Physics:R-symmetry
In theoretical physics, an R-symmetry is an internal symmetry present in some supersymmetric theories. It is often described as a symmetry which "rotates the supercharges", meaning that it commutes with the generators of spacetime symmetries but not with the supersymmetry generators.
The group of R-symmetry transformations is highly constrained by the spacetime symmetry and the structure of the spin representation in which the supersymmetry generators transform. For Poincaré-invariant theories in four spacetime dimensions, this group is determined by the Haag–Łopuszański–Sohnius theorem: for minimal supersymmetry, this R-symmetry group is ; for theories with -extended supersymmetry, it is the non-abelian group ; in the special case of , it may also be .[1][2][3]
A theory which is supersymmetric need not possess R-symmetry, and R-symmetry may be broken at the quantum level even where it is present classically, i.e. the symmetry may be anomalous. Both of these aspects are exhibited by the Wess-Zumino model, which is only invariant under R-symmetry when the superpotential is monomial, and the R-symmetry is always anomalous.[4][5] A theory may only be invariant under a subgroup of the R-symmetry; for example, the minimal Supersymmetric Standard Model has only R-parity, i.e. invariance under the discrete subgroup of .[6]
History and terminology
In 1974, Wess and Zumino introduced their aforementioned model in which a transformation appears in the commutator of two supersymmetry transformations.[4] More realistic models with such a symmetry were introduced in 1975 by Salam and Strathdee and by Fayet, with the latter using the notation "" for the associated tranformation.[7][8] Parallel to this, Haag, Łopuszański, Sohnius and Dondi showed that in the -extended case, the internal symmetry group acting on the supercharges is enlarged to include .[1][9][10] The term "R-invariance" was used in early instances, but "R-symmetry" was in use by at least 1978.[11]
Fayet later showed that in physically realistic models, in which the superpartners of known particles must be very massive, the R-symmetry must be broken to the subgroup (R-parity) to allow the gluinos and gravitino to acquire mass. The minimal supersymmetric standard model therefore does not possess R-symmetry, but only R-parity.[6]
Relation to supersymmetry algebras
The R-symmetry group can be considered as a group of (external) automorphisms of the Poincaré superalgebra, and its Lie algebra is contained in more general supersymmetry algebras.[3]
A model invariant under both supersymmetry and conformal transformations (such as the massless Wess-Zumino model) must also be R-invariant because the closure of the superconformal algebra requires R-symmetry terms in the odd-odd brackets.[1][2]
Gauged R-symmetry
In rigid supersymmetric theories, any R-symmetry is global and cannot be gauged. However, in supergravity theories, in which the supersymmetry is local, R-symmetry can be gauged, though this has to be achieved by "mixing" it with another global symmetry group.[12][13]
Other dimensions and signatures
The definition of the R-symmetry group depends on the reality structure of the spinor representation, hence it varies with dimension and signature. The R-symmetry group is always isomorphic to a classical Lie group or a product of two such groups.
In Lorentzian signature, we have the following. In dimensions 2, 6 and 10, where the reality condition is Weyl-compatible, there are two inequivalent real spinor modules distinguished by chirality, and the amount of supersymmetry is characterised by a pair of integers indicating the number of left- and right-handed sets of supercharges.[3][12]
| Spacetime dimension | Spinor type | R-symmetry group |
|---|---|---|
| 2 | Majorana-Weyl | |
| 3 | Majorana | |
| 4 | Majorana | |
| 5 | Symplectic Majorana | |
| 6 | Symplectic Majorana-Weyl | |
| 7 | Symplectic Majorana | |
| 8 | Majorana | |
| 9 | Majorana | |
| 10 | Majorana-Weyl |
Theories with extended supersymmetry generally have a number of scalar fields described by a sigma model, meaning that they can be considered as being coordinates on an abstract homogeneous Riemannian manifold. The R-symmetry can be identified as part of the isotropy of this "scalar manifold", which in turn is often viewed as the internal geometry in a compactification from a higher-dimensional theory (usually string theory or M-theory).[12][13]
With more general signature or with non-standard choices of squaring map (or equivalently "twisted" reality conditions on spinors) there are more possibilities for the R-symmetry group. They are relevant to the study of time-like T-duality.[14]
References
- ↑ 1.0 1.1 1.2 Haag, R.; Łopuszański, J.T.; Sohnius, M. (1975). "All possible generators of supersymmetries of the S-matrix". Nuclear Physics B 88 (2): 257–274. doi:10.1016/0550-3213(75)90279-5. Bibcode: 1975NuPhB..88..257H. https://dx.doi.org/10.1016/0550-3213%2875%2990279-5.
- ↑ 2.0 2.1 West, P. C. (2002). "Introduction to Rigid Supersymmetric Theories". Confinement, Duality, and Non-Perturbative Aspects of QCD. NATO Science Series: B. 368. pp. 453–476. doi:10.1007/0-306-47056-X_17. ISBN 0-306-45826-8.
- ↑ 3.0 3.1 3.2 Antoine Van Proeyen (1999). "Tools for supersymmetry". arXiv:hep-th/9910030.
- ↑ 4.0 4.1 Wess, J.; Zumino, B. (1974). "Supergauge transformations in four dimensions". Nuclear Physics B 70 (1): 39–50. doi:10.1016/0550-3213(74)90355-1. Bibcode: 1974NuPhB..70...39W. https://cds.cern.ch/record/201649.
- ↑ José M. Figueroa-O'Farrill (2001). "BUSSTEPP lectures on supersymmetry". arXiv:hep-th/0109172v1.
- ↑ 6.0 6.1 Fayet, Pierre (August 2001). "About the origins of the Supersymmetric Standard Model". Nuclear Physics B - Proceedings Supplements 101 (1-3): 81–98. doi:10.1016/S0920-5632(01)01495-5. https://arxiv.org/abs/hep-ph/0107228.
- ↑ Fayet, P. (September 1976). "Supersymmetry and weak, electromagnetic and strong interactions". Physics Letters B 64 (2): 159–162. doi:10.1016/0370-2693(76)90319-1.
- ↑ Salam, Abdus; Strathdee, J. (February 1975). "Supersymmetry and fermion-number conservation". Nuclear Physics B 87 (1): 85–92. doi:10.1016/0550-3213(75)90253-9.
- ↑ Dondi, P.H.; Sohnius, M. (November 1974). "Supersymmetry transformations with isospin". Nuclear Physics B 81 (2): 317–329. doi:10.1016/0550-3213(74)90171-0.
- ↑ Dondi, P H (August 1975). "Superfields with internal symmetry". Journal of Physics A: Mathematical and General 8 (8): 1298–1309. doi:10.1088/0305-4470/8/8/014.
- ↑ Clark, T.E.; Piguet, O.; Sibold, K. (October 1978). "Supercurrents, renormalization and anomalies". Nuclear Physics B 143 (3): 445–484. doi:10.1016/0550-3213(78)90064-0.
- ↑ 12.0 12.1 12.2 Antoine Van Proeyen (2003). "Structure of supergravity theories". arXiv:hep-th/0301005.
- ↑ 13.0 13.1 Bernard de Wit (2002). "Supergravity". arXiv:hep-th/0212245.
- ↑ Gall, L.; Mohaupt, T. (October 2021). "Supersymmetry algebras in arbitrary signature and their R-symmetry groups". Journal of High Energy Physics 2021 (10). doi:10.1007/JHEP10(2021)203.
 |  |
