Solution to the Balitsky-Kovchegov equation with the collinearly improved kernel including impact-parameter dependence

The solution to the impact-parameter dependent Balitsky-Kovchegov equation with the collinearly improved kernel is studied in detail. The solution does not present the phenomenon of Coulomb tails at large impact parameters that have affected previous studies. The origin of this behavior is explored numerically. It is found to be linked to the fact that this kernel suppresses large daughter dipoles. Solutions based on a physics motivated form of the initial condition are used to compute predictions for structure functions of the proton and the exclusive photoproduction and electroproduction of vector mesons. A reasonable agreement is found when comparing to HERA and LHC data.

Figure 1

Comparison between the behavior of ${\overline{\alpha }}_s$ computed from Eqs. (9) and (11) with $C=2.586$ (red) and $\mathrm{C}=9$ (blue).

Figure 2

Schematic picture of the variables that enter the initial condition presented in Eq. (15).

Figure 3

Absolute value of the ratio $K_{\mathrm{ci}}/K_{\mathrm{rc}}$ at a fixed dipole size $r=1{\mathrm{GeV}}^{-1}$ and orientation with respect to the daughter dipole ${\theta }_{r{r}_1}=\pi /2$ as a function of the daughter dipole size.

Figure 4

The three constituent terms of the BK kernel for the running coupling (left) and collinearly improved cases (right) at a fixed dipole size $r=1{\mathrm{GeV}}^{-1}$ and orientation with respect to the daughter dipole ${\theta }_{r{r}_1}=\pi /2$.

Figure 5

The scattering amplitude evolved to $Y=3$ with various kernels illustrates the effect of the different terms in the evolution and demonstrates that the computation based on the $K_{\mathrm{ci}}$ kernel does not develop the Coulomb tails seen when the $K_{\mathrm{rc}}$ kernel is used.

Figure 6

The scattering amplitude as a solution to the BK equation with the collinearly improved kernel as a function of $r$ for $b={10}^{-6}{\mathrm{GeV}}^{-1}$ (upper left) and $b=4{\mathrm{GeV}}^{-1}$ (upper right), and as a function of $b$ at $r=0.1{\mathrm{GeV}}^{-1}$ (lower left) and at $r=1{\mathrm{GeV}}^{-1}$ (lower right).

Figure 7

Evolution of the scattering amplitude from the initial condition at $Y=0$ (left) to $Y=10$ (right).

Figure 8

Growth of the dipole-target amplitude integrated over impact parameter as a function of rapidity for solutions of the BK equations with the running coupling and the collinearly improved kernel.

Figure 9

Comparison of the structure function data from HERA [55] (solid circles) to the prediction of the impact-parameter dependent BK equation with the collinearly improved kernel (lines).

Figure 10

Close-up comparison of the structure function data from HERA [55] (blue points) to the b-dependent prediction (red line) for $Q^2=8.5{\mathrm{GeV}}^2$ (left) and $Q^2=12{\mathrm{GeV}}^2$ (right).

Figure 11

The percentage pulls for various values of $Q^2$ and their average value.

Figure 12

The comparison of the prediction for the reduced cross section for charm to data from HERA [55].

Figure 13

Comparison of the predictions of the model (solid lines) with HERA data from H1 [68] and ZEUS [69] for the $|t|$ dependence (left) and the $W_{\gamma \mathrm{p}}$ dependence (right) of the exclusive electroproduction cross section of the $\varphi$ meson.

Figure 14

Comparison of the predictions of the model (solid lines) with HERA data from H1 [70, 71] for the $|t|$ dependence of the exclusive photoproduction (left) and electroproduction (right) cross sections of the $\mathrm{J}/\psi$ meson.

Figure 15

Comparison of the predictions of the model (solid lines) with HERA data from H1 [70] for the $W$ dependence of the exclusive photoproduction (left) and electroproduction (right) cross sections of the $\mathrm{J}/\psi$ meson at fixed $|t|$ values.

Figure 16

Comparison of the predictions of the model (solid lines) with HERA data from H1 [70, 71] and LHC data from ALICE [72, 73] for the $W_{\gamma \mathrm{p}}$ dependence of the exclusive photoproduction and electroproduction cross section of the $\mathrm{J}/\psi$ meson (left) and with HERA data from H1 [74] and ZEUS [75] for the $W_{\gamma \mathrm{p}}$ dependence of the exclusive photoproduction and electroproduction cross section $\mathrm{J}/\psi /\psi (2\mathrm{S})$ ratio (right).

Figure 17

Comparison of the predictions of the model (solid lines) with the HERA data from H1 [76] and ZEUS [77], and LHC data from LHCb [78] and CMS [79] for the $W_{\gamma \mathrm{p}}$ dependence of the exclusive photoproduction cross section of the $\mathrm{\Upsilon }(1\mathrm{S})$ meson.