Dressing phase in the $SL(2)$ Bethe ansatz

In this paper we study the function $\chi (x_1,x_2,g)$ that determines the dressing phase that appears in the all-loop Bethe ansatz equations for the $SL(2)$ sector of $\mathcal{N}=4$ super Yang-Mills theory. First, we consider the coefficients $c_{r,s}(g)$ of the expansion of $\chi (x_1,x_2,g)$ in inverse powers of $x_{1,2}$. 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 $\chi (x_1,x_2,g)$ itself and, using the same method, expand it for small and large coupling. All small and large coupling coefficients ${\chi }^{(n)}(x_1,x_2)$, 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.

Figure 1

Complex $s$ plane. The contour of integration is $(c-i\infty ,c+i\infty )$. Shifting it to the right we obtain the weak coupling expansion and, shifting it to the left, the strong coupling expansion.