Abstract
Applying the Casimir operator to four-point functions in CFTs allows us to find the conformal blocks for any external operators. In this work, we initiate the program to find the superconformal blocks, using the super Casimir operator, for 4D\( \mathcal{N}=1 \) SCFTs. We begin by finding the most general four-point function with zero U(1)R-charge, including all the possible nilpotent structures allowed by the superconformal algebra. We then study particular cases where some of the operators satisfy shortening conditions. Finally, we obtain the super Casimir equations for four point-functions which contain a chiral and an anti-chiral field. We solve the super Casimir equations by writing the superconformal blocks as a sum of several conformal blocks.
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References
J.-H. Park, N = 1 superconformal symmetry in four-dimensions, Int. J. Mod. Phys. A 13 (1998) 1743 [hep-th/9703191] [INSPIRE].
H. Osborn, N = 1 superconformal symmetry in four-dimensional quantum field theory, Annals Phys. 272 (1999) 243 [hep-th/9808041] [INSPIRE].
J.-F. Fortin, K. Intriligator and A. Stergiou, Current OPEs in Superconformal Theories, JHEP 09 (2011) 071 [arXiv:1107.1721] [INSPIRE].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D Conformal Field Theories with Global Symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].
I.A. Ramírez, Mixed OPEs in \( \mathcal{N}=2 \) superconformal theories, JHEP 05 (2016) 043 [arXiv:1602.07269] [INSPIRE].
M. Berkooz, R. Yacoby and A. Zait, Bounds on \( \mathcal{N}=1 \) superconformal theories with global symmetries, JHEP 08 (2014) 008 [Erratum ibid. 01 (2015) 132] [arXiv:1402.6068] [INSPIRE].
Z.U. Khandker, D. Li, D. Poland and D. Simmons-Duffin, \( \mathcal{N}= 1 \) superconformal blocks for general scalar operators, JHEP 08 (2014) 049 [arXiv:1404.5300] [INSPIRE].
Z. Li and N. Su, The Most General 4D \( \mathcal{N}=1 \) Superconformal Blocks for Scalar Operators, JHEP 05 (2016) 163 [arXiv:1602.07097] [INSPIRE].
P. Liendo, I. Ramirez and J. Seo, Stress-tensor OPE in \( \mathcal{N}=2 \) superconformal theories, JHEP 02 (2016) 019 [arXiv:1509.00033] [INSPIRE].
A. Manenti, A. Stergiou and A. Vichi, R-current three-point functions in 4d \( \mathcal{N}=1 \) superconformal theories, JHEP 12 (2018) 108 [arXiv:1804.09717] [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].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [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].
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. 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].
A. Dymarsky, J. Penedones, E. Trevisani and A. Vichi, Charting the space of 3D CFTs with a continuous global symmetry, arXiv:1705.04278 [INSPIRE].
G.F. Cuomo, D. Karateev and P. Kravchuk, General Bootstrap Equations in 4D CFTs, JHEP 01 (2018) 130 [arXiv:1705.05401] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial wave expansions for N = 4 chiral four point functions, Annals Phys. 321 (2006) 581 [hep-th/0412335] [INSPIRE].
F.A. Dolan, L. Gallot and E. Sokatchev, On four-point functions of 1/2-BPS operators in general dimensions, JHEP 09 (2004) 056 [hep-th/0405180] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The \( \mathcal{N}=2 \) superconformal bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE].
N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping SCFTs with Four Supercharges, JHEP 08 (2015) 142 [arXiv:1503.02081] [INSPIRE].
M. Lemos and P. Liendo, Bootstrapping \( \mathcal{N}=2 \) chiral correlators, JHEP 01 (2016) 025 [arXiv:1510.03866] [INSPIRE].
R. Doobary and P. Heslop, Superconformal partial waves in Grassmannian field theories, JHEP 12 (2015) 159 [arXiv:1508.03611] [INSPIRE].
A. Bissi and T. Lukowski, Revisiting \( \mathcal{N}=4 \) superconformal blocks, JHEP 02 (2016) 115 [arXiv:1508.02391] [INSPIRE].
M. Lemos, P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping \( \mathcal{N}=3 \) superconformal theories, JHEP 04 (2017) 032 [arXiv:1612.01536] [INSPIRE].
D. Li, D. Meltzer and A. Stergiou, Bootstrapping mixed correlators in 4D \( \mathcal{N}=1 \) SCFTs, JHEP 07 (2017) 029 [arXiv:1702.00404] [INSPIRE].
N. Bobev, E. Lauria and D. Mazac, Superconformal Blocks for SCFTs with Eight Supercharges, JHEP 07 (2017) 061 [arXiv:1705.08594] [INSPIRE].
Z. Li, Superconformal Partial Waves for Stress-tensor Multiplet Correlator in 4D \( \mathcal{N}=2 \) SCFTs, arXiv:1806.11550 [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, Z.U. Khandker, D. Li, D. Poland and D. Simmons-Duffin, Covariant Approaches to Superconformal Blocks, JHEP 08 (2014) 129 [arXiv:1402.1167] [INSPIRE].
J. Wess and J.A. Bagger, Supersymmetry and supergravity, 2nd edition Princeton Series in Physics, Princeton University Press, Princeton, NJ (1992) [INSPIRE].
J.-H. Park, Superconformal symmetry in six-dimensions and its reduction to four-dimensions, Nucl. Phys. B 539 (1999) 599 [hep-th/9807186] [INSPIRE].
J.-H. Park, Superconformal symmetry and correlation functions, Nucl. Phys. B 559 (1999) 455 [hep-th/9903230] [INSPIRE].
S.M. Kuzenko and S. Theisen, Correlation functions of conserved currents in N = 2 superconformal theory, Class. Quant. Grav. 17 (2000) 665 [hep-th/9907107] [INSPIRE].
D. Li and A. Stergiou, Two-point functions of conformal primary operators in \( \mathcal{N}=1 \) superconformal theories, JHEP 10 (2014) 37 [arXiv:1407.6354] [INSPIRE].
M. Lemos and P. Liendo, \( \mathcal{N}=2 \) central charge bounds from 2d chiral algebras, JHEP 04 (2016) 004 [arXiv:1511.07449] [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].
S. Ferrara and G. Parisi, Conformal covariant correlation functions, Nucl. Phys. B 42 (1972) 281 [INSPIRE].
S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products, Lett. Nuovo Cim. 4S2 (1972) 115 [INSPIRE].
S. Ferrara, A.F. Grillo and G. Parisi, Nonequivalence between conformal covariant Wilson expansion in euclidean and Minkowski space, Lett. Nuovo Cim. 5S2 (1972) 147 [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. B 53 (1973) 643] [INSPIRE].
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Ramírez, I.A. Towards general super Casimir equations for 4D\( \mathcal{N}=1 \) SCFTs. J. High Energ. Phys. 2019, 47 (2019). https://doi.org/10.1007/JHEP03(2019)047
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DOI: https://doi.org/10.1007/JHEP03(2019)047