Unitarity of the tree approximation to the Glauber AA amplitude for large A
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NUCLEI Theory
Unitarity of the Tree Approximation to the Glauber AA Amplitude for Large A * M. A. Braun and A. V. Krylov Saint-Petersburg State University, Russia Received July 20, 2010
Abstract—The nucleus–nucleus Glauber amplitude in the tree approximation is studied for heavy participant nuclei. It is shown that, contrary to previous published results, it is not unitary for realistic values of nucleon–nucleon cross sections. DOI: 10.1134/S1063778811020086
INTRODUCTION Scattering on the nuclei is commonly studied in the Glauber approximation, which can be rigorously derived in Quantum Mechanics, provided the transverse momenta transferred to the projectile are much smaller than its longitudinal momentum. With certain reservations it can be generalized to the high-energy region, where the elementary nucleon– nucleon (N N ) amplitudes become predominantly inelastic. For the nucleon–nucleus (N A) scattering the Glauber approximation has a transparent probabilistic interpretation. If the target nucleus is heavy, with atomic number A 1, the Glauber formula acquires a simple eikonal form, which clearly shows that the resulting amplitude is unitary, that is its modulus is smaller than unity at fixed impact parameter. With the advent of collider experiments nucleus– nucleus (AB) scattering becomes an important physical object. The Glauber approximation can be easily generalized to the AB case and it was in fact done very long ago. The Glauber formula for AB scattering looks very similar to the hA case. At fixed impact parameter b the scattering matrix S is assumed to be a product of N N scattering matrices s averaged over the transverse distributions of nucleons in both nuclei: A B s(xi − xk ) , (1) S(b) = i=1 k=1
A,B
xk
are the transverse coordinates of the where xi and nucleons in the projectile and target nuclei, respectively, and in absence of correlations in both nuclei averaging . . . A,B means (2) F (xi , xk ) A,B ∗
=
B A i
d2 xi d2 xk TA (xi )TB (xk − b)
k
× F (x1 , . . . , xA , x1 , . . . , xB ). Here, TA (x) and TB (x ) are the standard nuclear profile functions normalized to unity. However, in contrast to the N A case the content of the Glauber formula for AB scattering turns out to be much more complicated. Presenting in the standard manner the N N scattering matrix s(b) = 1 + ia(b), where a is the N N scattering amplitude, one obtains from (1) a set of terms corresponding to different ways the nucleons from the projectile and target may interact with each other. Each of these terms may be illustrated by simple diagrams indicating this interaction. Some examples are shown in the figure for two pairs of interacting nucleons in the projectile and target. One observes that in contrast to hadron– nucleus (hA) case the diagrams may contain disconnected parts (the figure a) and, most important, loops (the figure c), which involve internal integrations over transferred transverse momenta and thus N N amplitudes for nonzero transferred momenta. Loop contributions depend not only on t
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