Abstract
We discuss $\mathcal{P}\mathcal{T}$-symmetric Abelian gauge field theories as well as their extension to the Englert-Brout-Higgs mechanism for generating a mass for a vector boson. Gauge invariance is not straightforward, and we discuss the different related problems as well as a solution which consists in coupling the gauge field to a current that is not conserved. Non-Hermiticity then necessarily precludes the Lorenz gauge condition but nevertheless allows for a consistent formulation of the theory. We therefore generalize the Englert-Brout-Higgs mechanism to $\mathcal{P}\mathcal{T}$-symmetric field theories, opening the way to constructing non-Hermitian extensions of the Standard Model and new scenarios for particle model building.
Highlights
There has been much work in recent years on quantummechanical models with non-Hermitian, PT -symmetric Hamiltonians [1,2,3], which have become an important area of research in integrated photonics and other fields [4,5,6]— see Ref. [7] for a review of relations to conventional models with Hermitian Hamiltonians
Among applications to particle physics, the possibility of using the non-Hermitian term μψγ5ψ to describe neutrino masses was considered in Refs. [17,18,19], and the application of non-Hermitian Quantum field theories (QFTs) to neutrino oscillations was considered in Ref. [20]
We have shown in this paper how the Englert-BroutHiggs mechanism [26,27] for generating masses for gauge bosons can be generalized from the familiar case of Hermitian QFTs to the more general framework of PT symmetric field theories
Summary
There has been much work in recent years on quantummechanical models with non-Hermitian, PT -symmetric Hamiltonians [1,2,3], which have become an important area of research in integrated photonics and other fields [4,5,6]— see Ref. [7] for a review of relations to conventional models with Hermitian Hamiltonians. We show how the gauge symmetries of non-Hermitian and PT -symmetric theories may be broken via a generalization of the Englert-Brout-Higgs mechanism [26,27], opening the way to significant extensions of the Standard Model and other particle physics theories This extension is nontrivial, as it was discovered in Ref. PT -symmetric QFT with a quartic scalar potential in which the scalar fields have symmetry-breaking vacuum expectation values that are accompanied by a massless Goldstone mode The existence of the latter follows from current conservation, even though the Lagrangian is not invariant under the corresponding field transformations. The existence of this Goldstone mode was confirmed by an explicit calculation of the effective potential at the tree and one-loop levels Our analysis of these questions was based on a formulation of a non-Hermitian QFT that included a consistent quantization of the path integral.
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