ABJM quantum spectral curve and Mellin transform

The present techniques for the perturbative solution of quantum spectral curve problems in N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=4 $$\end{document} SYM and ABJM models are limited to the situation when the states quantum numbers are given explicitly as some integer numbers. These techniques are sufficient to recover full analytical structure of the conserved charges provided that we know a finite basis of functions in terms of which they could be written explicitly. It is known that in the case of N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=4 $$\end{document} SYM both the contributions of asymptotic Bethe ansatz and wrapping or finite size corrections are expressed in terms of the harmonic sums. However, in the case of ABJM model only the asymptotic contribution can still be written in the harmonic sums basis, while the wrapping corrections part can not. Moreover, the generalization of harmonic sums basis for this problem is not known. In this paper we present a Mellin space technique for the solution of multiloop Baxter equations, which is the main ingredient for the solution of corresponding quantum spectral problems, and provide explicit results for the solution of ABJM quantum spectral curve in the case of twist 1 operators in sl(2) sector for arbitrary spin values up to four loop order with explicit account for wrapping corrections. It is shown that the result for anomalous dimensions could be expressed in terms of harmonic sums decorated by the fourth root of unity factors, so that maximum transcendentality principle holds.