Abstract
Basso-Dixon integrals evaluate rectangular fishnets — Feynman graphs with massless scalar propagators which form a m × n rectangular grid — which arise in certain one-trace four-point correlators in the ‘fishnet’ limit of \( \mathcal{N} \) = 4 SYM. Recently, Basso et al. explored the thermodynamical limit m → ∞ with fixed aspect ratio n/m of a rectangular fishnet and showed that in general the dependence on the coordinates of the four operators is erased, but it reappears in a scaling limit with two of the operators getting close in a controlled way. In this note I investigate the most general double scaling limit which describes the thermodynamics when one of two pairs of operators become nearly light-like. In this double scaling limit, the rectangular fishnet depends on both coordinate cross ratios. I show that all singular limits of the fishnet can be attained within the double scaling limit, including the null limit with the four points approaching the cusps of a null square. A direct evaluation of the fishnet in the null limit is presented any m and n.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N.I. Usyukina and A.I. Davydychev, An approach to the evaluation of three and four point ladder diagrams, Phys. Lett. B 298 (1993) 363 [INSPIRE].
A.B. Zamolodchikov, ‘Fishnet’ diagrams as a completely integrable system, Phys. Lett. B 97 (1980) 63 [INSPIRE].
Ö. Gürdoğan and V. Kazakov, New Integrable 4D Quantum Field Theories from Strongly Deformed Planar \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 117 (2016) 201602 [Addendum ibid. 117 (2016) 259903] [arXiv:1512.06704] [INSPIRE].
J. Caetano, Ö. Gürdoğan and V. Kazakov, Chiral limit of \( \mathcal{N} \) = 4 SYM and ABJM and integrable Feynman graphs, JHEP 03 (2018) 077 [arXiv:1612.05895] [INSPIRE].
D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, Strongly γ-Deformed \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory as an Integrable Conformal Field Theory, Phys. Rev. Lett. 120 (2018) 111601 [arXiv:1711.04786] [INSPIRE].
N. Gromov et al., Integrability of Conformal Fishnet Theory, JHEP 01 (2018) 095 [arXiv:1706.04167] [INSPIRE].
B. Basso and D.-L. Zhong, Continuum limit of fishnet graphs and AdS sigma model, JHEP 01 (2019) 002 [arXiv:1806.04105] [INSPIRE].
N. Gromov and A. Sever, Quantum fishchain in AdS5, JHEP 10 (2019) 085 [arXiv:1907.01001] [INSPIRE].
N. Gromov and A. Sever, Derivation of the Holographic Dual of a Planar Conformal Field Theory in 4D, Phys. Rev. Lett. 123 (2019) 081602 [arXiv:1903.10508] [INSPIRE].
N. Gromov and A. Sever, The holographic dual of strongly γ-deformed \( \mathcal{N} \) = 4 SYM theory: derivation, generalization, integrability and discrete reparametrization symmetry, JHEP 02 (2020) 035 [arXiv:1908.10379] [INSPIRE].
B. Basso, G. Ferrando, V. Kazakov and D.-L. Zhong, Thermodynamic Bethe Ansatz for Biscalar Conformal Field Theories in any Dimension, Phys. Rev. Lett. 125 (2020) 091601 [arXiv:1911.10213] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].
R. Frassek, N. Kanning, Y. Ko and M. Staudacher, Bethe Ansatz for Yangian Invariants: Towards Super Yang-Mills Scattering Amplitudes, Nucl. Phys. B 883 (2014) 373 [arXiv:1312.1693] [INSPIRE].
D. Chicherin et al., Yangian Symmetry for Fishnet Feynman Graphs, Phys. Rev. D 96 (2017) 121901 [arXiv:1708.00007] [INSPIRE].
D. Chicherin and G.P. Korchemsky, The SAGEX review on scattering amplitudes Chapter 9: Integrability of amplitudes in fishnet theories, J. Phys. A 55 (2022) 443010 [arXiv:2203.13020] [INSPIRE].
B. Basso et al., Fishnet four-point integrals: integrable representations and thermodynamic limits, JHEP 07 (2021) 168 [arXiv:2105.10514] [INSPIRE].
B. Basso and L.J. Dixon, Gluing Ladder Feynman Diagrams into Fishnets, Phys. Rev. Lett. 119 (2017) 071601 [arXiv:1705.03545] [INSPIRE].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 superYang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions, JHEP 01 (2017) 130 [arXiv:1611.05577] [INSPIRE].
B. Eden and A. Sfondrini, Tessellating cushions: four-point functions in \( \mathcal{N} \) = 4 SYM, JHEP 10 (2017) 098 [arXiv:1611.05436] [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions II: Two-Particle Contributions, JHEP 02 (2018) 177 [arXiv:1711.05327] [INSPIRE].
B. Basso, J. Caetano and T. Fleury, Hexagons and Correlators in the Fishnet Theory, JHEP 11 (2019) 172 [arXiv:1812.09794] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Spacetime and Flux Tube S-Matrices at Finite Coupling for N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 111 (2013) 091602 [arXiv:1303.1396] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and Flux tube S-matrix II. Extracting and Matching Data, JHEP 01 (2014) 008 [arXiv:1306.2058] [INSPIRE].
S. Derkachov and E. Olivucci, Exactly solvable single-trace four point correlators in χCFT4, JHEP 02 (2021) 146 [arXiv:2007.15049] [INSPIRE].
S. Derkachov and E. Olivucci, Conformal quantum mechanics & the integrable spinning Fishnet, JHEP 11 (2021) 060 [arXiv:2103.01940] [INSPIRE].
D. Chicherin, S. Derkachov and A.P. Isaev, Conformal group: R-matrix and star-triangle relation, JHEP 04 (2013) 020 [arXiv:1206.4150] [INSPIRE].
S. Derkachov, V. Kazakov and E. Olivucci, Basso-Dixon Correlators in Two-Dimensional Fishnet CFT, JHEP 04 (2019) 032 [arXiv:1811.10623] [INSPIRE].
D.J. Broadhurst and A.I. Davydychev, Exponential suppression with four legs and an infinity of loops, Nucl. Phys. B Proc. Suppl. 205-206 (2010) 326 [arXiv:1007.0237] [INSPIRE].
V. Korepin and P. Zinn-Justin, Thermodynamic limit of the six-vertex model with domain wall boundary conditions, J. Phys. A 33 (2000) 7053.
V.A. Kazakov and K. Zarembo, Classical/quantum integrability in non-compact sector of AdS/CFT, JHEP 10 (2004) 060 [hep-th/0410105] [INSPIRE].
P.Y. Casteill and C. Kristjansen, The Strong Coupling Limit of the Scaling Function from the Quantum String Bethe Ansatz, Nucl. Phys. B 785 (2007) 1 [arXiv:0705.0890] [INSPIRE].
A.V. Belitsky, A.S. Gorsky and G.P. Korchemsky, Logarithmic scaling in gauge/string correspondence, Nucl. Phys. B 748 (2006) 24 [hep-th/0601112] [INSPIRE].
S. Frolov and A.A. Tseytlin, Semiclassical quantization of rotating superstring in AdS5×S5, JHEP 06 (2002) 007 [hep-th/0204226] [INSPIRE].
I.K. Kostov and M. Staudacher, Multicritical phases of the O(n) model on a random lattice, Nucl. Phys. B 384 (1992) 459 [hep-th/9203030] [INSPIRE].
S. Derkachov and E. Olivucci, Exactly solvable magnet of conformal spins in four dimensions, Phys. Rev. Lett. 125 (2020) 031603 [arXiv:1912.07588] [INSPIRE].
F. Coronado, Perturbative four-point functions in planar \( \mathcal{N} \) = 4 SYM from hexagonalization, JHEP 01 (2019) 056 [arXiv:1811.00467] [INSPIRE].
I. Kostov and V.B. Petkova, Octagon with finite bridge: free fermions and determinant identities, JHEP 06 (2021) 098 [arXiv:2102.05000] [INSPIRE].
N. Arkani-Hamed, A. Hillman and S. Mizera, Feynman polytopes and the tropical geometry of UV and IR divergences, Phys. Rev. D 105 (2022) 125013 [arXiv:2202.12296] [INSPIRE].
P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2-D Gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
F. Coronado, Bootstrapping the Simplest Correlator in Planar \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory to All Loops, Phys. Rev. Lett. 124 (2020) 171601 [arXiv:1811.03282] [INSPIRE].
A.V. Belitsky and G.P. Korchemsky, Exact null octagon, JHEP 05 (2020) 070 [arXiv:1907.13131] [INSPIRE].
A.V. Belitsky and G.P. Korchemsky, Octagon at finite coupling, JHEP 07 (2020) 219 [arXiv:2003.01121] [INSPIRE].
A.V. Belitsky and G.P. Korchemsky, Crossing bridges with strong Szegő limit theorem, JHEP 04 (2021) 257 [arXiv:2006.01831] [INSPIRE].
A.V. Belitsky, Null octagon from Deift-Zhou steepest descent, Nucl. Phys. B 980 (2022) 115844 [arXiv:2012.10446] [INSPIRE].
T. Bargheer, F. Coronado and P. Vieira, Octagons I: Combinatorics and Non-Planar Resummations, JHEP 08 (2019) 162 [arXiv:1904.00965] [INSPIRE].
T. Bargheer, F. Coronado and P. Vieira, Octagons II: Strong Coupling, arXiv:1909.04077 [INSPIRE].
E. Olivucci and P. Vieira, Stampedes I: fishnet OPE and octagon Bootstrap with nonzero bridges, JHEP 07 (2022) 017 [arXiv:2111.12131] [INSPIRE].
C. Ahn, L. Corcoran and M. Staudacher, Combinatorial solution of the eclectic spin chain, JHEP 03 (2022) 028 [arXiv:2112.04506] [INSPIRE].
C. Ahn and M. Staudacher, Spectrum of the hypereclectic spin chain and Pólya counting, Phys. Lett. B 835 (2022) 137533 [arXiv:2207.02885] [INSPIRE].
D. Allison and N. Reshetikhin, Numerical study of the 6-vertex model with domain wall boundary conditions, Annales Inst. Fourier 55 (2005) 1847 cond-mat/0502314.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2211.15056
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Kostov, I. Light-cone limits of large rectangular fishnets. J. High Energ. Phys. 2023, 156 (2023). https://doi.org/10.1007/JHEP03(2023)156
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2023)156