Gluon splitting at small $x$: a unified derivation for the JIMWLK, DGLAP and CSS equations

TL;DR

The paper addresses back-to-back dijet production in $eA$ DIS at small $x$ by performing a complete real-NLO calculation within the CGC/TMD framework. It shows that TMD factorisation with the WW gluon TMD persists at NLO, and the NLO corrections can be absorbed into a corrected WW TMD via a universal $\Delta \mathcal{W}_{\mathcal{R}}^{mn}$, while exposing three key evolutions: B-JIMWLK/DMMX at small $x$, CSS Sudakov evolution, and DGLAP evolution of the gluon PDF. By analyzing distinct regions in gluon-kinematics $(z_g,\boldsymbol{k}_g)$, the authors derive the real contributions that reproduce the Dominguez–Marquet–Munier–Xiao evolution, the Sudakov resummation, and the DGLAP splitting, and they obtain a closed form for the dilute limit in terms of a transverse-m momentum dependent gluon-gluon splitting function $P_{g^*\to g^*g}$. The work provides a unified, first-principles link between high-energy small-$x$ evolution and TMD-based parton dynamics, with practical implications for EIC phenomenology and potentially Monte Carlo implementations that incorporate the full set of small-$x$ and transverse-momentum logarithms.

Abstract

We revisit the calculation of the next-to-leading order (NLO) corrections to dijet production in electron-ion collisions at small $x$. We focus on the back-to-back configuration where the relative transverse momentum $P_\perp$ of the measured jets is much larger than both their momentum imbalance $K_\perp$ and the target saturation momentum $Q_s(x,A)$. In this regime, we present for the first time a complete calculation of the real NLO corrections at leading power in $1/P_\perp$. Our result exhibits TMD factorisation, with the same hard factor as at tree-level and a NLO correction to the Weiszäcker-Williams (WW) gluon transverse momentum dependent (TMD) distribution which involves four Wilson-line operators. By studying different kinematical regimes for $K_\perp$ and for the radiated gluon, we recover all the quantum evolutions that were previously identified for this process at NLO: the B-JIMWLK high-energy evolution and the CSS evolution of the gluon WW TMD, and the DGLAP evolution of the gluon PDF. When both $K_\perp$ and the transverse momentum transferred by the target are large compared to $Q_s$, all the Wilson-line operators boil down to the unintegrated gluon distribution and our NLO result for the gluon TMD can be used to isolate the transverse-momentum dependent gluon splitting function.

Figures (3)

• Figure 1: (Left) CGC dipole picture for the $\gamma^*$-nucleus reaction producing two jets at leading order in the colour dipole frame and including multiple gluon scattering. (Right) Target picture of the same process in the conventional TMD factorisation approach. The $t$-channel gluon knocked out of the collision has intrinsic transverse momentum $K_\perp$, much smaller than the transverse momentum $\sim P_\perp$ of each individual jet produced in the final state.
• Figure 2: CGC (left) vs target (right) pictures of the real NLO corrections (to be compared with Fig. \ref{['fig:s-vs-t-channel-picture']}). Depending on the kinematics $(z_g,k_{g\perp})$ of the radiated gluon, it contributes either to the high-energy or CSS evolution of the WW gluon TMD, or to the $\mathcal{O}(\alpha_s)$ finite corrections (without logarithms) to the WW gluon TMD.
• Figure 3: Real NLO amplitudes, labelled $R_1$ and $R_2$, for the $\gamma^*A\to q\bar{q}X$ process. The gluon is emitted either before (left) or after (right) the shock-wave, represented by the vertical gluons. The amplitudes $R_3$ and $R_4$ where the gluon is emitted by the anti-quark are not shown.