Abstract
We complete the proof of “Feynman rules” for constructing M-point conformal blocks with external and internal scalars in any topology for arbitrary M in any spacetime dimension by combining the rules for the blocks (based on their Witten diagram interpretation) with the rules for the construction of conformal cross ratios (based on the OPE and “flow diagrams”). The full set of Feynman rules leads to blocks as power series of the hypergeometric type in the conformal cross ratios. We then provide a proof by recursion of the Feynman rules which relies heavily on the first Barnes lemma and the decomposition of the topology of interest in comb structures. Finally, we provide a nine-point example to illustrate the rules.
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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 [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].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [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].
K.B. Alkalaev and V.A. Belavin, From global to heavy-light: 5-point conformal blocks, JHEP 03 (2016) 184 [arXiv:1512.07627] [INSPIRE].
V. Rosenhaus, Multipoint Conformal Blocks in the Comb Channel, JHEP 02 (2019) 142 [arXiv:1810.03244] [INSPIRE].
S. Parikh, Holographic dual of the five-point conformal block, JHEP 05 (2019) 051 [arXiv:1901.01267] [INSPIRE].
J.-F. Fortin and W. Skiba, New methods for conformal correlation functions, JHEP 06 (2020) 028 [arXiv:1905.00434] [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. Ma and W. Skiba, Higher-Point Conformal Blocks in the Comb Channel, JHEP 07 (2020) 213 [arXiv:1911.11046] [INSPIRE].
N. Irges, F. Koutroulis and D. Theofilopoulos, The conformal N-point scalar correlator in coordinate space, arXiv:2001.07171 [INSPIRE].
J.-F. Fortin, W.-J. Ma, V. Prilepina and W. Skiba, Efficient rules for all conformal blocks, JHEP 11 (2021) 052 [arXiv:2002.09007] [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, Nucl. Phys. B 968 (2021) 115433 [arXiv:2005.12523] [INSPIRE].
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. Hoback and S. Parikh, Towards Feynman rules for conformal blocks, JHEP 01 (2021) 005 [arXiv:2006.14736] [INSPIRE].
J.-F. Fortin, W.-J. Ma and W. Skiba, All Global One- and Two-Dimensional Higher-Point Conformal Blocks, arXiv:2009.07674 [INSPIRE].
I. Buric, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, From Gaudin Integrable Models to d-dimensional Multipoint Conformal Blocks, Phys. Rev. Lett. 126 (2021) 021602 [arXiv:2009.11882] [INSPIRE].
S. Hoback and S. Parikh, Dimensional reduction of higher-point conformal blocks, JHEP 03 (2021) 187 [arXiv:2009.12904] [INSPIRE].
D. Poland and V. Prilepina, Recursion relations for 5-point conformal blocks, JHEP 10 (2021) 160 [arXiv:2103.12092] [INSPIRE].
I. Buric, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, Gaudin models and multipoint conformal blocks: general theory, JHEP 10 (2021) 139 [arXiv:2105.00021] [INSPIRE].
I. Buric, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D, JHEP 11 (2021) 182 [arXiv:2108.00023] [INSPIRE].
I. Buric, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, Gaudin models and multipoint conformal blocks III: comb channel coordinates and OPE factorisation, JHEP 06 (2022) 144 [arXiv:2112.10827] [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].
A.L. 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].
W. Skiba and J.-F. Fortin, A Recipe for Conformal Blocks, LHEP 2022 (2022) 293 [arXiv:1905.00036] [INSPIRE].
G. Lauricella, Sulle funzioni ipergeometriche a piu variabili, Rend. Circ. Mat. Palermo 7 (1893) 111.
H.M. Srivastava and P.W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood, (1985) [ISBN: 9780853126027].
R.M. Aarts, Lauricella Functions, from MathWorld — A Wolfram Web Resource, created by Eric W. Weisstein [http://mathworld.wolfram.com/LauricellaFunctions.html].
J. Horn, Über die Konvergenz der hypegeometrischen Reihen zweier und drier Verländerlichen, Math. Ann. 34 (1889) 544.
V. Comeau, J.-F. Fortin and W. Skiba, Further Results on a Function Relevant for Conformal Blocks, SIGMA 16 (2020) 124 [arXiv:1902.08598] [INSPIRE].
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Fortin, JF., Hoback, S., Ma, WJ. et al. Feynman rules for scalar conformal blocks. J. High Energ. Phys. 2022, 97 (2022). https://doi.org/10.1007/JHEP10(2022)097
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DOI: https://doi.org/10.1007/JHEP10(2022)097