Testing the black disk limit inppcollisions at very high energy

Abstract

We use geometric scaling invariant quantities to measure the approach, or not, of the imaginary and real parts of the elastic scattering amplitude, to the black disk limit, in pp collisions at very high energy. Keywords:Cross-sections, Differential elastic cross-section, pp collisions at high energy, Evolution equation, Geometric scaling.Saturation phenomena are expected to dominate QCD physics at high energy and high matter density [1,2]. At LHC and at ultra high energy cosmic rays these effects should become easily detected. One of the possible consequences of such physics is the setting up of the black disk regime in the description of soft physics, namely total cross-section, elastic cross-section, elastic differential cross-section, etc., as discussed for instance in [3,4] (see also [5]).Non linear differential equations include, in a natural way, saturation effects. This happens with the well known logistic equation, which can be seen as a simplified version of the B-K equation [6]. For a general discussion on evolution and saturation, in the present context see [7] and [8]. In this paper we apply the logistic equation to the evolution of the imaginary part of the impact parameter elastic amplitude, ImG(s, b2 ).Email addresses: pedro.brogueira@ist.utl.pt (P. Brogueira), jorge.dias.de.deus@ist.utl.pt (J. Dias de Deus) Letters B June 14, 2018 In [9] we have argued that the evolution with the energy, √ s, of ImG(s, b 2 ) -or of the Fourier-Bessel transform, ImF (s, t) -qualitatively describe the evolution of the differential elastic cross-section, dσ/dt, in particular in the small |t| region. Here we present quantitative tests for the approach of ImF (s, t) to the black disk in the region |t| |t 0 |, t 0 being the position of the first diffractive zero. As most of the cross-sections are concentrated in the small |t| region, any test for the approach to the black disk has, at least, to work at small |t|. Preprint submitted to PhysicsWe study next the evolution of the profile function Γ(s, b 2 ), or the imaginary part of the impact parameter elastic amplitude,and we consider the logistic equationwhere γ > 0 is a positive constant, and b ≃ 2 √ s ℓ, ℓ being the angular momentum and √ s the center of mass energy. As, from unitarity,∂Γ/∂b ≤ 0, which means that Γ is a decreasing function of b, becoming eventually constant at small b. This means that saturation occurs first at small b. At large b, Γ decreases exponentially (Γ ∼ exp(−b/γ)). The parameter γ, which we take as a constant, controls the long range behaviour of the strong forces and can be associated to the two pion exchange diagram. A solution of (2) -not the most general one -isR being a positive quantity, R > 0. In comparisons with data, for √ s 50 GeV, we shall fix γ, γ = 1.1 mb 1/2 , which means that the dependence of Γ on √ s is exclusively contained in R → R(s). One sees that R is a radial scale parameter, in fact the only relevant parameter in the black disk limit. It is then clear, from (4), that there is also an evolution equation in R...

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