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Evolution kernels of skewed parton distributions: method and two-loop results

A. V. Belitsky, A. Freund, D. Müller

TL;DR

The paper develops a formalism to construct two-loop flavor-singlet evolution kernels for skewed parton distributions within the $\overline{\mathrm{MS}}$ scheme by leveraging conformal symmetry breaking patterns and ${\cal N}=1$ SUSY constraints to avoid full multiloop calculations. It reconstructs the non-forward ER-BL kernels through a decomposition into dotted, crossed-ladder ($G$), and diagonal ($D$) parts, with the diagonal piece fixed by forward-limit correspondence to known two-loop DGLAP kernels. The authors provide explicit representations for all channels, including parity-odd and parity-even sectors, and validate the results via Gegenbauer moments and quark-bubble (beta-function) checks. This work enables efficient numerical evolution of SPDs and yields deeper insight into the structure of exclusive two-loop kernels and their relation to forward evolution.

Abstract

We present a formalism and explicit results for two-loop flavor singlet evolution kernels of skewed parton distributions in the minimal subtraction scheme. This approach avoids explicit multiloop calculations in QCD and is based on the known pattern of conformal symmetry breaking in this scheme as well as constraints arising from the graded algebra of the $\cN = 1$ super Yang-Mills theory. The conformal symmetry breaking part of the kernels is deduced from commutator relations between scale and special conformal anomalies while the symmetric piece is recovered from the next-to-leading order splitting functions and $\cN = 1$ supersymmetry relations.

Evolution kernels of skewed parton distributions: method and two-loop results

TL;DR

The paper develops a formalism to construct two-loop flavor-singlet evolution kernels for skewed parton distributions within the scheme by leveraging conformal symmetry breaking patterns and SUSY constraints to avoid full multiloop calculations. It reconstructs the non-forward ER-BL kernels through a decomposition into dotted, crossed-ladder (), and diagonal () parts, with the diagonal piece fixed by forward-limit correspondence to known two-loop DGLAP kernels. The authors provide explicit representations for all channels, including parity-odd and parity-even sectors, and validate the results via Gegenbauer moments and quark-bubble (beta-function) checks. This work enables efficient numerical evolution of SPDs and yields deeper insight into the structure of exclusive two-loop kernels and their relation to forward evolution.

Abstract

We present a formalism and explicit results for two-loop flavor singlet evolution kernels of skewed parton distributions in the minimal subtraction scheme. This approach avoids explicit multiloop calculations in QCD and is based on the known pattern of conformal symmetry breaking in this scheme as well as constraints arising from the graded algebra of the super Yang-Mills theory. The conformal symmetry breaking part of the kernels is deduced from commutator relations between scale and special conformal anomalies while the symmetric piece is recovered from the next-to-leading order splitting functions and supersymmetry relations.

Paper Structure

This paper contains 17 sections, 189 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of evolution kernels.
  3. Support properties of evolution kernels.
  4. Conformal properties of evolution kernels.
  5. ${\cal N}=1$ supersymmetric constraints.
  6. Method of reconstruction of two-loop kernels.
  7. Reconstruction of evolution kernels in NLO.
  8. Construction of $\hbox{\boldmath$\dot V$}$.
  9. Construction of $\hbox{\boldmath$G$}$ kernels.
  10. Restoration of remaining diagonal terms.
  11. Parity odd sector.
  12. Parity even sector.
  13. Explicit representations of the kernels.
  14. Conclusions.
  15. Uniqueness of extension.
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Support of the singlet anomalous dimensions $\hbox{\boldmath $\gamma$}$ in light-cone position (a) and fraction (b) representation. In (b) we only show the support which arises from $1-y-z \ge 0$ in the light-cone position representation. A second contribution that comes from the region $1-y-z \le 0$ can be formally obtained by $t'\to -t'$, however, in the flavor singlet case it can be reduced to the first one by means of symmetry. Here the entries of $\hbox{\boldmath $f$}_{\pm\pm} = \hbox{\boldmath $f$}(\pm t,\pm t')$ are defined by Eqs. (\ref{['INTRE-AA']}) and (\ref{['INTRE-AB']}).