1. Introduction
Despite the success of the standard model (SM) of particle physics [1] as a low-energy effective field theory (EFT), it is incomplete in describing the puzzles of dark energy, dark matter (DM), cosmic inflation and baryon asymmetry of our Universe (BAU). The proposed solutions might call for a larger symmetry group for ultraviolet (UV) completion, which should be broken into the SM symmetry group in our current epoch. Some of these symmetry breakings would trigger cosmic first-order phase transitions (FOPTs) (see [2] for a comprehensive review and [3] for a pedagogical lecture) proceeding by the bubble nucleations, bubble expansion and bubble collisions, which would generate a stochastic background of gravitational waves (GWs) (see [4, 5] for recent reviews from LISA Collaboration [6] and [7, 8] for earlier reviews from eLISA/NGO mission [9]; see also [10] for a brief review) transparent to our early Universe that is otherwise opaque to light for us to probe via electromagnetic waves if the FOPTs occur before the recombination epoch. Therefore, the GWs' detection serves as a promising and unique probe [11, 12] for the new physics [13, 14] beyond SM (BSM) with FOPTs.
Since the SM admits no FOPT but a cross-over transition due to a relatively heavy Higgs mass [15], any model buildings with FOPTs should go beyond SM. However, a clean separation for BSM new physics with FOPTs from those without FOPTs turns out to be difficult, so does a clear classification for various FOPT models. Usually the FOPT models could be naively classified into models extended with higher-dimensional operators [16–39], scalar singlet [40–73]/doublet [74–91]/triplet [92–104]/quadruplet [30], composite Higgs [18, 19, 105–117], supersymmetry (SUSY) [118–144], warp extra-dimensions [116, 117, 145–184], and dark/hidden sectors [113, 164, 171, 185–242], which, however, are actually overlapping with each other when focusing on the sector that actually induces an FOPT. Nevertheless, most of the FOPT models could be regarded effectively as some kind of scalar extensions of the SM, while other FOPT models with fermion extensions [243–247] are special on their own for triggering a PT, we therefore only focus on the scalar extensions of the SM.
On the other hand, some of the scalar extensions of the SM could be described and parametrized in the effective field theory (EFT) framework, in which the new particles are integrated out and only the SM degrees of freedom are kept. The EFT description was adopted before, particularly for the higher-dimensional-operator extensions of the SM, which characterize the effect on the low-energy degrees of freedom when we integrate out the heavy degrees of freedom. However, the EFT description is only valid in the presence of a clear separation of scales, which is in conflict with the relatively low scale of the new degrees of freedom so as to introduce a large correction to the SM Higgs potential [248] in order to trigger a PT. Exceptions could be made for the Higgs-singlet extension with tree-level matching, though the EFT description is at its most qualitative for the dimension-six extension. Recently, a new perspective on this SM EFT description is made if the potential barrier separating the two minimums is generated radiatively instead of the tree-level barrier [249]. Nevertheless, we have found in this paper that the difficulty of an EFT description for the FOPT models could be circumvented by introducing a large number of scalar fields in the N-plet scalar field model. Therefore, for the electroweak phase transition process, the EFT description for some scalar extensions is not enough since in many cases the new light degree of freedom would contribute to the thermal plasma and thus one cannot integrate it out during phase transition. See also [250–252] for the dimensionally reduced effective field theory and its applications on reducing the uncertainties from the renormalization scale dependence [253, 254] and the thermal bubble nucleation calculation [255–257].
In this paper we instead take an intermediate strategy that lays between the specific new physics model and EFT treatment. We utilize a simplified model to illustrate the features of the electroweak phase transition (EWPT), which we call the effective model description on the phase transition. In this description, to capture different patterns of the EWPT and to compare the difference between the new physics model and the EFT description, we propose a specific effective model description: the general model extends the SM with an isospin N-plet scalar field, of which the light scalar case consists of a model with classical scale invariance (CSI) (model I) and a model without CSI (model II), while the heavy scalar case is simply a model with higher-dimensional operators, for example, a dimension-six operator (model III). The above cases could describe the patterns of the electroweak symmetry breaking (EWSB) via, for example, (1) radiative symmetry breaking, (2) Higgs mechanism, and (3) EFT description of EWSB. Our effective model description already covers those scalar models with N-plet on the market, such as (1) singlet models including a real scalar singlet extension of SM (xSM), composite Higgs model like SO(6)/SO(5) model, extra dimension model like radion model, dilaton model; (2) doublet models including SUSY model like minimal supersymmetry model (MSSM), two Higgs doublet model (2HDM), minimal dark matter model; (3) triplet models including left-right model, Type-II seesaw model. In other perspective, our effective model description consists of the simplified models (effective models I and II) for the realistic models (SUSY, composite Higgs, etc.) and an EFT model (model III). Therefore, our effective model provides an effective description for the EWSB that could admit a FOPT with associated GWs.
Although the effective scalar models describe different EWSB patterns, they share the same form of the Higgs potential. Thus, utilizing the polynomial potential form, we could analyze how the FOPT is realized in different cases. To show the source of a sizable barrier for realizing the first-order EWPT, we consider the following polynomial potential form in each of the three models,
$\begin{eqnarray}{V}_{p}={C}_{2}{\phi }^{2}+{C}_{3}{\phi }^{3}+{C}_{4}{\phi }^{4}+{C}_{6}{\phi }^{6},\end{eqnarray}$
where φ is the order parameter in the effective potential, and Cn are the effective couplings of φn. The φ2 and φ4 term can appear in it at tree-level, on the other hand, the φ6 term comes from the high dimensional operator in the model (III). The φ3 is the source of the first-order phase transition in the models (I) and (II), and it can be produced by the thermal loop effects. Let us summarize the main features and also the main results of this work on realizing the FOPT in the forementioned three models: For model I, there are no massive parameters, and we consider the EWPT along a flat direction in the tree-level potential to avoid invalidating the perturbative analysis, and thus the potential form with finite temperature effects are roughly given by Vp = C2φ2 + C3φ3 + C4φ4, where all terms are one-loop level effects coming from thermal loop effects and radiative corrections.
For model II, there are massive parameters in the Lagrangian unlike the model I. The potential of this model is still roughly given by Vp = C2φ2 + C3φ3 + C4φ4 but the tree-level effects are now in φ2 and φ4 terms.
For model III, the high dimensional operator shows up in the potential Vp = C2φ2 + C3φ3 + C4φ4 + C6φ6, where the tree-level effects are in φ2, φ4 and φ6 terms.
The main contributions to Cnφn in each of these models are summarized in Table 1. In the case of model I, not only the C3 term but also the C2 and C4 terms are from loop level effects, and then the strongly first-order EWPT can be easily realized. On the other hand, the model II receives tree-level effects from the C2 and C4 terms. The source of the barrier for models I and II is the negative C3 term from the thermal loop effects of bosons. Model III has the high dimensional operator C6, and thus we can have a negative C4 term to generate a sizable barrier. That is a different point from models I and II.Table 1. The potential forms in the three types of the models where the EWSB occurs via (I) radiative symmetry breaking; (II) Higgs mechanism; (III) EFT description. Here, φ is order parameter, and Cn is the effective coupling of φn. The cubic term can be produced by thermal loop effects of bosons. To generate a sizable barrier, the cubic term will be negative in models I and II. In the model III, the quartic term can be negative to generate the barrier. |
| Model | C2φ2 | C3φ3 | C4φ4 | C6φ6 |
|---|---|---|---|---|
| I | Loop | Loop | Loop | None |
| II | Tree | Loop | Tree | None |
| III | Tree | Loop | Tree | Tree |
The outline of this paper is as follows: in section 2 , we introduce our effective model description, whose effective potentials are detailed in section 3 . The resulted GWs from the FOPT models described above are extracted in a way depicted in section 4 . The FOPT predictions are summarized in section 5 . Lastly, section 6 is devoted for conclusions and discussions. Appendix A provides some details on the model without CSI.
2. Effective models
We focus on the model with an additional isospin N-plet scalar field ${{\rm{\Phi }}}_{2}\unicode{x0007E}({I}_{{{\rm{\Phi }}}_{2}},{Y}_{{{\rm{\Phi }}}_{2}})$ charged under SU(2)I × U(1)Y gauge symmetry, where ${I}_{{{\rm{\Phi }}}_{2}}$ is the isospin and ${Y}_{{{\rm{\Phi }}}_{2}}$ is the hypercharge. The scalar boson fields in the model are given by
$\begin{eqnarray}{{\rm{\Phi }}}_{1}=\left(\begin{array}{c}{G}^{\pm },\\ \frac{h+i{G}^{0}}{\sqrt{2}}\end{array}\right),\quad {{\rm{\Phi }}}_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}{\phi }_{1}+{\rm{i}}{\phi }_{1,i}\\ {\phi }_{2}+{\rm{i}}{\phi }_{2,i}\\ \vdots \\ {\phi }_{N}+{\rm{i}}{\phi }_{N,i}\end{array}\right),\end{eqnarray}$
where Φ1 is the SM-like double scalar field and φn (φn,i) is the real (imaginary) part of the additional isospin N-plet scalar field with $N\equiv 2{I}_{{{\rm{\Phi }}}_{2}}+1$. These scalar bosons Φ1, Φ2 have classical fields: $\left\langle {{\Phi }}_{1}\right\rangle /\sqrt{2}$ and $\left\langle {{\rm{\Phi }}}_{2}\right\rangle /\sqrt{2}$, which are related to the real part of the neutral scalar field. We will discuss the testability from the GW detection for three types of the extended models instead on different patterns of the EWSB. These models are illustrated in figure 1. Two types of them are the model with a light scalar field: (I) the model with classical scale invariance (CSI), (II) the model without CSI. The last type of the model is (III) the model with the N-plet scalar field with TeV scale. These models can realize the EWSB via (I) radiative symmetry breaking, (II) Higgs mechanism and (III) EFT description of EWSB, respectively. For the simplicity of excluding the mixing terms, we assume Z2 symmetry in such a way that the new scalar field is Z2 odd while the others are Z2 even.
Figure 1. Our effective model description for some BSM models with FOPTs from a general model with an additional scalar field, which is either light or heavy consisting of (I) the light scalar model with CSI, (II) the light scalar model without CSI, (III) the heavy scalar model with high dimensional operator. |
2.1. The model with classical scale invariance
In the first type of the model, we impose CSI on the tree-level potential without any dimensional parameters, then the spontaneous EWSB is generated by radiative corrections [258] given later in (23 ). The Lagrangian of this model is 3 .
$\begin{eqnarray}{ \mathcal L }={{ \mathcal L }}_{{\rm{SM}}}+{\left|{D}_{\mu }{{\rm{\Phi }}}_{2}\right|}^{2}-{V}_{0}({{\rm{\Phi }}}_{1},{{\rm{\Phi }}}_{2}),\end{eqnarray}$
where ${D}_{\mu }={\partial }_{\mu }-{\rm{i}}g{T}^{a}{W}_{\mu }^{a}-ig^{\prime} {Y}_{{{\rm{\Phi }}}_{2}}{B}_{\mu }/2$, $g^{\prime} $ and g are U(1)Y and SU(2)I gauge couplings, respectively, and Ta is the matrix for the generator of SU(2)I. The tree-level potential V0(Φ1, Φ2) is given by $\begin{eqnarray}{V}_{0}({{\rm{\Phi }}}_{1},{{\rm{\Phi }}}_{2})={\lambda }_{1}| {{\rm{\Phi }}}_{1}{| }^{4}+{\lambda }_{2}| {{\rm{\Phi }}}_{2}{| }^{4}+{\lambda }_{12}| {{\rm{\Phi }}}_{1}{| }^{2}| {{\rm{\Phi }}}_{2}{| }^{2}.\end{eqnarray}$
If the isospin ${I}_{{{\rm{\Phi }}}_{2}}$ is 1/2, then there are some mixing terms in the potential, such as $| {{\rm{\Phi }}}_{1}^{\dagger }{{\rm{\Phi }}}_{2}{| }^{2}$. For simplicity, we neglect such terms in the potential. According to [259, 260], there may be a flat direction in the tree-level potential to assure the valid perturbative analysis. If not, the large logarithmic term may show up in the one-loop correction via the renormalization scale, for example, this scale in the φ4 theory is $Q\unicode{x0007E}3\lambda v{e}^{16{\pi }^{2}}$ [258]. Therefore, we assume a flat direction in the tree-level potential to avoid invalidating the perturbative analysis. The details of the effective potential will be discussed in section We mention here the constraints on this model. For simplicity, we assume that the additional scalar field Φ2 does not couple to the SM fermions and does not have the vacuum-expectation-value (VEV). On the other hand, the isospin N-plet scalar field Φ2 can interact with the SM gauge bosons. Then, the model is constrainted by the perturbative unitarity bound. According to [261], the isospin should be less than 7/2. We assume that the additional scalar field does not couple to the SM-like fermion in our strategy. Therefore, the typical collider constraints on our model is not much tight. For the neutral heavy Higgs, since it does not mix with the SM Higgs and does not couple to the SM fermions, the only experimental constraint comes from the vector boson scattering process, and thus it is very loose. For the charged Higgs, due to their degenerate masses to the neutral one, experimental constraints can be relaxed except for the hγγ measurement. The additional charged scalar fields contribute to the Higgs coupling hγγ [262], which is given by 1.05 ± 0.09 [263]. We may distinguish models I and II by the results of the measurements and observation of the GW spectrum. In our work, we do not take into account the experimental constraints in the numerical analysis of the PT.
2.2. The model without classical scale invariance
In the model without CSI, there are mass parameters in the tree-level potential. The tree-level potential in this model is given as 39 ).
$\begin{eqnarray}\begin{array}{rcl}{V}_{0}({{\rm{\Phi }}}_{1},{{\rm{\Phi }}}_{2}) & = & -{\mu }_{1}^{2}| {{\rm{\Phi }}}_{1}{| }^{2}-{\mu }_{2}^{2}| {{\rm{\Phi }}}_{2}{| }^{2}+{\lambda }_{12}| {{\rm{\Phi }}}_{1}{| }^{2}| {{\rm{\Phi }}}_{2}{| }^{2}\\ & & +{\lambda }_{1}| {{\rm{\Phi }}}_{1}{| }^{4}+{\lambda }_{2}| {{\rm{\Phi }}}_{2}{| }^{4},\end{array}\end{eqnarray}$
where ${\mu }_{1}^{2}\gt 0$ and λ1, λ2 > 0 for the stability of the tree-level potential. The effective potential of this model will be given later in (Before discussing the effective potential with loop-corrections, we show the possible PT paths from the tree-level potential. At first, the extremal values in the potential are obtained by
$\begin{eqnarray}\frac{\partial {V}_{0}(\left\langle {{\rm{\Phi }}}_{1}\right\rangle ,\left\langle {{\rm{\Phi }}}_{2}\right\rangle )}{\partial \left\langle {{\rm{\Phi }}}_{1}\right\rangle }=\frac{\partial {V}_{0}(\left\langle {{\rm{\Phi }}}_{1}\right\rangle ,\left\langle {{\rm{\Phi }}}_{2}\right\rangle )}{\partial \left\langle {{\rm{\Phi }}}_{2}\right\rangle }=0,\end{eqnarray}$
which are solved by the following nine points in the field space as shown in figure 2, $\begin{eqnarray}\begin{array}{l}\left(\left\langle {{\rm{\Phi }}}_{1}\right\rangle ,\left\langle {{\rm{\Phi }}}_{2}\right\rangle \right)=(0,0),\quad \left(0,\pm \sqrt{\frac{{\mu }_{2}^{2}}{{\lambda }_{2}}}\right),\quad \left(\pm \sqrt{\frac{{\mu }_{1}^{2}}{{\lambda }_{1}}},0\right),\\ \left(\pm \sqrt{2\frac{{\lambda }_{12}{\mu }_{2}^{2}-2{\lambda }_{2}{\mu }_{1}^{2}}{{\lambda }_{12}^{2}-4{\lambda }_{1}{\lambda }_{2}}},\pm \sqrt{2\frac{{\lambda }_{12}{\mu }_{2}^{2}-2{\lambda }_{2}{\mu }_{1}^{2}}{{\lambda }_{12}^{2}-4{\lambda }_{1}{\lambda }_{2}}}\right),\\ \left(\pm \sqrt{2\frac{{\lambda }_{12}{\mu }_{2}^{2}-2{\lambda }_{2}{\mu }_{1}^{2}}{{\lambda }_{12}^{2}-4{\lambda }_{1}{\lambda }_{2}}},\mp \sqrt{2\frac{{\lambda }_{12}{\mu }_{2}^{2}-2{\lambda }_{2}{\mu }_{1}^{2}}{{\lambda }_{12}^{2}-4{\lambda }_{1}{\lambda }_{2}}}\right).\end{array}\end{eqnarray}$
The green point is the origin in the potential, the blue points are $\left\langle {{\rm{\Phi }}}_{2}\right\rangle \ne 0$ at zero temperature and the red and magenta points are along the $\left\langle {{\rm{\Phi }}}_{1}\right\rangle $ and $\left\langle {{\rm{\Phi }}}_{2}\right\rangle $ axes, respectively. When the red or blue point is minimum, the scalar field Φ1 can have a VEV. In this work, we especially focus on the one-step phase transition along $\left\langle {{\rm{\Phi }}}_{1}\right\rangle $ axis, because this path is the same as the CSI case. To realize the phase transition, we assume that the red point is at global minimum and the blue point is not minimum. We can assure such a situation by using the conditions for the determinants of the Hesse matrix and the height of the potential. The determinants of the Hesse matrix at the red, magenta and blue points are given by $\begin{eqnarray}{\rm{d}}{\rm{e}}{\rm{t}}[{{ \mathcal H }}_{{\rm{r}}{\rm{e}}{\rm{d}}}]={\mu }_{1}^{2}\left(\frac{{\lambda }_{12}{\mu }_{1}^{2}}{{\lambda }_{1}}-2{\mu }_{2}^{2}\right),\end{eqnarray}$
$\begin{eqnarray}{\rm{d}}{\rm{e}}{\rm{t}}[{{ \mathcal H }}_{{\rm{m}}{\rm{a}}{\rm{g}}}]={\mu }_{2}^{2}\left(\frac{{\lambda }_{12}{\mu }_{2}^{2}}{{\lambda }_{2}}-2{\mu }_{1}^{2}\right),\end{eqnarray}$
$\begin{eqnarray}{\rm{d}}{\rm{e}}{\rm{t}}[{{ \mathcal H }}_{{\rm{b}}{\rm{l}}{\rm{u}}{\rm{e}}}]=-4\frac{\left({\lambda }_{12}{\mu }_{2}^{2}-2{\lambda }_{2}{\mu }_{1}^{2}\right)\left({\lambda }_{12}{\mu }_{1}^{2}-2{\lambda }_{1}{\mu }_{2}^{2}\right)}{{\lambda }_{12}^{2}-4{\lambda }_{1}{\lambda }_{2}}.\end{eqnarray}$
Also, the height of the potential at these points are given by $\begin{eqnarray}{V}_{0}\left(0,\sqrt{{\mu }_{2}^{2}/{\lambda }_{2}}\right)=-{\mu }_{2}^{4}/4{\lambda }_{2},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{V}_{0}\left(\sqrt{{\mu }_{1}^{2}/{\lambda }_{1}},0\right)=-{\mu }_{1}^{4}/4{\lambda }_{1}\\ \times \,{V}_{0}\left(\sqrt{2\frac{{\lambda }_{12}{\mu }_{2}^{2}-2{\lambda }_{2}{\mu }_{1}^{2}}{{\lambda }_{12}^{2}-4{\lambda }_{1}{\lambda }_{2}}},\sqrt{2\frac{{\lambda }_{12}{\mu }_{2}^{2}-2{\lambda }_{2}{\mu }_{1}^{2}}{{\lambda }_{12}^{2}-4{\lambda }_{1}{\lambda }_{2}}}\right)\\ =\,\frac{{\lambda }_{1}{\mu }_{2}^{4}+{\lambda }_{2}{\mu }_{1}^{4}-{\lambda }_{12}{\mu }_{1}^{2}{\mu }_{2}^{2}}{{\lambda }_{12}^{2}-4{\lambda }_{1}{\lambda }_{2}}.\end{array}\end{eqnarray}$
The red point should be the lowest points among them. From that, we can obtain the following conditions $\begin{eqnarray}\frac{{\lambda }_{12}{\mu }_{1}^{2}}{{\lambda }_{1}}\gt 2{\mu }_{2}^{2},\quad \frac{{\lambda }_{12}{\mu }_{2}^{2}}{{\lambda }_{2}}\gt 2{\mu }_{1}^{2},\quad {\mu }_{2}^{4}/{\lambda }_{2}\lt {\mu }_{1}^{4}/{\lambda }_{1}.\end{eqnarray}$
Since the potential with these conditions have two minima at magenta and red points, a two-step phase transition may be realized. In our numerical analysis, we distinguish the phase transition pattern and focus on the one-step phase transition to compare differences of the results between the models with and without CSI.
Figure 2. The extreme values of the tree-level potential of the model without CSI given by equation ( |
2.3. The model with dimension-six operator from Φ2
In the last type of model, Φ2 is assumed at the TeV scale and then we integrate out the additional scalar field. The tree-level potential in this model is given by 51 ).
$\begin{eqnarray}\begin{array}{rcl}{V}_{0}({{\rm{\Phi }}}_{1},{{\rm{\Phi }}}_{2}) & = & -{\mu }_{1}^{2}| {{\rm{\Phi }}}_{1}{| }^{2}-{\mu }_{2}^{2}| {{\rm{\Phi }}}_{2}{| }^{2}+{\lambda }_{1}| {{\rm{\Phi }}}_{1}{| }^{4}+{\lambda }_{2}| {{\rm{\Phi }}}_{2}{| }^{4}\\ & & +{\lambda }_{12}| {{\rm{\Phi }}}_{1}{| }^{2}| {{\rm{\Phi }}}_{2}{| }^{2},\end{array}\end{eqnarray}$
where μ2 is at TeV scale. Otherwise, it is the same as the model without CSI. Then, we integrate out the heavy Φ2 by loop-level matching [248, 264] in our analysis, and the effective Lagrangian for the SM-like Higgs boson h reads $\begin{eqnarray}{{ \mathcal L }}_{{\rm{EFT}}}^{(6)}\unicode{x0007E}\frac{1}{2}| {\partial }_{\mu }h{| }^{2}-\left(-\frac{1}{2}{a}_{2}{h}^{2}+\frac{1}{4}{a}_{4}{h}^{4}+\frac{1}{6}{a}_{6}{h}^{6}\right)\end{eqnarray}$
with $\begin{eqnarray}{a}_{2}={\mu }_{1}^{2},\end{eqnarray}$
$\begin{eqnarray}{a}_{4}={\lambda }_{1}-(1+2{I}_{{{\rm{\Phi }}}_{2}})\frac{{\lambda }_{12}^{2}{\mu }_{1}^{2}}{9{(4\pi )}^{2}{m}_{{{\rm{\Phi }}}_{2}}^{2}},\end{eqnarray}$
$\begin{eqnarray}{a}_{6}=(1+2{I}_{{{\rm{\Phi }}}_{2}})\frac{1}{{(4\pi )}^{2}{m}_{{{\rm{\Phi }}}_{2}}^{2}}\left(\frac{{\lambda }_{12}^{3}}{8}+\frac{{\lambda }_{12}^{2}{\lambda }_{1}}{6}\right),\end{eqnarray}$
where ${m}_{{{\rm{\Phi }}}_{2}}^{2}={\mu }_{2}^{2}+{\lambda }_{12}{v}^{2}/2$ is the mass of the additional scalar field Φ2. The effective potential with radiative corrections of this model is given later in (We note that [248] suggests FOPT may be difficult in the model with a loop-level matching. According to their work, the FOPT requires the balancing between the dimension-four and dimension-six terms via $\frac{1}{4}\frac{U}{{M}^{2}}\frac{{c}_{i}{\kappa }^{2}}{16{\pi }^{2}}\unicode{x0007E}\frac{{m}_{h}^{2}}{{v}^{2}}\unicode{x0007E}0.12$ with ci = 1/2 (1) for a real (complex) scalar, where U and M correspond to the Higgs boson coupling to the heavy field and the mass parameter of the heavy field, respectively. On the other hand, the parameter region for a valid EFT expansion requires 2ckinv2 < 1/2 and ∣a8∣v2/a6 < 1, where ckin, a6 and a8 are high dimensional operators for kinetic term, dimension 6 and 8 terms, respectively. Since the conditions also limit to U/M2 and ciκ2/16π2. Therefore, they concluded that the FOPT cannot be generated in the model. At this time, we do not take into account other higher-dimensional operators involving the kinetic term.
3. Effective potentials
In this section, we discuss the forms of the effective potentials for our three effective model descriptions. To obtain the effective potential, we use the $\overline{{\rm{MS}}}$ scheme to absorb the divergence parts. Typically, the effective potential at the one-loop level reads 19 ) and thermal correction Πi in equation (22 ) depending on the details of model. Therefore we will discuss the form of the effective potential in each type of the model in the following.
$\begin{eqnarray}\begin{array}{l}{V}_{{\rm{eff}}}\left(\left\langle {{\rm{\Phi }}}_{1}\right\rangle ,\left\langle {{\rm{\Phi }}}_{2}\right\rangle ,T\right)={V}_{0}+{\sum }_{i}\frac{{n}_{i}}{64{\pi }^{2}}\,{M}_{i}^{4}\left(\left\langle {{\rm{\Phi }}}_{1}\right\rangle ,\left\langle {{\rm{\Phi }}}_{2}\right\rangle \right)\\ \quad \quad \times \,\left(\mathrm{ln}\left(\frac{{M}_{i}^{2}\left(\left\langle {{\rm{\Phi }}}_{1}\right\rangle ,\left\langle {{\rm{\Phi }}}_{2}\right\rangle \right)}{{Q}^{2}}\right)-{c}_{i}\right)+{\rm{\Delta }}{V}_{T},\end{array}\end{eqnarray}$
where ${M}_{i}^{2}\left(\left\langle {{\rm{\Phi }}}_{1}\right\rangle ,\left\langle {{\rm{\Phi }}}_{2}\right\rangle \right)$ is the field-dependent mass, ni is the number of the degree of freedom, Q is the renormalization scale and ci is 3/2 (i = boson, fermion) or 5/6 (i = gauge boson). The one-loop thermal contribution to the potential [265] is $\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{V}_{T} & = & \frac{{T}^{4}}{2{\pi }^{2}}\left\{{\sum }_{i={\rm{bosons}}}{n}_{i}{\displaystyle \int }_{0}^{\infty }{\rm{d}}x\,{x}^{2}\right.\\ & & \times \mathrm{ln}\left[1-\exp \left(-\sqrt{{x}^{2}+{M}_{i}^{2}(\langle {{\rm{\Phi }}}_{1}\rangle ,\langle {{\rm{\Phi }}}_{2}\rangle ,T)/{T}^{2}}\right)\right]\\ & & +{\sum }_{i={\rm{fermions}}}{n}_{i}{\displaystyle \int }_{0}^{\infty }{\rm{d}}x\,{x}^{2}\\ & & \times \left.\mathrm{ln}\left[1+\exp \left(-\sqrt{{x}^{2}+{M}_{i}^{2}(\langle {{\rm{\Phi }}}_{1}\rangle ,\langle {{\rm{\Phi }}}_{2}\rangle ,T)/{T}^{2}}\right)\right]\right\}.\end{array}\end{eqnarray}$
Here, we take into account the resummation effect ${V}_{T}^{{\rm{ring}}}$ obtained by [266] $\begin{eqnarray}\begin{array}{rcl}{V}_{T}^{{\rm{ring}}} & = & \frac{T}{12\pi }{\sum }_{i={\rm{b}}osons}{n}_{i}\left({({M}_{i}^{2}(\langle {{\rm{\Phi }}}_{1}\rangle ,\langle {{\rm{\Phi }}}_{2}\rangle ,0))}^{3/2}\right.\\ & & \left.-{({M}_{i}^{2}(\langle {{\rm{\Phi }}}_{1}\rangle ,\langle {{\rm{\Phi }}}_{2}\rangle ,T))}^{3/2}\right),\end{array}\end{eqnarray}$
where the thermal mass ${M}_{i}^{2}(\langle {{\rm{\Phi }}}_{1}\rangle ,\langle {{\rm{\Phi }}}_{2}\rangle ,T)={M}_{i}^{2}(\langle {{\rm{\Phi }}}_{1}\rangle ,\langle {{\rm{\Phi }}}_{2}\rangle )+{{\rm{\Pi }}}_{i}$ receives the thermal correction from the thermal self-energy Πi. Finally, the effective potential with finite temperature effects is obtained by $\begin{eqnarray}{V}_{{\rm{eff}}}\left(\left\langle {{\rm{\Phi }}}_{1}\right\rangle ,\left\langle {{\rm{\Phi }}}_{2}\right\rangle ,T\right)={V}_{0}+{V}_{1-{\rm{loop}}}+{\rm{\Delta }}{V}_{T}+{V}_{T}^{{\rm{ring}}}.\end{eqnarray}$
The field-dependent masses in the effective potential of equation (3.1. The model with classical scale invariance
For the model (4 ) with CSI, the spontaneous EWSB occurs on the flat direction, which is assumed along ⟨Φ1⟩. Then, the effective potential is simply
$\begin{eqnarray}{V}_{{\rm{eff}}}(\varphi ,T=0)=A{\varphi }^{4}+B{\varphi }^{4}\mathrm{ln}\frac{{\varphi }^{2}}{{Q}^{2}},\end{eqnarray}$
with A and B terms given by $\begin{eqnarray}A={\sum }_{i}\frac{{n}_{i}}{64{\pi }^{2}{v}^{4}}{m}_{i}^{4}\left(\mathrm{ln}\frac{{m}_{i}^{2}}{{v}^{2}}-{c}_{i}\right),\quad B={\sum }_{i}\frac{{n}_{i}}{64{\pi }^{2}{v}^{4}}{m}_{i}^{4},\end{eqnarray}$
where mi is the mass of the field species i excluding the Higgs boson since this effect is at the one-loop level in this model case. ni and ci in the potential are given by $({n}_{W},{n}_{Z},{n}_{t},{n}_{{{\rm{\Phi }}}_{2}})=(6,3,-12,2(2{I}_{{{\rm{\Phi }}}_{2}}+1))$ and (cW, cZ, ct) = (5/6, 5/6, 3/2, 3/2), respectively. Using the stationary condition, we have $\begin{eqnarray}{\left.\frac{\partial {V}_{{\rm{eff}}}(\varphi ,T=0)}{\partial \varphi }\right|}_{\varphi =v}=\mathrm{ln}\frac{{v}^{2}}{{Q}^{2}}+\frac{1}{2}+\frac{A}{B}=0,\end{eqnarray}$
where the renormalization scale Q is fixed. Because A and B are the loop effects, v can be large in contrast to the case of φ4 theory. Also, the Higgs boson mass is obtained as $\begin{eqnarray}{m}_{h}^{2}={\left.\frac{{\partial }^{2}{V}_{{\rm{eff}}}(\varphi ,T=0)}{\partial {\varphi }^{2}}\right|}_{\varphi =v}=8B{v}^{2},\end{eqnarray}$
from which we can obtain the additional scalar boson mass ${m}_{{{\rm{\Phi }}}_{2}}$ as $\begin{eqnarray}{m}_{{{\rm{\Phi }}}_{2}}=\frac{C}{{(2(2{I}_{{{\rm{\Phi }}}_{2}}+1))}^{1/4}},\end{eqnarray}$
$\begin{eqnarray}{C}^{4}\equiv 8{\pi }^{2}{v}^{2}{m}_{h}^{2}-3{m}_{Z}^{4}-6{m}_{W}^{4}+12{m}_{t}^{4}\end{eqnarray}$
with C = 543 GeV determined by equation (10) of [267], and λ12 coupling is obtained via ${m}_{{{\rm{\Phi }}}_{2}}^{2}={\lambda }_{12}{v}^{2}/2$ as $\begin{eqnarray}{\lambda }_{12}=\frac{\sqrt{2}{C}^{2}}{{v}^{2}\sqrt{2{I}_{{{\rm{\Phi }}}_{2}}+1}}.\end{eqnarray}$
With the help of equations (25 ) and (26 ), the one-loop effective potential at zero temperature could be obtained in terms of the renormalized mass of the Higgs boson as
$\begin{eqnarray}{V}_{{\rm{eff}}}(\varphi ,T=0)=\frac{{m}_{h}^{2}}{8{v}^{2}}{\varphi }^{4}\left(\mathrm{ln}\frac{{\varphi }^{2}}{{v}^{2}}-\frac{1}{2}\right).\end{eqnarray}$
Since the loop effects are renormalized into the Higgs boson mass, the value of hhh coupling with one-loop effects does not depend on the model extension [267]. The field-dependent masses and resummation effects in the effective potential with finite temperature effects are $\begin{eqnarray}\begin{array}{rcl}{M}_{{{\rm{\Phi }}}_{2}}^{2}(\varphi ,T) & = & \frac{{m}_{{{\rm{\Phi }}}_{2}}^{2}}{{v}^{2}}{\varphi }^{2}+{T}^{2}\left(({I}_{{{\rm{\Phi }}}_{2}}+1)\frac{{\lambda }_{2}}{6}+\frac{{m}_{{{\rm{\Phi }}}_{2}}^{2}}{3{v}^{2}}\right.\\ & & \left.+{I}_{{{\rm{\Phi }}}_{2}}({I}_{{{\rm{\Phi }}}_{2}}+1)\frac{{g}^{2}}{4}+{Y}_{{{\rm{\Phi }}}_{2}}^{2}\frac{g{{\prime} }^{2}}{16}\right).\end{array}\end{eqnarray}$
Similarly, the thermally corrected field-dependent masses of gauge bosons in the (W1, W2, W3, B) basis are $\begin{eqnarray}\begin{array}{rcl}{M}_{g}^{2(L,T)}(\varphi ,T) & = & \frac{{\varphi }^{2}}{4}\left(\begin{array}{cccc}{g}^{2} & & & \\ & {g}^{2} & & \\ & & {g}^{2} & gg^{\prime} \\ & & gg^{\prime} & g{{\prime} }^{2}\end{array}\right)\\ & & +{a}_{g}^{(L,T)}{T}^{2}\left(\begin{array}{cccc}{\pi }_{W} & & & \\ & {\pi }_{W} & & \\ & & {\pi }_{W} & \\ & & & {\pi }_{B}\end{array}\right),\end{array}\end{eqnarray}$
where $\begin{eqnarray}{M}_{W}^{2}=\frac{{g}^{2}}{4}{\varphi }^{2},\quad {M}_{WB}^{2}=\frac{gg^{\prime} }{4}{\varphi }^{2},\quad {M}_{B}^{2}=\frac{g{{\prime} }^{2}}{4}{\varphi }^{2},\end{eqnarray}$
$\begin{eqnarray}{\pi }_{W}=\frac{{g}^{2}}{9}{I}_{{{\rm{\Phi }}}_{2}}\left(1+{I}_{{{\rm{\Phi }}}_{2}}\right)\left(1+2{I}_{{{\rm{\Phi }}}_{2}}\right)+\frac{11}{6}{g}^{2},\end{eqnarray}$
$\begin{eqnarray}{\pi }_{B}=\frac{g{{\prime} }^{2}{T}^{2}}{12}\left(1+{I}_{{{\rm{\Phi }}}_{2}}\right){Y}_{{{\rm{\Phi }}}_{2}}^{2}+\frac{11}{6}g{{\prime} }^{2},\,{a}_{g}^{L}=1,\,{a}_{T}^{T}=0.\end{eqnarray}$
The field-dependent mass of the fermion does not receive thermal correction in T2 so that $\begin{eqnarray}{M}_{t}^{2}=\frac{{m}_{t}^{2}}{{v}^{2}}{\varphi }^{2}.\end{eqnarray}$
In this model, there are three parameters in the tree-level potential, 26 ), the isospin number ${I}_{{{\rm{\Phi }}}_{2}}$ is related to the mass ${m}_{{{\rm{\Phi }}}_{2}}$. Therefore, the free parameters in the effective potential with finite temperature effects are in fact
$\begin{eqnarray}{\lambda }_{1},{\lambda }_{2},{\lambda }_{12},\end{eqnarray}$
where λ1 is zero in order to assume a flat direction along ⟨Φ1⟩ axis. According to equation ( $\begin{eqnarray}{I}_{{{\rm{\Phi }}}_{2}},{Y}_{{{\rm{\Phi }}}_{2}},{\lambda }_{2}.\end{eqnarray}$
See Appendix B for the Landau pole in this model with CSI.
3.2. The model without classical scale invariance
For the model (5 ) without CSI, the effective potential is given by A .
$\begin{eqnarray}{V}_{{\rm{eff}}}\left({\varphi }_{1},{\varphi }_{2},T\right)={V}_{0}+\displaystyle \sum _{i}\frac{{n}_{i}}{64{\pi }^{2}}\,{M}_{i}^{4}\,\left(\mathrm{ln}\frac{{M}_{i}^{2}}{{Q}^{2}}-{c}_{i}\right)+{\rm{\Delta }}{V}_{T},\end{eqnarray}$
where $\left\langle {{\rm{\Phi }}}_{1}\right\rangle ={\varphi }_{1}/\sqrt{2}$, $\left\langle {{\rm{\Phi }}}_{2}\right\rangle ={\varphi }_{2}/\sqrt{2}$ and $\begin{eqnarray}{V}_{0}=-\frac{{\mu }_{1}^{2}}{2}{\varphi }_{1}^{2}-\frac{{\mu }_{2}^{2}}{2}{\varphi }_{2}^{2}+\frac{{\lambda }_{1}}{4}{\varphi }_{1}^{4}+\frac{{\lambda }_{2}}{4}{\varphi }_{2}^{4}+\frac{{\lambda }_{12}}{4}{\varphi }_{1}^{2}{\varphi }_{2}^{2}.\end{eqnarray}$
There are five model parameters in the tree-level potential and these parameters can be fixed in terms of the following parameters, $\begin{eqnarray}v,{m}_{h},{m}_{{{\rm{\Phi }}}_{2}},{\lambda }_{12},{\lambda }_{2},\end{eqnarray}$
by the stationary conditions and the second derivatives of the effective potential. The details of them are given in Appendix The field-dependent masses with resummation corrections from the finite temperature effects of the effective potential are
$\begin{eqnarray}\begin{array}{rcl}{M}_{h,{{\rm{\Phi }}}_{2}}^{2}({\varphi }_{1},{\varphi }_{2},T) & = & \frac{1}{2}\left({M}_{11}^{2}+{M}_{22}^{2}\right.\\ & \mp & \left.\sqrt{{({M}_{11}^{2}-{M}_{22}^{2})}^{2}+4{M}_{12}^{2}{M}_{21}^{2}}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{M}_{{N}_{1}}^{2}({\varphi }_{1},{\varphi }_{2},T)=-{\mu }_{1}^{2}+{\lambda }_{1}{\varphi }_{1}^{2}+\frac{{\lambda }_{12}}{2}{\varphi }_{2}^{2}\\ \quad +{T}^{2}\left((2{I}_{{{\rm{\Phi }}}_{2}}+1)\frac{{\lambda }_{12}}{12}\right.\\ \left.\quad +\frac{{\lambda }_{h}}{4}+\frac{3{g}^{2}}{16}+\frac{g{{\prime} }^{2}}{16}+\frac{{y}_{t}^{2}}{4}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{M}_{{N}_{2}}^{2}({\varphi }_{1},{\varphi }_{2},T)=-{\mu }_{2}^{2}+{\lambda }_{2}{\varphi }_{2}^{2}+\frac{{\lambda }_{12}}{2}{\varphi }_{1}^{2}\\ \quad +{T}^{2}\left(\frac{({I}_{{{\rm{\Phi }}}_{2}}+1){\lambda }_{2}}{6}+\frac{{\lambda }_{12}}{6}\right.\\ \left.\quad +\frac{{g}^{2}}{4}\left({I}_{{{\rm{\Phi }}}_{2}}^{2}+{I}_{{{\rm{\Phi }}}_{2}}\right)+\frac{g{{\prime} }^{2}{Y}_{{{\rm{\Phi }}}_{2}}^{2}}{16}\right),\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}\left(\begin{array}{cc}{M}_{11}^{2} & {M}_{12}^{2}\\ {M}_{21}^{2} & {M}_{22}^{2}\end{array}\right)=\left(\begin{array}{cc}-{\mu }_{1}^{2}+3{\lambda }_{1}{\varphi }_{1}^{2}+\frac{{\lambda }_{12}}{2}{\varphi }_{2}^{2} & {\lambda }_{12}{\varphi }_{1}{\varphi }_{2}\\ {\lambda }_{12}{\varphi }_{1}{\varphi }_{2} & -{\mu }_{2}^{2}+3{\lambda }_{2}{\varphi }_{2}^{2}+\frac{{\lambda }_{12}}{2}{\varphi }_{1}^{2}\end{array}\right)\\ \quad +{T}^{2}\left(\begin{array}{cc}(2{I}_{{{\rm{\Phi }}}_{2}}+1)\frac{{\lambda }_{12}}{12}+\frac{{\lambda }_{1}}{4}+\frac{3{g}^{2}}{16}+\frac{g{{\prime} }^{2}}{16}+\frac{{y}_{t}^{2}}{4} & 0\\ 0 & \frac{({I}_{{{\rm{\Phi }}}_{2}}+1){\lambda }_{2}}{6}+\frac{{\lambda }_{12}}{6}+\frac{{g}^{2}}{4}\left({I}_{{{\rm{\Phi }}}_{2}}^{2}+{I}_{{{\rm{\Phi }}}_{2}}\right)+\frac{g{{\prime} }^{2}{Y}_{{{\rm{\Phi }}}_{2}}^{2}}{16}\end{array}\right).\end{array}\end{eqnarray}$
Also the thermally corrected field-dependent masses of gauge bosons in the (W1, W2, W3, B) basis are $\begin{eqnarray}\begin{array}{rcl}{M}_{g}^{2(L,T)} & = & \left(\begin{array}{cccc}{M}_{W}^{2} & & & \\ & {M}_{W}^{2} & & \\ & & {M}_{W}^{2} & {M}_{WB}^{2}\\ & & {M}_{WB}^{2} & {M}_{B}^{2}\end{array}\right)\\ & & +{a}_{g}^{(L,T)}{T}^{2}\left(\begin{array}{cccc}{\pi }_{W} & & & \\ & {\pi }_{W} & & \\ & & {\pi }_{W} & \\ & & & {\pi }_{B}\end{array}\right),\end{array}\end{eqnarray}$
where $\begin{eqnarray}{M}_{W}^{2}=\frac{{g}^{2}}{4}\left({\varphi }_{1}^{2}+{I}_{W}^{2}{\varphi }_{2}^{2}\right),\quad {M}_{WB}^{2}=\frac{gg^{\prime} }{4}\left({\varphi }_{1}^{2}+{Y}_{{{\rm{\Phi }}}_{2}}^{2}{\varphi }_{2}^{2}\right),\end{eqnarray}$
$\begin{eqnarray}{M}_{B}^{2}=\frac{g{{\prime} }^{2}}{4}\left({\varphi }_{1}^{2}+{Y}_{{{\rm{\Phi }}}_{2}}^{2}{\varphi }_{2}^{2}\right),\quad {a}_{g}^{L}=1,\quad {a}_{T}^{T}=0,\end{eqnarray}$
$\begin{eqnarray}{\pi }_{W}=\frac{{g}^{2}}{9}{I}_{{{\rm{\Phi }}}_{2}}\left(1+{I}_{{{\rm{\Phi }}}_{2}}\right)\left(1+2{I}_{{{\rm{\Phi }}}_{2}}\right)+\frac{11}{6}{g}^{2},\end{eqnarray}$
$\begin{eqnarray}{\pi }_{B}=\frac{g{{\prime} }^{2}{T}^{2}}{12}\left(1+{I}_{{{\rm{\Phi }}}_{2}}\right){Y}_{{{\rm{\Phi }}}_{2}}^{2}+\frac{11}{6}g{{\prime} }^{2}.\end{eqnarray}$
In this model, the EWPT can be generated in multi-field space (φ1, φ2). We will focus on the path of the phase transition along φ1 axis. Therefore, we will discuss the possibility of one-step and two-step PTs, and will especially focus on the path of the phase transition along φ1 axis to discuss the distinction among the three types of models with one-step PT.3.3. The model with dimension-six operator from Φ2
For the effective model (15 ) after integrating out the heavy scalar sector from (14 ), the effective potential is 16 ), and the field-dependent mass of Higgs boson along with its thermal correction are
$\begin{eqnarray}\begin{array}{rcl}{V}_{{\rm{eff}}}\left({\varphi }_{1},T\right) & = & -\frac{1}{2}{a}_{2}{\varphi }_{1}^{2}+\frac{1}{4}{a}_{4}{\varphi }_{1}^{4}+\frac{1}{6}{a}_{6}{\varphi }_{1}^{6}\\ & & +\displaystyle \sum _{i}\frac{{n}_{i}}{64{\pi }^{2}}\,{M}_{i}^{4}\,\left(\mathrm{ln}\left(\frac{{M}_{i}^{2}}{{Q}^{2}}\right)-{c}_{i}\right)+{\rm{\Delta }}{V}_{T},\end{array}\end{eqnarray}$
where a2, a4 and a6 are given in equation ( $\begin{eqnarray}{M}_{h}^{2}({\varphi }_{1})=-{a}_{2}+3{a}_{4}{\varphi }_{1}^{2}+5{a}_{6}{\varphi }_{1}^{4},\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Pi }}}_{h}={T}^{2}\left(\frac{{a}_{4}}{4}+\frac{3{g}^{2}}{16}+\frac{g{{\prime} }^{2}}{16}+\frac{{y}_{t}^{2}}{4}\right).\end{eqnarray}$
The thermally corrected field-dependent masses of gauge bosons in the (W1, W2, W3, B) basis are $\begin{eqnarray}\begin{array}{rcl}{M}_{g}^{2(L,T)}(\varphi ,T) & = & \frac{{\varphi }^{2}}{4}\left(\begin{array}{cccc}{g}^{2} & & & \\ & {g}^{2} & & \\ & & {g}^{2} & gg^{\prime} \\ & & gg^{\prime} & g{{\prime} }^{2}\end{array}\right)\\ & & +{a}_{g}^{(L,T)}{T}^{2}\left(\begin{array}{cccc}{\pi }_{W} & & & \\ & {\pi }_{W} & & \\ & & {\pi }_{W} & \\ & & & {\pi }_{B}\end{array}\right),\end{array}\end{eqnarray}$
where $\begin{eqnarray}{M}_{W}^{2}=\frac{{g}^{2}}{4}{\varphi }_{1}^{2},\quad {M}_{WB}^{2}=\frac{gg^{\prime} }{4}{\varphi }_{1}^{2},\quad {M}_{B}^{2}=\frac{g{{\prime} }^{2}}{4}{\varphi }_{1}^{2},\end{eqnarray}$
$\begin{eqnarray}{\pi }_{W}=\frac{11}{6}{g}^{2},\quad {\pi }_{B}=\frac{11}{6}g{{\prime} }^{2},\quad {a}_{g}^{L}=1,\quad {a}_{T}^{T}=0\end{eqnarray}$
and the field-dependent mass of top quark is $\begin{eqnarray}{M}_{t}^{2}=\frac{{m}_{t}^{2}}{{v}^{2}}{\varphi }_{1}^{2}.\end{eqnarray}$
There are three new parameters in this potential, namely, $\begin{eqnarray}{m}_{{{\rm{\Phi }}}_{2}}^{2},{\lambda }_{12},{I}_{{{\rm{\Phi }}}_{2}}.\end{eqnarray}$
4. Gravitational waves
The GWs from the FOPTs serve as the promising probe for the new physics of BSM, including our EFT models of EWSB. In this section, we briefly outline the computation procedures to carry out our model constraints.
4.1. Phase-transition parameters
For an effective potential Veff exhibiting a false vacuum φ+ and a true vacuum φ− separated by a potential barrier, a cosmological FOPT proceeds via stochastic nucleations of true vacuum bubbles followed by a rapid expansion until a successful percolation to fully complete the phase transition process. In this section, we describe the phase transition dynamics consisting of the bubble nucleation and bubble percolation, which could be determined by the thermodynamics of effective potential alone without reference to the microscopic physics that leads to the macroscopic hydrodynamics of bubble expansion.
4.1.1. Bounce action
The bubble nucleations of true vacuum bubbles at finite temperature admit stochastic emergences of the field configuration φ(r) connecting a true vacuum region φ(r = 0) ≡ φ0 ≲ φ− (assuming φ+ < φ−) to the asymptotic false vacuum region φ(r → ∞) ≡ φ+ in an O(4)-symmetric manner ${\rm{d}}{s}^{2}={\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{3}^{2}$ or an O(3)-symmetric manner ${\rm{d}}{s}^{2}={\rm{d}}{\tau }^{2}+{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{2}^{2}$ depending on their maximum value of the nucleation rates [268–271]
$\begin{eqnarray}{\rm{\Gamma }}(T)={\rm{\max }}\left[{T}^{4}{\left(\frac{{S}_{3}}{2\pi T}\right)}^{\frac{3}{2}}{{\rm{e}}}^{-\frac{{S}_{3}}{T}},\frac{1}{{R}_{0}^{4}}\left(\frac{{S}_{4}}{2\pi }\right){{\rm{e}}}^{-{S}_{4}}\right],\end{eqnarray}$
where the O(4) bounce action $\begin{eqnarray}\begin{array}{rcl}{S}_{4} & = & 2{\pi }^{2}{\displaystyle \int }_{0}^{\infty }{\rm{d}}r\,{r}^{3}\\ & & \times \left[\frac{1}{2}{\left(\frac{{\rm{d}}{\phi }_{B}}{{\rm{d}}r}\right)}^{2}+{V}_{{\rm{eff}}}({\phi }_{B},T)-{V}_{{\rm{eff}}}({\phi }_{+},T)\right]\end{array}\end{eqnarray}$
is evaluated at the solution φB of the equation-of-motion $\begin{eqnarray}\frac{{{\rm{d}}}^{2}\phi }{{\rm{d}}{r}^{2}}+\frac{3}{r}\frac{{\rm{d}}\phi }{{\rm{d}}r}=\frac{\partial {V}_{{\rm{eff}}}}{\partial \phi },\quad \phi ^{\prime} (0)=0,\quad \phi (\infty )={\phi }_{+},\end{eqnarray}$
while the O(3) bounce action $\begin{eqnarray}\begin{array}{rcl}\frac{{S}_{3}}{T} & = & \frac{4\pi }{T}{\displaystyle \int }_{0}^{\infty }{\rm{d}}r\,{r}^{2}\\ & & \times \left[\frac{1}{2}{\left(\frac{{\rm{d}}{\phi }_{B}}{{\rm{d}}r}\right)}^{2}+{V}_{{\rm{eff}}}({\phi }_{B},T)-{V}_{{\rm{eff}}}({\phi }_{+},T)\right]\end{array}\end{eqnarray}$
is evaluated at the solution φB of the equation-of-motion $\begin{eqnarray}\frac{{{\rm{d}}}^{2}\phi }{{\rm{d}}{r}^{2}}+\frac{2}{r}\frac{{\rm{d}}\phi }{{\rm{d}}r}=\frac{\partial {V}_{{\rm{eff}}}}{\partial \phi },\quad \phi ^{\prime} (0)=0,\quad \phi (\infty )={\phi }_{+}.\end{eqnarray}$
In the realistic estimations, the vacuum decay rate from S4 usually dominates over the thermal decay rate from S3/T at extremely low temperature when the potential barrier does not vanish even at T = 0. R0 is the bubble radius defined by φB(r = R0) = (φ0 − φ+)/2.4.1.2. Nucleation temperature
During the whole process of bubble nucleation, the false vacuum becomes unstable once the temperature drops below the critical temperature defined by
$\begin{eqnarray}{V}_{{\rm{eff}}}({\phi }_{+},{T}_{c})={V}_{{\rm{eff}}}({\phi }_{-}({T}_{c}),{T}_{c}).\end{eqnarray}$
However, the bubble nucleation is only possible when the temperature further drops down to T = Ti < Tc defined $\begin{eqnarray}{\phi }_{B}(r=0,{T}_{i})={\phi }_{-}({T}_{i})\end{eqnarray}$
due to the presence of the Hubble friction term in the bounce equation. Since then, one can count the number of nucleated bubbles in one Hubble volume during a given time elapse by $\begin{eqnarray}N(T(t))={\int }_{{t}_{i}}^{t}{\rm{d}}t\frac{{\rm{\Gamma }}(t)}{{H}^{3}}={\int }_{T}^{{T}_{i}}\frac{{\rm{d}}T}{T}\frac{{\rm{\Gamma }}(T)}{H{(T)}^{4}},\end{eqnarray}$
until the nucleation temperature Tn < Ti defined by the moment when there is exactly one bubble nucleated in one Hubble volume, $\begin{eqnarray}N({T}_{n})=1.\end{eqnarray}$
4.1.3. Percolation temperature
The progress of the phase transition could be described by the expected volume fraction of the true-vacuum regions at time t [272–274],
$\begin{eqnarray}I(t)=\frac{4\pi }{3}{\int }_{{t}_{i}}^{t}{\rm{d}}t^{\prime} \,{\rm{\Gamma }}(t^{\prime} )a{(t^{\prime} )}^{3}r{(t,t^{\prime} )}^{3},\end{eqnarray}$
where $r(t,t^{\prime} )$ is the comoving radius of a bubble at t nucleated at an earlier time $t^{\prime} $, $\begin{eqnarray}r(t,t^{\prime} )\equiv \frac{{R}_{0}}{a(t^{\prime} )}+{\int }_{t^{\prime} }^{t}\frac{{v}_{w}(\tilde{t}){\rm{d}}\tilde{t}}{a(\tilde{t})}\approx {v}_{w}{\int }_{t^{\prime} }^{t}\frac{{\rm{d}}\tilde{t}}{a(\tilde{t})}.\end{eqnarray}$
Here, we omit the initial bubble radius R0 and fix the time-dependent bubble wall velocity vw(t) at its terminal vw. When the effects for the overlapping of true-vacuum bubbles and ‘virtual bubbles' nucleated in the true-vacuum regions are taken into account, the fraction of regions that are still sitting at the false vacuum at time t could be approximated by the exponentiation of I(t), $\begin{eqnarray}P(t)={{\rm{e}}}^{-I(t)}.\end{eqnarray}$
A percolation temperature Tp < Tn is therefore defined by a conventional estimation [24, 122, 274–276] $\begin{eqnarray}P({T}_{p})=1/e.\end{eqnarray}$
In what follows, the strength factor and characteristic length scale for determining the peak amplitude and frequency in the GW spectrum will be evaluated at the percolation temperature.4.1.4. Strength factor
To characterize the total released latent heat into the plasma, a parameter α is defined by [274]
$\begin{eqnarray}\alpha (T)=\frac{{\rm{\Delta }}{\rho }_{{\rm{vac}}}(T)}{{\rho }_{{\rm{rad}}}(T)},\end{eqnarray}$
with the total released vacuum energy defined by $\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{\rho }_{{\rm{vac}}}(T) & = & \rho ({\phi }_{+},T)-\rho ({\phi }_{-}(T),T)\\ & = & {\rm{\Delta }}{V}_{{\rm{eff}}}(T)-T\frac{\partial {\rm{\Delta }}{V}_{{\rm{eff}}}(T)}{\partial T}\end{array}\end{eqnarray}$
due to $\rho ={ \mathcal F }+Ts=-p+T\frac{\partial p}{\partial T}={V}_{{\rm{eff}}}-T\frac{\partial {V}_{{\rm{eff}}}}{\partial T}$ similar to the usual definition of latent heat L(Tc) = ρ(φ+, Tc) − ρ(φ−(Tc), Tc) at critical temperature. Here, ΔVeff(T) ≡ Veff(φ+(T), T) − Veff(φ−(T), T) for short.4.1.5. Characteristic length scale
The characteristic length scale R* for the peak position of the GW spectrum is estimated from the inverse duration β of the phase transition via the average number density of bubbles as
$\begin{eqnarray}{R}_{* }={n}_{B}^{-1/3}={(8\pi )}^{1/3}{v}_{w}{\beta }^{-1},\end{eqnarray}$
where the inverse duration β of the phase transition is computed by expanding the nucleation rate around the percolation time, $\begin{eqnarray}{\rm{\Gamma }}={{\rm{e}}}^{\beta (t-{t}_{p})+\cdots },\quad P({t}_{p})=1/e,\end{eqnarray}$
$\begin{eqnarray}\beta ={\left.-\frac{{\rm{d}}}{{\rm{d}}t}\frac{{S}_{3}(t)}{T(t)}\right|}_{t={t}_{p}}={\left.H(T)T\frac{{\rm{d}}}{{\rm{d}}T}\frac{{S}_{3}(T)}{T}\right|}_{T={T}_{p}}.\end{eqnarray}$
4.2. Gravitational wave spectrum
The GW spectrum from a FOPT consists of three contributions: the bubble wall collisions, the sound waves, and the magnetohydrodynamic (MHD) turbulences. The contribution from the MHD turbulences is usually sub-dominated and hence omitted in the current study. The contribution from the bubble wall collisions [277–281] only dominates the total GW spectrum when the bubble walls collide with each other while they are still rapidly accelerating. In most of the cases, the runaway wall expansion is highly unlikely, and they usually collide with each other long after having approached to the terminal velocity. Therefore, we only consider the contribution from the sound waves, which is well fitted numerically by [274, 282–286]
$\begin{eqnarray}\begin{array}{rcl}{h}^{2}{{\rm{\Omega }}}_{{\rm{sw}}} & = & 8.5\times 1{0}^{-6}{\left(\frac{100}{{g}_{{\rm{dof}}}}\right)}^{\frac{1}{3}}{\left(\frac{{\kappa }_{v}\alpha }{1+\alpha }\right)}^{2}\left(\frac{{H}_{* }}{\beta }\right){v}_{w}\\ & & \times \,{\frac{{7}^{\frac{7}{2}}{(f/{f}_{{\rm{sw}}})}^{3}}{\left(4+3{\left(f/{f}_{{\rm{sw}}}\right)}^{2}\right)}}^{\frac{7}{2}}\Upsilon ({\tau }_{{\rm{sw}}}),\end{array}\end{eqnarray}$
with the peak frequency $\begin{eqnarray}{f}_{{\rm{sw}}}=1.9\times 1{0}^{-5}\,{\rm{Hz}}{\left(\frac{{g}_{{\rm{dof}}}}{100}\right)}^{\frac{1}{6}}\frac{{T}_{\star }}{100{\rm{GeV}}}\frac{1}{{v}_{w}}\frac{\beta }{{H}_{* }}.\end{eqnarray}$
Here, the suppression factor ϒ is only important for sufficiently long duration comparable to or even slightly larger than the Hubble time scale, which is also neglected in our current study. Furthermore, both the wall velocity vw and efficiency factor of bulk fluid motions κv are also set to be unity for simplicity.To quickly locate the parameter space with a promising detectability of the GW signals, we will first use the ratio φC/TC at the critical temperature as a roughly estimation for the size of the strength factor [4, 287],
$\begin{eqnarray}\alpha \unicode{x0007E}{\alpha }_{\infty }\simeq 4.9\times 1{0}^{-3}{\left(\frac{{\phi }_{* }}{{T}_{* }}\right)}^{2},\end{eqnarray}$
which is a typically reliable estimation for most of the non-supercooling electroweak PTs since there are no new relativistic degrees of freedom or new particles with couplings to the Higgs comparable to those of the SU(2)L gauge bosons or top quark. To qualify the model detectability, the GWs signals h2ΩGW(f) are compared to the sensitivity curves of some GW detectors h2Ωsen(f) during the mission year ${ \mathcal T }$ by the signal-to-noise ratio (SNR), $\begin{eqnarray}{\rm{SNR}}=\sqrt{{ \mathcal T }{\int }_{{f}_{{\rm{\min }}}}^{{f}_{{\rm{\max }}}}{\rm{d}}f{\left[\frac{{h}^{2}{{\rm{\Omega }}}_{{\rm{GW}}}(f)}{{h}^{2}{{\rm{\Omega }}}_{{\rm{sen}}}(f)}\right]}^{2}}.\end{eqnarray}$
5. Summary of results
In this section, we show the results for the effective model descriptions with three types of the EWSB: (I) the light scalar model with CSI, (II) the light scalar model without CSI, and (III) the heavy scalar model with a higher-dimensional operator after integrating out the additional heavy scalar. Besides the common SM parameters $(v,{m}_{h},{m}_{W},{m}_{Z},{m}_{t},g,g^{\prime} ,{\alpha }_{s},{y}_{t})$, the new parameters in these three models are summarized below:
For model I, there are five parameters (${I}_{{{\rm{\Phi }}}_{2}},{Y}_{{{\rm{\Phi }}}_{2}},{\lambda }_{1},{\lambda }_{2},{\lambda }_{12}$) from the effective potential (
For model II, there are seven parameters $({I}_{{{\rm{\Phi }}}_{2}},{Y}_{{{\rm{\Phi }}}_{2}},{\mu }_{1},{\mu }_{2},{\lambda }_{1},{\lambda }_{2},{\lambda }_{12})$ in the effective potential. With the stationary condition, the massive parameter μ1 can be replaced by the EW VEV v. From the second derivatives of the potential, λ1 and μ2 parameters are given by the Higgs boson mass mh and the additional scalar boson mass ${m}_{{{\rm{\Phi }}}_{2}}$. Eventually, there are five free parameters: $({I}_{{{\rm{\Phi }}}_{2}},{Y}_{{{\rm{\Phi }}}_{2}},{\lambda }_{2},{m}_{{{\rm{\Phi }}}_{2}},{\lambda }_{12})$.
For model III, there are five parameters $({I}_{{{\rm{\Phi }}}_{2}},{\mu }_{1},{m}_{{{\rm{\Phi }}}_{2}},{\lambda }_{1},{\lambda }_{12})$ from the tree-level potential (
Note that in models I and II, although the hypercharge ${Y}_{{{\rm{\Phi }}}_{2}}$ is a free parameter, it does not significantly change the PT results since its effect is proportional to (mW − mZ) via Daisy diagram contributions. Thus, we take ${Y}_{{{\rm{\Phi }}}_{2}}$ = 1 as an illustrative example in the following PT analysis. For consistent comparison, we consider the models with common values for the shared free parameters. For example, the model I and model II can be compared in the parameter space of ${I}_{{{\rm{\Phi }}}_{2}}$ and λ2 for given choices of ${m}_{{{\rm{\Phi }}}_{2}}$ and λ12 as illustrated in figure 3. The black plane in figure 3 corresponds to the parameter region for model I. The ${m}_{{{\rm{\Phi }}}_{2}}$ is proportional to the λ12 in this model. Such a relation is given by a line in the plane of (${m}_{{{\rm{\Phi }}}_{2}}$, λ12. Since the λ2, which corresponds to the z axis in this figure, is independent free parameter, the black plane in figure 3 corresponds to the parameter region for model I. The green lines are the ${I}_{{{\rm{\Phi }}}_{2}}$ in model I. Since model II has five free parameters, the all free parameters in this model cannot be determined, even if the parameters have the same values as the CSI model. Such parameter regions correspond to the red and blue planes.
Figure 3. The presentation for a part of the effective models of interest in the parameter space supported by $({m}_{{{\rm{\Phi }}}_{2}},{\lambda }_{12},{\lambda }_{2})$ with a fixed ${Y}_{{{\rm{\Phi }}}_{2}}=1$. The black strip is model I with both ${m}_{{{\rm{\Phi }}}_{2}}$ and λ12 determined by ${I}_{{{\rm{\Phi }}}_{2}}$ labeled in green. For model II, we take two illustrative examples with (1) the same ${m}_{{{\rm{\Phi }}}_{2}}$ as model I but a free λ12 detached from ${I}_{{{\rm{\Phi }}}_{2}}$ presented as red slices, and (2) the same λ12 as model I but a free ${m}_{{{\rm{\Phi }}}_{2}}$ detached from ${I}_{{{\rm{\Phi }}}_{2}}$ presented as blue slices. |
5.1. The model with classical scale invariance
In this section, we first show the value of φC/TC for model I in the first panel of figure 4 to see which part of parameter space could produce detectable GWs. The horizontal axis is the quartic coupling λ2 and the vertical axis is the isospin ${I}_{{{\rm{\Phi }}}_{2}}$ of the additional scalar boson fields. It is easy to see that the value of φC/TC is large for either small isospin ${I}_{{{\rm{\Phi }}}_{2}}$ or small λ2 coupling. This result of φC/TC is different from our intuition since the strongly FOPT can be typically realized by adding a large number of additional scalar boson fields into the model, which corresponds to a large isospin ${I}_{{{\rm{\Phi }}}_{2}}$. In the model with CSI, the number of additional scalar boson fields is related to the Higgs boson mass via equation (26 ). Then, the φC/TC without ring diagram contribution is roughly estimated by 45 ), therefore, φC/TC becomes small through large ring diagram contributions from large ${I}_{{{\rm{\Phi }}}_{2}}^{2}$ value.
$\begin{eqnarray}\frac{{\phi }_{C}}{{T}_{C}}=\frac{E}{{\lambda }_{2}}\propto {I}_{{{\rm{\Phi }}}_{2}}^{1/4}.\end{eqnarray}$
On the other hand, the ring diagram contribution has ${I}_{{{\rm{\Phi }}}_{2}}^{2}$-dependence in equation (
Figure 4. The values of φC/TC, Tn, α and β/H for model I with respect to λ2 coupling and isospin ${I}_{{{\rm{\Phi }}}_{2}}$ of the additional scalar boson fields. |
The PT parameters Tn, α and β/H are shown in the last three panels of figure 4, respectively. From these parameters, we can describe the GW spectrum from the FOPT. The SNR for the testability of this model with CSI is shown in figure 5 with respect to the future space-borne GW detectors LISA, the DECi-hertz Interferometer Gravitational Wave Observatory (DECIGO), and the Big Bang Observer (BBO). When evaluating the SNR, we simply take the mission duration to be one year and the bubble wall velocity to be one. The red dashed curves in the panels of DECIGO and BBO represent SNR=10. In the parameter regions to the left of the red dashed curves, the SNR is larger than 10, therefore, these parameter regions could be tested at future GW detectors DECIGO and BBO. From these figures, most of parameter regions for the model with CSI could be tested by the DECIGO and BBO missions.
Figure 5. The values of SNR for the model with CSI at LISA, DECIGO and BBO. The experimental period is one year and bubble wall velocity is 1. The red dashed line represents SNR=10. |
5.2. The model without classical scale invariance
In this section, we show the results for model II, which admits two more free parameters than model I, namely, ${m}_{{{\rm{\Phi }}}_{2}}$ and λ12. To compare with model I/III with only one-step PT (the additional VEV does not appear in the potential at zero temperature), we will focus on the parameter regions where the same one-step PT for model II as model I/III (green → red in figure 2) can be realized. We also neglect the parameter regions with other paths of PT, for example, two-step PT (namely the path along green → magenta → red, where the red is a global minimum). We numerically examine the parameter regions where one-step EWPT can be generated by using following four parameter regions: (i) the same ${m}_{{{\rm{\Phi }}}_{2}}$ and λ12 as the model I, (ii) the same ${m}_{{{\rm{\Phi }}}_{2}}$ as model I, (iii) the same ${m}_{{{\rm{\Phi }}}_{2}}$ as model III, and (iv) different ${m}_{{{\rm{\Phi }}}_{2}}$ and λ12 from other models.
In parameter region (i), there are the minimum points along φ1 and φ2 axes at the black-point region in figure 6, but the minimum points along φ1 and φ2 axis are local and global minima, respectively. According to our numerical results, the two-step PT (green → red → magenta) is generated at the black-point region, and the magenta point becomes the true vacuum. In such a case, we cannot explain the fermion mass since the additional scalar field Φ2 does not couple to the fermion fields anymore. Therefore, in parameter (i) for model II, we cannot realize a proper PT which can make up the current universe. Although model I can generate the detectable GW from the one-step EWPT in figure 5, model II cannot realize the correct PT between green and red points in parameter region (i), which shares the same parameter values as model I. Therefore, we can distinguish the models with and without CSI by the GW detection.
Figure 6. The parameter region can have a stable minimum point along φ1 and φ2 axes. At these black points, there is global minimum along φ2 axis. |
In parameter case (ii), model II admits the same ${m}_{{{\rm{\Phi }}}_{2}}$ given by equation (27 ) as model I but with λ12 = 1.5, 2.0, 2.5, 3.0 decoupled from ${I}_{{{\rm{\Phi }}}_{2}}$ by equation (29 ) as shown in figure 7, respectively. Note that in the case of model I, ${m}_{{{\rm{\Phi }}}_{2}}$ and λ12 are related by ${m}_{{{\rm{\Phi }}}_{2}}^{2}={\lambda }_{12}{v}^{2}/2$, and thus we can get almost the same results for a similar case with the same λ12 as model I by equation (29 ) but decoupled ${m}_{{{\rm{\Phi }}}_{2}}$ from ${I}_{{{\rm{\Phi }}}_{2}}$ by equation (27 ). All colored points in figure 7 admit their global minimum along φ1 (namely the red point in figure 2). To show the parameter region where one-step PT along φ1 axis can be generated, we compare the critical temperatures for the PT along φ1 and φ2 axes. From the numerical results, the blue points in figure 7 have a higher critical temperature for the PTs along φ2 than the PT for φ1. On the other hand, the green and red points in figure 7 can realize the one-step PT (green → red in figure 2), especially, the red points can realize the strong FOPT along φ1 axis where the detectable GWs may be generated. At these red points in the model with λ12 = 2 and 2.5, the SNR for BBO with mission time ${ \mathcal T }$=1 yr and vb = 1 is larger than 10. Furthermore, the SNR for DECIGO with mission time ${ \mathcal T }$=1 yr and vb = 1 at (λ2, ${I}_{{{\rm{\Phi }}}_{2}}$)=(0.25,0.5) is about 10.87, while other red points have their SNR no more than 10 for DECIGO. In short summary, a small λ2 is necessary for detectable GWs in the model II, which is not necessary for model I as shown in figure 5.
Figure 7. Different PT regions in the parameter space supported by λ2 and Φ2 for model II with the same ${m}_{{{\rm{\Phi }}}_{2}}$ as model I but allowing for different λ12 = 1.5, 2.0, 2.5, 3.0. The blue points represent the two-step PT. The red and green points represent the one-step PT. The detectable GWs at the BBO detection can be generated for the red points. For the rest of the regions in this figure, the global minimum point does not appear along φ1 axis otherwise the potential is unstable. |
To compare the results between model II and model III, we show the results in parameter region (iii) with heavy ${m}_{{{\rm{\Phi }}}_{2}}$ as model III. For example, the results for heavy ${m}_{{{\rm{\Phi }}}_{2}}=650$ GeV with some λ2 values in model II are shown in figure 8, however, the effects from the additional heavy scalar field decouple. From the additional scalar mass ${M}_{{{\rm{\Phi }}}_{2}}{(\varphi )}^{2}\unicode{x0007E}-{\mu }_{2}^{2}+{\lambda }_{12}{\varphi }^{2}$ with large ${\mu }_{2}^{2}$ value, the cubic term of field-dependent mass in the effective potential can be expanded like ${({M}_{{{\rm{\Phi }}}_{2}}^{2})}^{3/2}\unicode{x0007E}| {\mu }_{2}^{3}| +{\lambda }_{12}{\mu }_{2}{\varphi }^{2}+{\lambda }_{12}^{2}{\varphi }^{4}/{\mu }_{2}$, but the cubic φ3 term does not appear in the potential. On the other hand, the ${M}_{{{\rm{\Phi }}}_{2}}{(\varphi )}^{3/2}$ with small ${\mu }_{2}^{2}$ value can have ${\lambda }_{12}^{3/2}{\varphi }^{3}$ as the source of a barrier. Therefore, it is difficult to generate a sizable barrier to realize first-order EWPT in model II with a much heavier additional scalar field. For the large negative ${\mu }_{2}^{2}$ term, the one-step PT occurs between green and red points in figure 2. In this case, λ2, which is the coefficient of ${{\rm{\Phi }}}_{2}^{4}$, does not much affect the EWPT, and the results are not very different between two figures of figure 8.
Figure 8. The model parameters for model II with ${\lambda }_{2}=0.1,\sqrt{4\pi }$ and ${m}_{{{\rm{\Phi }}}_{2}}$ = 650 GeV. The horizontal axis in left and right figures is λ12. Otherwise, the same as figure 7. |
The first two panels in figure 9 represent the results in the model with λ12=3 and ${m}_{{{\rm{\Phi }}}_{2}}=650$ and 425 GeV, which correspond to the parameter regions (iii) and (iv), respectively. With these parameters, the mass parameter μ2 becomes small (non-decoupling) and the detectable GW spectrum can be generated. Especially, in the model with ${m}_{{{\rm{\Phi }}}_{2}}=$ 425 GeV, the red (magenta) marks can be described in the upper right figure, and the SNR for DECIGO (BBO) is larger than 10. The last panel in figure 9 represents the results with respect to ${I}_{{{\rm{\Phi }}}_{2}}$ and ${m}_{{{\rm{\Phi }}}_{2}}$, which are related to parameter region (iv) in the model with benchmark parameters λ2 = 0.5 and λ12 = 2.5. The blue marks in this model can realize two-step PT. To summarize in short, the detectable GWs, which are represented by the red and magenta marks, can be produced in the massive model with small isospin of Φ2 and small λ2 value. However, for model II, such parameter regions with detectable GWs are not the same as those in model I and model III. Therefore we may distinguish the three types of models by the GW observations.
Figure 9. (First two panels) The model parameters for model II with λ12 = 3 and ${m}_{{{\rm{\Phi }}}_{2}}$ = 650, 425 GeV in the upper left and upper right panels, respectively. At the red (magenta) marks, the detectable GWs at DECIGO (BBO) can be generated. Otherwise, the same as figure 7. (Last panel) The results in the model with λ2 = 0.5 and λ12 = 2.5. The horizontal axis is the additional scalar boson mass. The two-step PT occurs at blue marks. |
5.3. The model with dimension-six operator from Φ2
In this section, we discuss the testability for model III. At first, the parameter region of a6 is shown in figure 10 for detectable GWs at DECIGO and BBO with mission duration ${ \mathcal T }=1$ yr and vb = 1. In the dark regions, the SNR is smaller than 10, hence we require a large a6 value. Recall that with equation (16 ), ${a}_{6}=(1+2{I}_{{{\rm{\Phi }}}_{2}})\frac{1}{{(4\pi )}^{2}{m}_{{{\rm{\Phi }}}_{2}}^{2}}\left(\frac{{\lambda }_{12}^{3}}{8}+\frac{{\lambda }_{12}^{2}{\lambda }_{1}}{6}\right)$, a large a6 value generally prefers a large isospin ${I}_{{{\rm{\Phi }}}_{2}}$, namely a large number of the additional scalar boson fields. We should further check the conditions for a valid EFT expansion ∣a8∣v2/a6 < 1 and 2ckinv2 < 1/2, which was discussed in [248]. In this model III, these conditions are
$\begin{eqnarray}\frac{{\lambda }_{12}{v}^{2}}{2{m}_{{{\rm{\Phi }}}_{2}}^{2}}\lt 1,\quad (1+2{I}_{{{\rm{\Phi }}}_{2}})\frac{{\lambda }_{12}^{2}{v}^{2}}{12{(4\pi )}^{2}{m}_{{{\rm{\Phi }}}_{2}}^{2}}\lt 1.\end{eqnarray}$
The left and right panels of figure 11 represent the values of the right hand side of these conditions. The left panel is the comparison between dimension 6 and 8 operators. Since the operators have one-loop level effects, the isospin ${I}_{{{\rm{\Phi }}}_{2}}$ cancels in this relation. On the other hand, the right panel is the comparison between the tree-level and one-loop level, therefore the value of this relation depends on the isospin ${I}_{{{\rm{\Phi }}}_{2}}$. For the presented parameter space, the conditions for a valid EFT expansion can be satisfied.
Figure 10. The SNR value from the model with a higher-dimensional operator for DECIGO and BBO with mission year ${ \mathcal T }=1$ yr and vb = 1. |
Figure 11. The valid EFT expansion 2ckinv2 < 1/2 and ∣a8∣v2/a6 < 1 in the model with a higher-dimensional operator with respect to ${I}_{{{\rm{\Phi }}}_{2}}$ and λ12 and fixed ${m}_{{{\rm{\Phi }}}_{2}}=650$ GeV. Since the first condition comes from the comparison between one-loop dimension six and eight operators, the values do not depend on the isospin ${I}_{{{\rm{\Phi }}}_{2}}$. For the presented parameter regions, the conditions for a valid EFT expansion can be satisfied. |
The values of φC/TC, Tn, α and β/H are then shown in figure 12 and the corresponding SNR at DECIGO and BBO are shown in figure 13, where the parameter regions to the right of the red dashed lines have their SNR larger than 10. However, SNR is less than 10−5 for LISA, hence LISA may be difficult to test the parameter region of the model III. For detectable GWs from the model III, we require a large number of additional scalar fields (namely a large isospin ${I}_{{{\rm{\Phi }}}_{2}}$) and large coupling λ12, which can be distinguished from models I and II by detectable GWs in the parameter regions of small isospin ${I}_{{{\rm{\Phi }}}_{2}}$. In particular, model I cannot take the heavy scalar field with ${m}_{{{\rm{\Phi }}}_{2}}=650$ GeV since the additional scalar mass should be less than 543 GeV from equation (27 ) [267]. Therefore, we may distinguish model III from model I and model II by constraining the isospin of additional scalar fields with the GWs background observed at future GW detectors.
Figure 12. The value of φC/TC, Tn, α and β/H in the EFT model with ${m}_{{{\rm{\Phi }}}_{2}}=650$ GeV. Otherwise, the same as figure 11. |
Figure 13. The values of SNR in the EFT model with ${m}_{{{\rm{\Phi }}}_{2}}=650$ GeV at DECIGO and BBO. The ratio of SNR is larger than 10 in the parameter region to right of the red dashed lines. |
6. Conclusions and discussions
The predictions for the GW background vary sensitively among different concrete models but also share a large degeneracy in the model buildings. From that, in this time, we take into account an EFT treatment for three BSM models based on different patterns of the EWSB: (I) classical scale invariance, (II) generic scalar extension, and (III) higher-dimensional operators. In these EFTs, the EWSB can be realized by (I) radiative symmetry breaking, (II) Higgs mechanism, and (III) EFT description of EWSB, respectively.
These three models can realize the strongly first-order EWPT and can produce the detectable GW spectrum by the effects summarized in Table 2. Here, Cn are the effective couplings of the order parameter operators φn in the effective potential. The dominant contributions of λ2 and ${I}_{{{\rm{\Phi }}}_{2}}$ in models (I) and (II) show up through the ring diagram effects as shown in equations (31 ), (34 ), (45 ) and (47 ). Thus, a small λ2 and a small ${I}_{{{\rm{\Phi }}}_{2}}$ are required to produce the detectable GW spectrum. The differences between models (I) and (II) are the C2 and C4 terms and the number of free parameters in the model. In model (II), these terms have tree-level contribution, and thus we need to tune model parameters to have a sizable C3 comparable with these C2 and C4 terms. Unlike model (I), we can use the λ12 parameter to realize such a situation and could generate the first-order EWPT as shown in figure 7. On the other hand, the λ2 in model (III) does not contribute to the effective potential after integrating out the Φ2 scalar field, and the high dimensional operator ${a}_{6}^{-1/2}$ in figure 10 is inversely proportional to λ12 and ${I}_{{{\rm{\Phi }}}_{2}}$ as shown in equation (16 ). Therefore, a small ${a}_{6}^{-1/2}$ can be realized by a large λ12 and a large ${I}_{{{\rm{\Phi }}}_{2}}$.
Table 2. The potential forms in the three types of the models where the EWSB can be realized by (I) radiative symmetry breaking, (II) Higgs mechanism and (III) EFT description of EWSB, respectively. The last column shows the source of detectable GW spectrum in these models. Otherwise, the same as Table 1. |
| Model | C2φ2 | C3φ3 | C4φ4 | C6φ6 | GW features |
|---|---|---|---|---|---|
| I | Loop | Loop | Loop | None | Small λ2 and small ${I}_{{{\rm{\Phi }}}_{2}}$ |
| II | Tree | Loop | Tree | None | Small λ2 and small ${I}_{{{\rm{\Phi }}}_{2}}$ |
| III | Tree | Loop | Tree | Tree | Large λ12 and large ${I}_{{{\rm{\Phi }}}_{2}}$ |
These three types of models might be distinguished by investigating the detectable GWs in the parameter regions with overlapping parameters: (1) model I and model II might be distinguished by the detection of GWs in the parameter regions as shown in figures 5 and 6. When taking the same ${m}_{{{\rm{\Phi }}}_{2}}$ and λ12 as model I by equations (27 ) and (29 ), model II cannot generate the correct one-step PT where the red point in figure 2 is the local minimum. However, when just taking the same ${m}_{{{\rm{\Phi }}}_{2}}$ as model I by equation (27 ) but decoupling λ12 from ${I}_{{{\rm{\Phi }}}_{2}}$ by equation (29 ), model II could produce detectable GWs in the parameter regions of small λ2 and small ${I}_{{{\rm{\Phi }}}_{2}}$, which is partially overlapping with model I with detectable GWs. In this case, although we cannot fully distinguish models I and II by the GW detections alone, we may use other observations, such as hhh coupling [172] and hγγ coupling [262, 267], to do that. (2) Model II and model III can be distinguished by the detection of GWs in the parameter regions as shown in figures 8 and 13 since detectable GWs are produced for model II in the parameter regions of small λ2 and small ${I}_{{{\rm{\Phi }}}_{2}}$, while model III favors a large ${I}_{{{\rm{\Phi }}}_{2}}$ to generate detectable GWs. (3) Similarly, model I could also be distinguished from model III by detectable GWs in the parameter regions of small ${I}_{{{\rm{\Phi }}}_{2}}$ (model I) and large ${I}_{{{\rm{\Phi }}}_{2}}$ (model III). Furthermore, the additional scalar mass should be less than 543 GeV in the model I, different from the case of model III with a heavy scalar field.
Therefore, we may distinguish these three effective model descriptions of EWPT by future GW detections in space. Nevertheless, our PT analysis for the effective model descriptions is preliminary in surveying part of the parameter space, and it only depicts those scalar extensions (with Z2 symmetry) of fundamental Higgs models [288–290] and Coleman–Weinberg Higgs models [267, 291, 292], but by no means covers all the effective models of EWPT. We will investigate the PT dynamics in a more general classifications of EWSB [293] in the ever-enlarging parameter space in future works.
Acknowledgments
This work is mainly supported by the National Key Research and Development Program of China under Grant Nos. 2021YFC2203004, 2021YFA0718304, and 2020YFC2201501. RGC is supported by the National Natural Science Foundation of China under Grants Nos. 11947302, 11991052, 11690022, 11821505 and 11851302, the Strategic Priority Research Program of the Chinese Academy of Sciences (CAS) under Grant Nos. XDB23030100 and XDA15020701, the Key Research Program of the CAS under Grant No. XDPB15, and the Key Research Program of Frontier Sciences of CAS. SJW is supported by the National Key Research and Development Program of China under Grant Nos. 2021YFC2203004 and 2021YFA0718304, the National Natural Science Foundation of China under Grant Nos. 12422502 and 12105344, the China Manned Space Project under Grant No. CMS-CSST-2021-B01. JHY is supported by the National Science Foundation of China under Grant Nos. 12022514, 11875003 and 12047503, and the National Key Research and Development Program of China under Grant Nos. 2020YFC2201501 and 2021YFA0718304, and the CAS Project for Young Scientists in Basic Research under Grant No. YSBR-006, and the Key Research Program of the CAS under Grant No. XDPB15.
Appendix A Stationary condition and CP-even boson masses in the model without CSI
We use the stationary condition at (φ1, φ2) = (v, 0) and the second derivatives of the potential with one-loop effects,
$\begin{eqnarray}{\left.\frac{\partial {V}_{\mathrm{eff}}}{\partial \left\langle {{\rm{\Phi }}}_{1}\right\rangle }\right|}_{\left\langle {{\rm{\Phi }}}_{1}\right\rangle =v,\left\langle {{\rm{\Phi }}}_{2}\right\rangle =0}=0,\quad {\left.\frac{{\partial }^{2}{V}_{\mathrm{eff}}}{\partial {\left\langle {{\rm{\Phi }}}_{1}\right\rangle }^{2}}\right|}_{\left\langle {{\rm{\Phi }}}_{1}\right\rangle =v,\left\langle {{\rm{\Phi }}}_{2}\right\rangle =0}={m}_{h}^{2},\end{eqnarray}$
where we take Q = v = 246 GeV. With tadpole condition ${{ \mathcal M }}_{{{\rm{\Phi }}}_{2}{{\rm{\Phi }}}_{2}}^{2}\gt {{ \mathcal M }}_{{{\rm{\Phi }}}_{1}{{\rm{\Phi }}}_{1}}^{2}$, one arrives at $\begin{eqnarray}\begin{array}{rcl}{\left.\frac{\partial {V}_{{\rm{eff}}}}{\partial \left\langle {{\rm{\Phi }}}_{1}\right\rangle }\right|}_{\left\langle {{\rm{\Phi }}}_{1}\right\rangle =v,\left\langle {{\rm{\Phi }}}_{2}\right\rangle =0} & = & -{\mu }_{1}^{2}+{\lambda }_{1}{v}^{2}+\frac{1}{16{\pi }^{2}}\\ & & \times \left[\right.\frac{6{\lambda }_{1}+{\lambda }_{12}}{4}{f}_{+}({m}_{{{\rm{\Phi }}}_{1,r}}^{2},{m}_{{{\rm{\Phi }}}_{2,r}}^{2})\\ & & +\frac{1}{4}(6{\lambda }_{1}-{\lambda }_{12}){f}_{-}({m}_{{{\rm{\Phi }}}_{1,r}}^{2},{m}_{{{\rm{\Phi }}}_{2,r}}^{2})\\ & & \quad +(2(2{I}_{{{\rm{\Phi }}}_{2}}-1)-1)\frac{{\lambda }_{12}{m}_{{N}_{2,r}}^{2}}{2}\\ & & \times \left(\mathrm{ln}\frac{{m}_{{N}_{2,r}}^{2}}{{Q}^{2}}-1\right)+\frac{6{m}_{W}^{4}}{{v}^{2}}\left(\mathrm{ln}\frac{{m}_{W}^{2}}{{Q}^{2}}-\frac{1}{3}\right)\\ & & \quad +\frac{3{m}_{Z}^{4}}{{v}^{2}}\left(\mathrm{ln}\frac{{m}_{Z}^{2}}{{Q}^{2}}-\frac{1}{3}\right)\\ & & -\frac{12{m}_{t}^{4}}{{v}^{2}}\left(\mathrm{ln}\frac{{m}_{t}^{2}}{{Q}^{2}}-1\right)\left]\right.=0,\end{array}\end{eqnarray}$
where A2 ), (A8 ) and (A9 ), we can replace the model parameters:
$\begin{eqnarray}{f}_{\pm }({m}_{i}^{2},{m}_{j}^{2})={m}_{i}^{2}\left(\mathrm{ln}\frac{{m}_{i}^{2}}{{Q}^{2}}-1\right)\pm {m}_{j}^{2}\left(\mathrm{ln}\frac{{m}_{j}^{2}}{{Q}^{2}}-1\right),\end{eqnarray}$
$\begin{eqnarray}{\rm{\Delta }}{m}_{{\rm{\Phi }}}^{2}={m}_{{{\rm{\Phi }}}_{2}}^{2}-{m}_{{{\rm{\Phi }}}_{1}}^{2}=\sqrt{{\left({{ \mathcal M }}_{{{\rm{\Phi }}}_{1}{{\rm{\Phi }}}_{1}}^{2}-{{ \mathcal M }}_{{{\rm{\Phi }}}_{2}{{\rm{\Phi }}}_{2}}^{2}\right)}^{2}+4{{ \mathcal M }}_{{{\rm{\Phi }}}_{1}{{\rm{\Phi }}}_{2}}^{4}},\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal M }}_{\mathrm{neutral}\,\,\mathrm{Higgs}}^{2}=\left(\begin{array}{cc}{{ \mathcal M }}_{{{\rm{\Phi }}}_{1}{{\rm{\Phi }}}_{1}}^{2} & {{ \mathcal M }}_{{{\rm{\Phi }}}_{1}{{\rm{\Phi }}}_{2}}^{2}\\ {{ \mathcal M }}_{{{\rm{\Phi }}}_{2}{{\rm{\Phi }}}_{1}}^{2} & {{ \mathcal M }}_{{{\rm{\Phi }}}_{2}{{\rm{\Phi }}}_{2}}^{2}\end{array}\right)\end{eqnarray}$
and the masses in the tadpole conditions of red point case are $\begin{eqnarray}{m}_{{{\rm{\Phi }}}_{1,r}}^{2}=-{\mu }_{1}^{2}+3{\lambda }_{1}{v}^{2},\quad {m}_{{{\rm{\Phi }}}_{2,r}}^{2}=-{\mu }_{2}^{2}+{\lambda }_{12}{v}^{2}/2,\end{eqnarray}$
$\begin{eqnarray}{m}_{{N}_{1,r}}^{2}=-{\mu }_{1}^{2}+{\lambda }_{1}{v}^{2},\quad {m}_{{N}_{2,r}}^{2}=-{\mu }_{2}^{2}+{\lambda }_{12}{v}^{2}/2,\end{eqnarray}$
with ${m}_{{{\rm{\Phi }}}_{2}}^{2}\gt {m}_{h}^{2}$. In order to replace two of the input parameters in the potential in terms of the Higgs boson mass mh and additional neutral CP-even boson mass ${m}_{{{\rm{\Phi }}}_{2}}$ 9, we use $\begin{eqnarray}\begin{array}{rcl}{\left.\frac{{\partial }^{2}{V}_{{\rm{eff}}}}{\partial {\left\langle {{\rm{\Phi }}}_{1}\right\rangle }^{2}}\right|}_{\left\langle {{\rm{\Phi }}}_{1}\right\rangle =v,\left\langle {{\rm{\Phi }}}_{2}\right\rangle =0} & = & 2{\lambda }_{1}{v}^{2}+\frac{{v}^{2}}{32{\pi }^{2}}\\ & & \times \left[\right.{A}_{2}^{r}\mathrm{ln}\frac{{m}_{{{\rm{\Phi }}}_{2}}^{2}{m}_{h}^{2}}{{Q}^{4}}-{A}_{3}^{r}\mathrm{ln}\frac{{m}_{h}^{2}}{{m}_{{{\rm{\Phi }}}_{2}}^{2}}\\ & & +4{\lambda }_{12}^{2}(2(2{I}_{{{\rm{\Phi }}}_{2}}-1)-1)\mathrm{ln}\frac{{m}_{{N}_{2,r}}^{2}}{{Q}^{2}}\\ & & \quad +12\frac{{m}_{Z}^{4}}{{v}^{4}}\left(\mathrm{ln}\frac{{m}_{Z}^{2}}{{Q}^{2}}+\frac{2}{3}\right)\\ & & +24\frac{{m}_{W}^{4}}{{v}^{4}}\left(\mathrm{ln}\frac{{m}_{W}^{2}}{{Q}^{2}}+\frac{2}{3}\right)-48\frac{{m}_{t}^{4}}{{v}^{4}}\mathrm{ln}\frac{{m}_{t}^{2}}{{Q}^{2}}\left]\right.\\ & \equiv & {m}_{h}^{2}\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\left.\frac{{\partial }^{2}{V}_{{\rm{eff}}}}{\partial {\left\langle {{\rm{\Phi }}}_{2}\right\rangle }^{2}}\right|}_{\left\langle {{\rm{\Phi }}}_{1}\right\rangle =v,\left\langle {{\rm{\Phi }}}_{2}\right\rangle =0}=-{\mu }_{2}^{2}+{\lambda }_{12}{v}^{2}+\frac{1}{64{\pi }^{2}}\left[\right.{B}_{1}{f}_{-}({m}_{{{\rm{\Phi }}}_{2}}^{2},{m}_{h}^{2})\\ \,+{B}_{2}{f}_{+}({m}_{{{\rm{\Phi }}}_{2}}^{2},{m}_{h}^{2})+4(2(2{I}_{{{\rm{\Phi }}}_{2}}-1)-1){\lambda }_{2}{m}_{{N}_{2,r}}^{2}\\ \,\times \left(\mathrm{ln}\frac{{m}_{{N}_{2,r}}^{2}}{{Q}^{2}}-1\right)+12\frac{{m}_{Z}^{4}{Y}_{{{\rm{\Phi }}}_{2}}^{2}}{{v}^{2}}\left(\mathrm{ln}\frac{{m}_{Z}^{2}}{{Q}^{2}}-\frac{1}{3}\right)\\ \,+24\frac{{m}_{W}^{4}{I}_{W}^{2}}{{v}^{2}}\left(\mathrm{ln}\frac{{m}_{W}^{2}}{{Q}^{2}}-\frac{1}{3}\right)\left]\right.\equiv {m}_{{{\rm{\Phi }}}_{2}}^{2},\end{array}\end{eqnarray}$
where $\begin{eqnarray}{A}_{2}^{r}=\frac{1}{4}\left({(6{\lambda }_{1}+{\lambda }_{12})}^{2}+{(6{\lambda }_{1}-{\lambda }_{12})}^{2}\right),\end{eqnarray}$
$\begin{eqnarray}{A}_{3}^{r}=-\frac{36{\lambda }_{1}^{2}-{\lambda }_{12}^{2}}{2},\end{eqnarray}$
$\begin{eqnarray}{B}_{1}=\frac{2}{{\rm{\Delta }}{m}^{2}}\left[\left(-6{\lambda }_{2}+{\lambda }_{12}\right)\left({{ \mathcal M }}_{{{\rm{\Phi }}}_{1}{{\rm{\Phi }}}_{1}}^{2}-{{ \mathcal M }}_{{{\rm{\Phi }}}_{2}{{\rm{\Phi }}}_{2}}^{2}\right)+4{\lambda }_{12}^{2}{v}^{2}\right],\end{eqnarray}$
$\begin{eqnarray}{B}_{2}=6{\lambda }_{2}+{\lambda }_{12}.\end{eqnarray}$
By using equations ( $\begin{eqnarray}({\mu }_{1}^{2},{\mu }_{2}^{2},{\lambda }_{1},{\lambda }_{2},{\lambda }_{12})\to (v,{m}_{{{\rm{\Phi }}}_{2}}^{2},{m}_{h},{\lambda }_{2},{\lambda }_{12}).\end{eqnarray}$
Appendix B Landau pole in the model with CSI
In this section, we show the Landau pole in the model with CSI. The model with CSI typically has the large values of couplings, which can be obtained by equation (29 ). The results of the Landau pole is given by figure 14. According to figures 5 and 14, the parameter region with detectable GW spectrum has Landau pole, which is O(1 TeV). When the value of λ2 will be large, the Landau pole also will be small.
Figure 14. The values of the Landau pole in the CSI model. |