Gravitational waves from first-order phase transition in an electroweakly interacting vector dark matter model

TL;DR

This paper investigates gravitational waves produced by a strong first-order phase transition in an extended electroweak symmetry model with vector dark matter, predicting detectable signals for future space-based interferometers.

Contribution

It demonstrates that the model's phase transition can generate observable gravitational waves, linking dark matter properties with gravitational wave signals.

Findings

The phase transition is strongly first-order across a wide parameter space.

The gravitational wave spectrum is within reach of future detectors.

Detection can probe scalar masses up to a few TeV.

Abstract

We discuss gravitational waves in an electroweakly interacting vector dark matter model. In the model, the electroweak gauge symmetry is extended to SU(2)$_0 \times$ SU(2)$_1 \times$SU(2)$_2 \times$ U(1)$_Y$ and spontaneously broken into SU(2)$_L \times$ U(1)$_Y$ at TeV scale. The model has an exchange symmetry between SU(2)$_0$ and SU(2)$_2$. This symmetry stabilizes some massive vector bosons associated with the spontaneous symmetry breaking described above, and an electrically neutral one is a dark matter candidate. In the previous study, it was found that the gauge couplings of SU(2)$_0$ and SU(2)$_1$ are relatively large to explain the measured value of the dark matter energy density via the freeze-out mechanism. With the large gauge couplings, the gauge bosons potentially have a sizable effect on the scalar potential. In this paper, we focus on the phase transition of SU(2)$_0 \times$ SU(2)$_1 \times$ SU(2)$_2 \to$ SU(2)$_L$. We calculate the effective potential at finite temperature and find that the phase transition is first-order and strong in a wide range of the parameter space. The strong first-order phase transition generates gravitational waves. We calculate the gravitational wave spectrum and find that it is possible to detect the gravitational waves predicted in the model by future space-based gravitational wave interferometers. We explore the regions of the parameter space probed by the gravitational wave detection. We find that the gravitational wave detection can probe the region where the mass of $h'$, a CP-even scalar in the model, is a few TeV.

Tables

Table 1. Table 1: Charge assignment under the SU(3) × C {}_{C}\times SU(2) × 0 {}_{0}\times SU(2) × 1 {}_{1}\times SU(2) 2 × \times U(1) Y gauge symmetry of the model.

field	spin	SU(3)c	SU(2)0	SU(2)1	SU(2)2	U(1)Y
$q_L$	$\frac{1}{2}$	3	1	2	1	$\frac{1}{6}$
$u_R$	$\frac{1}{2}$	3	1	1	1	$\frac{2}{3}$
$d_R$	$\frac{1}{2}$	3	1	1	1	-$\frac{1}{3}$
${\ell }_L$	$\frac{1}{2}$	1	1	2	1	-$\frac{1}{2}$
$e_R$	$\frac{1}{2}$	1	1	1	1	-1
$H$	0	1	1	2	1	$\frac{1}{2}$
${\Phi }_1$	0	1	2	2	1	0
${\Phi }_2$	0	1	1	2	2	0

Equations

$$V_{\mathrm{eff}}(\phi ,T)\simeq D(T^2-T_0^2){\phi }^2-ET{\phi }^3+{\lambda }_T{\phi }^4,$$

$${\Phi }_1\to U_0{\Phi }_1{U}_1^{†},{\Phi }_2\to U_2{\Phi }_1{U}_1^{†},$$

$$\Phi_{j}=-\epsilon\Phi_{j}^{*}\epsilon,\ \ \ \text{where }\epsilon=\matrixquantity(0&1\\ -1&0).$$

$$W_{0\mu }^a\to W_{2\mu }^a,W_{2\mu }^a\to W_{0\mu }^a,{\Phi }_1\to {\Phi }_2,{\Phi }_2\to {\Phi }_1,$$

$$-\frac{1}{4}\sum _{j=0}^2{W}_{j\mu \nu }^a{W}_j^{a\mu \nu }-\frac{1}{4}{B}_{\mu \nu }{B}^{\mu \nu }+\sum _{j=1}^2\frac{1}{2}\text{tr}(D^{\mu }{\Phi }_j^{†}{D}_{\mu }{\Phi }_j)-V_{\text{scalar}},$$

$$V_{\text{scalar}}=$$

$$+{\lambda }_{\Phi }{\left(\text{tr}\left({\Phi }_1^{†}{\Phi }_1\right)\right)}^2+{\lambda }_{\Phi }{\left(\text{tr}\left({\Phi }_2^{†}{\Phi }_2\right)\right)}^2+{\lambda }_{h\Phi }{H}^{†}H\text{tr}\left({\Phi }_1^{†}{\Phi }_1\right)$$

$$+{\lambda }_{h\Phi }{H}^{†}H\text{tr}\left({\Phi }_2^{†}{\Phi }_2\right)+{\lambda }_{12}\text{tr}\left({\Phi }_1^{†}{\Phi }_1\right)\text{tr}\left({\Phi }_2^{†}{\Phi }_2\right).$$

$$H=$$

$$m^2=$$

$$m_{\Phi }^2=$$

$$V_{\mu }^0$$

$$V_{\mu }^{\pm }$$

$$Z_{\mu }^{\prime }$$

$$W_{\mu }^{\prime \pm }$$

$$W_{\mu }^{\pm }$$

$$Z_{\mu }$$

$$A_{\mu }$$

$$m_{V^0}^2$$

$$m_{Z^{\prime }}^2$$

$$m_{W^{\pm }}^2$$

$$m_Z^2$$

$$g_0$$

$$g_1$$

$$g_W={\left(\frac{2}{g_2^2}+\frac{1}{g_1^2}\right)}^{-1/2}.$$

$$\text{evaluated}{V_{\text{eff}}}_{T=0}=$$

$$+9\frac{m_V^4}{32{\pi }^2}\frac{{\phi }^2}{2v_{\Phi }^2}+9\frac{m_{Z^{\prime }}^4}{32{\pi }^2}\frac{{\phi }^2}{2v_{\Phi }^2}$$

$$+9\frac{m_V^4}{64{\pi }^2}\frac{{\phi }^4}{4v_{\Phi }^4}\left(\ln \frac{{\phi }^2}{2v_{\Phi }^2}-\frac{3}{2}\right)+9\frac{m_{Z^{\prime }}^4}{64{\pi }^2}\frac{{\phi }^4}{4v_{\Phi }^4}\left(\ln \frac{{\phi }^2}{2v_{\Phi }^2}-\frac{3}{2}\right)$$

$$+\frac{1}{64{\pi }^2}\text{expectationvalue}{m_{h^{\prime }}^2}^2\left(\ln \frac{\text{expectationvalue}{m}_{h^{\prime }}^2}{m_{h^{\prime }}^2}-\frac{3}{2}\right)+\frac{1}{32{\pi }^2}\text{expectationvalue}{m}_{h^{\prime }}^2{m}_{h^{\prime }}^2$$

$$+\frac{1}{64{\pi }^2}\text{expectationvalue}{m_{h_D}^2}^2\left(\ln \frac{\text{expectationvalue}{m}_{h_D}^2}{m_{h_D}^2}-\frac{3}{2}\right)+\frac{1}{32{\pi }^2}\text{expectationvalue}{m}_{h_D}^2{m}_{h_D}^2$$

$$-6\frac{({\lambda }_{12}+2{\lambda }_{\Phi }{)}^2}{128{\pi }^2}3{\phi }^2(-4v_{\Phi }^2+{\phi }^2)$$

$$+6\frac{({\lambda }_{12}+2{\lambda }_{\Phi }{)}^2({\phi }^2-2v_{\Phi }^2{)}^2}{64{\pi }^2}\ln \frac{({\lambda }_{12}+2{\lambda }_{\Phi })({\phi }^2-2v_{\Phi }^2)}{m_G^2}$$

$$+\text{(φ-independent terms)},$$

$$m_G^2=$$

$$m_{\Phi }^2=$$

$$\text{expectationvalue}{m}_{h^{\prime }}^2=$$

$$\text{expectationvalue}{m}_{h_D}^2=$$

$$\Delta {\Sigma }_{h^{\prime }{h}^{\prime }}=$$

$$+\frac{3m_{\Phi }^4}{2{\pi }^2{v}_{\Phi }^2}{B}_0(0,0,0)+\frac{({\lambda }_{12}-6{\lambda }_{\Phi }{)}^2{v}_{\Phi }^2}{4{\pi }^2}{B}_0(0,m_{h_D}^2,m_{h_D}^2)$$

$$+\frac{9m_{\Phi }^4}{4{\pi }^2{v}_{\Phi }^2}{B}_0(0,m_{h^{\prime }}^2,m_{h^{\prime }}^2)+\frac{9m_V^4}{16{\pi }^2{v}_{\Phi }^2}{B}_0(0,m_V^2,m_V^2)$$

$$+\frac{9m_{Z^{\prime }}^4}{16{\pi }^2{v}_{\Phi }^2}{B}_0(0,m_{Z^{\prime }}^2,m_{Z^{\prime }}^2)$$

$$-\frac{3(-m_{h^{\prime }}^4+16m_{\Phi }^4)}{32{\pi }^2{v}_{\Phi }^2}{B}_0(m_{h^{\prime }}^2,0,0)-\frac{({\lambda }_{12}-6{\lambda }_{\Phi }{)}^2{v}_{\Phi }^2}{4{\pi }^2}{B}_0(m_{h^{\prime }}^2,m_{h_D}^2,m_{h_D}^2)$$

$$-\frac{9m_{\Phi }^4}{4{\pi }^2{v}_{\Phi }^2}{B}_0(m_{h^{\prime }}^2,m_{h^{\prime }}^2,m_{h^{\prime }}^2)$$

$$-\frac{3(m_{h^{\prime }}^4-4m_{h^{\prime }}^2{m}_V^2+12m_V^4)}{64{\pi }^2{v}_{\Phi }^2}{B}_0(m_{h^{\prime }}^2,m_V^2,m_V^2)$$

$$-\frac{3(m_{h^{\prime }}^4-4m_{h^{\prime }}^2{m}_{Z^{\prime }}^2+12m_{Z^{\prime }}^4)}{64{\pi }^2{v}_{\Phi }^2}{B}_0(m_{h^{\prime }}^2,m_{Z^{\prime }}^2,m_{Z^{\prime }}^2)$$

$$-m_{h^{\prime }}^2{\delta }_Z.$$

Full text

1 Introduction

Many astrophysical observations show the existence of dark matter (DM). DM constitutes approximately 26% of the energy in the universe Planck:2018vyg . However, the nature of DM remains unclear. Models beyond the standard model (SM) of particle physics often predict new particles that are DM candidates. In many particle DM models, the freeze-out mechanism Lee:1977ua is used to explain the measured value of the DM energy density. The mechanism requires a pair of DM particles to annihilate into other particles in the thermal bath in the early Universe. The canonical value of the annihilation cross section, which can explain the measured value of the DM energy density, is $\expectationvalue{\sigma v}\simeq 3\times 10^{-26}$ cm3 s${}^{-1}\simeq 1$ pb $c$. This value is of the same order as the cross section of the electroweak interaction, which implies that DM particles interact with the SM particles via the electroweak interaction. An example of such DM is the wino DM, which is an SU(2)L triplet spin-$\frac{1}{2}$ Majorana fermion, and the mass prediction of the thermally produced wino DM is approximately 3 TeV Hisano:2006nn ; Cirelli:2007xd .

An electroweakly interacting spin-1 vector DM model was proposed in Abe:2020mph . In this model, the electroweak symmetry is extended to ${\rm SU}(2)_{0}\times{\rm SU}(2)_{1}\times{\rm SU}(2)_{2}\times{\rm U}(1)_{Y}$. This extended electroweak symmetry is spontaneously broken by scalar fields $\Phi_{1}$ and $\Phi_{2}$ into the electroweak symmetry, ${\rm SU}(2)_{0}\times{\rm SU}(2)_{1}\times{\rm SU}(2)_{2}\times$U(1)${}_{Y}\to{\rm SU}(2)_{L}\times$U(1)Y. The Lagrangian is invariant under the exchange between ${\rm SU}(2)_{0}$ and ${\rm SU}(2)_{2}$. This exchange symmetry is the $Z_{2}$ symmetry, which stabilizes the DM. A linear combination of SU(2)0 and SU(2)2 gauge bosons denoted as $V_{\mu}^{a}$ is $Z_{2}$-odd under the exchange symmetry. It is an approximately SU(2)L triplet, and thus is similar to the wino DM but has spin-1. The electrically neutral component of $V_{\mu}^{a}$ is slightly lighter than its charged component owing to radiative corrections at the loop level; thus, it is a DM candidate. The mass of the spin-1 DM was predicted to be between $3$ and $19$ TeV to obtain the right amount of DM energy density via the freeze-out mechanism Abe:2020mph . The allowed mass range is broader than one in the wino DM model because DM particles can annihilate into extra gauge bosons, $W^{\prime}$ and $Z^{\prime}$. The constraints of new particle searches in collider experiments and indirect searches of the DM were studied in Abe:2020mph ; Abe:2021mry . However, a wide range of parameter spaces remains unexplored.

In this paper, we focus on the phase transition of scalar potential. In particular, we investigate gravitational wave (GW) generated during the first-order phase transition of ${\rm SU}(2)_{0}\times{\rm SU}(2)_{1}\times{\rm SU}(2)_{2}\to{\rm SU}(2)_{L}$. In general, the approximate effective potential at high temperature is described as

[TABLE]

where $\varphi$ is the order parameter for the phase transition and $T$ is the temperature. $D$, $T_{0}$, $E$, and $\lambda_{T}$ are model dependent parameters. The cubic term $ET\varphi^{3}$ is necessary to realize the first-order phase transition and is generated by the thermal loop effects of bosons even if the cubic term is absent at zero temperature. In the electroweakly interacting vector DM model, all new particles are bosons, and the new gauge couplings can be relatively large, although they remain within the perturbative regime Abe:2020mph . Thus, the cubic term is expected to be sufficiently large to realize a first-order phase transition.

First-order phase transitions generate gravitational waves (GWs) Grojean:2006bp . Future space-based GW interferometers, such as LISA LISA , DECIGO DECIGO , and BBO BBO , can be used to observe GW spectra; thus, a DM model with a first-order phase transition can be tested through GW observational experiments Schwaller:2015tja ; Chala:2016ykx ; Baldes:2017rcu ; Chao:2017vrq ; Beniwal:2017eik ; Addazi:2017gpt ; Tsumura:2017knk ; Huang:2017rzf ; Huang:2017kzu ; Hektor:2018esx ; Hashino:2018zsi ; Baldes:2018emh ; Madge:2018gfl ; Beniwal:2018hyi ; Bian:2018mkl ; Bai:2018dxf ; Bian:2018bxr ; Shajiee:2018jdq ; Mohamadnejad:2019vzg ; Bertone:2019irm ; Kannike:2019mzk ; Paul:2019pgt ; Croon:2019rqu ; Hall:2019ank ; Chen:2019ebq ; Hall:2019rld ; Barman:2019oda ; Chiang:2019oms ; Borah:2020wut ; Kang:2020jeg ; Pandey:2020hoq ; Hong:2020est ; Alanne:2020jwx ; Bhoonah:2020oov ; Han:2020ekm ; Wang:2020wrk ; Ghosh:2020ipy ; Huang:2020crf ; Deng:2020dnf ; Chao:2020adk ; Azatov:2021ifm ; Zhang:2021alu ; Davoudiasl:2021ijv ; Reichert:2021cvs ; Mohamadnejad:2021tke ; Bian:2021dmp ; Costa:2022oaa ; Liu:2022jdq ; Shibuya:2022xkj ; Costa:2022lpy ; Kierkla:2022odc ; Morgante:2022zvc ; Chakrabarty:2022yzp ; Arcadi:2022lpp ; Frandsen:2022klh .

We found that GWs can be generated in the electroweakly interacting vector DM model with large gauge couplings of SU(2)0 and SU(2)1 symmetries. If the mass of $h^{\prime}$, which is the $Z_{2}$-even under exchange symmetry, is a few TeV, the model can predict the detectable GWs in future experiments. Some regions of the parameter space can be tested by both GWs and $W^{\prime}$ searches in the collider experiments.

The rest of this paper is organized as follows. In Section 2, we briefly introduce the vector DM model with ${\rm SU}(2)_{0}\times{\rm SU}(2)_{1}\times{\rm SU}(2)_{2}\times{\rm U}(1)_{Y}$ gauge symmetry. In Section LABEL:sec:potential, we show the effective potential with finite-temperature effects for this model. We clarify the parameter region with a first-order phase transition, which can produce a detectable GW spectrum. The formula for the GW spectrum from the first-order phase transition is presented in Section LABEL:sec:GW. We discuss in Section LABEL:sec:numerical the testability of the model at the GW observation experiments, such as the LISA, DECIGO, and BBO experiments. Finally, Section 6 presents the conclusions of this study.

2 The model

In this section, we briefly introduce the electroweakly interacting vector DM model proposed in Abe:2020mph .

The model exhibits gauge symmetry, described as $\text{SU(3)}_{c}\times\text{SU(2)}_{0}\times\text{SU(2)}_{1}\times\text{SU(2)}_{2}\times\text{U(1)}_{Y}$. Here, $\text{SU(3)}_{c}$ and $\text{U(1)}_{Y}$ correspond to the gauge symmetries governing the quantum chromodynamics (QCD) and hypercharge, respectively. Because the QCD sector is the same as that in the SM, we focus on the electroweak sector, denoted as $\text{SU(2)}_{0}\times\text{SU(2)}_{1}\times\text{SU(2)}_{2}\times\text{U(1)}_{Y}$. For this electroweak sector, we use the notation $W^{a}_{j\mu}$ for gauge bosons associated with $\text{SU(2)}_{j}$ and $B_{\mu}$ for those linked to $\text{U(1)}_{Y}$. Here, $j$ can take values of 0, 1, or 2, and $a$ can take values of 1, 2, or 3. $g_{j}$ and $g^{\prime}$ are the gauge couplings for $\text{SU(2)}_{j}$ and $\text{U(1)}_{Y}$, respectively.

We introduce two scalar fields, $\Phi_{1}$ and $\Phi_{2}$, expressed in two-by-two matrices. They transform under gauge transformation as

[TABLE]

where $U_{0},U_{1}$, and $U_{2}$ represent two-by-two unitary matrices for $\text{SU(2)}_{0},\text{SU(2)}_{1}$, and $\text{SU(2)}_{2}$, respectively. In addition, we impose the following conditions for $\Phi_{j}$ to reduce their degrees of freedom,

[TABLE]

Hence, each $\Phi_{j}$ consists of four real scalar fields.

All other fields remain identical to those in the SM, except that they are charged under $\text{SU(2)}_{1}$ instead of $\text{SU(2)}_{L}$. The charge assignments for the matter fields are summarized in Table 1.

In addition to gauge symmetry, this model exhibits exchange symmetry. The Lagrangian is invariant under the following field transformations:

[TABLE]

whereas all other fields remain unchanged. This symmetry is equivalent to the exchange between $\text{SU(2)}_{0}$ and $\text{SU(2)}_{2}$, implying that the gauge couplings of $\text{SU(2)}_{0}$ and $\text{SU(2)}_{2}$ must identical. It is important to note that under this symmetry, $\frac{W^{a}_{0\mu}-W^{a}_{2\mu}}{\sqrt{2}}$ and $\frac{\Phi_{1}-\Phi_{2}}{\sqrt{2}}$ change sign, whereas $\frac{W^{a}_{0\mu}+W^{a}_{2\mu}}{\sqrt{2}}$, $\frac{\Phi_{1}+\Phi_{2}}{\sqrt{2}}$, and the other fields remain unchanged. Therefore, the symmetry described in Eq. (4) is equivalent to the $Z_{2}$ symmetry commonly used in DM models. The lightest particle among $\frac{W^{a}_{0\mu}-W^{a}_{2\mu}}{\sqrt{2}}$ and $\frac{\Phi_{1}-\Phi_{2}}{\sqrt{2}}$ is stable and is a dark matter candidate for this model.

The Lagrangian of the scalar and the electroweak gauge sectors are described as

[TABLE]

where

[TABLE]

Some couplings are equal owing to the exchange symmetry described in Eq. (4).

We assume that the scalar fields develop the following vacuum expectation values (VEVs),

[TABLE]

These VEVs do not break the exchange symmetry and maintain the $Z_{2}$ symmetry, which stabilizes the DM candidate. We parametrized the component fields of each scalar field as

[TABLE]

where $\pi^{\pm}_{3},\pi^{0}_{3},\pi^{\pm}_{j},\pi^{\pm}_{j}$, and $\pi^{0}_{j}$ are would-be Nambu-Goldstone (NG) bosons. Based on the stationary condition, we obtain the followings:

[TABLE]

2.1 Scalar boson masses

The mass terms for scalar fields other than the would-be NG bosons are described as

Figure 6: Detectability of the GW in the $m_{h^{\prime}}$-$m_{Z^{\prime}}$ plane for $m_{V}=$ 7 TeV and $m_{h_{D}}=1.2m_{V}$. The upper (lower) two panels are for $v_{b}=$ 0.3 ($v_{b}=1$). In the left (right) panels, $\mathcal{T}$ = 4 (10) yrs. In the light red regions, SNR $>10$ in the BBO, DECIGO, and LISA experiments. In the standard red regions, SNR $>10$ in the DECIGO and LISA experiments. In the dark red regions, SNR $>10$ only in the LISA experiment. The black dashed lines indicate the regions where the measured value of the DM energy density is explained by the freeze-out mechanism.

The heavy vector and scalar bosons in the parameter region in this figure cannot be explored through the direct $W^{\prime}$ search in the current and future collider experiments. The upper (lower) two panels correspond to $v_{b}=0.3$ ($v_{b}=0.1$). Dark red, standard red, and light red represent the regions where the SNR exceeds 10 in each experiment. In the light red region, the SNR is larger than ten in the BBO experiment. In contrast, the standard red (dark red) regions are tested using DECIGO and BBO (LISA, DECIGO, and BBO) experiments. A strong first-order phase transition is not realized in the white regions to the right of the light-red region; thus, detectable GW spectra are not generated. In the regions to the left of the dark red regions, the phase transition is not completed in the current universe, namely $\Gamma/H^{4}<1$. The black dashed lines indicate the regions where the measured value of the DM energy density is explained by the freeze-out mechanism. We find that if $m_{V}=7$ TeV and $m_{h^{\prime}}$ is within the range of 2.5 TeV $\lesssim m_{h^{\prime}}\lesssim 3.5$ TeV, the model can explain the DM energy density and can be tested using the GW observational experiments. It is difficult to produce such a heavy $h^{\prime}$ in collider experiments. However, we can test the heavy $h^{\prime}$ regime using the GW signals.

Figure 2.1 is similar to Fig. 2.1, except for $m_{V}$ = 5 TeV.

Figure 7: Detectability of the GW for $m_{V}=$ 5 TeV. The blue-hatched regions are explored using the HL-LHC. The other color notations are the same as in Fig. 2.1.

The HL-LHC can test the model using the $W^{\prime}$ search if the model parameters are within the blue-hatched regions. The blue-hatched and red regions are overlapped when $1.6\text{~{}TeV}\lesssim m_{h^{\prime}}\lesssim 2.5$ TeV. In this mass range, the model can be tested using both the $W^{\prime}$ search at the HL-LHC and GW observational experiments. Because it is difficult to produce a heavy $h^{\prime}$ in collider experiments, the GW signal is a useful tool to determine the value of $m_{h^{\prime}}$.

For $m_{V}$ = 3 TeV, which is shown in Fig. 2.1, some regions of the parameter space are excluded by the direct search of $W^{\prime}$ in the ATLAS experiment.

Figure 8: Detectability of the GW for $m_{V}=$ 3 TeV. The $W^{\prime}$ search at the ATLAS experiment excludes the black-hatched region. The other color notation is the same as in Fig. 2.1.

For the parameter points that can explain the measured value of the DM energy density, the HL-LHC is used to test the model for $m_{h^{\prime}}\gtrsim 4$ TeV, and the GW observational experiments are used for $2.8$ TeV $\lesssim m_{h^{\prime}}\lesssim$ 3.5 TeV. Therefore, the collider and GW observational experiments complement each other.

6 Conclusion

We have discussed the GWs in an electroweakly interacting vector DM model, which has an extended gauge symmetry: SU(3)C$\times$ SU(2)0$\times$ SU(2)1$\times$ SU(2)${}_{2}\times$U(1)Y. To ensure the stability of the DM, we assume exchange symmetry between SU(2)0 and SU(2)2, and a new gauge boson $V^{0}$ is a DM candidate. Above the electroweak scale, a phase transition ${\rm SU}(2)_{0}\times{\rm SU}(2)_{1}\times{\rm SU}(2)_{2}\to{\rm SU}(2)_{L}$ occurs. All new particles in the model are bosons, and the new gauge couplings can be relatively large within the perturbative regime. Thus, a potential barrier is easily produced during the phase transition. Consequently, the phase transition is strongly first order and produces detectable gravitational waves.

We found that the effective potential with finite temperature is sensitive to $m_{h^{\prime}}$. When $m_{h^{\prime}}$ value is large, the corrections by the gauge bosons are relatively smaller than the contribution of $m_{h^{\prime}}$ at the tree level; thus, the phase transition is not strongly first-order. For smaller $m_{h^{\prime}}$, the loop effects of the gauge bosons are significant; thus, $\varphi_{C}/T_{C}$ becomes larger. However, when $m_{h^{\prime}}$ is small, the gauge boson contributions generate a local minimum at the potential origin. For relatively smaller $m_{h^{\prime}}$, the local minimum at the origin becomes the global minimum of the potential, and the phase transition does not occur. Even when the global minimum is at $\varphi\neq 0$, the tunneling rate is low, and the phase transition does not occur when $m_{h^{\prime}}$ is small. Therefore, there is a lower bound on $m_{h^{\prime}}$ for the phase transition.

The model predicts a detectable GW spectrum in the LISA, DECIGO, and BBO experiments. With the assumption that the model explains the measured value of the DM energy density through the freeze-out mechanism, the model predicts detectable GW spectra when 2.5 TeV $\lesssim m_{h^{\prime}}\lesssim$ 3.5 TeV for $m_{V}=$7 TeV, 1.6 TeV $\lesssim m_{h^{\prime}}\lesssim$ 2.5 TeV for $m_{V}=$5 TeV, and 2.8 TeV $\lesssim m_{h^{\prime}}\lesssim$ 3.5 TeV for $m_{V}=3$ TeV. Because it is difficult to produce heavy $h^{\prime}$ in collider experiments, it is crucial to use GW signals in determining $m_{h^{\prime}}$.

Acknowledgements.

This work was supported by JSPS KAKENHI Grant Number 19H04615 and 21K03549 [T.A.]. We would like to thank Editage (www.editage.jp) for English language editing.

Appendix A Approximated expressions of the gauge bosons for $v_{\Phi}\gg v$

For $v_{\Phi}\gg v$, the approximate mass eigenstates are described as

[TABLE]

where $e$ is the electric charge, expressed as $e=\sqrt{2/g_{0}^{2}+1/g_{1}^{2}+1/g^{\prime 2}}$. The masses of these gauge bosons at the tree level are described as follows:

[TABLE]

At the 1-loop level, the mass difference between $V^{\pm}$ and $V^{0}$ is $m_{V^{\pm}}-m_{V^{0}}\simeq 166$ MeV Abe:2020mph . Using the masses of the gauge bosons, the gauge couplings $g_{0}$ and $g_{1}$ for $v_{\Phi}\gg v$ can be expressed as

[TABLE]

where $g_{W}$ is

[TABLE]

Appendix B $V_{\text{CW}}+\delta V$

Defining $\varphi=\sqrt{\varphi_{1}^{2}+\varphi_{2}^{2}}$, then the renormalized effective potential at $T=0$ as a function of $\varphi$ can be expressed as follows:

[TABLE]

where

[TABLE]

Here, $\delta_{Z}$ is the counter term for the wave-function renormalization. When the MS bar scheme is applied, the IR divergences originated from the would-be NG boson contributions in $V_{\text{CW}}$ and $\Delta\Sigma$ cancel each other. After the cancelation, the dominant contribution of $\Delta\Sigma$ to the potential comes from the terms depending on the gauge couplings, described as

[TABLE]