ARS Leptogenesis

ARS Leptogenesis

We review the current status of the leptogenesis scenario originally proposed by Akhmedov, Rubakov and Smirnov. It takes place in the parametric regime where the right-handed neutrinos are at the electroweak scale or below and the CP-violating effects are induced by the coherent superposition of different right-handed mass eigenstates. Two main theoretical approaches to derive quantum kinetic equations, the Hamiltonian time evolution as well as the Closed-Time-Path technique are presented, and we discuss their relations. For scenarios with two right-handed neutrinos, we chart the viable parameter space. Both, a Bayesian analysis, that determines the most likely configurations for viable leptogenesis given different variants of flat priors, and a determination of the maximally allowed mixing between the light, mostly left-handed, and heavy, mostly right-handed, neutrino states are discussed. Rephasing invariants are shown to be a useful tool to classify and to understand various distinct contributions to ARS leptogenesis that can dominate in different parametric regimes. While these analyses are carried out for the parametric regime where initial asymmetries are generated predominantly from lepton-number conserving, but flavor violating effects, we also review the contributions from lepton-number violating operators and identify the regions of parameter space where these are relevant.

One. Introduction

One of the most interesting implications of the extensions of the Standard Model with massive neutrinos is the possibility to explain the baryon asymmetry in the Universe via leptogenesis. This mechanism has been shown to be robust and generic in seesaw models that involve a very high scale of new physics, much higher than the electroweak scale, M sub N much greater than V. In the standard scenarios, leptogenesis takes place during the freeze-out of some heavy states that can decay violating charge-parity CP and lepton number L. These high-scale scenarios have been studied extensively. For comprehensive reviews see.

Akhmedov, Rubakov and Smirnov studied the possibility to generate a baryon asymmetry in type I seesaw models at a much lower scale, M sub N less than or equal to V. The key observation is that the small Yukawa couplings required to explain neutrino masses in this low-scale scenario could be small enough to ensure that some of the sterile states, also referred to as right-handed RH neutrinos, might not reach thermal equilibrium before the electroweak phase transition, when sphaleron processes are switched off. ARS leptogenesis therefore is a freeze-in scenario. Pending on the parametric regime, lepton-number violating LNV processes may be negligible both in the generation of the asymmetries as well as in the washout because the Majorana mass of the RH neutrinos is small compared to the temperature. In that situation, the initial asymmetries in active leptons are purely flavored and lepton-number conserving, and eventually total asymmetries in the active sector arise and are approximately counterbalanced by those in the sterile sector. If this situation survives until the electroweak phase transition, a net baryon asymmetry results, since the eventual equilibration later on can no longer be transmitted to the baryons in the absence of efficient sphaleron transitions. In other regions of parameter space LNV contributions may be relevant or even dominating in the source as well as washout terms and then must be accounted for. The aim of this chapter is to review the ARS mechanism of leptogenesis.

The model involves the simplest extension of the Standard Model with N sub R heavy Majorana singlets.

The Lagrangian equals the Standard Model Lagrangian plus the term bar N sub k i V N sub k minus the term one half times M sub N sub j k bar N sub j superscript C N sub k plus lambda sub alpha k bar ell sub alpha phi superscript C N sub k plus H. C..

where N sub k are right-handed spinors, such that P sub R N sub k equals N sub k, lambda is a three times N sub R complex matrix, M sub N is an N sub R-dimensional complex symmetric matrix, and phi superscript C equals epsilon phi superscript star. The spectrum of this theory contains three lighter states with a mass matrix given by the famous seesaw formula

M sub nu equals negative V squared over two lambda M sub N superscript negative one lambda transposed.

where V equals two four six GeV, and N sub R heavy ones with masses of the order of M sub N. The naive seesaw scaling, exact for one family, relating Yukawas with the light and heavy masses is therefore M sub nu is similar to the order of V squared y squared over two M sub N, where y is specified through the next equation.

In most of this work we will assume the minimal scenarios where N sub R equals two, three. Different parametrizations of the Yukawa matrices have been used in the literature. For some purposes the parametrization in terms of the two unitary matrices that bi-diagonalize lambda is useful:

lambda equals V dagger diagonal y sub one, y sub two, y sub three W.

For the purpose of parameter scanning however the Casas-Ibarra parametrization is most appropriate. For N sub B equals two it reads:

lambda equals negative i U sub nu star square root of M sub nu diagonal P sub N O R transposed Z square root of M sub N times the square root of two over V.

where U sub nu is the PMNS matrix, M sub nu superscript diag is the diagonal matrix of the light neutrino masses (note that the lightest neutrino is massless because only two Majorana singlets are included), M sub N equivalent to M sub N superscript diag equals diag (M sub one, M sub two), where M sub one, M sub two are the heavy neutrino masses and without loss of generality, we choose the mass basis for the RH neutrinos, P sub NO is a three by two matrix that depends on the neutrino ordering (NH, IH).

P sub NH equals left parenthesis zero, zero; one, zero; zero, one right parenthesis, P sub IH equals left parenthesis one, zero; zero, one; zero, zero right parenthesis, (five)

and finally R left parenthesis z right parenthesis is a generic two dimensional orthogonal complex matrix that depends on one complex angle z equivalent to theta plus i gamma. For n sub R equals three, the parametrization is lambda equals negative i U sub nu superscript star sqrt M sub nu superscript diag R transposed left parenthesis z sub one, z sub two, z sub three right parenthesis sqrt M sub N frac sqrt two over v, (six)

where R left parenthesis z sub one, z sub two, z sub three right parenthesis is a three dimensional complex orthogonal matrix that depends on three complex angles,