Fishnet four-point integrals: integrable representations and thermodynamic limits

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 $Ad{S}_3\times S^1$, in a generalized scaling combining the thermodynamic and short-distance limits.

arXiv comment44 pages, 8 figures

References in corpus (60)

• Strings in flat space and pp waves from $\mathcal{N}=4$ Super Yang Mills
• 2D Gravity and Random Matrices
• Quantum Inverse Scattering Method and Correlation Functions
• Semiclassical quantization of rotating superstring in AdS_5 x S^5
• Stringing Spins and Spinning Strings
• Space-time S-matrix and Flux-tube S-matrix at Finite Coupling
• Comments on operators with large spin
• The shape of a typical boxed plane partition
• Precision Spectroscopy of AdS/CFT
• An Operator Product Expansion for Polygonal null Wilson Loops
• New integrable non-gauge 4D QFTs from strongly deformed planar N=4 SYM
• Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions
• Logarithmic scaling in gauge/string correspondence
• Exact solution for the stationary Kardar-Parisi-Zhang equation
• Hexagonalization of Correlation Functions
• D-particles, Matrix Integrals and KP hierachy
• Classical/quantum integrability in non-compact sector of AdS/CFT
• Tessellating cushions: four-point functions in N=4 SYM
• Multicritical Phases of the O(n) Model on a Random Lattice
• Gluing Ladders into Fishnets
• The Strong Coupling Limit of the Scaling Function from the Quantum String Bethe Ansatz
• The Index Distribution of Gaussian Random Matrices
• Strongly γ-deformed N=4 SYM as an integrable CFT
• Bootstrapping the simplest correlator in planar N = 4 SYM at all loops
• Integrability of Conformal Fishnet Theory
• How many eigenvalues of a Gaussian random matrix are positive?
• Yangian Symmetry for Fishnet Feynman Graphs
• Perturbative Four-Point Functions In Planar N=4 SYM From Hexagonalization
• Bi-scalar integrable CFT at any dimension
• Conformal algebra: R-matrix and star-triangle relation
• Yangian Symmetry for Bi-Scalar Loop Amplitudes
• Determinant formula for the octagon form factor in $\mathcal{N}$=4 SYM
• Exact null octagon
• Basso-Dixon Correlators in Two-Dimensional Fishnet CFT
• Octagon at finite coupling
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• On a CFT limit of planar ${\gamma }_i$-deformed $\mathcal{N}=4$ SYM theory
• Exactly solvable magnet of conformal spins in four dimensions
• Emptiness formation probability in the domain-wall six-vertex model
• Hexagonalization of Correlation Functions II: Two-Particle Contributions
• Generalized Fishnets and Exact Four-Point Correlators in Chiral CFT${}_4$
• The Holographic Fishchain
• Integrable Fishnet from $\gamma$-Deformed $\mathcal{N}=2$ Quivers
• Massive Conformal Symmetry and Integrability for Feynman Integrals
• Thermodynamic Bethe Ansatz for Fishnet CFT
• The Octagon as a Determinant
• Continuum limit of fishnet graphs and AdS sigma model
• Crossing bridges with strong Szego limit theorem
• Truncated linear statistics associated with the top eigenvalues of random matrices
• RG stability of integrable fishnet models
• Quantum Fishchain in $Ad{S}_5$
• Exponential suppression with four legs and an infinity of loops
• Exactly solvable single-trace four point correlators in $\chi$CFT${}_4$
• Hexagons and Correlators in the Fishnet Theory
• Half-space stationary Kardar-Parisi-Zhang equation
• On Complex Gamma-Function Integrals
• The Holographic Dual of Strongly γ-deformed N=4 SYM Theory: Derivation, Generalization, Integrability and Discrete Reparametrization Symmetry
• Massive Fishnets
• Octagon with finite bridge: free fermions and determinant identities
• Spin Chains and Gustafson's Integrals

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