Renormalisation group improved small-x Green's function

TL;DR

This work develops a renormalisation group–improved small‑$x$ framework that merges leading‑collinear DGLAP dynamics with the NL BFKL kernel, including running coupling, to compute the two‑scale gluon Green's function and the corresponding gluon splitting function. The authors implement a practical resummed kernel via an ${\omega}$‑expansion, analyze the Green's function numerically, and extract high‑energy exponents $ω_s(α_s)$ and $ω_c(α_s)$ along with diffusion corrections, finding an extended perturbative domain and notable preasymptotic effects. They also obtain a resummed anomalous dimension and a robust small‑$x$ splitting function that exhibits a dip at moderate $x$ and a power return at very small $x$, with momentum conservation enforced in one scheme. The paper further extends the formalism to include phase‑space effects and two‑scale impact factors, laying groundwork for phenomenology of processes with two hard scales (e.g., DIS, $\gamma^*\gamma^*$, Mueller–Navelet jets). Overall, the RG‑improved approach stabilises high‑energy predictions, broadens the perturbative window, and provides a coherent framework for connecting small‑$x$ evolution to observable cross sections at current and future colliders.

Abstract

We investigate the basic features of the gluon density predicted by a renormalisation group improved small-x equation which incorporates both the gluon splitting function at leading collinear level and the exact BFKL kernel at next-to-leading level. We provide resummed results for the Green's function and its hard Pomeron exponent $ω_s(α_s)$, and for the splitting function and its critical exponent $ω_c(α_s)$. We find that non-linear resummation effects considerably extend the validity of the hard Pomeron regime by decreasing diffusion corrections to the Green's function exponent and by slowing down the drift towards the non-perturbative Pomeron regime. As in previous analyses, the resummed exponents are reduced to phenomenologically interesting values. Furthermore, significant preasymptotic effects are observed. In particular, the resummed splitting function departs from the DGLAP result in the moderate small-x region, showing a shallow dip followed by the expected power increase in the very small-x region. Finally, we outline the extension of the resummation procedure to include the photon impact factors.

Figures (19)

• Figure 1: $\omega_s$ as a function of $\alpha_s$ for different subtraction schemes together with the original result for the $\omega$-expansion. The calculation is done in the fixed coupling case.
• Figure 2: $\bar{\alpha}_s \chi_{{\mathrm{eff}}}(\gamma,\bar{\alpha}_s)$ as a function of $\gamma$ in different schemes for different values of $\alpha_s$: $\alpha_s=0.1$ (dash-dotted line), $\alpha_s=0.2$ (solid line), $\alpha_s=0.3$ (dashed line). The calculation is done in the fixed coupling case.
• Figure 3: $\chi"(\gamma_m,\bar{\alpha}_s)$ as a function of $\alpha_s$ for two different subtraction models and the $\omega$-expansion scheme.
• Figure 4: Gluon Green's function as a function of rapidity $Y$ for three different grid spacings $\Delta \tau=0.05,0.1,0.2$. LL evolution is used with a fixed coupling, $\bar{\alpha}_s = 0.2$; $\epsilon \simeq 0.2k_0$.
• Figure 5: Gluon Green's function $G(Y;k_0+\epsilon,k_0)$ as a function of rapidity $Y$: (a) for LL and the two RGI schemes A and B; (b) for scheme B and two variants of pure NLL evolution. The parameters are $k_0=20 \,\,\mathrm{GeV}$ and $\epsilon\simeq 0.2 k_0$.