On the dressing phase in the SL(2) Bethe Ansatz

TL;DR

This paper analyzes the dressing phase in the SL(2) Bethe Ansatz for N=4 super Yang-Mills, providing integral expressions and expansions that confirm conjectured equivalences and describe behavior at different coupling strengths.

Contribution

It derives a single integral expression for the dressing phase coefficients valid for all couplings and confirms the equivalence of small and large coupling expansions.

Findings

Integral expression for coefficients c_{r,s}(g) valid for all g

Confirmed the equivalence of small and large coupling expansions

Provided explicit formulas for small and large coupling coefficients

Abstract

In this paper we study the function chi(x1,x2,g) that determines the dressing phase that appears in the all-loop Bethe Ansatz equations for the SL(2) sector of N=4 super Yang-Mills theory. First, we consider the coefficients c_{r,s}(g) of the expansion of chi(x1,x2,g) in inverse powers of x1, x2. We obtain an expression in terms of a single integral valid for all values of the coupling g. The expression is such that the small and large coupling expansion can be simply computed in agreement with the expected results. This proves the, up to now conjectured, equivalence of both expansions of the phase. The strong coupling expansion is only asymptotic but we find an exact expression for the value of the residue which can be seen to decrease exponentially with g. After that, we consider the function chi(x1,x2,g) itself and, using the same method, expand it for small and large coupling. All small and large coupling coefficients chi^(n)(x1,x2), for even and odd n, are explicitly given in terms of finite sums or, alternatively, in terms of the residues of generating functions at certain poles.