Towards Baxter equation in supersymmetric Yang-Mills theories

TL;DR

This paper investigates the two-loop dilatation operator in N=2 and N=4 supersymmetric Yang-Mills theories, revealing integrable structures and proposing a deformed Baxter equation to encode the spectrum of anomalous dimensions.

Contribution

It introduces a deformed Baxter equation that captures the two-loop spectrum and uncovers integrability features in supersymmetric Yang-Mills theories.

Findings

Double degeneracy of eigenstates suggests integrability.

Two-loop anomalous dimensions differ by a normalization factor.

Proposed Baxter equation encodes the spectrum exactly.

Abstract

We perform an explicit two-loop calculation of the dilatation operator acting on single trace Wilson operators built from holomorphic scalar fields and an arbitrary number of covariant derivatives in N=2 and N=4 supersymmetric Yang-Mills theories. We demonstrate that its eigenspectrum exhibits double degeneracy of opposite parity eigenstates which suggests that the two-loop dilatation operator is integrable. Moreover, the two-loop anomalous dimensions in the two theories differ from each other by an overall normalization factor indicating that the phenomenon is not sensitive to the presence of the conformal symmetry. Relying on these findings, we try to uncover integrable structures behind the two-loop dilatation operator using the method of the Baxter Q-operator. We propose a deformed Baxter equation which exactly encodes the spectrum of two-loop anomalous dimensions and argue that it correctly incorporates a peculiar feature of conformal scalar operators -- the conformal SL(2) spin of such operators is modified in higher loops by an amount proportional to their anomalous dimension. From the point of view of spin chains this property implies that the underlying integrable model is ``self-tuned'' -- the all-loop Hamiltonian of the spin chain depends on the total SL(2) spin which in its turn is proportional to the Hamiltonian.