Gravitational waves from the minimal gauged $U(1)_{B-L}$ model

An additional $U(1)$ gauge interaction is one of promising extensions of the standard model of particle physics. Among others, the $U(1)_{B-L}$ gauge symmetry is particularly interesting because it addresses the origin of Majorana masses of right-handed neutrinos, which naturally leads to tiny light neutrino masses through the seesaw mechanism. We show that, based on the minimal $U(1)_{B-L}$ model, the symmetry breaking of the extra $U(1)$ gauge symmetry with its minimal Higgs sector in the early Universe can exhibit the first-order phase transition and hence generate a large enough amplitude of stochastic gravitational wave radiation which is detectable in future experiments.


I. INTRODUCTION
The standard model (SM) of particle physics based on the gauge group SU(3) C ×SU(2) L × U(1) Y successfully describes most of the elementary particle phenomena below the TeV scale.
Nevertheless, an additional U(1) gauge interaction is one of promising extensions of the SM.
As a minimal extension of the SM, we may consider models based on the gauge group [1,2], 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 model with a conventional U(1) B−L charge assignment, the gauge and gravitational anomaly cancellations require us to introduce three right-handed (RH) neutrinos whose Majorana masses are generated by the spontaneous breakdown of the U(1) B−L gauge symmetry. With Yukawa interactions between left-handed and RH neutrinos, the observed tiny neutrino masses is naturally explained by the so-called seesaw mechanism with the heavy Majorana RH neutrinos [3][4][5]. If the energy scale of the U(1) B−L symmetry breaking is much higher than the TeV scale as is often assumed, it is very difficult for any collider experiments to test the mechanism of the symmetry breaking and the RH neutrino mass generation.
In Ref. [6], two of the present authors showed that the U(1) B−L symmetry breaking with a simple extended U(1) B−L Higgs sector at such a high energy below about 10 7 GeV scale can be probed by the detection of gravitational wave (GW) and its energy scale dependence in the spectrum. For a similar work, see e.g., Ref. [7]. Although it has been pointed out that GWs generated by a first-order phase transition at a temperature about 10 7 GeV could be in reach of future experiments [8][9][10], the possibility has not been explicitly demonstrated for the specific high scale U(1) B−L model well-motivated by neutrino physics before the work in Ref. [6] 1 . In this previous work, the Higgs sector was extended so that a trilinear interaction of scalar fields in the tree level potential can be introduced as in Ref. [14], which play a crucial role to trigger the first-order phase transition.
In this paper, we investigate GWs from the first-order phase transition associated with the spontaneous U(1) B−L gauge symmetry breaking within the minimal model context. If a first-order phase transition happens in the early Universe, the dynamics of bubble collision [16][17][18][19][20] followed by turbulence of the plasma [21][22][23][24][25] and sonic waves generate GWs [26][27][28], which can be detected by the future experiments, such as the eLISA [29], the Big Bang Observer (BBO) [30], DECi-hertz Interferometer Observatory (DECIGO) [31] and Advanced LIGO (aLIGO) [32]. This paper is organized as follows: In the next section, we introduce formulas we adopt to describe the spectrum of GWs generated by a cosmological first-order phase transition.
In Sec. III, we describe the minimal U(1) B−L model and then derive the resultant GWs spectrum by estimating the latent heat and the transition time scale of the phase transition.
We also discuss model parameter dependence of the GWs spectrum. In the last section, we summarize our results.

II. GW GENERATION BY COSMOLOGICAL FIRST-ORDER PHASE TRANSI-TION
In this section, we briefly summarize the properties of GWs produced by three main GW production processes and mechanisms: bubble collisions, turbulence [21] and sound waves after bubble collisions [26]. See, for instance, Ref. [33,34] for a recent review. The GW spectrum generated by a first-order phase transition of a Higgs field critically depends on two quantities: the ratio of the latent heat energy to the radiation energy density ρ rad , which is expressed by a parameter α and the transition speed β defined below. In this section, we give definitions of those parameters and the fitting formula of the GW spectrum.
A. Scalar potential parameters related to the GW spectrum A phase transition is induced by a scalar field φ in the radiation dominated Universe with temperature T . At the moment of a first-order phase transition, the latent energy density is given by where φ high(low) denotes the field value of φ at the high (low) vacuum. Here and hereafter, quantities with the subscript ⋆ denote those at the time when the phase transition takes place [35]. On the other hand, the radiation energy density is given by with g * being the total number of relativistic degrees of freedom in the thermal plasma. The parameter α is defined by The bubble nucleation rate per unit volume at a finite temperature is given by Here, 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 [19]. The transition timescale is characterized by a dimensionless parameter with B. GW spectrum

Bubble collisions
The peak frequency and the peak amplitude of GWs generated by bubble collisions are, under the envelope approximation 2 and for β/H ⋆ ≫ 1 [20], expressed by Hz, with 2 For a recent development beyond the envelope approximation, see Ref. [36].

Sound waves
The peak frequency and the peak amplitude of GWs generated by sound waves are expressed by [26,27] f peak ≃ 19 Hz, The efficiency factor (κ v ) is given by [37] κ where c s denotes the sonic speed. The spectrum shape is well fitted by [38]

Turbulence
The peak frequency and amplitude of GWs generated by turbulence are expressed by [21] f peak ≃ 27 Hz, In our analysis, we follow Ref. [38] and conservatively set the efficiency factor for turbulence to be κ turb ≃ 0.05κ v . The spectrum shape is well fitted by [25,38,39]  ( with h ⋆ = 17 T ⋆ 10 8 GeV g * 100 Hz. 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 vacuum expectation value (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 tree level 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 VEV of the B − L Higgs field are much larger than that of the SM Higgs field.
In the minimal B − L model, one-loop quantum corrections to the scalar potential for both zero-and finite-temperature are essential for realizing the first-order phase transition.
For our numerical calculations, we have implemented our minimal U(1) B−L model into the public code CosmoTransitions [47], where both zero-and finite-temperature one-loop effective potentials, with Φ 2 = ϕ/ √ 2, have been calculated in the MS renormalization scheme at a renormaliza- In the following calculations, we assume Y N i ≪ g B−L , for simplicity, and neglect quantum corrections through neutrino Yukawa couplings Y N i . Thus, the effective potential (26) is described by only three free parameters, g B−L , λ 2 and M Φ 2 . In our analysis,

B. Parameter dependence
We now show a dependence of our results on three free parameters, g B−L , λ 2 and v 2 . At first, we focus on the gauge coupling dependence of the resultant GWs spectrum. The GWs spectrum for various values of the B − L gauge coupling constant for the fixed value of v 2 = 10 TeV and λ 2 = 0.002 is shown in Fig. 1. We have found a mild dependence for the frequency but the amplitude being quite sensitive. The largest amplitude is obtained for 0.35 g B−L 0.4.
Next, we focus on the VEV dependence of the resultant GWs spectrum. We show in We have found a very mild dependence for the amplitude but the strong dependence of the peak frequency. We have found an approximate relation of f peak ∝ v 2 .
At last, we focus on the λ 2 dependence of the resultant GWs spectrum. We show the GWs spectrum for various λ 2 values for g B−L = 0.4 and v 2 = 10 TeV in Fig. 3. As λ 2 decreases, the peak frequency also decreases while the peak amplitude increases. We approximate relations such as Ω GW h 2 (f peak ) ∝ λ −1/4 2 and f peak ∝ λ 2 . We also find a lower bound on λ 2 . As will be listed in Table II, α is as large as O(1) for λ 2 ∼ 10 −4 , which means that the energy density of background radiation and that of the latent heat is comparable. For λ 2 ≪ 10 −4 , the latent heat becomes too large. Indeed, we find the effective equation of state parameter w becomes smaller than −1/3 for such a too small λ 2 . In this case, the Universe would inflate as in the old inflation model [48,49].   curves from left to right correspond to points A, B and C, respectively. The future experimental sensitivity curves of LISA [50], DECIGO and BBO [51], aLIGO [52] and Cosmic Explore (CE) [53] are also shown in black.
a minimal Higgs sector, as long as Yukawa coupling effects on the effective Higgs potential are negligible.