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The BFKL equation, Mueller-Navelet jets and single-valued harmonic polylogarithms

Vittorio Del Duca, Lance J. Dixon, Claude Duhr, Jeffrey Pennington

TL;DR

This work addresses analytic access to the leading-logarithmic BFKL Green's function in transverse momentum space for Mueller-Navelet jets by introducing a generating function built from single-valued harmonic polylogarithms ($\mathcal{L}_{\omega}$). The authors derive fully analytic azimuthal-angle and transverse-momentum distributions, along with a generating function for the total cross section, order by order in $\alpha_s$, and provide explicit results up to high loop orders. The approach leverages SVHPLs to encode the perturbative coefficients, revealing structured, symmetry-respecting representations and enabling direct comparison with Mellin-Fourier formulations. These results advance analytic control over high-energy QCD observables and lay groundwork for extensions to NLL accuracy and to related gauge theories such as ${\cal N}=4$ super-Yang-Mills.

Abstract

We introduce a generating function for the coefficients of the leading logarithmic BFKL Green's function in transverse-momentum space, order by order in alpha_s, in terms of single-valued harmonic polylogarithms. As an application, we exhibit fully analytic azimuthal-angle and transverse-momentum distributions for Mueller-Navelet jet cross sections at each order in alpha_s. We also provide a generating function for the total cross section valid to any number of loops.

The BFKL equation, Mueller-Navelet jets and single-valued harmonic polylogarithms

TL;DR

This work addresses analytic access to the leading-logarithmic BFKL Green's function in transverse momentum space for Mueller-Navelet jets by introducing a generating function built from single-valued harmonic polylogarithms (). The authors derive fully analytic azimuthal-angle and transverse-momentum distributions, along with a generating function for the total cross section, order by order in , and provide explicit results up to high loop orders. The approach leverages SVHPLs to encode the perturbative coefficients, revealing structured, symmetry-respecting representations and enabling direct comparison with Mellin-Fourier formulations. These results advance analytic control over high-energy QCD observables and lay groundwork for extensions to NLL accuracy and to related gauge theories such as super-Yang-Mills.

Abstract

We introduce a generating function for the coefficients of the leading logarithmic BFKL Green's function in transverse-momentum space, order by order in alpha_s, in terms of single-valued harmonic polylogarithms. As an application, we exhibit fully analytic azimuthal-angle and transverse-momentum distributions for Mueller-Navelet jet cross sections at each order in alpha_s. We also provide a generating function for the total cross section valid to any number of loops.

Paper Structure

This paper contains 14 sections, 107 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The BFKL equation and Mueller-Navelet jets
  3. The saddle-point approximation
  4. The azimuthal angle distribution
  5. The Mueller-Navelet dijet cross section
  6. Single-valued harmonic polylogarithms
  7. The BFKL equation and single-valued harmonic polylogarithms
  8. Analytic distributions for Mueller-Navelet jets
  9. The azimuthal-angle distribution
  10. The transverse-momentum distribution
  11. The total cross section
  12. Conclusions
  13. A recursive formula for the Fourier coefficients $\chi_{\nu,n}$
  14. The small $\nu$ expansion of the BFKL eigenvalue $\chi_{\nu,n}$

Figures (2)

  • Figure 1: The transverse momentum distribution $b(\rho;\eta)$ in the high-energy limit as a function of the rapidity for $\rho = 0.5$ evaluated perturbatively through twelve loops (red), and compared to the exact Mellin integral (blue) and its saddle-point approximation (green).
  • Figure 2: Same as Fig. \ref{['fig:Bk_plot_0.5']}, but for $\rho=0.7$.