Exploring the hyperchargeless Higgs triplet model up to the Planck scale
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Abstract
We examine an extension of the SM Higgs sector by a Higgs triplet taking into consideration the discovery of a Higgslike particle at the LHC with mass around 125 GeV. We evaluate the bounds on the scalar potential through the unitarity of the scattering matrix. Considering the cases with and without \(\mathbb {Z}_2\)symmetry of the extra triplet, we derive constraints on the parameter space. We identify the region of the parameter space that corresponds to the stability and metastability of the electroweak vacuum. We also show that at large field values the scalar potential of this model is suitable to explain inflation.
1 Introduction
The revelation of the Higgs boson [1, 2, 3] in 2012 at the Large Hadron Collider (LHC) confirmed the existence of all the Standard Model (SM) particles and showed the Higgs mechanism to be responsible for electroweak symmetry breaking (EWSB). So far, the LHC, operated with pp collision energy at \(\sqrt{s}\sim \) 8 and 13 TeV, has not found any signature of new physics beyond the standard model (BSM). However, various theoretical issues, such as the hierarchy problem related to the mass of the Higgs, mass hierarchy and mixing patterns in the leptonic and quark sectors, suggest the need for new physics beyond the SM. Different experimental observations, such as the nonzero neutrino mass, the baryon–antibaryon asymmetry in the Universe, the mysterious nature of dark matter (DM) and dark energy, and inflation in the early Universe indicate the existence of new physics. Moreover, the measured properties of the Higgs boson with mass \(\sim \)125 GeV are consistent with those of the scalar doublet as predicted by the SM. However, the experimental data [4] still comfortably allow for an extended scalar sector, which may also be responsible for the EWSB.
The present experimental values of the SM parameter of the Lagrangian indicate that if the validity of the SM is extended up to the Planck mass \((M_{\mathrm{Pl}}=1.2\times 10^{19}~\mathrm{GeV})\), a second, deeper minimum is located near the Planck mass such that the EW vacuum is metastable. The transition lifetime of the EW vacuum to the deeper minimum is finite \(\tau _\mathrm{EW} \sim 10^{300}\) years [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The EW vacuum remains metastable even after adding extra scalar particles to the SM, which has been discussed in Refs. [15, 16, 17, 18, 19].
In this work, we add a real hypercharge \(Y=0\) scalar triplet to the SM. In the literature, this model is termed the hyperchargeless Higgs triplet model, HTM \((Y=0)\) [20]. We consider both the neutral CPeven component of the SM doublet and the extra scalar triplet take part in the EWSB. Including radiative corrections, we check the validity of the parameters of the model up to the Planck mass \(M_{\mathrm{Pl}}\). We review various theoretical and experimental bounds of this model. In this work, we especially discuss the unitary bounds of the quartic couplings of the scalar potential. To the best of our knowledge, the unitary bounds of this model were not discussed in the literature. Next, we impose a \(\mathbb {Z}_2\)symmetry such that an odd number of scalar particles of the triplet do not couple with the SM particles. The lightest neutral scalar particle does not decay and becomes stable. This scalar field can be taken as a viable DM candidate which may fulfill the relic abundance of the Universe. In this context, it is instructive to explore whether these extra scalars can also prolong the lifetime of the Universe. In this model, we find new regions in the parameter space of this model in which the EW vacuum remains metastable. We also consider that the extra neutral scalar field (also compatible as a viable dark matter candidate) can act as an inflaton. We show that this scalar field is able to explain the inflationary observables.
A detailed study of the HTM \((Y=0)\) parameter space, which is valid up to 1 TeV, has been performed in Refs. [21]. Two different renormalization schemes, electroweak precision, and decoupling of Higgs triplet scenario have been discussed in Ref. [22]. Using the electroweak precision test (EWPT) data and a oneloop correction to the \(\rho \) parameter, the Higgs mass range has been predicted in Refs. [23, 24, 25, 26, 27]. The detailed structure of the vacuum of the scalar potential at tree level has been studied in Ref. [28]. The constraints on the parameter spaces from the recent LHC \(\mu _{\gamma \gamma }\) and \(\mu _{Z\gamma }\) data have been discussed in Ref. [29]. The LHC and future collider experiments with high luminosity can be used as an useful tool to detect these extra scalar particles through vector bosons scatterings [30]. More recently, the inert scalar triplet has been investigated in the context of dark matter direct and indirect detection [31, 32, 33]. The heavier inert fields can decay through one loop via extra Majorana fermions [34, 35]. This model has the required ingredients to realize a successful leptogenesis which can explain the matter asymmetry in the Universe [34, 35]. Multicomponent dark matter has been investigated [36, 37] in HTM with extra scalar multiplets of the SU(2) representation.
The paper is organized as follows. Section 2 starts with a detailed description of the HTM (\(Y=0\)) model. We discuss detailed constraints in Sect. 3. Considering the lightest \(\mathbb {Z}_2\)odd neutral particle as a viable DM, we analyze the scalar potential up to the Planck mass and identify regions of parameter space corresponding to the stable and metastable EW vacuum in Sect. 4. We explain inflation as well in Sect. 5. Finally we conclude in Sect. 6.
2 Model
The SM gauge symmetry, \(SU(2)_L\), prohibits direct coupling of the SM fermions with the scalar fields of the triplet. The couplings of the new scalar fields \((H,H^\pm )\) with SM fermions are generated after the EWSB. The strengths of \(H\bar{f}f\) (the f are the up, down quarks and charged leptons) are proportional to \(\sin \gamma \). The couplings \(H^+ \bar{\nu _l}l^\) and \(H^+ \bar{u}d\) are proportional to \(\sin \beta \).
3 Constraints on the hyperchargeless Higgs triplet model
The parameter space of this model is constrained by theoretical considerations like the absolute vacuum stability, perturbativity, and unitarity of the scattering matrix. In the following, we will discuss these theoretical bounds and the constraints of the Higgs to diphoton signal strength from the LHC and the electroweak precision measurements.
3.1 Vacuum stability bounds
3.2 Perturbativity bounds
3.3 Unitarity bounds
The scalar quartic couplings in the physical bases \(G^\pm ,~G^0~,H^\pm ,~h\) and H are complicated functions of \(\lambda \)’s,\(~\gamma ,~ {\beta }\). The hhhh vertex is 6(\( \lambda _1 \cos ^4\gamma + \lambda _3 \cos ^2\gamma \sin ^2\gamma + \lambda _2 \sin ^4\gamma \)). It is difficult to calculate the unitary bounds in the physical bases. One can consider the nonphysical scalar fields bases, i.e., \(G_1^\pm ,~\eta ^\pm ,~ G^0,~h^0\) and \(\eta ^0\) before the EWSB. Here the crucial point is that the Smatrix, which is expressed in terms of the physical fields, can be transformed into a Smatrix for the nonphysical fields by making a unitary transformation [44, 45].
The first \(6\times 6\) submatrix, \(\mathcal{M}_1\), corresponds to scattering processes whose initial and final states are one of \(h^0~G_1^+,~G^0~G_1^+\), \(\eta ^0~G_1^+,~h^0~G_1^+,~G^0~\eta ^+,\) and \(\eta ^0~\eta ^+\). Using the Feynman rules in Eq. (3.5), one can obtain \(\mathcal{M}_1\) \(=\) diag\((~2\lambda _1,~2\lambda _1, ~2\lambda _1\), \(~\lambda _3\), \(\lambda _3,~\lambda _3\)).
The submatrix \(\mathcal{M}_2\) corresponds to scattering processes with one of the following initial and final states: \(h^0~G^0,~G_1^+~\eta ^,~\eta ^+~G_1^, ~\eta ^0~G^0,\) and \(h^0~\eta ^0\). Similarly, one can calculate \(\mathcal{M}_2\) \(=\) diag\((~2\lambda _1,~\lambda _3, ~\lambda _3,~\lambda _3, ~\lambda _3)\).
Unitary constraints of the scattering processes demand that the eigenvalues of the Smatrix should be less than \(8\pi \).
3.4 Bounds from electroweak precision experiments
3.5 Bounds from LHC diphoton signal strength
Recently, the ATLAS [50] and CMS [51] collaborations have measured the ratio of the prediction of the diphoton rate \(\mu _{\gamma \gamma }\) of the observed Higgs to the SM prediction. The present combined value of \(\mu _{\gamma \gamma }\) is \(1.14^{+0.19}_{0.18}\) from these experiments [4].
In \(\Gamma (h\rightarrow \gamma \gamma )_\mathrm{HTM}\) (see Eq. (3.13)), a positive \(\lambda _3\) leads to a destructive interference between HT and SM contributions and vice versa. One can see from Eq. (3.13) that the contribution to the Higgs diphoton channel is proportional to \(\frac{\lambda _3}{M_{{H}^\pm }^2}\). If the charged scalar mass is greater than 300 GeV, then the contribution of \(H^\pm \) to the diphoton signal is negligibly small.
Now we present our results for the central values of the SM parameters such as the Higgs mass \(M_{h}=125.7\) GeV, the top mass \(M_t=173.1\) GeV, the Z boson mass \(M_Z=91.1876\) GeV, and the strong coupling constant \(\alpha _\mathrm{s}=0.1184\). We take the triplet vev \(v_2\), \(\lambda _4\) and the other quartic couplings \(\lambda _{1,2,3}\) as input parameters. Hence, depending on these parameters the mixing angle \(\gamma \) can vary in between 0 and \(\pi /2\). The triplet scalar masses also become arbitrarily heavy. Here, we assume that no new physics shows up below the Planck mass \(M_{\mathrm{Pl}}\). We examine the renormalization group (RG) flow of all couplings and establish bounds on the heavy scalar masses under the assumption that the parameters are valid up to the Planck mass \(M_{\mathrm{Pl}}\). In this calculation, we use the SM RGEs up to three loops [52, 53, 54, 55] and the triplet contributions up to two loops. We first calculate all couplings at \(M_t\). To find their values at \(M_t\), one needs to take into account different threshold corrections up to \(M_t\) [5, 6, 15, 16, 76, 77]. Using the RGEs, we evolve all the coupling constants from \(M_t\) to the Planck mass \(M_{\mathrm{Pl}}\). By this procedure we obtain new parameter regions which are valid up to the Planck mass \(M_{\mathrm{Pl}}\).
In Fig. 1, we show the allowed region for fixed central values of all the SM parameters. In the left panel, we present the plot for the choice of the quartic couplings \(\lambda _{2,3}=0.1\) and triplet VEV \(v_2=3\) GeV. In the right panel, we use the value of the triplet VEV \(v_2=1\) GeV. We vary the quartic coupling \(\lambda _1\) and dimensionful mass parameter \(\lambda _4\) to calculate the neutral CPeven Higgs mass \(M_{H}\), the charged Higgs mass \(M_{{H}^\pm }\) and the mixing angle \(\gamma \). These scalar masses increase, whereas the mixing angle decreases with \(\lambda _4\). We find that the EW vacuum becomes unbounded from below for \(\lambda _1\lesssim 0.128\). The theory also violates unitarity bounds for \(\lambda _1 \gtrsim 0.238\) before the Planck mass \(M_{\mathrm{Pl}}\). One can see from Fig. 1a, the allowed region becomes smaller for the larger values of heavy scalar masses. In most of the parameter space the running couplings either violate unitary or perturbativity bounds before the Planck mass \(M_{\mathrm{Pl}}\).
As \(\lambda _{2,3}\) stabilize the scalar potential, we will get a wider green region for smaller scalar masses, but this will violate the unitarity bound in the higher mass region. We find that the EW vacuum becomes unbounded from below for the values of the quartic couplings \(\lambda _1\lesssim 0.027\) and \(\lambda _{2,3} = 0.285\). We also check that the choice of the quartic couplings \(\lambda _1 \gtrsim 0.05\) and \(\lambda _{2,3} = 0.285\) will violate unitary and perturbativity bounds before the Planck mass \(M_{\mathrm{Pl}}\). One can also understand from the expressions of Eq. (2.13) that if we decrease the value of \(v_2\), the area of the allowed region from the stability, unitary and perturbativity bounds will increase. We show the plot in Fig. 1b for the choice of \(v_2=1\) GeV.
If the vacuum expectation value of the scalar triplet becomes zero, then the minimization condition of the scalar potential given in Eq. (2.8) is no longer valid. The mass parameter \(\mu _2\) becomes free and the parameter \(\lambda _4\) does not play any role in the stability analysis. In the next section, we will show the detailed stability analysis in the presence of extra \(\mathbb {Z}_2\)symmetry in this model.
4 Dark matter in HTM (\(Y=0\))
In Fig. 2b, we present the plot for the relic density as a function of the DM mass for the fixed Higgs portal coupling \(\lambda _3(M_Z)=0.10\). The lightred band is excluded from the Higgs invisible decay width [58]. There are two deep regions in the relic density band (red line). The first one is situated near the DM mass \(M_\mathrm{DM}\approx 45\) GeV. It is due the resonance of the schannel \(H H^\pm \rightarrow \text {SM fermions}\) processes, mediated by the vector bosons \(W^\pm \). The second one is situated near the DM mass \(M_\mathrm{DM}\approx M_{h}/2\) for the Higgsmediated \(H H \rightarrow \text {SM fermions}\) processes. There is another, shallower region located around the DM mass \(M_\mathrm{DM}=100\) GeV, which is due to the dominant contributions coming from \(H H^\pm , H H \rightarrow \text {gauge bosons}\) channels.
For 500 GeV, we find that the total crosssection \(\left\langle \sigma v \right\rangle \sim 10^{25}\) \(\mathrm{cm^3 s^{1}}\), so the relic density becomes \(\sim 0.01\). In this region, the dominant channels are \(H,H^\pm \rightarrow Z W^\pm , \gamma W^\pm \) (\(\sim 35\), \(\sim 10\%\)) and \(H^\pm ,H^\pm \rightarrow Z W^\pm \) (\(\sim 25\%\)). We also check that the smaller dark matter mass along with the Higgs portal coupling \(\lambda _3\) (within the perturbative limit) does alter the relic density only in the third decimal place. If we increase the DM masses, then the effective annihilation crosssection decreases. This is mainly due to the mass suppression. We get a DM relic density in the right ballpark for DM masses greater than 1.8 TeV.
One can see that the mass splitting \(\Delta M\) attains saturation for \(M_\mathrm{DM} > 700\) GeV. Hence, the relic density is mainly regulated by the Higgsmediated schannel processes, although the contributions are small. We check that the Higgs portal coupling \(\lambda _3\) can be varied in between 0 to 1 for the DM mass 1850–2200 GeV to get the right relic density. For example, we obtain the relic density \(\Omega h^2 =0.1198\) for \(\lambda _3=0.001\) and \(M_\mathrm{DM}=1894.5\) GeV. We get the same relic density for \(\lambda _3=0.8\) and \(M_\mathrm{DM}=2040\) GeV. However, the running couplings will violate the unitary and perturbativity bounds for \(\lambda _3 \gtrsim 0.6\).
The nonobservation of DM signals in direct detection experiments at XENON 100 [64, 65], LUX [66] and LUX2016 [67] put severe restrictions [33] on the Higgs portal coupling \(\lambda _3\) for a given DM mass. In this model, we check the parameter regions which are satisfying the relic density and are allowed by the recent LUX2016 [67] and XENON1T2017 [68] data.
4.1 Metastability in ITM (\(Y=0\))
As in the SM the EW vacuum is metastable, it is important to explore if ITM has any solution in its reserve. As the scalar WIMP H protected by \(\mathbb {Z}_2\)symmetry can serve as a viable DM candidate, it is interesting to explore if they help prolong the lifetime of the Universe. The effective Higgs potential gets modified in the presence of these new extra scalars.
A set of values of all quartic coupling constants at \(M_t\) and \(M_{\mathrm{Pl}}\) for \(M_\mathrm{DM}=1897\) GeV
\(\lambda _1\)  \(\lambda _2\)  \(\lambda _3\)  

\(M_{t}\)  0.127054  0.10  0.10 
\(M_{\mathrm{Pl}}\)  \(\) 0.00339962  0.267706  0.206306 
4.2 Tunneling probability
In the ITM, the additional scalar fields give a positive contribution to \(\beta _{\lambda _1}\) (see Eqs. (A1) and (A2). Due to the presence of these extra scalars, a metastable EW vacuum goes towards the stability, i.e., the tunneling probability \(\mathcal{P}_0\) becomes smaller. We first calculate the minimum value of \(\lambda _{1,\mathrm eff}\) of Eq. (4.8). Putting this minimum value in Eq. (4.10), we compute the tunneling probability \(\mathcal{P}_0\). As the stability of the EW vacuum is very sensitive to the top mass \(M_t\), we show the variation of the tunneling probability \(\mathcal{P}_0\) as a function of \(M_t\) in Fig. 4a. The right band in Fig. 4a corresponds to the tunneling probability for our benchmark point. We present \(\mathcal{P}_0\) for the SM as the left band to see the effect of the additional IT scalar. We also display 1\(\sigma \) error bands in \(\alpha _\mathrm{s}\) (lightgray) and \(M_{h}\) (lightred). One can see from this figure that the effect of \(\alpha _\mathrm{s}\) on the tunneling probability is larger than the effect of \(M_{h}\). To see the effect of the ITM parameter spaces, we plot \(\mathcal{P}_0\) as a function of the Higgs portal coupling \(\lambda _3(M_Z)\) in Fig. 4b for different choices of \(\lambda _2(M_Z)\). We keep the fixed central values of all SM parameters. Here, the DM mass \(M_\mathrm{DM}\) is also varied with \(\lambda _3\) to get the DM relic density \(\Omega h^2=0.1198\).

If \(0>\lambda _1(\Lambda _B)>\lambda _\mathrm{1,min}(\Lambda _B)\), then the vacuum is metastable.

If \(\lambda _1(\Lambda _B)<\lambda _\mathrm{1,min}(\Lambda _B)\), then the vacuum is unstable.

If \(\lambda _2<0\), the potential is unbounded from below along the H and \(H^\pm \)direction.

If \(\lambda _3(\Lambda _\mathrm{I})<0\), the potential is unbounded from below along a direction in between H and h and also \(H^\pm \) and h.
4.3 Phase diagrams
In Fig. 5a, we calculate the confidence level for our bench mark points \(M_\mathrm{DM}=1897\) GeV, \(\lambda _2(M_Z)=0.10\) and \(\lambda _3(M_Z)=0.10\) by drawing an ellipse passing through the stability line \(\lambda =\beta _{\lambda }=0\) in the \(M_t\)–\(M_{h}\) plane. If the area of the ellipse is \(\chi \) times the area of the ellipse, it represents the \(1\sigma \) error in the same plane. This factor \(\chi \) is the confidence level of the stability of EW vacuum. We develop a proper method to calculate this factor and the tangency point for the stability line. In this case, the confidence level of metastability is decreased (onesided) with \(\alpha _\mathrm{s}(M_Z)\), i.e., the EW vacuum moves towards the stability region. We obtain the similar factor in the \(\alpha _\mathrm{s}(M_Z)\)–\(M_t\) plane. In this case, the confidence level decreases with \(M_{h}\). One can see from the phase diagrams in Fig. 5 that the stable EW vacuum is excluded at 1.2 \(\sigma \) (onesided).
5 Inflation in HTM (\(Y=0\))
Observations of superhorizon ansiotropies in the CMB data, measured by various experiments such as WMAP and Planck, have established that the early Universe underwent a period of rapid expansion. This is known as inflation. This can solve a number of cosmological problems, such as the horizon problem, the flatness problem and the magnetic monopole problem of the present Universe. If the electroweak vacuum is metastable, then the Higgs is unlikely to play the role of an inflaton [84, 85, 86, 87, 88, 89, 90, 91, 92] in the SM. Therefore, extra new degrees of freedom are needed in addition to the SM ones to explain inflation in the early Universe [93, 94, 95, 96, 97, 98].
Here, we study an extension of the Higgs sector with a real triplet scalar T in the presence of large couplings \(\zeta _{h,H}\) to the Ricci scalar curvature R. This theory can explain inflation in the early Universe at the large field values in the scale invariance Einstein frame.
6 Discussion and conclusions
The measurements of the properties of the Higgslike scalar boson detected at the Large Hadron Collider on 4th July 2012 are consistent with the minimal choice of the scalar sector. But the experimental data of the Higgs signal strengths and the uncertainties in the measurement of other standard model parameters still allow for an extended scalar sector. We have taken an extra hyperchargeless scalar triplet as new physics. First, we have assumed that the extra neutral CPeven component of the scalar triplet has also participated in the EWSB. We have shown the detailed structure of the treelevel scalar potential and mixing of the scalar fields. We have also discussed the bounds on the VEV (\(v_2\)) of the neutral CPeven component of the scalar triplet from the \(\rho \)parameter. To the best of our knowledge the full expressions of the unitary bounds on the quartic couplings of the scalar potential in this model have not yet been presented in the literature. We have shown these unitary bounds in this model. As the SM gauge symmetry \(SU(2)_L\) prohibits the coupling of SM neutrinos with the neutral CPeven component (\(\eta ^0\)) of the scalar triplet, the model does not lead to neutrino masses. But the model is still interesting, as it can play a role in improving the stability of the Higgs potential. We have taken into account various threshold corrections to calculate all the couplings at \(M_t\). Then using threeloop SM RGEs and twoloop triplet RGEs, we have evolved all the couplings up to the Planck mass \(M_{\mathrm{Pl}}\). We have shown the allowed region in the \(M_{{H}^\pm }\)–\(M_{H}\) plane. We have demanded that the EW vacuum of the scalar potential remain absolutely stable and do not violate the perturbative unitarity up to the Planck mass \(M_{\mathrm{Pl}}\). We have discussed the constraints on the parameter spaces from the recent LHC \(\mu _{\gamma \gamma }\) and \(\mu _{Z\gamma }\) data. Furthermore, only a very small region of the parameter space is shown to survive on imposing the EWPT constraints.
Astrophysical observations of various kinds, such as anomalies in the galactic rotation curves and gravitational lensing effects in bullet clusters, have indicated the existence of DM in the Universe. In the ITM, the extra scalar fields are protected by a discrete \(\mathbb {Z}_2\)symmetry which ensures the stability of the lightest neutral particle. We have verified that the mass of the neutral scalar particle (H) is slightly lighter than the mass of the charged particle (\(H^\pm \)) so that the contributions coming from coannihilation between H and \(H^\pm \) play a significant role in the relic density calculation. In the low mass region, the coannihilation rates are quite high so that the dark matter density is found to be much smaller than the right relic density \(\Omega h^2=0.1198 \pm 0.0026\) of the Universe. We have obtained the relic density in the right ballpark for a DM mass greater than 1.8 TeV. In this context, we have shown how the presence of an additional hyperchargeless scalar triplet improves the stability of the Higgs potential. In this study, we have used state of the art nexttonextto leading order (NNLO) for the SM calculations. We have used the SM Higgs scalar potential up to twoloop quantum corrections which is improved by threeloop renormalization groups of the SM couplings. We have taken into account the contributions to the effective Higgs potential of the new scalars at one loop only. These contributions are improved by twoloop renormalization groups of the new parameters. In this paper, we have explored the stability of the EW minimum of the new effective Higgs potential up to the Planck mass \(M_{\mathrm{Pl}}\). We have presented new modified stability conditions for the metastable EW vacuum. We have also shown various phase diagrams in various parameter spaces to show the explicit dependence of the EW (meta)stability on various parameters. For the first time, we have identified new regions of parameter space that correspond to the stable and metastable EW vacuum, which also provides the relic density of the DM in the Universe as measured by the WMAP and Planck experiments. In the present paper, we have also shown that the extra neutral scalar field H can play a role in inflation and can serve as a dark matter candidate. The scalar potential can explain inflation for large scalar field values. We have obtained the inflationary observables as observed by the experiments.
Footnotes
 1.
For \(v_2=0\), the notation in Eq. (2.1) \( H \equiv {\eta ^{0}} \) and \( H^\pm \equiv {\eta ^{\pm }} \) are the physical scalar fields.
 2.
As the \(\beta \)function of the Higgs quartic coupling, \(\lambda _1\) contains \(\frac{6 y_t^4}{16 \pi ^2}\) (see Eq. (A1)), the values of the Higgs quartic couplings \(\lambda _{1}\) at very high energies are extremely sensitive to \(M_t\).
Notes
Acknowledgements
This work is supported by a fellowship from the University Grants Commission. This work is partially supported by a Grant from the Department of Science and Technology, India, via Grant no. EMR/2014/001177. I would like to thank Subhendu Rakshit, Amitava Raychaudhuri, Amitava Datta, Subhendra Mohanty and Girish K. Chakravarty for useful discussions.
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