CP violation in mixing and oscillations in a toy model for leptogenesis with quasi-degenerate neutrinos

Abstract

We study the sources of CP violation for baryogenesis models with quasi-degenerate neutrinos. Our approach is to use the renormalized propagator in a quantum field theory model of neutrino oscillations, paying close attention to unitarity requirements. From the probabilities of lepton number violating processes obtained in this way, we derive a source term for the time evolution of the lepton asymmetry. The source term has contributions that can be identified with CP violation from mixing, oscillations and interference between both. Given that this source term does not involve processes with unstable particles in the initial or final states, neither does it require to calculate number densities of neutrinos, no subtraction of real intermediate states must be performed. In equilibrium the source term is null, as demanded by unitarity and CPT invariance, due to a cancellation between the terms coming from CP violation in mixing and oscillations. The calculations are done in a simple scalar toy model, and the resummed propagator is diagonalized at first order in the decay widths over the mass difference. We also comment on the effect of the interference term, which is mild at the order we work, but seems to become more important with increasing degeneracy.

CP violation in mixing and oscillations for leptogenesis. Part II. The highly degenerate case

“…Even more recently, another set of non-equilibrium quantum transport equations for flavor-mixing fermions was derived in [27] using the Schwinger-Keldysh closed time path formalism and applied to resonant leptogenesis. Although the scope of the equations in [27] is much bigger than the one of [26], it is interesting to note that the results of [27] also do not support some of the findings of [16][17][18][19] related to the mixing and oscillation terms and, like in [26], the lepton asymmetry in [27] converges to the usual Boltzmann result in the limit of large mass splittings.…”

“…Moreover, in the limit of large mass splittings, this source tends to the one of the standard classical Boltzmann equations appropriate for non-oscillating neutrinos. Therefore, the findings of [26] do not seem to agree with those of [16][17][18][19] regarding the relative sign of the contributions from mixing and oscillations.…”

“…Those works elucidated various issues, but a complete implementation of the method to baryogenesis remained open. Motivated by the simplicity and transparency of this formalism, we followed a quantum field theory approach in [26], that up to some point and under certain approximations, can be matched to the effective Hamiltonian formalism. In this approach the renormalized propagator is used in a quantum field theory model of neutrino oscillations in order to obtain probabilities of lepton number violating processes involving only stable particles in the initial and final states.…”

“…The source term obtained in [26] at first order in the decay widths over the mass splitting has contributions that can be identified with CP violation from mixing, oscillations and interference between both. In equilibrium the terms coming from CP violation in mixing and oscillations cancel, yielding a null source as required by unitarity and CPT invariance.…”

“…All in all, what seems clear is that the subject of CP violation in leptogenesis with quasi-degenerate neutrinos is not trivial and therefore different looks at the problem can be instructive. With this motivation, here we extend the approach introduced in [26] to the more involved highly degenerate case, with mass splittings similar or smaller than the decay widths. The work is organized as follows: in section 2 we calculate a time dependent CP asymmetry between lepton number violating processes involving only stable states, by performing an expansion around the poles of the resummed one-loop propagator and using this expansion in a quantum field theory model for oscillations.…”