Shadowing correction to the gluon distribution behavior at small x

We determined the saturation exponent of the gluon distribution using the solution of the QCD nonlinear Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (NLDGLAP) evolution equation at small x . The very small-x behavior of the gluon distribution is obtained by solving the Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) evolution equation with the nonlinear shadowing term incorporated. The form of the initial condition for the equation is determined. We find, with decreasing x , the emergence of a singular behavior and the eventual taming (at R = 5 GeV^-1) and the essential taming (at R = 2 GeV^-1) of this singular behavior by the shadowing term. The nonlinear gluon density functions are calculated and compared with the results for the integrated gluon density from the Balitsky-Kovchegov (BK) equation for the different values of Q². It is shown that the results for the gluon density function are comparable with the results obtained from the BK equation solution. Also we show that for each x , the Q²-dependence of the data is well described by gluon shadowing corrections to the GLR-MQ equation. The resulting analytic expression allows us to predict the logarithmic derivative \( {\frac{{\partial F^{s}_{2} (x,Q^{2})}}{{\partial \ln Q^{2}}}}\) and to compare the results with H1 data and a QCD analysis fit.