ELASTIC pp- AND pp-SCATTERING AND THE INELASTIC

OVERLAP FUNCTION MODEL

G. G. Arushanov, Sh. Kh. Dzhuraev, E. Karabaev and A. I. ~rgashev UDC 539.101

A combined description of the pp- and pp-scatterings is provided in the wide range of energies /~ = 4-1800 GeV and for a momentum transfer -t between 0 and i0 GeV 2, on the basis of the inelastic overlap function model. The calculated cross sections are compared with the experimental measurements and theoretical predictions are obtained for large values of the momentum transfer.

INTRODUCTION

Among the experimental aspects of pp- and pp-scattering established recently, and still not understood theoretically, are the diffraction picture with one dip in the differential cross section and the evolution of this picture with energy, particularly the transition from the "minimum-maximum" structure to the "shoulder" one, as well as the rise of the cross section and of the ratio between the real and imaginary parts of the forward scattering ampli- tude with energy [i]. As the energy increases, the nucleon becomes darker, sharper and larger (the so-called BEL effect [2]), and the profile function comes closer to a step- function, reflecting the Froissart-Martin bound [i].

At small momentum transfer -t, the structure of the differential cross section is deter- mined by the collective diffraction effect which, in turn, is determined by the many-particle unitarity condition. As the momentum transfer increases, the dynamics of the constituent interaction becomes important, and for very large -t, the hadron interaction reduces to the interaction of their constituents. From a theoretical viewpoint, the major difficulties arise in the domain of intermediate values of the momentum transfer. In this domain various mechanisms come into play. However, to date no consistent explanation of the differential cross section behavior exists in a large domain of values of the momentum transfer [i, 3]. The most complete explanation of the experimental dependence of the differential cross sec- tion on t is provided by eikonal models, of the inelastic overlap function (IOF) type we pre- viously used [4-6] to successfully describe the experimental data at certain energies (for pp-scattering at ISR energies, and for pp-scattering at the SPS-collider energy /~= 546 GeV). Since then, the old experimental results were sharpened, and new ones have been obtained. In particular, the first measurements of the differential cross section for elastic pp-scattering have been obtained at ~ = 1800 GeV at the Fermilab tevatron collider [7]. In this work we examine systematically the differential cross sections for elastic pp- and p-p-scattering in the wide range of energies ~ = 4-1800 GeV and values between 0 and i0 GeV 2 of the momentum transfer -t. We shall show that the model describes consistently well all available experi- mental data, and predicts the cross section at larger, not yet measured values of the momen- tum transfer.

i. BASIC FORMULAS OF THE MODEL

The dependence of the scattering amplitude on momentum transfer is given by the Fourier- Bessel transform of the amplitude ~(s, p) in the impact parameter p representation

( F(s, i V-- 1 ) t) = (s, p) 4 t) pap,

0

where J0(x) is the Bessel function. The scattering amplitude is normalized such that the differential cross section for elastic scattering is

A. P. Beruni Tashkent Technical University. Translated from Izvestiya Vysshikh Ucheb- nykh Zavedenii, Fizika, No. 12, pp. 78-82, December, 1992. Original article submitted November 25, 1991.

1064-8887/92/3512-1161512.50 © 1993 Plenum Publishing Corporation 1161

d,~/dt----- IP (s, t) 12,

and the total interaction cross section ot(s) is, by the optical theorem,

(~t(s) =4]/~ ImF (s, 0).

(2)

(3)

The unitarity relation, in terms of the overlap functions in the t-representation, is

4 ~ ImF(s, t )=E(s , t)+G(s, t), (4)

where E(s , t ) and G(s, t ) a re t h e e l a s t i c and i n e l a s t i c over lap f u n c t i o n s (EOF) and (IOF), r e s p e c t i v e l y . In t he impact pa ramete r r e p r e s e n t a t i o n , t he u n i t a r i t y c o n d i t i o n r eads :

2~e~(s, o)=,~e(s, o )+~(s , o), (5)

where ~¢~[~12 and ~g are the E0F and I0F in the impact parameter representation. The physical solution of Eq. (5) has the form

~(s, p)=l--exp(2ia(s , O))]/1--Wg(s, p), (6)

where = is the phase shift for the purely elastic scattering. The solution (6) shows that the elastic diffraction is the absorption shadow due to the existence of many open inelastic reaction channels. Thus, the absorption corrections, i.e., the effects of cross-scattering (the first term in Eq. (5)), play an important role in the diffraction elastic scattering. It can be seen from Eqs. (5) and (6) that the quadratic term in Eq. (5) can be neglected only for small ~g , i.e., for small I~I. The cross-scattering effect can be easily seen from the solution (6), if we set ~ = 0 and rewrite it as

~(S, p ) ~ l - - U l - - ~ g ( s , p ) = ~ ~g(S, p) + ~,}(s, p) (6*) 2 2 ( 1 + V 1 - wg(s, p))2

The relation (5) can be transcribed as

0~ (1--Re~(s, p))2+(Im~(s, 0))2= 1--q~g(s, 0)~1. (7)

The IOF ~g(s, 9) is restricted by the inequalities

0 ~ g ( s , 0) 41, (8)

and characterizes blackness and inelasticity.

We proceeded further from the following expression of the IOF in terms of the impact parameter [4]:

• e (s, p) : 2aexp (--p2/2bl) --ca2exp (--p2/bl). (9)

Here, a, b z, and c a r e model pa rame te r s , depending in gene ra l on energy , and the phase s h i f t of the purely elastic scattering will be written as [4]

2~(s, 9) = dexp(--P2/2b2), (10)

where d and b 2 are parameters, also energy dependent. For c = 0, the expression (9) reduces to the van Hove model, taking into account only the kinematic correlations of the secondary particles (the energy-momentum conservation). For c ~ 0, the expression (9) describes pheno- menologically the effects of absorption corrections, the dynamic correlations of secondary particles, and the destructive interference due to the phases of the matrix elements of in- elastic processes [4]. In Eq. (9), p appears in the combination p/bl, which through Fourier- Bessel transformation leads to a dependence of G(s, t) on the momentum transfer of the form blt. If the parameters a and c from (9) do not depend on energy, the IOF possesses the pro- perty of geometric scaling, the scaling variable being b1(s)t.

For given ~g(s, p) and ~(s, p), the imaginary and real parts of the scattering amplitude are given by the Fourier-Bessel integrals

1162

Ge

0

e,~'-

Fig. 1

dO, k mb~ [~ GeV 5~

/0-8~ - , , , , o a 8'-,, o ,,2

Fig. 2

oo

Im F ( s , t) = ].F~ j' (1 - - V 1 - ~g (s, p)) 4 (p g l ~/2) l~dp + 0

oo

q- 2 l/g. f V ] - - ,% (s, p) sin ~ ~ (s, p) Jo (P!t] 1'2) pdp, g (11)

ReF(s , t ) = ] / g f ] f l - - ~ g ( S , p) sin2~(s, p) 4 ( p l t t I2) pdp. (12) o

Let us mention some restrictions on the model parameters. From the relation (9) it f o l l o w t h a t f o r 0 = 0

~g(P--~-O)------g°<2a" (13)

Using the physical condition that ~(p)-+~ when p + ~ and the obvious inequality ~x > x valid for x < i, we find that

l ~ g o > a ; ca<l; cgo<l. (14)

From the inequalities (13) and (14) and from the fact that at t = 0 the total cross section of inelastic processes reduces in our normalization to IOF, we obtain

3a < ¢i,/~b i < 4a,

2go < oin/~bl < 3g0.

From the inequalities (14) and (16) we find that

(15)

(16)

c < 3rtb 1/oi,~. ( 17 )

2. COMPARISON WITH EXPERIMENTAL DATA

Let us apply now our model to describe the experimental differential cross sections for pp-scattering at ~s = 6.2; 13.8; 19.4; 31; 53; 62 GeV and for pp-scattering at /~ = 4.6;

1163

TABLE i. Values of the Model Parameters for pp- and ~p-Scattering Considered in This Work

Inter- I IbL, I L action ~s, GeV a GeV- 2 b2, GeV-2 c d

PP

6,2 0,62 11,7 3,7 0,86 0,64 13,8 0,77 9,9 3,5 0,99 --0,18

19,4 0,76 10,2 3,2 1,01 --0,22 31 0,81 10,2 3,2 1,01 --0,12

53 0,70 12,2 4,1 1,05 --0,26 62 0,69 12,9 4,4 1,07 --0,31

?p

4,6 0,80 13,6 4,1 0,94 1,3.10 -7

7,6 0,81 11,7 3,4 1,08 --0,63

9,8 0,92 10,l 2,7 1,003 --0,093

53 0,73 12,1 3,8 1,05 --0,32

546 0,78 16,8 6,8 1,22 --0,98 630 0,66 18,1 6,6 1,04 --0,62

1800 0,66 20,0 6,7 1,10 --1,40

7.6; 9.8; 53; 546; 630; 1,800 GeV [i, 3, 7]. The values of the model parameters found by numerical analysis are given in Table i. They differ somewhat from those we used previously [4, 6]. The measured angular distributions at high energies are adequately described for c > i. This solution of the unitarity condition is in agreement with the theory of complex angular momenta. At low energies, c < i. The striking distinction between these two solu- tions is due to the fact that for c < 1 the amplitude is given by a series of terms with the same signs, whereas for c > 1 the signs alternate. Hence, there is a correspondence be- tween the results obtained from the unitarity condition for elastic scattering at large values of the momentum transfer and the consequences of the model of multiple exchange of reggeons. These cases go into one another as c is varied (from c < 1 to c > i).

In Figs. 1 and 2 we reproduce the experimental differential cross sections for pp- and ~p-scattering, respectively, taken from [i, 3, 7]. The theoretical curves correspond to the IOF model. The agreement between theory and experiment for pp-scattering is very good for all six v_alues of the energy, from /~ = 6.2 to 62 GeV. A similar excellent agreement can be noted for pp-scattering at all six values of the energy from ~ss = 7.5 to 1,800 GeV, with the excep- tion of the lowest energy value /~ = 4.6 GeV. It is true however, that for pp-scattering there are few measurements at small momentum transfer. Moreover, in our model, unlike the case for many other models [i, 3], the imaginary part of the scattering amplitude vanishes not at the diffraction minimum (there the real part vanishes), but significantly before. T~ere- fore, the customary assumption that the imaginary part exceeds the real part of the amplitude, at the available accelerator energies, is valid only in a narrow range of values of the momen- tum transfer.

The authors thank N. B. Eroshkina for performing the numerical calculations.

.

2. 3.

.

.

.

7.

LITERATURE CITED

R. Castaldi and G. Sanguinetti, Ann. Rev. Nucl. Part. Sci., 35, 331 (1985). R. Henzi and P. Valin, Phys. Lett. B, 132, No. 4-6, 443 (1983). N. P. Zotov, S. V. Rusakov, and V. A. Tsarev, Fiz. Elem. Chast. At. Yadra, ii, No. 5, 1160 (1980) . G. G. Arushanov, E. I. Ismatov, and M. S. Yakubov, et al., Yad. Fiz., "38, No. 8, 420 (1983) .

G. G. Arushanov, I. M. Kirson, A. Yulchiev, and M. S. Yakubov, Yad. Fiz., 42, No. 12, 1495 (1985) .

G. G. Arushanov, I. M. Kirson, and M. S. Yakubov, Izv. Vyssh. Ucheb. Zaved., Fiz., 33, No. 5 , 86 (1988) . N. A. Amos, C. A v i l a , W. F. Baker , e t a l . , Phys. L e t t . B, 247, No. 1, 127 (1990) .

1164