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Dimensional Regularisation and Factorisation Schemes in the BFKL Equation at Subleading Leve

M. Ciafaloni, D. Colferai

TL;DR

This paper analyzes BFKL small-x evolution in 4+2ε dimensions to illuminate renormalisation-group factorisation between Q0 and MSbar schemes, incorporating running coupling and full NLx kernels. Using a γ-representation, it separates the gluon density into an anomalous-dimension exponential and fluctuation factors, identifying a universal R-factor and its dependence on ε. It establishes explicit relationships between scheme densities, derives NLx corrections to the MSbar anomalous dimension, and provides a resummed, universal expression for γ_qg^(MSbar) via off-shell splitting functions and operator methods. The results clarify how ε-dependence translates into subleading αs/ω effects, offering a framework for improved resummations and potential phenomenological applications in single- and singlet-channel evolution. Overall, the work strengthens the theoretical underpinning of scheme changes in high-energy QCD and guides future extensions to fully resummed schemes and data comparisons.

Abstract

We study the anomalous dimensions and coefficient functions generated by the BFKL equation in 4+2 epsilon dimensions, by investigating both running coupling effects, and the inclusion of the full next-to-leading kernel. After generalising the Fourier representation of the solutions to this case, we analyse the beta-dependent renormalisation-group factorisation of anomalous dimension and coefficient contributions to the gluon density. We derive on this basis the normalisation factor of the Q0-scheme with respect to the MSbar-scheme, including beta-dependent corrections to it, and we outline the derivation of the full next-to-leading contributions. We also provide an expression for the resummed gamma_qg in the MSbar-scheme which exhibits its universality and is explicit up to quadratures.

Dimensional Regularisation and Factorisation Schemes in the BFKL Equation at Subleading Leve

TL;DR

This paper analyzes BFKL small-x evolution in 4+2ε dimensions to illuminate renormalisation-group factorisation between Q0 and MSbar schemes, incorporating running coupling and full NLx kernels. Using a γ-representation, it separates the gluon density into an anomalous-dimension exponential and fluctuation factors, identifying a universal R-factor and its dependence on ε. It establishes explicit relationships between scheme densities, derives NLx corrections to the MSbar anomalous dimension, and provides a resummed, universal expression for γ_qg^(MSbar) via off-shell splitting functions and operator methods. The results clarify how ε-dependence translates into subleading αs/ω effects, offering a framework for improved resummations and potential phenomenological applications in single- and singlet-channel evolution. Overall, the work strengthens the theoretical underpinning of scheme changes in high-energy QCD and guides future extensions to fully resummed schemes and data comparisons.

Abstract

We study the anomalous dimensions and coefficient functions generated by the BFKL equation in 4+2 epsilon dimensions, by investigating both running coupling effects, and the inclusion of the full next-to-leading kernel. After generalising the Fourier representation of the solutions to this case, we analyse the beta-dependent renormalisation-group factorisation of anomalous dimension and coefficient contributions to the gluon density. We derive on this basis the normalisation factor of the Q0-scheme with respect to the MSbar-scheme, including beta-dependent corrections to it, and we outline the derivation of the full next-to-leading contributions. We also provide an expression for the resummed gamma_qg in the MSbar-scheme which exhibits its universality and is explicit up to quadratures.

Paper Structure

This paper contains 19 sections, 179 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $\boldsymbol\gamma$-representation and $\boldsymbol R$-factor in the LL$\boldsymbol x$ case
  3. The running coupling equation and its factorisation properties
  4. Factorisation at subleading level: $\boldsymbol{b}$-dependent corrections
  5. On the treatment of the full NL$\boldsymbol{x}$ corrections to the $\boldsymbol{\mathfrak{R}}$ factor
  6. Universality of $\boldsymbol{\gamma_{\mathsf{q}\mathsf{g}}^{(\overline{\mathrm{MS}})}}$ and its resummation formula
  7. Discussion
  8. Solutions of the difference equations in $\boldsymbol{\gamma}$-space
  9. Asymptotic solution
  10. Iterative solution
  11. Calculation of the $\boldsymbol{\rho}$-function
  12. Characteristic functions of the quark kernels $\boldsymbol{H_{\varepsilon}^{(p)}}$
  13. The Catani-Hautmann kernel $\boldsymbol{H^{(\mathrm{CH})}}$
  14. The transverse kernel $\boldsymbol{H^{(T)}}$
  15. Universal collinear behaviour and off-shell splitting function
  16. ...and 4 more sections

Figures (2)

  • Figure 1: The leading small-$x$ contribution to the quark density.
  • Figure 2: The process $\gamma^* \mathsf{g} \to \mathsf{q} \bar{\mathsf{q}} \mathsf{g}$ used for computing $\mathcal{H}_{\varepsilon}^{(T)}$.