No abstract

Anti-Hermitian mass terms are considered, in addition to Hermitian ones, for PT -symmetric complexscalar and fermionic field theories. In both cases, the Lagrangian can be written in a manifestly symmetric form in terms of the PT -conjugate variables, allowing for an unambiguous definition of the equations of motion. After discussing the resulting constraints on the consistency of the variational procedure, we show that the invariance of a non-Hermitian Lagrangian under a continuous symmetry transformation does not imply the existence of a corresponding conserved current. Conserved currents exist, but these are associated with transformations under which the Lagrangian is not invariant and which reflect the well-known interpretation of PT -symmetric theories in terms of systems with gain and loss. A formal understanding of this unusual feature of non-Hermitian theories requires a careful treatment of Noether's theorem, and we give specific examples for illustration.

Vanilla leptogenesis within the type I seesaw framework requires the mass scale of the right-handed neutrinos to be above 10 9 GeV. This lower bound can be avoided if at least two of the sterile states are almost mass degenerate, which leads to an enhancement of the decay asymmetry. Leptogenesis models that can be tested in current and upcoming experiments often rely on this resonant enhancement, and a systematic and consistent description is therefore necessary for phenomenological applications. In this review article, we give an overview of different methods that have been used to study the saturation of the resonant enhancement when the mass difference becomes comparable to the characteristic width of the Majorana neutrinos. In this limit, coherent flavor transitions start to play a decisive role, and off-diagonal correlations in flavor space have to be taken into account. We compare various formalisms that have been used to describe the resonant regime and discuss under which circumstances the resonant enhancement can be captured by simplified expressions for the CP asymmetry. Finally, we briefly review some of the phenomenological aspects of resonant leptogenesis.

We demonstrate the extension to PT -symmetric field theories of the Goldstone theorem, confirming that the spontaneous appearance of a field vacuum expectation value via minimisation of the effective potential in a non-Hermitian model is accompanied by a massless scalar boson.Laying a basis for our analysis, we first show how the conventional quantisation of the path-integral formulation of quantum field theory can be extended consistently to a non-Hermitian model by considering PT conjugation instead of Hermitian conjugation. The extension of the Goldstone theorem to a PT -symmetric field theory is made possible by the existence of a conserved current that does not, however, correspond to a symmetry of the non-Hermitian Lagrangian. In addition to extending the proof of the Goldstone theorem to a PT -symmetric theory, we exhibit a specific example in which we verify the existence of a massless boson at the tree and one-loop levels.