Gravitational waves from breaking of an extra $U(1)$ in $SO(10)$ grand unification

In a class of gauged $U(1)$ extended Standard Models (SMs), the breaking of the $U(1)$ symmetry is not only a source for Majorana masses of right-handed (RH) neutrinos crucial for the seesaw mechanism, but also a source of stochastic gravitational wave (GW) background. Such $U(1)$ extended models are well-motivated from the viewpoint of grand unification. In this paper, we discuss a successful ultraviolet completion of a $U(1)$ extended SM by an $SO(10)$ grand unified model through an intermediate step of $SU(5) \times U(1)$ unification. With a parameter set that is compatible with the $SO(10)$ grand unification, we find that a first-order phase transition associated with the $U(1)$ symmetry breaking can be strong enough to generate GWs with a detectable size of amplitude. We also find that the resultant GW amplitude reduces and its peak frequency becomes higher as the RH neutrino masses increase.


I. INTRODUCTION
An extra U(1) gauge interaction is one of promising and interesting extensions of the standard model (SM) of particle physics. Since tiny but non-vanishing neutrino masses are a clear evidence for the existence of the beyond the SM, one of the simplest and the most interesting models is the one based on the gauge group SU(3) C ×SU(2) L ×U(1) Y ×U(1) B−L [1,2], where the additional interaction is from the gauged U(1) B−L (baryon number minus lepton number) symmetry. In the standard U(1) B−L charge assignment, three right-handed (RH) neutrinos have to be introduced to fulfill the gauge and gravitational anomaly cancellation conditions. After Majorana masses of RH neutrinos are generated by the spontaneous U(1) B−L gauge symmetry breaking at a high energy scale, the observed tiny neutrino masses are naturally explained by the so-called seesaw mechanism with the heavy Majorana RH neutrinos through their Yukawa interactions with the SM left-handed neutrinos [3][4][5][6]. In addition, one of the three RH neutrinos can be a candidate for the dark matter in our universe [7][8][9][10][11][12].
In this paper, we consider an ultraviolet (UV) completion of such an extra U(1) extended SM. A primary candidate scenario for the completion is the Grand Unified Theory (GUT), in which all the SM gauge interactions are unified into a single gauge interaction at a high energy scale. In this paper, we consider an SO(10) GUT model, in which the extra U (1) gauge group along with the SM gauge group is embedded, and all the SM fermions and RH neutrinos in each generation are also unified into a single 16 representation of SO(10) [43]. Among several possible paths of symmetry breaking from the SO(10) to the SM gauge group, we consider the following: First, SO(10) breaks to SU(5) × U(1) at a very high scale M SO (10) . Next, the SU(5) breaks to the SM gauge group at a scale M SU(5) ≃ 10 16 GeV. The extra U(1) is essentially the gauged B − L symmetry and its breaking can take place at any scale below M SO (10) . If the extra U(1) symmetry breaking scale is very high, cosmic strings can be the dominant source for stochastic GWs [44,45]. Another promising GW source is the first-order phase transition in the early universe associated with the U(1) symmetry breaking at a scale lower than about 10 7 GeV [46][47][48]. In previous work on the U(1) extended SMs [17,21], we have treated the U(1) gauge coupling as a free parameter and have shown that with its suitable choice the first-order phase transition can generate the GWs large enough to be tested in the future experiments. However, once we consider the UV completion by the SO(10) GUT, the U(1) gauge coupling is no longer a free parameter and its low energy value is determined by the condition of the gauge coupling unification. In this paper, we will examine whether the parameter set compatible with the SO(10) unification can generate a detectable size of the GW spectrum.
This paper is organized as follows: In the next section, we describe the outline for the SO(10) unification of the U(1) extended SM based on the gauge group of Towards SO(10) unification, we consider an intermediate path with the SU(5) × U(1) X unification and show the successful embedding into the SO(10) model with unified gauge couplings. In Sec. III, we describe the system of the extra U(1) X breaking and discuss the first-order phase transition in the early universe by employing the finitetemperature effective potential for the U(1) X Higgs field. In Sec. IV, we introduce the formulas that we adopt to compute the GW spectrum generated by the first-order phase transition and present the resultant GW spectrum for various sets of the model parameters.
We also discuss the model-parameter dependence of the GW spectrum. The last section is devoted to summary.  [49,50]. The particle content of this model is listed in Table I. Except for the introduction of new parameter x, the model properties are quite similar to those of the minimal B − L model, 1 which is realized as the special case of x = 0. We now consider the embedding, As in the standard SU(5) GUT [54], anti right-handed down quarks and left-handed leptons are embedded in 5 * representation of SU(5), while left-handed quarks, anti right-handed up quarks, anti right-handed charged leptons are embedded in 10 representation: This quark and lepton unification requires the following two conditions, which should be satisfied with a unique x value. The solution is x = − 4 5 and hence the SU(5) unification leads to a quantization of U(1) X charge [55]. As is well known, the SU (5) GUT normalization for the SM U(1) Y coupling and rescaled charges are The SU(5) × U(1) X can be embedded into SO (10). In the following, we list the decom- For simplicity, we assume the SO(10) symmetry breaking to the U(1) X extended SM by non-zero VEVs of 1(0) and 24(0) in a 45-representation Higgs field: The final U(1) X breaking can be realized by a non-zero VEV of Φ † 2 = 1(2) ⊂ 126 Higgs field.
In the SO(10) GUT, the Yukawa interactions for the SM fermions are given by where 16 f is a fermion multiplet (the generation index is suppressed), and 10 H and 126 H are Higgs fields. Referring the above decomposition, one can see that the VEV of 1(2) ⊂ 126 Higgs breaks the U(1) X symmetry and generates Majorana masses of RH neutrinos in 16 f through the Yukawa coupling Y 126 in Eq. (7). In the SM gauge group decomposition, the Yukawa interactions include the neutrino Dirac Yukawa couplings of l L HN R .
B. Gauge coupling unification to SU (5) In non-supersymmetric framework, a simple setup to achieve the unification of the three SM gauge couplings is to introduce two pairs of vector-like quarks (Q +Q and D +D) with TeV scale masses, M Q and M D , respectively. Their representations are listed in Table   II. It has been shown in Refs. [57][58][59][60][61][62][63] that in the presence of the exotic quarks, the SM gauge couplings are successfully unified at M SU(5) ≃ 10 16 GeV. This unification scale corresponds to the proton lifetime of τ p ≃ 10 38 years, which is much longer than the current experimental lower bound of τ (p → π 0 e + ) ≃ 10 34 years reported by the Super-Kamiokande collaboration [64]. The presence of the exotic quarks can also work for stabilizing the SM Higgs potential [63].
In the SU (5) where 45 H is the 45-representation Higgs field in the SO(10) GUT, and the second and third lines are the expression under SU(5) × U(1) X . The SO(10) symmetry breaking down (6)) generates new mass terms, and we have with We set the parameters, Y ∼ 1,  (10) ). Next, we tune Y 24 H ≃ M 5 diag(−1, −1, −1, 3/2, 3/2) so as to make only D +D in the 5 + 5 * multiplet to be light. This procedure is analogous to the triplet-doublet splitting of the 5-plet Higgs field in the standard SU(5) GUT. We apply the same procedure to the 16 + 16 * multiplets including Q+Q to leave only them light. This is done by a tuning to realize (10) ).
Let us now discuss the gauge coupling unification. At the one-loop level, the renormalization group (GR) equations of the SM gauge couplings and U(1) χ gauge couplings are given where α i = g 2 i 4π (i =1, 2, 3, and χ). The beta function coefficients are expressed as where C 2 (G) is the casimir operator of the group G, T (R f (s) ) is the trace of the product of , and N f (s) is the number of fermions (scalars). For each gauge coupling constant in the energy range of M Q , M D < µ < M SU(5) , we have Higgs quartic couplings, respectively. We find that three SM gauge couplings are successfully unified at M SU(5) ≃ 2.24 × 10 16 GeV. Our results are shown in Fig. 1. We will discuss the RG evolution for g χ in the next subsection. To calculate the RG evolutions of g 5 and g χ in the energy range of M SU(5) < µ < M SO(10) , we need to know the particle spectrum to determine the beta functions of g 5 and g χ . We assume the minimal particle content for the SU(5) × U(1) theory connecting to the particle contents of the U(1) X extended SM. All the SM fermions and RH neutrinos which are embedded into the three generations of 16-plets of SO(10) contribute to the beta functions.
Under the gauge group SU(5) × U(1) X , the SM Higgs field is embedded into (5, −2/5) and the U(1) X Higgs field is in (1, −2). In addition, we have an SU(5) adjoint Higgs field with a vanishing U(1) X charge (24, 0), whose VEV breaks the SU(5) gauge group to the SM gauge group. Lastly, as we have discussed in Sec. II B, the vector-like quarks,Q + Q and D +D, respectively, are embedded in the full SU(5) multiplets of 10 * + 10 and 5 + 5 * at M SU (5) . Taking all these fields into account, the beta function coefficient of the SU(5) gauge coupling at the one-loop level is given by The beta function coefficient of the U(1) χ gauge coupling at the one-loop level is given by With these beta function coefficient and the boundary condition g 5 (M SO(10) ) = g χ (M SO(10) ), we find the solutions for the RG equations. Our results are shown in the right panel of Fig. 1.

III. EXTRA U (1) BREAKING
In the low energy effective theory based on where the first term is the neutrino Dirac Yukawa coupling, and the second is the Majorana Yukawa couplings. Once the Higgs field Φ 2 develops a nonzero vacuum expectation value (VEV), the U(1) gauge symmetry is broken and the Majorana mass terms of the RH neutrinos are generated. After the electroweak symmetry breaking, tiny neutrino masses are generated through the seesaw mechanism.
In the effective theory, 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 its little importance in the following discussion, since we are interested in the case that the VEV of the U(1) X Higgs field is much larger than that of the SM Higgs field.
The U(1) X Higgs field is expanded around its VEV (v 2 ) as The scalar masses are expressed as At the classical minimum with v 2 = 2M 2 Φ 2 /λ 2 , χ 2 is the would-be Nambu-Goldstone mode eaten by the U(1) X gauge boson (Z ′ boson) and m 2 φ 2 = λ 2 v 2 2 . The RH neutrinos N i R and the Z ′ boson acquire their masses as One-loop corrections to the scalar potential for both zero and finite temperatures are essential for realizing the first-order phase transition. One-loop correction is given by Here, g i , with i = s (scalars), f (fermions) and v (vectors) denotes the number of internal degrees of freedom, c i = 5/6 (3/2) are constant for a vector boson (a scalar or a fermion), and Q is the renormalization scale. The finite temperature correction to the effective potential is expressed by where J B(F ) is an auxiliary function in thermal corrections (see e.g., Ref. [68,69]).
The thermal correction to masses of φ 2 , χ 2 and the longitudinal mode of the Z ′ boson are given by where q Φ denotes the U(1) X charge of Φ = H, Φ 2 respectively, q L and q R are those of leftand right-handed fermion f , N Φ is the number of degrees of freedom in Φ, N c is the color factor, and f denotes the summation for all fermion flavors. We have the sum of charges where the contribution of the second line in Eq. (31) comes from the vector-like quarks.
For our numerical calculations, we have implemented our model into the public code CosmoTransitions [70], 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- Here, as a caveat, we note that there is a long-standing open problem of gauge dependence on the use of the effective Higgs potential. Our results are also subjects of this issue [71,72] and should be regarded as a reference value.

A. GW generation
There are three mechanisms generating GWs by a first-order phase transition: bubble collisions, sound waves, and turbulence after bubble collisions. The resultant spectrum of GW background produced by each three mechanisms is expressed as in terms of the density parameter Ω GW . Here, three terms in the right hand side denote the GW generated by bubble collisions, sound waves, and turbulence, respectively.

Bubble collisions
The GW spectrum generated by bubble collisions for a case of β/H ⋆ ≫ 1 is fitted with with the peak amplitude [30] the peak frequency and the spectral function [73] S (a, b) ≃ (2.7, 1.0).
The efficiency factor for bubble collisions is given by with A = 0.715 and denotes the bubble wall velocity v b dependence [31].

Sound waves
The GW spectrum generated by sound waves is fitted by with the peak amplitude [36,37,74] the peak frequency [73] f peak ≃ 19 Hz, (46) and the spectral function [75] S The active period of sound waves is expressed as where R ⋆ ≃ (8π) 1/3 v b /β is the average separation of between bubbles and U f is the rootmean square of fluid velocity, which can be approximated as [37] The last factor in Eq. (44) represents the suppression effect due to the short-lasting sonic wave as a source of gravitational wave generation compared with the Hubble time scale (H ⋆ ) for the case of H ⋆ τ sw < 1, as pointed out in Refs. [76,77] (see also Refs. [78][79][80][81]).

Turbulence
The GW spectrum generated by turbulence is fitted by with the peak amplitude [31]  the peak frequency Hz, (52) and the spectral function [35,75,82] S turb (f ) = f f peak
We set the efficiency factor for turbulence to be κ turb ≃ 0.05κ v .

B. Predicted spectrum for benchmark points
At first, we show the dependence of the resultant GW spectrum on the energy scale of symmetry breaking, or equivalently, the VEV (v 2 ) scale. In Fig. 2   We have set λ 2 = 6 × 10 −4 in this calculation.
U(1) B−L model that the peak amplitude decreases with the peak frequency getting higher as λ 2 increases, and this dependence is approximated as Ω GW h 2 (f peak ) ∝ λ −1/4 2 and f peak ∝ λ 2 .
We do not only reconfirm this λ 2 dependence in the SO(10) completed model but also find that non-vanishing Yukawa coupling has a similar effect on the GW spectrum. As a Yukawa coupling increases, the peak amplitude decreases with the peak frequency increasing.
Assuming the hierarchy among the Yukawa couplings as we show in Fig. 3 the GWs spectrum for various values of Y N and λ 2 for g χ = 0.463 and v 2 = 1 PeV (see the third benchmark in Table III). In the figure, we see that two different parameter sets, (Y N , λ 2 ) = (0, 0.002) and (Y N , λ 2 ) = (1, 0.001), predict almost the same GW spectrum. This is because the dependence of the resultant GW spectrum on Y N is quite similar to that of λ 2 . We have found that for λ 2 0.006, the dependence of GW spectrum on Y N is week. We list our results for five benchmark points in Table IV.

V. SUMMARY
In this paper, we have considered the U(1) X extended SM and studied the spectrum of stochastic GWs generated by the first-order phase transition associated with the extra U(1) X symmetry breaking in the early universe. This breaking is responsible for the generation unification condition. We have found that the first-order phase transition triggered by this extra U(1) symmetry breaking can be strong enough to generate GWs with a detectable size of amplitude if the U(1) X Higgs quartic coupling is small enough and the symmetry breaking scale (the bubble nucleation temperature T ⋆ ) is smaller than about 10 5 (10 4 ) PeV.
We have also clarified the dependence of the resultant GW spectrum on the RH neutrino Majorana Yukawa couplings, in other words, the mass scale of RH neutrinos. As the Yukawa couplings increase, the amplitude of GW background reduces and the peak frequency slightly increases. We have found a similar behavior of the GW spectrum as we change the U(1) X Higgs quartic coupling. Thus, different combinations of the Yukawa and the quartic couplings can result in almost the same GW spectrum. In order to extract the information about RH neutrino masses from the spectral shape of GW background, the information of the U(1) X Higgs quartic coupling is necessary.