Gauge invariance and the Englert-Brout-Higgs mechanism in non-Hermitian field theories

We discuss $\mathcal{PT}$-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 generalise the Englert-Brout-Higgs mechanism to $\mathcal{PT}$-symmetric field theories, opening the way to constructing non-Hermitian extensions of the Standard Model and new scenarios for particle model-building.