Probing the seesaw scale with gravitational waves

The $U(1)_{B-L}$ gauge symmetry is a promising extension of the standard model of particle physics, which is supposed to be broken at some high energy scale. Associated with the $U(1)_{B-L}$ gauge symmetry breaking, right-handed neutrinos acquire their Majorana masses and then tiny light neutrino masses are generated through the seesaw mechanism. In this paper, we demonstrate that the first-order phase transition of the $U(1)_{B-L}$ gauge symmetry breaking can generate a large amplitude of stochastic gravitational wave (GW) radiation for some parameter space of the model, which is detectable in future experiments. Therefore, the detection of GWs is an interesting strategy to probe the seesaw scale which can be much higher than the energy scale of collider experiments.


I. INTRODUCTION
The nonvanishing neutrino masses have been established through various neutrino oscillation phenomena. The most attractive idea to explain the tiny neutrino masses is the so-called seesaw mechanism with heavy Majorana right-handed (RH) neutrinos [1]. Then, the origin of neutrino masses is ultimately reduced to questions on the origin of RH neutrino masses. It is natural to suppose that masses of RH neutrinos are also generated associated with developing the vacuum expectation value (VEV) of a Higgs field which breaks a certain (gauge) symmetry at a high energy scale.
As a promising and minimal extension of the standard model (SM), we may consider models based on the gauge group SU(3) C × SU(2) L × U(1) Y × U(1) B−L [2] where the U(1) B−L (baryon number minus lepton number) gauge symmetry is supposed to be broken at a high energy scale. In this class of models with a natural/conventional U(1) B−L charge assignment, the gauge and gravitational anomaly cancellations require us to introduce three RH neutrinos whose Majorana masses are generated by the spontaneous breakdown of the U(1) B−L gauge symmetry. In the case that the U(1) B−L symmetry breaking takes place at an energy scale higher than the TeV scale, it is very difficult for any collider experiments to address the mechanism of the symmetry breaking and the RH neutrino mass generation.
The spectrum of stochastic GWs produced by the first-order phase transition in the early Universe, in particular, by the SM Higgs doublet field, has been investigated in the literature.
Here, the phase transition occurs at the weak scale. See, for instance, Ref. [25] for a recent review.
In this paper, we focus on GWs from the first-order phase transition associated with the spontaneous U(1) B−L gauge symmetry breaking at a scale higher than the TeV scale. GWs generated by a U(1) B−L extended model with the classical conformal invariance [26,27], where its phase transition takes place around the weak scale, have been studied in Ref. [28].
GWs from a second-order B − L phase transition during reheating have been studied in Ref. [29]. In this paper, we consider a slightly extended Higgs sector from the minimal model and introduce an additional U(1) B−L charged Higgs field with its charge +1. This is one of the key ingredients in this paper. GWs generated by a phase transition in this extended scalar potential, but at TeV scale, have been studied in Ref. [30]. As we will show below, the new Higgs field plays a crucial role in causing the first-order phase transition of the U(1) B−L symmetry breaking and the amplitude of resultant GWs generated by the phase transition can be much larger than the one we naively expect.

SITION
In this section, we briefly summarize the properties of GWs produced by a first-order phase transition in the early Universe. There are three main GW production processes and mechanisms: bubble collisions, turbulence [11] and sound waves after bubble collisions [16].
The GW spectrum generated by a first-order phase transition is mainly characterized by two quantities: the ratio of the latent heat energy to the radiation energy density, which is expressed by a parameter α and the transition speed β defined below. In this section, we introduce those parameters and the fitting formula of the GW spectrum.
A. Scalar potential parameters related to the GW spectrum We consider the system composed of radiation and a scalar field φ at temperature T .
The energy density of radiation is given by with g * being the number of relativistic degrees of freedom in the thermal plasma. At the moment of a first-order phase transition, the potential energy of the scalar field includes the latent energy density given by where φ high(low) denotes the field value of φ at the high (low) vacuum. Here and hereafter, quantities with the subscript ⋆ stand for those at the time when the phase transition takes place [32]. Then, a parameter α is defined as The bubble nucleation rate per unit volume at a finite temperature is given by where Γ 0 is a coefficient of the order of the transition energy scale, S is the action in the four-dimensional Minkowski space, and S 3 E is the three-dimensional Euclidean action [9]. The inverse of the transition timescale can be defined as Its dimensionless parameter β/H ⋆ can be expressed as B. GW spectrum

Bubble collisions
Under the envelope approximation 1 and for β/H ⋆ ≫ 1 [10], the peak frequency and the peak amplitude of GWs generated by bubble collisions are given by Hz, 1 For a recent development beyond the envelope approximation, see Ref. [31].
with the following fitting functions where v b denotes the bubble wall velocity. The efficiency factor (κ) is given by [11] with A = 0.715. The full GW spectrum is expressed as [32] Ω

Sound waves
The peak frequency and the peak amplitude of GWs generated by sound waves are given by [16,17] f peak ≃ 19 Hz, The efficiency factor (κ v ) is given by [35] κ with c s being the sonic speed. The spectrum shape is expressed as [34] The peak frequency and amplitude of GWs generated by turbulence are given by [11] f peak ≃ 27 Hz, In our analysis, we conservatively set the efficiency factor for turbulence to be κ turb ≃ 0.05κ v as in Ref. [34]. The spectrum shape is given by [15,33,34]  ( with Hz.  Table I. The Yukawa sector of the SM is extended to have where the first term is the neutrino Dirac Yukawa coupling, and the second is the Majorana Yukawa couplings. Once the U(1) B−L Higgs field Φ 2 develops a nonzero VEV, the U(1) B−L gauge symmetry is broken and the Majorana mass terms of the RH neutrinos are generated. Then, the seesaw mechanism is automatically implemented in the model after the electroweak symmetry breaking.
We consider the following scalar potential: Here, we omit the SM Higgs field (H) part and its interaction terms for not only simplicity but also little importance in the following discussion, since we are interested in the case that the VEVs of B − L Higgs fields are much larger than that of the SM Higgs field. 2 All parameters in the potential (22) are taken to be real and positive. At the U(1) B−L symmetry breaking vacuum, the B − L Higgs fields are expanded around those VEVs v 1 and v 2 , as Here, φ 1 and φ 2 correspond to two real degrees of freedom as CP -even scalars, one linear combination of χ 1 and χ 2 is the Nambu-Goldstone mode eaten by the U(1) B−L gauge boson (Z ′ boson) and the other is left as a physical CP -odd scalar. Mass terms of particles are expressed as With the U(1) B−L symmetry breaking, the RH neutrinos N i R and the Z ′ boson acquire their masses, respectively, as where g B−L is the U(1) B−L gauge coupling. The mass matrix of CP -even Higgs bosons (φ 1 and φ 2 ) and the mass of the physical CP -odd scalar P can be, respectively, simplified as and by eliminating M 2 Φ 2 and M 2 Φ 1 under the stationary conditions, Let us here note the LEP constraint m Z ′ /g B−L = 4v 2 2 + v 2 1 6 TeV [37,38] and the constraint from the LHC Run-2 (see, for example, Refs. [39][40][41][42]) for g B−L ≃ 0.7.
With a suitable choice of parameters, we find that the phase transition of the B −L gauge symmetry breaking by the Higgs fields Φ 1 and Φ 2 becomes of the first order in the early  Universe. In the following analysis, we set λ 1 = 0.1, λ 2 = 0.1, and λ 3 = 0.3 and all corrections through neutrino Yukawa couplings Y N i have been neglected, assuming Y N i ≪ g B−L , for simplicity. We show in Fig. 1 the shape of the one-loop scalar potential (22).
Implementing our model into the public code CosmoTransitions [43], we have evaluated the parameters α, β and T ⋆ at a renormalization scale Q 2 = (v 1 2 + v 2 1 )/2. We list our results for four benchmark points in Table II. In Table III  In Fig. 2, we show predicted GW spectra for our benchmark points along with expected sensitivities of future interferometer experiments. Here, the resultant spectra have been calculated with a bubble wall speed of v b = 0.6. We have confirmed that the results are not so significantly changed for other v b values of O(0.1). Green, blue, red and purple curves from left to right correspond to points A, B, C and D, respectively. Black solid curves denote the expected sensitivities of each indicated experiment, according to Ref. [45] for LISA, Ref. [46] for DECIGO and BBO, Ref. [47] for aLIGO and Ref. [48] for Cosmic Explore (CE). Curves are drawn by gwplotter [44]. The sensitivities of DECIGO and BBO reach the results of points A and B. Point C is an example which is not marginally able to be detected by DECIGO/BBO but its peak is within the reach of CE.

IV. SUMMARY
The origin of heavy Majorana RH neutrino masses is one of the essential pieces to understand the origin of neutrino masses through the seesaw mechanism. Gauged B −L symmetry and its breakdown are a natural framework to introduce the RH neutrinos into the SM and The future experimental sensitivity curves of LISA [45], DECIGO and BBO [46], aLIGO [47] and Cosmic Explore (CE) [48] are also shown as black curves.
of the effective Higgs potential. See, for example, Ref. [52] for recent discussions. So far, we have no clear resolution to this issue. According to Ref. [52], the resultant GW spectrum has one order of magnitude uncertainties under a specific gauge choice. Thus, even for the worst case, our benchmark points A and B can still be within the reach of future experiments.
Once a better prescription has been developed, we will reevaluate the amplitude of GWs.