New analytical solution of the nonlinear Gribov-Levin-Ryskin-Mueller-Qiu equation

The GLR-MQ equation is a nonlinear evolution equation that takes into account the shadowing effect, which tames the growth of the gluon at small- x . In this study, we analytically solve for the first time the nonlinear GLR-MQ equation using the homogeneous balance method. The definite solution of the GLR-MQ equation is obtained by fitting the MSTW2008LO gluon distribution data. We find that the geometric scaling is an inherent property of our analytical solution. Furthermore, the gluon distribution functions derived from our solution are capable of accurately reproducing the MSTW2008LO data across most regions. These results indicate that our analytical solution from the homogeneous balance method is valid to describe the gluon behavior at small- x and moderate Q 2 regions. Moreover, the saturation scale Q s has been extracted from our analytical solution, we find that the energy-dependent saturation scale obeys the exponential law Q s 2 ∝ Q 0 2 e λ Y . We also perform a numerical simulation using the analytical results at Q 2 = 5 GeV 2 as the initial condition. We find that our analytical results are consistent with the numerical results in small- x .