Loop equation and exact soft anomalous dimension in N $$ \mathcal{N} $$ = 4 super Yang-Mills
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Springer
Received: March 31, 2020 Accepted: May 11, 2020 Published: June 10, 2020
Simone Giombia and Shota Komatsub a
Department of Physics, Princeton University, Washington Road, Princeton, NJ 08540, U.S.A. b School of Natural Sciences, Institute for Advanced Study, 1 Einstein Dr, Princeton, NJ 08544, U.S.A.
E-mail: [email protected], [email protected] Abstract: BPS Wilson loops in supersymmetric gauge theories have been the subjects of active research since they are often amenable to exact computation. So far most of the studies have focused on loops that do not intersect. In this paper, we derive exact results for intersecting 1/8 BPS Wilson loops in N = 4 supersymmetric Yang-Mills theory, using a combination of supersymmetric localization and the loop equation in 2d gauge theory. The result is given by a novel matrix-model-like representation which couples multiple contour integrals and a Gaussian matrix model. We evaluate the integral at large N , and make contact with the string worldsheet description at strong coupling. As an application of our results, we compute exactly a small-angle limit (and more generally near-BPS limits) of the cross anomalous dimension which governs the UV divergence of intersecting Wilson lines. The same quantity describes the soft anomalous dimension of scattering amplitudes of W -bosons in the Coulomb branch. Keywords: AdS-CFT Correspondence, Supersymmetric Gauge Theory, Wilson, ’t Hooft and Polyakov loops ArXiv ePrint: 2003.04460
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)075
JHEP06(2020)075
Loop equation and exact soft anomalous dimension in N = 4 super Yang-Mills
Contents 1
2 1/8 BPS Wilson loop, 2d YM and matrix model 2.1 1/8 BPS Wilson loops and matrix model 2.2 A new integral representation for Wilson loop correlators 2.2.1 A single fundamental loop 2.2.2 Multiple fundamental loops
4 4 6 6 8
3 Loop equation and intersecting Wilson loops 3.1 Review of loop equation in 2d YM 3.2 Intersecting BPS loops in N = 4 SYM 3.2.1 Figure eight loop 3.2.2 Two-intersection loop
12 12 17 17 22
4 Cross anomalous dimension at small angle 4.1 Cross anomalous dimension in N = 4 SYM 4.2 Two-point function of intersections 4.3 Cross anomalous dimension from localization 4.4 U(1) factor and weak- and strong-coupling expansions
24 24 26 28 31
5 Conclusion
33
A Infinite sum of modified Bessel functions
34
B Cross anomalous dimension of two touching lines
36
1
Introduction
The loop equation was proposed initially in [1, 2] as an alternative way to formulate, and possibly solve, the gauge theories (see e.g. [3] for a review). It has the conceptual advantage that it directly constrains the most basic observables, namely the Wilson loops. In lowerdimensional theories such as matrix models [4] and two-dimensional Yang-Mills theory [5– 9], it has proven to be a powerful tool for solving the theories exactly. Unfortunately, solving the loop equation is much harder in higher dimensions and progress remains to be made. In this paper
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