Supported by European Research Council under ERC Grant NuMass (FP7-IDEAS-ERC ERC-CG 617143), H2020 funded ELUSIVES ITN (H2020-MSCA-ITN-2015, GA-2015-674896-ELUSIVES), InvisiblePlus (H2020-MSCA-RISE-2015, GA-2015-690575-InvisiblesPlus), and Fermi Research Alliance, LLC under Contract (DE-AC02-07CH11359)

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

In this paper, we introduce leptogenesis via a varying Weinberg operator from a semi-classical perspective. This mechanism is motivated by the breaking of an underlying symmetry which triggers a phase transition that causes the coupling of the Weinberg operator to become dynamical. Consequently, a lepton anti-lepton asymmetry arises from the interference of the Weinberg operator at two different spacetime points. Using the semi-classical approach, we treat the Higgs as a background field and show that a reflection asymmetry between leptons and anti-leptons is generated in the vicinity of the bubble wall. We solve the equations of motion of the lepton and anti-lepton quasiparticles to obtain the final lepton asymmetry.

Article funded by SCOAP^{3}

The origin of tiny neutrino masses and the asymmetry between baryons and anti-baryons in the Universe are two fundamental and open questions in particle physics. An important theoretical development linking both is baryogenesis via leptogenesis [

Recently, we proposed a new mechanism to generate the lepton asymmetry via the Weinberg operator [

where

● The Weinberg operator violates the lepton number by two units and triggers lepton-number-violating (LNV) processes, including

and their CP conjugate processes, where

● After electroweak symmetry breaking (EWSB), the Higgs acquires a VEV

This operator violates the lepton number and generates Majorana masses for neutrinos. As the primary motivation for the Weinberg operator is the generation of tiny neutrino masses, all processes triggered by this operator are very weak [

where

In our mechanism, CP violation is provided by a CP-violating phase transition (CPPT) in the very early Universe. This phase transition causes the coefficient of the Weinberg operator to be dynamically realised and to contain irremovable complex phases. Such a phase transition is strongly motivated by a variety of new symmetries such as

In this paper, we present a simplified and intuitive description of this mechanism based on the semi-classical approximation. In order to do so, we follow the method introduced in [^{①)}

This work, along with several others [

We emphasise that the CPPT mechanism works only if the UV-completion scale,

We organise the remainder of this paper as follows: we first review the mechanism in Section 2; we then state the main assumptions of the semi-classical description in Section 3. Finally, we present the calculation of lepton asymmetry in Section 4 and make concluding remarks in Section 5.

The Weinberg operator of Eq. (1) is the simplest higher-dimensional operator needed to explain tiny neutrino masses. As discussed in Refs. [

Assuming a first order phase transition, bubbles of the leptonically CP-violating broken phase nucleate. We denote the bubble wall width and bubble wall velocity as

Before CPPT is triggered, there are equal amounts of leptons and anti-leptons in the thermal plasma and they are thermally distributed. Once CPPT begins, a bubble nucleates with the bubble wall separating the symmetric and broken phases, which are denoted in

Lepton and antilepton reflection off the bubble wall during a phase transition from Phase I (

We note that the coefficient of the Weinberg operator varies only along the

We denote the amplitude for a transition from lepton to anti-lepton at

The interference of the Weinberg operators at different

where

In the following, we carry out the semi-classical approximation to relate

The CPPT mechanism shares a common feature with EWBG, namely that a phase transition is necessary to drive the generation of a baryon asymmetry. However, the two mechanisms differ markedly and it is worthwhile to remark on the features which distinguish them. First, in EWBG, the baryon number violation is provided by sphaleron transitions in the symmetric phase. Both the out-of-equilibrium condition and the C/CP violations are induced by the EW phase transition. Therefore, in EWBG, the phase transition is key to the generation of the non-equilibrium evolution. In order to achieve this, rapidly expanding bubble walls are required, such that the back-reactions are not efficient in washing out the generated baryon asymmetry. In the CPPT mechanism, the

In this section, we introduce the semi-classical approximations we use for the lepton asymmetry calculation. Firstly, we introduce the equations of motion (EOM) for the leptonic doublets and the effective mass-like matrix which parametrises the lepton anti-lepton transitions. Secondly, we review our treatment of the Higgs as a background field.

We begin from the well-known equation of motion for Majorana neutrinos at low energy. It is expressed as

where

In the early Universe, when the Higgs is in its symmetric phase, the Higgs field may fluctuate. Such fluctuations can be enhanced by temperature, and influence the behaviour of neutrinos as well as of the charged leptons. For this reason, we treat the Higgs as a background field. Taking into account the

In the

where we have made the

As the Majorana mass-like matrix,

As a complex field, the mean value is given by

where we have ignored the effective thermal masses and chemical potential of the Higgs. It is worth noting that the mean values of

Another interesting property is that the mean value
^{②)}

It is proved in the following. For a real scalar

The expectation values for

For the detailed discussion of correlations between lepton asymmetry and Wightman propagators, please see Ref. [

The concept of quasiparticles has been known for many decades [

where

Here, we have required

with

The energy-dependent term does not contribute to the CP violation in the wall rest frame, and thus we do not include it in the following discussion.

The calculation of lepton asymmetry generated by CPPT follows from solving the EOM for the leptonic doublet quasiparticles.

We now consider the amplitude of

The first step is to consider the propagation of quasiparticles in Phase I, where we restrict our discussion to the

In order that there are no sources of quasiparticles at spatial infinity, the boundary conditions

are necessary.

The solution of the Green functions with these boundary conditions is given by

The lepton quasiparticle will propagate from Phase I into Phase II. For this purpose, we may consider leptons with a

Since the Weinberg operator is relatively weakly coupled to the thermal plasma, we ignore all corrections

We can calculate the amplitude for

Finally, we obtain the CP asymmetry of the amplitude (defined in Eq. (5)) as

This quantity is determined by: 1) the momentum change

The momentum difference

We now discuss in detail the term

The total contribution with all flavour summed together is given by

For CP asymmetry between

Ignoring the energy-momentum exchange between leptons and the Higgs, and taking mean values on the right-hand-side and using (11) and (12), we obtain

The average among gauge components is given by

Taking into account the results for

which is qualitatively the same as our previous result [

In this paper, we apply the semi-classical approximation to calculate the lepton asymmetry generated by a varying Weinberg operator. Firstly, we approximate the Higgs field as a background field. Following this treatment, we can effectively regard the Weinberg operator as an effective “Majorana mass term” for the leptonic doublet. Then, we write out the EOM for both lepton and anti-lepton quasiparticles, in which the “Majorana mass term” results in lepton anti-lepton transition. During the CP-violating phase transition, the “Majorana mass term” varies with spacetime, and the transitions from lepton to anti-lepton and from anti-lepton to lepton are not equal. This treatment is analogous to the approximation used in EWBG, where the varying fermion mass results in asymmetric transitions between left-handed and right-handed components.

In the semi-classical approximation, we do not try to provide quantitatively precise results for lepton asymmetry as the energy-momentum transfer with the Higgs has been ignored. However, this simplified treatment allows to present the mechanism more intuitively. Moreover, one of the main results of this paper is that, in the single scalar case, the number density asymmetry between leptons and anti-leptons