Beyond the black disk limit

Physics Letters B 316 (1993) 175-177 North-Holland

PHYSICS LETTERS B

Beyond the black disk limit S.M. T r o s h i n a n d N.E. T y u r i n

Institute for High Energy Physics, 142284 Protvino, Moscow Region, Russia Received 23 July 1993 Editor: P.V. Landshoff We consider consequences of violation of the black disk limit possibly revealed by the new CDF measurements of the total, elastic and diffractive cross-sections.

The new C D F data [1] on the measurements of the total, elastic and diffractive pff-scattering crosssections at Tevatron-Collider ( v ~ = 1.8 TeV) reveal interesting and unexpected results: a large total cross-section O-tot 80.6+2.3 mb which is consistent with an In 2 s-rise of O-tot; - a large elastic cross-section O-el = 20.0 4-0.9 mb and a large ratio of elastic to total cross-section O-el/o-tot = 0.248 + 0.005 which shows that the greater the energy the larger both absolute and relative probabilities of elastic collisions; the scattering amplitude in the impact parameter representation has the value Im f (s, b = 0) = 0.50+ 0.01. With the data on the slope parameter of the diffraction cone and the single diffractive cross-section also measured by CDF, these results point out that the black disk limit is almost reached at small impact parameters, the scattering amplitude at b = 0 is probably beyond this limit, and the Pumplin bound for diffractive cross-section [2] is violated at b = 0 [3,4]. O f course, these conclusions should be drawn with certain precautions since the E710 gives different figures for the cross-sections [ 5 ]. However, the possibility that the scattering amplitude exceed the black disk limit at b = 0 seems worth discussing. First of all, let us remind that the unitary equation for the scattering amplitude -

=

-

t/(s,b) = E o - , ( s , b ) ,

(1 cont'd)

n

implies the constraint I f (s, b)l ~< 1 while the black disk limit assumes that I f (s, b)l ~< 1/2. The equality I f (s, b)l = 1/2 corresponds to the maximal absorption at given values of s and b. The Pumplin bound for the diffractive cross-section O-cliff(s,b) ~< ½0-tot(S,b) - o'el(s , b)

(2)

is also based on the assumption that the diffractive eigenamplitudes in the Good-Walker picture [6 ] do not exceed the black disk limit. Indeed this limit is a priori chosen by the use of the imaginary eikonal g2 = i% when the amplitude is written in the following form to ensure unitarity:

f ( s , b ) = 1i(1 - e x p [ i l 2 ( s , b ) ] ) .

(3)

There is another possibility to provide the direct channel unitarity. It is based on the following representation for an amplitude [7]:

f ( s , b ) = U ( s , b ) [ l - i U ( s , b ) ] -1.

(4)

In the latter case the inelastic channel contribution takes the following form: r/(s,b) = Im U(s,b)l 1 - i U ( s , b ) 1 - 2 .

(5)

Eq. (5) ensures s-channel unitarity provided that Im f (s,b) = If (s,b)l 2 + q(s,b), Elsevier Science Publishers B.V.

(1)

Im U(s,b) >t O. 175

Volume 316, number 1

PHYSICS LETTERS B

Usually eikonal or U-matrix are considered to be input dynamical quantities similar to the Born term (however, the expansion in these terms is not always possible) and there are a number of dynamical models used for the construction of the explicit form of the eikonal or U-matrix. Most of the QCD-inspired, Regge-type and geometrical models lead to a parameterization of the above quantities in the form

~ , U(s, b) o~ isa e x p [ - b 2 / a ( s ) ] ,

a(s) ,,~ Ins

or

t2, U (s, b) o¢ is a exp(-/zb), where/~ is a constant. The latter form respects analytical properties of the scattering amplitude in the complex cos 0-plane. These parameterizations of 12 and U and unitarization methods lead to growth of the total and elastic cross-sections, trtot(S) ,-~ gel(S) "~ l n 2 s .

However, the above two unitarization methods are different, e.g. they give different behavior of the ratio of elastic to total cross-sections:

ael(S)/atot(S) ---, 1/2 for eikonal unitarization and ael(S)/atot(S ) ~

l

for the U-matrix unitarization. These asymptotical regimes reflect the limitations for the scattering amplitude I f ( s , b ) l <~ 1/2 and I f ( s , b ) I ~< 1 which are imposed by the imaginary eikonal and U-matrix respectively. Therefore the U-matrix unitarization implies the two modes in hadron scattering: shadow and antishadow, while imaginary eikonal corresponds to the shadow scattering only. To clarify the difference between the two scattering modes we consider pure imaginary amplitude. The unitary equation has two solutions:

f(s,b)

= ½i[1 ± x/1 - 4 y ( s , b ) ] .

(6)

In the case of shadow scattering an elastic amplitude increases with increasing contribution of inelastic channels. This mode corresponds to the choice of 176

14 October 1993

the minus sign. In the antishadow mode (plus sign) the amplitude increases with decrease of the inelastic channel contribution. The shadow scattering mode is often considered as the only possible one; however, it should be noted that these two solutions have an equal meaning and at high energies the other mode could be realized. Let us consider such a transition from the shadow to antishadow scattering mode. In the framework of the U-matrix unitarization scheme the inelastic overlap function rl(s,b = O) increases with energy. It gets its maximum r/(s, b = 0) = 1/4 at some energy s = So and beyond that point the transition to the antishadow mode occurs when Im f ( s , b = 0) > 1/2 and rl(s,b = 0) < 1/4. Note that the value of x/~ = 2 TeV was predicted in ref. [8]. The function r/(s, b) becomes peripheral when the energy grows. At s > So the maximum of r/(s, b) is reached at b = R (s) where R (s) is the interaction radius [ 9]. This picture corresponds to the antishadow scattering at b < R ( s ) and to the shadow scattering at b > R ( s ) . At b = R ( s ) the maximal absorption takes place. The transition to the antishadow mode in the Umatrix method naturally occurs when U (s, b = 0) becomes greater than unity and in this case one cannot expand the amplitude into series over U. Of course, such a transition to the antishadow mode, in principle, could be realized in the eikonal approach also. However, this transition in particular implies that the real part of eikonal t2 gains an abrupt increase equal to 7r at s = So. The commonly accepted models for the eikonals do not foresee such a behavior. It is to be emphasized that the CDF data indicate that vrg = 1.8 TeV is in some sense threshold a energy, namely, that the amplitude pergaps goes beyond the black disk limit at zero impact parameter and that the antishadow scattering mode starts to develop in the central hadron collisions. The hadron scattering picture may be described as the transition from the grey to the black disk and then to the black ring with growing energy. The black ring picture is in complete agreement with unitarity. We would like to thank M.M. Islam, A.D. Krisch and V.A. Petrov for useful comments.

Volume 316, number 1

PHYSICS LETTERS B

References [1] CDF Collab., talk presented at the Vth Blois Conference on Elastic and Diffractive Scattering, Providence, 8-12 June, 1993. [2] J. Pumplin, Phys. Rev. D 8 (1973) 2899. [3] S.M. Troshin and N.E. Tyurin, Phys. Lett. B 208 (1988) 517. [4]L.L. Frankfurt, talk presented at the Vth Blois Conference on Elastic and Diffractive Scattering, Providence, 8-12 June, 1993. [5] R. Rubinshtein, talk presented at the Vth Blois Conference on Elastic and Diffractive Scattering, Providence, 8-12 June, 1993.

14 October 1993

[6] M.M. Good and W.D. Walker, Phys. Rev. 120 (1960) 1857. [7] A.A. Logunov, V.I. Savrin, N.E. Tyurin and O.A. Khrustalev, Teor. Mat. Fiz. 6 (1971) 157. [8] S.M. Troshin, N.E. Tyurin and O.P. Yuschenko, Nuovo Cim. A 91 (1986) 23; S.M. Troshin and N.E. Tyurin, paper presented at XXIII International Conference on High Energy Physics, Berkeley, 1986; preprint IHEP 86-232, Serpukhov, 1986. [9] N.E. Tyurin, Nucl. Phys. B (Proc. Suppl.) 25 B (1992) 91.

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