Abstract
We introduce analytic functionals which act on the crossing equation for CFTs in arbitrary spacetime dimension. The functionals fully probe the constraints of crossing symmetry on the first sheet, and are in particular sensitive to the OPE, (double) lightcone and Regge limits. Compatibility with the crossing equation imposes constraints on the functional kernels which we study in detail. We then introduce two simple classes of functionals. The first class has a simple action on generalized free fields and their deformations and can be used to bootstrap AdS contact interactions in general dimension. The second class is obtained by tensoring holomorphic and antiholomorphic copies of d = 1 functionals which have been considered recently. They are dual to simple solutions to crossing in d = 2 which include the energy correlator of the Ising model. We show how these functionals lead to optimal bounds on the OPE density of d = 2 CFTs and argue that they provide an equivalent rewriting of the d = 2 crossing equation which is better suited for numeric computations than current approaches.
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References
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP08 (2011) 130 [arXiv:0902.2790] [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].
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].
B. Mukhametzhanov and A. Zhiboedov, Modular invariance, tauberian theorems and microcanonical entropy, JHEP10 (2019) 261 [arXiv:1904.06359] [INSPIRE].
B. Mukhametzhanov and A. Zhiboedov, Analytic Euclidean Bootstrap, JHEP10 (2019) 270 [arXiv:1808.03212] [INSPIRE].
J. Qiao and S. Rychkov, A tauberian theorem for the conformal bootstrap, JHEP12 (2017) 119 [arXiv:1709.00008] [INSPIRE].
S. Pal and Z. Sun, Tauberian-Cardy formula with spin, JHEP01 (2020) 135 [arXiv:1910.07727] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, JHEP05 (2019) 212 [arXiv:1811.05710] [INSPIRE].
L.F. Alday and A. Zhiboedov, An Algebraic Approach to the Analytic Bootstrap, JHEP04 (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, JHEP09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz.66 (1974) 23 [INSPIRE].
K. Sen and A. Sinha, On critical exponents without Feynman diagrams, J. Phys.A 49 (2016) 445401 [arXiv:1510.07770] [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, JHEP05 (2017) 027 [arXiv:1611.08407] [INSPIRE].
R. Gopakumar and A. Sinha, On the Polyakov-Mellin bootstrap, JHEP12 (2018) 040 [arXiv:1809.10975] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
D. Mazac, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, JHEP04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, JHEP02 (2019) 162 [arXiv:1803.10233] [INSPIRE].
A. Kaviraj and M.F. Paulos, The Functional Bootstrap for Boundary CFT, arXiv:1812.04034 [INSPIRE].
D. Mazáč, A Crossing-Symmetric OPE Inversion Formula, JHEP06 (2019) 082 [arXiv:1812.02254] [INSPIRE].
D. Mazáč, L. Rastelli and X. Zhou, An analytic approach to BC F Td , JHEP12 (2019) 004 [arXiv:1812.09314] [INSPIRE].
M.F. Paulos and B. Zan, A functional approach to the numerical conformal bootstrap, arXiv:1904.03193 [INSPIRE].
T. Hartman, D. Mazáč and L. Rastelli, Sphere Packing and Quantum Gravity, JHEP12 (2019) 048 [arXiv:1905.01319] [INSPIRE].
S. El-Showk and M.F. Paulos, Bootstrapping Conformal Field Theories with the Extremal Functional Method, Phys. Rev. Lett.111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].
F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett.111 (2013) 161602 [arXiv:1307.3111] [INSPIRE].
S. El-Showk and M.F. Paulos, Extremal bootstrapping: go with the flow, JHEP03 (2018) 148 [arXiv:1605.08087] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [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 four point functions and the operator product expansion, Nucl. Phys.B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N ) vector models, JHEP06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-matrix, JHEP10 (2012) 032 [arXiv:1112.4845] [INSPIRE].
J. Qiao and S. Rychkov, Cut-touching linear functionals in the conformal bootstrap, JHEP06 (2017) 076 [arXiv:1705.01357] [INSPIRE].
Y. Kusuki, Light Cone Bootstrap in General 2D CFTs and Entanglement from Light Cone Singularity, JHEP01 (2019) 025 [arXiv:1810.01335] [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev.D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
I. Heemskerk and J. Sully, More Holography from Conformal Field Theory, JHEP09 (2010) 099 [arXiv:1006.0976] [INSPIRE].
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Paulos, M.F. Analytic functional bootstrap for CFTs in d > 1. J. High Energ. Phys. 2020, 93 (2020). https://doi.org/10.1007/JHEP04(2020)093
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DOI: https://doi.org/10.1007/JHEP04(2020)093