Abstract
At the classical level, the SU(2/1) superalgebra offers a natural description of the elementary particles: leptons and quarks massless states, graded by their chirality, fit the smallest irreducible representations of SU(2/1). Our new proposition is to pair the left/right space-time chirality with the superalgebra chirality and to study the model at the one-loop quantum level. If, despite the fact that they are non-Hermitian, we use the odd matrices of SU(2/1) to minimally couple an oriented complex Higgs scalar field to the chiral Fermions, novel anomalies occur. They affect the scalar propagators and vertices. However, these undesired new terms cancel out, together with the Adler-Bell-Jackiw vector anomalies, because the quarks compensate the leptons. The unexpected and striking consequence is that the scalar propagator must be normalized using the anti-symmetric super-Killing metric and the scalar-vector vertex must use the symmetric d_aij structure constants of the superalgebra. Despite this extraordinary structure, the resulting Lagrangian is actually Hermitian.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Y. Ne’eman, Irreducible gauge theory of a consolidated Weinberg-Salam model, Phys. Lett. B 81 (1979) 190 [INSPIRE].
D.B. Fairlie, Higgs’ fields and the determination of the Weinberg angle, Phys. Lett. B 82 (1979) 97 [INSPIRE].
V.G. Kac, Lie superalgebras, Adv. Math. 26 (1977) 8 [INSPIRE].
S. Weinberg, A model of leptons, Phys. Rev. Lett. 19 (1967) 1264 [INSPIRE].
S.L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426 [INSPIRE].
J.S. Bell and R. Jackiw, A PCAC puzzle: π0 → γγ in the σ model, Nuovo Cim. A 60 (1969) 47 [INSPIRE].
P.H. Dondi and P.D. Jarvis, A supersymmetric Weinberg-Salam model, Phys. Lett. B 84 (1979) 75 [Erratum ibid. 87 (1979) 403] [INSPIRE].
Y. Ne’eman and J. Thierry-Mieg, Geometrical gauge theory of ghost and Goldstone fields and of ghost symmetries, Proc. Natl. Acad. Sci. U.S.A. 77 (1980) 720.
M. Scheunert, W. Nahm and V. Rittenberg, Irreducible representations of the OSP(2, 1) and SPL(2, 1) graded Lie algebras, J. Math. Phys. 18 (1977) 155 [INSPIRE].
M. Marcu, The representations of Spl(2, 1): an example of representations of basic superalgebras, J. Math. Phys. 21 (1980) 1277 [INSPIRE].
G. Götz, T. Quella and V. Schomerus, Representation theory of sl(2/1), J. Algebra 312 (2007) 829.
Y. Su, Classification of finite dimensional modules of the Lie superalgebra sl(2/1), Commun. Algebra 20 (2007) 3259.
R. Coquereaux, Elementary fermions and SU(2/1) representations, Phys. Lett. 261 (1991) 449.
R. Coquereaux, G. Esposito-Farese and F. Scheck, Noncommutative geometry and graded algebras in electroweak interactions, Int. J. Mod. Phys. A 7 (1992) 6555 [INSPIRE].
R. Haussling and F. Scheck, Triangular mass matrices of quarks and Cabibbo-Kobayashi-Maskawa mixing, Phys. Rev. D 57 (1998) 6656 [hep-ph/9708247] [INSPIRE].
R. Haussling, M. Paschke and F. Scheck, Leptonic generation mixing, noncommutative geometry and solar neutrino fluxes, Phys. Lett. B 417 (1998) 312 [hep-ph/9709466] [INSPIRE].
R. Coquereaux, R. Haussling, N.A. Papadopoulos and F. Scheck, Generalized gauge transformations and hidden symmetry in the standard model, Int. J. Mod. Phys. A 7 (1992) 2809 [INSPIRE].
Y. Ne’eman, S. Sternberg and D. Fairlie, Superconnections for electroweak SU(2/1) and extensions and the mass of the Higgs, Phys. Rept. 406 (2005) 303 [INSPIRE].
D.S. Hwang, C.-Y. Lee and Y. Ne’eman, BRST quantization of SU(2/1) electroweak theory in the superconnection approach and the Higgs meson mass, Int. J. Mod. Phys. A 11 (1996) 3509 [INSPIRE].
C. Bouchiat, J. Iliopoulos and P. Meyer, An anomaly free version of Weinberg’s model, Phys. Lett. B 38 (1972) 519 [INSPIRE].
J.A. Minahan, P. Ramond and R.C. Warner, Comment on anomaly cancellation in the standard model, Phys. Rev. D 41 (1990) 715 [INSPIRE].
J. Thierry-Mieg and Y. Ne’eman, Exterior gauging of an internal supersymmetry and SU(2/1) quantum asthenodynamics, Proc. Nat. Acad. Sci. 79 (1982) 7068 [INSPIRE].
D. Quillen, Superconnections and the Chern character, Topology 24 (1985) 89 [INSPIRE].
A. Connes and J. Lott, Particle models and noncommutative geometry, Nucl. Phys. B Proc. Suppl. 18 (1990) 29 [INSPIRE].
A. Connes, Non-commutative geometry year 2000, in Visions in Mathematics, Birkhüser, Basel Switzerland (2000), pg. 481 [math/0011193].
A. Connes, Noncommutative geometry and the standard model with neutrino mixing, JHEP 11 (2006) 081 [hep-th/0608226] [INSPIRE].
L.V. Avdeev and M.V. Chizhov, Antisymmetric tensor matter fields: an Abelian model, Phys. Lett. B 32 (1994) 212 [hep-th/9312062].
C. Wetterich, Chiral freedom and the scale of weak interactions, Mod. Phys. Lett. A 23 (2008) 677 [hep-ph/0503164] [INSPIRE].
N.I. Stoilova, J. Thierry-Mieg and J. Van der Jeugt, Extension of the osp(m/n) ∼ so(m − n) Correspondence to the Infinite-Dimensional Chiral Spinors and Self Dual Tensors, J. Phys. A 50 (2017) 155201 [arXiv:1609.06350] [INSPIRE].
J. Thierry-Mieg, Chirality, the missing key to the definition of the connection and curvature of a Lie-Kac super-algebra, [arXiv:2003.12234] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2005.04754
Rights and permissions
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.
About this article
Cite this article
Thierry-Mieg, J. Scalar anomaly cancellation reveals the hidden superalgebraic structure of the quantum chiral SU(2/1) model of leptons and quarks. J. High Energ. Phys. 2020, 167 (2020). https://doi.org/10.1007/JHEP10(2020)167
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2020)167