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- Title
Fishnet four-point integrals: integrable representations and thermodynamic limits.
- Authors
Basso, Benjamin; Dixon, Lance J.; Kosower, David A.; Krajenbrink, Alexandre; Zhong, De-liang
- Abstract
We consider four-point integrals arising in the planar limit of the conformal "fishnet" theory in four dimensions. They define a two-parameter family of higher-loop Feynman integrals, which extend the series of ladder integrals and were argued, based on integrability and analyticity, to admit matrix-model-like integral and determinantal representations. In this paper, we prove the equivalence of all these representations using exact summation and integration techniques. We then analyze the large-order behaviour, corresponding to the thermodynamic limit of a large fishnet graph. The saddle-point equations are found to match known two-cut singular equations arising in matrix models, enabling us to obtain a concise parametric expression for the free-energy density in terms of complete elliptic integrals. Interestingly, the latter depends non-trivially on the fishnet aspect ratio and differs from a scaling formula due to Zamolodchikov for large periodic fishnets, suggesting a strong sensitivity to the boundary conditions. We also find an intriguing connection between the saddle-point equation and the equation describing the Frolov-Tseytlin spinning string in AdS3 × S1, in a generalized scaling combining the thermodynamic and short-distance limits.
- Subjects
INTEGRAL representations; FISHING nets; FEYNMAN integrals; ELLIPTIC integrals; DENSITY matrices; CAHN-Hilliard-Cook equation; SCATTERING amplitude (Physics); NUMERICAL integration
- Publication
Journal of High Energy Physics, 2021, Vol 2021, Issue 7, p1
- ISSN
1126-6708
- Publication type
Article
- DOI
10.1007/JHEP07(2021)168