[Paper Review] Conformal kernel for NLO BFKL equation in ${\cal N}$=4 SYM

Ian Balitsky, Giovanni Antonio Chirilli|OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information)|Dec 18, 2008

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper reconstructs the next-to-leading order (NLO) BFKL kernel in ${\cal N}=4$ SYM theory using Möbius invariance, deriving a conformally invariant kernel from known forward NLO eigenvalues. The key result is a closed-form expression for the NLO kernel in coordinate space that preserves conformal symmetry, resolving inconsistencies in prior non-conformal formulations and providing a consistent framework for high-energy scattering in the Regge limit.

ABSTRACT

Using the requirement of Möbius invariance of ${\cal N}$=4 SYM amplitudes in the Regge limit we restore the conformal NLO BFKL kernel out of the eigenvalues known from the forward NLO BFKL result.

Motivation & Objective

• To restore the conformal NLO BFKL kernel in ${\cal N}=4$ SYM from known forward NLO eigenvalues using Möbius invariance.
• To resolve discrepancies in the literature where the NLO BFKL kernel in coordinate space was found to break conformal invariance.
• To construct a Möbius-invariant rapidity cutoff for the evolution of color dipoles, ensuring consistency with conformal symmetry.
• To provide a conformally invariant formulation of the NLO BFKL kernel that matches known eigenvalues and can be used to derive the QCD NLO kernel with proper regularization.

Proposed method

• Imposes Möbius invariance (SL(2,C) symmetry) on the NLO BFKL kernel in ${\cal N}=4$ SYM to constrain its form.
• Uses known eigenvalues of the forward NLO BFKL kernel in momentum space to fix the structure of the coordinate-space kernel.
• Applies conformal invariance to relate non-forward to forward scattering amplitudes via conformal ratios $R$ and $r$.
• Performs explicit integrals over coordinate space using conformal symmetry and known integrals involving logarithms and dilogarithms.
• Derives the kernel in coordinate space by matching the eigenvalue structure to the conformal partial wave decomposition.
• Constructs a conformally invariant rapidity cutoff by analyzing the Fourier transform of the non-forward kernel, ensuring consistency with Möbius symmetry.

Experimental results

Research questions

• RQ1 Can the NLO BFKL kernel in ${\cal N}=4$ SYM be reconstructed from its forward eigenvalues while preserving Möbius invariance?
• RQ2 Why does the standard NLO BFKL kernel in coordinate space break conformal invariance, and how can this be corrected?
• RQ3 What is the correct form of the NLO BFKL kernel in ${\cal N}=4$ SYM that respects conformal symmetry in the Regge limit?
• RQ4 How can a conformally invariant rapidity cutoff be defined for the evolution of color dipoles in the NLO BFKL framework?
• RQ5 Can the conformal NLO kernel in ${\cal N}=4$ SYM be used to derive a consistent NLO kernel in QCD with proper regularization?

Key findings

• The NLO BFKL kernel in ${\cal N}=4$ SYM is reconstructed as a Möbius-invariant operator in coordinate space, ensuring conformal symmetry in the Regge limit.
• The kernel is fully determined by the known forward eigenvalues $\omega(n,\nu)$, with the NLO correction expressed via $\chi(n,\gamma)$, $\Phi(n,\gamma)$, and $\zeta(3)$.
• The kernel satisfies the condition $\omega_{\rm NLO}(n,\nu) = \alpha_s N_c / \pi \left[ \chi(n,\frac{1}{2}+i\nu) + \frac{\alpha_s N_c}{4\pi} \delta(n,\frac{1}{2}+i\nu) \right]$, matching known results.
• The conformal kernel is derived via integrals over coordinate space involving $\ln^2(z^2/z_1^2)$ and $\Phi(z_1,z)$, which reproduce the correct eigenvalue structure.
• A conformally invariant rapidity cutoff is implicitly defined through the Fourier transform of the non-forward kernel, resolving prior inconsistencies in the literature.
• The resulting kernel in ${\cal N}=4$ SYM is used to derive a consistent NLO QCD kernel with a conformally invariant cutoff, differing from previous formulations that used a non-conformal 'rigid cutoff'.