Three-Flavoured Non-Resonant Leptogenesis at Intermediate Scales

Three-Flavoured Non-Resonant Leptogenesis at Intermediate Scales

Leptogenesis can successfully explain the matter-antimatter asymmetry via out-of-equilibrium decays of heavy Majorana neutrinos in the early Universe. In this article, we focus on non-resonant thermal leptogenesis and the possibility of lowering its scale. In order to do so, we calculate the lepton asymmetry produced from the decays of one and two heavy Majorana neutrinos using three-flavoured density matrix equations in an exhaustive exploration of the model parameter space. We find regions of the parameter space where thermal leptogenesis is viable at intermediate scales, T is approximately ten to the power of six gigaelectronvolts. However, the viability of thermal leptogenesis at such scales requires a certain degree of cancellation between the tree and one-loop level contribution to the light neutrino mass matrix and we quantify such fine-tuning.

One. INTRODUCTION

There is overwhelming experimental evidence for an excess of matter over antimatter in the Universe. This asymmetry remains a fundamental and unresolved mystery whose explanation demands new physics beyond the Standard Model. The baryon asymmetry may be parametrised by the baryon-to-photon ratio, eta sub B, which is defined to be eta sub B equivalent to the fraction of n sub B minus n sub anti-B over n sub gamma, where n sub B, n sub anti-B and n sub gamma are the number densities of baryons, anti-baryons and photons, respectively. This quantity can be measured using two independent methods that probe the Universe at different stages of its evolution. Big-Bang nucleosynthesis, BBN, and Cosmic Microwave Background radiation, CMB, data give eta sub BBN equals left parenthesis five point eight zero minus six point six zero right parenthesis times ten to the power of negative ten, eta sub CMB equals left parenthesis six point zero two minus six point one eight right parenthesis times ten to the power of negative ten,

at ninety-five percent confidence level, respectively. As the uncertainties of the CMB measurement are smaller than those from BBN, we shall apply the CMB value throughout this work.

In order to dynamically produce the observed baryon asymmetry in the early Universe, the mechanism of interest must satisfy the Sakharov conditions: B (or left parenthesis L right parenthesis violation; C over C P violation and a departure from thermal equilibrium. Leptogenesis satisfies these conditions and produces a lepton asymmetry which is subsequently partially converted to a baryon asymmetry via B plus L violating sphaleron processes.

Leptogenesis is particularly appealing as it typically takes place in models of neutrino masses, simultaneously explaining the baryon asymmetry and the smallness of the neutrino masses. In its simplest realisation, the lepton asymmetry is generated via out-of-equilibrium decays of heavy Majorana neutrinos. This process occurs approximately when the temperature, T, of the Universe equals the mass scale of the decaying heavy Majorana neutrino.

In general, the scale of thermal leptogenesis is not explored below the Davidson-Ibarra bound, M sub one approximately equals ten to the power of nine gigaelectronvolts. Davidson and Ibarra found an upper bound, proportional to M sub one, on the absolute value of the C P-asymmetry of the decays of the lightest heavy Majorana neutrino. This constrains the regions of parameter sin w which successful leptogenesis may occur as a function of M sub one. This translates into the DI bound on M sub one itself as the minimum value required for successful leptogenesis. There have been a number of in-depth numerical studies which support this bound and require M sub one greater than ten to the power of nine gigaelectronvolts in conjunction with a bound on the lightest neutrino mass, m sub one less than or equal to zero point one electronvolt.

The original derivation of this bound makes some simplifying analytical assumptions and hence is subject to three caveats: only the lightest heavy Majorana neutrino decays; the heavy Majorana neutrino mass spectrum is hierarchical; and flavour effects, which account for the differing interaction rates of the charged-lepton decay products of the heavy Majorana neutrinos, are ignored. In this work, we shall investigate scenarios of three-flavoured thermal leptogenesis in a more general setting than these conditions allow. We shall then consider lower heavy Majorana neutrino masses at scales M one approximately equals ten to the power of six gigaelectronvolts. Given the existence of low-scale leptogenesis models at the TeV scale, we shall refer to this as "intermediate" scale leptogenesis.

There are several reasons to explore leptogenesis at intermediate scales. Firstly, the introduction of heavy neutrinos to the Standard Model leads to a correction to the Higgs mass which may potentially be unnaturally large. This is because the correction to the electroweak parameter p two (the negative of the coefficient in the quadratic term of the Higgs potential), is proportional to the light neutrino masses and to M cubed, with M the heavy Majorana neutrino mass scale. In order to avoid corrections to p two larger than say one TeV squared one requires the lightest pair of Majorana neutrino masses to have M one less than or equal to four times ten to the power of seven gigaelectronvolts and M two less than or equal to seven times ten to the power of seven gigaelectronvolts. Secondly, there is a tendency for baryogenesis models to reside at the GeV- or GUT-scales which leaves intermediate scales relatively unexplored. Finally, thermal leptogenesis at intermediate scales may resolve a problem that arises in the context of supersymmetric models which include gravitinos in their particle spectrum. Gravitinos have interaction strengths that are suppressed by the Planck scale and consequently are long-lived and persist into the nucleosynthesis era. The decay products of the gravitinos can destroy helium four and deuterium nuclei and ruin the successful predictions of nucleosynthesis. Thus, in order to reduce the number of gravitinos present at this stage, one requires a reheating temperature less than a few times ten to the power of nine gigaelectronvolts depending on the gravitino mass.

The scale of leptogenesis may be lowered through the introduction of a symmetry to the SM. Non-resonant thermal leptogenesis is explored at intermediate scales in the context of small B minus L violation. It is shown that the DI bound may be evaded because, in the context of this near-symmetry, the lepton number conserving part of the CP asymmetries can be enhanced as they are not connected to light neutrino masses. It is shown that the scale may be lowered to one hundred six GeV. An alternative symmetry-based approach is to introduce supersymmetry in which one may also reduce the scale of leptogenesis to intermediate scales. In this context, the bound on the absolute value of the CP-asymmetry found by Davidson and Ibarra is greatly enhanced. Consequently, the DI bound is lowered thus allowing for the possibility of intermediate scale leptogenesis.

Beyond the application of supersymmetry and heavy pseudo-Dirac neutrinos, there are other means of lowering the scale of leptogenesis; if the decaying heavy Majorana neutrinos are near-degenerate in mass, the indirect CP-violation may be resonantly enhanced and subsequently the mechanism may be lowered to the TeV scale. This has been explored in the context of type-One, Two, and Three seesaw mechanisms. Another mechanism, proposed by, is one in which leptogenesis is achieved via CP-violating heavy Majorana neutrino oscillations. The generation of the lepton asymmetry takes place close to the electroweak scale and the associated GeV-scale heavy Majorana neutrinos may be searched for at a variety of experiments such as LHCb, BELLE Two, and the proposed facility, SHIP. Although, leptogenesis via oscillations is a testable and plausible explanation of the baryon asymmetry, it has been shown its simplest formulations require a certain amount of fine-tuning.

In this article, we revisit the question: how low can the scale of thermal leptogenesis go? We focus solely on the possibility that the heavy neutrinos are Majorana in nature and find thermal leptogenesis is possible at intermediate scales without resonant effects. In addition, we present an in-depth numerical study of the dependence of the baryon asymmetry produced from non-supersymmetric thermal leptogenesis on the low and high-scale model parameters.

The work presented in this paper is structured as follows: in Section Two we review the origins of light neutrino masses in the type-One seesaw framework, further we review the Casas-Ibarra parametrisation of the Yukawa matrix and then introduce a modification of this parametrisation in the presence of large radiative corrections. We end this section by introducing a measure of fine-tuning in the context of the neutrino masses. In Section Three we discuss the motivations for and some theoretical aspects of thermal leptogenesis. We follow in Section Three A with the density matrix equations we shall solve to calculate the lepton asymmetry. We demonstrate in Section Three B, that the fully flavoured Boltzmann equations, which do not incorporate flavour oscillations, may significantly qualitatively differ from the lepton asymmetry calculated from the density matrix equations and justify the use of semi-classical density matrix equations rather than kinetic equations derived from first principles non-equilibrium quantum field theory. Our numerical methods are described in Section Four. The results of our numerical study for one and two decaying heavy Majorana neutrinos are presented in Section Five A and Section Five B respectively. In Section Six we explore the analytical consequences of the numerical results and provide an explanation for the fine-tuning. Finally, we summarise and make some concluding remarks in Section Seven.