Abstract
We study five-point correlation functions of scalar operators in d-dimensional conformal field theories. We develop a new approach to computing the five-point conformal blocks for exchanged primary operators of arbitrary spin by introducing a generalization of radial coordinates, using an appropriate ansatz, and perturbatively solving two quadratic Casimir differential equations. We then study five-point correlators 〈σσϵσσ〉 in the critical 3d Ising model. We truncate the operator product expansions (OPEs) in the correlator by including a finite number of primary operators with conformal dimension below a cutoff ∆ ⩽ ∆cutoff. We then compute several OPE coefficients involving ϵ and two spinning operators by demanding that the truncated correlator approximately satisfies the crossing relation.
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References
S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
S. El-Showk et al., Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D Fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D Fermions with Global Symmetries, JHEP 01 (2018) 036 [arXiv:1705.03484] [INSPIRE].
A. Dymarsky, J. Penedones, E. Trevisani and A. Vichi, Charting the space of 3D CFTs with a continuous global symmetry, JHEP 05 (2019) 098 [arXiv:1705.04278] [INSPIRE].
A. Dymarsky et al., The 3d Stress-Tensor Bootstrap, JHEP 02 (2018) 164 [arXiv:1708.05718] [INSPIRE].
D. Karateev, P. Kravchuk, M. Serone and A. Vichi, Fermion Conformal Bootstrap in 4d, JHEP 06 (2019) 088 [arXiv:1902.05969] [INSPIRE].
M. Reehorst, E. Trevisani and A. Vichi, Mixed Scalar-Current bootstrap in three dimensions, JHEP 12 (2020) 156 [arXiv:1911.05747] [INSPIRE].
R.S. Erramilli et al., blocks_3d: software for general 3d conformal blocks, JHEP 11 (2021) 006 [arXiv:2011.01959] [INSPIRE].
R.S. Erramilli et al., The Gross-Neveu-Yukawa archipelago, JHEP 02 (2023) 036 [arXiv:2210.02492] [INSPIRE].
Y.-C. He, J. Rong, N. Su and A. Vichi, Non-Abelian currents bootstrap, arXiv:2302.11585 [INSPIRE].
V. Rosenhaus, Multipoint Conformal Blocks in the Comb Channel, JHEP 02 (2019) 142 [arXiv:1810.03244] [INSPIRE].
S. Parikh, Holographic dual of the five-point conformal block, JHEP 05 (2019) 051 [arXiv:1901.01267] [INSPIRE].
S. Parikh, A multipoint conformal block chain in d dimensions, JHEP 05 (2020) 120 [arXiv:1911.09190] [INSPIRE].
S. Hoback and S. Parikh, Towards Feynman rules for conformal blocks, JHEP 01 (2021) 005 [arXiv:2006.14736] [INSPIRE].
J.-F. Fortin et al., Feynman rules for scalar conformal blocks, JHEP 10 (2022) 097 [arXiv:2204.08909] [INSPIRE].
S. Hoback and S. Parikh, Dimensional reduction of higher-point conformal blocks, JHEP 03 (2021) 187 [arXiv:2009.12904] [INSPIRE].
A. Pal and K. Ray, Conformal Correlation functions in four dimensions from Quaternionic Lauricella system, Nucl. Phys. B 968 (2021) 115433 [arXiv:2005.12523] [INSPIRE].
I. Buric et al., From Gaudin Integrable Models to d-dimensional Multipoint Conformal Blocks, Phys. Rev. Lett. 126 (2021) 021602 [arXiv:2009.11882] [INSPIRE].
I. Buric et al., Gaudin models and multipoint conformal blocks: general theory, JHEP 10 (2021) 139 [arXiv:2105.00021] [INSPIRE].
I. Buric et al., Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D, JHEP 11 (2021) 182 [arXiv:2108.00023] [INSPIRE].
I. Buric et al., Gaudin models and multipoint conformal blocks III: comb channel coordinates and OPE factorisation, JHEP 06 (2022) 144 [arXiv:2112.10827] [INSPIRE].
W. Skiba and J.-F. Fortin, A Recipe for Conformal Blocks, LHEP 2022 (2022) 293 [arXiv:1905.00036] [INSPIRE].
J.-F. Fortin and W. Skiba, New methods for conformal correlation functions, JHEP 06 (2020) 028 [arXiv:1905.00434] [INSPIRE].
J.-F. Fortin, W. Ma and W. Skiba, Higher-Point Conformal Blocks in the Comb Channel, JHEP 07 (2020) 213 [arXiv:1911.11046] [INSPIRE].
J.-F. Fortin, W.-J. Ma, V. Prilepina and W. Skiba, Efficient rules for all conformal blocks, JHEP 11 (2021) 052 [arXiv:2002.09007] [INSPIRE].
J.-F. Fortin, W.-J. Ma and W. Skiba, Six-point conformal blocks in the snowflake channel, JHEP 11 (2020) 147 [arXiv:2004.02824] [INSPIRE].
J.-F. Fortin, W.-J. Ma and W. Skiba, Seven-point conformal blocks in the extended snowflake channel and beyond, Phys. Rev. D 102 (2020) 125007 [arXiv:2006.13964] [INSPIRE].
V. Gonçalves, R. Pereira and X. Zhou, 20′ Five-Point Function from AdS5 × S5 Supergravity, JHEP 10 (2019) 247 [arXiv:1906.05305] [INSPIRE].
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].
A. Antunes, M.S. Costa, V. Gonçalves and J.V. Boas, Lightcone bootstrap at higher points, JHEP 03 (2022) 139 [arXiv:2111.05453] [INSPIRE].
T. Anous, A. Belin, J. de Boer and D. Liska, OPE statistics from higher-point crossing, JHEP 06 (2022) 102 [arXiv:2112.09143] [INSPIRE].
A. Kaviraj, J.A. Mann, L. Quintavalle and V. Schomerus, Multipoint lightcone bootstrap from differential equations, JHEP 08 (2023) 011 [arXiv:2212.10578] [INSPIRE].
V. Gonçalves et al., Kaluza-Klein five-point functions from AdS5 × S5 supergravity, JHEP 08 (2023) 067 [arXiv:2302.01896] [INSPIRE].
D. Poland and V. Prilepina, Recursion relations for 5-point conformal blocks, JHEP 10 (2021) 160 [arXiv:2103.12092] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
A.L. Fitzpatrick and E. Katz, Snowmass White Paper: Hamiltonian Truncation, arXiv:2201.11696 [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Radial expansion for spinning conformal blocks, JHEP 07 (2016) 057 [arXiv:1603.05552] [INSPIRE].
F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett. 111 (2013) 161602 [arXiv:1307.3111] [INSPIRE].
F. Gliozzi and A. Rago, Critical exponents of the 3d Ising and related models from Conformal Bootstrap, JHEP 10 (2014) 042 [arXiv:1403.6003] [INSPIRE].
W. Li, New method for the conformal bootstrap with OPE truncations, arXiv:1711.09075 [INSPIRE].
G. Kántor, V. Niarchos and C. Papageorgakis, Solving Conformal Field Theories with Artificial Intelligence, Phys. Rev. Lett. 128 (2022) 041601 [arXiv:2108.08859] [INSPIRE].
G. Kántor, V. Niarchos and C. Papageorgakis, Conformal bootstrap with reinforcement learning, Phys. Rev. D 105 (2022) 025018 [arXiv:2108.09330] [INSPIRE].
A. Laio, U.L. Valenzuela and M. Serone, Monte Carlo approach to the conformal bootstrap, Phys. Rev. D 106 (2022) 025019 [arXiv:2206.05193] [INSPIRE].
G. Kántor, V. Niarchos, C. Papageorgakis and P. Richmond, 6D (2, 0) bootstrap with the soft-actor-critic algorithm, Phys. Rev. D 107 (2023) 025005 [arXiv:2209.02801] [INSPIRE].
D. Meltzer, Higher Spin ANEC and the Space of CFTs, JHEP 07 (2019) 001 [arXiv:1811.01913] [INSPIRE].
D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
J. Henriksson, The critical O(N) CFT: Methods and conformal data, Phys. Rept. 1002 (2023) 1 [arXiv:2201.09520] [INSPIRE].
C. Cordova, J. Maldacena and G.J. Turiaci, Bounds on OPE Coefficients from Interference Effects in the Conformal Collider, JHEP 11 (2017) 032 [arXiv:1710.03199] [INSPIRE].
L. Hu, Y.-C. He and W. Zhu, Operator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Spheres, Phys. Rev. Lett. 131 (2023) 031601 [arXiv:2303.08844] [INSPIRE].
W. Zhu et al., Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization, Phys. Rev. X 13 (2023) 021009 [arXiv:2210.13482] [INSPIRE].
Acknowledgments
We thank Alexandre Belin, Aleksandar Bukva, Rajeev Erramilli, Walter Goldberger, Yin-Chen He, Murat Koloğlu, Matthew Mitchell, Vasilis Niarchos, Costis Papageorgakis, Slava Rychkov, Witold Skiba, Ning Su, and Yuan Xin for discussions. The work of D.P. and P.T. is supported by U.S. DOE grant DE-SC00-17660 and Simons Foundation grant 488651 (Simons Collaboration on the Nonperturbative Bootstrap). The work of V.P. is supported by the Perimeter Institute for Theoretical Physics. Computations in this work were performed on the Yale Grace computing cluster, supported by the facilities and staff of the Yale University Faculty of Sciences High Performance Computing Center.
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Poland, D., Prilepina, V. & Tadić, P. The five-point bootstrap. J. High Energ. Phys. 2023, 153 (2023). https://doi.org/10.1007/JHEP10(2023)153
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DOI: https://doi.org/10.1007/JHEP10(2023)153