fourteen) needed to fit the data on irp, pp, Kp and j5p interactions; its notable success has been the prediction of the rise of GT(PP) before it was experimentally observed. As far as the s-dependence of b, the model predicts that in the asymptotic region it should follow (in s)2 just as UT. The shrinkage seen in reference [6]is faster than their model would predict indicating that at these energies the exchange of Regge trajectories lower than that of the pomeron are important. This suggests a combination of the Barger—Philips and Geer approach (which has a sharp ~o.6 behavior) with a more gradual asymptotic shrinkage. Our discussion of the various theoretical models has been brief and far from exhaustive. We have given the main conclusions of only a few representative models in order to show that b(s) can be adequately fit under a variety of assumptions. This in itself leads us to believe that p—p with

~AB(x±)

=

constj’DA(xI

—

—

A. C. Melissinos, S. L. Olsen, Physics (and technique) of gas jet experiments

I

I

I

I

I

107

I

pp.—pp

./~=53GeV

10

-

4 BK5HMETAL.

10’

D

:::

(0

-

-

-

~

10’ ~

I

234567

It (0eV/c)2

Fig. 16. The differential cross-section for p—p elastic scattering for large ti and at an incident c.m. energy squared 5 = 2800 (GeV)2 [13]. The solid line is a fit using an eikonal model with Reggeized energy dependence and regeneration (S. Olsen [30]). The dot— dashed line is the fourth power of the electromagnetic form factor G4(t) [28] while the dashed curve is the prediction of a simple eikonal model (Durand and Lipes [30]). Table 6 Asymptotic behavior in various models Model Regge poles and cuts (with ap(O) Cheng, Wu and Walker [29], or Saturation of Froissart bound Regge dipole Observed so far

°T

b

Gel/aT

const.

ln s

(In s)~

(lns)2

(lns)2

1)

(ln sy2 —~0from positive values

1/2 -

ln s

ln s

(lns)2

(lns) or (const — s~~2)

const. 0.18

(lnsi’ 0 from positive values const.

crosses zero atE

-~

300 GeV

108

A. C

Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

elastic scattering is still a very complex process, which may or may not develop into a simpler regime. In table 6 we summarize the asymptotic features of the various classes of models which nevertheless must wait for their confirmation at substantially higher energies. 2.4. Bounds from local field thearl: From the basic axioms of local field theory (QFT) several bounds on elastic scattering crosssections can be derived, and we refer the reader to the detailed review by Eden [32]. The implications of these axioms on the scattering amplitude are (i) Analyticity (ii) Unitarity (iii) Polynomial bounds as s, t Most of the bounds are asymptotic, in the sense that they are valid for .s ~: furthermore, the energy scale is not fixed in the sense that the scale factor s0 is, a priori, unknown. This makes the —~

—~

comparison with experimental data difficult and one is more concerned with the form of the energy dependence of the scattering amplitude, rather than in its absolute magnitude. The best known of these bounds is the one due to Froissart [33] on the total cross section 2 as s-~~ (.15) ‘U [ln(s/s~)] where ~i is the mass of the pion, the lightest particle that can be exchanged in the t-channel. If we ignore the scale factors and thethe constant = 60that mb,all eq.total (15)cross statessections that UTtend cannot faster 2. Since until0recently belief ~/‘U2 was held to arise constant, than (lns) the Froissart bound was acadelnic. We now know that not only U 1 for pp rises [34] with s, but the same holds for the other hadrons [16]; thus one asks whether the bound of eq. (15) is violated or not. To examine this problem we can consider the following parametrization of UT for pp scattering at high energies 2 (mb). (15’) UT 38.4 + 0.49 [ln(s/122)] One may take the point of view that eq. (15’) saturates the Froissart bound in view of the [ln(s/122)]2 dependence. On the other hand the coefficient (0.5 mb) in the experimental data is much smaller than 7r/’U2 60 mb so that there is ample room for an even faster s-dependence at present energies which will eventually conform to eq. (15). Furthermore, the parametrization of eq. (15’) is not unique and UT(PP) can also be fit with a linear dependence in ins (UT = 38.44 + 0.9 ln (s/ 180); s>350). In any case, the choice of the scale factor s 0 is crucial in any attempt to test QFT by the available data [35]. One may even argue that the scale factor s0 is of fundamental nature, however results discussed in part 3 tend to indicate that asymptotia is still far away. We now turn to bounds which relate elastic scattering to the total cross section. In many cases it is necessary to assume that the scattering amplitude is almost purely imaginary, and spin independent. Both of these conditions seem to be quite valid in our present energy range. We start with the obvious relation ~

-~-

Gel~UT

which for purely diffractive scattering becomes

(16)

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

109

(16’)

UeI~~GT•

Next, it is convenient to define an effective slope parameter, beff, through Gel =

(dG/dtlt=O)/beff

(17)

.

Namely, beff is the average exponential slope of the elastic scattering, and from figs. 9, 10, we know that beff b. If one further assumes that the elastic scattering is purely diffractive, the dptical theorem gives (da/dt)~...0= o~./i6ir

(6’)

or beff =

(~)

~

(17’)

.

Combining this result with eq. (16’) we have that for diffractive scattering

(18)

87T beff>GT/

where ~fbeff is expressed in (GeV/c)2 and a 1 in (mb), beff> cj1/lO, a bound amply satisfied. We can also insert in2eq.which (17’)isthe observed values Of (Gel/UT) ~ 0.18, UT 40mb to obtain very close to the measured value of b. What is of interest in eq. beff = 11.1 (GeV/c) (17’) is that if Gel/UT remains constant, then beff must have the same s-dependence as UT. A lower bound on the slope parameter b has been given by McDowell and Martin [361 in the form 2~j[lnImF(s~t_0)] b~ 0>

(j~j-)1/(~)

(19)

so that b~.0is the slope parameter of the diffractive cross section at t = 0. The physical origin of this bound is based on the unitarity limit of the partial wave expansion coefficients (see section 2.2). If the total cross section rises, this implies the presence of more partial waves, which in turn implies a growth of the interaction radius. But a larger radius results in a sharper diffraction peak and thus larger b. The result is valid at high energies which however need not be asymptotic as in the case of the Froissart bound. Combining eq. (19) with eq;(l7’) we obtain the bound b~0> ~beff

(19’)

which is stringent. For instance, it shows that the elastic slope parameter cannot have a turnover as t -÷ 0 (but it is allowed to grow). Furthermore, as shown in ref. [36], the fact that experimentally bt=O/beff 1 implies that the absorption profile of the proton is close to Gaussian and not a black disk, or exponential; this is in complete agreement with the conclusions of section 2.2. There are also upper bounds on b. It holds [32]

c4

<~~_Uei[ln(5/So)]2

p 2. Making use of eq. (17’) we can which implies that Gel cannot drop faster than [UT/ln (s/s0)] write instead

(20)

110

A.C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

beff <

~

1 6p

[ln(s/s0)12.

(20’)

Finally, it holds [32] [ln(s/s

d

2 -i-- [lnF(s,

t = 0)] =

3 0)]

b~ 0<

C\/UeI/UT

(21)

—------

where C is a constant. We do not discuss the bounds at fixed t or for the ratio of real to imaginary part since they are in general weak [32]. Perhaps most all important conclusion is that ifUTtheis Froissart bound 0e1 andthe b must grow as (lns)2. At present, compatible withisthis saturated then both UT, behavior, the elastic scattering is purely absorbtive and it appears that uCl/GT 0. 18. On the other hand, the observed behavior b(s) ‘x ln (s/s 0) is not compatible with the above statements. The problem lies in the fact that the parametrization of eq. (l~5’)is not unique; as usual, more precise or higher energy data could decide between these alternatives. -~

3. The real part of the forward nuclear amplitude 3.1. Determination of the real part The nuclear elastic scattering amplitude, F(s, t), is a complex function of the two real variables s and t. For each value of s, t, we define the ratio p p(s,

t) =

Re F(s, t)/Im F(s,

t)

.

(22)

In the forward direction, p can be determined through the optical theorem (eq. (6)) if both UT and (dU/dt)eiastic (extrapolated to t = 0) are known to sufficient accuracy. A more sensitive approach is to observe in the near forward direction the interference between the Coulomb and nuclear scattering amplitudes. The Coulomb scattering is almost purely real, its phase being calculable, so that the interference contribution is a direct measure of ReF(s, t). Let us estimate the t-value at which the Coulomb and nuclear amplitudes are equal 2V~~—~--exp(—bt/2). 2, namely a recoil proton of kinetic energy For 40mb obtain t (GeV/c) T UT 1 MeV. This we energy range is0.002 ideally suited for the solid state detectors and since the density of the hydrogen jet is low, recoils of such energy can leave the target with only small multiple scattering. Furthermore, the detection conditions are again almost independent of the incident energy, making possible a precise determination of p(s). It is assumed that over the small t-range covered by the interference region, the t-dependence of p can be neglected. The experimental apparatus was identical to that used for the measurement of the slope parameter except that the t-range was reduced by placing all detectors closer to 90°.The total range covered was 0.0011 < tI < 0.04 (GeV/c)2. At the lowest t values the fractional t-range spanned by a single detector becomes sizeable. A detector centered at It) = 0.0011 (GeV/c)2 has an elastic t-acceptance of ~t/t 20%. The variation in da/d~over this range is even larger, ±30%. Elastic

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

ill

~±‘‘‘‘~,‘cc:

~:

0 360

300

(b)

240 80 20 60

p.1_In

fl,_~

20

40

60

80

00

p (MeV/c)

20

40

60

Fig. 17. The recoil proton momentum spectrum for a detector positioned at 56 mrad forward of 90°,at an incident energy of 400 GeV. The mean energy of the elastic peak is 3.2 MeV and the shape of the peak reflects the density distribution of the gas jet. (a) The “foreground” spectrum, (b) the spectrum after subtraction of the background.

peaks plotted versus the recoil momentum were adequately fitted by convoluting a Gaussian beam-jet profile with da/d~2.Fig. 17 shows an elastic peak in momentum space centered at I tI = 0.012 (GeV/c)2. In the small-t region covered in the measurements of ref. [7] the corrections for nuclear interactions in the detector material were always < 1%. The dead time corrections for these data were 1.5%. The t-value inferred from the kinetic energy of the elastic peak correlated well with the surveyed angular position of the detector. Nevertheless the final momentum transfer scale was set by minimizing the overall x2 of the fit. This involved overall shifts in the angular position of the detector carriage of 0.3 mrad. A total of 6 million events were obtained a 17 distinct beam energies. Typical differential cross-sections are shown in fig. 18 where the Coulomb peak can be prominently distinguished in comparison to the nuclear slope. In general, data in this region are of great statistical accuracy because the detectors are best matched to the energy of the recoil protons. Note, however, that the data are not normalized but give only the shape of the angular distribution. One could attempt to normalize the data by using the known Coulomb cross-section and the measured values of It. This is, however, impractical because of the very rapid t-dependence of the cross section. For instance a 2% relative accuracy in the cross section, at ItI = 0.002 implies ~ t 2 X 10~ (GeV/c)2, namely an absolute measurement of Trecoil to 10 keV. This is beyond the capabilities of the system, and should be contrasted with the width in T-space of the elastic peak due to the finite extent of the jet. However, the use of a continuous detector, or of a position sensitive detector [8] could perhaps make such a normalization possible. The alternative approach is to assume that UT ~5 known, and through the optical theorem normalize the extrapolated nuclear cross section; this was the method adopted for the analysis of the

A. C. Melissinos, S. L. Olsen, Physics (and technique) of gas jet experiments

112

500r~~ I— E=5OGeV 0

400

2

~~—--~

1

-—

Ill (GeV/c)’—

4

6

8

500 ~----—-~

I0x10°

-----—-

~—

~——------~

Er 400 0eV ~

400L

2

-

ItI (GeV/cI2

8

F 300I

200

38 44mb

200

~iX/~68

I

I

I

.005

I

I

/

FIT FOR C~’ 4O=b~

IOO\\~\1

~

I

.010

I

I

X/df’09 FIT FOR 0 ~38 44mb

I

I

I

I

p~-oI78

I

0I5

.020

025

________________

.030 .035

005

.010

.015

ti (GeV/c)2—-

020

025

030

.035

it) (0eV/c)2—

(a)

(b)

Fig. 18. The measured differential cross-section du/dt for p—p small angle elastic scattering at incident energies (a) 50 GeV, (b) 400 GeV. The solid curve is the best fit to the data which are normalized as discussed in the text. The dashed curve is the resulting fit if one assumes 0T 38.44 mb and (a) p 0, (b) p — 0.025; however for this case the da/dt scale must be multiplied (a) by 0.91, (b) by 1.06.

=

=

=

data. The differential cross section can be represented by the Bethe interference formula [37] -~ ~

=

K [(2

where: K a G(t) a~

)2

G4(t) — (p +~) ~

IOxIO°

300-

--

BEST FIT FOR 0

—~

G~(t)ebt/2

+ (U~2

(1

+ p2)

ebt]

(23)

is an overall normalization factor, is the fine structure constant, is the proton form factor = (1 + It 1/0.71 )_2, is the phase of the Coulomb amplitude. The phase calculated by Yennie and West [38] wasused, whereaØ =a[ln(t 2, C 0.577. These 0/ItI)—C], intoa~to = 0.08 authors estimate the theoretical uncertainty be (GeV/c) ±0.015, comparable to the precision of these measurements. b is the nuclear slope parameter. In principle one could fit the data to eq. (23) treating K, UT, p and b as free parameters. However, for these data sets, this procedure does not result in convergent fits. Therefore the total cross sections were fixed according to the fit of Leader and Maor [39], (see also fig. 13)

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments 0T = 38.4 +

0.49 [ln(s/122)]2

ii ~

(mb)

(24)

and the slope parameter b was constrained to within a Gaussian error of ±0.2 (GeV/c)2 about the values given by eqs. (8) and (8’), b = 8.23

+2 X

0.278 (lns) (GeV/c)2.

Then the overall normalization K and the ratio p were left as free parameters. The CERN minimization program MINUITS was used and the errors were obtained from the covariance matrix. Differential cross sections at 50 and 400 GeV are shown in fig. 18. The solid curve is the best fit result whereas the dashed curve results if one sets p = 0, and p = 0.025. However one must bear in mind that the overall normalization for the solid and dashed curves is different, the purpose of the figure being to show the change in shape due to the presence of the interference term. The results of the fit as well as the input a 1 are given in table 7. Also included are the correlations between p and UT, b, and ~ (the angular position of the detectors). The uncertainty in these variables has been included in the listed values of ~p. In addition there exists a possible overall systematic uncertainty which can shift p by ±0.0 15. —

Table 7A The ratio p(s) = ReF(s)/ImF(s) of the p—p forward scattering amplitude 2)

E (a) (GeV) 51.5 94.5 145.0 174.6 185.4 215.5 244.1 269.2 348.7 393.0

(GeV 98 178 273 329 349 405 459 506 656 739

~ (b) —0.157 ±0.012 —0.098 ±0.012 —0.064 ±0.010 —0.039 ±0.012 —0.038 ±0.014 —0.020 ±0.012 —0.013 ±0.010 +0.022 ±0.015 +0.025 ±0.015 +0.039 ±0.012

(GeV/cY2 b (c)

(mb) °T(d)

10.80 11.13 11.36 11.46 11.48 11.55 11.62 11.69 11.86 11.90

38.44 38.46 38.71 38.88 38.94 39.11 39.27 39.43 39.79 39.98

(a) The energy bins are centered at the value indicated and are typically 20 GeV wide. (b) The data are subject to an overall energy independent systematic uncertainty of ±0.015. (c) These values are obtained from the fit when b is constrained with a Gaussian error of ±0.2 (GeV/cY2 about the value given by eqs. (8) and (8’). (d) The values of a 1 used as an input to the fit. Table 7B

Systematic errors in the determination of p(s) Source of error

2

(a) Error in b; ~p/~b~+0.04/(GeV/cY (b) Error in UT; ~p/~aT —0.025/mb

Typical contribution to ~p <± 0.008 <±0.006

(c) Error in overall detector angular positions;

~p/~

—0.04/mrad

<± 0.012

114

AC’. Melissinos. SI.. Olsen, Physics (and technique) of gas jet experiments

II

r~I

~~L—

I

.

-

,

1.7

1000 “0

o

‘

.5

.5

inn “.5

0

.2’

0.005

0.02 —‘

~

0.03

(GeV/c)~

Fig. 19. The differential cross section do/dt for p—p small angle elastic at all energies but after subtraction of 2 ofscattering the fits are alsomeasured shown. Note the effect of the nuclearthe square of the Coulomb amplitude. The fits to the data and the x Coulomb interference which at low energies is constructive and eventually becomes destructive.

We note from table 7 that the ratio p changes sign in the vicinity of 300 GeV and becomes positive. A positive p implies destructive interference because the electromagnetic p---p force is repulsive. This behavior of the interference term can be seen in fig. 19 where the data points and the corresponding fit at each incident energy are shown. However, the pure Coulomb cross-section has been subtracted so that we plot only the sum of the nuclear contribution and of the interference term. The quality of the fit is indicated by giving the x2, where typically every distribution contains of the order of 70 data points. The values of p so obtained are shown in fig. 20 together with other existing data. The dashed curve gives the expected behavior of p determined from a dispersion relation using a total pp crosssection which tends to a constant as was widely believed until recently. On the other hand if the total cross section rises, p must become positive and tend to zero from above as discussed in the following section. These data confirm this prediction strongly because the correlation of p and a1in eq. (23) is such that ~P/~UT —0.020 (mbY’. Namely, if a flat ~T had been used as the input to the fit the value of p at the highest energy would have been increased by ~p = + 0.03. In turn such a rapid rise in p would be completely inconipatible with the flat UT dispersion relation result. In any ca~e,the recent measurements of UT in this energy range [16] are in agreement with eq. (24) and are sufficiently accurate so as to contribute negligibly to the error in p.

A.C. Melissinos, S.L. Olsen, Physics (and technique) of gas/st experiments

+0.2

I

TI

lilT)

I

I

IT

11111

I

I

1111111

115

I

I

111111

+

0-Ii

f

~

J

4i_O_ —0.5

I

I

2

——

REF.[7]

liii)

4

7

x AMALDI et.aI. ~ BEZNOGIKH et.al. 0 KIRILLOVA et.oI. • FOLEY *1.01. 0 BELLETTINI et.aI. ~ VOROBYOV et.al. I TAYLOR et.ol.

I’

I

I * I~ I i~lI1~’ I J.klW ‘--4—-— ~1 I~

-

—

.

-

-03

—

a

-

I ‘1’ I

I

10

i

~

20

111111)

40

I

I

70 100

1111111

400

I

1000

-

I

I

11111

10

LAB. ENERGY {GeV] Fig. 20. The real part of the p—p elastic scattering at t = 0 as a function of incident energy. The dashed curve is the result of a dispersion relation calculation assuming that the total cross section remains constant at °T= 38.4 mb forE > 50 GeV. The dot— dash curve is a dispersion relation calculation from ref. 141 for the lower energy data. The data are from refs. 17,34,4], L.F. Kivillova Ct al., Zh. Eksper. i Teor. Fiz. 50 (1966) 76; K.J. Foley et al., Phys. Rev. Lett. 19 (1967) 857; C. Bellettini et al., Phys. Lett. 14 (1965) 164; A.A. Vorobyov et al., Phys. Lett. 41B (1972) 639; and A.E. Taylor et al., Phys. Lett. 14 (1964) 64.

The use of eq. (23) is justified only if, in this energy range, the nuclear scattering amplitude is spin-independent. In general this is believed to be true and in agreement with a recent analysis by Bourrely et al. [401. These authors use three spin-dependent amplitudes to make a fit to all available small angle data and conclude that the contribution of spin effects to the differential cross section is of the order of 1 —4%. Furthermore, the values of p are not affected. It is our opinion, however, that the data are not of sufficient accuracy to unambigously conclude the absence of spin-dependent effects from the forward scattering amplitude. As is well known, the real part of the scattering amplitude is related to the absorptive part through a dispersion relation. The present data permit a strict test of these relations and this is the subject of the following section. In addition the data demonstrate the smallness of the real part of the nuclear amplitude, namely that at high energies elastic scattering is almost purely diffractive. As we saw in part 2, this is basic to many models of the strong interaction. Finally, in analogy with potential theory, where the sign of the real part determines the sign of the p—p force, one can infer that the forces between nucleons which at low energy are attractive, and soon become repulsive, revert again to attractive at an energy of 300 GeV. ‘—‘

AC Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

116

3.2. Interpretation of the results In this section we interpret the results on the real part of the forward scattering amplitude with the help of dispersion relations which express the real part of an analytic function as an integral over its imaginary part. Dispersion relations were first applied to nuclear scattering amplitudes by Goldberger et al. [411, and have been widely used since then. In particular, when the energy is far removed from the unphysical region (i.e., the threshold for scattering) the predictions of the dispersion relations become more reliable and easier to calculate. If the scattering amplitude is analytic in the cut complex E-plane, Cauchy’s formula is applicable Ref(E, t) = lmf(E’,t) dE’ (25) L

L

where the contour is as shown in fig. 21 a. If one considers the forward amplitude (I = 0) then unitarity relates the imaginary part of f(E, t = 0) to the total cross-section via the optical theorem: here we used the normalization IrnJ(E) with the

s 167r

Em 8ir

UT(E)_UT(L)

energy E expressed in the laboratory frame. Finally, assuming the boundedness of the

scattering amplitude as E -÷oo, the contour integral can be reduced to a principal value integral on the real axis by usual techniques.

Im(E)

(eshold __-Re(E)

—

Rising ~

__-E

E’~E

(b)

Jig. 21. (a) The contour in the complex E-plane used for the evaluation of the dispersion relations. (b) The principal value integral which is the dominant contribution to the dispersion relations at large energies E.

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

117

There are, however, still two problems left. One is the fact that in the region m
m Ref±(E)constant+_—f dE

UT(E’)~T(E’)

(26)

~E’-÷E ~E’±E

where the principal value of the integral is implied, and we have neglected the nucleon mass. The physical content of eq. (26) is best seen by expressing it in the following form; for simplicity we use only the f~ amplitude

~

Ref~(E)constant+ 8ir2 E2 +11±Ef

2

m

f

dE’

m

±—~I~—— [c~~(E’)—~

(26’)

1(E’)]

E’2E2

8ir

where the ratio p(E) is given by ~

E - Ref÷(E)- 8irRef~(E) ~Imf+(E) EmaT(E)

At high energies we expect [aT(E) —~ p(E) =

constant EUT(E) +

1(E)1-÷ 0 so that p takes the form 2 —E2) UT(E). ~a1(E) 2E ~ (E’dE’ ,

As E oc the first term vanishes and the major contribution to the integral comes from the region E’ E. This integral is represented in fig. 21 b where it can be seen that the positive contributions come from E’> E while forE’ < E the contribution is negative. Thus, a constant cross section implies p = 0 while a rising UT results in p> 0 and a decreasing UT in p < 0. Furthermore, for any rise in UT slower than Elab, the ratio p will go to zero from above as E -÷ o~,provided UT(E) aT(E) 0. A rigorous proof of these conditions on p was first given in 1965 by Khuri and Kinoshita [43]. The integral of eq. (26’) was evaluated numerically by using a parametrization for the difference in the pp and j5p cross-sections and for the energy dependence of the sum of the cross sections based on fits to the existing data [44], -~

—

-~

= 27.8E°602

(mb),

~(a+a)

= 38.4+ 0.49

[ln(2Em/122)]2

(mb)

The range of integration was from E’ = 10 GeV to infinity and evaluated for 50 < E < 106 GeV. The subtraction constant was fixed by making a one parameter fit to the data and the result is shown in fig. 22.

.4. C. Meljssjnos, S.L. Olsen, Physics (and technique) of gas jet experiments

118

I

Ii~ -04

I

~,

~‘~~‘I

•I!T

11111

-

-

05

‘‘‘‘‘‘‘

I 00

0

I 000

~

I

‘‘‘‘‘i’~

0000

jail

00000

1000000

ELAB (GeV) —‘ Fig. 22. Results of dispersion relation calculations for the real part of the p—p elastic scattering in the forward direction. The p—p and ~—ptotal cross-sections were parametrized as discussed in the text. Results are shown for cases where °Tbecomes constant at the indicated energies.

We also show curves for the case where the growth of the total cross section is presumed to stop at energies of 120, 1020, 2000 GeV, etc., as shown in the figure. The purpose of these curves is to indicate that the measurement of the real part at some energy E can foretell the behavior of °T at higher energies. For instance, the results indicate that the total cross section continues to rise at least up to equivalent laboratory energies of 2000 GeV. Since at high energies the cross sections are generally smooth, one can also obtain a quasiocal connection between the ratio p and the total cross-section. The idea is that if the cross section and its derivatives at E are known, this suffices to evaluate the dispersion integral and determine p(E). Bronzan et al. [45] have derived the following “derivative analyticity relation” Refe(s,

t) =

=

sa [tan [~ (~ 1 —

tan

[~(~ 1)1 —

2

+

d1ns)]~n)

Imfe(s,

t) +

~

2

(

[~(a

sec~

L2

—

1)]

d Imf5(s, t) + dlns ~o

L

which is valid for amplitudes that are even under crossing symmetry; a corresponding relation is found for odd amplitudes, and Ref (pip) = ~(Ref~± Ref 0). The parameter a was chosen a = 1.0 for the even amplitude, and a = 0.39 for the odd amplitude by fitting (~T UT) = CEU~in agreement with our previous choice. The energy dependence of (a-1- + ~T) was obtained from the data and thus in agreement with the parametrization used to derive the curves of fig. 22. The predictions of ref. [45] for the ratio p for pp and ~p are shown in fig. 23 and as expected agree with the asymptotic curve of fig. 22. This indicates that a very precise measurement of UT and of its derivatives contains information on the behavior of UT at higher energies, as it must be true for any analytic function. —

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas/ct experiments

~l~O.2

-0.2

I

I

I I I I

II

I

.

I

I

I

I I I

I

I

I

I

.

~JJ

I

I

I

I I I I)

I

I

I

I

I I I

X AMALDI ET AL. ~ BEZNOGIKK ET AL. 0 KIRILLOVA

•

-

El AL.

FOLEY El AL.

~ VOROBYOV

-

El AL

PP

-0.4-

V FOLEY

I

I

I

• REF. [7]

Q. —0.3

119

2

I

I

111111

4

7

10

I

20

I

1111111

40

70 100

I

I

-

El AL.

1111111

400

1000

I

I

111111

l0~

LAB. ENERGY [GeV] Fig. 23. Calculation of the real part of p—p and ~—pelastic scattering in the forward direction using derivative analyticity relations [451. The ~p point is from K.J. Foley et al., Phys. Rev. Lett. 19 (1967) 857.

A different technique for calculating p was introduced by Bourrely and Fisher [46]. These authors begin by constructing a scattering amplitude which is analytic and crossing symmetric and contains the appropriate singularities that correspond to pp and ~p scattering~In addition, the amplitude so constructed has the correct asymptotic behavior, and the free parameters are adjusted so as to obtain a good fit to the total cross-section data. Then p is obtained by simply taking the ratio of the real to the imaginary part of the fitted amplitude. Again the results are very similar to those of figs. 22 and 23. We conclude that the calculations of p from the known and expected behavior of UT and UT are on a very sound basis. This is to be compared with the attempts to predict b(s), described in part 2, which were strongly model dependent. Perhaps the most important conclusion from the data is that the forward dispersion relations for pp scattering are valid in this energy domain. While this is not surprising it nevertheless implies the validity of the basic axioms of local field theory, namely unitarity, analyticity, crossing and causality. Finally, the fact that p has been obtained through the analysis of interference data, once more assures us about the basic beliefs on the behavior of quantum mechanical amplitudes.

120

A. C. Melissinos, S.L. Olsen, Physics (and technique) ofgas jet experiments

4. Inelastic scattering 4.]. Introduction In addition to the elastic scattering of protons by protons and deuterons, the gas jet technique has been used in the study of reactions in which the incident proton was excited to a higher mass with the target particle staying intact. In particular, the inclusive reactions p+p—*X+p

(27)

and p+d-÷X+d.

(27’)

Here X denotes a forward undetected system of particles with invariant mass M~.These reactions can be described by the three Lorentz invariants s, t and M,~.Equation (2’) can be re-expressed in these variables to indicate their relation to the recoil angle ~ M,~ m~+ -~--\/i7~ (sin~

s+2m~ ~

—-

-~-~-

~

_y_L_ I.

(2”)

2m~/

Alternatively, in place of M,~one can use the Feynman scaling variable x defined as P11 /Pmax of the detected particle in the c.m. system. For the case where t is small 2 m2 M2 x~l—--—~--——---~ >1——---~-. (28) M ----

S

M~large

An important feature of the kinematics for these reactions is the minimum four momentum transfer tm~,, at which the system X can be produced. At high energies and moderate masses M2—m2 (—~~-~~) 2

tmin

(29)

-

Thus at a beam momentum of 400 GeV/c and forM~= 50 (GeV/c2)2, tmin = 0.004 (GeV/c)2. This corresponds to a three-momentum transfer of only 60 MeV/c which is quite small. At these very small momentum transfers it is possible to excite the proton’s internal degrees of freedom without breaking up the system. These processes, generally referred to as diffraction dissociation [471are expected to be similar to elastic scattering, having characteristic forward diffraction peaks and little variation with incident energy; the reactions are coherent and proceed with no exchange of quantum numbers other than orbital angular momentum. Indeed, it was for the last reason that much effort has been devoted to the study of the coherent deuteron reaction (eq. (27’)). As the deuteron has zero isotopic spin, the detection of a deuteron in the final state insures that the total undetected system X has the same isospin as the incident proton hopefully enhancing the diffraction dissociation signal. At lower energies (typically 10—30 GeV) reaction (27) has been extensively studied for low values of M,~.At these energies [48] the mass spectrum has considerable structure and a t behavior which varies drastically as M~increases. A broad resonance-like bump is observed at M~ 1.4 GeV/c2 whose production on protons is damped in t as exp (—20 It I). Other structures with —

A.C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

121

considerably smaller t dependence are reported atM~= 1.52, 1.69 and 2.19 GeV/c2. The latter bumps are generally considered to be due to the excitation of proton isobars, while the enhancement at 1.4 GeV/c2 is somewhat more controversial. It has been variously interpreted as being either the effect of other resonant states, or as threshold effects associated with 7T and ~.ir production.

~

2

~

p

p1 2

,

= I~i~rY(

P2

2’

(t

t

)

Fig. 24. The diffractive dissociation process and its connections through the generalized optical theorem of MOeller [49] to the imaginary part of the triple Regge amplitude.

The higher M~region is of interest as it corresponds to the kinematic region where the triple Regge phenomenology is expected to apply, i.e., where s and M~are large compared to m~,t and where M~/s ~ 1. Using the generalized optical theorem of Mueller [49], these processes can be related to three body scattering amplitudes as shown in fig. 24. Interpreting the three body scattering as proceeding via Reggeon exchange one finds that in this kinematic region the reaction is dominated by Reggeon—proton scattering with center of mass energy M~.How do high energy Reggeons scatter from protons? They exchange another Reggeon of course, hence the name triple Regge. Most of the theoretical work in this area has been concentrated on generalizing the Regge model for particle—particle scattering to the case of Reggeon particle scattering [50]. These reggeized models give an expression for the double differential cross section

d 2a dtdM,~

1 S

~G..k(t)

(30)

(—~-\ag(t)+o~(t) (M2)ak(o) x S

1/

Or, in terms of the scaling variable x d2a dtdx

(

d2 a ~ Gj.k(t) 1 ) 1a1(t)+c51(t) [s(l—x)] dtdM~ / ~1—x S

ak(O)

-

(30’)

Here i and / refer to the incoming and outgoing Reggeons and k corresponds to the one that is being exchanged. The functions GZIk(t) are the so-called triple Regge couplings and are a function oft only.

122

A. C Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

For the coherent deuteron processes only the pomeron and isospin zero Regge trajectories have to be included. Taking for the pomeron trajectory cs~(t)=

ItI

1.0—0.28

and for the Reggeon trajectory = 0.5

—

ti

we list the small t behavior for some of the terms in the expansion (see ref. [50]). 2o/dtdM,~

d2o/dtdx

PP P

l/M~

1/(1

PPR

l/M~

RRP

I/s

RRR

1/Ms

Triple Regge term

d

--I

--

--

x)312

x)”2

We note that the pure diffraction term P P P corresponds to a I /M~behavior for d2 o/d t dM~and energy independence in both d2u/dt dM,~and d2o/dtdx. Other work, particularly appropriate to the data, are the so-called Finite Mass Sum Rules (FMSR) [51]. These come from the generalization of the Finite Energy Sum Rules (FESR) which follow from Regge theory for particle—particle scattering, to the case of Reggeon—particle scattering under consideration here. In terms of the antisymmetric variable i = M,~ t m~,these relations for reaction (27) are given by --

0

/

,,

d2 a

n+1 _~o d t dM~d~ 2 -

Gijk~ijk’5~1-’O ~ ( / ____ (ak(O)

—

_____

~k(O)

—

a 1(t) —a1(t) +

n + 1)

‘

~

1, 3,

-

(31)

...

where = ~ (1 + r1 r1 Tk), r1 being the signature of the ith Regge trajectory. The value z.’~is arbitrary so long as it is in the triple Regge domain. Inclusive pp experiments in the triple Regge region have been performed at the TSR using a small angle spectrometer and at the Fermilab both in bubble chambers and in counter experiments. These experiments can roughly be summarized as follows: 2 Usb ISR Fefmilab Fermilab

Ref.

(521 1531 154]

Technique small angle spectrometer

Plab (GeV/c) 250--bOO

bubble chamber

100—-400

1551

~ 100

t (GeV/cY2

0.15 0.1

.

-‘

~

1.75 1.0

(GeV)2 5

0.7 Plats/100

CH 2 foil

Fermilab

M~(GeV) 140

gas jet

300 56—400

60

x

—0.8

0.019 0.16

-

0.19

---

0.33

-

0.5

These results are in general agreement with the triple Regge formulae. A recent fit by Field and Fox [561 indicates that aside from a few discrepancies all the data, so far, can be fit if one includes ir ir R and ir ir P together with P P P, P P R, R R P and R R R. The iT iT P and ir iT R term

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

123

should be absent in the coherent deuteron reaction (eq. (27)). Using low energy isobar data, these authors also find that the FMSR sum rule is valid at least for n = 1. Here we discuss results from measurements in the small mass region using the gas jet target [57, 58]. A program of measuring the higher mass region for reactions (27) and (27’) is currently in progress, preliminary results having been reported recently [59]. 4.2. Experimental details In the measurements of elastic scattering, the correlation between the recoil angle and kinetic energy proved to be extremely useful for separating the signal and the background. As discussed in part 1, simply moving the detectors to smaller recoil angles was sufficient for continuously monitoring the background in each detector. This constraint is missing for inelastic scattering, so that more elaborate techniques are needed to eliminate, or monitor backgrounds. At a particular laboratory angle the recoil kinetic energy corresponding to the production of an inelastic state with mass M~ is always lower than for those recoils which corresponds to elastic scattering. This makes measurements of inelastic processes with M~near m~susceptible to background caused by recoils from the (dominant) elastic reaction that loose kinetic energy by scraping the edge of a collimator or interacting in the material of the detector. As indicated in eqs. (5) and (5’) the difference in recoil kinetic energy between elastic and inelastic scattering with the same recoil angle falls inversely with the incident momentum, so that at high energies the inelastic recoils crowd near and into the elastic peak. The number of counts .~Nin a recoil energy interval ~ T is related to the invariant cross section by iT 4-~L1 32 d2a — dtdM~ - p 2 ~T ~ ( ) 1~4m~ It] + ~~j L is a factor which contains the beam-jet luminosity and ~M2is the solid angle subtended by the detector. L can be determined by simultaneously measuring elastic scattering. Then da 1

~

L=~j~~~.iM2ei,

(33)

where Nei is the number of elastic counts and ~~el is the solid angle of the monitor detector. The small angle differential cross section used is the well established relation da

02

2) 2mrx./T~Texp(—bIte

(1 +p

1I).

(33’)

is the (mean) t value for these elastic events. The values of p used are those given in part 3 and of b for reaction (27): tel

~

= 8.5± 0.281ns.

For reaction (27’) the same slope was used modified by the deuteron form factor as determined by Nikitin et aT. [60] bpd = 26

+

60 t + ~

.4. C. Melissinos, S.L. Olsen, Physics (and technique) ofgas jet experiments

124

For UT(PP) and UT(pd) the results of Baker et al. [16]were used. Uncertainties in L due to uncertainties in the input values of 0T’ b and p are negligible when compared with the typical statistical errors on~N.For the low mass cross sections the elastic counts in the same detectors were used as a monitor, hence the solid angle factor cancelled out. For small missing masses the mass resolution for the geometry of these experiments is almost totally due to the uncertainty in the determination of the recoil angle 0. For an uncollimated jet ~ ±3 mrad) at p 2 it holds (from eq. (5’)) ~ = 0.36 0 = 300 GeV/c and it] = 0.04 (GeV/c) (GeV)2 so that atM~= 1.4 (GeV)2, z~M~ = ±130 MeV. This resolution is comparable to the width and separation of the bumps reported in this mass region.

BEAM

~

A

~~T~GETHW

IT~

I I I~ fl I

~, B

\,~SLIT~,,/

m

n

RECOIL

ENERGY

RECOIL

ENERGY

COLLIMATOR RECOIL

n~ri II fri AREAHCn,2 I~Inn

.2mm

1.5mm

Fig. 25. Schematic drawing of the experimental arrangement, indicating the parallax effects caused by the slit,

flIT,,

Fig. 26. Elastic peaks observed with the slit; (a) wide open and (b) closed to a 4 mm aperture.

In order to improve the resolution in the low mass region, data were taken with a system of moveable slits installed near the gas jet. The arrangement is shown in fig. 25. In this case the slit and the detector collimator widths were both 4 mm, giving an angular resolution of ±I mrad. This resulted in a threefold improvement of the mass resolution. The effect of the slits is demonstrated by a distinct narrowing of the elastic peak as is shown in fig. 26. For the slit data it was crucial to normalize on elastic scatters in the same detector since in this case each detector sees a somewhat different region of the gas jet target. This limits the mass and t-range over which this technique is useful. Recoil energy histograms of coherent deuterons for three different energies in the same detector are shown in fig. 27. The 50 GeV histogram should have no real inelastic counts, so it serves as a monitor of background originating from the elastic peak or other processes. The inelastic signal in the higher energy histograms is quite clear, usually a factor of 10 above the background.

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

125

RECOIL ENERGY SPECTRA

/~~\ SI

_,~

e

50GeV/c

• I8OGeV/c

M

2’l.9

~\

(180GeV)

~&~:f M~’2.4 I2~0GeV)

(50GeV)

! I

I

I

.... ‘ix

RECOIL KINETIC ENERGY Fig. 27. Recoil energy spectra observed in a detector stack with the three incident energies superimposed. The pion threshold at 50 GeV is indicated together with the corresponding masses at the higher energies.

4.3. Low mass results Recently published results [57] for the p—p reaction (27)), beam momenta 175, 260 2 were from the(eq. same set ofatdata used for the of elastic and 400 GeV and for ItI < 0.05 (GeV/c) scattering determination discussed in part 2 of this report. These data were taken without the slit arrangement discussed in the previous section. A point by point comparison was made of the recoil energy histogram for a jet pulsed at high energy with the histogram accumulated concurrently in the same detector for a jet pulsed at p 0 = 50 GeV/c. The 50 GeV data were stretched and shifted slightly to correct for the small kinematic differences at the two energies, and then normalized to the high energy data so that the number of elastic counts were equal. These histograms were subtracted to give an inelastic spectrum which was converted using eq. (32). 2a/dt dM,~for four separate to ItI cross valuessection is shown in fig. 28. The double differential d A large enhancement at cross M~ section 1.9 (GeV/c2)2 which falls rapidly with increasing It] is apparent. If this peak is approximated by a Breit—Wigner resonance form with a width F — 200 MeV, a fit to the data yields da/d t — 8 exp mb/(GeV/c)2. The magnitude of this cross-section and its dependence on t are nearly the same as the results reported at lower energies for the N*(l400) l/2~ resonance [48]. Data for the p—d reaction (eq. (27’)) were taken with stacks of detectors to identify deuterons, and with the slit geometry described above. Results for d2a/dt dM,~as a function of M,~for representative t values are shown in fig. 29. Again an enhancement is observed at M~ 1.9 which falls rapidly in t when compared to the cross section at higher masses. Now, presumably because of the improved mass resolution, a second bump or shoulder is observed near M~ 3. This is most likely due to the N*(1688). A fit was made to the mass spectra (simultaneously at all values oft) —~

(—

15

It I)

AC. Meljssinos, SI,. Olsen, Physics (and technique) of gas let experiments

126

2GeV’2

-

mb’(GeV/cj”

6

~‘‘~

-~‘~

p

PLAB 1GeV/ri —0---~ 75

—0—.

‘—ô-—’ 0

_—~-—.

~p

p

260 400

[ O.OkItkO02

002~ItR003

C

I 3

2

I 2

I

I

003~Itk004 I 3

Q04~ltl~005

I

I

I

2

3

2

I

I

3

M 2 —GeV2 5

Fig. 28. The excitation cross section d2u/dt dM~for p + p ent incident energies.

X + p as a function of M~for different fixed values oft, and differ-

d20 80

--

ftdM~

270 GeV/c

180 GeV/c .0271GeV/c) 2

2~ 2

E5

tI

L._

.039 1GeV/c

_~

M 2 1GeV)2 5 I’ig. 29. The excitation cross-Section d2o/dtdM~ as a function of M~for the reaction p + d ‘X + d. Spectra are shown at fixed (-values ti = 0.027, 0.039 and 0.051 (GeV/c)2, and at two incident energies. The curves are Breit---Wigner resonance fits to the higher energy data as described in the text.

using a Breit—Wigner resonance centered near M~ = 2 (GeV/c2 )2 1 utJ

—bt

j-.2~’

e

dtdM,~ M~ 4(Mx~Mr)2+F2

(34)

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

1 27

whereMr and F are the “resonance” position and width to be determined by the fit. Similarly the constants C and b are determined by the fit. The isobar excitation cross section at t = 0 is obtained by integrating eq. (34) and is related to the fit parameters through (34’) (da/dt)~_0 FC[7r/2 + tan1(2(MT—MT)/F)J where MT = 1.08 (GeV/c2) is the pion production threshold. Non-resonance background was not included but in order to obtain a satisfactory fit it was necessary to include a second Breit—Wigner resonance centered in the vicinity of M~= 2.9 (GeV/c2)2. The parameters of this resonance were also determined by the fit [6 11. The parameters of the M~= 2.0 (GeV/c2)2 enhancement obtained through this procedure for an incident energy of 270 GeV are shown below. The quality of the fit is characterized by x2 124 for 117 degrees of freedom. N*(1400) parameters as determined from

Mr (MeV) F(MeV)

180GeV

270GeV

1392

1388

± 15

274±40

(da/dt)~

2

0mb/(GeV/c) b(GeV/c~2

fit

± 10

315±25

26.0

± 1.7

24.4

±1.82

45.1

± 1.5

43.0

± 1.7

The curves in fig. 29 represent the result of this fit. The data can also be parametrized in the form d2a/dtdM,~=Ae~tI =(da/dM~)be_bItI.

(35)

Best fit values for b, A and (du/dM~)are shown in fig. 30. One notes that in general the s-dependence is weak whereas both the slope b, and the excitation cross-section du/dM,~are strong functions of M,~.One must keep in mind that the data of fig. 30 are deuteron cross-sections and therefore the large values of b reflect the steep t-dependence of the deuteron form factor. To compare these results with p—p data we have assumed factorization and divided the deuterium data by the square of the form factor Fd(t). We used the value given by Nikitin et al. [601

IFd(t)12

=

(Upd/Upp)2

exp{—261t1

+

601t12}

(36)

which is valid for small t. Indeed the deuterium data so reduced are in good agreement with the p—p data shown in fig. 28. This agreement of the factorized p—d data with the p—p data can also be seen in fig. 31 where p—p data obtained with the slit geometry at p 0 385 (GeV/c) are shown [621; the curves are obtained from the fit to the p—d data discussed previously have 2 as given by eq. (36). One concludes that factorization isand valid at been least at only divided by IFd (t)1 a level of 20%. Finally in fig. 32 are shown the differential cross section for the excitation of the N* (1400) “resonance” (or enhancement) as obtained at low energies and at Fermilab energies. In the latter case both p—p and factorized p—d data have been included. The lack of s-dependence to within the experimental errors is rather striking and an overall fit to the usual form du/dt = A exp(—b ti) yields (da/dt)~ 2, and b = 19 ±2 (GeV/c)2. 0= 7.7 mb/(GeV/c)

128

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

50

0

180GeV

40

•

270GeV

~30 (0

i

T

(a)

20 10

I

I

I

‘!L-. 25 >20 (b)

.r5

5

)‘

I

UI

I

~0.5 p

(c) 0.3

3 S •~

eIO2

U

~0.I

U.

I

I

I 2 I

II, 3

4

2(GeV)5

M~

Fig. 30. The excitation cross Sections at fixed M~for p + d —+ X + d are fit to the form d2o/d t dM~= A exp (— bt) so that da/dM~= A/b. (a)The slope parameter b versusM~.The observed variation of b withM,~may be less fast than in reality due to the finite mass resolution. (b) The differential cross section extrapolated to t ‘~0. (c) The integrated excitation cross-Section da/dM~under the above assumptions.

p+p~X+p I

385 GeV/c ELASTIC

Itk.02 (GeV/c)2

JFwHM)\

M 2)GeV/c2 (2 Fig. 31. The excitation cross section d2a/d t dM,~for the reaction p5+ p —~ X + p at an incident energy of 385 GeV and at an average Iti = 0.02 (GeV/c). Note however that these are fixed angle data and thus ti varies with M~.The curves are from the fit to the data of fig. 29 assuming factorization of the pd data.

A. C. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

I

N (1400)

I

I

I

I

I

• pd 270 GeV’) o pd ISO GeV~ REF. [58] * pp 385 GeVJ

•

pp

165,270 GeV-BARTENEV ET AL

pp I~6GeV

-

BELLETTINI El AL

~ pp IO,I5,20 GeV- EDELSTEIN ET Al

-

1

~-

2-

I

129

I

.01

.02

I

I

.03

.04

I

.05

I

.06

.07

2

III (0eV/c)

Fig. 32. Small-t data on the reaction p +p - N* + p compared with the 270 GeV fit for N*(1400) production. The pd points on the plot are results of separate fits with two Breit—Wigner amplitudes at each value of t, and assuming factorization. The other data are from refs. [57,48].

5. Current and future program As is often the case in physics, the internal gas jet target, developed in particular for Coulomb interference experiments, has been found to be extremely useful for a broad range of experiments. These experiments include more small t work with a variety of targets as well as pp missing mass experiments out to considerable t values It I 3 or 4 (GeV/c)2. The advantages of the gas jet for low t measurements have been emphasized throughout this report. The low multiple scattering and energy loss coupled with continuously variable energies was essential to the work reviewed here. Further measurements are now in progress to study the small t behavior of coherent proton dissociation on hydrogen, deuterium and also helium. Elastic p—He scattering measurements are in preparation [101. Furthermore, more precise work on isobar production and elastic scattering is also being planned. The gas jet target is also ideal for extending large-t missing mass experiments to high energies. Missing mass experiments are characterized by a mass resolution which is proportional to the incident beam momentum. Thus to extend the 25 GeV, AGS and CERN PS missing mass experiments through the Fermilab energies, the measurement errors on the particle’s angle and momentum have to be reduced by about a factor of 20; this without a loss in luminosity. This is feasible if the low mass and high luminosity of the gas jet target is exploited. In the planned measurements [8] a quadrupole magnet lens system focuses parallel recoils from the jet onto a proportional wire plane positioned at the exit of the vaccum system. Thus the recoil angle is measured by

130

.4.C Melissinos, S.L. Olsen. Physics (and technique) ofgas jet experiments

one point with essentially no multiple scattering error introduced. The recoil momentum is then accurately determined by bending the (slow) recoil through a large angle I rad). With this system a mass resolution of— 150 MeV/c2 is expected at Pin = 400 GeV/c. A carbon polarimeter placed behind the spectrometer will measure proton’s polarization both for elastic and inelastic scattering [63]. An expansion of the Fermilab internal target area to accommodate these efforts is currently underway. ~-=

6. Acknowledgments This review has been motivated by our participation in the USA—USSR collaboration and we sincerely thank our colleagues V. Bartenev. A. Kuznetsov, B. Morozov, V. Nikitin, Y. Pilipenko. V. Popov, L. Zolin and Y. Akimov, L. Golovanov, S. Mukhin. (i. Takhtamyshev. V. Tsarev of the JINR, Dubna, USSR; R.A. Carrigan Jr., E. Malamud, R. Yamada, P. Zimmerman of the Fermilab: R.L. Cool, K. Goulianos, H. Sticker of Rockefeller University: I-Hung Chiang, D. Gross, D. Nitz of the University of Rochester who participated at various stages of these experiments. It was a pleasure to have been members of these collaborations and we benefited from many interesting discussions and a variety of points of view. These experiments could not have succeeded without the spirit and efforts of the staff of the Fermi National Accelerator Laboratory. Many individuals not named in this report have made significant contributions and we take this opportunity to acknowledge our indebtedness to them. In particular we thank the members of the Internal Target Laboratory who operated the gas jet target and T. Haelen of the University of Rochester for mechanical construction. Some of the figures in this report are from the thesis of D.A. Gross [15] to whom we extend special thanks. References and notes [1] E. Rutherford, Philos. Magaz. 6th Series 21(1911)669. [2] L.F. Kirillova et al., Yadern. Fiz. (USSR) 1(1965)533 [Soviet J. Nucl. Phys. 1(1965) 379). [3] Y.K. Akimov et al., J. Exptl. Theoret. Phys. (USSR) 48 (1965) 767—769; Y.K. Akimov et al., J. NucI. Phys. (USSR) 4 (1966) 88—92; G. Beznogikh et al., Phys. Lett. 30B (1969) 274; Phys. Lett. 43B (1973) 85. [4] G. Beznogikh et al., Phys. Lett. 39B (1972)411. [5] V. Bartenev et al., Phys. Rev. Lett. 29 (1972) 1755. [6] V. Bartenev et al, Phys. Rev. Lett. 31(1973)1088. [7] V. Bartenev et al., Phys. Rev. Lett. 31(1973)1367. [8] It is possible to use a continuous solid state detector such as “position sensitive detectors” which are now commercially available. Another possibility is the use of a focusing system to determine the angle of the recoils combined with a proportional wire chamber detector; Fermi National Accelerator Laboratory Proposal 198, Magnetic recoil Spectrometer for the Gas Jet Target, S. Olsen et al. and proposal 289 see ref. [10]. One can also attempt to measure the luminosity of the jet target system by optical methods or other techniques; as yet, this approach has not been proven successful. [9] V. Bartenev et al., Adv. Cryog. Eng. 18 (1973) 460. [10] It has been proposed to trap the helium by using cryogenic adsorption. Fermi National Accelerator proposal 289, Small Angle Proton Helium elastic and inelastic scattering from 8 to 500 GeV, F. Malamud et al. [11] R.M. Edelstein et al., Phys. Rev. D5 (1972) 1073. [12]G. Barbiellini et al., Phys. Lett. 39B (1972) 663. [13]A. Bôhm et al., Phys. Lett. 49B (1974) 491. [14] See for instance, U. Amaldi, CERN Internal Report NP 73-5 (1973) fig. 4.

A. C. Melissinos, S.L. Olsen, Physics (and techniqi~e)of gas jet experiments

131

[151 An unpublished set of data from the USA—USSR Collaboration yields similar values: b

0= 8.02 ±0.18, a’ 0.299 ±0.016; see D.A. Gross, Low Momentum Transfer Proton—proton Elastic Scattering up to 400 GeV, Ph.D. Thesis, University of Rochester (1974).

[16] W.F. Baker et al., Phys. Rev. Lett. 33 (1974) 928. [17] We are indebted to Dr. L. Zolin for communicating these results of the USA—USSR Collaboration to us. In evaluating °el’ eq. (7’) was used to parametrize the elastic da/dt assuming that the energy dependence of b2(s) is similar to that of b1(s). To be published. [18] L. Van Hove, Rev. Mod. Phys. 36 (1964) 655. [19] E.H. de Groot and N.I. Miettinen, Rutherford High Energy Laboratory Report No. RL-73-003 (1973). [20] R. Heinzi and P. Valin, McGill University Report (1973). [21] V. Barger, R.J.N. Phillips and K. Geer, NucI. Phys. B47 (1972) 29. [221 The exact parameters of this fit are obsoletesince they do not include a rise in °T Such behavior can, however, be accommodated in the fran~workof the Regge model as shown for instance by V. Barger et al., Nucl. Phys. B40 (1972) 205. [231V. Barges, K. Geer and R.J.N. Phillips, Phys. Lett. 36B (1971) 343; also Phys. Lett. 36B (1971) 350. [24] H. Fujisaki and E. Akiba, Progr. Theor. Phys. 48 (1972) 1294. Note that the parameters of the fit are obsolete because they fail to give the correct behavior of °Tand p for pp scattering. [25] A.D. Krisch, Phys. Rev. Lett. 19 (1967) 1149; Phys. Lett. 44B (1913) 71. [26] E. Leader and M.R. Pennington, Phys. Rev. Lett. 27 (1971) 1325. [27] S. Barshay and Y.A. Chao, Phys. Lett. 38B (1972) 225. [28] T.T. Chou and C.N. Yang, Phys. Rev. 170 (1968) 1591. [29] H. Cheng, J.K. Walker and T.T. Wu, Phys. Lett. 44B (1973) 97; Phys. Lett. 44B (1973) 283; H. Cheng and T.T. Wu, Phys. Lett. 45B (1973) 367. [30] L. Durand and R. Lipes, Phys. Rev. Lett. 20 (1968) 637; M. Elitzer and R. Lipes, Phys. Rev. D7 (1973) 1420; S.L. Olsen, Elastic and Inelastic Diffraction Scattering, Univ. of Rochester preprint (unpublished). [311 B. Carreras and J.N.J. White, Nuci. Phys. B32 (1971) 319. [32]. R.J. Eden, Rev. Mod. Phys. 43 (1971) 15. [33] M. Froissart, Phys. Rev. 123 (1961) 1053. We recall that if the Froissart bound as given in the text is saturated, then °eI= Such behavior is contrary to our intuitive feeling that at high energies the elastic scattering is purely diffractive, i.e., ~e1= ~0T. Imposition of this condition lowers the bound by a factor off. [34] U. Amaldi et al., Phys. Lett. 44B (1973) 112; S.R. Amendolia et al., Phys. Lett. 44B 2 as (1973) in most 119. applications of Regge theory, then the bound becomes very weak. If [35] SO> If in eq. we chooses0 (GeV) i03 (15) (GeV)2 the bound=is1 irrelevant at present energies. One cannot avoid the influence of ~ because of the quadratic dependence on ln (s/s 0);2)[ln(s for instance, the difference of the cross section at two values of s is bounded by ~T(~1) — aT(52) < (ir/p 1/s2)] [ln(s1s2/s~)]. [361 S.W. McDowell and A. Martin, Phys. Rev. 135 (1964) B960. [37] H.A. Bethe, Ann. Phys. (N.Y.) 3 (1958) 190. [38] G.B. West and DR. Yennie, Phys. Rev. 172 (1968) 1413. [39] E. Leader and U. Maor, Phys. Lett. 43B (1973) 505. [40] C. Bourrely, J. Soffer and D. Wray, Preprint 74/P.1957 Centre de Physique Theorique, Marseille, France; J. Soffer and D.Wray, Phys. Lett. 43B (1973) 514. [41] ML. Goldberger, Y. Nambu and R. Oheme, Ann. Phys. (N.Y.) 2(1957) 226. [42] P. Söding, Phys. Lett. 8 (1964) 285. We note that the expressions derived by Sóding assume that p —~ 0 as E — °o. This condition is used to fix the subtraction constant of the odd amplitude. [43] N.N. Khuri and T. Kinoshita, Phys. Rev. 137B (1965) 720; 140B (1965) 706. We can see this behavior ofp in a simple way 50Tderivative = Im [f(s)]). Thus, ifrelation the total by considering that at high energies the p—p scattering amplitude is even, in which case the analyticity of eq. (26”) cross section reads behaves Re [f(s)] as (lns)?~we /s = tan have ir(d/dp In s~ ~Xir Im(lnsY’ [f(s)] /s} namely wherep we —~0have from setabove a = 1 or (and below depending on the sign of X. See, for instance, J.D. Jackson, Introduction to Hadronic Interactions at High Energies, Scottish Universities Summer School (1973). [44] W. Barte~land AN. Diddens, CERN NP Internal Report 73-4 (1973). [45] J.B. Bronzan, G.L. Kane and U.P. Sukhatme, Preprint NAL-Pub-74/23-THY (1974). [46] C. Bourrely and J. Fischer, CERN Preprint TH 1652 (1973). [47] E.L. Feinberg and I. Pomeranchuk, Suppl. Nuovo Cimento 3 (1956) 652; M.L. Good and W.D. Walker, Phys. Rev. 120 (1960) 1857.

{~

132

AC. Melissinos, S.L. Olsen, Physics (and technique) of gas jet experiments

[48] R.M. Edelstein et al., Phys. Rev. D5 (1972) 1073; G. Belletini et a!., Phys. Lett. 18 (1965) 167; J.V. Allaby et al., NucI. Phys. B52 (1973) 416. This article summarizes previous work on this subject. [49] A.H. Mueller, Phys. Rev. D2 (1970) 2963. [50] See, for instance, G.C. Fox, Inclusive Structure of Diffraction Scattering, in: Proc. Fifth Intern. Conf. on High Energy Collisions, Stony Brook 1974, p. 180; AlP Conf. Proc. No. 15. [51] M.B. Einhorn, J. Ellis and J. Finkelstein, Phys. Rev. D5 (1972) 2063. [52] M.G. Albrow et al., Nucl. Phys. B54 (1973)6; B51 (1973) 388. [53] See, for instance, J. Chapman et al., Phys. Rev. Lett. 32 (1974) 257; H. Boggild and T. Ferbel, Inclusive Reactions, Ann. Rev. Nuci. Science, to be published. [54] S. Childress et al., Phys. Rev. Lett. 32 (1974) 389. [55]K. Abe et al., Phys. Rev. Lett. 31(1973)1527; 31(1973)1530. [56] R.D. Field and G.C. Fox, Cal Tech Preprint (CALT-68-434), Triple Regge and Finite Mass Sum Rule Analysis of the Inclusive Reaction pp —o px’°. [57] V. Bartenev et al., Phys. Lett. SiB (1974) 299. [58] Y. Akimov etal., Excitation of the Proton to Low Mass States at 180 and 270 GeV, Univ. of Rochester preprint, COO-3065-88 (UR-495) and contributed to the XVIIth Intern. Conf. on High Energy Physics, London (1974). [59] Y. Akimov et al., Proton—Deuteron Elastic Scattering and Diffraction Dissociation from 50 to 400 GeV, Rockefeller Univ. preprint, COO-2232A-1 and contributed to the XVIlth Intern. Conf. on High Energy Physics, London (1974). [60] V.A. Nikitin et al., Preprint No. E1-7207 JINR, Dubna, USSR (1973). [611 The data were not sensitive enough to give convergent fits when a third resonance was included. If some N*(1520) is present its effect most likely would be in broadening the width of the 1688. For this reason the results of the fit for the 1688 are probably not too reliable. The cross sections given in the table and in figs. 29 and 30 are deuteron cross sections derived from the proton cross sections of ref. [58]. [62] Note, however, that the data in fig. 31 are displayed at fixed angle so that different t-values correspond to different masses as shown. [63] Fermi National Accelerator Proposal 313, Polarization in p~—pElastic, Inelastic and Inclusive Reactions at NAL Energies, HA. Neal et a!.