Large impact parameter behavior in the CGC/saturation approach: A new nonlinear equation

Abstract

In this paper we propose a solution to the long standing problem present in the color glass condensate/ saturation approach: i.e., the powerlike fall off of the scattering amplitudes at large b. We propose a new nonlinear equation, which takes into account random walks both in transverse momenta of the produced gluons, and in their impact parameter. We demonstrate, that this equation is in accord with previous attempts to include diffusion in impact parameters in the Balitsky-Fadin-Kuraev-Lipatov evolution equation. We show in the paper, that the solution to a new equation results in the exponential decrease of the scattering amplitude at large impact parameter, and in the restoration of the Froissart theorem.

Evolution equation for elastic scattering of hadrons

“…Furthermore, we show that at LHC energies the non-linear effects play an important role as the profiles approach the unitarity bound for small impact parameter and that the diffusion of the profiles in b-space follows a reaction-diffusion equation. We also observe that at 13 TeV the so-called hollowness effect appears in the inelastic profiles and persists for larger energies. The obtained real and imaginary amplitudes in t-space also behave in agreement with general theoretical expectations and, interestingly, possess stationary fixed points in t.…”

“…Recently, in Ref. [13] the Authors extended nonlinear QCD evolution equations for the Pomeron amplitude to incorporate diffusion in the impact parameter space, but the equations have not been tested phenomenologically.…”

Non-linear equation in the re-summed next-to-leading order of perturbative QCD: the leading twist approximation

“…We believe that this is an interesting result since Eq. (52b) follows from the first attempt to take into account both the diffusion on ln(k T ), which stems from perturbative QCD, and the Gribov's diffusion in b [71].…”

“…in Refs. [62][63][64][65][66][67][68][69][70][71] it is made an attempt to incorporate in the BFKL equation the Gribov's diffusion [61] in impact parameter (b). As the result of this tthe following formula for the saturation scale was suggested [69,71]:…”