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Corresponding author.

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We analyze the feasibility of low-scale leptogenesis where the inverse seesaw (ISS) and linear seesaw (LSS) terms are not simultaneously present. In order to generate the necessary mass splittings, we adopt a minimal lepton flavor violation (MLFV) hypothesis where a sterile neutrino mass degeneracy is broken by flavor effects. We find that resonant leptogenesis is feasible in both scenarios. However, because of a flavor alignment issue, MLFV-ISS leptogenesis succeeds only with a highly tuned choice of Majorana masses. For MLFV-LSS, on the other hand, a large portion of parameter space is able to generate sufficient asymmetry. In both scenarios we find that the lightest neutrino mass must be of order

The type-I seesaw model

This scenario has been extensively studied and it has long been known that in the standard hierarchical

Hierarchical is typically taken to mean

We define low-scale leptogenesis to occur at

Lowering the thermal leptogenesis scale is an attractive and obvious way to resolve the above tension, and forms the basis of the present analysis. The scale of leptogenesis can be significantly lowered if a quasidegenerate (meaning not exactly degenerate) spectrum of masses for the SNs is assumed

A possible method of lowering the scale of thermal leptogenesis with potential cLFV signals is by introducing two SN states (with opposite lepton number assignments) per light neutrino and simultaneously promoting lepton number to being a “good symmetry” of the theory. For certain regimes of the couplings a double suppression of the active neutrino masses occurs, allowing significantly larger Yukawa couplings for SNs at TeV-scale masses compared to the minimal scenario. Additionally, due to the weakly broken lepton number symmetry, the heavy SNs form pseudo-Dirac states with mass splittings proportional to the small lepton number breaking terms. For specific choices of parameters, two popular limiting cases constitute the “inverse seesaw” (ISS)

In the specific case of the LSS above the electroweak phase transition, the heavy SNs are degenerate in mass and therefore self-energy diagrams vanish, leaving only the highly suppressed vertex contributions to produce the

In addition to the intrafamily SN degeneracies, there is also the possibility of a quasidegeneracy between SNs from different families, a flavor degeneracy in other words. To that end we explore a scenario in which an

We work in the framework of an extended version of the minimal lepton flavor violation (MLFV)

In the absence of neutrino mass. If neutrino mass is included more rotations are present.

flavor space rotations which leave the Lagrangian invariant. Therefore an MFV or MLFVUnder this

Because of the multiple distinct ways in which neutrino masses can be incorporated into the SM there is no unique way of defining an MLFV

See

Recently it was found that flavor effects can still be relevant at much higher temperatures than thought previously, potentially reducing the allowed parameters space even further

The minimal flavor violation

Representations of the SM quark fields under the Abelian and non-Abelian parts of

The Yukawa terms in the SM are not invariant under these flavor transformations, but can be made invariant if the Yukawa matrices are “promoted” to be

In this work, promoting a Yukawa coupling to be a dynamical field will be represented by

Representation assignments for Yukawa spurions such that the Yukawa terms in the MFV effective theory respect the flavor symmetry which exists in their absence.

Quark masses are generated in this framework when the spurionic fields acquire nonzero VEVs alongside the SM Higgs doublet. These background values relate to the

The MFV

The flavor symmetry present in the SM quark kinetic terms is a “good” symmetry which the high-scale sector respects. The

Therefore under this

The SM is an effective theory for which all renormalizable and nonrenormalizable operators must respect both the gauge and

SM effective field theory operators which describe flavor changing processes require insertions of spurion combinations in order to become flavor invariant. For example, the four fermion operator

Of particular importance when adapted to resonant leptogenesis, which will be discussed in more detail later, all terms in the lowest-order MFV Lagrangian will receive corrections to their couplings from spurion terms which transform in the same way as the basic spurion under the flavor symmetry. For example, in

The motivation behind adopting the MFV

Approximate lower bounds on the scale

A key advantage of the former assumption is that strict relations between different flavor changing processes arise due to precise measurements of the CKM parameters. Therefore MFV-like theories have a degree of predictivity which can aid experimental searches. In the case of an exact MFV

Predictions for some rare hadronic observables under the MFV

Most channels have measurements (or upper limits) in agreement with MFV predictions; however, a discrepancy exists for

While the MFV

Similar to the quarks and MFV, the gauge sector of the leptons obey a flavor-transformation invariance which can be defined analogously to Eq.

Predictivity can be recovered if only one of the spurion fields is taken to have nontrivial flavor transformations, with the other not acting as a source of flavor-symmetry breaking. Usually, it is assumed that the Majorana mass term does not act as a source of lepton-flavor breaking and, and as in the quark sector, only the Yukawa couplings are responsible. If it is assumed that the Majorana mass term is lepton-flavor blind, then

An alternative approach is to assume

Allowing for leptonic

As already emphasized, MFV and MLFV should be understood as hypotheses motivated by experiment and do not provide a complete UV description of the flavor sector. Attempts have been made in moving from this

It was noted in

Scale separation, however, does not occur in the minimal version of the type-I and type-III seesaw mechanisms where only one new scale is introduced such that

The simplest

In

As before,

As with the quark sector and the example of the minimal type-I seesaw model, the kinetic terms in Eq.

The leading-order flavor degeneracy amongst the heavy SNs is built into the

The case of the simplest MLFV scenario has been explored in detail

In the case of the MLFV

In order to calculate the

Alternatively, the consideration of higher-order corrections to the Majorana mass which are not all aligned in flavor space will cause nonzero real entries in the relevant term of Eq.

Because of the extra fields introduced, the above flavor alignment problem does not occur in general for the extended seesaw models. Consider the full

In this section we briefly describe the procedure and formalism we employ for estimating the efficiency of asymmetry generation. A more detailed explanation of our conventions, definitions and numerical calculations can be found in

As is conventional, we work in the heavy sterile neutrino mass basis, i.e., the bottom-right submatrix of Eq.

The combination

This would imply that

We consider a low-scale scenario in which the sterile masses are set to

We choose the regulator of

Specifically for the LSS scenario, it is important to note that the second term in Eq.

We will work under the condition that, due to the strong washout nature of the temperature regime we favor

In full analogy to the type-I scenario, all appropriate flavor invariant operators must be included. This leads to corrections to the Majorana mass terms from spurion combinations transforming the appropriate way such that when coupled with the heavy SN fields, the term is flavor invariant at the high scale. Now in the case of the extended seesaw both Majorana masses receive corrections, as per

The coefficients

We consider the general scenario where one copy of

In the limiting case of the ISS, only the first line of Eq.

At one-loop order however, additional terms are generated for the active neutrino masses which can be important

Combining the tree-level and one-loop contributions at leading order leads to a general light-neutrino mass matrix in our MLFV

For both the ISS and LSS we fix

We fit active-neutrino data by fixing the Dirac mass matrix

The parametrization is only approximate as we ignore the corrections to the mass terms e.g.,

The matrix

In order to reduce the number of free parameters we fix the real components of the mixing angles to

By contrast to the complex-orthogonal

The active-neutrino masses

List of experimental measurements of the parameters in the PMNS matrix and the light neutrino mass splittings fixed by active neutrino oscillation experiments. The light neutrino mass differences were allowed to vary within

We work in the perturbative regime of the Yukawa couplings generated by the Casas-Ibarra parametrizations such that

To check our numerical solutions to the Boltzmann equations, we compare the results to a known approximate analytic expression of the baryon asymmetry that is valid in the strong washout regime (

The asymmetry is approximated by

Comparison between the numerically computed asymmetry

Finally, the small mass differences between the heavy sterile states generating the resonant enhancement could also lead to coherent oscillations between the SNs. The dynamics of the coherent oscillations will alter the evolution of the lepton asymmetry and could potentially significantly impact the net asymmetry generated for some region of parameter space. To properly account for their effects would require a flavor-covariant set of transport equations, as opposed to the semiclassical Boltzmann equations we employ. We will therefore estimate the impact of coherent sterile neutrino oscillations on the final baryon asymmetry by employing an analytic estimate derived specifically for resonant scenarios of leptogenesis

For the ISS (

We separately consider three scenarios for the ISS: first where no corrections are introduced, second where corrections are introduced at the next lowest order, labeled

For the ISS as

This is because

As only the off-diagonal blocks of

In Fig.

Plot of the asymmetry generated in the ISS for the individual cases where no radiative corrections are considered (orange), where only

Figure

Plot of the asymmetry generated in the ISS when both

Additionally, the effects of coherent oscillations are estimated (in orange) in the bottom plot of Fig.

Summarizing, we find two criteria for successful MLFV-ISS resonant leptogenesis: (1) large values of the Majorana mass

Plot of the decay width

These conclusions where drawn for fixed values of certain parameters. Most importantly the Wilson coefficients were fixed such that

Variation in the baryon asymmetry as a function of the lightest active neutrino mass

Asymmetry generation is sufficient for large Majorana masses not simply because of the enhancement in the mass splitting. Due to the relationship between the Yukawa couplings required to satisfy active neutrino mixing data and the input parameters from Eq.

Plot of the washout as a function of

While the naïve washout grossly overestimates the efficiency of washout, it is clear that for large values of the LNV parameters palatable values of washout (albeit still very much in the strong washout regime) of

Figures.

Due to the anarchic nature of our scenario there is no preference for a specific lepton flavor.

Plot of the

Plot of the

Finally in Fig.

Plot of the baryon asymmetry as a function of the complex angle

Here we briefly discuss the consequences of MLFV-ISS on low-energy cLFV processes, specifically the impact that the introduction of

In the SM effective field theory framework the process

A simple expression for the branching ratio is obtained

In Eq.

Many detailed explorations of cLFV in MLFV have been made for the minimal type-I seesaw scenario

Figure

Plot of the ratios

Similar plots are presented in Fig.

Plot of the ratios

Finally in Fig.

Plot of the branching ratio

Successful MLFV-ISS resonant leptogenesis, however, requires very large values of

For the LSS (

For the ISS scenario the corrections to the Majorana mass

Here the dimensionful parameters

We fix these parameters to

Figure

Plot of the asymmetry generated in the LSS for the three scenarios (left) and comparing the asymmetry generated from mixing to oscillations (right) as a function of

Plot of the effective washout to a specific lepton flavor

Points in orange correspond to spurion insertions at lowest order and points in cyan include all relevant terms. Now, due to the inclusion of the radiative Majorana masses, mass splittings occur between the six heavy SNs. The self-energy component of Eq.

The right side of Fig.

Figure

Plot of the

To illustrate this, Fig.

Plot of the mass splitting

In Fig.

Variation in the baryon asymmetry as a function of the lightest active neutrino mass

Based on these two scenarios, we conclude that successful MLFV resonant leptogenesis will also occur if the ISS and LSS were operative together. Appropriate choices for the now three LNV parameters based on the two scenarios here will allow for minimized washout with mass splittings related to the heavy SN decay widths for the necessary resonance to occur. However, as resonant leptogenesis is already feasible

Finally, in Fig.

Plot of the baryon asymmetry as a function of the complex parameter

Once again we briefly consider the prospects of detection of MLFV-LSS through cLFV processes. Unlike in the MLFV-ISS scenario a much less tuned region of parameter space is required in order to generate the necessary baryon asymmetry which may lead to improved detection prospects.

As before we require insertions of spurions transforming as a

Figure

Plot of the ratios

Figure

Plot of the ratios

Finally, Fig.

Plot of the branching ratio of

We have studied a well-motivated way in which small mass splittings between heavy SNs from different families may arise, within both the ISS and LSS frameworks, such that leptogenesis is possible despite the strong washout present in the theory. Previously it was found that while a mass splitting naturally exists for the ISS it is not sufficient in order for resonant leptogenesis to occur. For the LSS the degeneracy amongst the SNs at the high scale prevents significant asymmetry generation. While leptogenesis is feasible when all LNV terms are switched on (the

In the context of broken flavor symmetries and the MLFV hypothesis, a degeneracy amongst the heavy SNs is naturally produced, for the purposes of having a predictive theory. The degeneracy is then broken by higher-order spurion VEV contributions, leading to a parameter region consistent with resonant asymmetry generation. In order for the desired splitting the occur in the intended way during cosmological evolution, the critical temperature at which the spurions acquire their nonzero VEVs must be assumed to be above the scale of thermal leptogenesis.

We found that for MLFV-ISS only a small region of very large Majorana masses is able to generate the required asymmetry. Here asymmetry generation requires next-to-leading order corrections to be included, therefore suppressing the overall size of the

For MLFV-LSS a large region of parameter space is capable of satisfying the resonance condition simultaneously with the minimized washout required for successful asymmetry generation. Here corrections at lowest order are not flavor aligned and therefore much larger values of the

In both cases we estimated the impact of the lightest neutrino mass

This work was supported in part by the Australian Research Council. T. P. D. thanks Vincenzo Cirigliano and Luca Merlo for clarifications related to MLFV. T. P. D. is grateful to Uli Felzmann for access to computing hardware on which our scans were conducted.