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DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory

A. V. Kotikov, L. N. Lipatov

TL;DR

The paper presents a detailed analysis of DGLAP and BFKL evolution in N=4 supersymmetric Yang-Mills theory up to next-to-leading order. It demonstrates that the BFKL kernel eigenvalue is analytic in the conformal spin |n| and can be continued to negative values to recover residues of twist-2 anomalous dimensions, aligning with DGLAP results. A key finding is the emergence of hermitian separability at NLO, alongside a partial violation of holomorphic separability due to double-logarithmic effects, and the results point to an underlying integrability in the multi-color limit. The work also provides explicit LO and NLO anomalous dimension matrices, constructs multiplicatively renormalizable twist-2 operators, and develops DGLAP evolution within this supersymmetric framework, highlighting a deep connection between high-energy Regge behavior and parton evolution in a highly symmetric theory.

Abstract

We derive the DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory in the next-to-leading approximation. The eigenvalue of the BFKL kernel in this model turns out to be an analytic function of the conformal spin |n|. Its analytic continuation to negative |n| in the leading logarithmic approximation allows us to obtain residues of anomalous dimensions γof twist-2 operators in the non-physical points j=0,-1,... from the BFKL equation in an agreement with their direct calculation from the DGLAP equation. Moreover, in the multi-color limit of the N=4 model the BFKL and DGLAP dynamics in the leading logarithmic approximation is integrable for an arbitrary number of particles. In the next-to-leading approximation the holomorphic separability of the Pomeron hamiltonian is violated, but the corresponding Bethe-Salpeter kernel has the property of a hermitian separability. The main singularities of anomalous dimensions γat j=-r obtained from the BFKL and DGLAP equations in the next-to-leading approximation coincide but our accuracy is not enough to verify an agreement for residues of subleading poles.

DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory

TL;DR

The paper presents a detailed analysis of DGLAP and BFKL evolution in N=4 supersymmetric Yang-Mills theory up to next-to-leading order. It demonstrates that the BFKL kernel eigenvalue is analytic in the conformal spin |n| and can be continued to negative values to recover residues of twist-2 anomalous dimensions, aligning with DGLAP results. A key finding is the emergence of hermitian separability at NLO, alongside a partial violation of holomorphic separability due to double-logarithmic effects, and the results point to an underlying integrability in the multi-color limit. The work also provides explicit LO and NLO anomalous dimension matrices, constructs multiplicatively renormalizable twist-2 operators, and develops DGLAP evolution within this supersymmetric framework, highlighting a deep connection between high-energy Regge behavior and parton evolution in a highly symmetric theory.

Abstract

We derive the DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory in the next-to-leading approximation. The eigenvalue of the BFKL kernel in this model turns out to be an analytic function of the conformal spin |n|. Its analytic continuation to negative |n| in the leading logarithmic approximation allows us to obtain residues of anomalous dimensions γof twist-2 operators in the non-physical points j=0,-1,... from the BFKL equation in an agreement with their direct calculation from the DGLAP equation. Moreover, in the multi-color limit of the N=4 model the BFKL and DGLAP dynamics in the leading logarithmic approximation is integrable for an arbitrary number of particles. In the next-to-leading approximation the holomorphic separability of the Pomeron hamiltonian is violated, but the corresponding Bethe-Salpeter kernel has the property of a hermitian separability. The main singularities of anomalous dimensions γat j=-r obtained from the BFKL and DGLAP equations in the next-to-leading approximation coincide but our accuracy is not enough to verify an agreement for residues of subleading poles.

Paper Structure

This paper contains 23 sections, 189 equations.

Table of Contents

  1. Introduction
  2. Anomalous dimensions of twist-2 operators and their singularities
  3. NLO corrections to the BFKL kernel in $N=4$ SUSY
  4. The set of eigenvalues
  5. Hermitian separability of the Bethe-Salpeter kernel
  6. The violation of a generalized holomorphical separability
  7. Asymptotics of "cross-sections" at $s\to \infty$
  8. Non-symmetric choice of the energy normalization
  9. Limit $\gamma \to 0$ for $n=0$
  10. Hermitian separability for a non-symmetric normalization and the symmetry $\gamma \leftrightarrow J-\gamma$
  11. Anomalous dimension matrix in the $N=4$ SUSY
  12. LLA results for the anomalous dimension matrix in $N=4$ SUSY
  13. Anomalous dimensions and twist-2 operators with a multiplicative renormalization
  14. NLO anomalous dimensions and twist-two operators with the multiplicative renormalization
  15. DGLAP evolution
  16. ...and 8 more sections