One-loop non-planar anomalous dimensions in super Yang-Mills theory
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Received: July 7, 2020 Accepted: September 15, 2020 Published: October 20, 2020
Tristan McLoughlin, Raul Pereira and Anne Spiering School of Mathematics & Hamilton Mathematics Institute, Trinity College Dublin, College Green, Dublin, Ireland
E-mail: [email protected], [email protected], [email protected] 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/N 2 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 . Keywords: 1/N Expansion, Integrable Field Theories, Supersymmetric Gauge Theory ArXiv ePrint: 2005.14254
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)124
JHEP10(2020)124
One-loop non-planar anomalous dimensions in super Yang-Mills theory
Contents 1 Introduction 1.1 Non-planar dilatation operator 1.2 Planar theory and integrability
1 5 6 9 11 12 17
3 β-deformed SYM theory 3.1 β-deformed dilatation operator 3.2 Deformed planar theory 3.3 Matrix elements and dimensions 3.4 BMN limit
20 21 22 24 27
4 Level-crossing and spectral statistics
33
5 Conclusions
39
A Overlaps from the algebraic Bethe ansatz
41
B Unfolding procedure
43
1
Introduction
The eigenvalue problem for the dilatation operator, D, acting on the set of gauge-invariant local operators, Oi , in N = 4 super Yang-Mills (SYM) theory, D · O i = ∆ i Oi
(1.1)
has been of continued interest due to its role as a proving ground for novel calculational techniques and because of its importance in the AdS/CFT correspondence. The operator dimensions, ∆i = ∆i (gYM , N ), are non-trivial functions of the coupling gYM and N , the rank of the gauge group. In perturbation theory we can expand the dilatation operator in 2 N powers of the ’t Hooft coupling λ = gYM D=
X
g 2k D2k
with
k
g2 =
λ 16π 2
(1.2)
and at each order in g 2 we can further consider the large-N expansion of the operator dimensions. A key development [1] was the insight that for the so(6) sector of operators
–1–
JHEP10(2020)124
2 Perturbative non-planar anomalous dimensions 2.1 Matrix elements from spin-chain scalar products 2.2 A hexagon-like formulation 2.3 Anomalous dimensions from overlaps
–2–
JHEP10(2020)124
the one-loop, O(g 2 ), leading large-N anomalous dimensions can be computed by means of an integrable spin chain. Single-trace operators composed of L scalar fields were ident
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