Gravitational wave and collider probes of a triplet Higgs sector with a low cutoff

TL;DR

This paper investigates a scalar triplet extension of the Standard Model with a low cutoff, focusing on gravitational wave signals from electroweak phase transitions and collider signatures, offering new ways to probe the model.

Contribution

It analyzes the gravitational wave signals and collider phenomenology of a low-cutoff scalar triplet model, highlighting unique signatures and experimental prospects.

Findings

LISA can probe the parameter space with strong first-order phase transitions.

Collider searches, including pair-production of charged scalars, can test the model.

The model's collider signatures differ significantly from the renormalizable case.

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 first-order 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 HL-LHC. Likewise, we develop a dedicated analysis to test the key but still unexplored signature of pair-production of charged scalars decaying to third-generation 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.

Figures

Tables

Table 1. Table 1: Comparison of tree-level obtained parameters ( 0 ) versus the ones computed at one loop for different values of the three inputs.

$m_{\Phi }$	${\lambda }_{H\Phi }$	${\lambda }_{\Phi }$	${\left({\mu }_H^2\right)}^0$	${\left({\mu }_{\Phi }^2\right)}^0$	${\lambda }_H^0$	${\mu }_H^2$	${\mu }_{\Phi }^2$	${\lambda }_H$
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	$-0.024$

Table 2. Table 2: Top) Effective cross section in fb for the signal (for m Φ = 185 subscript 𝑚 Φ 185 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 HL-LHC 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
$3b$-tags	2.3	267.4	0.07531	45.45	0.0835
similar mass	0.93	81.1	0.0235	12.35	0.0220

Equations

$$L=$$

$$\displaystyle+\frac{1}{f}\bigg{\{}\text{i}\,\overline{q_{L}^{i}}\bigg{[}c^{u}_{ij}(\tilde{H}\Phi)u^{j}_{R}+c^{d}_{ij}(H\Phi)d_{R}^{j}\bigg{]}+\text{h.c.}\bigg{\}}~{},$$

$$\tilde{H}\Phi=\bigg{(}\begin{array}[]{c}\phi_{0}h_{0}^{*}-\sqrt{2}\phi^{+}h^{-}\\ \phi_{0}h^{-}-\sqrt{2}\phi^{-}h_{0}^{*}\end{array}\bigg{)}~{},$$

$$L\supset \frac{v}{\sqrt{2}}\frac{c}{f}$$

$$\displaystyle\bigg{[}-\sqrt{2}\text{i}\phi^{-}\overline{b}(y^{t}P_{R}+y^{b}P_{L})t+\text{h.c.}\bigg{]}\bigg{\}}~{},$$

$$\Gamma \left({\varphi }_0\to q\overline{q}\right)$$

$$\Gamma \left({\varphi }^{+}\to q\overline{q^{\prime }}\right)$$

$$\Gamma \left(t\to {\varphi }^{+}b\right)$$

$$\Delta W=\mu H_1{H}_2+\lambda H_1\Sigma H_2+\frac{1}{2}{\mu }_{\Sigma }\text{tr}{\Sigma }^2,$$

$$L_{\text{SB}}=m_{4}^{2}\text{tr}\Sigma^{\dagger}\Sigma+\bigg{[}B_{\Sigma}\text{tr}\Sigma^{2}+\lambda A_{\lambda}H_{1}\Sigma H_{2}+\text{h.c.}\bigg{]}~{}.$$

$${\mu }_{\Phi }^2$$

$${\kappa }_{\Phi }$$

$${\lambda }_{H\Phi }\sim \frac{(\lambda {\mu }_{\Sigma }{)}^2}{m_2^2},\frac{c^{u(d)}}{f}\sim y_2^{u(d)}\frac{\lambda {\mu }_{\Sigma }}{m_2^2},$$

$$L\sim y_{u}\overline{q_{L}}\tilde{H}\bigg{[}1+\frac{\gamma}{f}\Phi+\mathcal{O}(1/f^{2})\bigg{]}u_{R}~{},$$

$$V=V_{\text{tree}}+\Delta V_{\text{CW}}+\Delta V_T+C.$$

$$\Delta V_{\text{CW}}$$

$$m_W^2$$

$$m_Z^2$$

$$m_{G^{0,\pm }}^2$$

$$m_{{\varphi }^{\pm }}^2$$

$$m_t^2$$

$${\mathcal{M}}^2=$$

$$\displaystyle\bigg{[}\begin{array}[]{cc}-\mu_{H}^{2}+3\lambda_{H}h^{2}+\frac{1}{2}\lambda_{H\Phi}\phi_{0}^{2}{}&\lambda_{H\Phi}h\phi_{0}\\ \lambda_{H\Phi}h\phi_{0}&\mu_{\Phi}^{2}+3\lambda_{\Phi}\phi_{0}^{2}+\frac{1}{2}\lambda_{H\Phi}h^{2}\end{array}\bigg{]}$$

$$\Delta V_T=$$

$$\frac{\partial V}{\partial h}=0,\frac{{\partial }^{2}V}{\partial h^2}=m_h^2,\frac{{\partial }^{2}V}{\partial {\varphi }_0^2}=m_{\Phi }^2.$$

$$\alpha \sim \frac{ϵ(T_n)}{35T_n^4};$$

$$\frac{\beta }{H}\sim T_n\frac{d}{dT}\frac{S_3}{T}\sim T_n\frac{\Delta (S_3/T)}{\Delta T}.$$

$$\displaystyle\Gamma(h\rightarrow\gamma\gamma)=\frac{\alpha^{2}m_{h}^{3}}{1024\pi^{3}}\bigg{\{}$$

$$\displaystyle+\lambda_{H\Phi}\frac{v}{m_{\Phi}^{2}}A_{0}(\tau_{\Phi})\bigg{\}}^{2}~{},$$

$$\bigg{|}\frac{d_{n}}{e}\bigg{|}\sim 20~{}\text{MeV}\frac{g_{3}^{3}}{(4\pi)^{4}}\frac{y_{t}^{2}v^{2}}{m_{t}^{2}m_{\Phi}^{2}}\frac{c\kappa}{f}h(m_{t}^{2}/m_{h}^{2})~{},$$

$$h(z)$$

$$=z^2{\int }_0^{1}dx{\int }_0^{1}dy\frac{x^3{y}^3(1-x)}{[zx(1-xy)+(1-x)(1-y){]}^2}.$$

$$\frac{d}{e}\sim\frac{\alpha v^{2}c}{6\pi^{3}m_{t}f}y_{t}\frac{\kappa y_{e}}{m_{\Phi}^{2}}\bigg{[}f(m_{t}^{2}/m_{h}^{2})+g(m_{t}^{2}/m_{h}^{2})\bigg{]}~{},$$

$$f(z)$$

$$g(z)$$

$$A_0(x)$$

$$A_{1/2}(x)$$

Full text

††preprint: IPPP/18/104

1. I Introduction
2. II Model
3. III The electroweak phase transition
4. IV Collider signatures
5. V LHC sensitivity
6. VI Conclusions
7. A Loop functions

I Introduction

Hyperchargeless triplet scalars $\Phi$ arise in a variety of models of new physics. They include

(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 Espinosa:1991wt ; DiChiara:2008rg .

(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. $SU(5)/SO(5)$ Vecchi:2013bja ; Ferretti:2016upr . Moreover, models based on $SO(7)/G_{2}$ Chala:2012af 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 CP-odd 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 long-lived. The corresponding phenomenology has been studied in Refs. Cirelli:2005uq ; FileviezPerez:2008bj ; Carmona:2015haa . 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. 111(The fine-tuning scales roughly as $\Delta\sim m_{h}^{2}/f^{2}$, with $m_{h}$ the Higgs mass Panico:2015jxa . $f=1$ TeV gives already $\Delta\sim 1\,\%$, and this falls below the permile level for $f>4$ TeV.) 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 Blinov:2015sna ; Inoue:2015pza computing the reach of future detectors for gravitational waves that originate in the production, evolution and eventual collisions of bubbles of vacuum in a first-order phase transition. The paper is organized as follows. We introduce the model in section II. We discuss the dynamics of the EWPT in section III. We explore collider signatures in section IV. In section V 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 HL-LHC, namely the LHC running at a center of mass energy (c.m.e) $\sqrt{s}=13$ TeV with integrated luminosity $\mathcal{L}=3$ ab*-1*. We conclude in section VI.

II Model

The Lagrangian of the CP-odd scalar triplet when it is assumed embedded in an UV theory, takes the form:

[TABLE]

where $H=(h^{+},h_{0}=(h+v)/\sqrt{2})$ and $\Phi=(\phi^{+},-\phi^{0},\phi^{-})$. We will assume flavour diagonal couplings: $c^{u(d)}_{ij}\sim cy_{u(d)}\delta_{ij}$ with $c$ a constant and $y$ Yukawa. We will comment on departures from this assumption in the conclusions. The relevant parameter is therefore the ratio $c/f$. The product $\tilde{H}\Phi$ $(H\Phi)$ stands for the doublet in the $SU(2)_{L}\times U(1)_{Y}$ decomposition $\mathbf{2}_{-(+)1/2}\times\mathbf{3}_{0}=\mathbf{2}_{-(+)1/2}+\mathbf{4}_{-(+)1/2}$. Explicitly:

[TABLE]

and analogously for $H\Phi$ upon the replacement $h_{0}^{*}\rightarrow h^{+}$ and $h^{-}\rightarrow-h_{0}$. Therefore, in the unitary gauge after EWSB, we obtain:

[TABLE]

with $v\sim 246$ GeV and the sum extends to the first and second families of quarks, that we will denote collectively by $q$. Couplings to the leptons could be also present. We neglect them in this analysis.

$\phi^{0(\pm)}$ can decay into SM quarks. Likewise, for $m_{t}>m_{\Phi}$, the top quark can decay into the triplet and a bottom quark. ($m_{t}$ stands for the top quark mass, whereas $m_{\Phi}$ is the physical mass of $\Phi$; at tree level $m_{\Phi}^{2}=\mu_{\Phi}^{2}+\lambda_{H\Phi}v^{2}/2$.) The following relations hold:

[TABLE]

In the second equation we are assuming $m_{q}\gg m_{q^{\prime}}$, which is normally the case if the former is an up quark and the later a down quark. Note that decays into the light quarks are dominant if channels involving the top quark are kinematically closed. Partial widths as a function of $m_{\Phi}$ are depicted in Fig. -659.

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 Tanabashi:2018oca . Therefore, the corresponding decay would still be subdominant with respect to those suppressed by $v/f$ even for $f\sim 10$ TeV.

This setup can be easily accommodated in a SUSY framework. The minimal model consists of the MSSM extended with a supermultiplet $\Sigma$ with quantum numbers $(1,\mathbf{3})_{0}$; see Ref. Delgado:2013zfa . The most general and renormalizable superpotential is the MSSM superpotential extended by

[TABLE]

with $H_{1}$ and $H_{2}$ the two doublet superfields. Likewise, the extra soft-breaking Lagrangian reads

[TABLE]

In the limit $\lambda\rightarrow 0$, $\lambda\mu_{\Sigma}\rightarrow\text{finite}$, the fermionic partner in $\Sigma$ decouples and the model is approximately inert with

[TABLE]

In the previous expression, $\kappa_{\Phi}$ stands for the trilinear coupling in $\kappa_{\Phi}\Phi HH_{2}$. (With a slight abuse of notation, we are denoting here by $H_{2}$ the scalar component of this superfield.) After integrating $H_{2}$ out, we get

[TABLE]

with $m_{2}$ the mass of $H_{2}$ and $y_{2}^{u(d)}$ its Yukawa couplings.

The Lagrangian above can also arise naturally in CHMs, in which both $H$ and $\Phi$ are pNGBs originated in the (approximate) symmetry breaking pattern $\mathcal{G}\rightarrow\mathcal{H}$ at a scale $f$ at which a new strong sector confines. The smallest realization of this setup relies on $SO(7)\rightarrow G_{2}$. The global symmetry is only approximate because it is explicitly broken by loops of SM gauge bosons, as well as by linear mixings between the left- and right-handed top quark fields and composite operators $\mathcal{O}_{L,R}$. The latter transform in representations of $SO(7)$. If $\mathcal{O}_{L}\sim\mathbf{35}$ and $\mathcal{O}_{R}\sim 1$, one obtains the Lagrangian Ballesteros:2017xeg

[TABLE]

and similarly for other quarks. $\gamma$ parametrises the degree of mixing of $q_{L}$ with the two doublets in the $\mathbf{35}$. (Under $G_{2}$, $\mathbf{35}=1+\mathbf{7}+\mathbf{27}$, whereas $\mathbf{7}=(\mathbf{2},\mathbf{2})+(\mathbf{3},1)$ and $\mathbf{27}=(1,1)+(\mathbf{2},\mathbf{2})+(\mathbf{3},\mathbf{3})+(\mathbf{4},\mathbf{2})+(\mathbf{5},1)$ under the custodial symmetry group $SU(2)_{L}\times SU(2)_{R}$.)

The elementary ITM can also get $1/f$ corrections provided it is extended with new vector-like 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. II is then generated after integrating the heavy modes at the mass scale $M\sim f$; see Fig. -658. This list exhausts the possible tree-level weakly-coupled UV completions of the ITM that fit into our phenomenological framework.

LEP operated at $\sqrt{s}=209$ GeV, excluding $\Phi$ masses below $\sim 100$ GeV. (This limit is however slightly model dependent; other sources of new physics could weaken it to even $\sim 75$ GeV Egana-Ugrinovic:2018roi .) We will therefore restrict our analysis to the mass range $100<m_{\Phi}<500$ GeV.

III The electroweak phase transition

The new scalar potential modifies the EWPT, which in the SM is a cross over. In the region of the parameter space where it is first order and strong, gravitational waves can be produced via nucleation and eventual collision of bubbles of symmetry-breaking vacuum. In order to explore this phenomenology, we study the evolution of the one-loop effective potential at finite temperature:

[TABLE]

$C$ is a constant fixed so that $V$ vanishes at the origin of the field space. $V_{\text{tree}}$ stands for the tree-level potential. $V_{\text{CW}}$ is the one-loop correction at zero temperature in $\overline{\text{MS}}$ and Landau gauge, namely

[TABLE]

where $i$ runs over all bosons ($+$) and fermions $(-)$. The factor $n_{i}$ denotes de number of degrees of freedom of the field $i$, while $c_{i}$ is $5/6$ for gauge bosons and $3/2$ otherwise. The field-dependent masses squared of the spectator fields are

[TABLE]

In addition, we have two more field dependent masses squared, $m_{1}^{2}$ and $m_{2}^{2}$, given by the eigenvalues of the mixing matrix

[TABLE]

Finally, the finite temperature corrections read

[TABLE]

As input parameters, we take $m_{\Phi}$, $\lambda_{H\Phi}$ and $\lambda_{\Phi}$. The remaining three parameters in the tree level potential are numerically obtained after requiring $V_{\text{tree}}+\Delta V_{\text{CW}}$ to have a extreme at $\langle h\rangle=v,\langle\phi_{0}\rangle=0$, at which the physical Higgs and $\Phi$ masses are $m_{h}\sim 125$ GeV and $m_{\Phi}$, respectively. In other words:

[TABLE]

At tree level, $(v,0)$ is guaranteed to be an extreme provided $\lambda_{\Phi},\lambda_{H\Phi}>0$ and $\mu_{\Phi}^{2}>-1/2v^{2}\lambda_{H\Phi}$. A comparison between tree and loop level values of $\mu_{H}$, $\mu_{\Phi}$ and $\lambda_{H}$ in a set of benchmark inputs can be seen in Tab. 1.

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. -657. In this case, the EWPT proceeds in two steps. An example of a first-order EWPT in one step is shown in Fig. -656. 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 Quiros:1999jp ; Laine:2016hma .

In the region enclosed by the dashed green line in the plane ($\lambda_{H\Phi},m_{\Phi})$ of Fig. -655, $v_{n}/T_{n}>1$; i.e. the phase transition is said to be strong. The nature of the strongest phase transition (one or two steps) is also labelled. The way we performed the scan is as follows: We varied $m_{\Phi}$ in the range $[100,500]$ GeV in steps of $20$ GeV. We varied $\lambda_{H\Phi}$ in the range $[0.1,10]$ in steps of $0.1$. For each pair $(m_{\Phi},\lambda_{H\Phi})$, we found the value of $\lambda_{\Phi}$ in $[0.1,0.3,\cdots,10]$ maximizing $v_{n}/T_{n}$. The points with smallest value of $\lambda_{H\Phi}$ are interpolated using straight lines. Likewise for those with largest value of this coupling. The resulting lines are further smoothed according to the bezier method using Gnuplot.

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 2–step 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.

It is also well known that strong first order phase transition, resulting from non-standard Higgs sectors, produce gravitational waves Choi:1993cv ; Ashoorioon:2009nf ; Enqvist:2014zqa ; Caprini:2015zlo ; Kakizaki:2015wua ; Schwaller:2015tja ; Jaeckel:2016jlh ; Huang:2016odd ; Hashino:2016rvx ; Tenkanen:2016idg ; Huang:2016cjm ; Artymowski:2016tme ; Huang:2017laj ; Hashino:2016xoj ; Vaskonen:2016yiu ; Chao:2017vrq ; Hindmarsh:2017gnf ; Kobakhidze:2016mch ; Beniwal:2017eik ; Kobakhidze:2017mru ; Cai:2017tmh ; Bian:2017wfv ; Huang:2017kzu ; Croon:2018erz ; Mazumdar:2018dfl ; Hashino:2018wee ; Ahriche:2018rao ; Beniwal:2018hyi ; Huang:2017rzf ; Breitbach:2018ddu . They are roughly characterized by the normalized latent heat of the phase transition

[TABLE]

(with $\epsilon(T_{n})$ the latent heat at $T_{n}$), and by the inverse duration time of the phase transition,

[TABLE]

We computed these quantities using CosmoTransitions Wainwright:2011kj . $\Delta(S_{3}/T)$ is estimated finding the two values of $T$ for which $S_{3}/T=100,200$ GeV, respectively. (Therefore, $\Delta(S/T)=100$ GeV.) We warn that, due to the rapid growth of $S_{3}/T$ with $T$, the linear estimation of the derivative can be sensibly overestimated. Given that small values of $\beta/H$ give rise to stronger gravitational waves, our results are conservative. The region of the parameter space that we estimate it can be tested by the future gravitational wave observatory LISA lies above the dotted green line in Fig. -655. The points in this are lead to $\alpha,\beta$ within the region “C1” of Ref. Caprini:2015zlo for $T_{n}=100$ GeV. (The bubble velocity is close to unity in good approximation. Also, we have neglected the effect that sounds waves might be not “long-lasting”, what could weaken the gravitational wave signal Ellis:2018mja .)

Large values of $\lambda_{H\Phi}$ can be also probed in the $h\rightarrow\gamma\gamma$ channel. Indeed, the width of the later in this case reads

[TABLE]

with $\alpha$ the electromagnetic constant and $\tau_{i}=4m_{i}^{2}/m_{h}^{2}$. The last ATLAS+CMS combined measurement of the Higgs decay into photons was provided in Ref. Sirunyan:2018koj , $\Gamma(h\rightarrow\gamma\gamma)/\Gamma(h\rightarrow\gamma\gamma)_{\text{SM}}=1.14^{+0.19}_{-0.18}$. The region in the plane $(m_{\Phi},\lambda_{H\Phi})$ that is consequently excluded at the $95$ % CL is enclosed by the solid orange line in Fig. -655. The expectation at the HL-LHC is that ratios outside the range $1.0\pm 0.1$ will be excluded ATL-PHYS-PUB-2013-014 . The corresponding region is enclosed by the dashed orange line. It is clear that, if departures from the SM prediction on the Higgs to diphoton rate are not observed, only one-step EWPT (i.e. single peak signatures) could be detected by LISA.

Finally, let us very briefly comment on the possibility of EW baryogenesis Kuzmin:1985mm ; Cohen:1993nk ; Riotto:1999yt ; Morrissey:2012db . In our scenario, CP is violated spontaneously during the second transition in the two-step case, when both $h$ and $\phi_{0}$ change VEV and therefore the top mass acquires a CP violating phase. In related models Espinosa:2011eu (see also Ref. Vaskonen:2016yiu ; Huang:2018aja ; Grzadkowski:2018nbc ), 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. -657) and therefore $c\Delta v/f\gtrsim 0.1$ for $c/f\sim 1$ 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 H, is only needed to avoid domain wall problems Espinosa:2011eu . In our setup, this can be triggered by a small CP-violating term in the potential, $\sim\kappa\Phi H^{2}$. At leading order in $\kappa$, it reflects in the (finite-temperature) potential as $V\sim\kappa T^{3}/(4\pi)^{2}$. Avoiding domain walls then implies $\kappa\gtrsim 10^{-12}$ GeV.

Let us show that this amount of CP violation evades easily neutron and electron dipole moment (EDM) constraints. Indeed, the neutron EDM arises mainly from the running of the diagram on the right panel of Fig. -654. It gives Choi:2016hro

[TABLE]

with $g_{3}$ the QCD coupling at the scale $\sim 1$ GeV, and

[TABLE]

For the values of $c/f$ and $\kappa$ stated previously we obtain $|d_{n}|\sim 10^{-38}$ e cm, much smaller than the current 90 % CL bound $2.9\times\sim 10^{-26}$ e cm Baker:2006ts .

An electron EDM will be generated mainly via two-loop diagrams as that depicted in the left panel of Fig. -654. Using the expressions of Ref. Keus:2017ioh , we find that the value of the electron EDM in our case reads

[TABLE]

with

[TABLE]

We obtain $|d_{e}|\sim 10^{-42}$ e cm, much smaller than the latest measurement by ACME Andreev:2018ayy , $d<1.1\times 10^{-29}$ e cm.

Regarding $c/f$, more stringent bounds could be set at colliders. We dedicate next section to this point.

IV Collider signatures

The scalar triplet can be produced at $pp$ colliders in a variety of ways; see Fig. -653. The corresponding cross sections at $\sqrt{s}=8,13$ TeV are given in Fig. -652. For completeness, we also provide numbers for $27$ and $100$ TeV center-of-mass 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 $m_{\Phi}<2m_{t}$, the most constraining searches are those looking for single production of $b\overline{b}$ resonances. The most up-to-date such analysis was recently released by CMS; see Ref. Sirunyan:2018ikr . It is based on $35.9$ 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. -651. It is expected that they become a factor of $\sqrt{3\,\text{fb}/35.9\,\text{ab}}\sim 9$ stronger at the HL-LHC. 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. Aaboud:2018cwk , which uses $36.1$ fb*-1* of LHC data collected at 13 TeV. It combines both the semi- and di-leptonic 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. -651. A naive rescaling with the luminosity enhancement suggests that cross sections a factor of $\sim 0.1$ smaller can be tested at the HL-LHC. 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 $\mathcal{L}=19.7$ fb*-1* Khachatryan:2015uua . This latter reference sets an upper bound on this rare top decay of $\mathcal{B}(t\rightarrow\phi^{\pm}b,\phi^{\pm}\rightarrow jj)<$ 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. -651. The projected bound at the HL-LHC is depicted, too.

Finally, irrespectively of the value of $c/f$, the scalar triplets can be always pair-produced via EW charged currents (CC), $pp\rightarrow W^{\pm(*)}\rightarrow\phi^{\pm}\phi_{0}$, as well as via neutral currents (NC), $pp\rightarrow Z/\gamma\rightarrow\phi^{+}\phi^{-}$. (Note that $\phi_{0}$ does not interact with the $Z$ boson and therefore it can not be pair-produced via NCs.) For $m_{\Phi}<m_{t}$, the new charged and neutral scalars decay mainly into $q^{\prime}\overline{q}$ and $b\overline{b}$, respectively. Searches for pair-produced dijet resonances might therefore be sensitive to this regime. The most constraining such search is the CMS analysis presented in Ref. Sirunyan:2018rlj . At this mass scale, each pair of quarks is very collimated and manifests as a single jet. The experimental analysis uses boosted techniques, including jet grooming to remove QCD radiation. The analysis considers $\mathcal{L}=35.9$ fb*-1* at 13 TeV of c.m.e. The current limits on the total cross section range from $\sim 170$ pb (100 GeV) to $\sim 20$ pb (170 GeV). Therefore, the parameter space of our model is not constrained. Furthermore, a naive rescaling with the larger luminosity shows that this analysis will not be even constraining at the HL-LHC.

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. CMS:2013gqa . 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 HL-LHC. 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 section V.

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. Kribs:2018ilo 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 2-step EWPT, and therefore EW baryogenesis, can occur.

V LHC sensitivity

EW pair production of $t\overline{b}(\overline{t}b),b\overline{b}$ resonances occurs naturally in broadly-studied 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.

A plausible explanation is that the majority of these analyses are based on the 2HDM of the MSSM. In that case, one Higgs doublet gives mass to the up fermions, while the second gives mass to the down fermions. The physical charged components then couple to the top and bottom quarks with effective $c/f\sim(y_{t}\cot{\beta}+y_{b}\tan{\beta})/v$, with $\tan{\beta}$ the ratio of the two doublet VEVs. Therefore, $c/f\geqslant 2\sqrt{y_{b}y_{t}}/v\sim 1$ TeV*-1*, the inequality being saturated at $\tan{\beta}=\sqrt{y_{t}/y_{b}}\sim 1.3$. As it can be seen in Fig. -651, this value is at the reach of $b\overline{b}$ resonant searches. Therefore, there is a priori no need for further analyses. The situation in our model and in other versions of the 2HDM is very different, though, because the coupling of the new scalars to the SM fermions can be (and typically is) very small. In the composite Higgs scenario, the effective scalar-fermion coupling can also be small, i.e. $c/f\lesssim 1~{}\text{TeV}^{-1}$, according to equation 13 for natural values of $0<\gamma\leq 1$.

With the aim of filling this gap, we perform a dedicated analysis for the process $pp\rightarrow\phi^{\pm}\phi_{0}\rightarrow t\overline{b}(\overline{t}b),b\overline{b}$. We generate the hard processes for signal and background using MadGraph v5 Alwall:2014hca and shower them to a fully hadronised final state using Pythia v8 Sjostrand:2014zea . No parton-level cuts are applied. The main backgrounds are $t\overline{t}+\text{jets}$, $t\overline{t}b\overline{b}$, $t(\overline{t})+3b$ and $W+4b$.

At the reconstruction level, a lepton is considered isolated if the hadronic energy deposit within a cone of size $R=0.3$ is smaller than 10 % of the lepton candidate’s $p_{T}$. Jets are defined by the anti-$k_{t}$ algorithm with $R=0.4$. The following cuts are imposed:

Exactly one isolated lepton with $|y|<2.5$ and $p_{T}>15$ GeV; 2. 2.

At least four jets, with $p_{T}>30$ GeV.

The longitudinal component of the missing neutrino is reconstructed using the requirement $m_{W}^{2}=(p_{l}+p_{\nu})^{2}$, with $m_{W}$ the mass of the $W$ boson. The neutrino and lepton four-momenta, $p_{\nu}$ and $p_{l}$, are then added to the jet wich gives a total invariant mass closest to the top quark mass. After this,

The invariant mass of this top is required to be within $50$ GeV of the top mass; 2. 4.

Three $b$-tagged jets are to be found among the jets not coming from the leptonic top. 3. 5.

We require the reconstructed masses of $\phi_{0}$ and $\phi^{\pm}$ to be similar, i.e. $|m_{\phi_{0},\mathrm{rec}}-m_{\phi^{\pm},\mathrm{rec}}|\leq 50$ GeV.

To decide which $b$-tagged jet is assigned to $\phi^{\pm}$ and which two are assigned to $\phi_{0}$, we compute all possible combinations and choose the one resulting in the minimum difference between $m_{\phi_{0},\mathrm{rec}}$ and $m_{\phi^{\pm},\mathrm{rec}}$. The normalized distribution of this former variable in the main background ($t\overline{t}$+jets) and in the signal for $m_{\Phi}=185$ GeV and $m_{\Phi}=310$ GeV is depicted in Fig. -650. In Fig. -649, we also show the normalized distribution of the $p_{T}$ of the reconstructed $\phi^{0}$. However, cutting on this variable is costly in cross section, but would allow to improve the ratio of signal over background further.

Finally,

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. -650 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.

The cut-flow for the signal and the relevant backgrounds is given in Tab. 2. The sensitivity estimates are conservative, as the reconstruction relies on fairly inclusive cuts using large mass windows. Further, it is likely that some of the background was counted twice, as $b$-quarks from the parton shower in $t\bar{t}$ and from the matrix element in $t\bar{t}b\bar{b}$ are both contributing to the total background. Thus, we expect that the sensitivity can be improved further using a combination of multi-variate techniques Soper:2011cr ; Soper:2012pb ; Soper:2014rya and high-$p_{T}$ final states Abdesselam:2010pt .

We estimate the sensitivity at the HL-LHC 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. -651.

VI 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 pair-produced 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 HL-LHC.

For this analysis, we have neglected new scalar couplings to the leptons. Under the sort of Minimal Flavour Violation DAmbrosio:2002vsn assumed after Eq. II, 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. Akeroyd:2018axd .

We also stress that, had he assume flavour-violating couplings $c_{ij}^{u(d)}$, they would give rise to a plethora of signals in meson decays Harnik:2012pb . They could be also seen in top decays as in the singlet extension of the Higgs sector Banerjee:2018fsx .

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. -648 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 $\lambda_{H\Phi}=\lambda_{H\Phi}(c)$.

Acknowledgments

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, COMPETE2020-Portugal2020, FEDER, POCI-01-0145-FEDER-007334).

Appendix A Loop functions

In our case, $2m_{\Phi}>m_{h}$, and therefore

[TABLE]

with

[TABLE]