Prospects of gravitational waves in the minimal left-right symmetric model

The basic idea of LRSMs is to extend the EW sector of $SU(2)_{L}\times U(1)_{Y}$ of the SM gauge group to be left-right symmetric, i.e. $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$. Various LRSMs have been proposed to understand the parity symmetry and CP breaking of the SM, the origin of masses of matters or even DM candidates and the matter-antimatter asymmetry of the universe. The main differences between these LRSMs could be in the gauge structure, the scalar fields, the matter contents, and/or the seesaw mechanisms.

The most popular, or say conventional, LRSM is the version with a Higgs bidoublet $\Phi$, a left-handed triplet $\Delta_{L}$ and a right-handed triplet $\Delta_{R}$ Pati:1974yy ; Mohapatra:1974gc ; Senjanovic:1975rk

$\displaystyle\Phi=\left(\begin{array}[]{cc}\phi^{0}_{1}&\phi^{+}_{2}\\ \phi^{-}_{1}&\phi^{0}_{2}\end{array}\right),\;$

$\displaystyle\Delta_{L}=\left(\begin{array}[]{cc}\Delta^{+}_{L}/\sqrt{2}&\Delta^{++}_{L}\\ \Delta^{0}_{L}&-\Delta^{+}_{L}/\sqrt{2}\end{array}\right),\;\;\;\Delta_{R}=\left(\begin{array}[]{cc}\Delta^{+}_{R}/\sqrt{2}&\Delta^{++}_{R}\\ \Delta^{0}_{R}&-\Delta^{+}_{R}/\sqrt{2}\end{array}\right).$

(7)

When the right-handed triplet $\Delta_{R}$ acquires a vacuum expectation value (VEV) $v_{R}$, the gauge symmetry $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ in the LRSM is broken to the SM gauge group $SU(2)_{L}\times U(1)_{Y}$. Two triplets $\Delta_{L}$ and $\Delta_{R}$ are introduced to give Majorana masses to the active neutrinos and RHNs, respectively, which enables the type-I Minkowski:1977sc ; Mohapatra:1979ia ; Yanagida:1979as ; GellMann:1980vs ; Glashow:1979nm and type-II Mohapatra:1980yp ; Magg:1980ut ; Schechter:1980gr ; Cheng:1980qt ; Lazarides:1980nt seesaw mechanisms for the tiny neutrino masses.

The $SU(2)_{R}\times U(1)_{B-L}$ symmetry can also be broken only by a right-handed doublet $H_{R}$ Babu:1988mw ; Babu:1989rb . In this case, heavy vector-like fermions have to be introduced to generate the SM quark and lepton masses via seesaw mechanism (see also Mohapatra:2014qva ). There are also LRSM scenarios with inverse seesaw Mohapatra:1986aw ; Mohapatra:1986bd , linear seesaw Akhmedov:1995ip ; Malinsky:2005bi , or extended seesaw Gavela:2009cd ; Barry:2011wb ; Zhang:2011vh ; Dev:2012sg in the literature. Cold DM is not included in the conventional LRSM (a light RHN can only be a warm DM candidate Nemevsek:2012cd ), but it is easy to add a fermion or boson multiplet, where the lightest neutral component is naturally stabilized by the residual $Z_{2}$ symmetry from $U(1)_{B-L}$ breaking Heeck:2015qra ; Garcia-Cely:2015quu . Alternatively, based on the gauge group $SU(2)_{L}\times SU(2)_{R}\times U(1)_{Y_{L}}\times U(1)_{Y_{R}}$ (with $Y_{L}$ the hypercharge in the SM and $Y_{R}$ the “right-handed” counter partner), heavy RHNs can be the cold DM candidate Dev:2016qbd ; Dev:2016xcp ; Dev:2016qeb .

In this work, we focus on the minimal LRSM with one bidoublet $\Phi$ and two triplets $\Delta_{L}$ and $\Delta_{R}$ in the scalar sector. The most general scalar potential in the LRSM can be written as Deshpande:1990ip

	$\displaystyle{\cal V}$	$\displaystyle=$	$\displaystyle-\mu_{1}^{2}\operatorname{Tr}[\Phi^{\dagger}\Phi]-\mu_{2}^{2}\left(\operatorname{Tr}[\tilde{\Phi}\Phi^{\dagger}]+\operatorname{Tr}[\tilde{\Phi}^{\dagger}\Phi]\right)-\mu_{3}^{2}\left(\operatorname{Tr}[\Delta_{L}\Delta_{L}^{\dagger}]+\operatorname{Tr}[\Delta_{R}\Delta_{R}^{\dagger}]\right)$		(8)
			$\displaystyle+\rho_{1}\left(\operatorname{Tr}[\Delta_{L}\Delta_{L}^{\dagger}]^{2}+\operatorname{Tr}[\Delta_{R}\Delta_{R}^{\dagger}]^{2}\right)+\rho_{2}\left(\operatorname{Tr}[\Delta_{L}\Delta_{L}]\operatorname{Tr}[\Delta_{L}^{\dagger}\Delta_{L}^{\dagger}]+\operatorname{Tr}[\Delta_{R}\Delta_{R}]\operatorname{Tr}[\Delta_{R}^{\dagger}\Delta_{R}^{\dagger}]\right)$
			$\displaystyle+\rho_{3}\operatorname{Tr}[\Delta_{L}\Delta_{L}^{\dagger}]\operatorname{Tr}[\Delta_{R}\Delta_{R}^{\dagger}]+\rho_{4}\left(\operatorname{Tr}[\Delta_{L}\Delta_{L}]\operatorname{Tr}[\Delta_{R}^{\dagger}\Delta_{R}^{\dagger}]+\operatorname{Tr}[\Delta_{L}^{\dagger}\Delta_{L}^{\dagger}]\operatorname{Tr}[\Delta_{R}\Delta_{R}]\right)$
			$\displaystyle+\lambda_{1}\operatorname{Tr}[\Phi^{\dagger}\Phi]^{2}+\lambda_{2}\left(\operatorname{Tr}[\tilde{\Phi}\Phi^{\dagger}]^{2}+\operatorname{Tr}[\tilde{\Phi}^{\dagger}\Phi]^{2}\right)$
			$\displaystyle+\lambda_{3}\operatorname{Tr}[\tilde{\Phi}\Phi^{\dagger}]\operatorname{Tr}[\tilde{\Phi}^{\dagger}\Phi]+\lambda_{4}\operatorname{Tr}[\Phi^{\dagger}\Phi]\left(\operatorname{Tr}[\tilde{\Phi}\Phi^{\dagger}]+\operatorname{Tr}[\tilde{\Phi}^{\dagger}\Phi]\right)$
			$\displaystyle+\alpha_{1}\operatorname{Tr}[\Phi^{\dagger}\Phi]\left(\operatorname{Tr}[\Delta_{L}\Delta_{L}^{\dagger}]+\operatorname{Tr}[\Delta_{R}\Delta_{R}^{\dagger}]\right)+\alpha_{3}\left(\operatorname{Tr}[\Phi\Phi^{\dagger}\Delta_{L}\Delta_{L}^{\dagger}]+\operatorname{Tr}[\Phi^{\dagger}\Phi\Delta_{R}\Delta_{R}^{\dagger}]\right)$
			$\displaystyle+\left[\alpha_{2}e^{i\delta}\left(\operatorname{Tr}[\Delta_{L}\Delta_{L}^{\dagger}]\operatorname{Tr}[\tilde{\Phi}\Phi^{\dagger}]+\operatorname{Tr}[\Delta_{R}\Delta_{R}^{\dagger}]\operatorname{Tr}[\tilde{\Phi}^{\dagger}\Phi]\right)+\mathrm{H.c.}\right]$
			$\displaystyle+\beta_{1}\left(\operatorname{Tr}[\Phi\Delta_{R}\Phi^{\dagger}\Delta_{L}^{\dagger}]+\operatorname{Tr}[\Phi^{\dagger}\Delta_{L}\Phi\Delta_{R}^{\dagger}]\right)+\beta_{2}\left(\operatorname{Tr}[\tilde{\Phi}\Delta_{R}\Phi^{\dagger}\Delta_{L}^{\dagger}]+\operatorname{Tr}[\tilde{\Phi}^{\dagger}\Delta_{L}\Phi\Delta_{R}^{\dagger}]\right)$
			$\displaystyle+\beta_{3}\left(\operatorname{Tr}[\Phi\Delta_{R}\tilde{\Phi}^{\dagger}\Delta_{L}^{\dagger}]+\operatorname{Tr}[\Phi^{\dagger}\Delta_{L}\tilde{\Phi}\Delta_{R}^{\dagger}]\right),$

where $\tilde{\Phi}=\sigma_{2}\Phi^{\ast}\sigma_{2}$ (with $\sigma_{2}$ the second Pauli matrix). Required by left-right symmetry, all the quartic couplings in the potential above are real parameters. The CP violating phase $\delta$ associated with $\alpha_{2}$ is shown explicitly.

At the zero temperature, the neutral components of the scalar fields can develop non-zero VEVs, i.e.

$$\langle\Phi\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\kappa_{1}&0\\ 0&\kappa_{2}e^{i\theta_{\kappa}}\end{array}\right),\quad\langle\Delta_{L}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}0&0\\ v_{L}e^{i\theta_{L}}&0\end{array}\right),\quad\langle\Delta_{R}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}0&0\\ v_{R}&0\end{array}\right)\,,$$

(9)

where $\theta_{\kappa}$ and $\theta_{L}$ are CP violating phases. The two bidoublet VEVs are related to the EW VEV $v_{\rm EW}\simeq(\sqrt{2}G_{F})^{-1/2}\simeq 246$ GeV (with $G_{F}$ the Fermi constant) via $\sqrt{\kappa_{1}^{2}+\kappa_{2}^{2}}=v_{\rm EW}$. In light of the hierarchy of top and bottom quark masses $m_{b}\ll m_{t}$ in the SM, it is a reasonable assumption that $\kappa_{2}\ll\kappa_{1}$ Deshpande:1990ip . There are three key energy scales in the LRSM, i.e. the right-handed scale $v_{R}$, the EW scale $v_{\rm EW}$ and the scale $v_{L}$ which is relevant to tiny active neutrino masses via type-II seesaw. Furthermore, from the first-order derivative of the scalar potential (8), $v_{L}$ is related to the EW and right-handed VEVs via Mohapatra:1980yp ; Deshpande:1990ip ; Kiers:2005gh

$\displaystyle v_{L}=\frac{v_{\rm EW}^{2}/v_{R}}{(1+\xi^{2})(2\rho_{1}-\rho_{3})}\left[\beta_{1}\xi\cos(\alpha-\theta_{L})+\beta_{2}\cos\theta_{L}+\beta_{3}\xi^{2}\cos(2\alpha-\theta_{L})\right]\,,$

(10)

where $\xi=\kappa_{2}/\kappa_{1}$. Due to the tiny masses of active neutrinos, it is a good approximation to set $v_{L}=0$, therefore we will set $\beta_{i}=0$ so as to simplify our discussions below.

With $v_{L}=0$, there are only two energy scales in the LRSM, i.e. the EW scale $v_{\rm EW}$ and the right-handed scale $v_{R}$. In light of the hierarchy structure $v_{\rm EW}\ll v_{R}$, a two-step phase transition is supposed to occur in the LRSM. In the early universe, the temperature is so high $T\gg v_{R}$ that the symmetry $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ is restored. As the universe keeps expanding, the temperature decreases. When the temperature is lower than a critical temperature but much higher than EW scale, i.e. $v_{\rm EW}\ll T\sim v_{R}$, $\Delta_{R}^{0}$ develops a non-vanishing VEV and the gauge symmetry $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ is spontaneously broken to $SU(2)_{L}\times U(1)_{Y}$. When the temperature becomes lower than the EW scale $T\sim v_{\rm EW}$, $\Phi_{1,2}^{0}$ obtain their VEVs and the symmetry is further broken into the electromagnetic (EM) group $U(1)_{\rm EM}$.

After symmetry breaking at the $v_{R}$ scale, we can rewrite the bidoublet $\Phi$ in terms of two $SU(2)_{L}$ doublets, i.e. $\Phi=\big{(}i\sigma_{2}H_{1}^{\ast}|H_{2}\big{)}$. Then the bidoublet relevant terms in the potential (8) can be recast in terms of $H_{1,\,2}$:

	$\displaystyle{\cal V}(\Phi)\ \supset\$	$\displaystyle-m_{11}^{2}H_{1}^{\dagger}H_{1}+m_{22}^{2}H_{2}^{\dagger}H_{2}-m_{12}^{2}(H_{1}^{\dagger}H_{2}+\text{H.c.})$
		$\displaystyle+\lambda_{1}(H_{1}^{\dagger}H_{1})^{2}+\lambda_{1}(H_{2}^{\dagger}H_{2})^{2}+2\lambda_{1}H_{1}^{\dagger}H_{1}H_{2}^{\dagger}H_{2}+4\lambda_{3}H_{1}^{\dagger}H_{2}H_{2}^{\dagger}H_{1}$
		$\displaystyle+[4\lambda_{2}(H_{1}^{\dagger}H_{2})^{2}+2\lambda_{4}(H_{1}^{\dagger}H_{1}+H_{2}^{\dagger}H_{2})H_{1}^{\dagger}H_{2}+\text{H.c.}]\,,$		(11)

where the mass terms are respectively

	$\displaystyle m_{11}^{2}$	$\displaystyle\ =\$	$\displaystyle-\frac{\alpha_{3}}{2}\frac{\kappa_{2}^{2}v_{R}^{2}}{\kappa_{1}^{2}-\kappa_{2}^{2}}+\lambda_{1}v_{\rm EW}^{2}+2\lambda_{4}\kappa_{1}\kappa_{2}\,,$		(12)
	$\displaystyle m_{22}^{2}$	$\displaystyle\ =\$	$\displaystyle-m_{11}^{2}+\frac{\alpha_{3}}{2}v_{R}^{2}\,,$		(13)
	$\displaystyle m_{12}^{2}$	$\displaystyle\ =\$	$\displaystyle\frac{\alpha_{3}}{2}\frac{\kappa_{1}\kappa_{2}v_{R}^{2}}{\kappa_{1}^{2}-\kappa_{2}^{2}}+2(2\lambda_{2}+\lambda_{3})\kappa_{1}\kappa_{2}+\lambda_{4}v_{\rm EW}^{2}\,.$		(14)

Although the potential in Eq. (2.1) seems to be very similar to that in a general 2HDM Branco:2011iw , there are still some obvious differences: In presence of the scale $v_{R}$, all the states predominately from the heavy doublet $H_{2}$ are at the $v_{R}$ scale, and their masses are degenerate at the leading-order, which is clearly distinct from the 2HDMs, where all the scalars in the 2HDMs are at the EW scale, and the BSM scalar masses depend on different quartic couplings Branco:2011iw .

In the LRSM, the BSM particles include the heavy $W_{R}$ and $Z_{R}$ bosons, three RHNs $N_{i}$ (with $i=1,\,2,\,3$), neutral CP-even scalar $H_{1}^{0}$ and CP-odd $A_{1}^{0}$, singly-charged scalar $H_{1}^{\pm}$ predominately from the bidoublet $\Phi$, neutral CP-even scalar $H_{2}^{0}$ and CP-odd $A_{2}^{0}$, singly-charged scalar $H_{2}^{\pm}$ and doubly-charged scalar $H_{1}^{\pm\pm}$ mostly from the left-handed triplet $\Delta_{L}$, and the neutral CP-even scalar $H_{3}^{0}$ and doubly-charged scalar $H_{2}^{\pm\pm}$ mostly from the right-handed triplet $\Delta_{R}$. Thorough studies of the scalar sector of LRSM at future high-energy colliders can be found e.g. in Ref. Gunion:1989in ; Deshpande:1990ip ; Polak:1991vf ; Barenboim:2001vu ; Azuelos:2004mwa ; Zhang:2007da ; Jung:2008pz ; Bambhaniya:2013wza ; Dutta:2014dba ; Bambhaniya:2014cia ; Bambhaniya:2015ipg ; Maiezza:2015lza ; Bambhaniya:2015wna ; Bonilla:2016fqd ; Maiezza:2016ybz ; Maiezza:2016bzp ; Nemevsek:2016enw ; Chakrabortty:2016wkl ; Dev:2016dja ; Dev:2016vle ; Dev:2017dui ; Cao:2017rjr ; Dev:2018foq ; Dev:2018kpa ; Chauhan:2019fji . In this paper, we assume that the gauge coupling $g_{R}$ for $SU(2)_{R}$ can be different from the gauge coupling $g_{L}$ for $SU(2)_{L}$, which might originate from renormalization group running effects such as in the $D$-parity breaking LRSM versions Chang:1983fu .

For completeness, we collect all the theoretical constraints on the gauge and scalar sectors of the LRSM in the literature, which will be taken into consideration in the calculations of phase transition and GW production below.

• •

  Perturbativity limits: In some versions of the LRSM, the right-handed gauge coupling $g_{R}$ can be different from $g_{L}$ Chang:1983fu . As the gauge couplings have the relationship (with $g_{BL}$ the gauge coupling for $U(1)_{B-L}$)

  $\displaystyle\frac{1}{e^{2}}\ =\ \frac{1}{g_{L}^{2}}+\frac{1}{g_{Y}^{2}}\ =\ \frac{1}{g_{L}^{2}}+\frac{1}{g_{R}^{2}}+\frac{1}{g_{BL}^{2}}\,,$

  (15)

  the gauge couplings $g_{R}$ and $g_{BL}$ can not be either too large or too small if we want them to be perturbative. Renormalization group running these gauge couplings up to a higher energy scale put more stringent limits on them. Perturbativity up to the GUT scale requires that the ratio $r_{g}\equiv g_{R}/g_{L}$ to satisfy Chauhan:2018uuy ¹¹1Note that the perturbativity limits in Ref. Chauhan:2018uuy are on the LRSM without the left-handed triplet $\Delta_{L}$ at the TeV-scale. In presence of $\Delta_{L}$ at the TeV-scale, the perturbativity limits should be to some extent different. As an approximation we will adopt the limits from Ref. Chauhan:2018uuy .

  $\displaystyle 0.65<r_{g}<1.60\,.$

  (16)

  Furthermore, as the masses $\sqrt{\alpha_{3}/2}v_{R}$ of $H_{1}^{0}$, $A_{1}^{0}$ and $H_{1}^{\pm}$ (cf. Table 5 in Appendix A) are severely constrained by the neutral meson mixings (see Section 2.2 and Table 1), perturbativity also implies an lower bound on the $v_{R}$ scale Chauhan:2018uuy :

  $\displaystyle v_{R}\gtrsim 10\,{\rm TeV}\,.$

  (17)

  For $v_{R}$ below this value, $\alpha_{3}$ is so large that all the quartic and gauge couplings will hit the Landau pole very quickly before reaching the GUT or Planck scale Rothstein:1990qx ; Chakrabortty:2013zja ; Chakrabortty:2016wkl ; Maiezza:2016ybz .

• •

  Unitarity conditions: The parameters in the potential (8) should satisfy the unitarity conditions Chakrabortty:2016wkl when we consider the scattering amplitudes of the scalar fields at the high-energy scale $\sqrt{s}\gg\mu^{2}_{i}$ (for simplicity we neglect here the effects of all the scalar masses). In other words, the partial wave amplitudes should not violate the bound of unitarity so as to guarantee that the probability is conserved. The tree-level unitarity conditions turn out to be Chakrabortty:2016wkl

  	$\displaystyle\lambda_{1,\,4}<\frac{4\pi}{3}\,,\quad\lambda_{1}+4\lambda_{2}+2\lambda_{3}<4\pi\,,\quad\lambda_{1}-4\lambda_{2}+2\lambda_{3}<\frac{4\pi}{3}\,,$
  	$\displaystyle\rho_{1}<\frac{4\pi}{3}\,,\quad\rho_{1}+\rho_{2}<2\pi\,,\quad\rho_{2,\,4}<2\sqrt{2}\pi\,,\quad\rho_{3}<8\pi\,,$
  	$\displaystyle\alpha_{1}<8\pi\,,\quad\alpha_{2}<4\pi\,,\quad\alpha_{1}+\alpha_{3}<8\pi\,.$		(18)

• •

  Vacuum stability conditions: The vacuum stability conditions require that Chakrabortty:2013zja ; Chakrabortty:2013mha ; Chakrabortty:2016wkl (see also Kannike:2016fmd )

  $\displaystyle\lambda_{1}>0\,\,,\quad\rho_{1}>0\,\,,\quad\rho_{1}+\rho_{2}>0\,\,,\quad\rho_{1}+2\rho_{2}>0\,\,.$

  (19)

• •

  Correct vacuum criteria: After the spontaneous symmetry breaking, all the scalar fields have to form some specific structure in the phase space such that we reside in the correct vacuum, i.e. the vacuum with the lowest VEV in the potential Dev:2018foq ; Chauhan:2019fji . For completeness, the correct vacuum criteria have been collected in Appendix B, which are obtained with the assumption $\alpha_{2}=0$. Therefore, we will set $\alpha_{2}=0$ throughout this paper.

  In the limit of $\kappa_{2}\ll\kappa_{1}\ll v_{R}$, in Eq. (13), the quadratic coefficient of $H_{2}$ term is proportional to $\alpha_{3}v_{R}^{2}/2$, thus the heavy doublet scalars $H_{1}^{0},A_{1}^{0},H_{2}^{\pm}$ will obtain a mass of $\sqrt{\alpha_{3}/2}v_{R}$ at the leading-order. To get the correct EW vacuum, a necessary condition is $m_{11}^{2}>0$, i.e.

  $$-\frac{\alpha_{3}}{2}\frac{\kappa_{2}^{2}v_{R}^{2}}{\kappa_{1}^{2}-\kappa_{2}^{2}}+\lambda_{1}v_{\rm EW}^{2}+2\lambda_{4}\kappa_{1}\kappa_{2}>0.$$

  (20)

  This yields an upper bound of $\xi$. Approximately, we have

  	$\displaystyle\xi$	$\displaystyle\ \lesssim\$	$\displaystyle\frac{\sqrt{\lambda_{1}}v_{\rm EW}}{M_{H_{1}^{0}}}$		(21)
  		$\displaystyle\ \lesssim\$	$\displaystyle 8.9\times 10^{-3}\left(\frac{\lambda_{1}}{0.13}\right)^{1/2}\left(\frac{m_{H_{1}^{0}}}{10\,{\rm TeV}}\right)^{-1}\,.$

All the current LHC limits on the BSM particles in the LRSM are collected in Table 1 and also depicted in Fig. 1. Here are more details:

• •

  At the LHC, the $W_{R}$ boson in the LRSM can be produced via the right-handed charged quark currents. After its production, it can decay predominately into two quark jets (including the $\bar{t}b$ channel) and RHNs plus a charged lepton, i.e. $W_{R}\to jj,\,\bar{t}b,\,N_{i}^{(\ast)}\ell_{\alpha}$ (with $\alpha=e,\,\mu,\,\tau$). If the RHNs are lighter than the $W_{R}$ boson, as a result of the Majorana nature of RHNs, the same-sign dilepton plus jets $W_{R}\to N\ell\to\ell_{\alpha}\ell_{\beta}jj$ constitute a smoking-gun signal of the $W_{R}$ boson Keung:1983uu . Assuming $g_{R}=g_{L}$, the current most stringent LHC data require that the $W_{R}$ mass $m_{W_{R}}>(3.8-5)$ TeV for a RHN mass $100\,{\rm GeV}<m_{N}<1.8$ TeV Aaboud:2018spl ; Aaboud:2019wfg . The dijet Aad:2019hjw ; Sirunyan:2019vgj and $\bar{t}b$ Sirunyan:2017ukk ; Sirunyan:2017vkm limits are relatively weaker, which are respectively 4 TeV and 3.4 TeV. The strongest $W_{R}$ limit of $(3.8-5)$ TeV is presented in Fig. 1.

• •

  The most stringent limits on the $Z_{R}$ boson is from the dilepton data $pp\to Z_{R}\to\ell^{+}\ell^{-}$. The current dilepton limit on a sequential $Z^{\prime}$ boson is 5.1 TeV Aad:2019fac . Following e.g. Ref. Chauhan:2018uuy , one can rescale the production cross section times branching fraction $\sigma(pp\to Z^{\prime}\to\ell^{+}\ell^{-})$ for the sequential $Z^{\prime}$ model, which leads to the LHC dilepton limit of 4.82 TeV on the $Z_{R}$ boson in the LRSM. This is shown in Fig. 1 as the $Z_{R}$ limit. There are also dijet searches of the $Z^{\prime}$ boson, however the corresponding limits are relatively weaker Aad:2019hjw ; Sirunyan:2019vgj .

• •

  At the leading order, the scalars $H_{2}^{0}$, $A_{2}^{0}$, $H_{2}^{\pm}$ and $H_{1}^{\pm\pm}$ from the left-handed triplet $\Delta_{L}$ have the same mass Zhang:2007da (see Table 5). The doubly-charged scalar $H_{1}^{\pm\pm}$ can decay into either same-sign dilepton or same-sign $W$ bosons, i.e. $H_{1}^{\pm\pm}\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm},\,W^{\pm}W^{\pm}$, which constitute the most promising channels to probe $\Delta_{L}$ at the LHC, and the branching fractions ${\rm BR}(H_{1}^{\pm\pm}\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm})$ and ${\rm BR}(H_{1}^{\pm\pm}\to W^{\pm}W^{\pm})$ depend on the Yukawa coupling $f_{L}$ and the left-handed triplet VEV $v_{L}$. Assuming $H_{1}^{\pm\pm}$ decays predominately into electrons and muons, the current LHC limits are around 770 to 870 GeV, depending on the flavor structure Aaboud:2017qph . In the di-tauon channel $H_{1}^{\pm\pm}\to\tau^{\pm}\tau^{\pm}$, the LHC limit is relatively weaker, i.e. 535 GeV CMS:2017pet .²²2 As the singly-charged scalar $H_{2}^{\pm}$ and doubly-charged scalar $H_{1}^{\pm\pm}$ are mass degenerate at the leading order in the LRSM, here we have adopted the combined LHC limit from the pair production $pp\to H_{1}^{++}H_{1}^{--}$ and the associate production $pp\to H_{1}^{\pm\pm}H_{2}^{\mp}$. In these two channels, the separate channels are respectively 396 GeV and 479 GeV CMS:2017pet . If the doubly-charged scalar $H_{1}^{\pm\pm}$ decays predominately into same-sign $W$ bosons, the LHC limits are much weaker, around 200 to 220 GeV Aaboud:2018qcu . There are also some searches of singly-charged scalar $H_{2}^{\pm}\to\tau^{\pm}\nu$ at the LHC Aaboud:2016dig ; Aaboud:2018gjj ; Sirunyan:2019hkq . However these searches assume $H_{2}^{\pm}$ is produced from its interaction with top and bottom quarks, therefore these limits are not applicable to $H_{2}^{\pm}$ in the LRSM which does not couple directly to the SM quarks. The strongest same-sign dilepton limits of $(530-870)$ GeV on $H_{1}^{\pm\pm}$ (and also on other scalars from $\Delta_{L}$) is shown in Fig. 1.

• •

  As the $W_{R}$ boson is very heavy, the TeV-scale right-handed doubly-charged scalar $H_{2}^{\pm\pm}$ decays only into same-sign dileptons. The couplings of $H_{2}^{\pm\pm}$ to the photon and $Z$ boson have opposite signs, therefore the production cross section of $H_{2}^{\pm\pm}$ at the LHC is smaller than that for the left-handed doubly-charged scalar $H_{1}^{\pm\pm}$. Rescaling the LHC13 cross section of $H_{1}^{\pm\pm}$ by a factor of 1/2.4, The same-sign dilepton limits on $H_{2}^{\pm\pm}$ turn out to be 271 to 760 GeV for all the six combinations $ee,\,e\mu,\,\mu\mu,\,e\tau,\,\mu\tau,\,\tau\tau$ of lepton flavors, which is presented in Fig. 1.

• •

  The scalars $H_{1}^{0}$, $A_{1}^{0}$ and $H_{1}^{\pm}$ from the bidoublet $\Phi$ are degenerate in mass at the leading order. $H_{1}^{0}$ and $A_{1}^{0}$ has tree-level flavor-changing neutral-current (FCNC) couplings to the SM quarks, and contribute to $K-\overline{K}$, $B_{d}-\overline{B}_{d}$ and $B_{s}-\overline{B}_{s}$ mixings significantly. As a result, their masses are required to be at least $(10-25)$ TeV, depending on the nature of left-right symmetry (either generalized parity or generalized charge conjugation), the hadronic uncertainties Ecker:1983uh ; Zhang:2007da ; Maiezza:2010ic ; Bertolini:2014sua and the potentially large QCD corrections Bernard:2015boz . The stringent FCNC limits on the heavy bidoublet scalars is shown in Fig. 1.

• •

  The neutral scalar $H_{3}^{0}$ from the right-handed triplet $\Delta_{R}$ is hadrophobic, i.e. it does not couples directly to the SM quarks in the Lagrangian. It can be produced at the LHC and future higher energy colliders either in the scalar portal through coupling to the SM Higgs (and the heavy scalars $H_{1}^{0}$ and $A_{1}^{0}$), or in the gauge portal via coupling to the $W_{R}$ and $Z_{R}$ bosons. Therefore the direct LHC limits are very weak Dev:2016vle ; Dev:2017dui . However, when it is sufficiently light, say at the GeV-scale, $H_{3}^{0}$ can be produced from (invisible) decay of the SM Higgs or even from the meson decays Dev:2016vle ; Dev:2017dui . More details can be found in Section 5.2.

• •

  The RHNs in the LRSM can be either very light, e.g. at the keV scale to be a warm DM Nemevsek:2012cd candidate, or very heavy at the $v_{R}$ scale, and there are almost no laboratory limits on their masses, although their mixings with the active neutrinos are tightly constrained in some regions of the parameter space Bolton:2019pcu . For simplicity, in the following sections we will set the masses of RHNs to be free parameters and neglect their mixings with the active neutrinos.

To be complete, the masses of 100 GeV scale SM particles, i.e. the SM Higgs $h$, the top quark $t$ and the $W$ and $Z$ bosons, are depicted in Fig. 1 as horizontal black lines. See Fig. 7 for complementarity of GW prospects of the BSM particle masses and the current experimental limit.

To study phase transitions in the LRSM, we consider the effective potential at finite temperature, which includes contributions of the one-loop corrections and daisy resummations. Renormalized in the $\overline{\text{MS}}$ scheme, the effective potential can be cast into the following form Basler:2018cwe

	$\displaystyle{\cal V}_{\rm eff}(\phi_{i},v)$	$\displaystyle\ =\$	$\displaystyle V_{0}(\phi_{i},v)+V_{1}^{T=0}(\phi_{i},v)+V_{1}^{T\neq 0}(\phi_{i},v)+V_{D}(\phi_{i},v)$		(22)
		$\displaystyle\ =\$	$\displaystyle V_{0}(\phi_{i},v)+\frac{1}{64\pi^{2}}\sum_{i}g_{i}m_{i}^{4}(\phi_{i},v)\left(\log\frac{m_{i}^{2}(\phi_{i},v)}{\mu^{2}}-C_{i}\right)$
			$\displaystyle+\frac{T^{4}}{2\pi^{2}}\sum_{i}g_{i}J_{\pm}\left(\frac{m_{i}^{2}(\phi_{i},v)}{T^{2}}\right)$
			$\displaystyle-\frac{T}{12\pi}\sum_{i={\rm bosons}}\left[\left(m_{i}^{2}(\phi_{i},v)+\Pi_{i}(T)\right)^{3/2}-\left(m_{i}^{2}(\phi_{i},v)\right)^{3/2}\right],$

where $V_{0}(\phi_{i},v)$ is the tree-level potential, $V_{1}^{T=0}$ is the Coleman-Weinberg one-loop effective potential Coleman:1973jx , and $V_{1}^{T\neq 0}$ and $V_{D}$ are the thermal contributions at finite temperature. The $V_{1}^{T\neq 0}$ term includes only the one-loop contributions, and $V_{D}$ denotes the high-order contributions from daisy diagrams. In Eq. (22) the sum runs over all the particles in the model. The scalar mass matrices $m_{i}^{2}(\kappa_{i},v_{R})$ in the LRSM can be found in Ref. Deshpande:1990ip , and the corresponding thermal self-energies $\Pi_{i}(T)$ are provided in Appendix A. As for the fermions, we consider only the third generation quarks and three RHNs. In the LRSM their masses are respectively

$\displaystyle m_{t}=\frac{1}{\sqrt{2}}(y_{t}\kappa_{1}+y_{b}\kappa_{2})\,,\quad m_{b}=\frac{1}{\sqrt{2}}(y_{b}\kappa_{1}+y_{t}\kappa_{2})\,,\quad M_{N}=\sqrt{2}y_{N}v_{R}\,,$

(23)

with $y_{t,\,b}$ the Yukawa couplings for top and bottom quarks in the SM, $M_{N}$ the RHN masses and $y_{N}$ the corresponding Yukawa coupling. In the following study, for the sake of simplicity, we will assume three RHNs are mass degenerate and does not have any mixings among them. The degrees of freedom $g_{i}$ and constants $C_{i}$ in Eq. (22) are given by

$\displaystyle(g_{i},C_{i})=\left\{\begin{array}[]{ll}(1,\frac{3}{2}),&\mbox{for scalars}\,,\\ (-2\lambda,\frac{3}{2}),&\mbox{for fermions}\,,\\ (3,\frac{5}{6}),&\mbox{for gauge bosons}\,,\end{array}\right.$

(24)

with $\lambda=1\,(2)$ for Weyl (Dirac) fermions, and the functions $J_{-}(J_{+})$ for bosons (fermions) are defined as

$$J_{\pm}(x^{2})=\int_{0}^{\infty}dkk^{2}\log\left(1\pm e^{-\sqrt{x^{2}+k^{2}}}\right)\,.$$

(25)

In the limit of small $x^{2}=m^{2}/T^{2}$, we can use the approximations Basler:2018cwe :

	$\displaystyle J_{+}\left(x^{2}\right)$	$\displaystyle\ =\$	$\displaystyle\frac{7\pi^{4}}{360}-\frac{\pi^{2}}{24}x^{2}-\frac{1}{32}x^{4}\log\frac{x^{2}}{a_{F}}+\mathcal{O}(x^{4})\,,$		(26)
	$\displaystyle J_{-}\left(x^{2}\right)$	$\displaystyle\ =\$	$\displaystyle-\frac{\pi^{4}}{45}+\frac{\pi^{2}}{12}x^{2}-\frac{\pi}{6}\left(x^{2}\right)^{3/2}-\frac{1}{32}x^{4}\log\frac{x^{2}}{a_{B}}+\mathcal{O}(x^{4})\,,$		(27)

where

$\displaystyle a_{F}\ =\ \pi^{2}e^{3/2-2\gamma_{E}}\,,\quad a_{B}\ =\ 16\pi^{2}e^{3/2-2\gamma_{E}}\,.$

(28)

In this paper we focus on the phase transition at the $v_{R}$ scale, thus as an approximation all the effects of SM components on the symmetry breaking $SU(2)_{R}\times U(1)_{B-L}\to U(1)_{Y}$ can be neglected. Neglecting the daisy contributions, the effective potential ${\cal V}_{\rm eff}$ can be written down explicitly in the following form Cohen:1993nk :

$\displaystyle V_{eff}(v,\Pi_{i}=0)\ \simeq\ D\,(T^{2}-T_{0}^{2})\,v^{2}-E\,T\,v^{3}+\frac{\rho_{T}}{4}\,v^{4}\,,$

(29)

where $D$, $T_{0}$, $E$ and $\rho_{T}$ can be expressed by the model parameters as

	$\displaystyle D$	$\displaystyle\ =\$	$\displaystyle\frac{1}{8v_{R}^{2}}\left(M_{Z_{R}}^{2}+2M_{W_{R}}^{2}+M_{N}^{2}\right)+D_{H}\,,$		(30)
	$\displaystyle T_{0}^{2}$	$\displaystyle=$	$\displaystyle\frac{M_{H_{3}^{0}}^{2}}{4D}+T^{2}_{H}\,,$		(31)
	$\displaystyle E$	$\displaystyle\ =\$	$\displaystyle\frac{M_{Z_{R}}^{3}+2M_{W_{R}}^{3}}{4\pi v_{R}^{3}}+E_{H}\,,$		(32)
	$\displaystyle\rho_{T}$	$\displaystyle\ =\$	$\displaystyle\rho_{1}-\frac{3\left(M_{Z_{R}}^{4}+2M_{W_{R}}^{4}\right)}{16\pi^{2}v_{R}^{4}}\left(\frac{5}{6}+\log\frac{\mu^{2}}{a_{B}T^{2}}\right)$		(33)
			$\displaystyle+\frac{6M_{N}^{4}}{16\pi^{2}v_{R}^{4}}\left(\frac{3}{2}+\log\frac{\mu^{2}}{a_{F}T^{2}}\right)+\rho_{H}\,,$

where $M_{X}$ is the mass for the particle $X$, and $\mu$ is the renormalization scale. Since there are lots of scalars in the LRSM, we deliberately separate their contributions from the vector bosons and RHNs. The contributions of scalars for each of the terms in Eq. (30) to (33) can be written in terms of the scalar masses via

	$\displaystyle D_{H}$	$\displaystyle\ =\$	$\displaystyle\frac{1}{24v_{R}^{2}}\left(4M^{2}_{H_{1}^{0}}+6M_{H_{2}^{0}}^{2}+7M_{H_{3}^{0}}^{2}+2M_{H_{2}^{\pm\pm}}^{2}\right)\,,$		(34)
	$\displaystyle T^{2}_{H}$	$\displaystyle\ =\$	$\displaystyle\frac{M_{H_{3}^{0}}^{2}}{D}\frac{6M_{H_{2}^{0}}^{2}+7M_{H_{3}^{0}}^{2}+2M_{H_{2}^{\pm\pm}}^{2}}{64\pi^{2}v_{R}^{2}}\left(\frac{3}{2}+\log\frac{\mu^{2}}{a_{B}T^{2}}\right)\,,$		(35)
	$\displaystyle E_{H}$	$\displaystyle\ =\$	$\displaystyle\frac{1}{16\pi v_{R}^{3}}\left\{\frac{16}{3}M_{H_{1}^{0}}^{3}+\sqrt{2}M_{H_{3}^{0}}^{3}\left(1-r_{v}\right)^{3/2}+\sqrt{6}M_{H_{3}^{0}}^{3}\left(1-\frac{1}{3}r_{v}\right)^{3/2}\right.$		(36)
			$\displaystyle+\left.2\sqrt{2}\left[M_{H_{3}^{0}}^{2}\left(1-r_{v}\right)+2M_{A_{2}^{0}}^{2}\right]^{3/2}+\frac{2\sqrt{2}}{3}\left[M_{H_{3}^{0}}^{2}\left(1-r_{v}\right)+2M_{H_{2}^{\pm\pm}}^{2}\right]^{3/2}\right\}\,,$
	$\displaystyle\rho_{H}$	$\displaystyle\ =\$	$\displaystyle-\frac{4M_{H_{1}^{0}}^{4}+6M_{H_{2}^{0}}^{4}+5M_{H_{3}^{0}}^{4}+2M_{H_{2}^{\pm\pm}}^{4}+6M_{H_{2}^{0}}^{2}M_{H_{3}^{0}}^{2}+2M_{H_{3}^{0}}^{2}M_{H_{2}^{\pm\pm}}^{2}}{16\pi^{2}v_{R}^{4}}$		(37)
			$\displaystyle\quad\times\left(\frac{3}{2}+\log\frac{\mu^{2}}{a_{B}T^{2}}\right)\,,$

where we have defined $r_{v}\equiv v_{R}^{2}/v^{2}$. It should be pointed out that all the masses in Eqs. (34) to (37) depend upon the right-handed VEV $v_{R}$ instead of $v$. It is observed that the RHNs can also contribute to the symmetry breaking $SU(2)_{R}\times U(1)_{B-L}\to U(1)_{Y}$ via affecting the parameters $D$, $T_{0}$ and $\rho_{T}$, while the parameter $E$ receives only contributions from the scalars and gauge bosons.

As seen in Eqs. (33) and (37), the parameter $\rho_{T}$ receive not only tree-level contribution from the quartic coupling $\rho_{1}$ which corresponds to the $H_{3}^{0}$ mass via $\rho_{1}\simeq M_{H_{3}^{0}}^{2}/2v_{R}^{2}$ (see Table 5), but also loop-level contributions from the heavy scalars, gauge bosons and RHNs in the LRSM. In particular, when the quartic coupling $\rho_{1}$ is small, or equivalently the scalar $H_{3}^{0}$ is much smaller than the $v_{R}$ scale, which is the parameter space of interest for phase transition and GW production in the LRSM (cf. Figs. 2, 3 and 8), the loop-level contributions in Eq. (33) might dominate $\rho_{T}$. Furthermore, $\rho_{T}$ depends also on the gauge coupling $g_{R}$ via the heavy gauge boson masses $M_{W_{R}}$ and $M_{Z_{R}}$.

To have strong FOPT, the cubic terms proportional to $-ETv^{3}$ are crucial. In the limit of $E\to 0$, the phase transition is of second-order. In the SM, the effective coefficient $E$ of $\phi^{3}$ term is dominated by the gauge boson contributions, while in the LRSM, it receives contributions from both the scalars and gauge bosons, As a result of the large degree of freedom in the scalar sector of LRSM, it is remarkable that the scalar contributions to $E$ can even be much larger. The order parameter describing the FOPT is given by $v_{c}/T_{c}$, where $v_{c}$ is the non-vanishing location of the minimum at the critical temperature $T_{c}$ at which the effective potential ${\cal V}_{\rm eff}$ has two degenerate minima. In the EW baryogenesis Kuzmin:1985mm ; Shaposhnikov:1986jp ; Shaposhnikov:1987tw , to avoid the washout effects in the broken phase within the bubble wall, a strong FOPT is typically required to satisfy the following condition

$\displaystyle\frac{v_{c}}{T_{c}}=\frac{2E}{\rho_{T}}\geq 1\,.$

(38)

The effective potential (22) is a function of temperature $T$. Meanwhile, the minima of the effective potential vary when the temperature changes. In order to find the quantity $v_{c}/T_{c}$ which measures the strength of FOPT, we need to find both the critical temperature $T_{c}$ and the critical VEV $v_{c}$.³³3There might be some theoretical uncertainties in perturbative calculations of FOPTs and resultant GWs, which can be found, e.g. in Ref. Croon:2020cgk . In term of the parametrization given in Eq. (29), the critical temperature can be approximately expressed as

$\displaystyle T_{c}^{2}\simeq T_{0}^{2}\frac{\rho_{T}D}{\rho_{T}D-E^{2}}\,.$

(39)

Thus it is clear that $T_{c}\sim v_{R}\gg v_{\rm EW}$. Therefore, it is justified to neglect the contributions of SM particles to the phase transition at the right-handed scale $v_{R}$, since their masses $m_{\rm SM}$ are at most close to $v_{\rm EW}$ and their contributions are suppressed due to their tiny couplings to the right-handed triplet.

For given $v_{R}$ and heavy particle masses in the LRSM, the two key parameters $T_{c}$ and $v_{c}$ can be obtained from the effective potential (22) by requiring the two conditions $V_{eff}(T_{c};v_{c})=V_{eff}(T_{c};0)$ and $v_{c}\neq 0$. In the numerical evaluations, we change the temperature from a sufficiently highly energy scale, say $v_{R}$, toward lower values around the EW scale. A reasonable critical temperature $T_{c}$ for the phase transition $SU(2)_{R}\times U(1)_{B-L}\to\times U(1)_{Y}$ is assumed to be within this range. The dependence of $v_{c}/T_{c}$ on the parameters in the LRSM is exemplified in Fig. 2, where in the numerical calculations we have included all the contributions in Eq. (22).

Taking into account all the theoretical and experimental constraints in Section 2, we first consider scenarios with the simplifications $\lambda_{2}=\lambda_{3}=\lambda_{4}=\alpha_{1}=\alpha_{2}=0$. In order to identify the parameter space where the phase transition is of first-order, we calculate $v_{c}/T_{c}$ at the critical temperature $T_{c}$ with different values of the quartic couplings $\rho_{1}$, $\rho_{2}$, $\rho_{3}-2\rho_{1}$, $\alpha_{3}$. When we calculate the dependence of $v_{c}/T_{c}$ on two of the quartic couplings, all others are fixed in the way that their corresponding scalar masses equal the $W_{R}$ mass, and the gauge coupling $g_{R}=g_{L}$. To be concrete, we have set the renormalization scale $\mu$ to be the $v_{R}$ scale in Eq. (22). The corresponding results are shown in the first three panels of Fig. 2. The dependence of $v_{c}/T_{c}$ on the couplings $\rho_{1}$ and $\alpha_{3}$, $\rho_{3}-2\rho_{1}$ and $\rho_{1}$, and $\rho_{2}$ and $\rho_{1}$ are shown respectively in the upper left, upper right and lower left panels. The quantity $v_{c}/T_{c}$ is a dimensionless parameter and it is independent of the right-handed scale $v_{R}$ in the limit of $v_{R}\gg v_{\rm EW}$. As the quartic couplings $\rho_{1}$, $\rho_{2}$, $\rho_{3}-2\rho_{1}$, $\alpha_{3}$ are related directly to the scalar masses $M_{H_{3}^{0}}$, $M_{H_{2}^{\pm\pm}}$, $M_{H_{2}^{0}}$ and $M_{H_{1}^{0}}$ (cf. Table 5), the dependence of $v_{c}/T_{c}$ on the quartic couplings in Fig. 2 can also be understood effectively as the dependence of $v_{c}/T_{c}$ on the mass$-v_{R}$ ratios $M_{H_{1}^{0}}/v_{R}$, $M_{H_{2}^{0}}/v_{R}$, $M_{H_{3}^{0}}/v_{R}$ and $M_{H_{2}^{\pm\pm}}/v_{R}$. Through the gauge boson masses $M_{W_{R}}$ and $M_{Z_{R}}$, the parameter $v_{c}/T_{c}$ depends also on the gauge coupling ratio $r_{g}$, or equivalently on the right-handed gauge coupling $g_{R}$. This is shown in the lower right panel in Fig. 2; as seen in this figure, the $v_{c}/T_{c}$ limit on $\rho_{1}$ has a moderate or weak dependence on $r_{g}$, depending on the value of $\rho_{1}$.