PHYSICAL REVIEW C VOLUME 37, NUMBER 4 APRIL 1988

Neutron-proton final-state interaction in md breakup: DifFerential cross section

W. List, E. T. Boschitz, H. Garcilazo, * W. Gyles, C. R. Otterrnann, R. Tacik,M. Wessler, and U. Wiedner

Kernforschungszentrum Karlsruhe, Institut fu'r Kernphysik und Institut fii r Experimentelle Kernphysikder Uniuersitiit Karlsruhe, D 7500 -Karlsruhe, Federa/ Republic of Germany

R. R. JohnsonTRIUMF and Uniuersity ofBritish Columbia, Vancouver, Canada, V6T2A3

(Received 8 June 1987)

Threefold dift'erential cross sections o(H Hp Pp) have been measured in a kinematica11y completeexperiment. The experiment has been designed to extend earlier measurements into the kinematicregion of the np final-state interaction. The data are compared with the predictions from the newrelativistic Faddeev theory of Garcilazo.

I. INTRODUCTION

The final-state interaction between two nucleons hasbeen of considerable interest for a long time. In low ener-

gy nuclear physics experiments the main effort wasdirected towards the extraction of nucleon-nucleonscattering parameters. ' At higher energies the final stateinteraction has been considered responsible for discrepan-cies between quasifree NN scattering data and calcula-tions based on simple knockout models.

The first direct observation of the neutron-proton finalstate interaction above 200 MeV came from the pd break-up reaction at T =585 MeV. Later, this work was ex-tended to 800 MeV. In these kinematically complete ex-periments triple differential cross sections were measuredfor several proton-proton pairs, particularly in the regionof phase space where the relative energy of the neutron-proton subsystem was low. The strong enhancement ofthe cross section in this region was analyzed applying thefinal state interaction formalism of Goldberger and Wat-son. The spectra were fit with the incoherent sum of thefinal state interaction factor for singlet and triplet npfinal-state interaction plus a constant term. From thisthe authors extracted the individual contributions of theSp and S, np states and found a large S, /'Sp ratio.

Strong effects due to the nucleon-nucleon final state in-teraction were also observed in nucleon-nucleon col-lisions above the pion production threshold. In the zero-degree measurements of the inclusive reaction np~pXbetween 400 and 800 MeV, pronounced peaks were seenin the double differential cross section d o /d Q&dP& plot-ted as a function of the proton momentum. Similarly theexclusive cross section measurement of the pp —m+pn re-action displayed strong np final state interaction peaks.The first theoretical calculations of these reactions didnot include final state interaction effects with the excep-tion of the field-theoretical peripheral model of VerWest,which included the NN final state interaction only quali-tatively and phenomenologically. Recently, Dubach,Kloet, and Silbar extended their unitary, unified model of

NN and mNN interactions to include final state interac-tion corrections. This was done following the ~atson-Migdal final state interaction prescription by multiplyingtheir NN-vrNN reaction T matrix by a factor which in-corporates the final state interaction in relative S waves.The authors were able to explain the peaks in the crosssections using 'So and S, final state interactions. Morerecently, a unitary meson exchange calculation of theNN-mNN reaction was published by Matsuyama andLee in which the final state interaction was treatedrigorously, but it was limited to the region near the finalstate peak. There the relative motion of the outgoing NNsubsystem is low, and it is therefore sufficient to includeonly the S& —D, and 'So NN channels in defining thequasiparticle state "d." This calculation was able toreproduce both the shape and the position of the finalstate interaction peaks, but underestimated their magni-tudes. It must be noted, however, that these results werenot renormalized, as in the calculation of Dubach, Koet,and Silbar which involved an adjustable parameter to fitthe data.

The np final state interaction in the ~d breakup reac-tion was first observed by Bayukov et a1. ' Negativepions of 870 MeV were scattered by a CD2 target, andpions and protons were detected by optical sparkchambers. Due to the low count rate, the data were plot-ted as a single differential cross section as a function ofneutron momentum. At large neutron momenta a simplequasifree my scattering model failed, and the phenomeno-logical addition of a np final state interaction was re-quired. Similar effects were seen in a ~d breakup experi-ment by Dakhno et al. , "who utilized a liquid deuteriumbubble chamber for the detection of the pion and protonin the final state. The first kinematically complete mea-surement of ~d breakup was performed by Hoftiezeret a/. ' Two extreme kinematical situations were em-phasized in the experiment, one in which the outgoingneutron momentum is small and therefore the impulseterm is expected to be dominant, the other where the out-going neutron momentum is large in comparison to the

37 1587 1988 The American Physical Society

1588 %'. LIST et al.

deuteron internal momentum, and consequently process-es other than the impulse term are expected to be impor-tant. But, no pronounced final state interactions wereseen in this particular choice of phase space. Recently,Gyles et al. ' measured the m.d breakup cross sectionover a much larger region of phase space. The data werecompared with calculations by Garcilazo' which werebased on the relativistic Faddeev theory of Aaron, Ama-do, and Young. ' In general, good agreement was foundwithin the experimental uncertainties, even in the lowcross section regions. (Recently, a numerical error wasfound in the theoretical calculation reported in Ref. 13,and almost simultaneously an error in the data reductionprogram was discovered, both of which cancel to a largeextent in the OFS region. ) Since then a novel relativisticFaddeev theory of the mNN system was developed by oneof us, ' which uses a variable-mass ansatz for the m.N andNN amplitudes and the requirement that the spectatorparticles always be on the mass shell. Utilizing thistheory a number of recent calculations showed that thenp final state region is most sensitive to difFerent theoreti-cal descriptions. '

The motivation of the present work was to extendsome of the existing data at 294 MeV into the region oflow proton momentum where efFects due to np final stateinteractions (FSI) are expected to be large, and comparethem with theoretical predictions from Garcilazo. '

II. EXPERIMENTAL TECHNIQUE

The experimental setup is shown in Fig. 1. Pions fromthe ~M1 beam line of the Swiss Institute for NuclearResearch (SIN) impinged on a liquid deuterium target.The outgoing pions and protons were detected in coin-

S~ -Hodoscope

cidence by scintillation counters. The arrangementspecified the momentum of the incident pion, the azimu-thal and polar angles of the outgoing pion and proton,and the momentum of the proton by a time-of-flight mea-surement. Therefore, the experiment was kinematicallycomplete. The energy of the incident pions was T =294MeV. The average beam intensity during the experimentwas 1.3&10 pion/s. The incident pions were detectedby a scintillation counter hodoscope S1 and a beamdefining counter S2. The hodoscope consisted of fivescintillation strips (0.2&&1.0)&10.0 cm ) and was placed1.5 m upstream from the target. The beam definingcounter with a 3 cm diameter hole in the center wasoperated as veto counter. A beam event was defined byBEAM=S1 S2 rf where rf is a radio frequency pickupsignal from the accelerator. For a continuous stabilitycheck of the position, shape and intensity of the pionbeam, two types of monitors were used: a 10 cm)& 10 cmmultiwire chamber as a beam profile monitor and a set ofscintillation counters Ml-M3. The profile monitor withan integrating readout was connected to an oscilloscopeon which the X and Y profile of the beam was continu-ously displayed. The scintillation counters M1-M3 werearranged as a triple counter telescope and viewed thehodoscope Sl from an angle above the scattering plane.

The liquid deuterium target cell was disc shaped, 0.5cm thick, and 10.6 cm in diameter. Deuterium gas bal-last chambers on both sides of the cell separated theliquid target from the vacuum. The 36 pm Mylar foils ofthe target walls were prestretched. In a separate experi-ment it was shown that the deformation of the foils dueto the hydrostatic pressure of the liquid deuterium wasnegligible.

The outgoing pions were detected at 8 = 110.0',125.0', and 141.0' by scintillation counter telescopes.Each telescope consisted of a counter n 1 (0.3)(6.0)& 16.0cm ) at a distance of 50 cm and a counter n 2(0.5)&10.0)&30.0 cm ) at 100 cm from the target center.

& Monitor Telescope

Beam Profile ChamberAg

BiS~-Veto

C I -- ~--~L Target

lp ~

2 A

0 1m

FIG. 1. Schematic drawing of the experimental setup.

10

8

6-4

h 4-.O

n+d —2p —141'

Gp = —19.3'

~+a —a+pe

~+a —~+g

I

10

I

15

I

20 25 30

Profon TOF (ns)

FIG. 2. Density plot of the scattering events. The solid linepresents the kinematic locus for the ~d breakup reaction, thedashed lines show the kinematic cuts applied for the data reduc-tion.

37 NEUTRON-PROTON FINAL-STATE INTERACTION IN md. . . 1589

104

0. = I.4I'10 8 — I9 3O

10

10'

104

10 '

10IP&t 1&1 8! I . i . IP

200 300 400 500 600 700

Proton Momentum (Me V/'c)

FIG. 3. Comparison of the cross section data from this ex-periment (solid squares) with the ones from Ref. 13 (open cir-cles).

The latter one defined the time of flight. Therefore thisscintillator was viewed by two photomultipliers, one oneach end.

The outgoing protons were detected at 8„=—13.3',—19.3', and —25.3' (the conjugate angles for ~ kine-matics) by detectors each composed of two scintillation

counters. Counter pl (0.5&(20.0X70.0 cm ) positionedat a distance of 1.88 m from the target and counter p2(20.0X20.0X70.0 cm') 12 cm behind formed a detectionsystem with E and AE information in addition to thetime of flight. The counters were viewed by two pho-tomultipliers each. This system permitted a clean separa-tion of competing reactions like ~+d ~a.+d or m+d~2p.The detection angles for pions and protons were chosenin such a way as to produce an overlap with earlier datataken with a different setup, ' and to completely coverthe final state interaction region at low proton momenta.The electronics setup permitted coincidences betweeneach proton and each pion arm. The time of flight wascalibrated using the mp elastic scattering reaction fromthe hydrogen in the Mylar foils. The time-of-flight reso-lution amounted to 0.8 ns FWHM. The data analysiswas performed similar to the one described in the earlierpublication. ' Figure 2 shows a typical two-dimensionalevent plot. After extracting the yield for the appropriateproton momentum bins the cross section was calculatedfrom the expression

d 0' Yieldd QQ QpdP p Nb„N, s,FEQ+Q phP p

e„=-is.s' —19.3' —25.3'

10'

10'

10'

100

f t r I r I I ' I ' I 4 t ' I I i I ' I ' t t ' I ' I

141

10

10'

10

101 r

10-' r~rr

r

10

104 r II ~

/1

1I

I

'I

/r

~a ~

125'

110

t i I i I a

r~ r

I i I a I i I

200 400 600 200 400 600

Proton Momentum

a I'r

I s I a I ~

200 400

(me Vic)

a I a

600

FICx. 4. Triple differential cross sections obtained for nine proton-pion angle pairs. The solid line represents the prediction fromthe new theory of Garcilazo {Ref. 16), the dashed line corresponds to the impulse approximation.

1590 W. LIST et aI. 37

where Nb„ is the number of incident pions, Ntgt is thenumber of deuterons per cm in the target, e is the com-bined efficiency of the detectors and the data acquisitionsystem, measured during the experiment (0.7—0.8), b,Q„and bQ are the pion and proton solid angles (29.9 msrand 39.8 msr, respectively); and b,P is the protonmomentum bin (20 MeV jc).

The data were corrected with the use of a Monte Carlocalculation for the effect of averaging over the finitedetector sizes and the momentum binning of the data. Adetailed description of the Monte Carlo correction pro-cedure will be given in a forthcoming publication. Itraises the cross section in the quasifree ~ scattering re-gion, and lowers it on the sides. In the np final state in-teraction region the correction amounted to less than10%, again raising the peak and lowering the sides.

The systematic uncertainties in the measurement of thecross section consist of a 5% error from the determina-tion of the polynomial of the Monte Carlo correction, of4% in the target thickness, 2%%uo in the solid angles of thedetectors, and 1% in the incident beam intensity. Thisresults in a systematic error of 14% which is not includedin the statistical uncertainties presented in the figures.To show the typical consistency of this data with the ear-lier measurements' the cross sections from both experi-ments are compared in Fig. 3 for one angle pair. There isgood agreement in the quasifree my scattering region.

III. RESULTS AND DISCUSSIONS

Before comparing the experimental results with thetheoretical predictions, a brief description of the theoreti-cal developments previous to the present theory seemsappropriate.

One of the first things with which a theorist is con-fronted when dealing with the m.NN system at intermedi-ate energies is the fact that this system requires, one wayor another, a relativistic treatment. Due to its small

'mass, a pion of say 300 MeV kinetic energy is a highlyrelativistic particle, while if this same pion is absorbed,the two nucleons emitted have kinetic energies which arenot much smaller than the nucleon's rest mass. Thus, inthe development of the theoretical description of this sys-tern, one can notice a clear trend in which additional rela-tivistic features are put into the calculations. The start-ing point of this trend is, of course, the pioneering workof Afnan and Thomas' in which pion-deuteron elasticscattering, pion-deuteron absorption, and nucleon-nucleon scattering at energies not very much above thepion threshold, were all described simultaneously usingthe nonrelativistic Faddeev equations in which the nu-cleon was assumed to be a bound state of a pion and a nu-cleon.

Semirelativistic versions of the Faddeev equations werelater introduced in order to use them in the descriptionof pion-deuteron elastic scattering. For example,

e~ = -13.3' —19.3' —25.3

10'

10

10'

104

10

10'

M 10

~~ 10'

)00

Q ]0'

125

10

10'

104

10-'

200 400 600 200 400 600 200 400 600

Proton Momentum (He VYc)FIG. 5. Triple difFerential cross sections obtained for nine proton-pion angle pairs. The solid line represents the prediction from

the new theory of Garcilazo (Ref. 16), the dashed line corresponds to the prediction from his earlier theory (Refs. 13 and 14).

37 NEUTRON-PROTON FINAL-STATE INTERACTION IN m.d. . . 1591

Madelzweig, Garcilazo, and Eisenberg used a relativisticphase space and a relativistic Faddeev propagator butevaluated the three-body recoupling coe5cient with non-relativistic kinematics, ' while Thomas treated the twonucleons nonrelativistically and used relativistic kinemat-ics only for the pion.

In order to treat the kinematics of the three particlesfully relativistically, a number of calculations were per-formed ' in which the basic framework was the rela-tivistic three-body formalism of Aaron, Amado, andYoung, ' in which the kinematics of the three-body sys-tem corresponds to having all three particles on theirmass shells. A further advancement of this theoreticalframework was the inclusion of relativistic treatment notonly for the space variables but for the spin variables aswell, which followed from the application of Wick'sthree-body helicity formalism.

A very serious drawback, however, of using on-mass-shell kinematics in all intermediate states of three parti-cles is that in general one does not have conservation oftotal four-momentum at every vertex. In order toremedy this situation, one of us' has recently developeda new relativistic Faddeev theory of the mNN system inwhich all the spectator particles are on the mass shell andall exchanged particles are off the mass shell, while thepion-nucleon and nucleon-nucleon amplitudes with anynumber of particles off the mass shell are represented bysums of isobars of variable mass. As a result of having al-lowed some of the particles to go off the mass shell, oneguarantees conservation of total four-momentum, and inaddition, one is able to describe correctly processes wherethere is an emission or absorption of a real or virtualpion.

There are two stable isobars in the mNN system, thenucleon in the m.N subsystem and the deuteron in the NNsubsystem. Thus, the integral equations of the spectator-on-mass-shell theory' give the amplitudes for all the pro-cesses which start with one free particle and one of thestable isobars and end up with another free particle andeither a stable or an unstable isobar, so that they corre-spond, respectively, to two-body-two-body processes(like n d~n.d, m.d ~NN, and NN ~NN) or two-body- three-body processes (like m.d ~m NN andNN~mNN). The two-body input to these equations arethe off-shell amplitudes for the pion-nucleon S„,S»,P)), P)3, P3), and P33 channels and the nucleon-nucleonS&- D& and 'S0 channels as described in Ref. 16.

In Fig. 4 the md breakup cross section is plotted as afunction of the proton momentum for the nine proton-pion angle pairs. The cross section distribution is dom-inated by quasifree my scattering. In the region ofP =300 MeV/c a strong enhancement of the cross sec-

Ption is observed which is due to the np final state interac-tion. The solid and dashed lines are theoretical predic-tions of the new theory of Garcilazo. ' The solid line cor-responds to the full calculation, the dashed one to the im-pulse approximation. As one would expect, both predic-tions are identical in the quasifree md scattering region,but deviate drastically in the final state interaction re-gion. While the full calculation reproduces the final stateinteraction qualitatively, the impulse approximation falls

short by more than an order of magnitude.The disagreement between the full calculation and the

data in the region of the final state interaction is con-sistent with the discrepancies observed in large angle m.dscattering. We have no explanation for this effect at themoment.

In Fig. 5 we compare predictions from the earlier cal-culations of Garcilazo' ' —dashed lines —with the onesof his new theory Ref. 16)—solid lines. Differences be-tween the two theories are in the treatment of the two-body vertex functions and of the propagator of the ex-changed particle that is allowed to be off the mass shell.The differences between the two theories are expected tobe larger for the FSI region than for the quasifree (QF)region due to the fact that the QF process is a singlescattering mechanism while the FSI process is a doublescattering mechanism. Thus, while for the QF diagramthe exchanged particle is reasonably close to the mass

10'0, = -253'

10

10

1O'.r

10

10',~r0-2

200 30010 ~

~~

1O',

10

1O'

400 500 600 700

141'

125'

10

10' r

10

10

10'

I I I ~ I I I ~ I

300 400 500 600 700

110'

10

10'

1O'

10' r r

10-2

200

I a I L I i I a I

300 400 500 600 700

Proton Momentum (Me YYc)

FIG. 6. Investigation of the effect of the D state of the deute-ron on the calculated cross sections. The solid line representsthe full calculation, the dashed line is the extreme case in whichthe D state is not included at all. More realistic variations be-tween 4.25% D state (Bonn potential) and 5.8% (Paris potential)show differences too sma11 to be seen in the figure.

1592 W. LIST et al.

shell, in the FSI diagram, on the other hand, one has tocarry out a loop integration in which one picks up contri-butions where the exchanged particles are very far off themass shell.

At this point it might be interesting to study the sensi-tivity of diferent parts of the mN or NN input in thetheory. For this study we arbitrarily restrict ourselves tothe right column of Fig. 4. The gross features of the ~dbreakup reaction are dominated by the deuteron wavefunction. It may be interesting, therefore, to see theeffect of the D state on the cross section. In Fig. 6 wefirst show the extreme case when the D state is complete-ly neglected in the deuteron wave function. The compar-ison with the full calculation shows that the effect is siz-able only in the np final state interaction region. It issmallest where the relative proton-neutron momentumapproaches a minimum. A more realistic comparison ofdiferent D-state components can be achieved by compar-ing the Paris potential with the Bonn potential as NN in-

put in the calculation. While the D-state component inthe Paris potential amounts to 5.8%, it is only 4.25% inthe Bonn potential. The predictions show essentially nodifference.

A further comparison shows the infiuence of the 'Soand ( S,- D&) NN channels on the cross section in thefinal state interaction region (Fig. 7). The dotted line cor-responds to the case where no final state interaction is in-cluded in the calculation, the dashed line corresponds toonly including the So channel, while the solid line corre-sponds to including also the S, - D, channel. As onewell sees, the effect of the 'So channel is negligible incomparison with that of the S]- D& channel. The verysmall sensitivity to the nucleon-nucleon final-state in-teraction in the 'So channel is the special case of a gen-eral feature of the md system. It was first noticed in theelastic channel by Rivera and Garcilazo, while this be-havior was later confirmed by Rinat et al. A simple ex-planation for this behavior can be obtained if one

10'tL'

103 .124 86 58t

10

10'~

100

60 93

ep ———25.3'10'

I

10',

10

10',

10 ~le-+O

—25.3't e t ~

141

10',I

102 ) ~

200 30010' .

, -1.0663 1910 r

10'

10'

1OO

4X 500 600 700

3785 p" 125'L

10 I

1200

10~'

10' p

10

10'

1Oo

~ I k I ~ I

300 400 5|Xi 600 700

125'

~~10 r ~, .-'. - ~ .-10) I ~ t ~ I

200 300 4004]0 s ~ w g ~

'. 91 47 22 57 105103

a I

500

'~

I

600 700

11G

10'~10-

200)04

103

~ I ~ I ~ I ~ I ~ I

300 400 500 600 700

110'

10

10'

10

10

10'rI a I ~ I a I a I, I

Ã0 300 400 500 600 700

Proton Momentum (Me Vfc)

10

1P.1

I a I s I ~ t ~ I ~ I

2 300 400 SOO 600 700

Proton Momentum (Me Vf. c)

FIG. 7. Effects of the singlet and triplet NN channels on thecross sections in the final-state interaction region. The dottedline corresponds to "no FSI," the dot-dashed one to the 'Sochannel only, and the solid curve corresponds to both singletand triplet contributions included.

FIG. 8. Effects due to the nonresonant mN partial waves inthe two-body input into the theory. The solid line correspondsto the full calculation, the dashed one to the 5 dominance (P33n.N partial wave only).

37 NEUTRON-PROTON FINAL-STATE INTERACTION IN md. . . 1593

remembers that the pion-deuteron system is dominatedby the J =2+ state, in which the b(3,3) and the nucleonare in a relative S state while their spins are parallel toeach other. Thus, it is easy to see that this state will cou-ple very strongly to the three-body configuration wherethe two nucleons are in the S, channel (spins parallel)while the pion is in a relative P wave with respect to thetwo nucleons, since in order to get the quantum numbersof the J =2+ state, this configuration requires also thatall the spins and orbital angular momenta be parallel toeach other. On the other hand, if the two nucleons are inthe 'So channel (spins antiparallel), in order to have totalangular momentum J =2 the pion must be in a relative Dstate with respect to the two nucleons, and this impliesthat the state has quantum numbers J =2 since thestate J =2+ is not allowed. Thus, the small effect of thenucleon-nucleon 'So channel is due to the fact that thischannel does not contribute to the dominant

configuration of the md system, which is J =2+.Finally we show the effect of the small mN waves. In

Fig. 8 again the solid line is the full calculation, while thedashed one corresponds to the calculation with only theP33 partial wave taken for the m.N input (5 dominance).A sizable effect is observed over the entire momentumrange. The experimental data seem to prefer the full cal-culation.

ACKNOWLEDGMENTS

We thank Prof. W. Kluge from the University ofKarlsruhe for the loan of the liquid hydrogen target ap-paratus. This work would have been impossible withoutthe generous help and the considerable skills of the staffof SIN. This work was supported by the bundesmin-isterium fiir Forschung und Technologie of the FederalRepublic of Germany.

On leave from Escuela Superior de Ffsica y Matematicas, Insti-tuto Politecnico Nacional, Mexico 14 D.F., Mexico.

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