Abstract
In the study of conformal field theories, conformal blocks in the lightcone limit are fundamental to the analytic conformal bootstrap method. Here we consider the lightcone limit of 4-point functions of generic scalar primaries. Based on the nonperturbative pole structure in spin of Lorentzian inversion, we propose the natural basis functions for cross-channel conformal blocks. In this new basis, we find a closed-form expression for crossed conformal blocks in terms of the Kampé de Fériet function, which applies to intermediate operators of arbitrary spin in general dimensions. We derive the general Lorentzian inversion for the case of identical external scaling dimensions. Our results for the lightcone limit also shed light on the complete analytic structure of conformal blocks in the lightcone expansion.
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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 [Sov. Phys. JETP39 (1974) 9] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer, New York, NY, U.S.A. (1997) [INSPIRE].
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. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising model with the conformal bootstrap, Phys. Rev.D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, 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, JHEP11 (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, JHEP08 (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].
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].
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP11 (2007) 019 [arXiv:0708.0672] [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].
L.F. Alday, Solving CFTs with weakly broken higher spin symmetry, JHEP10 (2017) 161 [arXiv:1612.00696] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin, JHEP11 (2015) 083 [arXiv:1502.01437] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP11 (2015) 101 [arXiv:1502.07707] [INSPIRE].
L.F. Alday and A. Zhiboedov, Conformal bootstrap with slightly broken higher spin symmetry, JHEP06 (2016) 091 [arXiv:1506.04659] [INSPIRE].
D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
P. Dey, K. Ghosh and A. Sinha, Simplifying large spin bootstrap in Mellin space, JHEP01 (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, JHEP11 (2018) 052 [arXiv:1806.10919] [INSPIRE].
J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin, d-dimensional SYK, AdS loops and 6j symbols, JHEP03 (2019) 052 [arXiv:1808.00612] [INSPIRE].
C. Cardona, S. Guha, S.K. KaNuMIlli and K. Sen, Resummation at finite conformal spin, JHEP01 (2019) 077 [arXiv:1811.00213] [INSPIRE].
C. Sleight and M. Taronna, Anomalous dimensions from crossing kernels, JHEP11 (2018) 089 [arXiv:1807.05941] [INSPIRE].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP09 (2017) 078 [arXiv:1703.00278] [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].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP12 (2012) 091 [arXiv:1209.4355] [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].
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].
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].
M. Hogervorst, H. Osborn and S. Rychkov, Diagonal limit for conformal blocks in d dimensions, JHEP08 (2013) 014 [arXiv:1305.1321] [INSPIRE].
L.F. Alday, J. Henriksson and M. van Loon, Taming the ϵ-expansion with large spin perturbation theory, JHEP07 (2018) 131 [arXiv:1712.02314] [INSPIRE].
J.A. Wilson, Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal.11 (1980) 690.
W. Groenevelt, The Wilson function transform, Int. Math. Res. Not.2003 (2003) 2779 [math.CA/0306424].
M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP11 (2017) 193 [arXiv:1702.08471] [INSPIRE].
S. Albayrak, D. Meltzer and D. Poland, More analytic bootstrap: nonperturbative effects and fermions, JHEP08 (2019) 040 [arXiv:1904.00032] [INSPIRE].
M. Hogervorst, Dimensional reduction for conformal blocks, JHEP09 (2016) 017 [arXiv:1604.08913] [INSPIRE].
J. Henriksson and M. Van Loon, Critical O(N) model to order ϵ 4from analytic bootstrap, J. Phys.A 52 (2019) 025401 [arXiv:1801.03512] [INSPIRE].
O. Aharony, L.F. Alday, A. Bissi and R. Yacoby, The analytic bootstrap for large N Chern-Simons vector models, JHEP08 (2018) 166 [arXiv:1805.04377] [INSPIRE].
M. Isachenkov and V. Schomerus, Superintegrability of d-dimensional conformal blocks, Phys. Rev. Lett.117 (2016) 071602 [arXiv:1602.01858] [INSPIRE].
M. Isachenkov and V. Schomerus, Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory, JHEP07 (2018) 180 [arXiv:1711.06609] [INSPIRE].
W. Li, Lightcone expansions of conformal blocks in closed form, arXiv:1912.01168 [INSPIRE].
G.J. Turiaci and A. Zhiboedov, Veneziano amplitude of Vasiliev theory, JHEP10 (2018) 034 [arXiv:1802.04390] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
J. Penedones, E. Trevisani and M. Yamazaki, Recursion relations for conformal blocks, JHEP09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
M. Hogervorst, S. Rychkov and B.C. van Rees, Unitarity violation at the Wilson-Fisher fixed point in 4 − ϵ dimensions, Phys. Rev.D 93 (2016) 125025 [arXiv:1512.00013] [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, JHEP10 (2014) 042 [arXiv:1403.6003] [INSPIRE].
W. Li, New method for the conformal bootstrap with OPE truncations, arXiv:1711.09075 [INSPIRE].
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Li, W. Closed-form expression for cross-channel conformal blocks near the lightcone. J. High Energ. Phys. 2020, 55 (2020). https://doi.org/10.1007/JHEP01(2020)055
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DOI: https://doi.org/10.1007/JHEP01(2020)055