The Higgs Trilinear Coupling and the Scale of New Physics for the SM-Axion-Seesaw-Higgs Portal Inflation (SMASH) Model

Abstract

In the extended scalar sector of the SMASH (Standard Model - Axion-Seesaw-Higgs portal inflation) framework, we conduct a phenomenological investigation of the observable effects. In a suitable region of the SMASH scalar parameter spaces, we solve the vacuum metastability problem and discuss the one-loop correction to the triple Higgs coupling, ${\lambda }_{HHH}$. The ${\lambda }_{HHH}$ and SM Higgs quartic coupling ${\lambda }_H$ corrections are found to be proportional to the threshold correction. A large ${\lambda }_{HHH}$ correction (≳5%) implies vacuum instability in the model and thus limits the general class of theories that use threshold correction. We performed a full two-loop renormalization group analysis of the SMASH model. The SMASH framework has also been used to estimate the evolution of lepton asymmetry in the universe.

1. Introduction

After the discovery of the Standard Model (SM) Higgs boson [1,2], every elementary particle of the SM has been confirmed to exist. Even though the past forty years have been a spectacular triumph for the SM, the mass of the Higgs boson ($m_H=125.25\pm 0.17$ GeV) [3] poses a serious problem for the SM [4]. It is well-known that the SM Higgs potential is metastable [5], as the sign of the quartic coupling, ${\lambda }_H$, turns negative at instability scale ${\mathsf{\Lambda }}_{\mathrm{IS}}\sim {10}^{11}$ GeV. On the other hand, the SM is devoid of non-perturbative problems since the non-perturbative scale ${\mathsf{\Lambda }}_{\mathrm{NS}}\gg M_{Pl}$, where $M_{Pl}=1.22\times {10}^{19}$ GeV is the Planck scale, but still there are studies on non-perturbative effects of the SM [6,7,8,9,10]. In the post-Planckian regime, effects of quantum gravity are expected to dominate, and the non-perturbative scale is therefore well beyond the validity region of the SM, unlike the instability scale. The largest uncertainties in SM vacuum stability are driven by top quark pole mass and the mass of the SM Higgs boson [11]. The current data are in significant tension with the stability hypothesis, making it more likely that the universe is in a false vacuum state [12,13,14,15]. The expected lifetime of vacuum decay to a true vacuum is extraordinarily long, and it is unlikely to affect the evolution of the universe [16,17]. However, it is unclear why the vacuum state entered into a false vacuum to begin with during the early universe. In this post-SM era, the emergence of vacuum stability problems (among many others) forces the particle theorists to expand the SM in such a way that the ${\lambda }_H$ will stay positive during the run all the way up to the Planck scale.

It is possible that at or below the instability scale, heavy degrees of freedom originating from a theory beyond the SM start to alter the running of the SM parameters of renormalization group equations (RGE). It has been shown that incorporating the Type-I seesaw mechanism [18,19,20,21,22,23,24,25,26,27,28] will have a large destabilizing effect if the neutrino Yukawa couplings are large [29], and an insignificantly small effect if they are small. Thus, to solve the vacuum stability problem simultaneously with neutrino mass, a larger theory extension is required. Embedding the invisible axion model [30,31,32] together with the Type-I seesaw was considered in [33,34]. The axion appears as a phase of a complex singlet scalar field. This approach aims to solve the vacuum stability problem by proving that the universe is currently in a true vacuum. The scalar sector of such a theory may stabilize the vacuum with a threshold mechanism [35,36]. The effective SM Higgs coupling gains a positive correction $\delta \equiv {\lambda }_{H\sigma }^2/{\lambda }_{\sigma }$ at $m_{\rho }$, where ${\lambda }_{H\sigma }$ is the Higgs doublet-singlet portal coupling and ${\lambda }_{\sigma }$ is the quartic coupling of the new scalar.

Corrections altering ${\lambda }_H$ in such a model would also induce corrections to the triple Higgs coupling, ${\lambda }_{HHH}^{\mathrm{tree}}=3m_H^2/v$, where $v=246.22$ GeV is the SM Higgs vacuum expectation value (VEV) [37,38,39]. The triple Higgs coupling is uniquely determined by the SM but is unmeasured. In fact, the Run 2 data from the Large Hadron Collider (LHC) has only been able to determine the upper limit of the coupling to be 12 times the SM prediction [3]. Therefore, future prospects of measuring a deviation of triple Higgs coupling by the high-luminosity upgrade of the LHC (HL-LHC) [40,41] or by a planned next-generation Future Circular Collider (FCC) [42,43,44,45,46,47,48,49] give us hints of the structure of the scalar sector of a beyond-the-SM theory. Previous work has shown that large corrections to triple Higgs coupling might originate from a theory with one extra Dirac neutrino [50,51], inverse seesaw model [52], two Higgs doublet model [38,39,53,54], one extra scalar singlet [37,55,56] or in the Type II seesaw model [57].

The complex singlet scalar, and consequently, the corresponding threshold mechanism, is embedded in a recent SMASH [58,59,60] theory, which utilizes it at ${\lambda }_{H\sigma }\sim -{10}^{-6}$ and ${\lambda }_{\sigma }\sim {10}^{-10}$. The mechanism turns out to be dominant unless the new Yukawa couplings of SMASH are $\mathcal{O}\left(1\right)$. In addition to its simple scalar sector extension, SMASH includes electroweak singlet quarks Q and $\overline{Q}$ and three heavy right-handed Majorana neutrinos $N_1$, $N_2$ and $N_3$ to generate masses for neutrinos.

The structure of this paper is as follows: In Section 2, we summarize the SMASH model and cover the relevant details of its scalar sector. We also establish the connection between the threshold correction and the leading order ${\lambda }_{HHH}$ correction. In Section 3, we discuss the methods, numerical details, RGE running, and our choice of benchmark points. Our results are presented in Section 4, where the viable parameter space is constrained by various current experimental limits. In SMASH, one can obtain at most ∼5% correction to ${\lambda }_{HHH}$ while simultaneously stabilizing the vacuum. We give our short conclusions on Section 5.

2. Theory

The SMASH framework [58,59,60] expands the scalar sector of the SM by introducing a complex singlet field

$$\sigma =\frac{1}{\sqrt{2}}\left(v_{\sigma }+\rho \right)e^{iA/v_{\sigma }},$$

(1)

where $\rho$ and A (the axion) are real scalar fields, and $v_{\sigma }\gg v$ is the VEV of the complex singlet. The scalar potential of SMASH is then

$$\begin{array}{ccc}\hfill V(H,\sigma )& =& {\lambda }_H{\left(H^{†}H-\frac{v^2}{2}\right)}^2+{\lambda }_{\sigma }{\left({\left|\sigma \right|}^2-\frac{v_{\sigma }^2}{2}\right)}^2\hfill \\ & & +2{\lambda }_{H\sigma }\left(H^{†}H-\frac{v^2}{2}\right)\left({\left|\sigma \right|}^2-\frac{v_{\sigma }^2}{2}\right).\hfill \end{array}$$

(2)

Defining ${\varphi }_1=H$ and ${\varphi }_2=\sigma$, in basis ($H,\sigma$), the scalar mass matrix of this potential is

$$\begin{array}{cc}\hfill {\left(M_{ij}\right)}_{\mathrm{scalar}}& =\frac{1}{2}\frac{{\partial }^{2}V}{\partial {\varphi }_i\partial {\varphi }_j}\left|{}_{\begin{array}{c}H=v/\sqrt{2},\\ \sigma =v_{\sigma }/\sqrt{2}\end{array}}\right.=\left(\begin{array}{cc}2{\lambda }_H{v}^2& 2{\lambda }_{H\sigma }v{v}_{\sigma }\\ 2{\lambda }_{H\sigma }v{v}_{\sigma }& 2{\lambda }_{\sigma }{v}_{\sigma }^2\end{array}\right),\end{array}$$

(3)

which has eigenvalues

$$m_H^2=v^2{\lambda }_H+v_{\sigma }^2{\lambda }_{\sigma }-\sqrt{v^4{\lambda }_H^2+4v^2{v}_{\sigma }^2{\lambda }_{H\sigma }^2-2v^2{v}_{\sigma }^2{\lambda }_H{\lambda }_{\sigma }+v_{\sigma }^4{\lambda }_{\sigma }^2},$$

(4)

and

$$m_{\rho }^2=v^2{\lambda }_H+v_{\sigma }^2{\lambda }_{\sigma }+\sqrt{v^4{\lambda }_H^2+4v^2{v}_{\sigma }^2{\lambda }_{H\sigma }^2-2v^2{v}_{\sigma }^2{\lambda }_H{\lambda }_{\sigma }+v_{\sigma }^4{\lambda }_{\sigma }^2}.$$

(5)

At the heavy singlet limit ${\lambda }_{\sigma }{v}_{\sigma }^2\gg {\lambda }_H{v}^2$

$$m_H^2=2v^2\left({\lambda }_H-\frac{{\lambda }_{H\sigma }^2}{{\lambda }_{\sigma }}\right)+\mathcal{O}\left(\frac{v^2}{v_{\sigma }^2}\right),$$

(6)

and

$$m_{\rho }^2=2v_{\sigma }^2{\lambda }_{\sigma }-2v^2\frac{{\lambda }_{H\sigma }^2}{{\lambda }_{\sigma }}+\mathcal{O}\left(\frac{v^4}{v_{\sigma }^2}\right).$$

(7)

Defining threshold correction $\delta \equiv {\lambda }_{H\sigma }^2/{\lambda }_{\sigma }$ in Equation (13),

$$m_H^2\approx 2v^2({\lambda }_H-\delta )\equiv 2v^2{\lambda }_H^{\mathrm{SM}},$$

(8)

and

$$m_{\rho }^2\approx 2v_{\sigma }^2{\lambda }_{\sigma }-2v^2\delta .$$

(9)

The first term in the Equation (9) is the leading component.

The SMASH framework also includes a new quark-like field, Q, which has color but is an electro-weak singlet. It gains its mass via the Higgs mechanism, through a complex singlet $\sigma$. It arises from the Yukawa term

$${\mathcal{L}}_Q^Y=Y_Q\overline{Q}\sigma Q\Rightarrow m_Q\approx \frac{Y_Q{v}_{\sigma }}{\sqrt{2}}.$$

(10)

We will show later that $Y_Q=\mathcal{O}\left(1\right)$ is forbidden by the vacuum stability requirement. The hypercharge of Q is chosen to be $q=-1/3$, even though $q=2/3$ is possible. Our analysis is almost independent of the hypercharge assignment.

Threshold correction: Consider an energy scale below $m_{\rho }<{\mathsf{\Lambda }}_{\mathrm{IS}}$, where the heavy scalar $\rho$ is integrated out. The low-energy Higgs potential should match the SM Higgs potential

$$V\left(H\right)={\lambda }_H^{\mathrm{SM}}{\left(H^{†}H-\frac{v^2}{2}\right)}^2.$$

(11)

It turns out that the quartic coupling we measure has an additional term

$${\lambda }_H^{\mathrm{SM}}={\lambda }_H-\frac{{\lambda }_{H\sigma }^2}{{\lambda }_{\sigma }}.$$

(12)

Since the SM Higgs quartic coupling will be approximately ${\lambda }_H\left(M_{Pl}\right)\approx -0.02$, the threshold correction

$$\delta \equiv \frac{{\lambda }_{H\sigma }^2}{{\lambda }_{\sigma }}$$

(13)

should have a minimum value close to $|{\lambda }_H\left(M_{Pl}\right)|$ or slightly larger to push the high-energy counterpart ${\lambda }_H$ to positive value all the way up to $M_{Pl}$. A too-large correction will, however, increase ${\lambda }_H$ too rapidly, exceeding the perturbativity limit $\sqrt{4\pi }$. We demonstrate the conditions for $\delta$ in Section 4. Similar to ${\lambda }_H$, the SM Higgs quadratic parameter ${\mu }_H$ gains a threshold correction

$${\left({\mu }_H^{\mathrm{SM}}\right)}^2={\mu }_H^2-\frac{{\lambda }_{H\sigma }}{{\lambda }_{\sigma }}{\mu }_{\sigma }^2.$$

(14)

In the literature [35,36], there are two possible ways of implementing this threshold mechanism. One may start by solving the SM RGE’s up to $m_{\rho }$, where the new singlet effects kick in, and the quadratic and quartic couplings gain sudden increments. Continuation of RGE analysis to even higher scales then requires utilizing the new RGE’s up to the Planck scale.

Another approach is to only solve the new RGEs on the SM scale while ignoring the low-scale SM RGEs entirely. We will use the former approach.

One-loop correction to triple Higgs coupling: The portal term of the Higgs potential contains the trilinear couplings for $HH\rho$ and $H\rho \rho$ vertices. The vertex factors for $HH\rho$ and $H\rho \rho$ vertices are introduced in Figure 1. The one-loop diagrams contributing to SM triple Higgs coupling are in Figure 2. We denote the SM tree-level triple Higgs coupling as ${\lambda }_{HHH}$. The correction is gained by adding all the triangle diagrams (taking into account the symmetry factors)

$$\begin{array}{ccc}\hfill \mathsf{\Delta }{\lambda }_{HHH}& =& \left(2^2·{\lambda }_{HHH}{\lambda }_{H\sigma }^2{v}_{\sigma }^{2}I(m_H,m_H,m_{\rho };p,q)+2\mathrm{permutations}\right)\hfill \\ & & +\left(2^3·{\lambda }_{H\sigma }^{3}v{v}_{\sigma }^{2}I(m_H,m_{\rho },m_{\rho };p,q)+2\mathrm{permutations}\right)\hfill \\ & & +2^3·{\lambda }_{H\sigma }^3{v}^{3}I(m_{\rho },m_{\rho },m_{\rho };p,q).\hfill \end{array}$$

(15)

Here, p and q are the external momenta and the loop integral is defined as

$$I(m_A,m_B,m_C;p,q)=$$

$$\int \frac{d^{4}k}{{\left(2\pi \right)}^4}\frac{1}{(k^2-m_A^2)({(k-p)}^2-m_B^2)({(k+q)}^2-m_C^2)}.$$

(16)

The contribution from diagram Figure 3 is subleading, since it is proportional to ${\lambda }_{H\sigma }^{2}v\Rightarrow \delta {\lambda }_{\sigma }v\ll {\lambda }_{HHH}$.

The process $H^{*}\to HH$ is disallowed for on-shell external momenta, so at least one of them must be off-shell. Specifically, the momentum-dependent correction to the triple coupling at the tree-level is an effective coupling that enters the specific process with one off-shell Higgs decaying into two real Higgses. Note that the correction is dependent on the Higgs off-shell momentum $q\equiv q^{*}$, which we assume to be at $\mathcal{O}\left(1\right)$ TeV at the LHC and HL-LHC. The first diagram is dominant due to the heaviness of the $\rho$ scalar. Therefore, we may ignore the subleading contributions of diagrams involving two or more $\rho$ propagators. We integrate out the heavy scalar, causing the finite integral in Equation (16) to be logarithmically divergent. We calculate the finite part of it using dimensional regularization and obtain

$$\begin{array}{ccc}\hfill \mathsf{\Delta }{\lambda }_{HHH}& =& -4{\lambda }_{HHH}{\left(\frac{v_{\sigma }}{m_{\rho }}\right)}^2\left(\frac{{\lambda }_{H\sigma }^2}{16{\pi }^2}\right)\left(2+ln\frac{{\mu }^2}{m_H^2}-zln\frac{z+1}{z-1}\right)\hfill \\ & \simeq & -2{\lambda }_{HHH}\left(\frac{\delta }{16{\pi }^2}\right)\left(2+ln\frac{{\mu }^2}{m_H^2}-zln\frac{z+1}{z-1}\right),\hfill \end{array}$$

(17)

where $z\equiv \sqrt{1+(4m_H^2/q^2)}$ and $\mu =m_{\rho }$ is the regularization scale1. We have used the modified minimal subtraction scheme ($\overline{\mathrm{MS}}$), where the terms $ln4\pi$ and Euler–Mascheroni constant ${\gamma }_E\approx 0.57722$ emerging in the calculation are absorbed to the regularization scale $\mu$. For calculations, we use the value $q^{*}=1$ TeV. It is especially interesting to see that at the leading order, the triple Higgs coupling correction is proportional to the threshold corrections. This intimate connection forbids a too-large correction. In fact, the bound from vacuum stability turns out to constrain the triple Higgs coupling correction to ≲$5\%$, as we shall see in Section 4. Consequently, if LHC or HL-LHC manages to measure a correction to ${\lambda }_{HHH}$, this will rule out theories that utilize exclusively threshold correction mechanisms as a viable solution to the vacuum stability problem. Indeed, there are alternate ways to produce large $\mathsf{\Delta }{\lambda }_{HHH}$ without expanding the scalar sector [50,52].

It should be noted that loop corrections contributing to the final to-be-observed value are included in the SM. Indeed, experiments are measuring ${\lambda }_{HHH}^{\mathrm{SM}}={\lambda }_{HHH}^{\mathrm{SM}\left(\mathrm{tree}\right)}+{\lambda }_{HHH}^{\mathrm{SM}(1-\mathrm{loop})}\left(q^{*}\right)+\dots$, where the SM one-loop correction depends on the Higgs off-shell momentum. At the $\mathcal{O}\left(1\right)$ TeV scale we are considering, the SM 1-loop correction amounts to approximately $-7\%$ [50].

Light neutrino masses: The neutrino sector of SMASH is able to generate correct neutrino masses and observe the baryon asymmetry of the universe with suitable benchmarks. The relevant Yukawa terms for neutrinos in the model are

$${\mathcal{L}}_{\nu }^Y=-\frac{1}{2}{Y}_n^{ij}\sigma N_i{N}_j-Y_{\nu }^{ij}{L}_i\epsilon H{N}_j.$$

(18)

We take a simplified approach: Dirac and Majorana Yukawa matrices ($Y_{\nu }$ and $Y_n$, respectively) are assumed to be diagonal.

$$Y_{\nu }=\left(\begin{array}{ccc}{y}_1& 0& 0\\ 0& y_2& 0\\ 0& 0& y_3\end{array}\right),Y_n=\left(\begin{array}{ccc}{Y}_1& 0& 0\\ 0& Y_2& 0\\ 0& 0& Y_3\end{array}\right).$$

(19)

To generate baryonic asymmetry in the universe, SMASH utilizes the thermal leptogenesis scenario [61], which generates lepton asymmetry in the early universe and leads to baryon asymmetry. In the scenario, heavy neutrinos require a sufficient mass hierarchy [62,63] and one or more Yukawa couplings must have complex CP phase factors. We assume the CP phases are $\mathcal{O}\left(1\right)$ radians to near-maximize the CP asymmetry [64,65,66]

$${\epsilon }_{\mathrm{CP}}=\frac{\mathsf{\Gamma }(N_1\to H+{\ell }_L)-\mathsf{\Gamma }(N_1\to H^{†}+{\ell }_L^{†})}{\mathsf{\Gamma }(N_1\to H+{\ell }_L)+\mathsf{\Gamma }(N_1\to H^{†}+{\ell }_L^{†})}\lesssim \frac{3M_1{m}_3}{16\pi v^2}.$$

(20)

If the CP violation is maximal, the largest value is obtained. To produce matter–antimatter asymmetry in the universe, a large asymmetry is required. Following [58], we set the heavy neutrino mass hierarchy at $M_3=M_2=3M_1$, corresponding to $Y_3=Y_2=3Y_1$. These choices give the full $6\times 6$ neutrino mass matrix

$$M_{\nu }=\left(\begin{array}{cc}{\mathbf{0}}_{3\times 3}& m_D\\ m_D^T& M_M\end{array}\right),$$

(21)

which is in block form, and contains two free parameters: $v_{\sigma }$ and $Y_1$. Here, $m_D=Y_{\nu }v/\sqrt{2}$ is the Dirac mass term and $M_M=Y_n{v}_{\sigma }/\sqrt{2}$ is the Majorana mass term. Light neutrino masses are then generated via the well-known Type I seesaw mechanism [18,19,20,21,22,23,24,25,26,27,28], by block diagonalizing the full neutrino mass matrix $M_{\nu }$.

It is possible to obtain light neutrino masses consistent with experimental constraints from atmospheric and solar mass splittings $\mathsf{\Delta }{m}_{32}^2$ and $\mathsf{\Delta }{m}_{21}^2$ [67] and cosmological constraint $m_1+m_2+m_3<0.12$ eV [68,69,70,71,72,73] (corresponding to $m_1\lesssim 0.03$ (0.055) eV with normal (inverse) neutrino mass ordering, from Equations (10)–(12) and Figure 1 of [74] for upper bound), assuming the standard $\mathsf{\Lambda }$CDM cosmological model [74,75,76,77,78]. However, the total mass $m_1+m_2+m_3$ should not be less than 0.06 (0.10) eV for normal (inverse) hierarchy as per Equation (13) of [74].

The light neutrino mass matrix is

$$m_{\nu }=-\frac{v^2}{\sqrt{2}{v}_{\sigma }}{Y}_{\nu }{Y}_n^{-1}{Y}_{\nu }^T.$$

(22)

After removing the irrelevant sign via field redefinition,

$$\begin{array}{ccc}\hfill m_{\nu }& =& C\left(\begin{array}{ccc}{y}_1^2/Y_1& 0& 0\\ 0& y_2^2/Y_2& 0\\ 0& 0& y_3^2/Y_3\end{array}\right)\hfill \\ & =& \left(\begin{array}{ccc}{m}_1& 0& 0\\ 0& \sqrt{m_1^2+\mathsf{\Delta }{m}_{31}^2}& 0\\ 0& 0& \sqrt{m_1^2+\mathsf{\Delta }{m}_{21}^2+\mathsf{\Delta }{m}_{32}^2}\end{array}\right),\hfill \end{array}$$

(23)

where we have denoted $C=v^2/\left(\sqrt{2}{v}_{\sigma }\right)$ and assumed normal mass ordering $m_1<m_2<m_3$. This gives the neutrino masses $m_i=C{y}_i^2/Y_i$. We do not know the absolute masses, but the mass squared differences have been measured by various neutrino oscillation experiments [67,79]. Nevertheless, their values provide two constraints, leaving three free parameters. However, the heavy neutrino Yukawa couplings $Y_i$ must be no larger than $\mathcal{O}\left({10}^{-3}\right)$ to avoid vacuum instability [59].

In addition, an order-of-magnitude estimate of the generated matter–antimatter asymmetry (baryon-to-photon ratio) is directly proportional to the CP asymmetry

$$\eta \equiv \frac{n_B}{n_{\gamma }}=\mathcal{O}\left({10}^{-2}\right){\epsilon }_{\mathrm{CP}}\kappa ,$$

(24)

where $\kappa \sim$ 0.01–0.1 is an efficiency factor. We arrive at

$$\eta =\mathcal{O}\left({10}^{-10}\right)\times \frac{v_{\sigma }}{{10}^8\mathrm{GeV}}\times \frac{Y_1}{{10}^{-2}}\times \frac{\kappa }{0.1},$$

(25)

which, in principle, can be consistent with the observed $\eta$. To achieve successful resonance leptogenesis, $v_{\sigma }$ should be between ${10}^{10}$ and ${10}^{12}$ GeV (

Table 1. Used benchmark points (BP) in our analysis. Note that we assume specific texture to right-handed neutrino Yukawa matrix $Y_n$.

Benchmarks	BP1	BP2	BP3
$y_1$	$1.118\times {10}^{-7}$	$1.312\times {10}^{-5}$	$9.610\times {10}^{-6}$
$y_2$	$7.754\times {10}^{-4}$	$5.347\times {10}^{-4}$	$1.893\times {10}^{-3}$
$y_3$	$1.878\times {10}^{-3}$	$1.309\times {10}^{-3}$	$4.582\times {10}^{-3}$
$Y_1$	$9.947\times {10}^{-3}$	$9.614\times {10}^{-3}$	$8.423\times {10}^{-3}$
$Y_Q$	${10}^{-3}$	${10}^{-3}$	${10}^{-3}$
$v_{\sigma }$ (GeV)	${10}^{11}$	$5\times {10}^{10}$	$7\times {10}^{11}$
${\lambda }_{\sigma }$	$7.2\times {10}^{-9}$	$4.48\times {10}^{-7}$	$2.48\times {10}^{-7}$
${\lambda }_{H\sigma }$	$-3\times {10}^{-5}$	$-2.25\times {10}^{-4}$	$-1.67\times {10}^{-4}$

). We will provide suitable benchmark points in the next section. The estimation of lepton asymmetry, which is one of the crucial implications of SMASH as the framework claims to solve the matter-asymmetry issue since the scenario only consists of the decay and inverse decay of $N_2$ or $N_3$ to $N_1$. The leptogenesis evolution for the benchmark values shown in

Table 2. The computed values of neutrino masses for normal hierarchy ($m_1<m_2<m_3$), the sum of light neutrino masses, and light neutrino mass squared differences. These neutrino masses are within experimental limits [67,68,69,70,71,72,73,79]. meV is Milli(${10}^{-3}$) (symbol m) eV.

Benchmarks	BP1	BP2	BP3	Experimental Values
$m_1$ (meV)	$5.39\times {10}^{-7}$	0.015	$6.71\times {10}^{-4}$	≲55 (Equations (10) & (11) and Figure 1 of [74]
$m_2$ (meV)	8.64	8.50	8.68	with mass bound from [68])
$m_3$ (meV)	50.67	50.93	50.88	≲60 (Equation (12) and Figure 1 of [74]
				with mass bound from [68])
$m_1+m_2+m_3$ (meV)	59.30	59.45	59.57	<120 [68,70]
				but, ⪆60 (Equation (13) of [74])
$\mathsf{\Delta }{m}_{21}^2$ (${10}^{-5}$ eV${}^2$)	7.46	7.22	7.54	6.79–8.0 [67,79]
$|\mathsf{\Delta }{m}_{32}^2|$ (${10}^{-3}$ eV${}^2$)	2.57	2.59	2.59	2.412–2.625 [67,79]
$M_1$ (GeV)	$7.03\times {10}^8$	$3.40\times {10}^8$	$4.17\times {10}^9$	Unknown
$M_2,M_3$ (GeV)	$2.11\times {10}^9$	$1.02\times {10}^9$	$1.25\times {10}^{10}$

is in Figure 4.

We will investigate the influence of $N_1$, $N_2$, and $N_3$ oscillations (i.e., right-handed neutrino oscillations) on leptogenesis evolutions, predict baryon-to-photon ratios for different set masses of light active left-handed neutrinos, and evaluate a more precise value of $\kappa$ by solving complicated Boltzmann equations in the future course of analysis in the SMASH framework.

3. Methods

We generate the suitable benchmark points demonstrating different physics aspects of the model in the neutrino sector by fitting in the known neutrino mass squared differences $\mathsf{\Delta }{m}_{ij}^2$, assuming normal mass ordering $(m_1<m_2<m_3)$. This leaves three free neutrino parameters, the values of which we generate by logarithmically distributed random sampling. These are the candidates for benchmark points. We then require that the candidate points be consistent with the bounds for the sum of light neutrino masses [68,69,70,71,72,73,74,75,76,77,78]. The next step is to choose suitable values for other unknown parameters, using the stability of the vacuum as a requirement.

The authors of [58] have generated the corrections to the two-loop $\beta$ functions of SMASH. We solve numerically the full two-loop 14 coupled renormalization group differential equations with SMASH corrections with respect to Yukawa ($Y_u,Y_d,Y_e,Y_{\nu },Y_n,Y_Q$), gauge ($g_1,g_2,g_3$) and scalar couplings (${\mu }_H^2,{\mu }_S^2,{\lambda }_H,{\lambda }_{\sigma },{\lambda }_{H\sigma }$), ignoring the light SM degrees of freedom, from $M_Z$ to Planck scale. We assume Yukawa matrices are on a diagonal basis, with the exception of $Y_{\nu }$. We use the $\overline{\mathrm{MS}}$ scheme for the running of the RGE’s. Since the top quark $\overline{\mathrm{MS}}$ mass is different from its pole mass, the difference is taken into account via the relation [80]

$$m_t^{\mathrm{pole}}\approx m_t^{\overline{\mathrm{MS}}}\left(1+0.4244{\alpha }_3+0.8345{\alpha }_3^2+2.375{\alpha }_3^3+8.615{\alpha }_3^4\right),$$

(26)

where ${\alpha }_3\equiv g_3^2/4\pi \approx 0.1085$ at $\mu =m_Z$. We define the Higgs quadratic coupling as ${\mu }_H=m_H/\sqrt{2}$ and quartic coupling as ${\lambda }_H=m_H^2/2v^2$.

We use MATLAB R2019’s ode45-solver. See Table 1 for the used SMASH benchmark points and

Table 3. Used SM inputs in our analysis, at $\mu =m_Z=91.18$ GeV, with the exception of top mass, which is evaluated at $\mu =m_t$. Masses and vacuum expectation values are in GeV units [3].

Parameter	${\mathit{m}}_{\mathit{t}}^{\overline{\mathbf{MS}}}\left({\mathit{m}}_{\mathit{t}}\right)$	${\mathit{m}}_{\mathit{b}}$	${\mathit{m}}_{\mathit{H}}$	${\mathit{m}}_{\mathit{\tau }}$	v	${\mathit{g}}_1$	${\mathit{g}}_2$	${\mathit{g}}_3$	${\mathit{\lambda }}_{\mathit{H}}$
Value	164.0	4.18	125.25	1.777	246.22	0.357	0.652	1.221	0.126

for our SM input [3]. Our scale convention is $t\equiv {log}_{10}\mu /$ GeV.

In some papers, the running of SM parameters ($Y^t,Y^b,Y^{\tau },g_1,g_2,g_3,{\mu }_H^2,{\lambda }_H$) obeys the SM RGE’s without corrections from a more effective theory until some intermediate scale ${\mathsf{\Lambda }}_{\mathrm{BSM}}$ [35], after which the SM parameters gain threshold correction (where it is relevant) and the running of all SM parameters follows the new RGE’s from that point onwards. We choose to utilize this approach while acknowledging an alternative approach, where the threshold correction is applied at the beginning ($\mu =m_Z$) [36], and both approaches give almost the same results. As previously stated, SM Higgs quadratic and quartic couplings will gain the threshold correction.

Our aim is to find suitable benchmark points, which:

• Allow the quartic and Yukawa couplings of the theory to remain positive and perturbative up to the Planck scale;
• Utilize a threshold correction mechanism to ${\lambda }_H$ via $\delta \simeq 0.1$;
• Avoid the overproduction of dark radiation via the cosmic axion background (requiring ${\lambda }_{H\sigma }<0$);
• Produce a significant contribution matter–antimatter asymmetry via leptogenesis (requiring hierarchy between the heavy neutrinos);
• Produce a ∼5% correction to triple Higgs coupling ${\lambda }_{HHH}$.

4. Results

Stability of vacuum: We have plotted how the running of the SM quartic coupling, ${\lambda }_H$ changes with each benchmark point in Figure 5. Note that all the threshold corrections are utilized well before the SM instability scale ${\mathsf{\Lambda }}_{\mathrm{IS}}$. One can choose $v_{\sigma }>{\mathsf{\Lambda }}_{\mathrm{IS}}$ if $m_{\rho }<{\mathsf{\Lambda }}_{\mathrm{IS}}$ is ensured. This is the case with BP3.

We numerically scanned over the parameter space $m_t^{\mathrm{pole}}\in [164,182]$ GeV and $m_H\in [110,140]$ GeV to analyze vacuum stability in three different benchmark points BP1–BP3. Our results for the chosen benchmarks are in Figure 6, where the SM best fit is denoted by a red star. Clearly, the electroweak vacuum is stable with our benchmark points, and it is assigned to $m_t^{\mathrm{pole}}\simeq 172.69\pm 0.3$ GeV and $m_H\simeq 125.25\pm 0.17$ GeV [3]. For every case, we investigated the running of the quartic couplings of the scalar potential. We used the following stability conditions

$${\lambda }_H\left(\mu \right)>0,{\lambda }_{\sigma }\left(\mu \right)>0,{\lambda }_H\left(\mu \right){\lambda }_{\sigma }\left(\mu \right)>{\lambda }_{H\sigma }{\left(\mu \right)}^2,$$

(27)

and for ${\lambda }_{H\sigma }<0$ [35]

$$-{\lambda }_{H\sigma }\left(\mu \right)<\sqrt{{\lambda }_H\left(\mu \right){\lambda }_{\sigma }\left(\mu \right)}.$$

(28)

If one or more conditions are not met on the scale $\mu \in [m_Z,M_{Pl}]$, we denote this point as unstable. If any of the quartic couplings rises above $\sqrt{4\pi }$, we denote this point non-perturbative.

We chose the new scalar parameters in such a way that the threshold correction is large but allowed, $0.1<\delta <{\lambda }_H$. This changes the behavior of the coupling’s running so that after the correction, the ${\lambda }_H$increases in energy instead of decreasing, the opposite of the coupling’s running in a pure SM scenario. A too-large threshold correction will have an undesired effect, lowering the non-perturbative scale to energies lower than the Planck scale. These effects are visualized in Figure 7, where for each benchmark point kept ${\lambda }_{\sigma }$ at its designated value in Table 1. Instead, we let the portal coupling, ${\lambda }_{H\sigma }$, vary between 0 and $\sqrt{0.6{\lambda }_{\sigma }}$. This demonstrates the small range of viable parameters space.

We have also investigated the significance of $v_{\sigma }$ on the bounds of threshold correction $\delta$. A choice of $\delta$ is available as long as $v_{\sigma }\lesssim 2\times {10}^{13}$ GeV. This can be seen clearly from Figure 8. Given a fixed $\delta$, the result is independent of ${\lambda }_{H\sigma }$ and ${\lambda }_{\sigma }$. The lower and higher bound for $\delta$ increases as a function of $v_{\sigma }$. Instability bound increases, since the needed vacuum-stabilizing threshold effect increases as one approaches the SM instability scale ${\mathsf{\Lambda }}_{\mathrm{IS}}$. At $v_{\sigma }\gtrsim 2\times {10}^{13}$ GeV, the $m_{\rho }>{\mathsf{\Lambda }}_{\mathrm{IS}}$, so the quartic coupling ${\lambda }_H$ will turn negative before threshold correction is utilized. On the other hand, the non-perturbative scale increases, since, as the cutoff point $m_{\rho }$ increases, the quartic coupling ${\lambda }_H$ decreases and correspondingly, the largest possible threshold correction increases.

Our next scan was over the new quartic couplings, ${log}_{10}(-{\lambda }_{H\sigma })\in [-7,0]$ and ${log}_{10}{\lambda }_{\sigma }\in [-10,0]$. The scalar potential is stable and the couplings remain perturbative at only a narrow band, where $\delta \sim$ 0.01–0.1, see Figure 9. If one considers small $\delta$, the SM Higgs quartic coupling will decrease to near zero at $\mu =M_{Pl}$. This corresponds to a region near the left side of the stability band. In contrast, we chose our benchmarks with large $\delta$, placing it near the right side of the stability band, corresponding to the large value of ${\lambda }_H$ at $\mu =M_{Pl}$. This was a deliberate choice to maximize the correction to ${\lambda }_{HHH}$.

In addition, we have scanned the Dirac neutrino and new quark-like particle Yukawa couplings ($y_1$ and $Y_Q$, respectively) over $y_1\in [0,2]$ and $Y_Q\in [0,0.04]$, keeping $y_2$ and $y_3$ small, real2 and positive but non-zero. See Figure 10 for details corresponding to each benchmark point. There, we point to an area producing a stable vacuum. The Dirac neutrino Yukawa couplings may have a maximum value of $\mathcal{O}\left(1\right)$, but a more stringent constraint is found for $Y_Q$. It should be noted that even though, from the vacuum instability point of view, $Y_Q^{max}<y_1^{max}$, this does not imply $Y_Q<y_1$, since both are in principle free parameters. See Table 2 for computed values for neutrino masses for normal hierarchy ($m_1<m_2<m_3$) corresponding to each benchmark. Note that all BP1–BP3 produce a value of baryon-to-photon ratio comparable to experimental values and a mass of axion consistent with axion dark matter scenario, because it requires axion decay constant $f_A\equiv v_{\sigma }$ to be $\mathcal{O}\left({10}^{11}\right)$ GeV [30,31,32].

In Figure 4, we show the evolution of $N_2$ or $N_3$ abundance, as well as the lepton asymmetry generated by the CP violating decays and inverse decays of $N_2$ or $N_3$, divided by the CP asymmetry parameter ${\epsilon }_{\mathrm{CP}}$ as per [62]. The resulting lepton asymmetry is translated to baryon asymmetry via the sphaleron process with a $c_s$ fraction. We have also shown the $N_2$ or $N_3$ abundance in thermal equilibrium. The number density n of particles decreases in an expanding universe if there are now particle number-changing interactions. However, the ratio of number density n to entropy density s, that is, “abundance” $=n/s$, is constant. Changing “abundance” during the early universe thus indicates particle interactions, or in our case, $N_2$ or $N_3$ decays and inverse decays. A corresponding mass hierarchy for right-handed neutrinos implies an upper bound of ${\epsilon }_{\mathrm{CP}}\sim {10}^{-5}$ to ${10}^{-6}$ [63,81].

Correction to SM triple Higgs coupling: According to PDG [3], the largest possible experimental value for ${\lambda }_{HHH}$ is 12 times the SM prediction3, from Run 2 data for the $b\overline{b}\gamma \gamma$ channel alone. The real singlet scalar $\rho$ mixes with the SM Higgs, providing a one-loop correction to SM triple Higgs coupling ${\lambda }_{HHH}$. We scanned the parameter space with ${log}_{10}(-{\lambda }_{H\sigma })\in [-7,0]$ and ${log}_{10}{\lambda }_{\sigma }\in [-10,0]$. At each point, we calculated the correction to ${\lambda }_{HHH}$. See Figure 11 for details. We identified a section of parameter space excluded by triple Higgs coupling searches from LHC run 2 and determined the area sensitive to future experiments, namely, HL-LHC and FCC-hh. We assume HL-LHC uses 14 TeV center-of-mass energy and integrated luminosity $\mathcal{L}=3{\mathrm{ab}}^{-1}$, for FCC-hh we assume center-of-mass energy 100 TeV and integrated luminosity $\mathcal{L}=3a{b}^{-1}$. The relative correction in

Table 4. The computed values of threshold correction $\delta$, BSM scalar masses $m_A$ and $m_{\rho }$, baryon-to-photon ratio $\eta$, quartic self-couplings at $M_{Pl}$, correction to the triple Higgs coupling $\mathsf{\Delta }{\lambda }_{HHH}$ compared to the SM prediction.

Benchmarks	BP1	BP2	BP3	Experimental Values
$\delta (\mu =m_{\rho })$	0.125	0.113	0.113	None
$m_A$ (eV)	$5.7\times {10}^{-5}$	$1.1\times {10}^{-4}$	$8.1\times {10}^{-6}$	Model-dependent
$m_{\rho }$ (GeV)	$8.49\times {10}^6$	$3.34\times {10}^7$	$3.49\times {10}^8$
$\eta$	∼${10}^{-11}$	∼${10}^{-11}$	∼${10}^{-10}$	$(6.0\pm 0.2)\times {10}^{-10}$
${\lambda }_H\left(M_{Pl}\right)$	0.222	0.166	0.149	None
${\lambda }_{\sigma }\left(M_{Pl}\right)$	$5.44\times {10}^{-9}$	$4.5\times {10}^{-7}$	$2.47\times {10}^{-7}$
$\mathsf{\Delta }{\lambda }_{HHH}$	$-5$%	−5%	$-6$%	<1400%

is calculated with respect to the SM tree-level prediction. We chose our benchmark points in a way that their correction to triple Higgs coupling will be borderline observable at FCC-hh [82], that is, the correction will be ∼5%. Thus, $\eta$ in BP3 for a factor of 10 larger is necessary for stable vacuum and FCC-hh better detection shown in Figure 11. Future FCC-hh accelerator is sensitive to ∼5% deviation of the Standard Model prediction. This is demonstrated by the benchmark points we have chosen. Although the model’s stable region allows for even smaller deviations, part of the region is still accessible by FCC-hh.

This has implications for a general class of BSM theories that utilize complex singlet scalars and other new non-scalar fields. If the corrections from non-scalar contributions to SM triple Higgs and quartic couplings are tiny, any large correction to ${\lambda }_{HHH}$ (such as a discrepancy from a SM value measured by the HL-LHC) would rule out such a class of theories, including SMASH. It will be up to the HL-LHC experiment to determine whether this is the case.

5. Conclusions

We have investigated suitable benchmark scenarios for the simplest SMASH model regarding the scalars and neutrinos, constraining the new Yukawa couplings and scalar couplings via the vacuum stability and theory perturbativity requirements. The model can easily account for the neutrino sector, predicting the correct light neutrino mass spectrum while evading the experimental bounds for right-handed heavy sterile Majorana neutrinos. In [58], the authors of the SMASH model performed a one-loop RGE analysis of the model and presented the two-loop RGE’s. We have extended the analysis to two-loop to gain the increased precision needed for the combined achievement of a stabilized electroweak vacuum and a large-enough triple Higgs coupling correction to be sensitive at FCC-hh. To the best of the authors’ knowledge, this is the first report on the connection between threshold correction to ${\lambda }_H$ and one-loop correction to ${\lambda }_{HHH}$.

We found an interesting interplay between the triple Higgs coupling correction and the SM Higgs quartic coupling correction. A successful vacuum stabilization mechanism (threshold mechanism) in SMASH is consistent with small triple Higgs coupling corrections, requiring it to be at most ∼$5\%$. Since the $\mathsf{\Delta }{\lambda }_{HHH}$ is proportional to the threshold correction $\delta$, a large correction to $\mathsf{\Delta }{\lambda }_{HHH}$ inevitably leads to a large threshold correction. Detecting a ${\lambda }_{HHH}$ correction larger than ∼$35\%$ is within the sensitivity of a future high-luminosity upgrade of the LHC [40,41]. If detected, it would, therefore, rule out the simplest scalar sector of the model completely. This would force the model to develop non-minimal alternatives, such as an additional scalar doublet or triplet instead of a singlet. These alternatives were considered by the authors of the SMASH model in their recently updated study [60]. The lepton asymmetry $|\mathsf{\Delta }L/{\epsilon }_{\mathrm{CP}}|$ is around $6\times {10}^{-7}$ to ${10}^{-5}$ for the present-day scenario of the universe, which can be verified experimentally [83] at the FCC [47], LHC [84], and by the Circular Electron Positron Collider (CEPC) [85] for the SMASH framework.