Abstract
This work presents the building-blocks of an integrability-based representation for multi-point Fishnet Feynman integrals with any number of loops. Such representation relies on the quantum separation of variables (SoV) of a non-compact spin-chain with symmetry SO(1, 5) explained in the first paper of this series. The building-blocks of the SoV representation are overlaps of the wave-functions of the spin-chain excitations inserted along the edges of a triangular tile of Fishnet lattice. The zoology of overlaps is analyzed along with various worked out instances in order to achieve compact formulae for the generic triangular tile. The procedure of assembling the tiles into a Fishnet integral is presented exhaustively. The present analysis describes multi-point correlators with disk topology in the bi-scalar limit of planar γ-deformed \( \mathcal{N} \) = 4 SYM theory, and it verifies some conjectural formulae for hexagonalisation of Fishnets CFTs present in the literature. The findings of this work are suitable of generalization to a wider class of Feynman diagrams.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [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].
N. Gromov et al., Integrability of Conformal Fishnet Theory, JHEP 01 (2018) 095 [arXiv:1706.04167] [INSPIRE].
E. Olivucci, Hexagonalization of Fishnet integrals. Part I. Mirror excitations, JHEP 11 (2021) 204 [arXiv:2107.13035] [INSPIRE].
B. Basso, J. Caetano and T. Fleury, Hexagons and Correlators in the Fishnet Theory, JHEP 11 (2019) 172 [arXiv:1812.09794] [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].
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].
V. Kazakov, E. Olivucci and M. Preti, Generalized fishnets and exact four-point correlators in chiral CFT4, JHEP 06 (2019) 078 [arXiv:1901.00011] [INSPIRE].
L. Onsager, Crystal statistics. 1. A Two-dimensional model with an order disorder transition, Phys. Rev. 65 (1944) 117 [INSPIRE].
P.P. Kulish, N.Y. Reshetikhin and E.K. Sklyanin, Yang-Baxter Equation and Representation Theory. 1, Lett. Math. Phys. 5 (1981) 393 [INSPIRE].
R.J. Baxter, Exactly solved models in statistical mechanics, in G. D’Ariano, A. Montorsi and M. Rasetti eds., Integrable Systems in Statistical Mechanics, World Scientific Publishing Company (1982) pp.5–63 [https://doi.org/10.1142/9789814415255_0002] [INSPIRE].
M. D’Eramo, G. Parisi and L. Peliti, Theoretical predictions for critical exponents at the lambda point of bose liquids, Lett. Nuovo Cim. 2 (1971) 878 [INSPIRE].
H. Au-Yang and J.H.H. Perk, The large N limits of the chiral Potts model, Physica A 268 (1999) 175 [math/9906029] [INSPIRE].
B. Basso and L.J. Dixon, Gluing Ladder Feynman Diagrams into Fishnets, Phys. Rev. Lett. 119 (2017) 071601 [arXiv:1705.03545] [INSPIRE].
F. Aprile and E. Olivucci, Multipoint Feynman diagrams in the Fishnet CFT: sequential splitting, to appear.
C. Bercini, V. Gonçalves and P. Vieira, Light-Cone Bootstrap of Higher Point Functions and Wilson Loop Duality, Phys. Rev. Lett. 126 (2021) 121603 [arXiv:2008.10407] [INSPIRE].
E. Olivucci and P. Vieira, Null Polygons in Conformal Gauge Theory, Phys. Rev. Lett. 129 (2022) 221601 [arXiv:2205.04476] [INSPIRE].
I. Kostov, Light-cone limits of large rectangular fishnets, JHEP 03 (2023) 156 [arXiv:2211.15056] [INSPIRE].
V. Kazakov, F. Levkovich-Maslyuk and V. Mishnyakov, Integrable Feynman Graphs and Yangian Symmetry on the Loom, arXiv:2304.04654 [INSPIRE].
F. Loebbert, Integrability for Feynman Integrals, SciPost Phys. Proc. 14 (2023) 008 [arXiv:2212.09636] [INSPIRE].
S. Derkachov, G. Ferrando and E. Olivucci, Mirror channel eigenvectors of the d-dimensional fishnets, JHEP 12 (2021) 174 [arXiv:2108.12620] [INSPIRE].
S. Derkachov, V. Kazakov and E. Olivucci, Basso-Dixon Correlators in Two-Dimensional Fishnet CFT, JHEP 04 (2019) 032 [arXiv:1811.10623] [INSPIRE].
O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Integrability and the Conformal Field Theory of the Higgs branch, JHEP 06 (2015) 103 [arXiv:1411.3676] [INSPIRE].
C. Ahn and M. Staudacher, The Integrable (Hyper)eclectic Spin Chain, JHEP 02 (2021) 019 [arXiv:2010.14515] [INSPIRE].
G. Ferrando, A. Sever, A. Sharon and E. Urisman, A large twist limit for any operator, JHEP 06 (2023) 028 [arXiv:2303.08852] [INSPIRE].
Acknowledgments
I express my gratitude to B. Basso and S. Derkachov for inspiring discussions on related topics. I thank F. Aprile and V. Kazakov for useful comments about the draft. Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. I would like to thank Laboratoire de Physique - ENS of Paris for the kind hospitality in February 2023, where I held fruitful discussions about this work. Part of this work was carried out during the authors’ stay at the NCCR SwissMAP workshop ‘Integrability in Condensed Matter Physics and QFT’ (3rd to 12th of February 2023) which took place at the SwissMAP Research Station. The authors would like to thank the Swiss National Science Foundation, which funds SwissMAP (grant number 205607) and, in addition, supported the event via the grant IZSEZ0_215085. This work was additionally supported by a grant from the Simons Foundation (Simons Collaboration on the Nonperturbative Bootstrap #488661) and ICTP-SAIFR FAPESP grant 2016/01343-7 and FAPESP grant 2019/24277-8.
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: 2306.04503
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
Olivucci, E. Hexagonalization of Fishnet integrals. Part II. Overlaps and multi-point correlators. J. High Energ. Phys. 2024, 81 (2024). https://doi.org/10.1007/JHEP01(2024)081
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2024)081