Implications of tachyonic phase transition in classically scale invariant general $U(1)_{X}$ models

Author(s)

Das, Arindam, Hashino, Katsuya, Orikasa, Yuta, Tanaka, Masanori

Abstract

We investigate the complementarity between collider searches and gravitational-wave (GW) observations in the classically scale-invariant general $U(1)_X$ extension of the Standard Model. The $U(1)_X$ scale is generated radiatively through the Coleman-Weinberg mechanism, while the electroweak scale is induced through the Higgs portal, without explicit mass terms in the scalar potential. Three right-handed neutrinos are introduced to ensure anomaly cancellation and acquire Majorana masses upon $U(1)_X$ symmetry breaking, which also generates the mass of the neutral gauge boson $Z'$. In the strongly supercooled regime, a tachyonic $U(1)_X$ phase transition can produce a stochastic GW background. Assuming three heavy Majorana neutrinos, we study their effects on the phase-transition dynamics and reheating, and estimate the resulting GW signals. We identify the regions of the $U(1)_X$ gauge coupling and $Z'$ mass accessible to future GW observatories, including LISA and DECIGO, and compare them with existing LEP and LHC constraints. We find that GWs from tachyonic phase transitions can probe regions with small gauge couplings and heavy $Z'$ bosons that are difficult to access through collider searches, demonstrating the complementarity of these probes.

Figures

The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 1 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R)$, 0 (middle,B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for the QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
Caption The gauge coupling dependence of the characteristic temperatures relevant for the phase transition considering three degenerate heavy neutrinos of mass 10 TeV, two benchmark points of $m_{Z^\prime}=100$ (1000) TeV given in the left (right) panel for $x_H=-2$ (top, $U(1)_R$), 0 (middle, B$-$L) and 2 (bottom). The red, blue and green solid lines denote the percolation $(T_{\rm perc})$, the reheating $(T_{\rm re})$ and the critical $(T_{C})$ temperatures, respectively. In the orange region, the tachyonic phase transition can occur. The perturbativity excludes the gray region. The black dashed line represents the temperature $(T_{\rm QCD})$ for QCD phase transition.
Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=-2$ (top, $U(1)_R$), $-1$ (bottom) respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC as described in Sec.~\ref{secII}. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Caption Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=-2$ (top, $U(1)_R$), $-1$ (bottom) respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC as described in Sec.~\ref{secII}. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=-2$ (top, $U(1)_R$), $-1$ (bottom) respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC as described in Sec.~\ref{secII}. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Caption Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=-2$ (top, $U(1)_R$), $-1$ (bottom) respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC as described in Sec.~\ref{secII}. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=-2$ (top, $U(1)_R$), $-1$ (bottom) respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC as described in Sec.~\ref{secII}. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Caption Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=-2$ (top, $U(1)_R$), $-1$ (bottom) respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC as described in Sec.~\ref{secII}. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=-2$ (top, $U(1)_R$), $-1$ (bottom) respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC as described in Sec.~\ref{secII}. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Caption Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=-2$ (top, $U(1)_R$), $-1$ (bottom) respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC as described in Sec.~\ref{secII}. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=0$(top, B$-$L), $2$ (bottom), respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Caption Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=0$(top, B$-$L), $2$ (bottom), respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=0$(top, B$-$L), $2$ (bottom), respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Caption Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=0$(top, B$-$L), $2$ (bottom), respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=0$(top, B$-$L), $2$ (bottom), respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Caption Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=0$(top, B$-$L), $2$ (bottom), respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=0$(top, B$-$L), $2$ (bottom), respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Caption Parameter region explored by GW observatories and colliders for each benchmark point for $x_H=0$(top, B$-$L), $2$ (bottom), respectively for degenerate heavy neutrinos with masses 1 (10)TeV in the left (right) panel. In the colored regions, GWs can be detectable at each observatory. The dark gray region shows constraints by current collider searches from LEP and LHC. In the light gray regions, the perturbativity bound $|\lambda_{\Phi}|<4\pi$ or the requirements from Eq.~\eqref{eq:m_varphi_mh_mN} and Eq.~\eqref{eq:mQCD_Hubble} are not satisfied.
Predicted GWs in each benchmark point in Table~\ref{tab:BP_GW}. The sensitivity curves of LISA~\cite{Caprini:2024hue}, DECIGO~\cite{Kawamura:2006up,Yagi:2011wg}, Cosmic Explorer (CE)~\cite{Reitze:2019iox} and Einstein telescope (ET)~\cite{Punturo:2010zz, ET:2025xjr} are also shown. The gray dotted lines denote the background signal from galactic foreground and extra-galactic foreground~\cite{Caprini:2024hue}.
Caption Predicted GWs in each benchmark point in Table~\ref{tab:BP_GW}. The sensitivity curves of LISA~\cite{Caprini:2024hue}, DECIGO~\cite{Kawamura:2006up,Yagi:2011wg}, Cosmic Explorer (CE)~\cite{Reitze:2019iox} and Einstein telescope (ET)~\cite{Punturo:2010zz, ET:2025xjr} are also shown. The gray dotted lines denote the background signal from galactic foreground and extra-galactic foreground~\cite{Caprini:2024hue}.
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