Running coupling and power corrections in nonlinear evolution at the high--energy limit

TL;DR

This work extends nonlinear small-x QCD evolution (JIMWLK/BK) to incorporate running-coupling effects by deriving an all-orders, dispersively resummed kernel. It reveals infrared renormalon ambiguities and associated non-perturbative power corrections, and provides a PV-Borel sum to quantify their impact. Numerical BK evolution with the resummed kernel shows that both perturbative running-coupling corrections and power corrections are sizable at present energies, slowing the evolution and sharpening the saturation scale behavior. The results provide a first predictive framework for including running-coupling and power corrections in high-density QCD and highlight key directions for extending the formalism to full NLO and phenomenology.

Abstract

A main feature of high-energy scattering in QCD is saturation in the number density of gluons. This phenomenon is described by non-linear evolution equations, JIMWLK and BK, which have been derived at leading logarithmic accuracy. In this paper we generalize this framework to include running coupling corrections to the evolution kernel. We develop a dispersive representation of the dressed gluon propagator in the background of Weiszacker Williams fields and use it to compute O(beta_0^{n-1} alpha_s^n) corrections to the kernel to all orders in perturbation theory. The resummed kernels present infrared-renormalon ambiguities, which are indicative of the form and importance of non-perturbative power corrections. We investigate numerically the effect of the newly computed perturbative corrections as well as the power corrections on the evolution and find that at present energies they are both significant.

Figures (9)

• Figure 1: Generic evolution trend for a single--scale dipole correlator. (a) shows $N(Y,r)$ as a function of the dipole size $r=\vert \bm{x}-\bm{y}\vert$ for several values of the saturation scale $Q_s(Y)$. $Q_s$ increases with $Y$; saturation then sets in at smaller distances. (b) shows $\partial_Y N$ and thus the activity in a given evolution step as a function of the same variables. With increasing $Q_s(Y)$ contributions are centered at ever shorter distances.
• Figure 2: Virtual and real contributions at LO (a), (b) and NLO (c), (d) and (e). Vertically (as delineated by the background shading) they are grouped by the Wilson line structure induced by the target interaction. The latter are indicated on the diagrams via large solid dots, their analytical form is listed at the bottom. Diagram (e) induces a new structure via a $q\Bar q$ loop that interacts with the target.
• Figure 3: $R({\bm r}_1\Lambda,{\bm r}_2\Lambda)$ as defined with a PV regularization of the Borel sum (\ref{['eq:K-replacement']}), and the convergence of perturbation theory at the first few orders, according to \ref{['eq:pert-exp']}, with $\mu^2=4\,\exp{\left(-\frac{5}{3}-2\gamma_E\right)}/r^2$ (in (a)) and $\mu^2=8\, \exp{\left(-\frac{5}{3}-2\gamma_E\right)}/(r_1^2+r_2^2)$ (in (b)). $R$ is shown as a function of $r_1$, with $r\Lambda = 10^{-4}$, ${\bm r} \parallel {\bm r}_1 \parallel {\bm r}_2$ and $r_2=r_1+r$.
• Figure 4: Comparison of ad hoc implementations for effective couplings $R$ calculated with the principal value (PV) prescription, the "square root" prescription $[\alpha_s(c^2/r_1^2)\,\alpha_s(c^2/r_2^2)]^{1/2}/\pi$, and the parent dipole running $\alpha_s(c^2/r^2)/\pi$. On the left panel the size of the parent dipole is fixed, $r\Lambda = 0.1$ (and $r_2 = r_1+r$); on the right panel the ratio of the dipole sizes is fixed, $r_1 = 2r$, $r_2 = 3r$.
• Figure 5: $R_{\rm PV}$ and the expected range of power corrections for fixed $r\Lambda=0.1$ (left panel), and fixed ratio $r_1=2r$, $r_2=3r$ (right panel) with $r_1$ parallel to $r_2$. The central solid line corresponds to the principal value result. The shaded regions estimate the relevance of power corrections by adding and subtracting $\pi |\text{residue}|(u=m)$. The inner band takes the first residue at $u=1$, and the outer hashed band is the sum of absolute values of contributions from all residues. The dashed line shows the result when the $u$-integral has been cut at $u_{\rm max}=0.75$, before the first pole is encountered.