Direct numerical solution of the coordinate space Balitsky-Kovchegov equation at next-to-leading order

We present the first numerical solution to the next-to-leading-order Balitsky-Kovchegov equation in coordinate space in the large-$N_{\mathrm{c}}$ limit. In addition to the dipole operator, we also solve the evolution of the “conformal dipole” for which the conformal invariance breaking double logarithmic term is absent from the evolution equation. The next-to-leading-order corrections are shown to slow down the evolution. We show that the solution depends strongly on the details of the initial condition and that the solution to the equation is not positive definite with all initial conditions relevant for phenomenological applications.

Figure 1

Dipole amplitude and conformal dipole amplitude at initial condition and after evolution compared to the solution of the leading-order BK equation at rapidities $y=0,5$ and $y=10$ (from right to left).

Figure 2

Evolution speed for the conformal and nonconformal dipoles as a a function of the saturation scale compared to the leading-order BK equation solution.

Figure 3

Logarithmic derivative of the dipole amplitude (evolution speed) at initial condition with different values for the anomalous dimension. (a) $\gamma =0.6$ (b) $\gamma =0.8$ (c) $\gamma =1.0$.

Figure 4

Evolution speed of the dipole amplitude at $y=5$ with different values for the anomalous dimension at the initial condition. (a) $\gamma =0.6$ (b) $\gamma =0.8$ (c) $\gamma =1.0$.

Figure 5

Evolution speed of the conformal dipole amplitude at initial condition with different values for the anomalous dimension. (a) $\gamma =0.6$ (b) $\gamma =0.8$ (c) $\gamma =1.0$.

Figure 6

Evolution speed of the conformal dipole amplitude at $y=5$ with different values for the anomalous dimension at the initial condition. (a) $\gamma =0.6$ (b) $\gamma =0.8$ (c) $\gamma =1.0$.

Figure 7

Anomalous dimension $\gamma (r)=\mathrm{d}\ln N(r)/\mathrm{d}\ln r^2$ as a function of dipole size at different rapidities. The plots from right to left are for different rapidities $y=1,5,30$. Shown in each plot are the solutions to the evolution equation with different initial anomalous dimensions. The solid lines are the solutions of the LO equation and the dashed lines the nonconformal dipoles, with different dashed lines corresponding to different initial anomalous dimensions. (a) $\mathrm{y}\mathrm{=}1$ (b) $\mathrm{y}\mathrm{=}5$ (c) $\mathrm{y}\mathrm{=}30$.

Figure 8

Same as Fig. 7 for the conformal dipoles. (a) $\mathrm{y}\mathrm{=}1$ (b) $\mathrm{y}\mathrm{=}5$ (c) $\mathrm{y}\mathrm{=}30$.

Figure 9

Evolution speed of the dipole amplitude at the initial condition. Shown are separately the full NLO and LO evolution equation results and the contributions from the conformal (no double logarithmic term) and nonconformal (only double logarithmic term) parts of the NLO BK equation. To demonstrate the location of the saturation scale, the dipole amplitude is also shown as a thick line on the background. (a) $\gamma =0.6$ (b) $\gamma =0.8$ (c) $\gamma =1.0$.

Figure 10

Evolution speed of the conformal dipole amplitude at the initial condition. Shown are separately the full NLO and LO evolution equation results and the contributions from the $\ln r^2$ term and the rest of the evolution equation. To demonstrate the location of the saturation scale, the dipole amplitude is also shown as a thick line on the background. (a) $\gamma =0.6$ (b) $\gamma =0.8$ (c) $\gamma =1.0$.