IL NUOV0 CII~IENT0

VOL. 72 ~, N. 3

1 Dieembre 1982

Study of the Contribution of the Inelastic Correction to the Differential Cross-Section da/dt for p-~He Elastic Scattering at 200, 393 and 1000 GeV, with Different ~He Wave Functions. J. P~o~ioT, and B. JA~SEAIX

~aboratoire de Physique Corpusculai~e Univevsi~ de Clermon$ II-IN2P3, B.1). 45 - ~63170 Aubi~re, ~'rancc (ricevuto il 15 Giugno 1982)

S u m m a r y . - - We have computed the contribution of the inelastic correction to the differential cross-section da/dt in the p-4He elastic scattering at 200, 393 and 1000 GeV. We use different 4He wave functions and different nucleon-nucleon amplitudes: We compare the calculations with experimental results.

1. - I n t r o d u c t i o n .

R e c e n t experimental dat~ on p-4He elastic scattering (~-8) and recent e~lculations (4) have pointed out the necessity to include an inelastic contribution

(1) A. BUJAK, P. DEVENSKY, A. KUZNETSOV, B. ~OROSOV, V. NIKITIN, P. NOMOKONOV, Y. ~)ILIP]~NKO, V. SMIRNOV, E. JENKINS, ]~. ~r ~V~.MIYAJIMA an4 ~. YAMADA: Phys. t~vv. D, 23, 1895 (1981). (3) M. A~BROSIO, G. ANZ~VINO, G. BARB~a~INO, V. BECKE~, G. C~BO~I, V. CAVXSINNI, T. D E L PI~TE, G. KANTARDJIAN, D. LLOYD 0W~N, ~. I~ORGANTI, J. ~ARADISO, G. PATERNOSTER,S. PATRICELLI,~ . STEUR and V. VALDATA-NAPPI:Phys. SetS. B, 113, 347 (1982). (a) W. BELL, K. B~AVNE, G. O. CLAESSON,D. D~IJA~D, Yr. A. FXESSLE~,H. G. Flsem~R, It. F. F~ERSE, R. W. FREY, S. I. A. GARP~IAN,P. C. GUGELOT,P. HANKE, P. C. INNOCENTI, T. J, KETEL, E. E. KLUGE, G. I~ORNACCHI, IV[. T. I~AKADA, I. 0TTERLUND, B. POVH, A. PUTZER, E. STENLUND, T. J. N[. SYMONS, R. SZWED, 0. ULLALV~D and T. WALCHER: CERN-EP/82-64. (4) J. PRO~IOL, S. MARRY and B. JARGEAIX: Phys. ZeSt. B, 110, 95 (1982). 278

STUDY

OF THE

CONTRIBUTION

OF THE

INELASTIC CORRECTION

ETC.

279

to the Glauber amplitude to explain, in part, the experimental measurements of the differential cross-section da/dt. This m e t h o d was originally used, with great success, to explain the experim e n t a l d a t a of pD and DD elastic scattering a t high energy (5.9). I n this paper, we use the m e t h o d designed b y ALBEI~I (~-s) to s t u d y the influence of the 4He wave function on the contribution of the inelastic correction to elastic scattering. W e use the standard Glauber m e t h o d to study the elastic scattering, b u t we introduce the wave function via the Fourier t r a n s f o r m of IV[2. The different choices are given in sect. 2. I n sect. 3, we describe the inelastic contribution t h a t we introduce in our calculations and then we can compare the experimental results with theoretical calculations in computing, as in (6), the ratios

(1)

/~1

dt ~

da

dt al~ub~r

da

W e discuss the results obtained for these ratios at 200, 393 and 1000 GeV. I n the appendix, we recall the Glauber amplitude written with the Fourier transform of ]~12.

2. - T h e c h o i c e

of 4He wave

functions.

The success of the pD and of the DD analysis (5.8) is due, in part, to the accurate knowledge of the deuteron wave function. Unfortunately, we use in p - q ! e elastic-scattering problems v e r y simple wave functions fitted with the aHe charge form factor. Such a way ignores most of the nuclear properties o f t h e 4He nucleus as the binding energy, and the S' and D state contributions. Recent calculations (9) were done to compute the 4He wave function with recent nucleon-nucleon potentials. Unfortunately, this wave function is known numerically and it is difficult to include it in a general Glauber formalism; a s t u d y is in progress to overcome this difficulty (19).

(5) G. ALRERI and F. BALDRACCmNI: Nuvl Phys. B, 138, 164 (1979). (~) G. GOGOI, C. CAVALLI-S~ORZA,C. COSTA, M. FRATER~I, G. C. MA~TOVANI, F. PASTORE and G. ALBERI: Nq,cl. Phys. B, 149, 381 (1979). (7) G. ALRERI, 1~. BALDRACCHINIand V. ROBERTO: Nuovo Cimento A, 57, 249 (1980); (s) G. ALBERI and G. GoGoI: Phys. t~ep. 74, 1 (1981). (9) J . L . B~LOT: Z. Phys. A, 302, 347 (1981). (10) j. L. BAT,LOT: private communication.

280

J . PRORIOL

and

B. J A R G E A I X

R e c e n t l y (1~), we h a v e proposed a new m e t h o d to write the Glauber f o r m a l i s m a n d t o choose a w a v e function for 1)-dHe calculations. I n s t e a d of choosing the w a v e function ~, we choose the S(P, Q, R) function which is the Fourier t r a n s f o r m of I~P. I n the appendix, we give the definition of the S-function a n d the Glauber /~(q) a m p l i t u d e written for p-dHe elastic scattering with a nucleon-nucleon a m p l i t u d e ](q). I n p a p e r (z~) we h a v e pointed out the reasons to choose the P J M V function S of t h e f o r m

The p a r a m e t e r s ~ and fl are fitted with the 4I!e charge f o r m factor a n d we get

(

(3)

~ ~-- (3.33-k0.01) (GeV/c) -2 fl ~-- ( 1 3 , 0 9 • 1 6 5

-~ .

A n o t h e r choice of the S-function is

(4) S(P, Q, R) ---- exp [-- G(P 2 § Q2 §

with G ---- (12.30-k0.01) (GeV/c) -~ .

W e h a v e also used the Czyz and M a x i m o n m e t h o d (12), which is the s t a n d a r d m e t h o d used for the s t u d y of ~tte-~He elastic scattering. T h e nucleon-nucleon a m p l i t u d e is chosen of the f o r m

l(q)=-~6-~t~247

(5) TABLE I. /~(GeV)

-

Parameters ~sed ]or the nucleon-nucleon amplitude. atot(mb)

~o

B((GeV/c) -~)

r((GoV/c) -~)

5.785

0.183

200

38.97

--0.039

393

40.04

0.012

5.97

0.152

1000

41.79

0.062

6.46

0.151

T h e values of atot, ~, B for E ----%/~m~ § k ~ are given in t a b l e I. W h e n we use the P J ~ V S-function and the Czyz and M a x i m o n m e t h o d , we set ~----~o. W e s t u d y the influence of a possible variation of ~(q2) on / ~

(11) j . PRO~mL, B. JA~Gv.AIX, S. 1VIAVRY and F. VAZEIL~: ZYUOVOGimeuto A, 71, 149 (1982). (12) W, CzYz and L. C. l~_xiMoz(: Ann. Phys, (hr.:Y,), 52, 59 (1969).

STUDY 01~ THE COI~T!~IBUTION OF THe. INELASTIC CORR]~CTION ~ETC.

281

by choosing, for computing commodity, an exponential shape of e(q2):

9(q ~) = (~o -~ 3) exp [-- 7q ~] -- 3,

(6)

which is not far from the function ~ = ~ o - 0.44q 2. The ~ value is given in table I.

3. - T h e i n e l a s t i c correction.

To introduce the inelastic correction to the Glauber amplitude, we use Alberi's method (5.s). We consider the double-scattering amplitude ~i~(q) in the p-~He amplitude

~(q):

(l The inelastic contribution A~H to the amplitude 2F can be visualized as in fig. 1, where X is exchanged instead of a nucleon. x

4He

~He

l~ig. 1. - Inelastic correction A2/~ to ~he 2/~ amplitude.

The inelastic contribution to this amplitude is (~.8) (8)

A~F = ~ k J d 2 q ~ S

, v / 2 Q ]x(q/2 § q ' ) / x ( q / 2 - q ' ) 2

'

where X is a particle produced and absorbed in the collision; the vector Q has components q' and q l - ~ ( M E - M~)/2k. According to .~LBERI (s-s), w e separate the X contribution in two regions: the low-mass region M < M1 = 2 GeV, where the contribution comes mainly from resonances, and the triple-Regge region. To compute the resonance contribution, we suppose that Ix is mainly imaginary and we neglect the q' contribution. We set (9)

]~ : -- (k2/~)[d2a/dtdM ~ (t : -- ~q~)],, .

282

J. PEORIOL a n d B. J A R G E A I X

The resonance contribution to A:~ is

"1

~:

)

--~

--

4~

/Jres

9

The triple-Regge contribution is given by (~-s)

(n)

The function A/~ '~,~ is the Alberi function (6). The value of A/~6 is

with

(12)

dff~,t,m

dt d M 2

--

..

1 2

V xx~(~q )

I n the preceding relation, ~m(t) and fl~(t) are the ~egge trajectories and V are triple-Reggeon and Reggeon-nucleon vertex functions. These functions were fitted by RoY and R o B ] ~ S (~a). We use Hendrick's fit (14) for the function (d2a/dt dM~)~o.; we introduce also the ~ attenuation parameter used by ALBE~I in the pD case (7). We have estim a t e d a triple-scattering amplitude inelastic contribution A3/v: (13)

S AsF = - (2~ik) ~

I q, § q,,).

The vector Q has components q" and ql. The Aa_~ contribution was computed with the following simplification ]( 89 q ' ) ~ ](89 and b y replacing the Ix product b y ]x(89q) ]x(89q)" The contribution is less t h a n 15 % of the A~/7 contribution. (is) D. 1). ROY and R. G. ROBERTS" N~Cl. Phys. B, 77, 240 (1974). (1~) R. E. HESDRIOX: .Phys. Rev. D, 11, 536 (1975).

STUDY 017 T H E CONTRIBUTION OF T H E I N E L A S T I C CORRI~,CTION I~.TC.

283

4. - Values o f / ~ 1 and / ~ . W e h a v e c o m p u t e d t h e R1 a n d Rs r a t i o s a t E = 200 G e u f o r d i f f e r e n t ~He w a v e f u n c t i o n s . (See fig. 2-5.)

101

t

t

"I"

-I4-

10 0

...l.

0.1

I

I

I

I

0.2

0.3

0.4

0.5

Fig. 2. - i ~ values ( ), i~1 e x p e r i m e n t a l values for E -~ 200 GeV wi~h t h e Czyz a n d ~ a x i m o n w a v e function.

I

R

-If

+

"1-

i0 ~ § .t +-I-§

0

I

0.1

olz

o'.3 -t [~e~/~ ~]

' 0.4-

I

0.5

0,6

Fig. 3. - R s values ( ) , i~1 e x p e r i m e n t a l values for E = 200 GeV w i t h t h e exp o n e n t i a l S - f u n c t i o n (relation (4)) a n d ~ = ~o.

9.84

J.

PRORIOL

and

B. JARGEAIX

I n the figures, the continuous curve gives R~ ratios, the experimental ratios R1 are computed from the experimental differentiM cross-sections given in ref. (1). I n fig. 2, we have used the Czyz and Maximon method; in fig. 3, we have used the S-function of relation (&) with ~ ---- 60; in fig. ~, we used the S-function of relation (4) and Q of relation (6) and in fig. 5 we used the S-function of relation (2). W e see t h a t the Czyz and Maximon m e t h o d is not convenient; there are not great differences for the experimental agreement in fig. 3 and 4. The agreement is b e t t e r around the m i n i m u m of d~/dt in fig. 5.

§ t. -~

§

-t-

100

I

I

I

I

2

3

4

5

Fig. 4. - R 2 values ( ), R 1 experimental values for E = 200 GeV with the exponential S-function (relation (4)) and Q = ~(q~) (relation (6)).

:lit §

I 0.7

+

9

§

§

!

0.2 -t

o13

Fig, 5. - i~ values ( . ), i~1 experimental values for E ~- 200 GeV with the PJ1KV S-function (relation (2)).

STUDY

OF T H E

CONTRIBUTION

OF T H E

INELASTIC

CORRECTION

"P,,TC.

2~

t

--I--9

L..+.)

v

9

v

::>

-+..~

o, ,..~j I

+=om

d

I

6

+ . _ + _ + , , ~ ~

<~ d I

I I I I

I_

I

I

1

I

%

<~

II

1[

,.+,.+ d

I .p ..i+

.,,.+

...i.-

6

-I-

+::>.,

+

u

-----~"~ _

. ,..b

I

[.

I

4-

\ I

[

f

l

I

I

if'~J,

%

I

;

I

I

I

k

t;,.

I

I

286

J. PRORIOL a n d B . J ~ L R G E ~

B u t in all cases there is a disagreement for low t and for high t values. I n fig. 6, we used, at E - ~ 393 GeV, the S-function of relation (2). The agreement is similar as at 200 GeV. I n fig. 7, we have used, at /~ ~ 1000 GeV, the S-function of relation (2). T h e experimental values from ISR are from ref. (~,3). T h e disagreement is impressive. 101

I0 ~

I O.l

I 0.2

I

0.3

I 0.4-

I 0.5

I

0.6

I 0.7

0.8

), /~1 e x p e r i m e n t a l v a l u e s for E ~ 1000 G e V w i t h t h e exFig. 8. - /~2 values ( ponential S-function (relation (4)) and e = e(q~) (relation (6)).

To study, at very high energy, the influence of ~(q2), we have used, for fig. 8, the S-function of relation (4) and the e value of relation (6). The agreem e n t is better at high t values.

5. -

Conclusion.

W i t h different wave functions, we could not get full agreement for the R ratios. We have seen t h a t different wave functions (or S-functions) give some important differences. I t seems t h a t the agreement, at 200 and 393 GeV, is better with the P J M V S-function of relation (2).

STUDY

OF T H E C O N T R I B U T I O N OF T H E I N E L A S T I C C O R R E C T I O N E T C .

297

There is always disagreement at low and at high t values. For ISR results, it is difficult to conclude. I t seems t h a t a 0(q 2) function gives better results. $$$

We would like to t h a n k Prof. G. ALBEgI for several discussions and Dr. M . A . FAESSLEg for providing the ISR results before publication.

APPENDIX

The variables for the particles in 41=[e are r~ (i ---- 1, . . . , 4). We choose the new variables

R = (r~+ r~+ r~+ r,)/2, p~ = ~/~ ( - ~ +

~ (,-1+ r~ + r~))/2,

p~ = ~/] ( - r~ + 89

The

S(P, Q., R)

~)),

functions is defined as a Fourier transform of ]~vi2. Then

S(P, Q_,R) = f e x p [i(P.P1 -k Q'P~ + R . P3)] I~l~d~p1d3p2 d3~3. We call

](q) the

WJ~ amplitude. The Glauber amplitude for p-dHe is

(q q 5) --

F(q)=4t(q)S 2Vg,~/-~,

+(4/(2~ik)~

] +~-y / -x. )f,(q + ~ + y )(q )(q )

"S ( 2 ~ , ~/~ x, V-2 y) d2xd2y--

t~--3' ~ y' ~2 z) d2x d~Y d ~z 9

288

9

J. PRORIOL a n d B. JA.RG~,~X

RIASSUNTO

(*)

Si ealcola fl contributo della eorrezione anelas$iea alia sozione d'urto differenziale da/dt hello scattering elastieo p-4He a 200, 393 e 1000 GeV. Si usano diverse funzioni d ' o n d a di tHe e diverse ampiezze nueleone-nueleone. Si eonfrontano i ealeoli con i risultati sperimentali. (*)

Traduzione a cura della l~edazione.

Pe3ioMe He IIOny~e~ro.