Gravitational footprints of massive neutrinos and lepton number breaking

We investigate the production of primordial Gravitational Waves (GWs) arising from First Order Phase Transitions (FOPTs) associated to neutrino mass generation in the context of type-I seesaw schemes. We examine both"high-scale"as well as"low-scale"variants, with either explicit or spontaneously broken lepton number symmetry. In the latter case, a pseudo-Goldstone boson, dubbed majoron, may provide a candidate for warm or cold cosmological dark matter. We find that schemes without majoron lead to either no FOPTs or too weak FOPTs, precluding the detectability of GWs in present or near future experiments. Nevertheless, we found that, in the presence of majorons, one can have strong FOPTs and non-trivial primordial GW spectra which can fall well within the frequency and amplitude sensitivity of upcoming experiments, including LISA, BBO and u-DECIGO. We further analyze the associated types of FOPTs and show that in certain cases, the resulting GW spectra entail, as characteristic features, double or multiple peaks, which can be resolved in forthcoming experiments. We also found that the majoron variant of the low-scale seesaw mechanism implies a different GW spectrum than the one expected in the high-scale majoron seesaw. This feature will be testable in future experiments. Our analysis shows that GWs can provide a new and complementary portal to test the neutrino mass sector.

The detection of Gravitational Waves (GWs) by the LIGO team has opened an entirely novel method to probe the underlying new physics associated to neutrino mass generation. It was advocated that the spectrum of primordial GWs, potentially measurable at the currently planned GW interferometers, may represent an important cutting-edge probe for new physics. This follows from the fact that these interferometers can be sensitive enough to measure the echoes of the possible First Order Phase Transitions (FOPTs), which might have happened in the past cosmological history [28].
In this letter, we focus on possible gravitational footprints of the various variants of the popular type-I seesaw mechanism for Majorana neutrinos. The relevant part of the Lagrangian is given by Here, L = (ν, l) T are the SM lepton doublets, H is the SM Higgs doublet, ν c are the three SM singlet "righthanded" neutrinos.
The lightness of the left-handed neutrinos is then ascribed to the heaviness of the "right-handed" isosinglet partners e.g. for Another popular realization of this idea is the "low-scale" variant, in which two gauge singlet fermions ν c and S are added sequentially to the SM particle content [16][17][18][19][20]. The template of these schemes has exact conservation of lepton number and, as a result, strictly massless neutrinos. Yet flavor is violated to a potentially large degree, subjected only to constraints from weak interaction precision observables, such as universality tests [29][30][31][32][33]. To this template one adds a small seed of lepton number violation, leading to nonzero neutrino mass. One example is the so-called "inverse seesaw" mechanism, where the smallness of the neutrino mass is linked to the breaking of the lepton number symmetry U (1) L to its Z 2 subgroup, through the so called µ-term. The relevant part of Lagrangian in this case is given by where µ is also a 3 × 3 symmetric matrix. The light neutrino mass is then given by Note that small neutrino masses are "protected", since m ν → 0 as the lepton number symmetry gets restored by having µ → 0 [16][17][18][19][20]. In this case there can be sizable unitarity violation in neutrino propagation [34][35][36].
For both high-and low-scale seesaw, one can have spontaneous breaking of U (1) L → Z 2 , leading to the so-called majoron variants of the seesaw [17,37,38]. This is accomplished by adding the SM singlet scalar σ, which carries two units of lepton number charge. Then σ ≡ v σ spontaneously breaks U (1) L → Z 2 , leading to a dynamical explanation of the small neutrino masses. To get the majoron variants of minimal type-I and inverse seesaw one should replace (1) and (3), respectively. An additional attractive feature of majoron models is the existence of a pseudo Nambu-Goldstone boson providing a good [39][40][41], and testable [42,43] dark matter candidate.
Gravitational waves from FOPTs In order to characterize the features of the GWs originating from FOPTs in seesaw schemes, we calculate the saturated latent heat and compare it to the latent heat released by the transition. Bubbles run away if the amount of released latent heat is bigger than the saturated one. Usually, the bubbles do not run away, while for a smaller fraction of those that do run away, we use the proper procedure outlined in Ref. [44].
For the case of non-runaway nucleated bubbles, the intensity of the GW radiation grows with the strength of the transition, given by the ratio v n /T n , where v n ≡ v h (T n ) is the Higgs vev at the bubble nucleation temperature T n . From the discussion in Ref. [45,46], it follows that bubble wall collisions do not provide an efficient way of producing GWs in the models of interest to us here. As a result, GWs originate mainly from two sources: II. Sound shock waves (SW) of the early Universe plasma, generated by the bubble's violent expansion.
These contributions arise over transient times at the early Universe and get subsequently "redshifted" by the expansion. To a present observer this appears as a cosmic gravitational stochastic background. Intuitively, one expects that from any of these leading order contributions, a high wall velocity is necessary to generate detectable GWs. Our numerical analysis confirms the general expectation that the bubble wall velocities are close to the speed of light, i.e. v b 1. Besides, in our results the SW contribution dominates the peak frequency and the peak amplitude, while the tails are mostly set by the MHD turbulence term. For certain parameter configurations one also expects sequential phase transition patterns leading to potentially resolvable multi-peak GWs spectra studied for the first time in [47].

Seesaw-induced Gravitational Waves
First, we note that within the type-I seesaw mechanism with explicitly broken lepton number, no FOPTs are obtained. The heavy isosinglet neutrinos practically decouple at the EW scale, and do not alter the nature of the EW phase transition. In contrast, in the inverse seesaw mechanism the singlet neutrinos lie closer to the EW scale, and can have a sizable coupling to the Higgs boson. This can alter the EW phase transition, making it first-order. Indeed, we find FOPTs for many points in parameter space. Nevertheless, as shown in Fig. 1, the expected "intensity" parameter h 2 Ω peak GW lies far below the sensitivity of any conceivable experiment, rendering the testability of this scenario very remote. In Fig. 1, we have allowed the mass parameter M to vary between 50 GeV and 500 GeV, with the Yukawa coupling Y ν between 0.2 and 3.0. For completeness, we have studied the possibility of having more than three families of heavy singlet neutrinos. As mentioned, the lightness of m ν follows from the small µ-term. As seen from Fig. 1, even for large values of Y ν ∼ O(1), the FOPTs remain weak, leading to an undetectable GW signal. This follows from the fact that the fermions affect the phase transitions only at the loop level. Even if we increase their number up to 30 1 , the GW signal is not enhanced enough to be detectable.
When the seesaw mechanism is associated with the spontaneous breaking of the lepton number symmetry, the situation changes dramatically. First of all, the spontaneous breaking of U (1) L together with the EW symmetry allows for a richer pattern of FOPTs. Indeed, not only the heavy isosinglet fermions, but also the complex scalar field σ driving the spontaneous breaking of U (1) L , can couple substantially to the Higgs boson. This can generate two peaks in the GW spectrum and, depending on the parameter region, up to three peaks for a given parameter space choice. Fig. 2 shows some benchmark single and double peak GW-spectra for the majoron inverse seesaw. Notice that the single peak case (green) lies well within the LISA range, while both peaks in the double peak scenario (blue) are well within the sensitivity range of the u-DECIGO-corr measurement.
To understand these characteristic features, we must keep in mind that at the end of any FOPT the scalar potential minimization requires non-vanishing vevs (v h , v σ ) associated with the generation of both the EW and neutrino mass scales. The benchmark values for parameters corresponding to the three curves in Fig. 2 are given in Tab. I and Tab. II.    Fig. 2. Here, σ R is the CP even part of the σ-scalar after symmetry breaking, λ σh , while λσ denote the quartic couplings.
In Fig. 2 we show the GW energy density spectrum obtained for different nucleation temperatures. In all three cases we assumed that the singlet neutrino coupling to the majoron is of order one. The latter can be sizeably coupled to the Higgs boson, while consistent with the current LHC bounds from invisible Higgs decays [26,27]. The three cases shown in Fig. 2 correspond to two consecutive FOPTs. The green curve represents a scenario with   a single very strong EW phase transition, v n /T n = 119, and a nearly preserved lepton symmetry. This is possible due to the large quartic coupling λ σh , which makes the m/T -ratio sizable. Hence, the cubic (m/T ) 3 terms in the thermal expansion can produce a potential barrier between the vacua, inducing this type of transitions. The same effect generates the rightmost and the leftmost peaks in the red and blue curves, respectively. Once again, one sees that these results can be probed at BBO and u-DECIGO.   Note that the multi-peak configurations in the GW spectrum occur very frequently in the inverse seesaw with majoron. This fact is further highlighted in Fig. 3. Indeed, the double-peak feature of the GW spectrum is a generic prediction of our model, that can arise for many parameter choices, as shown in Fig. 3a. Configurations with larger peak multiplicities are also possible, as seen in Fig. 3b, where the color denotes the peak number, 1 (blue), 2 (cyan) or 3 (red). Such a rich peak structure is favoured for large quartic couplings involving σ. From Fig. 3b we also see that the GW spectra with three peaks are rarer than single or double GW-peak spectra. However, a significant fraction of the single peak cases are testable at LISA and BBO, while the double-peak cases may mostly be accessible to u-DECIGO, as shown in Fig. 4a.
The multi-peak feature of GWs provides a potentially viable way to distinguish amongst the basic underlying neutrino mass generation patterns, making the whole scenario potentially falsifiable at the fundamental level. To further illustrate its importance we take the type-I seesaw with majoron. As mentioned before, it is clear that, (1). In this limit, all the new particles can be integrated out for processes occurring at the EW scale, leading to no-FOPT solutions (the majoron couplings are highly suppressed, with no effect on EW scale physics). However, one can take Y ν ∼ O(10 −6 ), corresponding to M = Y σ v σ / √ 2 ∼ O(100) GeV. The fields do not decouple at the EW scale, and can still lead to FOPTs -hence to potentially observable primordial GW signals. One sees that this scenario requires tiny values of the neutrino "Dirac" Yukawa couplings Y ν to fit the small neutrino masses. Such "fake" low-scale seesaw contrasts with the "genuine" low-scale seesaw, which does not require this restriction.
Curve m h 2 /GeV λ h λ σh λσ cos θ vσ(T = 0) Mν /GeV Yσ   Fig. 4a.   Fig. 4a. The hidden peaks, Green 1 and Red 1, are seven and two orders of magnitude below the spectrum envelope, respectively. The vevs before (v i h,σ ) and after (v f h,σ ) the phase transition are given in GeV.
We find that, in a large region of parameters, both "fake" and "genuine" low-scale seesaw + majoron lead to the possibility of strong FOPTs. The corresponding GW spectra obtained for v σ ∼ O(100) GeV are shown in Fig. 4. Parameter values associated to Fig. 4a are given in Tab. III and Tab. IV. We can compare, for instance, the GW spectra featuring the double-peak structures with the sensitivity regions of the future experiments. In both "fake" and "genuine" lowscale seesaw we collected several possible double-peak spectra for the two cases. Comparing Figs. 4b and 3 one sees that double-peaks are more frequent in the "genuine" low-scale seesaw + majoron than in the "fake" one. Although "fake" low-scale seesaw leads to GW signals within the reach of upcoming experiments, double-peaks in this case are much rarer, and when present, one of the peaks typically lies beyond experimental reach, see Fig. 4b. This should be contrasted with the "genuine" low-scale seesaw, where not only double peaks are more frequent but, as shown in Fig. 2, they are often accessible to upcoming experiments 2 .
To conclude, we analysed the most popular implementations of the type-I seesaw mechanism for neutrino mass generation. We studied both the cases of explicit and spontaneous breakdown of the lepton number symmetry. The second, "dynamical" symmetry breaking implies the majoron field, which may also be interpreted as a viable [39][40][41] and testable [42,43] dark matter candidate. We have found that various scenarios lead to different patterns of phase transitions. We showed that explicit lepton number violation cannot induce any strong electroweak phase transition. Therefore, it does not lead to any gravitational-wave background signal testable by next-generation satellite interferometers.
The case when neutrino masses emerge from a dynamical mechanism in which lepton number violation happens spontaneously leads to much clearer gravitational footprints. Within such majoron seesaw case, we found that both the standard type-I seesaw (taken at a low scale) and the "genuine" low-scale type-I seesaw (like the inverse seesaw) predict a strong gravitational wave signal, testable in the 0.1 − 100 mHz frequency range. This highly motivates future experimental proposals, including LISA, u-DECIGO and BBO, accessing to the mHz frontier, as an indirect and complementary probe of neutrino mass generation, providing important information on the electroweak phase transition.
While "genuine" low-scale seesaw predicts large charged lepton flavor violation [29][30][31][32][33], as well as unitarity violation in neutrino propagation [34][35][36], this feature is absent in the "fake" low-scale seesaw. This makes the two schemes distinguishable in high-intensity and highenergy frontier setups. Here we have shown that "fake" and "genuine" schemes also have potentially distinct gravitational footprints. We saw explicitly that they produce different gravitational-wave spectra, potentially testable in the upcoming gravitational-wave experiments. As we stand right now, the new unexpected channel provided by the gravitational-wave physics in the multi-messenger era may contribute to shed light on the mystery of neutrino mass generation.