Abstract
In \( \mathcal{N} \) =1 superconformal theories in four dimensions the form of two-point functions of superconformal multiplets is known up to an overall constant. A superconformal multiplet contains several conformal primary operators, whose two-point function coefficients can be determined in terms of the multiplet’s quantum numbers. In this paper we work out these coefficients in full generality, i.e. for superconformal multiplets that belong to any irreducible representation of the Lorentz group with arbitrary scaling dimension and R- charge. From our results we recover the known unitarity bounds, and also find all shortening conditions, even in non-unitary theories. For the purposes of our computations we have developed a Mathematica package for the efficient handling of expansions in Grassmann variables.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
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].
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].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
J. Erdmenger and H. Osborn, Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions, Nucl. Phys. B 483 (1997) 431 [hep-th/9605009] [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].
D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
M. Berkooz, R. Yacoby and A. Zait, Bounds on \( \mathcal{N} \) =1 superconformal theories with global symmetries, JHEP 08 (2014) 008 [arXiv:1402.6068] [INSPIRE].
J.-F. Fortin, K. Intriligator and A. Stergiou, Current OPEs in Superconformal Theories, JHEP 09 (2011) 071 [arXiv:1107.1721] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, Z.U. Khandker, D. Li, D. Poland et al., Covariant Approaches to Superconformal Blocks, JHEP 08 (2014) 129 [arXiv:1402.1167] [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].
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.-H. Park, Superconformal symmetry and correlation functions, Nucl. Phys. B 559 (1999) 455 [hep-th/9903230] [INSPIRE].
S.M. Kuzenko, On compactified harmonic/projective superspace, 5 −D superconformal theories and all that, Nucl. Phys. B 745 (2006) 176 [hep-th/0601177] [INSPIRE].
W.D. Goldberger, W. Skiba and M. Son, Superembedding Methods for 4d N = 1 SCFTs, Phys. Rev. D 86 (2012) 025019 [arXiv:1112.0325] [INSPIRE].
W. Siegel, Embedding versus 6D twistors, arXiv:1204.5679 [INSPIRE].
S.M. Kuzenko, Conformally compactified Minkowski superspaces revisited, JHEP 10 (2012) 135 [arXiv:1206.3940] [INSPIRE].
W.D. Goldberger, Z.U. Khandker, D. Li and W. Skiba, Superembedding Methods for Current Superfields, Phys. Rev. D 88 (2013) 125010 [arXiv:1211.3713] [INSPIRE].
Z.U. Khandker and D. Li, Superembedding Formalism and Supertwistors, arXiv:1212.0242 [INSPIRE].
J.-F. Fortin, K. Intriligator and A. Stergiou, Superconformally Covariant OPE and General Gauge Mediation, JHEP 12 (2011) 064 [arXiv:1109.4940] [INSPIRE].
J.-F. Fortin and A. Stergiou, Field-theoretic Methods in Strongly-Coupled Models of General Gauge Mediation, Nucl. Phys. B 873 (2013) 92 [arXiv:1212.2202] [INSPIRE].
P. Kumar, D. Li, D. Poland and A. Stergiou, OPE Methods for the Holomorphic Higgs Portal, JHEP 08 (2014) 016 [arXiv:1401.7690] [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [INSPIRE].
M. Flato and C. Fronsdal, Representations of Conformal Supersymmetry, Lett. Math. Phys. 8 (1984) 159 [INSPIRE].
V.K. Dobrev and V.B. Petkova, All Positive Energy Unitary Irreducible Representations of Extended Conformal Supersymmetry, Phys. Lett. B 162 (1985) 127 [INSPIRE].
S. Ferrara and B. Zumino, Transformation Properties of the Supercurrent, Nucl. Phys. B 87 (1975) 207 [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, 1992.
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
ArXiv ePrint: 1407.6354
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Li, D., Stergiou, A. Two-point functions of conformal primary operators in \( \mathcal{N} \) = 1 superconformal theories. J. High Energ. Phys. 2014, 37 (2014). https://doi.org/10.1007/JHEP10(2014)037
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2014)037