energy dependence of proton-proton differential cross section

4
ENERGY DEPENDENCE OF PROTON-PROTON DIFFERENTIAL CROSS SECTION G. G. Arushanov, I. H. Kitsch, and M. S. Yakubov UDC 539.101 In recent years a large amount of experimental data has been obtained for elastic scattering of hadrons by nucleons, particularly for proton-proton interactions, over a wide range of energies and transferred momenta. Various theoretical models have been proposed to describe these experimental data; among these, the model based on the unitarity condition in the s-channel is widely use [i]. In this model, the absorptive nature of diffraction elastic scattering appears explicitly. According to this picture, diffraction scattering is the shadow of the absorption due to the existence at high energies of many inelastic reaction channels [i]. In [2, 3] it was shown that this model successfully describes experimental data on the differential cross section for proton-proton elastic scattering as a function of t, the square of the transferred momentum. In this paper, in order to obtain the energy dependence, we achieve a parametrization of the model with crossing symmetry accounted for, based on two fundamental points: CPT invariance and analytic properties of the scattering amplitude. We can thereby describe fairly well the differential cross sections for elastic proton-proton scattering for values of t up to -t ~ 15 (GeV/c) 2, and also the energy dependence for the range 400-2000 GeV. Calculated values o~ the ratio of the real to the imaginary part of the forward scattering amplitude are also in good agreement with experimental values in the range Plab = 10-10a GeV/c of momenta in the lab system. An elastic scattering model first proposed by one of us in [4] was used in [2, 3]. In this model the dependence of the inelastic overlap function ~g(p) on the impact parameter p is taken in the form [in (1)-(5) the energy variable is omitted, since at the moment, we are interested in the p or t dependence] 9e (9) = 2aexp (--Oa/2bl)--ca2exp (--92/b,), (1) where a, bl, and c are model parameters which, generally speaking, are energy dependent, If the geometric scaling hypothesis is valid, the parameters a and c do not depend on energy, while b I (the slope parameter of the diffraction cone) varies with energy in the same way that the total cross section of the interaction ot does. The phase shift for pure elastic scattering is taken in the form 2a (p) = --dexp (--62/262), (2) where d and b 2 are energy-dependent parameters. The elastic scattering amplitude in the impact parameter representation is (p) = 1 -- V 1 - ~g (p) exp (2ia(p)). (3) The amplitude as a function of transferred momentum is obtained by using a Fourier--Bessel transformation: co F (t) = il/~ t" :, (p) Jo(p I r I'~) W o, (~) 0 while the differential cross section for elastic scattering is d~/dt = IF(t)12 . (5) Let #(s, t) be the relativistically invariant elastic pp scattering amplitude (#(s, t) m sF(s, t)), and let #(s, t) be the pp scattering amplitude. The crossing-symmetry relation Physicotechnical Institute, Academy of Sciences of the Uzbek SSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 86-90, May, 1988. Original article submitted October 15, 1985. 416 0038-5697/88/3105-0416512.50 1988 Plenum Publishing Corporation

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Page 1: Energy dependence of proton-proton differential cross section

ENERGY DEPENDENCE OF PROTON-PROTON DIFFERENTIAL

CROSS SECTION

G. G. Arushanov, I. H. Kitsch, and M. S. Yakubov UDC 539.101

In recent years a large amount of experimental data has been obtained for elastic scattering of hadrons by nucleons, particularly for proton-proton interactions, over a wide range of energies and transferred momenta. Various theoretical models have been proposed to describe these experimental data; among these, the model based on the unitarity condition in the s-channel is widely use [i]. In this model, the absorptive nature of diffraction elastic scattering appears explicitly. According to this picture, diffraction scattering is the shadow of the absorption due to the existence at high energies of many inelastic reaction channels [i].

In [2, 3] it was shown that this model successfully describes experimental data on the differential cross section for proton-proton elastic scattering as a function of t, the square of the transferred momentum. In this paper, in order to obtain the energy dependence, we achieve a parametrization of the model with crossing symmetry accounted for, based on two fundamental points: CPT invariance and analytic properties of the scattering amplitude. We can thereby describe fairly well the differential cross sections for elastic proton-proton scattering for values of t up to -t ~ 15 (GeV/c) 2, and also the energy dependence for the range 400-2000 GeV. Calculated values o~ the ratio of the real to the imaginary part of the forward scattering amplitude are also in good agreement with experimental values in the range Plab = 10-10a GeV/c of momenta in the lab system.

An elastic scattering model first proposed by one of us in [4] was used in [2, 3]. In this model the dependence of the inelastic overlap function ~g(p) on the impact parameter p is taken in the form [in (1)-(5) the energy variable is omitted, since at the moment, we are interested in the p or t dependence]

9e (9) = 2aexp (--Oa/2bl)--ca2exp (--92/b,), (1)

where a, b l , and c a re model parameters which, g e n e r a l l y speaking , a re energy dependent , I f the geometric scaling hypothesis is valid, the parameters a and c do not depend on energy, while b I (the slope parameter of the diffraction cone) varies with energy in the same way that the total cross section of the interaction o t does. The phase shift for pure elastic scattering is taken in the form

2a (p) = --dexp (--62/262), (2)

where d and b 2 are energy-dependent parameters. The elastic scattering amplitude in the impact parameter representation is

(p) = 1 -- V 1 - ~g (p) exp (2ia (p)). (3)

The amplitude as a function of transferred momentum is obtained by using a Fourier--Bessel transformation:

co

F (t) = i l / ~ t" :, (p) Jo(p I r I'~) W o, (~) 0

while the differential cross section for elastic scattering is

d~/dt = IF(t)12 . (5)

Let #(s, t) be the relativistically invariant elastic pp scattering amplitude (#(s, t) m

sF(s, t)), and let #(s, t) be the pp scattering amplitude. The crossing-symmetry relation

Physicotechnical Institute, Academy of Sciences of the Uzbek SSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 86-90, May, 1988. Original article submitted October 15, 1985.

416 0038-5697/88/3105-0416512.50 �9 1988 Plenum Publishing Corporation

Page 2: Energy dependence of proton-proton differential cross section

asserts that

m(s, t ) = ~* (u, t), (6) where s, t, u are the Mandelshtam variables. We introduce, as usual, the linear combinations

~s=2 -~(~+~), ~ A = 2 - ' ( ~ - - ~ ) ,

which are symmetric and antisymmetric with respect to the interchange ~ ~ ~. dance with Eq. (6), we obtain

Then,

(7) in accor-

( - ~, t) = ~* (~, t),

a~s(-~, t ) = s(~, t), % , ( - - ,, t) = - ~'a (~, t) ,

(8)

(9) ( lO)

where

v=2- ' ( s - -u) .

The interchange s ~ u is equivalent to ~ ~ -~, and v ~ s as s ~ ~. is real, i.e., a real analytic function o~ v for fixed t:

(ll)

By assumption, r t)

Let

Substituting (13)

ms (~, t) = w~ (~*, t).

Ws (~, t) = Ys ( ,e% t).

in (9) , and t ak ing (12) i n to account , we o b t a i n

(12)

(13)

f s (,e,,+,.~, t ) = f s ( ~ # ' , t ) = f s (~e -~', t). (14)

From this we get ~ = --~/2, i.e., the variable v enters into ~S(V, t) in the form v exp(-i~/2). Thus

q~s( - - ~, O = f s ( ~ e -~''2, t) , (15)

and it can similarly be shown that

~D a (-- ~, t) = i/A (~e -i='2, t), (16)

where f s and fA are a r b i t r a r y r e a l f u n c t i o n s of v exp (__-i~/2) f o r f i x e d t . There fore , in accordance with crossing symmetry, the elastic pp and pp scattering amplitudes must have the form

�9 (s, t )=/s (~exp(- - i r , :"2) , t) i /a(vexp(-- ir . . '2) , t), (17)

~)(u, t ) = / s ( - - ~ e x p ( - - i = : : 2 ) , t ) - - i f A ( - - ~ e x p ( - - i r . 2), t), (18)

i.e., the energy variable enters into" the relativistically invariant amplitudes in the form v exp (-i~/2).

Using the complex elastic scattering phase 6(v,p) = ~(v, p) + iS(v, p), which determines the eikonal amplitude

(v, 9) = 1--exp (2i6 (% 9))

with the help of transformation (4), the cross-symmetry relation can be written as

(19)

6(v exp(ia), 9 ) = - - 6 " ( v, 9), (20)

i.e., the complex phase 6(v, p) is an arbitrary real function of v exp (-i~/2), multiplied by the imaginary unit i. Using the real and imaginary parts of the phase shift, Eq. (20) takes the form

a (~) q-~ (v exp (i~)) = i (8 (v)--8 (v exp (in)) ) (21)

or, splitting it into real and imaginary parts,

a(v)+Re e(~ exp(in)) -- Im 8(v exp(i.~)), (22)

417

Page 3: Energy dependence of proton-proton differential cross section

Im a(v exp (ia)) = ~ (v)- -Re ~ (v exp (in)), (23)

where we have omitted P for simplicity.

Let us go on to consider our model (i). The energy-dependent parameter b I will, as usual, be parametrized in the logarithmic form

bl (v) = b,o+bulnv, (24)

where b l0 and bx l a r e c o n s t a n t s . E q u a t i o n s (22) and (23) t a k e t h e form

= (~, p) + Re �9 (~ exp (i=), p) = -- 4 -~ arctg (B (,, ~)/A (~, p)), (25)

Im= (, exp (i=), p) = 8- ' In [(A ~ (~, p) + B 2 (v, ~))'(I -- ~, (,, p)):], (26)

where

A (~, p) = I -- 2a exp ( - - Fp=) cos (=p2) + ca2 exp (- - 2Fp z) cos (2~p~), B (~, p) = -- 2a exp ( _ ~p2) sin (~p2) + ca2 exp (-- 2 ~ 2) sin (2zp~),

b, ( , ) . �9 (~) = ~ b , , = ~ (~) =b , ,

(~) = 2 (b~ (~) + ~'b~,) 2 (b~ (*) + ='bf,) b, (v) ' (27)

and ?g(V, p) is g iven by (1) and (24) .

A numerical analysis shows that [B/A[ ~ 0.i, and the argument of the logarithm in (26) differs from unity by an amount on the order of less than one percent. Therefore, with a high degree of accuracy, we can write -4 -I B/A in the right-hand side of (25), and in the right-hand side of (26) we can substitute the logarithmic expression by the log argument itself minus one. We proceed to simplify still further by neglecting entirely the right- hand side of (26). For a(~, p) we wish to obtain a solution of the system of equations (25) and (26) which is similar to (2). It is not hard to verify that the required solution in a first approximation is

2a (% p) ~ - - d (v) exp (--T P2~Ot (v)) --4-1B (~, p)/A (~, p), (28)

where d(~) = -d(--v) is an odd function and the energy dependence of b2(v) in (2) is taken to be the same as the dependence of b1(v):

b2 (v) ----- T - 'b l (v), (29)

where y is independent of energy. Equation (29) follows, firstly, from the requirement for a good description of experimental data on elastic scattering of hadrons [2, 3] (7 ~ 3 and 4 for pp and ~p scattering, respectively); secondly, it is dictated by the idea of geometric scaling; thirdly, dispersion relation calculations together with the solution to the unitarity condition permit us to write, with sufficient accuracy,

a(% p)~"a(p/R(v))/K (30)

for the pp elastic interaction in the momentum interval Plab ~ 3-100 GeV/c [5], where R(9) is the effective interaction radius and R(v) % bz(v), while ~ is the wave number in the c.m.s. From (30), the parametrization

d (~)~,dol~}llZ/v (31)

also follows, which assumes that the first term of (28) predominates at low energies, while the second predominates at high energies. The solution of (28) automatically leads to a sign change in the real part of the forward scattering amplitude at some experimentally obtained energy [6]. To explain this experimental fact within the framework of the given model without accounting for crossing symmetry, it was necessary to artificially add a second term of opposite sign to Eq. (2) (see [7]). As we see, this difficulty automatically disappears when crossing symmetry is taken into account.

Figure 1 gives experimental data on the differential cross section for elastic proton- proton scattering at momenta 400, 500, 1000, 1500, and 2000 GeV/c, while Fig. 2 gives data for the ratio 6(/s) of the real to the imaginary part of the forward sattering amplitude at t = 0 for pp scattering, as a function of energy. The solid-line theoretical curves in these figures were plotted for the following parameter values:

418

Page 4: Energy dependence of proton-proton differential cross section

dOldt, mbl i~: (GeV/c) 2

I0

108 p p ~ p p

" ' .,, Vlc) 2,

'r Id'| , , .

o 2 ~ 6 8 to /2 -t,(GeVlc) =

Fig. 1

-o.,- :7

I"

I to too i ,

t0 I0 ~

F~. GeV

io ~ iO ~ e1~b,C~V/c

Fig. 2

a ~ 0 . 7 , c ~ - 1 . 0 7 , b i o , ~ 6 ( O e V / c ) "2,

. 2 bll~0.7 ( G e V / c ) , d o l l 6 GeV , 7 "~3 .

(32)

Thus, we have shown that an approach based on solving the unitarity condition in the direct channel while accounting for crossing symmetry permits describing the differential cross section for elastic pp scattering as a function of transferred momentum and energy over a wide interval of energies and momentum transfers.

LITERATURE CITED

i. N.P. Zotov, S. V. Rusakov, and V. A. Tsarev, Fiz. Elem. Chas~. At. Yadra, l_!l, 1160 (1980). 2. G.G. Arushanov, E. I. Ismatov, V. G. Arushanov, et al., Yad. Fiz., 3_88, 420 (1983). 3. G.G. Arushanov, E. I. Ismatov, V. G. Arushanov, et al., Ukr. Fiz. Zh., 2_88, 498 (1983). 4. G.G. Arushanov, Yad. Fiz., 15, 128 (1972). 5. G.G. Arushanov and A. Yulchiev, Yad. Fiz., 26, 188 (1977). 6. V. Bartenev et al., Phys. Rev. Lett., 3_~i, 1367 (1973). 7. S.A. Azimov, G. G. Arushanov, and I. I. Pirmatov, Izv. Vyssh. Uchebn. Zaved., Fiz.,

No. 6, 39 (1975).

419