The Chou-Yang model and π−p elastic scattering at 200 GeVc

Volume 173, number 2

PHYSICS LETTERS B

5 June 1986

THE CHOU-YANG MODEL AND "~-p ELASTIC SCATTERING AT 200 G e V / c Mujahid KAMRAN

Centre for High Energy Pk~vsics, Punjab Unit,ersiO,. Lahore-20, Pakistan and I.E. Q U R E S H 1 Nuclear P,~vsics Division, PINSTECH, P.O. Nilore, lslamubad, Pakistan Received 25 October 1985; revised manuscript received 17 March 1986

By considering a variety of pion and proton form factor combinations it is shown that the Chou-Yang model does not correctly describe the structure observed in ~r p differential cross sections at 200 GeV/c. The model yields either structureless cross sections or cross sections with incorrectly placed and/or spurious dips. The t range considered is 0 ~ - t < 8 (GeV/c) 2.

The connection between high-energy elastic hadronic cross sections and the form factors of the colliding objects was first exploited by Chou and Yang [ 1 ] and Durand and Lipes [2]. The basic assumption of the well-known C h o u - Y a n g model is that the eikonal is proportional to the Fourier-Bessel transform of the product of the form factors o f the colliding hadrons. The first application of this model to pp elastic scattering led to the famous prediction of the - t 1.4 (GeV/c) 2 dip seen subsequently at ISR. Various modifications of detail have been made from time to time in the C h o u - Y a n g model as data at higher and higher energies became available (c.f.e.g. ref. [3] for a brief survey) but the essential features of the model have not changed. The application o f the C h o u - Y a n g model to 7rp elastic scattering by Chan et al. [4] led to the prediction of a dip in the - t ~ 4 - 5 (GeV/e) 2 region. This dip has been subsequently observed in measurements at F N A L [5]. In this paper we take up 7rp elastic scattering (at 200 GeV/c) in the light of the C h o u Yang model by using a variety of pion and proton form factors apart from reconsidering the work of ref. [4]. Our main objective is to find out which combination, if any, o f the pion and proton form factors leads to better agreement with the data. We shall use

the same formulae as those of ref. [4] in our calculations. The relevant formulae are:

do/dr = NTI 2 , T(s, t) = i ; 0 ~2(s, b) = K f 0

(1)

bdb (1 - e - a ) J o ( b x / -

t),

(2)

(X/Z-{) d ( x / ~ ) F . ( t ) F p ( t )

x J0 (bx/-:-{),

(3)

where FTr and Fp are, respectively, the pion and proton form factors and K is a constant (denoted as/lxo in ref. [4]). The total cross section is given by

a T = 47rim T(s, 0) = 4rr f 0

bdb(1 - e - a )

.

(4)

In eq. (3) the choice of form factors will affect the results of eqs. (1) and (2). The usual practice has been to use the electric form factor G E for F p in calculalatior~s where G E = (1 - t/0.71) - 2 .

(5)

The pion form factor F~r (also denoted as F v ) is usual 205

Volmne 173, number 2

PHYSICS LETTERS B

ly given by V = (1 _ F~r -_ FTr

t/m2) - !

(6)

Apart from using F V Chan et al. also use a quark model based version of F,r (denoted as F~n)

= F2p/3 = (1

-

t/O.71) - 4 / 3

.

(7)

In order to facilitate integration Chan et al. have simulated each of the products FVFp and F~nFp by sums of gaussians which are quoted in ref. [4]. We find that if we use the sums of gaussians for the form factor products in (3) the calculated differential cross sections do exhibit dips as calculated by Chan et al. For instance for K = 5.78 and the sum of gaussians for FVFp one obtains a total cross section of 23.577 mb (in agreement with the value 23.6 mb of Chan et al.) hnd a dip at - t ~ 3.6. The corresponding calculation for F ~ F p yields at total cross section of 23.87 mb and a dip at - t ~ 3.9. Having reproduced the results of Chan et al. we then used the exact expressions given in eqs. ( 5 ) - ( 7 ) for estimating FVFp and F ~ F p to calculate the cross sections. To our surprise we found that in both cases no dip is obtained if one uses the exact expressions for the form factors. For instance for FVFp the use o f K = 5.96 gives a total cross section of 24.348 mb and a differential cross section which falls from 28.2 mb/(GeV/e) 2 at t = 0 to 4.185 X 10 . 8 mb/(GeV/c) 2 at - t = 8. Similar results are obtained for the choice of F~nFp instead of FVFp. Raising the total cross section substantially also does not produce the required dip. For instance for F ~ F . a o r of 30 mb is obtained

for K = 7.58 but there is no dip in da/dt upto t ~ 8. The cause of the failure to produce the observed dip by exact products FVFp and F ~ F p was finally traced out by taking the ratio of these products with the corresponding sums of gaussians used to simulate them. For __F~Fp the corresponding ratio oscillates around the value 1 (to within 10% to 15% generally) upto - t ~ 6.3 after which the ratio begins to depart significantly from the value 1, being 0.8 at - t = 6.8, 0.75 at - t = 7.4, etc. In the range - t ~ 8.0 to 13.0 this ratio ranges between 0.65 and 0.71, the value 0.65 occurring around t ~ 10.4. Between t ~ 13 to 15.4 it varies from 0.71 to 0.85 and so on. For the ratio of the exact product FVFn with the corresponding sum of gaussians used for it the situation is even worse. The ratio oscillates around the value i up to 206

5 June 1986

- t ~ 2.5 and then starts rising steadily. At t ~ 3 it has a value 1.11, at - t ~ 4 the value is 1.25, at - t ~ 5 it is 2.728, at - t ~ 6 it is 5.036 and so on. By - t = 10 the ratio has risen to 100.2 while at - t = 15 its value is 7741.6. It is therefore clear that the sums of gaussians for F VFp and F QFp do not properly simulate these functions. Having thus found that the exact products FVFp at F ~ F p do not give a dip in 7rp elastic scattering at high energies we then decided to try alternative forms for Fp in conjunction with F V and F~n. For Fp we used the following two choices to begin with: Fp = F 1 = (4m 2 - 2.79t) (4m 2 - t ) - l ( 1 - t/0.71) - 2 ,

(8) where F 1 is the Dirac form factor used extensively in the recent work of Dennachie and Landshoff [6]. The other form factor we chose is the one due to Felst [7] which has been used recently by Glauber and Vetasco in their work [8]. The Felst form factor (say F F ) is: Fp = F F =(1 - 3.04t + 1.54t 2 - 0.068t3) -1 .

(9)

We then carried out calculations using the products where F V and F ~ are given in eqs. (6) and (7), respectively. Since the highest energy at which np differential cross section data is available corresponds to Plab = 200 GeV/c we calculated our results for this energy. The value of K was adjusted in each case to reproduce the total cross section value of 24.3 mb which is the 200 GeV/c value for Oy(rr-p) measured by Carroll et al. [9] (exact value 24.34 -+ 0.04 mb). Some features of the results of our calculations are summarized in table 1. We find that the use of the Dirac form factor F 1 does not produce any dip upto - t = 8 (beyond which we did no calculations) for a T = 24.3 mb whether one uses it with F v or F ~ . The first and second columns of table 2 show the corresponding differential cross sections at several t values in the range - t = 0.0 to 8.0. A comparison of results shows that in both cases do/dt falls rapidly at first but beyond - t ~ 4 the rate of fall slows down. Also the cross sections calculated by using the product FQF 1 are lower than the corresponding cross sections calculated by using FVF1 . Fig. 1 gives a comparison of the model with experiment for the Dirac form factors. As may be seen the model predicts values well above the data.

FVF1,F~nF1,FVFF and F ~ F F

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Volume 173, number 2 Table 1 Product of form factor used

K (GeV-2)

Total cross section (mb)

Comments

FVF1 a)

6.15 6.042

24.346 24.30

no dip no dip

F~F F

5.93 5.84

24.344 24.34

no dip no dip

FVFBsww

5.96

24.348

two shallow dips at - t ~ 6.3 and 7.9

F~FBsww

5.835

24.34

three shallow dips at - t ~ 4.9, 6.2 and 7.9

F~F 1

FVFF

a) F1 is the Dirac form factor and F F is a parameterisation of the electric form factor due to Felst. FBSWW is a four pole parameterisation of the electric form factor due to Borkowski et al.

When we carried o u t calculations w i t h the parame t e r i s a t i o n o f the electric f o r m f a c t o r due to Felst in c o n j u n c t i o n w i t h the f o r m f a c t o r s F V and @ no dip was o b t a i n e d (fig. 2). Tables 1 and 2 give s o m e n u m b e r s related to these calculations. As in the

case (table 2). Having f o u n d o u t t h a t n o n e o f t h e f o r m f a c t o r p r o d u c t s gives a dip in the 7r-p differential cross sections at 200 G e V / c we c o n s i d e r e d a four pole p a r a m e t e r i s a t i o n o f the electric f o r m f a c t o r (say

previous case the cross sections for t h e p r o d u c t F ~v F F are higher t h a n the c o r r e s p o n d i n g cross sect i o n s for F ~ F F. Also the c o r r e s p o n d i n g cross s e c t i o n s for FVF1 are higher t h a n t h o s e o f FVFv . . . ~ Similarly the calculated cross sections for the F ~ F 1 case are

FBSWW ) due to B o r k o w s k i et al. [10] (used in ref. [81 for ~p data): 4 FBSWW = ~ ( 1 i=1

-

t/m2)-lai,

(10)

higher t h a n the c o r r e s p o n d i n g values for the F ~ F F

Table 2 Differential cross section in

mb/(GeV/c)2 obtained by using different products of pion and proton form factors.

- t (GeV/c) 2

FVF1

F~F1

FVFF

F~FF

0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

30.3 0.803 5.891 7.201 1.255 2.938 9.000 3.472 1.634 8.888 5.417 3.578 2.475 1.797 1.348 1.021 7.899

30.1 0.667 4.044 4.095 5.986 1.207 3.331 1.215 5.635 3.076 1.896 1.269 8.803 6.399 4.805 3.613 2.773

30.2 0.548 3.438 3.868 6.509 1.505 4.626 1.760 8.224 4.296 2.52 1.621 1.054 7.395 5.473 3.926 2.922

30.9 6.66 2.422 2.309 3.410 7.165 2.062 7.419 3.338 1.679 9.632 6.176 3.945 2.786 2.111 1.507 1.129

× x × x x x x x x x x x x x x

10 -z 10 .3 10 -3 10-4 10 .5 10 -s 10 -s 10 -6 10.6 10 -6 10.6 10.6 10 -6 10.6 10 .7

× x x × x x x x x x x x x x x

10 -2 10 -a 10-4 10-4 10 .5 10.6 10.6 10.6 10 .6 10 .6 10. 7 10 .7 10 .7 10 -7 10 .7

× x x × x x x x x × x x x x x

10 -z 10 .3 10 -4 10 .4 10 -s 10 -s 10 .6 10 .-6 10 .6 10 -6 10.6 10 -7 10 .7 10 .7 10 -7

× × x × x x x x x x x x x x x x

10 -1 10 -2 10 -3 10 -4 10 .5 10 -s 10 .6 10 .6 10 .6 10 -7 10 -7 10 .7 l0 -7 10 -7 10 .7 10 .7

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Volume 173, number 2 I

PHYSICS LETTERS B I

10-2

I

--

5 June 1986 I

I

i

I

•

200 GeV/c ~ p V F..rr F 1

i

200 6eV/c 7r-p

10 .2

\ 10-3

10-3 D ~

et

%',~\

10-4

~

10-4

~

lO-S

;> ,.Q

10-s

E .~

ir 10-6

"~ 1 0 - 6

10-7

10-7

10 - 8

10-8

0

I

I

2

6

I__ 8

[

I

10

12

- t [(GeV/c) 2 ]

0

2

iI' i I

I

t

I

4

6

8

10

12

- t [(GeV/c)2 ]

Fig. 1. Predictions of the Chou-Yang model for the Dirac form factor FI and the 200 GeV/c ~r- p data. The solid curve corresponds to the choice of the product EVE1 and the dashed curve to the product F ~ F 1 in the eikonal. No structure is predicted by the model.

Fig. 2. Comparison of the predictions of the Chou-Yang model for the proton electric form factor due to Felst (FF). The solid curve corresponds to the choice of the product FFFV and the dashed curve to the product FFF~ in the eikonal. No structure is predicted by the model.

with the parameters m/2 and ai given in ref. [ 10]. The resulting cross sections for both F V F B s w w and F ~ F B s w w give a multiple dip structure with two and three rather shallow dips respectively in the - t ~< 8 (GeV/c) 2 regions (fig. 3 and tables 1 and 3). As expected the cross section values given by the FVFB SWW case are higher than the F ~ F B s w w with the latter showing rather close and better agreement with the data in general. For the F V F B s w w case the dips arise at - t ~ 5.0, 6.3 and 7.9 whereas for F~nFBSWW the dips arise at - t ~ 4.9, 6.2 and 7.9. Experiment however shows a clear cut dip around - t ~ 4 so that the location of this dip is not correctly given by the use of the BSWW form factor.

The recently reported experimental values of n - p differential cross sections show that though there are rather large error bars in the 4 < - t < 8 region the data does not appear to contain three dips in the - t < 8 (GeV/c) 2 region. The possibility of two dips one around - t -~ 4 and the other around - t ~ 6 is however not inconsistent with the data. Even if the data contains two dips the calculations based on the choice of the BSWW form factor for Fp still contain an extra dip. The question therefore as to whether or not the C h o u - Y a n g model correctly describes high energy n - p data in terms of standard expressions for the pion and proton form factors is very much an open question at the moment. The

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Volume 173, number 2

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Table 3 Detailed values of do/dt [mb/(GeV/c) 2 ] in the dip regions for the BSWW form factors.

I

200 GeVlc n'-P

10-2 --

-

-t( GeV /c) 2

FV].r FBSWW

-

-3

10

>

&

-6

I0

1~7

II l I

-B

10

i

I

I

I

I

|

0

2

4

6

8

10

12

- t [(GeV/c) 2 ] Fig. 3. The structure predicted by the use of a four-pole formula for the proton electric form factor due to Borkowski et al. (FBsww). The solid line represents the case when FVFBsww is chosen and the dashed line the case when F ~ F B s w w is chosen in the eikonal.

F ~ FBSWW

FV FBsww

4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5

2.844 2.277 1.942 1.798 --* 1.797 1.889 2.018 2.124 --* 2.154 2.078 1.894

× x x x x × x × × × x

10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7

1.013 8.437 7.284 6.575 6.196 6.026 5.948 5.852 5.654 5.312 4.827

x × × × × x × x x x ×

10 -6 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7

6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0

6.674 5.659 --, 5.226 5.289 5.757 6.518 7.417 8.254 8.823 --, 8.963 8.606

x × × x x × X x x x x

10 -8 10 -8 10 -8 10 -8 10 -8 10 -8 10 -8 10 -8 10 -8 10 -8 10 -8

2.170 1.911 1.764 ~ 1.714 1.738 1.812 1.905 1.985 ~ 2.022 1.997 1.901

× x × X X X X X X X X

10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7 10 -7

7.6 7.7 7.8 7.9 8.0

2.558 2.082 1.839 --, 1.799 1.938

x x x × ×

10 -8 10 -8 10 -8 10 -8 10 -8

7.780 6.756 6.150 ~ 5.922 6.019

x × × x X

10 -8 10 -8 10 -8 10 -8 10 -8

aspects o f its e v o l u t i o n w i t h energy c o r r e c t l y . The dip c o m e s o u t to be d e e p e r , and the s e c o n d a r y maxim u m higher, t h a n the data. The filling in o f t h e dip at 546 GeV can be achieved b y i n t r o d u c i n g a real

analysis carried o u t in this p a p e r w o u l d suggest t h a t spurious dips, as in t h e p p case, do seem t o arise in np scattering as well. Or else the m o d e l implies t h a t if one uses t h e s t a n d a r d e x p r e s s i o n (5) for the electric f o r m f a c t o r G E in c o n j u n c t i o n w i t h F V or there are n o dips in this m o d e l for - t < 8 at a T = 24.3 m b i.e. at 200 G e V / c l a b o r a t o r y m o m e n t u m . The C h o u - Y a n g m o d e l has b e e n applied to t h e ~p data at collider energies in the last few years [8,11]. The u p s h o t o f these investigations is t h a t the C h o u - Y a n g m o d e l , in its pristine f o r m (eqs. ( 2 ) - ( 4 ) ) , is qualitatively c o r r e c t b u t q u a n t i t a t i v e l y inexact. It p r e d i c t s s t r u c t u r e in the c o r r e c t place b u t does n o t describe the details o f the s t r u c t u r e or

part [ 11 ] b u t n u m e r i c a l a g r e e m e n t w i t h e x p e r i m e n t is still n o t g o o d . In ref. [8] Glauber and Velasco have used a modified version o f the C h o u - Y a n g m o d e l in an a t t e m p t to describe the data at 546 G e V c o r r e c t l y . T h e y , in a sense, derive the C h o u - Y a n g m o d e l b y a p p l y i n g a p a r t o n picture in c o n j u n c t i o n w i t h t h e Glauber t e c h n i q u e . A p a r t f r o m i n t r o d u c i n g a real part o f t h e eikonal t h e y are led t o t h e i n t r o d u c t i o n o f an unk n o w n t - d e p e n d e n t factor w h i c h takes a c c o u n t o f the a n i s o t r o p y in the p a r t o n - p a r t o n i n t e r a c t i o n . T h o u g h these m o d i f i c a t i o n s lead to the i n t r o d u c t i o n o f n e w p a r a m e t e r s a very g o o d d e s c r i p t i o n o f the 546 G e V data is o b t a i n e d p r o v i d e d one e m p l o y s the 209

Volume 173, number 2

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electric form factor of Borkowski et al. [10]. The use of the parameterisation of Felst [7] in ref. [8] leads to reasonable agreement with the 546 GeV data. It would therefore appear that the form factor parameterisation of Borkowski et al. is slightly favoured in the gp case in the context of the C h o u - Y a n g model. Our investigations for 7r-p elastic scattering, although carried out at much lower energies compared to t h e ~ p case, also point, somewhat more conspicuously, in the same direction. In particular the description coming closest to the 7r-p data is the one employing the BSWW form factor. Other aspects of the C h o u - Y a n g model and np (and Kp) scattering are being investigated and will shortly be reported in detail elsewhere. One of us (M.K.) would like to thank Dr. Inam ur Rehman for the hospitality extended to him at the Centre for Nuclear Studies, Nilore. He would also like to thank the Director PINSTECH for allowing access to computing and library facilities.

Note added. After completing this paper our attention was drawn to a recent paper by Chua and Lai [12] where a new parameterisation of the pion form factor is produced in order to obtain a fit to the 200 GeV/c l r - p data. However this parameterisation is based on the assumption that the C h o u - Y a n g model is correct. We have on the other hand made no such

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5 June 1986

assumption using only the available parameterisations of the pion form factor to compare the model with experiment. However it is quite possible that the standard expression for F,r may not be correct at very large Itl.

References [1] T.T. Chou and C.N. Yang, Phys. Rev. Lett. 19 (1968) 3268. [2] L.J. Durand and R. Lipes, Phys. Rev. Lett. 20 (1968) 637. [3] R. Castaldi and G. Sanguinetti, CERN-EP/85-36 (March 1985), to be published in Ann. Rev. Nucl. Part. Sci.; M. Kamran, Phys. Rep. 108 (1984) 275. [4] S.C. Chan et al., Phys. Rev. D 17 (1978) 802; see also: C.H. Lai et al., Phys. Lett. B 122 (1983) 177; Phys. Rev. D 27 (1983) 2214. [5] R. Rubinstein et al., FERMI LAB-Pub-84/54-E (June 1984); W.F. Baker et al., Phys. Rev. Lett. 47 (1981) 1683. [6] A. Donnachie and P.V. Landshoff, Nucl. Phys. B 231 (1984) 189. [7] R. Felst, DESY 73/56 (Nov. 1973). [8] R. Glauber and J. Velasco, Phys. Lett. B 147 (1984) 380. [9] A.S. Carroll et al., Phys. Lett. B 80 (1979) 423. [10] F. Borkowski et al., Nucl. Phys. B 93 (1975) 461. [11] T.T. Chou and C.N. Yang, Phys. Lett. B 128 (1983) 457. [12] B.W. Chua and C.H. Lai, Phys. Lett. B 150 (1985) 455.