Asymptotic Behavior and Subtractions in the Mandelstam Representation

Abstract

This differs from the Nsnal expression, which includes the core interaction (st,gsl Uslg~rtv) as well T. his usual result is obtained if, in the first term of Eq. (10) (which we know to be identically zero), we replace X, t+& by its Born approximation p . In this approximation, this term is no longer zero, since rt, and P, are eigenfunctions of different Hamiltonians. The result is just the extra term found by perturbation techniques. It is clear that this term should rot be present, and that only the inadequacies of the perturbation approach have led to its appearance. Some of the virtues of an approach in which the exact expression is obtained before approximations are made can be seen from this example.We may also observe that the so-called "post-prior" discrepancy has evaporated. This discrepancy arises when the ordinary (but incorrect) perturbation result is used, since then either U or U& may enter the expression for M&~". While formally they give identical results, that is (rt y&l U I@ st&)= (st y&l && ly Its); when approximate bound-state wave functions are introduced into these matrix elements, the equality no longer holds.We now see that this term should not be present at all and that no problem exists.It. is proved that a two-body reaction amplitude involving scalar particles and satisfying Mandelstam's representation is bounded by expressions of the form Csin's at the forward and backward angles, and Cs4 ln&s at any other fixed angle in the physical region, C being a constant, s being the total squared c.m. energy. This corresponds to cross sections increasing at most like ln s. These restrictions limit the freedom of choice of the subtraction terms to six arbitrary single spectral functions and one subtraction constant.

Infrared gluons, intrinsic transverse momentum and rising total cross-sections

“…Since 1/2 < p < 1 [17], the above result shows that, with soft gluon momenta integrated into the IR region, k t < Λ, and a singular but integrable coupling to the quark current, our model leads to satisfaction of the Froissart-Martin bound [17,18]. This region, with the scale Λ ≃ O(Λ QCD ) is of course inaccessible to the perturbative coupling for α s , but it plays a crucial role in many inclusive low p t processes.…”

Strangeness production on the nucleon

“…Indeed, it was proven by Froissart [105] on the basis of unitarity, that the total cross section at high energies is constrained by:…”

Tachyon dynamics of the black disc limit in the hard small x processes in QCD

“…[1]. It follows from the unitarity of the S matrix and the analytic properties of amplitudes in the momentum transfer plane that the cross sections of hadron-hadron collisions can not exceed the limit σ ≤ c h ln 2 (s/s o ) [2,3], with the coefficient c h determined by the radius of the pion cloud of a hadron [4]. At achievable energies the contribution of the Pomeron exchange instead of the projectile scattering of the pion cloud dominates in the peripheral collisions because pion is a pseudo-goldstone meson of spontaneously broken chiral symmetry [5], cf.…”