[Paper Review] DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory

Kotikov, A. V., Lipatov, L. N.|arXiv (Cornell University)|Aug 23, 2002

Black Holes and Theoretical Physics332 citations

TL;DR

This paper derives the DGLAP and BFKL evolution equations in N=4 supersymmetric Yang-Mills theory at next-to-leading order (NLO), demonstrating that the BFKL kernel's eigenvalue is an analytic function of conformal spin |n|. It establishes a precise link between DGLAP and BFKL dynamics via analytic continuation, showing agreement in anomalous dimensions and residues at non-physical poles j = 0, -1, ..., and reveals that while holomorphic separability breaks in NLO, hermitian separability of the Bethe-Salpeter kernel is preserved.

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 \gamma of 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 \gamma at 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.

Motivation & Objective

• To derive the DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory at next-to-leading order (NLO).
• To investigate the analytic structure of the BFKL kernel eigenvalue as a function of conformal spin |n| in the N=4 model.
• To establish a correspondence between the DGLAP and BFKL equations via analytic continuation of the BFKL kernel to negative |n|.
• To examine the behavior of anomalous dimensions and their poles at non-physical points j = 0, -1, ... using both DGLAP and BFKL frameworks.
• To analyze the integrability and separability properties of the Pomeron Hamiltonian in the multi-color limit and in NLO.

Proposed method

• Derives the DGLAP and BFKL equations in the N=4 SUSY gauge theory using the dimensional reduction (DRED) scheme to preserve supersymmetry.
• Computes the eigenvalue of the BFKL kernel in NLO and demonstrates its analyticity in the conformal spin |n|, enabling analytic continuation to negative |n|.
• Uses Mellin transformation to relate parton distribution moments to anomalous dimension matrices γab(j), connecting DGLAP dynamics to twist-2 operator anomalous dimensions.
• Applies analytic continuation of the BFKL eigenvalue ω0(n, ν) to negative |n| to extract residues of anomalous dimensions at non-physical poles j = 0, -1, ..., and compares them with direct DGLAP calculations.
• Analyzes the structure of the Bethe-Salpeter kernel in NLO, showing that while holomorphic separability is violated, hermitian separability is preserved.
• Performs independent calculations of anomalous dimensions using the renormalization group in both leading and next-to-leading logarithmic approximations.

Experimental results

Research questions

• RQ1 How do the DGLAP and BFKL evolution equations interrelate in the N=4 supersymmetric gauge theory at next-to-leading order?
• RQ2 Does the analytic continuation of the BFKL kernel eigenvalue in conformal spin |n| reproduce the residues of anomalous dimensions at non-physical poles j = 0, -1, ... as computed from the DGLAP equation?
• RQ3 What is the structure of the BFKL kernel in the N=4 model in the next-to-leading approximation, and does it preserve separability properties such as holomorphic or hermitian separability?
• RQ4 In the multi-color limit of N=4 SUSY, is the BFKL and DGLAP dynamics integrable for an arbitrary number of particles in the leading logarithmic approximation?
• RQ5 To what extent do the main singularities of anomalous dimensions at j = -r, as derived from BFKL and DGLAP equations in NLO, agree?

Key findings

• The eigenvalue of the BFKL kernel in the N=4 model is an analytic function of the conformal spin |n|, enabling consistent analytic continuation to negative |n|.
• Residues of anomalous dimensions γ at non-physical poles j = 0, -1, ... extracted from the BFKL equation via analytic continuation agree exactly with direct calculations from the DGLAP equation.
• In the multi-color limit of the N=4 model, both BFKL and DGLAP dynamics are integrable for an arbitrary number of particles in the leading logarithmic approximation.
• In the next-to-leading order, holomorphic separability of the Pomeron Hamiltonian is violated, but the Bethe-Salpeter kernel retains the property of hermitian separability.
• The main singularities of anomalous dimensions at j = -r, as derived from both BFKL and DGLAP equations in NLO, coincide, although the accuracy is insufficient to verify agreement for subleading pole residues.
• The anomalous dimension matrices for twist-2 operators in both polarized and unpolarized cases are derived explicitly in terms of harmonic sums S1(j ± r), with results independent of arbitrary normalization parameters in the diagonalization procedure.