Abstract
We present new asymptotic formulas for the distribution of OPE coefficients in conformal field theories. These formulas involve products of four or more coefficients and include light-light-heavy as well as heavy-heavy-heavy contributions. They are derived from crossing symmetry of the six and higher point functions on the plane and should be interpreted as non-Gaussianities in the statistical distribution of the OPE coefficients. We begin with a formula for arbitrary operator exchanges (not necessarily primary) valid in any dimension. This is the first asymptotic formula constraining heavy-heavy-heavy OPE coefficients in d > 2. For two-dimensional CFTs, we present refined asymptotic formulas stemming from exchanges of quasi-primaries as well as Virasoro primaries.
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References
E. Shaghoulian, Modular forms and a generalized Cardy formula in higher dimensions, Phys. Rev. D 93 (2016) 126005 [arXiv:1508.02728] [INSPIRE].
A. Belin, J. de Boer, J. Kruthoff, B. Michel, E. Shaghoulian and M. Shyani, Universality of sparse d > 2 conformal field theory at large N, JHEP 03 (2017) 067 [arXiv:1610.06186] [INSPIRE].
E. Shaghoulian, Modular Invariance of Conformal Field Theory on S1 × S3 and Circle Fibrations, Phys. Rev. Lett. 119 (2017) 131601 [arXiv:1612.05257] [INSPIRE].
A. Belin, J. De Boer and J. Kruthoff, Comments on a state-operator correspondence for the torus, SciPost Phys. 5 (2018) 060 [arXiv:1802.00006] [INSPIRE].
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].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [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].
J.L. Cardy, Operator content and modular properties of higher dimensional conformal field theories, Nucl. Phys. B 366 (1991) 403 [INSPIRE].
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE Convergence in Conformal Field Theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].
P. Kraus and A. Maloney, A cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE].
D. Das, S. Datta and S. Pal, Charged structure constants from modularity, JHEP 11 (2017) 183 [arXiv:1706.04612] [INSPIRE].
J. Cardy, A. Maloney and H. Maxfield, A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance, JHEP 10 (2017) 136 [arXiv:1705.05855] [INSPIRE].
D. Das, S. Datta and S. Pal, Universal asymptotics of three-point coefficients from elliptic representation of Virasoro blocks, Phys. Rev. D 98 (2018) 101901 [arXiv:1712.01842] [INSPIRE].
J. Qiao and S. Rychkov, A tauberian theorem for the conformal bootstrap, JHEP 12 (2017) 119 [arXiv:1709.00008] [INSPIRE].
B. Mukhametzhanov and A. Zhiboedov, Analytic Euclidean Bootstrap, JHEP 10 (2019) 270 [arXiv:1808.03212] [INSPIRE].
S. Pal and Z. Sun, Tauberian-Cardy formula with spin, JHEP 01 (2020) 135 [arXiv:1910.07727] [INSPIRE].
Y. Gobeil, A. Maloney, G.S. Ng and J.-q. Wu, Thermal Conformal Blocks, SciPost Phys. 7 (2019) 015 [arXiv:1802.10537] [INSPIRE].
E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev. D 98 (2018) 126015 [arXiv:1804.07924] [INSPIRE].
A. Romero-Bermúdez, P. Sabella-Garnier and K. Schalm, A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled, JHEP 09 (2018) 005 [arXiv:1804.08899] [INSPIRE].
Y. Hikida, Y. Kusuki and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018) 026003 [arXiv:1804.09658] [INSPIRE].
S. Collier, A. Maloney, H. Maxfield and I. Tsiares, Universal dynamics of heavy operators in CFT2, JHEP 07 (2020) 074 [arXiv:1912.00222] [INSPIRE].
A. Belin, J. de Boer and D. Liska, Non-Gaussianities in the Statistical Distribution of Heavy OPE Coefficients and Wormholes, arXiv:2110.14649 [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. Goncalves and J.V. Boas, Lightcone bootstrap at higher points, JHEP 03 (2022) 139 [arXiv:2111.05453] [INSPIRE].
P. Saad, Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity, arXiv:1910.10311 [INSPIRE].
J. Pollack, M. Rozali, J. Sully and D. Wakeham, Eigenstate Thermalization and Disorder Averaging in Gravity, Phys. Rev. Lett. 125 (2020) 021601 [arXiv:2002.02971] [INSPIRE].
A. Belin and J. de Boer, Random statistics of OPE coefficients and Euclidean wormholes, Class. Quant. Grav. 38 (2021) 164001 [arXiv:2006.05499] [INSPIRE].
D. Stanford, More quantum noise from wormholes, arXiv:2008.08570 [INSPIRE].
A. Blommaert, Dissecting the ensemble in JT gravity, arXiv:2006.13971 [INSPIRE].
A. Belin, J. De Boer, P. Nayak and J. Sonner, Charged Eigenstate Thermalization, Euclidean Wormholes and Global Symmetries in Quantum Gravity, SciPost Phys. 12 (2022) 059 [arXiv:2012.07875] [INSPIRE].
A. Altland, D. Bagrets, P. Nayak, J. Sonner and M. Vielma, From operator statistics to wormholes, Phys. Rev. Res. 3 (2021) 033259 [arXiv:2105.12129] [INSPIRE].
B. Freivogel, D. Nikolakopoulou and A.F. Rotundo, Wormholes from Averaging over States, arXiv:2105.12771 [INSPIRE].
K. Goto, Y. Kusuki, K. Tamaoka and T. Ugajin, Product of random states and spatial (half-)wormholes, JHEP 10 (2021) 205 [arXiv:2108.08308] [INSPIRE].
A. Belin, J. de Boer, P. Nayak and J. Sonner, Generalized Spectral Form Factors and the Statistics of Heavy Operators, arXiv:2111.06373 [INSPIRE].
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.
L. Foini and J. Kurchan, Eigenstate thermalization hypothesis and out of time order correlators, Phys. Rev. E 99 (2019) 042139 [arXiv:1803.10658] [INSPIRE].
C. Murthy and M. Srednicki, Bounds on chaos from the eigenstate thermalization hypothesis, Phys. Rev. Lett. 123 (2019) 230606 [arXiv:1906.10808] [INSPIRE].
A. Chan, A. De Luca and J.T. Chalker, Eigenstate Correlations, Thermalization and the Butterfly Effect, Phys. Rev. Lett. 122 (2019) 220601 [arXiv:1810.11014] [INSPIRE].
A. Dymarsky, Bound on Eigenstate Thermalization from Transport, Phys. Rev. Lett. 128 (2022) 190601 [arXiv:1804.08626] [INSPIRE].
J. Richter, A. Dymarsky, R. Steinigeweg and J. Gemmer, Eigenstate thermalization hypothesis beyond standard indicators: Emergence of random-matrix behavior at small frequencies, Phys. Rev. E 102 (2020) 042127 [arXiv:2007.15070] [INSPIRE].
J. Wang, M.H. Lamann, J. Richter, R. Steinigeweg, A. Dymarsky and J. Gemmer, Eigenstate Thermalization Hypothesis and Its Deviations from Random-Matrix Theory beyond the Thermalization Time, Phys. Rev. Lett. 128 (2022) 180601 [arXiv:2110.04085] [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].
T. Anous and F.M. Haehl, On the Virasoro six-point identity block and chaos, JHEP 08 (2020) 002 [arXiv:2005.06440] [INSPIRE].
B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of Uq(sl(2, ℝ)), Commun. Math. Phys. 224 (2001) 613 [math/0007097] [INSPIRE].
B. Mukhametzhanov and A. Zhiboedov, Modular invariance, tauberian theorems and microcanonical entropy, JHEP 10 (2019) 261 [arXiv:1904.06359] [INSPIRE].
B. Mukhametzhanov and S. Pal, Beurling-Selberg Extremization and Modular Bootstrap at High Energies, SciPost Phys. 8 (2020) 088 [arXiv:2003.14316] [INSPIRE].
D. Das, Y. Kusuki and S. Pal, Universality in asymptotic bounds and its saturation in 2D CFT, JHEP 04 (2021) 288 [arXiv:2011.02482] [INSPIRE].
M. Abramowitz, I.A. Stegun and R.H. Romer, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Am. J. Phys. 56 (1988) 958.
A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 1088.
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE].
M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, JHEP 04 (2019) 018 [arXiv:1703.09805] [INSPIRE].
Y. Kusuki and M. Miyaji, Entanglement Entropy, OTOC and Bootstrap in 2D CFTs from Regge and Light Cone Limits of Multi-point Conformal Block, JHEP 08 (2019) 063 [arXiv:1905.02191] [INSPIRE].
G.W. Moore and N. Seiberg, Polynomial Equations for Rational Conformal Field Theories, Phys. Lett. B 212 (1988) 451 [INSPIRE].
G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].
Y. Kusuki, Light Cone Bootstrap in General 2D CFTs and Entanglement from Light Cone Singularity, JHEP 01 (2019) 025 [arXiv:1810.01335] [INSPIRE].
S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, JHEP 05 (2019) 212 [arXiv:1811.05710] [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].
I. Buric, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, 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, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, Gaudin models and multipoint conformal blocks: general theory, JHEP 10 (2021) 139 [arXiv:2105.00021] [INSPIRE].
J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin, d-dimensional SYK, AdS Loops, and 6j Symbols, JHEP 03 (2019) 052 [arXiv:1808.00612] [INSPIRE].
S. Caron-Huot, Y. Gobeil and Z. Zahraee, The leading trajectory in the 2 + 1D Ising CFT, arXiv:2007.11647 [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
F.M. Haehl, R. Loganayagam, P. Narayan, A.A. Nizami and M. Rangamani, Thermal out-of-time-order correlators, KMS relations, and spectral functions, JHEP 12 (2017) 154 [arXiv:1706.08956] [INSPIRE].
F.M. Haehl, R. Loganayagam, P. Narayan and M. Rangamani, Classification of out-of-time-order correlators, SciPost Phys. 6 (2019) 001 [arXiv:1701.02820] [INSPIRE].
F.M. Haehl and M. Rozali, Fine Grained Chaos in AdS2 Gravity, Phys. Rev. Lett. 120 (2018) 121601 [arXiv:1712.04963] [INSPIRE].
F.M. Haehl and M. Rozali, Effective Field Theory for Chaotic CFTs, JHEP 10 (2018) 118 [arXiv:1808.02898] [INSPIRE].
T. Anous and J. Sonner, Phases of scrambling in eigenstates, SciPost Phys. 7 (2019) 003 [arXiv:1903.03143] [INSPIRE].
P. Basu and K. Jaswin, Higher point OTOCs and the bound on chaos, arXiv:1809.05331 [INSPIRE].
S. Chaudhuri, C. Chowdhury and R. Loganayagam, Spectral Representation of Thermal OTO Correlators, JHEP 02 (2019) 018 [arXiv:1810.03118] [INSPIRE].
D.A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121 [arXiv:1610.04903] [INSPIRE].
A.M. García-García and V. Godet, Euclidean wormhole in the Sachdev-Ye-Kitaev model, Phys. Rev. D 103 (2021) 046014 [arXiv:2010.11633] [INSPIRE].
I. Esterlis, A.L. Fitzpatrick and D. Ramirez, Closure of the Operator Product Expansion in the Non-Unitary Bootstrap, JHEP 11 (2016) 030 [arXiv:1606.07458] [INSPIRE].
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Anous, T., Belin, A., de Boer, J. et al. OPE statistics from higher-point crossing. J. High Energ. Phys. 2022, 102 (2022). https://doi.org/10.1007/JHEP06(2022)102
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DOI: https://doi.org/10.1007/JHEP06(2022)102