TOTAL CROSS SECTIONS, REAL PART OF THE AMPLITUDE,

AND SLOPE OF DIFFRACTION CONES OF THE pp AND pp

FORWARD SCATTERING

G. G. Arushanov, E. I. Ismatov, I. M. Kirson, and M. S. Yakubov UDC 539.1.01

In the framework of the model described in [6], analytic expressions are obtained for the ratio of the real part of the forward scattering amplitude to its imaginary part, and for the total cross sections and the slopes of diffraction cones for hadron-hadron and hadron-anti~adron interactions. A comparison with the existing experimental data for pp and pp interaction shows that that the computed values of parameters of the elastic scattering are in good qualitative and fair quantitative agreement with their observed values in a wide range of energies v~s = 5 to 546 GeV.

Usually one computes the real part of the amplitude of the elastic forward pp and pp scattering using dispersion relations, which express the ratio ~(s) of the real and imaginary parts of the amplitude of the elastic scattering with no momentum transfer (t = 0) via the dispersion integral of the energy taken from the proton's rest energy to infinity. The

integrand depends on the total pp and p-p interaction cross-sections otPP and ot~P- These cross sections have been measured for finite energies only, so that in order to compute the dispersion integral, one has to use various extrapolations of the total cross sections to asymptotically high energy regions. As shown by Dremin and Nazirov [I], the choice of the extrapolation of the behavior of total cross sections of the pp interaction to the ultra- high energy region can be simplified after one obtains the information about the ratio of the real part of the forward scattering amplitude to its imaginary part on the SPS collider.

In the present paper we use the model of [2] in order to obtain analytic expressions for ~(s), total cross sections ot, and slope parameters of the diffraction cones in pp and UP scattering. The comparison with the existing experimental data shows that the obtained expressions describe the experimental data qualitatively well, and quantitatively fairly, in a wide energy range.

The model is based on the solution of the unitarity condition in the direct s-channel with a given form of the inelastic overlap function, which takes into account, in particular, the absorption corrections. It was shown in [3] and [4] that this model well describes the experimental data on the dependence of the differential cross section of the elastic pp scattering on the momentum transfer square t in a wide range of t and s. In [5], the same model was successfully applied to the description of the differential cross section of the elastic pp scattering at the SPS collider energy V~s = 546 GeV. Finally, in [6] in order to obtain the dependence of the elastic pp scattering on the energy V~s, the parametrization of the model was done taking into account the crossing symmetry; however, the differential cross sections do/dr in a wide range of t, as well as 6(s), were computed numerically.

i. The Model. In [6], an approximate expression was found for the real part of the elastic scattering phase shift =(s, p) in the framework of the model under consideration. This expression follows from the crossing symmetry requirement, and has the following form:

2~(s, p)_~_--d(s)exp(--%p2/(2bl(s)))--B(s, s p)), (1)

where s is the square of the total energy in the center-of-inertia (CO1) system, and p is the target parameter. In Eq. (i), the functions A(s, p) and B(s, p) are defined by the expres- sions

A (s, p) = 1 - -2a . exp (-- ~2) cos (~p2) + c a 2 exp (-- 2~ ~) cos (2~p2); (2)

Institute for Nuclear Physics, Academy of Sciences of the Uzbek SSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 65-70, May, 1990. Original article submitted May 23, 1988.

436 0038-5697/90/3305-0436512.50 �9 1990 Plenum Publishing Corporation

B (s, p) - - - - 2a.exp (-- t*fl) sin (~p2) + c a ~ exp (-- 2~p ~) sin (,.ap-).

In these formulas,

r162 ~b.~ (s) b, (s) ; ~- (s) --= (s) - 2 (0 ~, (s) + ~b~,) 2 (b~ (s) + ~'b~) o, (s)

(3)

(4)

The parameters of the model are c, a, and ~, which are assumed to be energy-independent, and the energy-dependent bi(s) and d(s). The energy dependence of the parameter bi(s), which practically coincides with the slope parameter of the diffraction cone, is chosen to be logarithmic (as in Regge models):

bi (s) "~bm+b111ns, ( 5 )

where bi0 and bii are constant. The constants c, a, y, bi0, bll are supposed to be identical for the pp and ~p scattering. The parameter d(s) depends on the energy as follows:

d (s) ~do/ VT, ( 6 )

where the constant d o is different for the pp and pp interactions. In order to describe the energy dependence of the elastic forward scattering amplitude for the pp and ~p interactions in a wide range of energies including the relatively low energy region ~ss % 5 GeV, we add to the expression (5) a term of the type (6):

bl (s) ----- bloq-b1,1nsmblrm/l ~ s, ( 7 )

where the energy-independent quantity bi(0) is different for the pp and ~p interactions.

In the aiming parameter representation, the scattering amplitude in the model under consideration is

(s, p) = 1 - - 1 /1 - - 2a exp ( - - p=12b~) + ca~e -#l*, exp (2i= (s, p)), (8)

where a(s, p) is defined by Eq. (i). The parameter c is slightly greater than one, so that in the present case of forward scattering we can set c % 1 [5]. Moreover, in this case, when expanding the exponential in Eq. (8) in powers of a(s, p), one can restrict oneself to leading terms and obtain for the imaginary and real parts of the function @(s, p):

~,(s, p) ~ - - ( l - - ae x p ( - - p2 / ( 2b l t ) )2rx(s, !_,)"

cpR(s, p)"~exp(--p*/(2bl) ) + (1--aexp)(--92/(2bi)))2a2(s, p). (9)

(lO)

A numerical evaluation shows that in this computation one has the right to substitute the co- sines and sines in Eqs. (2) and (3) by one and their arguments, accordingly.

The dependences of the scattering amplitude on the momentum transfer can be obtained by a Fourier-Bessel transform of the amplitude in the target parameter representation:

F(s , t) = i g ' ~ S g ( s , p)4(pltl'l~)pdp, ( 8 ' )

2. The Real Part of the Forward Scattering An~.plitude. It consists of two terms which correspond to the two terms of Eq. (i);

where FR~-FRI+FR2, (ii)

/ ] \ ] / g db,

a

t7 , o o

FR,~__~o; i~ ae-~'l,o,) e - ~ ~,3 a - -- , ' dp. - . j 1 - - a e - . ~ .

0

In order to compute this integral, we expand the integrand in powers of a-exp(--~p2), and neglect the terms of order -b/p << i, where

(12)

(13)

437

=~bi5 = ( 2 b , ) - ~ - - ,~ = -- (14)

20, (b~ + =~b~) '

which results in

F Ri~--n';/Zob11/2. (15)

3. Total Cross Sections. Computations similar to those described in the previous sec- tion lead to the following expression for the total cross section of the interaction:

v i+2~ +-~-,, \7+ ~ -~,(;+i)~ (16)

Here ~ and e are expressed via b I and bzl according to Eq. (4). The ratio of the real part of the forward scattering amplitude to its imaginary part is

6=F~/F~, (17)

where F R is defined from Eqs. (ii), (12), and (15), and F I is given by the second equality of Eq. (16).

The total cross section of the inelastic processes in the model under consideration equals [5]

aJ , ,~4nabl ( i - -ca l4) , (18)

while the total cross section of the elastic scattering can be found as the difference

c,~_l---- ae-ai , , . ( 19 )

4. The Slope of the Diffraction Cone. The computations lead to the following expression for the t-derivative of the real part of the amplitude at t = O:

F~= V;ab~(1 a \ ~.'/' 2 ~ (4 + ' ) f ) + ' - ~ - ab,,b,. (20)

The derivative with respect to t at t = 0 of the imaginary part of the amplitude can be com- puted in the same way as the imaginary part itself. One has to keep in mind that the corre- sponding integrands contain an additional factor of p2/4. The result is

F~-- 2 + ' - ~ -bl2 d= - + 6`x= . (21)

The slope of the differential cross section of the scattering at t = 0 is defined by the formula

b = 2 F t F 't + F R F 'R _ 2 F ', I + ~ F'R / F 't (223

5. Numerical Analysis and Comparison of the Obtained Result with the Experiment. A numerical analysis shows that in the first approximation, one can neglect the contributions of the expressions in square brackets in Eqs. (16) and (21). Then, in this approximation, the ratio 8 for pp and pp interactions is defined, according to Eqs. (ii), (12), (15), (6), (7), and (16), by the formula

~• 1 (1 a ) d + . r. b,, a 7 7 + l , V - - ~ - l - ~ b t o + b , , l n s + O ~ l V ~s' (23)

where the sign +(-) refers to the pp (pp) interaction. sults in

r. bl, rc d lnb , 2 b4 2 d i n s

Neglecting the power term, this re-

d lna , 2 d l n s (24)

438

f~;b' (C-eV/sec) -2

I

I

I

I

�9 i i pp

i I 1 I I I I I I 1 I l I

I0 2 r /0~ ~, GeV/sec

5

f

0,5

Fig. 1

~16~, mb

I I I t I I I I i I i t i , I

m so m" ~ v

Fig. 2

According to Eqs. (16) and (7), for the total cross sections in the approximation under con- sideration we obtain

:~_~4=a(b~o + b,~ Ins + b ~ ) / ~ s) , (25)

so that the difference between the total cross sections equals

A=,~=f-o~-----4=a - + r- (bl(o)- bl(o))/~ s (26)

and has a Regge behavior.

For the diffraction cone slope, according to Eq. (22), we obtain the approximative ex- pression

b ~ b l + ~ 2 b f l / 2 b l ~ b l , (27)

so that the difference between the slopes is approximately

Ab ---- b - - - b + ,'-" ( b ~ ) -- b ~ o ) ) / V s . (28)

The ratio of the difference of the total cross sections of the pp and pp interactions to the corresponding difference of the slopes equals

A m / A b ~ 4 n a (29)

and i s i n d e p e n d e n t o f e n e r g y in t h i s a p p r o x i m a t i o n .

We assume t h e v a l u e s o f t h e p a r a m e t e r s a , b l 0 , b l z , and ~ t o be

a~--0.73; bm~6.25 (~ev/sec)-2; bl l~0.7 (SeV/sec)-2; 7 ~ 3 . (30)

In o r d e r t o f i n d t h e v a l u e s o f t h e r e m a i n i n g f o u r p a r a m e t e r s d~, b~0) we use t h e e x p e r i m e n t a l

l o w - e n e r g y d a t a , as w e l l as t h e b e h a v i o r o f t h e r a t i o s 5 ( t h e i r v a n i s h i n g and s i g n change a t a c e r t a i n v a l u e o f s o ) . The c o n d i t i o n s o f v a n i s h i n g o f ~+ a t ~ss % 22-23 GeV, and 6- a t ~ss % 10 GeV [ 7 ] , g i v e t h e f o l l o w i n g n u m e r i c a l e s t i m a t e o f t h e p a r a m e t e r s d~:

d$- '~lO GeV; d~--~"5 C~V. (31)

439

o.1 Zz ~

-8,2

V,, ~ ii,,I I I r ill~ll

# 10 fO 2 Y~, GeV

5f~

50i ~ / i c o ~ c

P ~ p ~ , ~ , h l I l l ; i , r i l

10 ,o' z. &V

Fig. 3 Fig. 4

From the experimental data for the slope parameter at low energies [7] we can find an evalua- tion for the numerical values of the parameters b~0):

..~ ,-I (32) b~0)--4.3 (Gev/sea)-l; b~0) 21.5 (GeV]sec/ .

Then t h e formula (26) has t he f o l l o w i n g numer i ca l form:

~ , ..~ 57/F} , (33)

where ao i i s g iven in mb, and r in GeV.

F i g u r e 1 shows th~ e x p e r i m e n t a l v a l u e s o f the s l o p e pa r ame te r s o f the d i f f e r e n t i a l c ro s s s e c t i o n o f t he pp and pp s c a t t e r i n g wi th no momentum t r a n s f e r , t aken from the rev iew [7] , and t he v a l u e s computed a c c o r d i n g to Eqs. (27) and (7) wi th t he pa r ame te r va lue s of (30) and (32) . F igu re 2 shows the e x p e r i m e n t a l v a l u e s o f t he d i f f e r e n c e a o t , t aken from the same rev iew [7 ] , and t he computa t ion r e s u l t s u s ing t he fo rmula (33) . F i g u r e (3) g ive s a comparison between the c o m p u t e d ' v a i u e s o f the r a t i o s o and t he expe r imen ta~ d a t a , whi le Fig . 4 g ives a s i m i l a r com- p a r i s o n f o r t h e t o t a l c r o s s s e c t i o n s o f t he pp and pp s c a t t e r i n g . These f i g u r e s show a good q u a l i t a t i v e and a f a i r q u a n t i t a t i v e d e s c r i p t i o n o f the main pa r ame te r s of the e l a s t i c pp and ~p s c a t t e r i n g . However, t he above comparison of t he computed and e x p e r i m e n t a l r e s u l t s should be c o n s i d e r e d as j u s t an i l l u s t r a t i o n , s i n c e we did no t seek f o r t he be s t pa rame te r va lue s and compared wi th the expe r imen t on ly s i m p l i f i e d e x p r e s s i o n s . B e s i d e s , t he form (6) of the parameter d follows from dispersion relations and the unitarity condition at relatively low energies [8, 6], while at higher energies the parametrization (6) may have to be changed.

LITERATURE CITED

1. I.M. Dremin and M. T. Nazirov, Pis'ma Zh. Eksp. Teor. Fiz., 3__7, No. 3, 163 (1983). 2. G.G. Arushanov, Yad. Fiz., 15, No. l, 128 (1972). 3. G.G. Arushanov, E. I. Ismatov, M. S. Yakubov, et al., Yad. Fiz., 3_88, No. 8; 420 (1983). 4. G.G. Arushanov, E. I. Ismatov, M. S. Yakubov, et al., Ukr. Fiz. Zh., 2_88, No. 4, 498

(1983). 5. G.G. Arushanov, I. M. Kirson, A. Yulchiev, and M. S. Yakubov, Yad. Fiz., 4._22, No. 12,

1495 (1985). 6. G.G. Arushanov, I. M. Kirson, and M. S. Yakubov, Izv. Vyssh. Uchebn. Zaved., Fiz. No. 5,

86 (1988). 7. R. Castaldi and G. Sanguinetti, Ann. Rev. Nucl. Part. Sci., 3__5, 331 (1985). 8. G.G. Arushanov and A. Yulchiev, Yad. Fiz., 2__66, No. 8, 188 (1977).

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