Gravitational wave and collider probes of a triplet Higgs sector with a low cutoff
Abstract
We study the scalar triplet extension of the standard model with a low cutoff, preventing large corrections to the quadratic masses that would otherwise worsen the hierarchy problem. We explore the reach of LISA to test the parameter space region of the scalar potential (not yet excluded by Higgs to diphoton measurements) in which the electroweak phase transition is strongly firstorder and produces sizeable gravitational waves. We also demonstrate that the collider phenomenology of the model is drastically different from its renormalizable counterpart. We study the reach of the LHC in ongoing searches and project bounds for the HLLHC. Likewise, we develop a dedicated analysis to test the key but still unexplored signature of pairproduction of charged scalars decaying to thirdgeneration quarks: \(pp\rightarrow t\overline{b} (\overline{t}b), b\overline{b}\). These results apply straightforwardly to other extensions of the Higgs sector such as the 2HDM/MSSM.
1 Introduction
 (i)

theories of grand unification (GUT), where scalar multiplets, often transforming in the adjoint representation of the GUT group, break spontaneously the GUT symmetry. A simple example is the \(\mathbf {24}\) in SU(5), which decomposes as \((1,1)_0+(1,\mathbf {3})_0+\cdots \) under the Standard Model (SM) gauge group \(SU(3)_c\times SU(2)_L\times U(1)_Y\), therefore delivering a scalar triplet. Likewise, the \(\mathbf {45}\) representation of SO(10) contains the \(\mathbf {24}\) of SU(5) and therefore a SM triplet as well.
 (ii)

Supersymmetric (SUSY) models. As a matter of fact, the triplet extension of the MSSM is one of the simplest options to alleviate the little hierarchy problem [1, 2].
 (iii)

Composite Higgs models (CHM). The scalar sector of most CHMs is non minimal. It includes a hyperchargeless triplet in one of the two simplest cosets admitting an UV completion à la QCD in four dimensions, viz. SU(5) / SO(5) [3, 4]. Moreover, models based on \(SO(7)/G_2\) [5] provide exactly one triplet in addition to the Higgs boson.
Therefore, the phenomenology of such triplet is not dictated by the renormalizable Lagrangian. The latter has to be instead supplemented with effective operators encoding the effects of the heavier resonances (SUSY partners, composite states, etc.), which can modify drastically the dynamics of \(\Phi \). To demonstrate this, we will work under the assumption that the triplet does not get a (custodial symmetry breaking) vacuum expectation value (VEV). This limit can be naturally enforced assuming the triplet is a CPodd scalar and CP is conserved in the Higgs sector. At the renormalizable level, the Lagrangian becomes accidentally \(\mathbb {Z}_2\) symmetric, i.e. \(\Phi \rightarrow \Phi \), making the neutral component of the triplet a potential dark matter candidate. The charged components are in turn longlived. The corresponding phenomenology has been studied in Refs. [6, 7, 8]. However, the effective operators make all components decay promptly even if the cutoff is \(f\sim \) several TeV at which new resonances are out of the reach of current facilities. A much larger cutoff would introduce too large corrections also to the triplet mass, worsening the hierarchy problem.^{1} In this article, we study probes of current and future colliders to this more natural version of the inert triplet model (ITM).
The extended Higgs sector modifies also the electroweak (EW) phase transition (EWPT). Thus, we extend previous studies in this respect [10, 11] computing the reach of future detectors for gravitational waves that originate in the production, evolution and eventual collisions of bubbles of vacuum in a firstorder phase transition. The paper is organized as follows. We introduce the model in Sect. 2. We discuss the dynamics of the EWPT in Sect. 3. We explore collider signatures in Sect. 4. In Sect. 5 we propose an LHC analysis that has not been yet worked out experimentally for probing the key channel \(pp\rightarrow t\overline{b} (\overline{t}b), b\overline{b}\). We study signal and background and provide prospects for the HLLHC, namely the LHC running at a center of mass energy (c.m.e) \(\sqrt{s} = 13\) TeV with integrated luminosity \(\mathcal {L} = 3\) \(\hbox {ab}^{1}\). We conclude in Sect. 6.
2 Model
Note also that, had we assumed a CP violating trilinear term in the potential, \(\sim \kappa \Phi H^2\), this term would induce a VEV for the triplet, \(v^\prime \sim \kappa v^2/m_\Phi ^2\). The triplet could also decay into the Higgs degrees of freedom, the corresponding width scaling as \(\Gamma \sim (v^\prime /v^2)^2 m_\Phi ^3\). Given that \(v^\prime \) modifies the \(\rho \) parameter, it is bounded to be \(v^\prime \lesssim \) GeV [12]. Therefore, the corresponding decay would still be subdominant with respect to those suppressed by v / f even for \(f\sim 10\) TeV.
The elementary ITM can also get 1 / f corrections provided it is extended with new vectorlike quarks with quantum numbers \((\mathbf {3},\mathbf {3})_{2/3\,(1/3)}\), \((\mathbf {3},\mathbf {2})_{1/6}\); and/or with scalar doublets with quantum numbers \((1,\mathbf {2})_{1/2}\). The effective operator in Eq. 1 is then generated after integrating the heavy modes at the mass scale \(M\sim f\); see Fig. 2. This list exhausts the possible treelevel weaklycoupled UV completions of the ITM that fit into our phenomenological framework.
Comparison of treelevel obtained parameters (\(^0\)) versus the ones computed at one loop for different values of the three inputs
\(m_\Phi \)  \(\lambda _{H\Phi }\)  \(\lambda _\Phi \)  \((\mu _H^2)^0\)  \((\mu _\Phi ^2)^0\)  \(\lambda _H^0\)  \(\mu _H^2\)  \(\mu _\Phi ^2\)  \(\lambda _H\) 

120  1.1  1.1  7812.5  −18883.8at  0.13  9050.9  \(16600.6\)  0.127 
200  2.0  1.0  7812.5  −20516.0  0.13  8284.1  \(17069.8\)  0.125 
320  3.5  1.5  7812.5  −3503.0  0.13  5443.6  129.3  0.086 
400  4.3  0.1  7812.5  29890.6  0.13  3583.3  29765.1  0.032 
460  4.9  0.1  7812.5  63335.8  0.13  2693.4  61181.1  \(0.024\) 
3 The electroweak phase transition
At high temperatures, the EW symmetry is restored. For certain values of the model parameters, the transition between \(\langle h\rangle = 0\rightarrow \langle h \rangle = v_n\) is not smooth as in the SM, but rather first order. One example is given in Fig. 3. In this case, the EWPT proceeds in two steps. An example of a firstorder EWPT in one step is shown in Fig. 4. We will denote by \(T_n\) the nucleation temperature, namely the temperature at which the Higgs first order phase transition takes place. This is determined by the condition \(S_3/T_n \sim 100\), where \(S_3\) stands for the action of the thermal transition between vacua [16, 17].
Note that at \(T > 0\), the triplet squared term reads \(\sim \mu ^2_\Phi + T^2\), which can not be negative for any value of \(\mu ^2_\Phi > 0\). Therefore, the 2step EWPT can only occur if the triplet minimum is present at \(T=0\). Moreover, for a fixed \(m_\Phi \), there is a minimum \(\lambda _{H\Phi }\) below which \(\mu ^2_\Phi \) is not negative. Likewise, there is a maximum value of the coupling above which the potential at the triplet minimum, \(V(0, \langle \phi _0\rangle )\sim \mu _\Phi ^4/\lambda _\Phi \), is deeper than the Higgs one, the theory being therefore unstable. Altogether, they explain the bounded shape of the figure above.
Finally, let us very briefly comment on the possibility of EW baryogenesis [52, 53, 54, 55]. In our scenario, CP is violated spontaneously during the second transition in the twostep case, when both h and \(\phi _0\) change VEV and therefore the top mass acquires a CP violating phase. In related models [56] (see also Refs. [32, 57, 58]), EW baryogenesis has been shown successful provided \(c\Delta v/f \gtrsim 0.1\), with \(\Delta v\) the change in VEV during the EWPT. In our case, \(\Delta v\) can be easily \(\gtrsim 100\) GeV (see Fig. 3) and therefore \(c \Delta v/f \gtrsim 0.1\) for \(c/f\sim 1\) \(\hbox {TeV}^{1}\).
A small explicit CP violating potential \(\Delta V/T_n^4 \gg \text {H}/T_n\sim 10^{16} \), with the Hubble scale \(\text {H}\), is only needed to avoid domain wall problems [56]. In our setup, this can be triggered by a small CPviolating term in the potential, \(\sim \kappa \Phi H^2\). At leading order in \(\kappa \), it reflects in the (finitetemperature) potential as \(V\sim \kappa T^3/(4\pi )^2\). Avoiding domain walls then implies \(\kappa \gtrsim 10^{12}\) GeV.
4 Collider signatures
The scalar triplet can be produced at pp colliders in a variety of ways; see Fig. 7. The corresponding cross sections at \(\sqrt{s} = 8, 13\) TeV are given in Fig. 8. For completeness, we also provide numbers for 27 and 100 TeV centerofmass energy.
The triplet can be singly produced in \(q\overline{q}\) initiated processes. The Yukawa suppression, together with the 1 / f factor, makes the production cross section in this channel very small, though. Still, \(\phi _0\) can be singly produced in gluon fusion. In the regime \(m_\Phi < 2 m_t\), the most constraining searches are those looking for single production of \(b\overline{b}\) resonances. The most uptodate such analysis was recently released by CMS; see Ref. [66]. It is based on 35.9 \(\hbox {fb}^{1}\) of integrated luminosity collected at \(\sqrt{s} = 13\) TeV. The region of the plane \((m_\Phi , c/f)\) that is excluded by this analysis is enclosed by the solid blue line in Fig. 9. It is expected that they become a factor of \(\sqrt{3\,\text {fb}/35.9\,\text {ab}}\sim 9\) stronger at the HLLHC. The projected bound on the plane is enclosed by the dashed blue line in the same figure. For \(m_\Phi > 2m_t\), \(\phi _0\) decays mostly into \(t\overline{t}\). There are however no resonant searches for invariant masses below 500 GeV, neither at \(\sqrt{s}=8\) TeV nor \(\sqrt{s}=13\) TeV.
Moreover, the scalar triplet can be produced in association with top and bottom quarks, namely \(pp\rightarrow \phi ^{+} t \overline{b}\). For \(m_\Phi > m_t\), the most updated and constraining search is the ATLAS study of Ref. [67], which uses 36.1 \(\hbox {fb}^{1}\) of LHC data collected at 13 TeV. It combines both the semi and dileptonic channels. The limits on \(\sigma (pp\rightarrow tb\phi ^\pm )\times \mathcal {B}(\phi ^\pm \rightarrow tb)\) translate into the bounded region delimited by the green solid line in Fig. 9. A naive rescaling with the luminosity enhancement suggests that cross sections a factor of \(\sim 0.1\) smaller can be tested at the HLLHC. Translated to the plane \((m_\Phi , c/f)\), the corresponding bound is given by the region enclosed by the dashed line of the same colour.
In addition, for \(m_t > m_\Phi \), the triplet can be also produced in the decay of the top quark. Current searches for \(t\overline{t}\) production with \(t\rightarrow \phi ^\pm b, \phi ^\pm \rightarrow jj\) have been carried out in CMS at \(\sqrt{s}=8\) TeV with an integrated luminosity of \(\mathcal {L} = 19.7\) \(\hbox {fb}^{1}\) [65]. This latter reference sets an upper bound on this rare top decay of \(\mathcal {B}(t\rightarrow \phi ^\pm b, \phi ^\pm \rightarrow j j)<\) 1–2 % for \(m_\Phi \sim \) 100–160 GeV. Using Eq. 5, this constraint translates into the region enclosed by the solid red line in Fig. 9. The projected bound at the HLLHC is depicted, too.
For \(m_t<m_\Phi <2m_t\), the NC process gives rise to the final state \(t\overline{b}, \overline{t}b\). The latest analysis exploring this channel for masses below 500 GeV was performed by CMS at \(\sqrt{s} = 8\) TeV; see Ref. [69]. Unfortunately, the corresponding limits range from \(\sim 2.5\) pb (250 GeV) to \(\sim 0.5\) pb (500 GeV). No region in our parameter space can be even constrained at the HLLHC. Likewise, the CC gives \(t\overline{b}(\overline{t}b), b\overline{b}\). To the best of our knowledge, there is however no dedicated search for pair produced resonances decaying to these final states. Being this channel c / f independent, we perform a signal and background simulation of this process in Sect. 5.
For \(m_\Phi > 2m_t\), the NCs still give resonant \(t\overline{b}, b\overline{t}\). The CC channel instead results in \(t\overline{t}, t\overline{b} (b\overline{t})\). Once more, no dedicated analysis exists for this final state. (The lack of analyses sensitivity to similar final states has been also recently pointed out in Ref. [70] in the context of composite dark sectors.) However, in comparison to this one, the \(t\overline{b}(b\overline{t}),b\overline{b}\) analysis is much cleaner. Furthermore, it probes the mass range where the 2step EWPT, and therefore EW baryogenesis, can occur.
5 LHC sensitivity
EW pair production of \(t\overline{b} (\overline{t}b), b\overline{b}\) resonances occurs naturally in broadlystudied models, such as the 2HDM. Moreover, the cross section is independent of scalar to fermion couplings, provided the former decay promptly. It is therefore surprising that no experimental search has explored this channel at the LHC yet.
Top) Effective cross section in fb for the signal (for \(m_\Phi = 185\) GeV) and the backgrounds after each cut (1 – 5), as described in the text. Bottom) Effective cross section in fb after all cuts, including cut 6, for different signals and for the total background. The sensitivity at the HLLHC is also shown
Cuts  \(m_\Phi = 185\)  \(t\overline{t}+\) jets  \(t+3b\)  \(t\overline{t} b\overline{b}\)  \(W + 4b\) 

iso. lepton  72.94  96693.1  0.65632  326.72  1.327 
nr. jets  29.71  55288.5  0.6834  305.10  0.6147 
lep. top  17.69  32626.6  0.393  198.10  0.3647 
3 btags  2.3  267.4  0.07531  45.45  0.0835 
similar mass  0.93  81.1  0.0235  12.35  0.0220 
Final rec  Signal  Background  \(s/\sqrt{s+b}\)  

\(\Phi (185)\)  0.39  25.6  4.2  
\(\Phi (235)\)  0.78  33.5  7.3  
\(\Phi (285)\)  0.41  26.5  4.3  
\(\Phi (335)\)  0.22  19.0  2.7 
 1.
Exactly one isolated lepton with \(y < 2.5\) and \(p_T > 15\) GeV;
 2.
At least four jets, with \(p_T > 30\) GeV.
 3.
The invariant mass of this top is required to be within 50 GeV of the top mass;
 4.
Three btagged jets are to be found among the jets not coming from the leptonic top.
 5.
We require the reconstructed masses of \(\phi _0\) and \(\phi ^\pm \) to be similar, i.e. \(m_{\phi _0,\mathrm {rec}}  m_{\phi ^\pm ,\mathrm {rec}} \le 50\) GeV.
 6.
We reconstruct the resonances \(\phi _0\) and \(\phi ^{\pm }\) in the mass window of \(\pm 30\) and \(\pm 40\) GeV, respectively. As shown in Fig. 10 the experimental width of the resonances depends on their masses, thus, the central value of the mass window in the \((m_{\phi _0,\mathrm {rec}}, m_{\phi ^\pm ,\mathrm {rec}})\) plane has to be optimised for each \(m_\Phi \) separately.
We estimate the sensitivity at the HLLHC as \(\mathcal {S} = s/\sqrt{s+b}\), with s and b the number of signal and background events after all cuts, respectively. It ranges from 2.7 to 7.3 for \(m_\Phi \) between 185 and 340 GeV. Thus, at the LHC (\(\sqrt{s}=13\) TeV) we can probe the entire mass interval with \(3000~\mathrm {fb}^{1}\). The corresponding region in the plane \((m_\Phi , c/f)\) is therefore a vertical band, the one enclosed by the dashed orange line in Fig. 9.
6 Conclusions
Natural scalar extensions of the SM must have a low cutoff f preventing large corrections to the scalar masses. However, such models are usually studied neglecting 1 / f terms. Basing on the real triplet extension of the SM, we have highlighted that, if these terms are taken into account, the phenomenology can be drastically different. In particular, the only renormalizable interaction allowing the new scalars to decay is so suppressed by the measurement of the \(\rho \) parameter, that decays mediated by effective operators dominate.
We have studied the reach of current LHC analyses. We have found that, despite being f independent, searches for EW pairproduced charged scalars decaying to third generation quarks are absent. This is particularly surprising given that such signals appear in a plethora of new physics models, including the 2HDM/MSSM. Therefore, we have developed a dedicated analysis to probe the cleanest of these channels: \(pp\rightarrow \phi ^\pm \phi _0\rightarrow t\overline{b} (\overline{t} b), b\overline{b}\). We have shown that the whole range of masses \(\sim 185\)–340 GeV can be tested at the HLLHC.
For this analysis, we have neglected new scalar couplings to the leptons. Under the sort of Minimal Flavour Violation [77] assumed after Eq. 1, explicitly reproduced in concrete models as shown in Eq. 13, the only relevant lepton would be the tau. Still, the decay of \(\phi ^0\) into \(\tau ^+\tau ^\) would involve only a \(\sim m_\tau ^2/(m_b^2 N_c)\sim 5\,\%\) of its width; our results being effectively unaffected. If couplings to the leptons are accidentally larger, the scalar could be better seen elsewhere; see Ref. [78].
We also stress that, had he assume flavourviolating couplings \(c_{ij}^{u(d)}\), they would give rise to a plethora of signals in meson decays [79]. They could be also seen in top decays as in the singlet extension of the Higgs sector [80].
On another front, we have studied the reach of the future gravitational wave observatory LISA to the gravitational waves produced in the EWPT for certain region of the parameter space of the model. In particular, we have demonstrated that regions not yet excluded by Higgs to diphoton measurements will be testable. In this region, the EWPT proceeds mainly in one step, and therefore only one signal peak might be expected.
Finally, it is worth mentioning that in concrete models c and \(\lambda _{H\Phi }\) related; normally \(\lambda _{H\Phi }\propto c\). This is evident for example in CHMs, in which the former (latter) is induced by integrating out heavier resonances at tree level (one loop). To exhaust this point, we plot in Fig. 12 the current and future bounds on the plane \((m_{\Phi }, c)\) considering all collider searches and gravitational wave signatures for different simple assumptions on the relation \(\lambda _{H\Phi } = \lambda _{H\Phi }(c)\).
Footnotes
Notes
Acknowledgements
We would like to thank Nuno Castro, Marek Lewicki, Mariano Quirós and Carlos Tamarit for helpful discussions. MC is supported by the Royal Society under the Newton International Fellowship programme. MR is supported by Fundação para a Ciência e Tecnologia (FCT) under the Grant PD/BD/142773/2018 and also acknowledges financing from LIP (FCT, COMPETE2020Portugal2020, FEDER, POCI010145FEDER007334).
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