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The gluon splitting function at moderately small x

Marcello Ciafaloni, Dimitri Colferai, Gavin P. Salam, Anna M. Stasto

TL;DR

The paper investigates a surprising dip in the gluon-gluon splitting function $xP_{gg}(x)$ at moderately small $x$, challenging the notion that small-$x$ evolution is dominated by a straightforward power rise from BFKL resummation. It shows this dip arises from the interplay of low-order perturbative terms, with a leading negative NNLO contribution and higher-order positive logarithmic terms, and identifies a characteristic scale $\,\ln(1/x_{\min}) \sim 1/\sqrt{\bar{\alpha}_s}$ where the dip occurs. By combining a sqrt($\alpha_s$) expansion with a cut-representation resummation argument, the authors derive an upper bound on the dip position and a practical matching condition that delineates when fixed-order NNLO is reliable versus when small-$x$ resummation is necessary. Their analysis suggests NNLO may remain applicable down to the dip, but for αs beyond roughly 0.05–0.1, resummation becomes essential, informing phenomenology for HERA-era and future high-energy colliders.

Abstract

It is widely believed that at small x, the BFKL resummed gluon splitting function should grow as a power of 1/x. But in several recent calculations it has been found to decrease for moderately small-x before eventually rising. We show that this `dip' structure is a rigorous feature of the P_gg splitting function for sufficiently small alpha_s, the minimum occurring formally at ln 1/x of order 1/sqrt(alpha_s). We calculate the properties of the dip, including corrections of relative order sqrt(alpha_s), and discuss how this expansion in powers of sqrt(alpha_s), which is poorly convergent, can be qualitatively matched to the fully resummed result of a recent calculation, for realistic values of alpha_s. Finally, we note that the dip position, as a function of alpha_s, provides a lower bound in x below which the NNLO fixed-order expansion of the splitting function breaksdown and the resummation of small-x terms is mandatory.

The gluon splitting function at moderately small x

TL;DR

The paper investigates a surprising dip in the gluon-gluon splitting function at moderately small , challenging the notion that small- evolution is dominated by a straightforward power rise from BFKL resummation. It shows this dip arises from the interplay of low-order perturbative terms, with a leading negative NNLO contribution and higher-order positive logarithmic terms, and identifies a characteristic scale where the dip occurs. By combining a sqrt() expansion with a cut-representation resummation argument, the authors derive an upper bound on the dip position and a practical matching condition that delineates when fixed-order NNLO is reliable versus when small- resummation is necessary. Their analysis suggests NNLO may remain applicable down to the dip, but for αs beyond roughly 0.05–0.1, resummation becomes essential, informing phenomenology for HERA-era and future high-energy colliders.

Abstract

It is widely believed that at small x, the BFKL resummed gluon splitting function should grow as a power of 1/x. But in several recent calculations it has been found to decrease for moderately small-x before eventually rising. We show that this `dip' structure is a rigorous feature of the P_gg splitting function for sufficiently small alpha_s, the minimum occurring formally at ln 1/x of order 1/sqrt(alpha_s). We calculate the properties of the dip, including corrections of relative order sqrt(alpha_s), and discuss how this expansion in powers of sqrt(alpha_s), which is poorly convergent, can be qualitatively matched to the fully resummed result of a recent calculation, for realistic values of alpha_s. Finally, we note that the dip position, as a function of alpha_s, provides a lower bound in x below which the NNLO fixed-order expansion of the splitting function breaksdown and the resummation of small-x terms is mandatory.

Paper Structure

This paper contains 4 sections, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Low perturbative orders and $\boldsymbol{\sqrt{\alpha_s}}$-expansion
  3. Resummation and cut-representation argument
  4. Conclusions

Figures (3)

  • Figure 1: The $x P_{gg}(x)$ splitting function. The resummed (NLL$_\mathrm{B}$) curve corresponds to scheme B of CCSSkernel.
  • Figure 2: Representation of different classifications of logarithmically enhanced terms. Symbols 'x' indicate terms that are present; '$0$' indicates terms that could have been present but are zero; '$n_f$' indicates a term whose only non-vanishing part is proportional to $n_f$; a dash indicates terms which do not exist by definition.
  • Figure 3: Properties of the dip in the NLL$_\mathrm{B}$ model of CCSSkernel compared to our analytical predictions. See text for details.