Supported by the National Natural Science Foundation of China (12135006, 12075097, 12047527, 11775092), as well as by the Fundamental Research Funds for the Central Universities (CCNU20TS007, CCNU19TD012, CCNU22LJ004)
The Dirac neutrino masses could be simply generated by a neutrinophilic scalar doublet with a vacuum being dramatically different from the electroweak one. While the case with an eV-scale vacuum has been widely explored previously, we exploit in this work the desert where the scalar vacuum is of
Article funded by SCOAP3
The observations of neutrino oscillations have thus far prompted several puzzles about neutrinos in the Standard Model (SM), including their tiny mass origin, Dirac or Majorana nature, and distinctive Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing pattern from that of the quark Cabibbo–Kobayashi–Maskawa mixing matrix. Any attempt towards these problems has catalyzed a host of investigations from theoretical constructions to experimental searches, and from low-energy particle physics to high-temperature early Universe. Despite the leading interests in the Majorana neutrinos, the Dirac neutrino scenarios on their own right can help to explain some fundamental problems encountered in the SM, such as the baryon asymmetry of the Universe (BAU) [
A simple and testable scenario for generating the Dirac neutrino masses is obtained by introducing a new Higgs doublet with a much smaller vacuum than the electroweak one and an exclusive interaction to the neutrinos,
When the vacuum reaches the keV scale, a purely thermal Dirac leptogenesis mechanism [
In addition to providing a simple framework to explain the Dirac neutrino mass origin and the matter-antimatter asymmetry in the early Universe, the keV-vacuum neutrinophilic 2HDM can manifest itself via a significant contribution to the
The purpose of this work is, therefore, to explore the desert where the vacuum in the neutrinophilic 2HDM populates in the keV scale by focusing on its potential effects in the cosmological regime. We will assume that the new scalar mass is determined by the electroweak vacuum such that the perturbative unitarity condition sets the mass bounded from above. It will be shown that this scenario is more realistic and also more predictable than the one considered in Ref. [
In Sec. II, we revisit briefly the keV-vacuum neutrinophilic 2HDM with a quasi-degenerate scalar mass spectrum. Then, in Sec. III, we show the inert property of the scenario in the low-energy flavor physics with typical examples from the severely constrained LFV transitions. In Sec. IV, we illustrate the
The Dirac neutrino masses could be simply generated by coupling the right-handed Dirac neutrinos
in addition to the SM content. A global
After the spontaneously electroweak symmetry breaking specified by
where
The resulting hierarchy
where
and will be bounded from above owing to the perturbative unitarity condition [
Most previous studies on the neutrinophilic 2HDM focused on a vacuum
As investigated already in Ref. [
Here,
(color online) Current neutrino mass spectrum in the normal ordering. The lightest neutrino mass has a maximum value
For the LFV
Similar to that observed for the
While we have only demonstrated above that the keV-vacuum neutrinophilic 2HDM has negligible effects on the
In fact, similar inert effects are also expected in electroweak precision observables, such as the invisible
Finally, let us make a brief comment on the LHC phenomenology. The current LHC constraints on the keV-vacuum neutrinophilic 2HDM are weak. For instance, the
From the discussion in the last section, any observation of the LFV processes with the future forecast experimental sensitivities can rule out the neutrinophilic 2HDM with a keV-scale vacuum in explaining the Dirac neutrino masses. While being nearly secluded from the low-energy flavor physics, such a minimal framework with a keV-scale vacuum can successfully realize a purely thermal Dirac leptogenesis to explain the BAU problem, as detailed in our recent sequential studies [
where
with
The purely thermal Dirac leptogenesis applies, for the first time, the finite-temperature effect to the leptogenesis scenarios where the SM neutrinos are of Dirac type [
It is especially interesting to note that, after some simple basis transformations, the coupling phase is entirely provided by the Dirac CP-violating phase in the PMNS matrix, rendering therefore a direct link between the low-energy CP violation and the high-temperature CP asymmetry [
Concerning the first possibility that has not been discussed in Ref. [
with the summed index
with
With respect to the second possibility, as the lightest neutrino
Let us now proceed to compute the baryon asymmetry within the framework outlined in Sec. II. Our starting point is the general formula of the baryon asymmetry [
where the factor
where the indices
Note that the temperature integration in Eq. (12) is performed within the range
Before electroweak symmetry breaking, the decay rate with a final state
Here
In the finite-temperature regime, the thermal lepton mass
where
Within the temperature range considered in Eq. (12), we can approximately simulate the mass parameter
To evaluate the baryon asymmetry numerically, we use the central values of the PMNS mixing angles
where
(color online) Dependence of the function
To realize a successful
The out-of-equilibrium condition for realizing the
leading therefore to
Notice that, because
where we have replaced the Yukawa couplings
we can estimate Eq. (20) as
where
with the relativistic scalar distribution
which indicates that the left-hand (right-hand) side of the above condition is stronger (weaker) than that of Eq. (23). This is because
The above condition now implies an upper bound of
(color online) Upper bounds on the lightest neutrino mass
Finally, we would like to make a comparison between the analysis presented above and that in Ref. [
As discussed in Sec. III, there is rare hope to see the LFV signals in neutrinophilic 2HDM with a keV-scale vacuum. Nevertheless, the expected sensitivity is dramatically different in the cosmic regime, especially given the fact that the current precision of astrophysical and cosmological observations is now making the probes of feeble couplings and light species strikingly possible [
Owing to the feeble Yukawa coupling, the lightest right-handed Dirac neutrino is essentially produced via the freeze-in mechanism [
where the d.o.f
where we have neglected the inverse decay and the Pauli-blocking effects,
The phase-space integral of Eq. (28) can be calculated following Ref. [
we can rewrite Eq. (28) as
where
summing over the three lepton flavors. Integrating Eq. (30) first over
where the Boltzmann distribution
Note that the integration with a full Bose-Einstein statistics can also be performed analytically, which, nevertheless, does not cause significant numerical difference with respect to the Boltzmann distribution. This is expected because the potential Bose-Einstein enhancement at the soft momentum regime is now suppressed by the thermal mass correction, which can reach
The energy density at the frozen-in temperature
where we have taken the approximation
where the SM energy density is given by
For the two thermalized right-handed Dirac neutrinos, their energy densities have the thermal spectrum
where the energy density from the antiparticle has been taken into account. After the two heavier right-handed Dirac neutrinos freeze out, the energy densities at late times will be determined by Eq. (35).
The extra radiation contributing to the SM plasma in the early Universe can be parameterized by a shift of the effective neutrino number via
where
where
Applying the above equations, we find that the
(color online) Left:
Note that
with a nonrelativistic
with
Currently, a combined constraint from BBN and CMB, and, in particular,
We show in
(color online) Our prediction of the
The fact that there is no observation as yet of Dirac neutrino related signals in both direct collider detection and indirect low-energy flavor physics could just hint towards a feeble Dirac neutrino Yukawa regime. This motivates us to consider in this work the neutrinophilic 2HDM with a keV-scale vacuum. We have shown that, while the framework is inert in the low-energy flavor physics such that any observation of the LFV processes is not expected in the future experiments, the promising sensitivities from the cosmological regime can probe such an LFV-inert model. Besides being distinguishable from the lighter-vacuum cases with observable LFV processes, the scenario considered can also provide a correlative explanation of the BAU problem, the Dirac neutrino mass origin, and an observable
The relation
Here one should note that the decay rate is calculated in the basis where the charged lepton Yukawa matrix is diagonal, and we have used a thermal scalar mass
Since the neutrino Yukawa coupling is small, the decay rate induced by the effective four-fermion interaction mediated by the scalar is much smaller than that of the two-body scalar decay, and thus the frozen-out temperature of
With these factors factored out, the collision amplitude-squared signifies the decays of each scalar and its CP-conjugation in the doublet component, where each scalar has only one inertial d.o.f. If
For