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.
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Basso, B., Dixon, L.J., Kosower, D.A. et al. Fishnet four-point integrals: integrable representations and thermodynamic limits. J. High Energ. Phys. 2021, 168 (2021). https://doi.org/10.1007/JHEP07(2021)168
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DOI: https://doi.org/10.1007/JHEP07(2021)168