Analytical solution of Balitsky-Kovchegov equation with homogeneous balance method

Xiaopeng Wang,Xurong Chen,Wei Kou,Rong Wang,Yirui Yang

doi:10.1103/physrevd.103.056008

Abstract

Nonlinear QCD evolution equations are essential tools in understanding the saturation of partons at small Bjorken $x_{\rm B}$, as they are supposed to restore an upper bound of unitarity for the cross section of high energy scattering. In this paper, we present an analytical solution of Balitsky-Kovchegov (BK) equation using the homogeneous balance method. The obtained analytical solution is similar to the solution of a traveling wave. By matching the gluon distribution in the dilute region which is determined from the global analysis of experimental data (CT14 analysis), we get a definitive solution of the dipole-proton forward scattering amplitude in the momentum space. Based on the acquired scattering amplitude and the behavior of geometric scaling, we present also a new estimated saturation scale $Q_s^2(x)$.

Highlights

The powerful and sharpest way to resolve the proton structure is by lepton deep inelastic scattering (DIS) off the proton at high energy
We present an analytical solution of the Balitsky-Kovchegov equation using the homogeneous balance method
By matching the gluon distribution in the dilute region which is determined from the global analysis of experimental data (CT14 analysis), we get a definitive solution of the dipole-proton forward scattering amplitude in the momentum space

Summary

INTRODUCTION

The powerful and sharpest way to resolve the proton structure is by lepton deep inelastic scattering (DIS) off the proton at high energy. In the JIMWLK equation, the quantum fluctuation is added to the evolution for the strong gluon field at small x, while, the quantum corrections from resumming multiple rescatterings is implemented for the dipole forward amplitude Both equations are derived in the framework of the quantum evolution process. The geometric scaling [35] observed at small x can be explained with the traveling wave solution of the FKPP equation It is shown in a pioneering work that the transition to the parton saturation region in high-energy QCD is identical to the formation of the front of a traveling wave [26,27]. The definitive and analytical solutions of the BK equation are shown in Sec. IV, for the physical forward dipole-proton scattering amplitude in the momentum space.

BK EQUATION AND FKPP EQUATION

SOLUTIONS OF FKPP EQUATION WITH HOMOGENEOUS BALANCE METHOD

DISCUSSIONS AND SUMMARY