Extending QCD perturbation theory to higher energies

TL;DR

The paper addresses the instability of high-energy QCD perturbation theory due to large leading-log contributions and sizable subleading terms. It implements a renormalisation-group–improved (RGI) NLL BFKL resummation with a running-coupling, omega-dependent kernel to compute the gluon Green function $G(Y;k,k_0)$ and the resummed splitting function $P_{\mathrm{eff}}(z,Q^2)$ in $x$-space. The results show substantially reduced high-energy exponents and diffusion, a delayed onset of non-perturbative Pomeron effects, and a robust factorisation of the small-$x$ Green function, yielding a dip in $P_{\mathrm{eff}}$ at moderate small-$x$ followed by BFKL-like growth at very small-$x$. These findings extend the perturbative domain and provide a viable framework for predicting high-energy cross sections at future colliders with improved control over preasymptotic dynamics.

Abstract

On the basis of the results of a new renormalisation group improved small-x resummation scheme, we argue that the range of validity of perturbative calculations is considerably extended in rapidity with respect to leading log expectations. We thus provide predictions for the energy dependence of the gluon Green function in its perturbative domain and for the resummed splitting function. As in previous analyses, high-energy exponents are reduced to phenomenologically acceptable values. Additionally, interesting preasymptotic effects are observed. In particular, the splitting function shows a shallow dip in the moderate small-x region, followed by the expected power increase.

Figures (4)

• Figure 1: Green's function calculated with four different infrared regularisations of the coupling, shown for LL and RGI NLL ('NLL$_\mathrm{B}\!$') evolution. The bands indicate the sensitivity of the ${\bar{k}}=0.74\,\,\mathrm{GeV}$ results to a variation of $x^2_\mu$ in the range $\frac{1}{2}$ to $2$. The left and right hand plots differ only in their scales.
• Figure 2: Contour plots showing the sensitivity of $G(Y,k+\epsilon, k-\epsilon)$ to one's choice of non-perturbative regularisation, as obtained by examining the logarithm of the ratio of the regularisations giving the largest and smallest result for $G$. Darker shades indicate insensitivity to the NP regularisation, and contours have been drawn where the logarithm of the ratio is equal to $0.1$, $0.2$ and $0.4$. Plot (a) shows the result for LL evolution, while (b) shows RGI NLL evolution ($\mathrm{NLL_B}$). The regularisations considered are those of Fig. \ref{['f:manyReg']}.
• Figure 3: The small-$x$ exponents: the Green's function effective exponent $\omega_s$ is shown to first order in the $b$-expansion; the splitting function exponent $\omega_c$ is shown together with NP and renormalisation scale uncertainty bands, defined in figure \ref{['f:Peff']}. Also shown, for reference, is the result for $\omega_c$ using the method of CCS1, for $b(n_f = 4)$.
• Figure 4: Small-$x$$\mathrm{NLL_B}$ resummed splitting function, compared to the pure 1-loop DGLAP and the (fixed-coupling) LL BFKL splitting functions. The central $\mathrm{NLL_B}$ result corresponds to $x_\mu=1$, ${\bar{k}}=0.74\,\,\mathrm{GeV}$; the inner band is that obtained by varying ${\bar{k}}$ between $0.5$ and $1.0\,\,\mathrm{GeV}$, while the outer band corresponds to $\frac{1}{2} < x_\mu^2 < 2$.