Abstract
We present new closed-form expressions for 4-point scalar conformal blocks in the s- and t-channel lightcone expansions. Our formulae apply to intermediate operators of arbitrary spin in general dimensions. For physical spin ℓ, they are composed of at most (ℓ + 1) Gaussian hypergeometric functions at each order.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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 [Sov. Phys. JETP 39 (1974) 9] [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].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, Germany (1997).
S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, Covariant expansion of the conformal four-point function, Nucl. Phys. B 49 (1972) 77 [Erratum ibid. B 53 (1973) 643] [INSPIRE].
S. Ferrara, A.F. Grillo, R. Gatto and G. Parisi, Analyticity properties and asymptotic expansions of conformal covariant Green’s functions, Nuovo Cim. A 19 (1974) 667 [INSPIRE].
S. Ferrara, R. Gatto and A.F. Grillo, Properties of Partial Wave Amplitudes in Conformal Invariant Field Theories, Nuovo Cim. A 26 (1975) 226 [INSPIRE].
V.K. Dobrev, V.B. Petkova, S.G. Petrova and I.T. Todorov, Dynamical derivation of vacuum operator product expansion in Euclidean conformal quantum field theory, Phys. Rev. D 13 (1976) 887 [INSPIRE].
K. Lang and W. Rühl, The critical O(N) σ-model at dimensions 2 < d < 4: fusion coefficients and anomalous dimensions, Nucl. Phys. B 400 (1993) 597 [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.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].
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].
S. El-Showk et al., Solving the 3d Ising model with the conformal bootstrap II. c-minimization and precise critical exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision islands in the Ising and O(N) models, JHEP 08 (2016) 036 [arXiv:1603.04436] [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].
J.D. Qualls, Lectures on conformal field theory, arXiv:1511.04074 [INSPIRE].
S. Rychkov, EPFL lectures on conformal field theory in D ≥ 3 dimensions, arXiv:1601.05000.
D. Simmons-Duffin, The conformal bootstrap, arXiv:1602.07982 [INSPIRE].
S.M. Chester, Weizmann lectures on the numerical conformal bootstrap, arXiv:1907.05147 [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
J. Penedones, E. Trevisani and M. Yamazaki, Recursion relations for conformal blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [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].
M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
D. Simmons-Duffin, Projectors, shadows and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and D. Poland, Conformal blocks in the large D limit, JHEP 08 (2013) 107 [arXiv:1305.0004] [INSPIRE].
M. Hogervorst, H. Osborn and S. Rychkov, Diagonal limit for conformal blocks in d dimensions, JHEP 08 (2013) 014 [arXiv:1305.1321] [INSPIRE].
M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP 02 (2015) 151 [arXiv:1411.7351] [INSPIRE].
A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Deconstructing conformal blocks in 4D CFT, JHEP 08 (2015) 101 [arXiv:1505.03750] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
F. Rejon-Barrera and D. Robbins, Scalar-vector bootstrap, JHEP 01 (2016) 139 [arXiv:1508.02676] [INSPIRE].
L. Iliesiu et al., Fermion-scalar conformal blocks, JHEP 04 (2016) 074 [arXiv:1511.01497] [INSPIRE].
A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Seed conformal blocks in 4D CFT, JHEP 02 (2016) 183 [arXiv:1601.05325] [INSPIRE].
M. Isachenkov and V. Schomerus, Superintegrability of d-dimensional conformal blocks, Phys. Rev. Lett. 117 (2016) 071602 [arXiv:1602.01858] [INSPIRE].
J.-F. Fortin and W. Skiba, Conformal bootstrap in embedding space, Phys. Rev. D 93 (2016) 105047 [arXiv:1602.05794] [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP 07 (2016) 018 [arXiv:1603.05551] [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].
M. Hogervorst, Dimensional reduction for conformal blocks, JHEP 09 (2016) 017 [arXiv:1604.08913] [INSPIRE].
H.-Y. Chen and J.D. Qualls, Quantum integrable systems from conformal blocks, Phys. Rev. D 95 (2017) 106011 [arXiv:1605.05105] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
V. Schomerus and E. Sobko, From spinning conformal blocks to matrix Calogero-Sutherland models, JHEP 04 (2018) 052 [arXiv:1711.02022] [INSPIRE].
M. Isachenkov and V. Schomerus, Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory, JHEP 07 (2018) 180 [arXiv:1711.06609] [INSPIRE].
P. Kravchuk, Casimir recursion relations for general conformal blocks, JHEP 02 (2018) 011 [arXiv:1709.05347] [INSPIRE].
J.-F. Fortin and W. Skiba, A recipe for conformal blocks, arXiv:1905.00036 [INSPIRE].
W. Li, Closed-form expression for cross-channel conformal blocks near the lightcone, JHEP 01 (2020) 055 [arXiv:1906.00707] [INSPIRE].
R.S. Erramilli, L.V. Iliesiu and P. Kravchuk, Recursion relation for general 3d blocks, JHEP 12 (2019) 116 [arXiv:1907.11247] [INSPIRE].
I. Burić, V. Schomerus and M. Isachenkov, Conformal group theory of tensor structures, arXiv:1910.08099 [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].
P. Dey, A. Kaviraj and A. Sinha, Mellin space bootstrap for global symmetry, JHEP 07 (2017) 019 [arXiv:1612.05032] [INSPIRE].
P. Dey and A. Kaviraj, Towards a Bootstrap approach to higher orders of ϵ-expansion, JHEP 02 (2018) 153 [arXiv:1711.01173] [INSPIRE].
R. Gopakumar and A. Sinha, On the Polyakov-Mellin bootstrap, JHEP 12 (2018) 040 [arXiv:1809.10975] [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].
M.F. Paulos, Analytic functional bootstrap for CFTs in d > 1, JHEP 04 (2020) 093 [arXiv:1910.08563] [INSPIRE].
D. Mazáč, L. Rastelli and X. Zhou, A basis of analytic functionals for CFTs in general dimension, arXiv:1910.12855 [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].
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].
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [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].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP 11 (2007) 019 [arXiv:0708.0672] [INSPIRE].
G. Vos, Generalized additivity in unitary conformal field theories, Nucl. Phys. B 899 (2015) 91 [arXiv:1411.7941] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin, JHEP 11 (2015) 083 [arXiv:1502.01437] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Universal anomalous dimensions at large spin and large twist, JHEP 07 (2015) 026 [arXiv:1504.00772] [INSPIRE].
L.F. Alday and A. Zhiboedov, Conformal bootstrap with slightly broken higher spin symmetry, JHEP 06 (2016) 091 [arXiv:1506.04659] [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 and A. Bissi, Crossing symmetry and Higher spin towers, JHEP 12 (2017) 118 [arXiv:1603.05150] [INSPIRE].
L.F. Alday, Large spin perturbation theory for conformal field theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
L.F. Alday, Solving CFTs with weakly broken higher spin symmetry, JHEP 10 (2017) 161 [arXiv:1612.00696] [INSPIRE].
D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
P. Dey, K. Ghosh and A. Sinha, Simplifying large spin bootstrap in Mellin space, JHEP 01 (2018) 152 [arXiv:1709.06110] [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].
C. Cardona and K. Sen, Anomalous dimensions at finite conformal spin from OPE inversion, JHEP 11 (2018) 052 [arXiv:1806.10919] [INSPIRE].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
C. Sleight and M. Taronna, Anomalous dimensions from crossing kernels, JHEP 11 (2018) 089 [arXiv:1807.05941] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102 [arXiv:1805.00098] [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].
C. Cardona, S. Guha, S.K. KaNuMIlli and K. Sen, Resummation at finite conformal spin, JHEP 01 (2019) 077 [arXiv:1811.00213] [INSPIRE].
S. Albayrak, D. Meltzer and D. Poland, More analytic bootstrap: nonperturbative effects and Fermions, JHEP 08 (2019) 040 [arXiv:1904.00032] [INSPIRE].
G.J. Turiaci and A. Zhiboedov, Veneziano amplitude of Vasiliev theory, JHEP 10 (2018) 034 [arXiv:1802.04390] [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].
J. Henriksson and M. Van Loon, Critical O(N) model to order ϵ4 from analytic bootstrap, J. Phys. A 52 (2019) 025401 [arXiv:1801.03512] [INSPIRE].
L.F. Alday, J. Henriksson and M. van Loon, An alternative to diagrams for the critical O(N) model: dimensions and structure constants to order 1/N2 , JHEP 01 (2020) 063 [arXiv:1907.02445] [INSPIRE].
D. Carmi and S. Caron-Huot, A conformal dispersion relation: correlations from absorption, arXiv:1910.12123 [INSPIRE].
A. Bissi, P. Dey and T. Hansen, Dispersion relation for CFT four-point functions, JHEP 04 (2020) 092 [arXiv:1910.04661] [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].
H.-Y. Chen and H. Kyono, On conformal blocks, crossing kernels and multi-variable hypergeometric functions, JHEP 10 (2019) 149 [arXiv:1906.03135] [INSPIRE].
W. Li, Inverse bootstrapping conformal field theories, JHEP 01 (2018) 077 [arXiv:1706.04054] [INSPIRE].
W. Li, New method for the conformal bootstrap with OPE truncations, arXiv:1711.09075 [INSPIRE].
D. Li, D. Meltzer and D. Poland, Conformal collider physics from the lightcone bootstrap, JHEP 02 (2016) 143 [arXiv:1511.08025] [INSPIRE].
D.M. Hofman, D. Li, D. Meltzer, D. Poland and F. Rejon-Barrera, A proof of the conformal collider bounds, JHEP 06 (2016) 111 [arXiv:1603.03771] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1912.01168
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Li, W. Lightcone expansions of conformal blocks in closed form. J. High Energ. Phys. 2020, 105 (2020). https://doi.org/10.1007/JHEP06(2020)105
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2020)105