Anomalous dimensions of twist 2 operators and $\mathcal{N}=4$ SYM quantum spectral curve

Abstract

We present algorithmic perturbative solution of $\mathcal{N}=4$ SYM quantum spectral curve in the case of twist 2 operators, valid to in principle arbitrary order in coupling constant. The latter treats operator spins as arbitrary integer values and is written in terms of special class of functions -- products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions. It is shown that this class of functions is closed under elementary operations, such as shifts, partial fractions, multiplication by spectral parameter and differentiation. Also, it is fully sufficient to solve arising non-homogeneous multiloop Baxter and first order difference equations. As an application of the proposed method we present the computation of anomalous dimensions of twist 2 operators up to four loop order.