Fermion number violating effects in low scale leptogenesis

Fermion number violating effects in low scale leptogenesis

Abstract

The existence of baryon asymmetry and dark matter in the Universe may be related to CP-violating reactions of three heavy neutral leptons with masses well below the Fermi scale. The dynamical description of the lepton asymmetry generation, which is the key ingredient of baryogenesis and of dark matter production, is quite complicated due to the presence of many different relaxation time scales and the necessity to include quantum-mechanical coherent effects in heavy neutral lepton oscillations. We derive kinetic equations accounting for fermion number violating effects missed so far and identify one of the domains of heavy neutral lepton masses that can potentially lead to large lepton asymmetry generation boosting the sterile neutrino dark matter production.

One. Introduction

Though the canonical Standard Model has been completed by the discovery of the Higgs boson and may be a valid effective quantum field theory all the way up to the Planck scale it is inconsistent with a number of observations. They include the non-zero neutrino masses, the presence of Dark Matter in the Universe, and its baryon asymmetry. Perhaps, the most minimal way to address all these problems on the same footing is to extend the Standard Model by three right-handed neutrinos with masses below the Fermi scale. These new fermions N sub I, I equals one, two, three (following the Particle Data Group we will call them Heavy Neutral Leptons or HNLs for short) are singlets with respect to the Standard Model gauge group and thus are allowed to have Majorana neutrino masses. The lightest of these particles, N sub one, may play a role of Dark Matter. Two others (N sub two and N sub three), if (almost) degenerate, can produce the baryon asymmetry of the Universe and explain non-zero neutrino masses and mixings at the same time. This model was dubbed the vMSM for "Neutrino Minimal Standard Model". For a number of computations of baryon asymmetry in this model.

The most conservative scenario of the Universe evolution, which does not require any new physics beyond the vMSM, proceeds as follows. First, the Universe is inflated by the Standard Model Higgs field and heated up due to Higgs field oscillations to temperatures T approximately ten to the power of fourteen GeV. The Higgs inflation prepares the initial conditions for the Hot Big Bang at T approximately ten to the power of fourteen GeV: baryon and lepton numbers of the Universe are equal to zero, and the number densities of heavy neutral leptons at this time are zero as well. The particles N sub two and N sub three enter into thermal equilibrium below the sphaleron freeze-out temperature T sph approximately one hundred thirty GeV and produce baryon asymmetry of the Universe in a set of processes which include their coherent oscillations, transfer of lepton number from heavy neutral leptons to active leptons and back, and rapid anomalous sphaleron transitions. The lighter heavy neutral lepton - N sub one - Dark Matter sterile neutrino never equilibrates and is mainly produced at temperatures T DM approximately one hundred to three hundred mega-EV by transitions from the ordinary neutrinos to N sub one. The combination of X-ray and Lyman-alpha bounds on the Dark Matter sterile neutrino excludes the "non-resonant" Dodelson-Widrow mechanism for their production, which operates in the cosmic plasma with small lepton asymmetries. In other words, to get enough Dark Matter particles N sub one, the processes involving N sub two, three should produce sufficiently large lepton asymmetry A L over L greater than two times ten to the negative three which must be present at temperatures T DM. This is needed to boost the production of N sub one due to the resonant mechanism proposed by Shi and Fuller and developed in a rigorous way. The production of this large lepton asymmetry must take place below the sphaleron temperature T sph, otherwise the baryon asymmetry will be too large.

The estimates of the equilibration rates of N sub two, three in and in more recent works based on careful thermal field theory computations showed that for all parameter choices consistent with observed pattern of neutrino masses and oscillations the heavy neutral leptons N sub two, three enter in thermal equilibrium at some temperature T in exceeding tens of GeV and go out of thermal equilibrium at temperatures T out less than T in which can be as small as one GeV. This has led to the conclusion that the equilibrium period between T in and T out erases all the lepton asymmetry which could have been generated at freeze-in temperature T in, requiring that the large lepton asymmetry needed for effective dark matter production must be created at T less than T out.

The analysis made demonstrated that a large lepton asymmetry can indeed be generated in the scattering processes involving N sub two, three at the freeze-out temperature T out and below it in out-of-equilibrium decays of N sub two, three. This asymmetry does not exceed A L over L approximately three times ten to the negative two, leading to the conclusion that the mass of the Dark Matter sterile neutrino must lie in the interval from one to fifty kilo-EV, to be consistent with the Lyman-alpha and phase density constraints coming from observations of dwarf galaxies. As for N sub two, three, their physical masses should be between one point five GeV and approximately eighty GeV (the W-boson mass) and be extremely degenerate, A M phys over M less than ten to the negative fifteen. The latter condition comes from the requirement that the period of N sub two greater than N sub three oscillations should be comparable with the age of the Universe at the time of lepton asymmetry production, to insure the resonance. The minimal scenario that has been proven to work, albeit under the requirement of a strong fine-tuning (of the order of ten to the power of negative four) between two different contributions to the physical mass difference: one coming from the Yukawa couplings and the Higgs condensate, and another from Majorana masses of N sub two, three.

The aim of the present paper is to show that the part of the lepton asymmetry generated at T in can in fact survive until the temperatures of sterile neutrino Dark Matter production approximately one hundred mega-EV, in-spite of the fact that heavy neutral leptons are well in thermal equilibrium between T in and T out. Qualitatively, this comes about because of the following reasons. In the symmetric phase of the electroweak theory the transfer of asymmetry from active to sterile sector and back occurs mainly via the processes with fermion number conservation (we attribute positive fermion number to left-handed neutrinos and to right-handed heavy neutral leptons) with the rate T plus. The rate of fermion number non-conserving processes T sub minus is suppressed by a kinematic factor (M over k) squared, where M is the heavy neutral lepton mass, and k approximately three T is the typical momentum of fermions in the plasma.

On the contrary, in the Higgs phase, at temperatures of the order of tens gigaelectronvolts, the dominant reaction is induced by the mixing term between v's and N's and has a rate T exceeding that of the Universe expansion at Tout less than T less than Tin. It proceeds with fermion number non-conservation: left-handed neutrinos go into left-handed anti-HNLs and vice-versa.

In a large portion of the vMSM parameters the reactions with fermion number conservations are faster and give the main contribution to baryogenesis at the sphaleron freeze-out temperature T approximately one hundred thirty gigaelectronvolts. However, in a specific domain of the NHL masses and couplings, the rate of these processes never exceeds the Hubble rate. Thus, the asymmetry in this almost conserved number is protected from dilution, in-spite of the fact that HNLs are equilibrated due to the processes with fermion number violation. Moreover, at T approximately Tin the rate I+ can be close to the Hubble rate, meaning that large asymmetry in this number can be generated. To understand whether it is indeed produced would require numerical solution of our integro-differential kinetic equations for many parameters of the vMSM, which is not attempted here.

The paper is organised as follows. In Section Two we will derive kinetic equations accounting for helicity structure of HNL interactions. In Section Three we analyse the different rates and identify the range of vMSM parameters which may potentially lead to large lepton asymmetries surviving until small temperatures where the production of DM sterile neutrino takes place. In Section Four we summarise our results.