Next-to-leading order Balitsky-Kovchegov equation beyond large $N_c$

We calculate finite-$N_{\mathrm{c}}$ corrections to the next-to-leading order (NLO) Balitsky-Kovchegov (BK) equation. We find analytical expressions for the necessary correlators of six Wilson lines in terms of the two-point function using the Gaussian approximation. In a suitable basis, the problem reduces from the diagonalization of a six-by-six matrix to the diagonalization of a three-by-three matrix, which can easily be done analytically. We study numerically the effects of these finite-$N_{\mathrm{c}}$ corrections on the NLO BK equation. In general, we find that the finite-$N_{\mathrm{c}}$ corrections are smaller than the expected $1/N_{\mathrm{c}}^2\sim 10\%$. The corrections may be large for individual correlators, but have less of an influence on the shape of the amplitude as a function of the dipole size. They have an even smaller effect on the evolution speed as a function of rapidity.

Figure 1

Coordinates in the LO-like operators of the BK equation placed in a line configuration as a function of some value $a$.

Figure 2

Correlators in the LO-like piece of the BK equation (2), in the line configuration of coordinates as shown in Fig. 1.

Figure 3

Coordinates in the NLO-like operators of the BK equation placed in a line configuration as a function of some value $a$.

Figure 4

Six-point correlators present in the NLO BK equation (2), in the line configuration of coordinates as shown in Fig. 3.

Figure 5

Correlator factors $⟨D_{2,1}⟩$ and $⟨D_{2,2}⟩$ in the NLO-like part of the BK equation (2), in the line configuration of coordinates as shown in Fig. 3. Both factors reduce to the same expression $⟨D_2{⟩}_{\text{Large}{N}_{\mathrm{c}}}$ in the large-$N_{\mathrm{c}}$ limit, as also shown in the figure.

Figure 6

Evolution speed of the dipole amplitude at the initial condition at large-$N_{\mathrm{c}}$ and at finite-$N_{\mathrm{c}}$. We show separately the contribution from the LO- and NLO-like terms (note that the LO-like contribution includes the order ${\alpha }_{\mathrm{s}}^2$ contribution included in $K_1^{\mathrm{fin}}$).

Figure 7

Difference of the evolution speeds at finite $N_{\mathrm{c}}$ and at large $N_{\mathrm{c}}$, shown separately for the LO-like, NLO-like and total (LO-like $+$ NLO-like) contributions.

Figure 8

Evolution for the ratio of the dipole amplitudes obtained by performing the finite-$N_{\mathrm{c}}$ and large-$N_{\mathrm{c}}$ evolutions with the same initial condition.

Figure 9

Evolution speed of the saturation scale $Q_s^2$ as a function of rapidity at large $N_{\mathrm{c}}$ and at finite $N_{\mathrm{c}}$.