Abstract
We consider non-planar one-loop anomalous dimensions in maximally supersymmetric Yang-Mills theory and its marginally deformed analogues. Using the basis of Bethe states, we compute matrix elements of the dilatation operator and find compact expressions in terms of off-shell scalar products and hexagon-like functions. We then use non-degenerate quantum-mechanical perturbation theory to compute the leading 1/N2 corrections to operator dimensions and as an example compute the large R-charge limit for two-excitation states through subleading order in the R-charge. Finally, we numerically study the distribution of level spacings for these theories and show that they transition from the Poisson distribution for integrable systems at infinite N to the GOE Wigner-Dyson distribution for quantum chaotic systems at finite N.
Highlights
The one-loop, O(g2), leading large-N anomalous dimensions can be computed by means of an integrable spin chain
Single-trace operators composed of L scalar fields were identified with closed spin chains of length L and the planar dilatation operator with a spin-chain Hamiltonian that can be diagonalised by use of the Bethe ansatz
This prompted a great deal of work and lead to non-perturbative results for the spectrum of planar anomalous dimensions first through the thermodynamic Bethe ansatz and subsequently by means of the Quantum Spectral Curve (QSC), see [5,6,7,8] for reviews
Summary
N = 4 SYM theory contains six scalar fields, (φI )ab, I = 1, . These operators can be organised into SO(6) representations with Dynkin labels [M, L − 2M, M ] This sector is known to be closed under the action of the dilatation operator and does not mix with operators containing other scalars, field strengths or fermions. This is not important for N = 4 SYM and we could well consider a U(N ) gauge group, it will become relevant when we subsequently consider the β-deformed theory Using this notation, the tree-level dilatation operator in the su(2) sector can be written as. We can decompose the action of the one-loop dilatation operator on multi-trace operators into planar and non-planar pieces. In order to find the eigenvalues of D2, one can first solve the planar problem using integrability and attempt to use perturbation theory to find the 1/N k corrections
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