Abstract
In this paper we study a wide class of planar single-trace four point correlators in the chiral conformal field theory (χCFT4) arising as a double scaling limit of the γ-deformed \( \mathcal{N} \) = 4 SYM theory. In the planar (t’Hooft) limit, each of such correlators is described by a single Feynman integral having the bulk topology of a square lattice “fishnet” and/or of an honeycomb lattice of Yukawa vertices. The computation of this class of Feynmann integrals at any loop is achieved by means of an exactly-solvable spin chain magnet with SO(1, 5) symmetry. In this paper we explain in detail the solution of the magnet model as presented in our recent letter and we obtain a general formula for the representation of the Feynman integrals over the spectrum of the separated variables of the magnet, for any number of scalar and fermionic fields in the corresponding correlator. For the particular choice of scalar fields only, our formula reproduces the conjecture of B. Basso and L. Dixon for the fishnet integrals.
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B. Basso and L.J. Dixon, Gluing Ladder Feynman Diagrams into Fishnets, Phys. Rev. Lett. 119 (2017) 071601 [arXiv:1705.03545] [INSPIRE].
N.I. Usyukina and A.I. Davydychev, Exact results for three and four point ladder diagrams with an arbitrary number of rungs, Phys. Lett. B 305 (1993) 136 [INSPIRE].
D.J. Broadhurst, Summation of an infinite series of ladder diagrams, Phys. Lett. B 307 (1993) 132 [INSPIRE].
A.P. Isaev, Multiloop Feynman integrals and conformal quantum mechanics, Nucl. Phys. B 662 (2003) 461 [hep-th/0303056] [INSPIRE].
O. 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].
S. Derkachov and E. Olivucci, Exactly solvable magnet of conformal spins in four dimensions, Phys. Rev. Lett. 125 (2020) 031603 [arXiv:1912.07588] [INSPIRE].
E.K. Sklyanin, Separation of variables in the Gaudin model, J. Sov. Math. 47 (1989) 2473 [INSPIRE].
E.K. Sklyanin, Quantum inverse scattering method. Selected topics, hep-th/9211111 [INSPIRE].
E.K. Sklyanin, Separation of variables — new trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [INSPIRE].
E.K. Sklyanin, Separation of variables in the quantum integrable models related to the Yangian Y[sl(3)], J. Math. Sci. 80 (1996) 1861 [hep-th/9212076] [INSPIRE].
V.O. Tarasov, L.A. Takhtajan and L.D. Faddeev, Local Hamiltonians for integrable quantum models on a lattice, Theor. Math. Phys. 57 (1983) 1059 [INSPIRE].
L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Les Houches School of Physics: Astrophysical Sources of Gravitational Radiation, pp. 149–219 (1996) [hep-th/9605187] [INSPIRE].
L.N. Lipatov, Integrability of scattering amplitudes in N = 4 SUSY, J. Phys. A 42 (2009) 304020 [arXiv:0902.1444] [INSPIRE].
J. Bartels, L.N. Lipatov and A. Prygarin, Integrable spin chains and scattering amplitudes, J. Phys. A 44 (2011) 454013 [arXiv:1104.0816] [INSPIRE].
L.N. Lipatov, High-energy asymptotics of multicolor QCD and two-dimensional conformal field theories, Phys. Lett. B 309 (1993) 394 [INSPIRE].
L.N. Lipatov, Asymptotic behavior of multicolor QCD at high energies in connection with exactly solvable spin models, JETP Lett. 59 (1994) 596 [hep-th/9311037] [INSPIRE].
L.D. Faddeev and G.P. Korchemsky, High-energy QCD as a completely integrable model, Phys. Lett. B 342 (1995) 311 [hep-th/9404173] [INSPIRE].
S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Noncompact Heisenberg spin magnets from high-energy QCD: 1. Baxter Q operator and separation of variables, Nucl. Phys. B 617 (2001) 375 [hep-th/0107193] [INSPIRE].
M. D’Eramo, G. Parisi and L. Peliti, Theoretical predictions for critical exponents at the λ 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].
J. Caetano, O. 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].
N. Gromov, V. Kazakov, G. Korchemsky, S. Negro and G. Sizov, Integrability of Conformal Fishnet Theory, JHEP 01 (2018) 095 [arXiv:1706.04167] [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, V. Kazakov and G. Korchemsky, Exact Correlation Functions in Conformal Fishnet Theory, JHEP 08 (2019) 123 [arXiv:1808.02688] [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, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [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, JHEP 01 (2017) 130 [arXiv:1611.05577] [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions II: Two-Particle Contributions, JHEP 02 (2018) 177 [arXiv:1711.05327] [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].
I. Kostov, V.B. Petkova and D. Serban, Determinant Formula for the Octagon Form Factor in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 122 (2019) 231601 [arXiv:1903.05038] [INSPIRE].
T. Fleury and V. Goncalves, Decagon at Two Loops, JHEP 07 (2020) 030 [arXiv:2004.10867] [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].
B. Basso, J. Caetano and T. Fleury, Hexagons and Correlators in the Fishnet Theory, JHEP 11 (2019) 172 [arXiv:1812.09794] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
V. Kazakov, Quantum Spectral Curve of γ-twisted \( \mathcal{N} \) = 4 SYM theory and fishnet CFT, Rev. Math. Phys. 30 (2018) 1840010 [arXiv:1802.02160] [INSPIRE].
F. Levkovich-Maslyuk and M. Preti, Exploring the ground state spectrum of γ-deformed N = 4 SYM, arXiv:2003.05811 [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].
V. Kazakov and E. Olivucci, Biscalar Integrable Conformal Field Theories in Any Dimension, Phys. Rev. Lett. 121 (2018) 131601 [arXiv:1801.09844] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Relativistic Factorized S Matrix in Two-Dimensions Having O(N) Isotopic Symmetry, JETP Lett. 26 (1977) 457 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models, Annals Phys. 120 (1979) 253 [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].
A.N. Vasilev, The field theoretic renormalization group in critical behavior theory and stochastic dynamics, Chapman and Hall/CRC (2004) [INSPIRE].
L.N. Lipatov, High energy scattering in QCD, Surv. High Energy Phys. 10 (1997) 327.
L.N. Lipatov, Integrability properties of high energy dynamics in the multi-color QCD, Phys. Usp. 47 (2004) 325 [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].
E.S. Fradkin and M.Y. Palchik, Recent Developments in Conformal Invariant Quantum Field Theory, Phys. Rept. 44 (1978) 249 [INSPIRE].
M. Preti, STR: a Mathematica package for the method of uniqueness, Int. J. Mod. Phys. C 31 (2020) 2050146 [arXiv:1811.04935] [INSPIRE].
M. Preti, The Game of Triangles, J. Phys. Conf. Ser. 1525 (2020) 012015 [arXiv:1905.07380] [INSPIRE].
A.V. Kotikov, The Gegenbauer polynomial technique: The Evaluation of a class of Feynman diagrams, Phys. Lett. B 375 (1996) 240 [hep-ph/9512270] [INSPIRE].
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, New Approach to Evaluation of Multiloop Feynman Integrals: The Gegenbauer Polynomial x Space Technique, Nucl. Phys. B 174 (1980) 345 [INSPIRE].
A. Perelomov, Generalized coherent states and some of their applications, Springer (1986) [INSPIRE].
L.D. Faddeev and A.A. Slavnov, Gauge fields. Introduction to quantum theory, Front. Phys. 50 (1980) 1 [INSPIRE].
A.P. Isaev and V.A. Rubakov, Theory of Groups and Symmetries, WSP (2018) [INSPIRE].
S.E. Derkachov and A.N. Manashov, Iterative construction of eigenfunctions of the monodromy matrix for SL(2, ℂ) magnet, J. Phys. A 47 (2014) 305204 [arXiv:1401.7477] [INSPIRE].
R.J. Baxter, Exactly solved models in statistical mechanics, Elsevier (1982) [INSPIRE].
L.A. Takhtajan and L.D. Faddeev, The Quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surveys 34 (1979) 11 [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].
P.P. Kulish and E.K. Sklyanin, Quantum spectral transform method recent developments, in Integrable Quantum Field Theories, J. Hietarinta and C. Montonen eds., pp. 61–119, Springer Berlin Heidelberg (1982) [INSPIRE].
A.B. Zamolodchikov, ‘Fishnet’ diagrams as a completely integrable system, Phys. Lett. B 97 (1980) 63 [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, Non-conformality of γi-deformed N = 4 SYM theory, J. Phys. A 47 (2014) 455401 [arXiv:1308.4420] [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
S. Derkachov, V. Kazakov and E. Olivucci, Basso-Dixon Correlators in Two-Dimensional Fishnet CFT, JHEP 04 (2019) 032 [arXiv:1811.10623] [INSPIRE].
A. Pittelli and M. Preti, Integrable fishnet from γ-deformed \( \mathcal{N} \) = 2 quivers, Phys. Lett. B 798 (2019) 134971 [arXiv:1906.03680] [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
D.I. Kazakov, Multiloop Calculations: Method of Uniqueness and Functional Equations, Theor. Math. Phys. 62 (1985) 84 [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, Quantum fishchain in AdS5 , JHEP 10 (2019) 085 [arXiv:1907.01001] [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].
J.M. Maillet and G. Niccoli, On quantum separation of variables, J. Math. Phys. 59 (2018) 091417 [arXiv:1807.11572] [INSPIRE].
J.M. Maillet and G. Niccoli, On quantum separation of variables beyond fundamental representations, arXiv:1903.06618 [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains, JHEP 09 (2017) 111 [arXiv:1610.08032] [INSPIRE].
A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Separation of variables and scalar products at any rank, JHEP 09 (2019) 052 [arXiv:1907.03788] [INSPIRE].
P. Ryan and D. Volin, Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame, J. Math. Phys. 60 (2019) 032701 [arXiv:1810.10996] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk, P. Ryan and D. Volin, Dual Separated Variables and Scalar Products, Phys. Lett. B 806 (2020) 135494 [arXiv:1910.13442] [INSPIRE].
P. Ryan and D. Volin, Separation of variables for rational gl(n) spin chains in any compact representation, via fusion, embedding morphism and Backlund flow, arXiv:2002.12341 [INSPIRE].
Y. Jiang, S. Komatsu and E. Vescovi, Structure constants in \( \mathcal{N} \) = 4 SYM at finite coupling as worldsheet g-function, JHEP 07 (2020) 037 [arXiv:1906.07733] [INSPIRE].
E.K. Sklyanin, Classical limits of the SU(2)-invariant solutions of the Yang-Baxter equation, J. Sov. Math. 40 (1988) 93.
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Derkachov, S., Olivucci, E. Exactly solvable single-trace four point correlators in χCFT4. J. High Energ. Phys. 2021, 146 (2021). https://doi.org/10.1007/JHEP02(2021)146
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DOI: https://doi.org/10.1007/JHEP02(2021)146