Analytical solution of Balitsky-Kovchegov equation with homogeneous balance method

Nonlinear QCD evolution equations are essential tools in understanding the saturation of partons at small Bjorken $x_{\mathrm{B}}$, 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 the Balitsky-Kovchegov 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 $Q_s^2(x)$.

Figure 1

The traveling wave solution of the FKPP equation from the homogeneous balance method, with the coefficients $c_0=0$, $c_1=1/\sqrt{6}$, and $\theta =0$.

Figure 2

The BK forward scattering amplitude is fitted by the gluon distributions of CT14(NNLO) at high $k^2$. The curves show the fitted BK solutions, and the various markers show the calculations from CT14(NNLO) gluon distributions.

Figure 3

The BK forward amplitude in the wide $k$ range at different rapidities, with the parameters $A_0=33.3$, $A_1=-58.3$, $A_2=26.2$, and $\theta =-3.09$ by matching to CT14 PDFs in the nonsaturation region.

Figure 4

The saturation scales extracted from the solution of the BK equation in this work (dashed line), compared to the prediction by the GBW model (solid line) [43, 44, 45].

Figure 5

Comparison between our fit with the obtained saturation scale $Q_s$ (solid curve) and the experimental data of ZEUS [48, 49] (red dots) for the ${\gamma }^{*}p$ DIS cross section as a function of $\tau$.