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The 2-loop generalized scaling function from the BES/FRS equation

Dmytro Volin

TL;DR

The paper derives a strong-coupling expansion of the generalized scaling function $f^{FRS}[g,j]$ from the BES/FRS equation in the regime $\ell=\frac{j}{4g}$ fixed, obtaining tree-level, one-loop, and two-loop results. It demonstrates that the dressing phase drops out at finite $\ell$, enabling a recursive perturbative solution for the resolvent $S$ and the auxiliary function $R_h$, with explicit expressions for the first two orders and a detailed two-loop analysis. The two-loop result agrees with the BA calculation up to a $\ell$-dependent correction $\delta[\ell]$, and its large-$\ell$ expansion yields the BMN-like coefficient $c_{12}=\frac{16}{3}$, confirmed numerically at weak coupling. The analysis also clarifies the branch-point structure near $u=\pm a$ and fixes a key constant ${\cal Q}$ via double-scaling, reinforcing the consistency with string theory expectations in the appropriate limits.

Abstract

We formulate the BES/FRS equation as a functional equation in the rapidity space and perform its strong coupling expansion in the limit when $\ell=j/4g$ is kept finite. We obtain a result which is consistent with the previous calculations at tree level and one loop and which differs from the two-loop calculation in 0805.4615 by a term singular at $\ell=0$.

The 2-loop generalized scaling function from the BES/FRS equation

TL;DR

The paper derives a strong-coupling expansion of the generalized scaling function from the BES/FRS equation in the regime fixed, obtaining tree-level, one-loop, and two-loop results. It demonstrates that the dressing phase drops out at finite , enabling a recursive perturbative solution for the resolvent and the auxiliary function , with explicit expressions for the first two orders and a detailed two-loop analysis. The two-loop result agrees with the BA calculation up to a -dependent correction , and its large- expansion yields the BMN-like coefficient , confirmed numerically at weak coupling. The analysis also clarifies the branch-point structure near and fixes a key constant via double-scaling, reinforcing the consistency with string theory expectations in the appropriate limits.

Abstract

We formulate the BES/FRS equation as a functional equation in the rapidity space and perform its strong coupling expansion in the limit when is kept finite. We obtain a result which is consistent with the previous calculations at tree level and one loop and which differs from the two-loop calculation in 0805.4615 by a term singular at .

Paper Structure

This paper contains 9 sections, 84 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. BES/FRS equation
  3. Analytic properties of the resolvents and possible regimes at strong coupling
  4. Perturbative solution at strong coupling
  5. Disappearing of the dressing phase
  6. The perturbative procedure
  7. Tree and one loop level
  8. Two loops
  9. Behavior of the solution near the branch point.

Figures (1)

  • Figure 1: Left: Analytical structure of $R_m[u]$. Right: Analytical structure of $R_h[u]$. Solid line - the only cut on the physical sheet which corresponds to the roots/holes condensation. Dashed lines correspond to the cuts which appear if we analytically continue the resolvents from the upper half plane by the path shown by arrow.