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.

Highlights

  • This is a U.S government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020

  • 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)

  • The unexpected and striking consequence is that the scalar propagator must be normalized using the antisymmetric super-Killing metric and the scalar-vector vertex must use the symmetric d aij structure constants of the superalgebra

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Summary

The chiral scalar-Fermion minimal coupling

Let us assume the existence of an oriented complex scalar field Φ Φ coupled to the chiral Fermions ψψ via the odd generators λi of the superalgebra The scalars are oriented: they transport left spin states, they are emitted by left ψL Fermions (which become right) and absorbed by right ψR Fermions (which become left) according to the Feynman diagrams:. The first term of (2.5) gives, for any representation of the superalgebra, the symmetric structure constants of the superalgebra (appendix A, equations (A.3) and (A.6)):. Which are not well defined, because the commutators of the odd matrices do not close on the even matrices We call this second term anomalous, and generalizing the Adler-Bell-Jackiw condition (1.2) we request that:. We leave as an open problem the general classification of all the chiral representations of the simple superalgebras satisfying the four equations (1.2), (2.4), (2.8), (2.12) and conjecture that these anomalies play a role in the exponentiation of the superalgebra into a supergroup These results are unexpected and were not anticipated in the SU(2/1) literature. They each generate an anomaly, yet together they produce the desired symmetric vertex

Rediagonalization to an explicitly Hermitian Lagrangian
Generation mixing
Limitations of the model
Discussion
A Definition of a chiral superalgebra
The Adler-Bell-Jackiw vector anomaly
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