Abstract
We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition at infinity. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.
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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].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
S. M. Chester et al., Carving out OPE space and precise O(2) model critical exponents, JHEP 06 (2020) 142 [arXiv:1912.03324] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A. L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
L. F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].
L. F. Alday and A. Zhiboedov, An Algebraic Approach to the Analytic Bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].
L. F. Alday, Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Li, D. Meltzer and D. Poland, Conformal Bootstrap in the Regge Limit, JHEP 12 (2017) 013 [arXiv:1705.03453] [INSPIRE].
M. S. Costa, T. Hansen and J. Penedones, Bounds for OPE coefficients on the Regge trajectory, JHEP 10 (2017) 197 [arXiv:1707.07689] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, Shocks, Superconvergence, and a Stringy Equivalence Principle, JHEP 11 (2020) 096 [arXiv:1904.05905] [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].
B. Mukhametzhanov and A. Zhiboedov, Modular invariance, tauberian theorems and microcanonical entropy, JHEP 10 (2019) 261 [arXiv:1904.06359] [INSPIRE].
S. Ganguly and S. Pal, Bounds on the density of states and the spectral gap in CFT2, Phys. Rev. D 101 (2020) 106022 [arXiv:1905.12636] [INSPIRE].
S. Pal, Bound on asymptotics of magnitude of three point coefficients in 2D CFT, JHEP 01 (2020) 023 [arXiv:1906.11223] [INSPIRE].
D. Mazac, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
D. Mazac and M. F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, JHEP 02 (2019) 162 [arXiv:1803.10233] [INSPIRE].
D. Mazac and M. F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP 02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
D. Mazáč, A Crossing-Symmetric OPE Inversion Formula, JHEP 06 (2019) 082 [arXiv:1812.02254] [INSPIRE].
A. Kaviraj and M. F. Paulos, The Functional Bootstrap for Boundary CFT, JHEP 04 (2020) 135 [arXiv:1812.04034] [INSPIRE].
D. Mazáč, L. Rastelli and X. Zhou, An analytic approach to BCFTd, JHEP 12 (2019) 004 [arXiv:1812.09314] [INSPIRE].
T. Hartman, D. Mazáč and L. Rastelli, Sphere Packing and Quantum Gravity, JHEP 12 (2019) 048 [arXiv:1905.01319] [INSPIRE].
M. S. Viazovska, The sphere packing problem in dimension 8, Annals Math. 185 (2017) 991 [arXiv:1603.04246].
M. F. Paulos, Analytic functional bootstrap for CFTs in d > 1, JHEP 04 (2020) 093 [arXiv:1910.08563] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Dispersive CFT Sum Rules, JHEP 05 (2021) 243 [arXiv:2008.04931] [INSPIRE].
D. Carmi and S. Caron-Huot, A Conformal Dispersion Relation: Correlations from Absorption, JHEP 09 (2020) 009 [arXiv:1910.12123] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, AdS Bulk Locality from Sharp CFT Bounds, arXiv:2106.10274 [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, Conformal Bootstrap in Mellin Space, Phys. Rev. Lett. 118 (2017) 081601 [arXiv:1609.00572] [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP 05 (2017) 027 [arXiv:1611.08407] [INSPIRE].
R. Gopakumar and A. Sinha, On the Polyakov-Mellin bootstrap, JHEP 12 (2018) 040 [arXiv:1809.10975] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
G. Mack, D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [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].
J. Qiao and S. Rychkov, Cut-touching linear functionals in the conformal bootstrap, JHEP 06 (2017) 076 [arXiv:1705.01357] [INSPIRE].
M. S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
M. S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
C. Sleight and M. Taronna, Anomalous Dimensions from Crossing Kernels, JHEP 11 (2018) 089 [arXiv:1807.05941] [INSPIRE].
C. Sleight and M. Taronna, Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications, Fortsch. Phys. 66 (2018) 1800038 [arXiv:1804.09334] [INSPIRE].
X. Zhou, Recursion Relations in Witten Diagrams and Conformal Partial Waves, JHEP 05 (2019) 006 [arXiv:1812.01006] [INSPIRE].
L. F. Alday, Solving CFTs with Weakly Broken Higher Spin Symmetry, JHEP 10 (2017) 161 [arXiv:1612.00696] [INSPIRE].
S. Rychkov and Z. M. Tan, The ϵ-expansion from conformal field theory, J. Phys. A 48 (2015) 29FT01 [arXiv:1505.00963] [INSPIRE].
P. Basu and C. Krishnan, ϵ-expansions near three dimensions from conformal field theory, JHEP 11 (2015) 040 [arXiv:1506.06616] [INSPIRE].
F. Gliozzi, A. Guerrieri, A. C. Petkou and C. Wen, Generalized Wilson-Fisher Critical Points from the Conformal Operator Product Expansion, Phys. Rev. Lett. 118 (2017) 061601 [arXiv:1611.10344] [INSPIRE].
P. Dey, A. Kaviraj and A. Sinha, Mellin space bootstrap for global symmetry, JHEP 07 (2017) 019 [arXiv:1612.05032] [INSPIRE].
K. Roumpedakis, Leading Order Anomalous Dimensions at the Wilson-Fisher Fixed Point from CFT, JHEP 07 (2017) 109 [arXiv:1612.08115] [INSPIRE].
P. Liendo, Revisiting the dilatation operator of the Wilson-Fisher fixed point, Nucl. Phys. B 920 (2017) 368 [arXiv:1701.04830] [INSPIRE].
P. Dey, K. Ghosh and A. Sinha, Simplifying large spin bootstrap in Mellin space, JHEP 01 (2018) 152 [arXiv:1709.06110] [INSPIRE].
F. Gliozzi, Anomalous dimensions of spinning operators from conformal symmetry, JHEP 01 (2018) 113 [arXiv:1711.05530] [INSPIRE].
L. F. Alday, J. Henriksson and M. van Loon, Taming the ϵ-expansion with large spin perturbation theory, JHEP 07 (2018) 131 [arXiv:1712.02314] [INSPIRE].
L. Rastelli and X. Zhou, Mellin amplitudes for AdS5 × S5, Phys. Rev. Lett. 118 (2017) 091602 [arXiv:1608.06624] [INSPIRE].
L. F. Alday and A. Bissi, Loop Corrections to Supergravity on AdS5 × S5, Phys. Rev. Lett. 119 (2017) 171601 [arXiv:1706.02388] [INSPIRE].
F. Aprile, J. M. Drummond, P. Heslop and H. Paul, Quantum Gravity from Conformal Field Theory, JHEP 01 (2018) 035 [arXiv:1706.02822] [INSPIRE].
F. Aprile, J. M. Drummond, P. Heslop and H. Paul, Unmixing Supergravity, JHEP 02 (2018) 133 [arXiv:1706.08456] [INSPIRE].
L. Rastelli and X. Zhou, How to Succeed at Holographic Correlators Without Really Trying, JHEP 04 (2018) 014 [arXiv:1710.05923] [INSPIRE].
F. Aprile, J. M. Drummond, P. Heslop and H. Paul, Loop corrections for Kaluza-Klein AdS amplitudes, JHEP 05 (2018) 056 [arXiv:1711.03903] [INSPIRE].
L. F. Alday and S. Caron-Huot, Gravitational S-matrix from CFT dispersion relations, JHEP 12 (2018) 017 [arXiv:1711.02031] [INSPIRE].
F. Aprile, J. Drummond, P. Heslop and H. Paul, Double-trace spectrum of N = 4 supersymmetric Yang-Mills theory at strong coupling, Phys. Rev. D 98 (2018) 126008 [arXiv:1802.06889] [INSPIRE].
S. Caron-Huot and A.-K. Trinh, All tree-level correlators in AdS5 × S5 supergravity: hidden ten-dimensional conformal symmetry, JHEP 01 (2019) 196 [arXiv:1809.09173] [INSPIRE].
L. F. Alday, On genus-one string amplitudes on AdS5 × S5, JHEP 04 (2021) 005 [arXiv:1812.11783] [INSPIRE].
L. F. Alday, A. Bissi and E. Perlmutter, Genus-One String Amplitudes from Conformal Field Theory, JHEP 06 (2019) 010 [arXiv:1809.10670] [INSPIRE].
D. J. Binder, S. M. Chester, S. S. Pufu and Y. Wang, \( \mathcal{N} \) = 4 Super-Yang-Mills correlators at strong coupling from string theory and localization, JHEP 12 (2019) 119 [arXiv:1902.06263] [INSPIRE].
V. Gonçalves, R. Pereira and X. Zhou, 20′ Five-Point Function from AdS5 × S5 Supergravity, JHEP 10 (2019) 247 [arXiv:1906.05305] [INSPIRE].
J. M. Drummond, D. Nandan, H. Paul and K. S. Rigatos, String corrections to AdS amplitudes and the double-trace spectrum of \( \mathcal{N} \) = 4 SYM, JHEP 12 (2019) 173 [arXiv:1907.00992] [INSPIRE].
M. F. Paulos and B. Zan, A functional approach to the numerical conformal bootstrap, JHEP 09 (2020) 006 [arXiv:1904.03193] [INSPIRE].
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Mazáč, D., Rastelli, L. & Zhou, X. A basis of analytic functionals for CFTs in general dimension. J. High Energ. Phys. 2021, 140 (2021). https://doi.org/10.1007/JHEP08(2021)140
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DOI: https://doi.org/10.1007/JHEP08(2021)140