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Towards Baxter equation in supersymmetric Yang-Mills theories. (English) Zbl 1117.81100

Summary: 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 \(\mathcal N=2\) and \(\mathcal N=4\) 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 \(\mathbb Q\)-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 \(\text{SL}(2)\) spin of such operators is modified in higher loops by an amount proportional to their anomalous dimension.

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
47N50 Applications of operator theory in the physical sciences
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