On the running coupling in the JIMWLK equation

Abstract

We propose a new method to implement the running coupling constant in the JIMWLK equation, imposing the scale dependence on the correlation function of the random noise in the Langevin formulation. We interpret this scale choice as the transverse momentum of the emitted gluon in one step of the evolution and show that it is related to the “Balitsky” prescription for the BK equation. This slows down the evolution speed of a practical solution of the JIMWLK equation, bringing it closer to the x-dependence inferred from fits to HERA data. We further study our proposal by a numerical comparison of the BK and JIMWLK equations.

Appendix: Lattice effects

As test of the equivalence of the scales $k_{T}^{2}$ in the momentum space coupling and the scale $4e^{-2 \gamma_{\mathrm{E}}} /r_{T}^{2}$ in coordinate space we show in Fig. 9 a comparison between a solution of the rcJIMWLK equation using the two couplings. This can be done straightforwardly when one uses the “square root” coupling, where one can easily multiply either the kernel K _x or its Fourier-transform with $\sqrt{\alpha_{\mathrm{s}}}$. Thus the curves in Fig. 9 correspond to evolution using the kernels

$$ \sqrt{\alpha_{\mathrm{s}}({\mathbf{r}})} {\mathbf{K}}_{\mathbf{r}}, $$

(A.1)

labeled $r \sqrt{\alpha_{\mathrm{s}}}$, and

$$ i(2\pi)^2 \int\frac{\mathrm{d}^2{\mathbf{k}}}{(2\pi)^2} \sqrt{\alpha_{\mathrm{s}}({ \mathbf{k}})}\frac{e^{-i {\mathbf {k}}\cdot{\mathbf{x}} }}{{\mathbf{k}}^2}, $$

(A.2)

labeled $k \sqrt{\alpha_{\mathrm{s}}}$, with α _s(x) and α _s(k) given by Eqs. (39) and (40). They are seen to agree remarkably well, justifying our use of the constant $4e^{-2\gamma_{\mathrm{E}}}$ in comparing the momentum and position space formulations.

Fig. 9

Evolution of the scattering amplitude $N \equiv1-\langle \hat{D}_{{\mathbf{r}} }\rangle$ with the “square root” coupling rcJIMWLK equation, with coupling evaluated at the scale $k_{T}^{2}$ in the momentum space kernel, and at the scale $4e^{-2\gamma_{\mathrm{E}}}/r_{T}^{2}$ in the position space kernel

The BK equation is solved on a logarithmic grid in r _T , with the values of the scattering amplitude between the grid points obtained from interpolation. This allows one in practice to study as small or large r _T -values as desired. In the case of JIMWLK, on the other hand, one works on a fixed linear lattice with cutoffs ∼1/L in the infrared and ∼1/a in the ultraviolet. One expects large lattice effects when Q _s∼1/L and when Q _s∼1/a, and since Q _s changes exponentially during the evolution, this restricts the accessible rapidity range.

To illustrate the numerical uncertainties in our calculation we compare in Fig. 10 the rcJIMWLK calculation with a square root running coupling, shown also in Fig. 6, to the BK equation with the same running coupling prescription. One can see that the solutions agree very well for smaller rapidities, but start to deviate more later in the evolution. Also shown is a rcJIMWLK calculation with the same initial g ² μ/Λ _QCD (approximately the same Q _s/Λ _QCD) performed on a smaller 512²-lattice in such a way that either the IR ∼1/L or UV ∼1/a cutoff is different. We see that for the initial condition Q _s is so small that changing the IR cutoff starts to have an effect. More importantly, late in the evolution the solution becomes sensitive to the UV cutoff; an extrapolation to a→0 would bring the JIMWLK result closer to BK.

Fig. 10

Evolution of the scattering amplitude $N \equiv1-\langle \hat{D}_{{\mathbf{r}} }\rangle$ with the “square root” rcJIMWLK equation (dotted line) and with the BK equation (thinner dashed line) using the same “square root” running coupling. The initial condition is the same as in Fig. 6 and the amplitude is shown every 4 units in y. Also shown are JIMWLK simulations on a smaller 512²-lattice with the physical lattice size reduced (“small L”, thick dashed line) or lattice spacing increased (“large a”, thin solid line) by a factor 2

A further comparison of the evolution speed in the BK and rcJIMWLK codes is shown in Fig. 11. It can be seen that in BK the evolution is slightly faster, and increasingly so at large Q _s, when the lattice ultraviolet cutoff in the JIMWLK code slows down the evolution. In a JIMWLK simulation closer to the continuum limit the difference would be smaller. The figure also demonstrates the dependence on the parameter c controlling the smoothness of the infrared regularization of the coupling. As was seen previously in Fig. 8, the evolution speed is sensitive to the regularization for small Q _s/Λ _QCD.

Fig. 11

Evolution speed of the saturation scale λ with the “square root” coupling. Shown are the results from the BK and JIMWLK simulations with the same kernel, using infrared regularizations of the running coupling with different values of the parameter c

In addition, the BK and JIMWLK equations are not equivalent even in the continuum, because of the mean field approximation. Making a precise statement about this difference would require a more systematical extrapolation of the lattice calculation to the continuum and infinite volume limits than is done here. For fixed coupling the difference in the evolution speed between the BK and JIMWLK equations has been studied in Ref. [43], where JIMWLK evolution was found to be slower by few per cent, while an even smaller difference was seen for running coupling in Ref. [52].