Abstract
We conjecture a simple set of “Feynman rules” for constructing n-point global conformal blocks in any channel in d spacetime dimensions, for external and exchanged scalar operators for arbitrary n and d. The vertex factors are given in terms of Lauricella hypergeometric functions of one, two or three variables, and the Feynman rules furnish an explicit power-series expansion in powers of cross-ratios. These rules are conjectured based on previously known results in the literature, which include four-, five- and six-point examples as well as the n-point comb channel blocks. We prove these rules for all previously known cases, as well as two new ones: the seven-point block in a new topology, and all even-point blocks in the “OPE channel.” The proof relies on holographic methods, notably the Feynman rules for Mellin amplitudes of tree-level AdS diagrams in a scalar effective field theory, and is easily applicable to any particular choice of a conformal block beyond those considered in this paper.
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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. JETP 39 (1974) 9] [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].
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. Ferrara, A.F. Grillo and R. Gatto, Manifestly conformal covariant operator-product expansion, Lett. Nuovo Cim. 2 (1971) 1363 [INSPIRE].
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. 53 (1973) 643] [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].
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].
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].
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].
L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Fermion-scalar conformal blocks, JHEP 04 (2016) 074 [arXiv:1511.01497] [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.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].
P. Kravchuk, Casimir recursion relations for general conformal blocks, JHEP 02 (2018) 011 [arXiv:1709.05347] [INSPIRE].
X. Zhou, Recursion relations in Witten diagrams and conformal partial waves, JHEP 05 (2019) 006 [arXiv:1812.01006] [INSPIRE].
R.S. Erramilli, L.V. Iliesiu and P. Kravchuk, Recursion relation for general 3d blocks, JHEP 12 (2019) 116 [arXiv:1907.11247] [INSPIRE].
D. Simmons-Duffin, Projectors, shadows, and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [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].
A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Seed conformal blocks in 4D CFT, JHEP 02 (2016) 183 [arXiv:1601.05325] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
G.F. Cuomo, D. Karateev and P. Kravchuk, General bootstrap equations in 4D CFTs, JHEP 01 (2018) 130 [arXiv:1705.05401] [INSPIRE].
H. Isono, On conformal correlators and blocks with spinors in general dimensions, Phys. Rev. D 96 (2017) 065011 [arXiv:1706.02835] [INSPIRE].
J.-F. Fortin and W. Skiba, Conformal bootstrap in embedding space, Phys. Rev. D 93 (2016) 105047 [arXiv:1602.05794] [INSPIRE].
J.-F. Fortin and W. Skiba, A recipe for conformal blocks, arXiv:1905.00036 [INSPIRE].
J.-F. Fortin and W. Skiba, New methods for conformal correlation functions, JHEP 06 (2020) 028 [arXiv:1905.00434] [INSPIRE].
J.-F. Fortin, V. Prilepina and W. Skiba, Conformal four-point correlation functions from the operator product expansion, JHEP 08 (2020) 115 [arXiv:1907.10506] [INSPIRE].
C. Sleight and M. Taronna, Spinning Witten diagrams, JHEP 06 (2017) 100 [arXiv:1702.08619] [INSPIRE].
M.S. Costa and T. Hansen, AdS weight shifting operators, JHEP 09 (2018) 040 [arXiv:1805.01492] [INSPIRE].
M. Hogervorst, Dimensional reduction for conformal blocks, JHEP 09 (2016) 017 [arXiv:1604.08913] [INSPIRE].
A. Kaviraj, S. Rychkov and E. Trevisani, Random field Ising model and Parisi-Sourlas supersymmetry. Part I. Supersymmetric CFT, JHEP 04 (2020) 090 [arXiv:1912.01617] [INSPIRE].
M. Besken, A. Hegde, E. Hijano and P. Kraus, Holographic conformal blocks from interacting Wilson lines, JHEP 08 (2016) 099 [arXiv:1603.07317] [INSPIRE].
A. Bhatta, P. Raman and N.V. Suryanarayana, Holographic conformal partial waves as gravitational open Wilson networks, JHEP 06 (2016) 119 [arXiv:1602.02962] [INSPIRE].
A. Bhatta, P. Raman and N.V. Suryanarayana, Scalar blocks as gravitational Wilson networks, JHEP 12 (2018) 125 [arXiv:1806.05475] [INSPIRE].
M. Isachenkov and V. Schomerus, Superintegrability of d-dimensional conformal blocks, Phys. Rev. Lett. 117 (2016) 071602 [arXiv:1602.01858] [INSPIRE].
V. Schomerus, E. Sobko and M. Isachenkov, Harmony of spinning conformal blocks, JHEP 03 (2017) 085 [arXiv:1612.02479] [INSPIRE].
I. Buric, V. Schomerus and E. Sobko, Superconformal blocks: general theory, JHEP 01 (2020) 159 [arXiv:1904.04852] [INSPIRE].
I. Burić, M. Isachenkov and V. Schomerus, Conformal group theory of tensor structures, JHEP 10 (2020) 004 [arXiv:1910.08099] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten diagrams revisited: the AdS geometry of conformal blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
M. Nishida and K. Tamaoka, Geodesic Witten diagrams with an external spinning field, PTEP 2017 (2017) 053B06 [arXiv:1609.04563] [INSPIRE].
A. Castro, E. Llabrés and F. Rejon-Barrera, Geodesic diagrams, gravitational interactions & OPE structures, JHEP 06 (2017) 099 [arXiv:1702.06128] [INSPIRE].
E. Dyer, D.Z. Freedman and J. Sully, Spinning geodesic Witten diagrams, JHEP 11 (2017) 060 [arXiv:1702.06139] [INSPIRE].
H.-Y. Chen, E.-J. Kuo and H. Kyono, Anatomy of geodesic Witten diagrams, JHEP 05 (2017) 070 [arXiv:1702.08818] [INSPIRE].
S.S. Gubser and S. Parikh, Geodesic bulk diagrams on the Bruhat-Tits tree, Phys. Rev. D 96 (2017) 066024 [arXiv:1704.01149] [INSPIRE].
P. Kraus, A. Maloney, H. Maxfield, G.S. Ng and J.-Q. Wu, Witten diagrams for torus conformal blocks, JHEP 09 (2017) 149 [arXiv:1706.00047] [INSPIRE].
K. Tamaoka, Geodesic Witten diagrams with antisymmetric tensor exchange, Phys. Rev. D 96 (2017) 086007 [arXiv:1707.07934] [INSPIRE].
M. Nishida and K. Tamaoka, Fermions in geodesic Witten diagrams, JHEP 07 (2018) 149 [arXiv:1805.00217] [INSPIRE].
S. Das, Comments on spinning OPE blocks in AdS3/CFT2, Phys. Lett. B 792 (2019) 397 [arXiv:1811.09375] [INSPIRE].
V. Rosenhaus, Multipoint conformal blocks in the comb channel, JHEP 02 (2019) 142 [arXiv:1810.03244] [INSPIRE].
D. Meltzer, E. Perlmutter and A. Sivaramakrishnan, Unitarity methods in AdS/CFT, JHEP 03 (2020) 061 [arXiv:1912.09521] [INSPIRE].
S. Parikh, Holographic dual of the five-point conformal block, JHEP 05 (2019) 051 [arXiv:1901.01267] [INSPIRE].
V. Gonçalves, R. Pereira and X. Zhou, 20′ five-point function from AdS5 × S5 supergravity, JHEP 10 (2019) 247 [arXiv:1906.05305] [INSPIRE].
C.B. Jepsen and S. Parikh, Propagator identities, holographic conformal blocks, and higher-point AdS diagrams, JHEP 10 (2019) 268 [arXiv:1906.08405] [INSPIRE].
S. Parikh, A multipoint conformal block chain in d dimensions, JHEP 05 (2020) 120 [arXiv:1911.09190] [INSPIRE].
J.-F. Fortin, W.-J. Ma, V. Prilepina and W. Skiba, Efficient rules for all conformal blocks, arXiv:2002.09007 [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].
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].
A. Pal and K. Ray, Conformal correlation functions in four dimensions from quaternionic Lauricella system, arXiv:2005.12523 [INSPIRE].
S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes, JHEP 02 (2019) 040 [arXiv:1810.12459] [INSPIRE].
S. Albayrak, C. Chowdhury and S. Kharel, New relation for Witten diagrams, JHEP 10 (2019) 274 [arXiv:1904.10043] [INSPIRE].
S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes. Part II. Gravitons, JHEP 12 (2019) 135 [arXiv:1908.01835] [INSPIRE].
S. Albayrak, C. Chowdhury and S. Kharel, Study of momentum space scalar amplitudes in AdS spacetime, Phys. Rev. D 101 (2020) 124043 [arXiv:2001.06777] [INSPIRE].
A. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A natural language for AdS/CFT correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].
M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 [arXiv:1107.1504] [INSPIRE].
D. Nandan, A. Volovich and C. Wen, On Feynman rules for Mellin amplitudes in AdS/CFT, JHEP 05 (2012) 129 [arXiv:1112.0305] [INSPIRE].
G. Mack, D-dimensional conformal field theories with anomalous dimensions as dual resonance models, Bulg. J. Phys. 36 (2009) 214 [arXiv:0909.1024] [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].
E.W. Barnes, A new development of the theory of the hypergeometric functions, Proc. Lond. Math. Soc. s2-6 (1908) 141.
J.-F. Fortin, W.-J. Ma and W. Skiba, Seven-point conformal blocks in the extended snowflake channel and beyond, Phys. Rev. D 102 (2020) 125007 [arXiv:2006.13964] [INSPIRE].
S. Kharel and G. Siopsis, Tree-level correlators of scalar and vector fields in AdS/CFT, JHEP 11 (2013) 159 [arXiv:1308.2515] [INSPIRE].
V. Gonçalves, J. Penedones and E. Trevisani, Factorization of Mellin amplitudes, JHEP 10 (2015) 040 [arXiv:1410.4185] [INSPIRE].
J. Faller, S. Sarkar and M. Verma, Mellin amplitudes for fermionic conformal correlators, JHEP 03 (2018) 106 [arXiv:1711.07929] [INSPIRE].
H.-Y. Chen, E.-J. Kuo and H. Kyono, Towards spinning Mellin amplitudes, Nucl. Phys. B 931 (2018) 291 [arXiv:1712.07991] [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].
G. Lauricella, Sulle funzioni ipergeometriche a piu variabili (in Italian), Rend. Circ. Matem. Palermo 7 (1893) 111 [Erratum ibid. 7 (1893) 158].
H.M. Srivastava and P.W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood, (1985).
R.M. Aarts, Lauricella functions, from MathWorld — a Wolfram web resource, created by E.W. Weisstein, http://mathworld.wolfram.com/LauricellaFunctions.html.
H. Exton, Multiple hypergeometric functions and applications, Ellis Horwood, (1976).
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Hoback, S., Parikh, S. Towards Feynman rules for conformal blocks. J. High Energ. Phys. 2021, 5 (2021). https://doi.org/10.1007/JHEP01(2021)005
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DOI: https://doi.org/10.1007/JHEP01(2021)005