Testing the electroweak phase transition in scalar extension models at lepton colliders^*

The dim-6 operator $| H{| }^{6}$ in the Higgs potential,

$$\begin{eqnarray}&&V(H)={\mu }^{2}| H{| }^{2}+\lambda | H{| }^{4}-{c}_{6}| H{| }^{6},\end{eqnarray} \tag{ 10 }$$

introduces a tree-level barrier so as to realize the SFOPT [20–24]; see Refs. [18, 19, 48–53] for recent studies. Note that the values of ${{\rm{\mu }}}^{2}$ and λ are no longer the SM values, in order to satisfy the SFOPT condition and give a 125.09 GeV Higgs boson. The contributions from other dim-6 operators to the EW phase transition are negligible, for the following reasons. In this type of tree-level barrier SFOPT induced by the potential in Eq. (10), the ${c}_{6}| H{| }^{6}$ term directly contributes to the SFOPT. Other dim-6 operators contribute to the EW phase transition mainly through modifying the masses of the particles which can contribute to the SFOPT. However, the mass modifications are negligible from current data, and the coefficient c₆ can be rather large, since there is nearly no constraint on c₆ from current data.

When the SFOPT is considered, one can simplify the potential by substituting H with $h/\sqrt{2}$:

$$\begin{eqnarray}&&{V}_{\mathrm{tree}}(h)=\displaystyle \frac{1}{2}{\mu }^{2}{h}^{2}+\displaystyle \frac{\lambda }{4}{h}^{4}-\displaystyle \frac{{c}_{6}}{8}{h}^{6}.\end{eqnarray} \tag{ 11 }$$

The corresponding finite-temperature effective potential up to one-loop level can be written as [54, 55],

$$\begin{eqnarray}&&{V}_{\mathrm{eff}}(h,T)={V}_{{\rm{tree}}}(h)+{V}_{1}^{T=0}(h)+{\rm{\Delta }}{V}_{1}^{T\ne 0}(h,T),\end{eqnarray} \tag{ 12 }$$

where ${V}_{{\rm{tree}}}(h)$ is the tree-level potential in Eq. (11), ${V}_{1}^{T=0}(h)$ is the Coleman-Weinberg potential at zero temperature, and ${\rm{\Delta }}{V}_{1}^{T\ne 0}(h)$ represents the leading thermal effects with daisy resummation. After including the full one-loop results given in Refs. [48, 51], the washout condition for the SFOPT, $v({T}_{c})/{T}_{c}\gtrsim 1$, can easily be satisfied [51]. More precise washout conditions, based on a detailed study on the sphaleron process with the dim-6 effective operators, are given in Refs. [56, 57]. If the EW phase transition is a SFOPT, then, at one-loop level, the SFOPT and vacuum stability give the following constraints [58]

$$\begin{eqnarray}&&\displaystyle \frac{1}{(0.89\mathrm{TeV}{)}^{2}}\lt -{c}_{6}\lt \displaystyle \frac{1}{(0.55\mathrm{TeV}{)}^{2}}.\end{eqnarray} \tag{ 13 }$$

The SFOPT condition modifies the trilinear Higgs boson as follows:

$$\begin{eqnarray}&&{{\mathscr{L}}}_{{hhh}}=-\displaystyle \frac{1}{3!}(1+{\delta }_{h}){A}_{h}{h}^{3},\end{eqnarray} \tag{ 14 }$$

where ${A}_{h}=3{m}_{h}^{2}/v$ is the trilinear Higgs boson coupling in the SM, and δ_h is the modification of the trilinear Higgs coupling induced by the dim-6 operator. In this scenario,

$$\begin{eqnarray}&&{\delta }_{h}\approx -0.468{c}_{6}\times {{\rm{TeV}}}^{2},\end{eqnarray} \tag{ 15 }$$

and roughly varies from 0.6 to 1.5 in the allowed parameter space. The $| H{| }^{6}$ operator yields a distinctive signal at the Large Hadron Collider (LHC), e.g. two peaks in the invariant mass distribution of the Higgs boson pairs [24]. Unfortunately, due to its low experimental precision, the LHC is not capable of testing this type of EW phase transition scenario. However, precise information on the triplet Higgs coupling might be obtained at future lepton colliders [59–61].¹⁾

In order to describe EW observables, we use the Z-scheme in which three of the most precisely measured values {α, m_Z, ${G}_{{\rm{F}}}$} are chosen as the input parameters. Other EW observables, e.g. m_W and ${{{\varGamma }}}_{Z}$, are expressed as a function of the three input parameters (together with Higgs mass or fermion masses, if necessary). We first consider the EW precision measurements at the Tevatron and the Large Electron-Positron Collider (LEP), including

$$\begin{eqnarray}&&{m}_{W},{N}_{\nu },{A}_{{FB}}^{b},{R}_{b},{R}_{\mu },{R}_{\tau },{\sin }^{2}{\theta }_{\mathrm{eff}}^{l}.\end{eqnarray} \tag{ 16 }$$

We check that, for the NP models considered here, the constraints from triple gauge coupling measurements at the LEP and LHC are quite weak. Thus, we do not consider these here. The scalar extended models can have a remarkable impact on Higgs-relevant processes since they often have sizable interactions with the Higgs field. Therefore, we also consider the measurement of Higgs decay branching ratios at the LHC, e.g.

$$\begin{eqnarray}\begin{array}{lrl}{{\rm{Br}}}_{h\to {WW}},\quad &{{\rm{Br}}}_{h\to {ZZ}},\quad& {{\rm{Br}}}_{h\to \gamma \gamma },\\ {{\rm{Br}}}_{h\to {gg}},\quad &{{\rm{Br}}}_{h\to \tau \tau },\quad &\,{{\rm{Br}}}_{h\to \mu \mu }.\end{array}\end{eqnarray} \tag{ 17 }$$

In comparison with the low energy precision observables listed in Eq. (16), the Higgs data impose rather weak constraints on NP model parameters, but they can offer constraints on some specific operators that are nearly free of EW precision measurements. For example, at tree level, the contributions of ${{\mathscr{O}}}_{{WW}}$ and ${{\mathscr{O}}}_{{BB}}$ to EW observables can be absorbed into the redefinition of gauge boson field and gauge coupling, leaving no physics effects. Such operators can be probed only through Higgs-involved process. Their contributions to EW observables are through Higgs loops, and hence are tiny due to loop suppression. However, the observables listed in Eq. (17) can probe such operators at tree level.

One way to test the EW phase transition scenario at future lepton colliders is through the Z-boson and Higgs boson associated production (the Zh channel) [63, 64]. The cross section of the Zh channel $\sigma ({Zh})$ could be measured with an accuracy of ${\mathscr{O}}(0.1 \% )\sim {\mathscr{O}}(1 \% )$ at future lepton colliders [65, 66]. For example, an accuracy of 0.5% for σ_Zh measurement could be achieved at the CEPC with an integrated luminosity of $5{\mathrm{ab}}^{-1}$ [28, 67], while the FCC-ee is expected to do a better job [30, 31]. The operator ${{\mathscr{O}}}_{6}=| {\mathscr{H}}{| }^{6}$ contributes to the Zh cross section through a triangle loop. Other dim-6 operators, e.g. ${{\mathscr{O}}}_{{\rm{H}}}=\tfrac{{\mathscr{1}}}{{\mathscr{2}}}{({\partial }_{\mu }| {\mathscr{H}}{| }^{2})}^{2}$ and ${{\mathscr{O}}}_{T}=\tfrac{{\mathscr{1}}}{{\mathscr{2}}}{({{\mathscr{H}}}^{\dagger }{\mathop{{\mathscr{D}}}\limits^{\leftrightarrow }}_{\mu }{\mathscr{H}})}^{2}$, contribute to the Zh production at tree-level. We define the deviation of cross section of the Zh production, normalized to the SM cross section, as follows:

$$\begin{eqnarray}&&{\delta }_{\sigma ({Zh})}\equiv \displaystyle \frac{{\sigma }_{{Zh}}}{{\sigma }_{{Zh}}^{\mathrm{SM}}}-1.\end{eqnarray} \tag{ 18 }$$

As the golden channel of Higgs physics, the Zh production rate has been calculated very precisely, including one-loop and two-loop quantum corrections [58, 68, 69]. Here, we refer ${\sigma }_{{Zh}}^{\mathrm{SM}}$ to the most state-of-art calculation of Zh production in the SM.

At a lepton collider with $\sqrt{s}=250\mathrm{GeV}$ ($\sqrt{s}$ being the center of mass energy of the collider), the high dimension operators' contribution to the Zh production is approximately given by [63, 64]

$$\begin{eqnarray}{\delta }_{\sigma ({Zh})} & \simeq & (0.26{c}_{{WW}}+0.01{c}_{{BB}}+0.04{c}_{{WB}}\\ & & -0.06{c}_{{\rm{H}}}-0.04{c}_{T}+0.74{c}_{{\rm{L}}}^{(3){\ell }}+0.28{c}_{\mathrm{LL}}^{(3){\ell }}\\ & & +1.03{c}_{{\rm{L}}}^{{\ell }}-0.76{c}_{{\rm{R}}}^{e})\times {{\rm{TeV}}}^{2}+0.016{\delta }_{h},\end{eqnarray} \tag{ 19 }$$

where δ_h is the deviation of the Higgs trilinear coupling defined in Eq. (14). The δ_h contribution, suffering from a loop suppression, is usually ignored in the operator analysis of the Zh channel [70, 71]. However, we argue that it cannot be ignored in our study of the SFOPT, owing to the following reasons. First, the SFOPT condition requires a large c₆ coefficient, resulting in a considerable contribution to ${\delta }_{\sigma ({Zh})}$ of $0.96 \% \sim 2.4 \%$. Note that there are nearly no constraints on c₆ from current experiments. Second, the constraints on the coefficients of other dim-6 operators are stronger than c₆. Hence, the δ_h contribution cannot be ignored.

We first consider a weak ${SU}{(2)}_{{\rm{L}}}$ triplet scalar extension model. For simplicity, we choose the triplet scalar not gauging under the $U{(1)}_{Y}$ group. We define the triplet scalar ${{\varSigma }}$ as

$$\begin{eqnarray}{{\varSigma }}(1,3,0)=\left[\begin{array}{cc}\displaystyle \frac{1}{2}{\delta }^{0} & \displaystyle \frac{1}{\sqrt{2}}{\delta }^{+}\\ \displaystyle \frac{1}{\sqrt{2}}{\delta }^{-} & -\displaystyle \frac{1}{2}{\delta }^{0}\end{array}\right]={{{\varSigma }}}^{\dagger }={{{\varSigma }}}^{a}\displaystyle \frac{{\sigma }^{a}}{2}.\end{eqnarray} \tag{ 20 }$$

The relevant Lagrangian involving the triplet scalar ${{\varSigma }}$ is

$$\begin{eqnarray}\delta {\mathscr{L}} & = & \mathrm{Tr}{[(({D}^{\mu }{{\varSigma }}{)}^{\dagger }{D}_{\mu }{{\varSigma }}]-{M}_{{{\varSigma }}}^{2}\mathrm{Tr}({{{\varSigma }}}^{2})-{\zeta }_{{{\varSigma }}}[\mathrm{Tr}({{{\varSigma }}}^{2})]}^{2}\\ & & +2{\xi }_{{{\varSigma }}}{H}^{\dagger }{{\varSigma }}H-2{\kappa }_{{{\varSigma }}}| H{| }^{2}\mathrm{Tr}({{{\varSigma }}}^{2}),\end{eqnarray} \tag{ 21 }$$

where the covariant derivative is given by ${D}_{\mu }{{\varSigma }}=[{D}_{\mu },{{\varSigma }}]$. The triplet scalar exists in many NP models, e.g. Grand Unification Theories, Little Higgs models, neutrino mass models, the Georgi-Machacek model [72], and the ${SU}{(3)}_{C}\otimes {SU}{(3)}_{{\rm{L}}}\otimes U{(1)}_{X}$ gauge extension model [73–75]. We keep ${\zeta }_{{{\varSigma }}}\gt 0$ and ${\kappa }_{{{\varSigma }}}\gt 0$, so that the scalar potential is bounded below.

Assuming the triplet scalar decouples, we use the CDE method to integrate out the ${{\varSigma }}$ field. We obtain the coefficients of those dim-6 operators mentioned in Section 2 as follows:

$$\begin{eqnarray}{c}_{{WW}} & = & \displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{\kappa }_{{{\varSigma }}}}{6{M}_{{{\varSigma }}}^{2}},\\ {c}_{2W} & = & {c}_{3W}=\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{g}^{2}}{30{M}_{{{\varSigma }}}^{2}},\\ {c}_{H} & = & \displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{\kappa }_{{{\varSigma }}}^{2}}{{M}_{{{\varSigma }}}^{2}},\\ {c}_{T} & = & \displaystyle \frac{{\xi }_{{{\varSigma }}}^{2}}{{M}_{{{\varSigma }}}^{4}}+\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{10{\zeta }_{{{\varSigma }}}{\xi }_{{{\varSigma }}}^{2}}{{M}_{{{\varSigma }}}^{4}},\\ {c}_{r} & = & \displaystyle \frac{2{\xi }_{{{\varSigma }}}^{2}}{{M}_{{{\varSigma }}}^{4}}+\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{20{\zeta }_{{{\varSigma }}}{\xi }_{{{\varSigma }}}^{2}}{{M}_{{{\varSigma }}}^{4}},\\ -{c}_{6} & = & \displaystyle \frac{{\kappa }_{{{\varSigma }}}{\xi }_{{{\varSigma }}}^{2}}{{M}_{{{\varSigma }}}^{4}}+\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{2{\kappa }_{{{\varSigma }}}^{3}}{{M}_{{{\varSigma }}}^{2}}+\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{10{\zeta }_{{{\varSigma }}}{\kappa }_{{{\varSigma }}}{\xi }_{{{\varSigma }}}^{2}}{{M}_{{{\varSigma }}}^{4}},\end{eqnarray} \tag{ 22 }$$

where those terms without (with) the factor of $1/(4{\rm{\pi }}{)}^{2}$ are induced by the tree (one-loop) level contributions, respectively. The coefficients are highly correlated to respect the weak quantum number of the triplet ${{\varSigma }}$. Our results are consistent with Refs. [27, 76]. The operator ${{\mathscr{O}}}_{r}$ can be reduced to ${{\mathscr{O}}}_{{\rm{H}}}$ and ${{\mathscr{O}}}_{6}$ by using Eq. (9). Note that the Wilson coefficients are determined completely by the model parameters ${\kappa }_{{{\varSigma }}}$, ${\xi }_{{{\varSigma }}}$, ${\zeta }_{{{\varSigma }}}$ and ${M}_{{{\varSigma }}}$. We require

$$\begin{eqnarray}&&{\kappa }_{{{\varSigma }}}\lt 4{\rm{\pi }},\quad {\zeta }_{{{\varSigma }}}\lt 4{\rm{\pi }},\quad \displaystyle \frac{{\xi }_{{{\varSigma }}}}{{M}_{{{\varSigma }}}}\lt 4{\rm{\pi }},\end{eqnarray} \tag{ 23 }$$

to ensure perturbativity and unitarity [77].

For simplicity, we set ${\zeta }_{{{\varSigma }}}=0$. Given the tiny coefficients of ${{\mathscr{O}}}_{2W}$ and ${{\mathscr{O}}}_{3W}$, their effects are negligible and we can omit them safely. Because the operators are induced at the scale of ${M}_{{{\varSigma }}}$, when we use them to discuss EW scale physics, the effects of renormalization group (RG) running should be studied carefully. After converting the operators into the Warsaw basis, following Refs. [41, 78, 79] we calculate the RG running effects and obtain the dim-6 operators and their Wilson coefficients at the EW scale. The condition of SFOPT is identified as

$$\begin{eqnarray}&&\displaystyle \frac{1}{(0.89\mathrm{TeV}{)}^{2}}\lt -{\left.{c}_{6}({\kappa }_{{{\varSigma }}},{\xi }_{{{\varSigma }}},{M}_{{{\varSigma }}})\right|}_{{\rm{EW}}{\rm{scale}}}\lt \displaystyle \frac{1}{(0.55\mathrm{TeV}{)}^{2}}.\end{eqnarray} \tag{ 24 }$$

Armed with the Wilson coefficients of dim-6 operators at the weak scale, we perform a global χ² fit to obtain constraints on the dim-6 operators from the EW precision test and Higgs branching ratio measurements; see Eqs. (16) and (17). The dependence of EW precision observables on the dim-6 operators is calculated in Ref. [39, 40, 46]. The SM predictions of those EW observables are given in Refs. [39, 80, 81], while the experimental measurements are presented in Refs. [4, 82–84]. We define χ² as

$$\begin{eqnarray}&&{\chi }^{2}=\mathop{\sum }\limits_{j}\displaystyle \frac{({\hat{O}}_{j}^{\exp }-{\hat{O}}_{j}^{\mathrm{theo}}{)}^{2}}{({\delta }_{j}^{\exp }{)}^{2}},\end{eqnarray} \tag{ 25 }$$

where ${\hat{O}}_{j}^{\exp }$ and ${\hat{O}}_{j}^{\mathrm{theo}}$ denote the experimental central value and theoretical prediction of the observable j, respectively, while ${\delta }_{j}^{\exp }$ represents the experimental error of the measurement of observable j. ${\hat{O}}_{j}^{\mathrm{theo}}$ is a function of the Wilson coefficients c_i. As the c_i depend on the model parameters, we derive the exclusion limit on $({\kappa }_{{{\varSigma }}},{\xi }_{{{\varSigma }}},{M}_{{{\varSigma }}})$ at the 95% confidence level. Figure 1 shows the allowed parameter space in the plane of (${\kappa }_{{{\varSigma }}}$, ${\xi }_{{{\varSigma }}}$) with ${M}_{{{\varSigma }}}=1\mathrm{TeV}$ (a), (${\kappa }_{{{\varSigma }}}$, ${M}_{{{\varSigma }}}$) with ${\xi }_{{{\varSigma }}}=0.05\mathrm{TeV}$ (b), and (${\xi }_{{{\varSigma }}}$, ${M}_{{{\varSigma }}}$) with ${\kappa }_{{{\varSigma }}}=5$ (c). The green regions satisfy the SFOPT condition while the blue regions are excluded by the EW precision test and Higgs branching ratios. In the model parameter space of interest to us, the heavy scale ${M}_{{{\varSigma }}}\sim (1-3)\mathrm{TeV}$. Running from ${M}_{{{\varSigma }}}$ down to the EW scale ($\sim 100\mathrm{GeV}$) modifies the Wilson coefficients slightly, therefore, the shape of the green and blue regions can easily be understood from the Wilson coefficients obtained at the scale of ${M}_{{{\varSigma }}}$; see Eq. (22).

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The constraint from the EW precision test is predominantly from the oblique T-parameter (or equivalently, the ρ-parameter). The ρ-parameter characterizes the weak isospin breaking that cannot be accounted for by the SM Higgs doublet, and has been measured very precisely, e.g. $\rho =1.00037\pm 0.00023$ [4]. As the operator ${{\mathscr{O}}}_{T}$ explicitly breaks the weak isospin, its coefficient ${c}_{T}\sim {\xi }_{{{\varSigma }}}^{2}/{M}_{{{\varSigma }}}^{4}$ is severely constrained, yielding a tiny value of ${\xi }_{{{\varSigma }}}$. For example, ${\xi }_{{{\varSigma }}}\le 0.1\mathrm{TeV}$ for ${M}_{{{\varSigma }}}=1\mathrm{TeV}$. Moreover, the c_T has nothing to do with ${\kappa }_{{{\varSigma }}}$, therefore, the EW precision test is not sensitive to ${\kappa }_{{{\varSigma }}}$. See the blue bands in Figs. 1(a) and 1(b).

The SFOPT condition is controlled solely by c₆,

$$\begin{eqnarray}&&-{c}_{6}\approx \displaystyle \frac{{\kappa }_{{{\varSigma }}}{\xi }_{{{\varSigma }}}^{2}}{{M}_{{{\varSigma }}}^{4}}+\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{2{\kappa }_{{{\varSigma }}}^{3}}{{M}_{{{\varSigma }}}^{2}}.\end{eqnarray} \tag{ 26 }$$

Since ${\xi }_{{{\varSigma }}}$ is tiny in comparison with ${M}_{{{\varSigma }}}$, the second term c₆ dominates over the first term such that the SFOPT condition is not sensitive to ${\xi }_{{{\varSigma }}}$. See the green bands in Figs. 1(a) and 1(c). In order to overcome the ${(4{\rm{\pi }})}^{2}$ suppression factor, a large value of ${\kappa }_{{{\varSigma }}}$ is needed. Such a large coupling in the triplet extension model may generate a fairly low Landau pole (LP) for the scalar coupling. For example, for the benchmark point

$$\begin{eqnarray}&&{\kappa }_{{{\varSigma }}}=5,\quad {\xi }_{{{\varSigma }}}=0.05{\rm{TeV,}}\quad {M}_{{{\varSigma }}}=1{\rm{TeV}},\end{eqnarray} \tag{ 27 }$$

the scalar coupling will break the perturbative limit at a scale around $1.1\times {10}^{2}$ TeV and blows up at a scale around $2.7\times {10}^{3}$ TeV. However, as pointed out in Ref. [85], the SM gauge coupling constants are free of the LP problem for triplet extension models. For a given ${\xi }_{{{\varSigma }}}$, the SFOPT condition requires ${\kappa }_{{{\varSigma }}}\propto {M}_{{{\varSigma }}}^{2/3}$; see Figs. 1(b).

Before upgrading to $\sqrt{s}=250\mathrm{GeV}$, the CEPC plans to operate at the EW precision energy (e.g. Z-pole and W pair threshold) in its early stages. The EW observables would be measured with an accuracy better than the LEP results. We adopt the CEPC projected uncertainties (${\delta }_{j}^{\exp }$) to estimate the sensitivities of the CEPC to the model parameters. If no anomalies are observed in CEPC EW precise operations, then the hatched regions in Fig. 1 are excluded at 95% confidence level.

The Zh channel can be measured very precisely at the CEPC with $\sqrt{s}=250\mathrm{GeV}$; a fraction deviation of cross section larger than 0.51%, i.e. ${\delta }_{\sigma ({Zh})}\ge 0.005$, can be detected [28]. The dependence of ${\delta }_{\sigma ({Zh})}$ on operators are given in Eq. (19). Those fermionic operators are generated through RG running and their coefficients are very small, e.g. ${c}_{{\rm{L}}}^{(3){\ell }}\simeq 8\times {10}^{-5}\,{\mathrm{TeV}}^{-2}$, ${c}_{{\rm{L}}}^{{\ell }}\simeq 2\times {10}^{-5}\,{\mathrm{TeV}}^{-2}$ and ${c}_{{\rm{R}}}^{e}\simeq 2\times {10}^{-5}$ TeV⁻² for the choice of Eq. (27). Numerically, ${\delta }_{\sigma ({Zh})}$ is dominated by ${{\mathscr{O}}}_{T}$, ${{\mathscr{O}}}_{{WW}}$ and ${{\mathscr{O}}}_{6}$. We plot the contours of ${\delta }_{\sigma ({Zh})}=1 \%,0.5 \%,0.25 \%$ in Fig. 1; see the red dashed curves.

Note that the value of δ_h in the parameter space of SFOPT is about $0.6\sim 1.5$. Even though suffering from small coefficient suppression, δ_h ($\propto {c}_{6}$) gives rise to a sizable contribution in ${\delta }_{\sigma ({Zh})}$, similar to other operators. For example,

$$\begin{eqnarray}&&{\delta }_{\sigma ({Zh})}=\mathop{\underbrace{0.001}}\limits_{{{\mathscr{O}}}_{WW}}-\mathop{\underbrace{0.009}}\limits_{{{\mathscr{O}}}_{{\rm{H}}}}+\mathop{\underbrace{0.013}}\limits_{{{\mathscr{O}}}_{6}}=0.005,\end{eqnarray} \tag{ 28 }$$

for the parameter choice of Eq. (27). Hence, the contour lines of ${\delta }_{\sigma ({Zh})}$ exhibit similar shapes to the SFOPT band. Especially, the red contour lines nearly coincide with the green band in the small ${\xi }_{{{\varSigma }}}$ region, where ${{\mathscr{O}}}_{6}$ dominates over ${{\mathscr{O}}}_{T}$ such that both ${\delta }_{\sigma ({Zh})}$ and SFOPT are determined by c₆.

Figure 1 shows that most of the parameter space allowed by the SFOPT condition and EW precision test can be well explored at the 250 GeV CEPC, given that ${\delta }_{\sigma ({Zh})}$ can be measured within an accuracy of 0.51% [67, 71]. The precision knowledge of EW observables achieved in the EW precise operation of CEPC can be used to cross-check the ${\delta }_{\sigma ({Zh})}$ measurement, and in addition, it can also probe model parameters insensitive to Zh production.

The second simplified model we investigated is the doublet extension model. For illustration, we consider a new ${SU}{(2)}_{{\rm{L}}}$ doublet scalar ${\rm{\Phi }}$ with hypercharge ${Y}_{{\rm{\Phi }}}=-1/2$. The relevant Lagrangian is

$$\begin{eqnarray}\delta {\mathscr{L}} & = & {D}_{\mu }{{\rm{\Phi }}}^{\dagger }{D}^{{\rm{\mu }}}{\rm{\Phi }}-{M}_{{\rm{\Phi }}}^{2}{{\rm{\Phi }}}^{\dagger }{\rm{\Phi }}-\displaystyle \frac{{\lambda }_{{\rm{\Phi }}}}{4}({{\rm{\Phi }}}^{\dagger }{\rm{\Phi }}{)}^{2}\\ & & -{\lambda }_{1}{{\rm{\Phi }}}^{\dagger }{\rm{\Phi }}{H}^{\dagger }H-{\lambda }_{2}| {\rm{\Phi }}\cdot H{| }^{2}-{\lambda }_{3}[({\rm{\Phi }}\cdot H{)}^{2}+h.c.]\\ & & +({\eta }_{{\rm{H}}}| H{| }^{2}+{\eta }_{{\rm{\Phi }}}| {\rm{\Phi }}{| }^{2})({\rm{\Phi }}\cdot H+h.c.)\,\end{eqnarray} \tag{ 29 }$$

where ${\rm{\Phi }}\cdot H={H}^{T}i{\sigma }^{2}{\rm{\Phi }}$. The scalar sector shown above mimics the well-studied Two Higgs Doublet Model. When ${M}_{{\rm{\Phi }}}\gg v$, ${\rm{\Phi }}$ can be integrated out to induce dim-6 operators. The Wilson coefficients of dim-6 operators matched at the scale ${M}_{{\rm{\Phi }}}$ in the doublet scalar extension model are given as follows:

$$\begin{eqnarray}{c}_{{WW}} & = & {c}_{{BB}}=\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{2{\lambda }_{1}+{\lambda }_{2}}{48}\displaystyle \frac{1}{{M}_{{\rm{\Phi }}}^{2}},\\ {c}_{{WB}} & = & \displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{\lambda }_{2}}{24}\displaystyle \frac{1}{{M}_{{\rm{\Phi }}}^{2}},\\ {c}_{2W} & = & {c}_{3W}=\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{g}^{2}}{60}\displaystyle \frac{1}{{M}_{{\rm{\Phi }}}^{2}},\\ {c}_{2B} & = & \displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{g}^{^{\prime} 2}}{60}\displaystyle \frac{1}{{M}_{{\rm{\Phi }}}^{2}},\\ {c}_{T} & = & \displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{\lambda }_{2}^{2}-4{\lambda }_{3}^{2}}{12}\displaystyle \frac{1}{{M}_{{\rm{\Phi }}}^{2}},\\ {c}_{r} & = & \displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\left(6{\eta }_{{\rm{\Phi }}}{\eta }_{{\rm{H}}}+\displaystyle \frac{{\lambda }_{2}^{2}+4{\lambda }_{3}^{2}}{6}\right)\displaystyle \frac{1}{{M}_{{\rm{\Phi }}}^{2}},\\ {c}_{H} & = & \displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\left(6{\eta }_{{\rm{\Phi }}}{\eta }_{{\rm{H}}}+\displaystyle \frac{4{\lambda }_{1}^{2}+4{\lambda }_{1}{\lambda }_{2}+{\lambda }_{2}^{2}+4{\lambda }_{3}^{2}}{12}\right)\displaystyle \frac{1}{{M}_{{\rm{\Phi }}}^{2}},\\ {c}_{6} & = & {\eta }_{{\rm{H}}}^{2}+\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\left[\displaystyle \frac{3}{2}{\lambda }_{{\rm{\Phi }}}{\eta }_{{\rm{H}}}^{2}+6{\eta }_{{\rm{\Phi }}}({\lambda }_{1}+{\lambda }_{2})\right.\\ & & \left.-\displaystyle \frac{2{\lambda }_{1}^{3}+3{\lambda }_{1}^{2}{\lambda }_{2}+3{\lambda }_{1}{\lambda }_{2}^{2}+{\lambda }_{2}^{3}}{6}-2({\lambda }_{1}+{\lambda }_{2}){\lambda }_{3}^{2}\right]\displaystyle \frac{1}{{M}_{{\rm{\Phi }}}^{2}}.\end{eqnarray} \tag{ 30 }$$

Similar to the triplet case, the terms with $1/(4{\rm{\pi }}{)}^{2}$ factor are induced at one-loop order. Again, we require ${\lambda }_{\mathrm{1,2,3}}\lt 4{\rm{\pi }}$ and ${\lambda }_{{\rm{\Phi }}}\lt 4{\rm{\pi }}$ to ensure perturbativity and unitarity.

For simplicity, we let ${\eta }_{{\rm{H}}}={\eta }_{{\rm{\Phi }}}=0$, which reduces the doublet scalar extension model to the so-called Inert Doublet Model [86–88]. After running down to the EW scale, one obtains those non-zero coefficients of dim-6 operators relevant to the SFOPT condition, EW precision tests, and the discovery potential of CEPC. Again, we note that the contribution of fermionic operators can be safely ignored, and we focus on bosonic operators below. Omitting the operators with negligible coefficients, we end up with a set of operators as

$$\begin{eqnarray}&&\{{{\mathscr{O}}}_{{WW}},{{\mathscr{O}}}_{{BB}},{{\mathscr{O}}}_{{WB}},{{\mathscr{O}}}_{T},{{\mathscr{O}}}_{{\rm{H}}},{{\mathscr{O}}}_{6}\}.\end{eqnarray} \tag{ 31 }$$

For illustration, we choose ${M}_{{\rm{\Phi }}}=1\mathrm{TeV}$ and scan over the model parameter space of the doublet extension model. Figure 2 shows the parameter space that is compatible with SFOPT (green) and current EW and Higgs boson data (blue) in the plane of (λ₁, λ₂) with ${\lambda }_{3}=0$ (a), (λ₁, λ₃) with ${\lambda }_{2}=0$ (b), and (λ₂, λ₃) with ${\lambda }_{1}=0$ (c).

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Similar to the case of the triplet scalar model, the shape of the parameter regions in Fig. 2 can be understood by those Wilson coefficients obtained at the scale of ${M}_{{\rm{\Phi }}}$ in Eq. (30). For example, the oblique T-parameter imposes severe constraints on the coefficient of the ${{\mathscr{O}}}_{T}$ operator, ${c}_{T}\sim ({\lambda }_{2}^{2}-4{\lambda }_{3}^{2})$. Hence, the blue shaded regions and blue hatched regions are insensitive to λ₁ in Figs. 2(a) and 2(b). However, the blue regions exhibit hyperbolic shapes in Fig. 2(c). Again, the T-parameter constraint also validates the assumption of heavy ${\rm{\Phi }}$ scalar decoupling we made above. Note that three massive scalars emerge from the doublet ${\rm{\Phi }}=[({{\mathscr{H}}}^{0}+i{{\mathscr{A}}}^{0})/\sqrt{2},{{\mathscr{H}}}^{-}]$ after symmetry breaking, ${H}^{T}\to (0,v/\sqrt{2})$. The masses of the resulting new scalar particles are given by

$$\begin{eqnarray}{m}^{2}({{\mathscr{H}}}^{\pm }) & = & {M}_{{\rm{\Phi }}}^{2}+{\lambda }_{1}{v}^{2},\\ {m}^{2}({{\mathscr{H}}}^{0}) & = & {M}_{{\rm{\Phi }}}^{2}+({\lambda }_{1}+{\lambda }_{2}+2{\lambda }_{3}){v}^{2};\\ {m}^{2}({{\mathscr{A}}}^{0}) & = & {M}_{{\rm{\Phi }}}^{2}+({\lambda }_{1}+{\lambda }_{2}-2{\lambda }_{3}){v}^{2}.\end{eqnarray} \tag{ 32 }$$

Vacuum stability demands [89]

$$\begin{eqnarray}&&\lambda,\ {\lambda }_{{\rm{\Phi }}}\gt 0,\quad {\lambda }_{1},\ {\lambda }_{1}+{\lambda }_{2}-\left|2{\lambda }_{3}\right|\gt -\sqrt{\lambda {\lambda }_{{\rm{\Phi }}}},\end{eqnarray} \tag{ 33 }$$

where λ is the Higgs quartic self coupling defined in Eq. (10). For large ${\lambda }_{\mathrm{1,2,3}}$, one of the neutral scalars, ${{\mathscr{H}}}^{0}$ or ${{\mathscr{A}}}^{0}$, could be lighter than ${M}_{{\rm{\Phi }}}$ and could potentially be around the EW scale. This would obviously violate the ${\rm{\Phi }}$ decoupling assumption. In the parameter space allowed by vacuum stability and the T-parameter (${c}_{T}\propto {\lambda }_{2}^{2}-4{\lambda }_{3}^{2}\sim {10}^{-4}{{\rm{}}{\rm{TeV}}}^{-2}$), the masses of the three new scalars are larger than ${M}_{{\rm{\Phi }}}$ as long as ${\lambda }_{1}\ge 0$,

The band of the SFOPT condition (green) is controlled by

$$\begin{eqnarray}&&-{c}_{6}=\left({\lambda }_{1}^{3}+({\lambda }_{1}+{\lambda }_{2}{)}^{3}+12({\lambda }_{1}+{\lambda }_{2}){\lambda }_{3}^{2}\right)\displaystyle \frac{1}{6{M}_{{\rm{\Phi }}}^{2}}.\end{eqnarray} \tag{ 34 }$$

In all three cases in Fig. 2, there is enough parameter space for SFOPT after considering the current experimental data. From the cross section deviation of Zh associated production (red dashed contours) and the region probed by EW precise measurements (blue dashed lines), we find that the CEPC is able to test this type of SFOPT.

Now consider the singlet scalar extension model [90]. We first investigate the case of a real scalar and then comment on the case of a complex scalar.

The relevant Lagrangian of a real singlet scalar is

$$\begin{eqnarray}\delta {\mathscr{L}} & = & \displaystyle \frac{1}{2}{\partial }_{{\rm{\mu }}}S{\partial }^{{\rm{\mu }}}S-\displaystyle \frac{{M}_{S}^{2}}{2}{S}^{2}-\displaystyle \frac{{\mu }_{S}}{3!}{S}^{3}-\displaystyle \frac{{\lambda }_{S}}{4!}{S}^{4}\\ & & -\displaystyle \frac{{\kappa }_{S}}{2}{S}^{2}| H{| }^{2}-{a}_{S}S| H{| }^{2}.\end{eqnarray} \tag{ 35 }$$

Since the S scalar is a gauge singlet, only two operators, ${{\mathscr{O}}}_{6}$ and ${{\mathscr{O}}}_{{\rm{H}}}$, are generated at the M_S scale. The corresponding dim-6 operators matched from the singlet scalar model up to next-to-leading order are

$$\begin{eqnarray}{{\mathscr{L}}}_{\mathrm{eff}} & \supset & \left(-\displaystyle \frac{{\kappa }_{S}{a}_{S}^{2}}{2{M}_{S}^{4}}-\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{\kappa }_{S}^{3}}{12{M}_{S}^{2}}+\displaystyle \frac{{\mu }_{S}{a}_{S}^{3}}{3!{M}_{S}^{6}}\right){{\mathscr{O}}}_{6}\\ & & +\left(\displaystyle \frac{{a}_{S}^{2}}{{M}_{S}^{4}}+\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{\kappa }_{S}^{2}}{12{M}_{S}^{2}}\right){{\mathscr{O}}}_{{\rm{H}}}.\end{eqnarray} \tag{ 36 }$$

Here, we consider ${\kappa }_{S}\lesssim 10$ to make sure the model is perturbatively calculable, following Ref. [61].¹⁾ Owing to RG running effects, new operators such as ${{\mathscr{O}}}_{T}$, ${{\mathscr{O}}}_{{\rm{L}}}^{l}$ and ${{\mathscr{O}}}_{{\rm{R}}}^{e}$ appear at the EW scale. However, their contributions to the EW precision observables and Higgs measurements are fairly small and can be safely ignored. The corrections to the renormalizable operators in the Higgs potential can be absorbed into the redefinition of the coefficients of the renormalizable operators, whose values are determined by the Higgs mass and the vacuum expectation value v.

The SFOPT condition is dominated by the leading term of the coefficient of ${{\mathscr{O}}}_{6}$, i.e. $-{c}_{6}\sim {\kappa }_{S}{a}_{S}^{2}/2{M}_{S}^{4}$; see the green bands in Fig. 3. We emphasize that a non-zero a_S in the real singlet scalar model is crucial to trigger SFOPT within the perturbative limit of κ_S. When the a_S term is absent, the ${{\mathscr{O}}}_{6}$ operator can be induced only at the one-loop level and thus suffers a large suppression from the factor of $1/(4{\rm{\pi }}{)}^{2}$. In order to achieve the SFOPT (i.e. sizable c₆), a huge κ_S is needed, e.g. ${\kappa }_{S}\sim 14$. Such a large coupling violates the perturbative limit mentioned above.

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A non-zero a_S term also yields a vacuum expectation value to the S field,

$$\begin{eqnarray}&&\langle S\rangle =\displaystyle \frac{-{a}_{S}{v}^{2}/2}{{M}_{S}^{2}+{\kappa }_{S}{v}^{2}/2},\end{eqnarray} \tag{ 37 }$$

after the Higgs field H develops the vacuum expectation value v. The term also induces a mixing between the SM Higgs boson h and S, which universally modifies the Higgs boson couplings to the SM particles such that it is strongly constrained by current LHC data. In the heavy S limit, the sine of the mixing angle is $\sin \theta \approx {{va}}_{S}/{M}_{S}^{2}$. The ratio of the hVV couplings in the real singlet scalar model and the SM is given by

$$\begin{eqnarray}&&{\kappa }_{V}=\displaystyle \frac{{g}_{{hVV}}}{{g}_{{hVV}}^{\mathrm{SM}}}\approx 1-\displaystyle \frac{{v}^{2}{a}_{S}^{2}}{2{M}_{S}^{4}}.\end{eqnarray} \tag{ 38 }$$

For the benchmark point ${a}_{S}={M}_{S}=1\,\mathrm{TeV}$, ${\kappa }_{V}\sim 0.97$, which is consistent with current experimental data [91, 92]. We scan over the whole parameter space and find the parameter spaces of interest to us are all consistent with experimental data.

The EW precision observables are dominated by the operator ${{\mathscr{O}}}_{{\rm{H}}}$; note that ${{\mathscr{O}}}_{6}$ is almost unconstrained at the LEP and LHC. As ${{\mathscr{O}}}_{{\rm{H}}}$ only modifies the Higgs boson interaction through the field redefinition of the Higgs field, it contributes a universal factor to the Higgs couplings without changing the Higgs-boson decay branching ratios. The low energy precision tests are nearly irrelevant to ${{\mathscr{O}}}_{{\rm{H}}}$, while the Higgs boson measurements at the LHC are not precise enough to constrain ${{\mathscr{O}}}_{{\rm{H}}}$. Therefore, most of the parameter space of the real singlet scalar model remains a blind region in current experimental searches and even at the CEPC in EW precision operation; see the solid blue and hatched blue regions in Fig. 3. The parameter space can be widely probed at the 250 GeV CEPC through Zh production; see the red contours. The shapes of ${\delta }_{\sigma ({Zh})}$ are quite different from those of the SFOPT condition, as the former is determined by both ${c}_{{\rm{H}}}$ and c₆ while the latter depends only on c₆. In Ref. [93], the authors claimed that the EFT approach for discussing the phase transition may lead to some mismatches and miss some parameter spaces which are allowed in the renormalizable singlet model. It is true that some allowed parameter spaces are missed in the EFT framework compared to the renormalizable model. However, we are only concerned with whether there are allowed parameter spaces for this type of tree-level barrier SFOPT induced by non-renormalizable operators (for a given renormalizable model, there may exist different types of SFOPT in different parameter spaces [18]; we will not consider these aspects in this work), and the allowed parameter spaces are consistent with the results in Ref. [93].

Let us proceed to the complex scalar extension model. The remarkable difference between the complex scalar and real scalar extension models is the former does not have the a_S term while the latter does. The general Lagrangian of the singlet complex scalar S is

$$\begin{eqnarray}&&\delta {\mathscr{L}}={D}_{{\rm{\mu }}}{S}^{\dagger }{D}^{\mu }S-{M}_{S}^{2}| S{| }^{2}-\displaystyle \frac{{\lambda }_{S}}{4}| S{| }^{4}-{\kappa }_{S}| S{| }^{2}| H{| }^{2},\end{eqnarray} \tag{ 39 }$$

where ${D}_{{\rm{\mu }}}S={\partial }_{{\rm{\mu }}}S-{{ig}}^{^{\prime} }{Y}_{S}{B}_{{\rm{\mu }}}S$, with Y_S being the hypercharge of the singlet scalar S. The $U{(1)}_{Y}$ symmetry of the complex scalar field forbids the linear and cubic terms of the complex scalar, i.e. no a_S or μ_S terms as in the singlet real scalar model. Using the CDE method, we obtain the dim-6 effective operators at the one-loop level as follows:

$$\begin{eqnarray}{{\mathscr{L}}}_{\mathrm{eff}} & \supset & -\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{\kappa }_{S}^{3}}{6{M}_{S}^{2}}{{\mathscr{O}}}_{6}+\displaystyle \frac{{\mathscr{1}}}{({\mathscr{4}}{\rm{\pi }}{)}^{2}}\displaystyle \frac{{\kappa }_{S}^{2}}{{\mathscr{6}}{{{M}}}_{S}^{2}}{{\mathscr{O}}}_{{\rm{H}}}\\ & & +\displaystyle \frac{1}{(4{\rm{\pi }}{)}^{2}}\displaystyle \frac{{\kappa }_{S}{Y}_{S}^{2}}{12{M}_{S}^{2}}{{\mathscr{O}}}_{{BB}}+\displaystyle \frac{{\mathscr{1}}}{({\mathscr{4}}{\rm{\pi }}{)}^{2}}\displaystyle \frac{{{\mathscr{g}}}^{^{\prime} 2}{{\mathscr{Y}}}_{S}^{2}}{{\mathscr{30}}{{{M}}}_{S}^{2}}{{\mathscr{O}}}_{2B}.\end{eqnarray} \tag{ 40 }$$

The coefficients of ${{\mathscr{O}}}_{6}$ and ${{\mathscr{O}}}_{{\rm{H}}}$ are twice those in the real scalar model (setting ${a}_{S}={\mu }_{S}=0$), because a complex scalar is equivalence to two real scalars. Besides those pure scalar operators, we have two more operators involving gauge bosons, ${{\mathscr{O}}}_{{BB}}$ and ${{\mathscr{O}}}_{2B}$.

As discussed above, it is hard to explain the SFOPT in the perturbative parameter space in the complex scalar model. In the complex scalar model, the dim-6 operators are suppressed by a factor of $1/{(4{\rm{\pi }})}^{2}$. The parameter κ_S has to be large enough to satisfy the SFOPT condition. In such a case, perturbativity may be violated; for example, for ${M}_{S}=1\,\mathrm{TeV}$, the SFOPT condition demands ${\kappa }_{S}\gtrsim 10.7$, which is beyond the perturbative region. Of course, slightly reducing M_S can render κ_S marginally below the perturbative limit, e.g. ${\kappa }_{S}\gtrsim 9.2$ for ${M}_{S}=0.8\mathrm{TeV}$, but it is not natural. In addition, if M_S is too small, the EFT description fails and one has to carry out the loop calculation in the UV-complete model. Therefore, we conclude that the $| H{| }^{6}$ SFOPT scenario is not favored in the singlet heavy complex-scalar model.

From the above study, it is clear that the scenario of SFOPT induced by $| H{| }^{6}$ can still be realistic and allowed by current experimental data. Our discussion on the dim-6 operators generated in the three scalar extension models can be widely extended to many NP models. Usually, the SFOPT needs the Higgs portal coupling to be of order one, and the large Higgs portal coupling may give a hint of the composite nature of the Higgs boson [49]. If the Higgs boson is a pseudo-Goldstone boson, from strong dynamics, the coefficients of dim-6 operators can be estimated by naive dimensional analysis (NDA) [94]. The coefficients of dominant CP-conserving operators, estimated from the NDA, are shown as follows:

$$\begin{eqnarray}&&\begin{array}{l}{c}_{{WW}}\sim {c}_{{BB}}\sim {c}_{{WB}}\sim 1/{{{\varLambda }}}^{2}\sim 1/(4{\rm{\pi }}f{)}^{2}\\ {c}_{H}\sim {c}_{T}\sim 1/{f}^{2}\\ {c}_{6}\sim -{{{\varLambda }}}^{2}/{f}^{4}=-1/(f/4{\rm{\pi }}{)}^{2}.\end{array}\end{eqnarray} \tag{ 41 }$$

If the EW phase transition is SFOPT, then one needs

$$\begin{eqnarray*}&&\displaystyle \frac{1}{(0.89\mathrm{TeV}{)}^{2}}\lt -{c}_{6}\lt \displaystyle \frac{1}{(0.55\mathrm{TeV}{)}^{2}},\end{eqnarray*}$$

which demands

$$\begin{eqnarray*}&&6.91\mathrm{TeV}\lt f\lt 11.18\mathrm{TeV}.\end{eqnarray*}$$

The coefficients ${c}_{{WW},{BB},{WB},H,T}$ are consistent with current experiments if the scale f is within the above range.

The cross section deviation of Zh production at the CEPC is approximately given by

$$\begin{eqnarray}{\delta }_{\sigma ({Zh})} & \approx & (0.26{c}_{{WW}}+0.01{c}_{{BB}}+0.04{c}_{{WB}}-0.06{c}_{{\rm{H}}}\\ & & -0.04{c}_{T})\times {{\rm{TeV}}}^{2}+0.016{\delta }_{h}.\end{eqnarray} \tag{ 42 }$$

Without the δ_h term, we find ${\delta }_{\sigma ({Zh})}\sim 0.1 \%$ when choosing those coefficients shown in Eq. (41). The SFOPT condition requires $0.6\lt {\delta }_{h}\lt 1.5$. Therefore, including the δ_h contribution yields ${\delta }_{\sigma ({Zh})}$ in the range of 0.96%–2.4%, which could be probed at future lepton colliders, such as the CEPC, ILC and FCC-ee.