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

Mikael Chalaa, Maria Ramosb and Michael Spannowskya
aInstitute of Particle Physics Phenomenology, Physics Department, Durham
University, Durham DH1 3LE, UK
bLaboratório de Instrumentação e Física Experimental de Partículas,
Departamento de Física da Universidade do Minho, Campus de Gualtar, 4710-057
Braga, Portugal

###### 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.

††preprint: IPPP/18/104

###### Contents

  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:

$\displaystyle L=$
$\displaystyle\frac{1}{2}|D_{\mu}\Phi|^{2}-\bigg{\\{}\frac{1}{2}\mu_{\Phi}^{2}|\Phi|^{2}+\frac{1}{2}\lambda_{H\Phi}|H|^{2}|\Phi|^{2}+\frac{1}{4}\lambda_{\Phi}|\Phi|^{4}\bigg{\\}}$
$\displaystyle+\frac{1}{f}\bigg{\\{}\text{i}\,\overline{q_{L}^{i}}\bigg{[}c^{u}_{ij}(\tilde{H}\Phi)u^{j}_{R}+c^{d}_{ij}(H\Phi)d_{R}^{j}\bigg{]}+\text{h.c.}\bigg{\\}}~{},$
(1)

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:

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

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:

$\displaystyle L\supset\dfrac{v}{\sqrt{2}}\frac{c}{f}$
$\displaystyle\bigg{\\{}\text{i}y^{t}\phi_{0}\overline{t}\gamma_{5}t-\text{i}y^{b}\phi_{0}\overline{b}\gamma_{5}b$
(3)
$\displaystyle\bigg{[}-\sqrt{2}\text{i}\phi^{-}\overline{b}(y^{t}P_{R}+y^{b}P_{L})t+\text{h.c.}\bigg{]}\bigg{\\}}~{},$
(4)

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.

Figure -1079: Exotic decay widths of the different heavy particles in our
model for $f=1$ TeV.

$\phi^{0(\pm)}$ 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:

$\displaystyle\Gamma\left(\phi_{0}\rightarrow q\overline{q}\right)$
$\displaystyle=\frac{3y_{q}^{2}v^{2}}{16\pi}\frac{c^{2}}{f^{2}}m_{\phi}\sqrt{1-\frac{4m_{q}^{2}}{m_{\phi}^{2}}}~{},$
(5) $\displaystyle\Gamma\left(\phi^{+}\rightarrow
q\overline{q^{\prime}}\right)$
$\displaystyle=\frac{3(y_{q}^{2}+y_{q^{\prime}}^{2})v^{2}}{16\pi}\frac{c^{2}}{f^{2}}m_{\phi}\left[1-\frac{m_{q}^{2}}{m_{\phi}^{2}}\right]^{2}~{},$
(6) $\displaystyle\Gamma\left(t\rightarrow\phi^{+}b\right)$
$\displaystyle=\frac{(y_{b}^{2}+y_{t}^{2})v^{2}}{32\pi}\frac{c^{2}}{f^{2}}m_{t}\left[1-\frac{m_{\phi}^{2}}{m_{t}^{2}}\right]^{2}~{}.$
(7)

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. -1079.

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

$\Delta W=\mu H_{1}H_{2}+\lambda H_{1}\Sigma
H_{2}+\frac{1}{2}\mu_{\Sigma}\text{tr}\Sigma^{2}~{},$ (8)

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

$L_{\text{SB}}=m_{4}^{2}\text{tr}\Sigma^{\dagger}\Sigma+\bigg{[}B_{\Sigma}\text{tr}\Sigma^{2}+\lambda
A_{\lambda}H_{1}\Sigma H_{2}+\text{h.c.}\bigg{]}~{}.$ (9)

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

$\displaystyle\mu_{\Phi}^{2}$ $\displaystyle\sim
m_{4}^{2}+\mu_{\Sigma}^{2}+B_{\Sigma}\mu_{\Sigma}~{},$ (10)
$\displaystyle\kappa_{\Phi}$
$\displaystyle\sim-\sqrt{2}\lambda\mu_{\Sigma}~{}.$ (11)

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

$\displaystyle\lambda_{H\Phi}\sim\dfrac{(\lambda\mu_{\Sigma})^{2}}{m_{2}^{2}}~{},\quad\frac{c^{u(d)}}{f}\sim
y_{2}^{u(d)}\frac{\lambda\mu_{\Sigma}}{m_{2}^{2}}~{},$ (12)

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

$L\sim
y_{u}\overline{q_{L}}\tilde{H}\bigg{[}1+\frac{\gamma}{f}\Phi+\mathcal{O}(1/f^{2})\bigg{]}u_{R}~{},$
(13)

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. -1078. 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.

Figure -1078: Generation of the effective operator in Eq. II after integrating
out a heavy scalar with with quantum numbers $(1,\mathbf{2})_{1/2}$ (left), a
heavy vector-like quark $\sim(\mathbf{3},\mathbf{3})_{{-2/3}\,(1/3)}$ (center)
or $\sim(\mathbf{3},\mathbf{2})_{-1/6}$ (right) at a mass scale $M\sim f$.
From left to right and top to bottom, the different fields are: $q_{L}$,
$t_{R}$, $H$, $\Phi$.

##  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:

$V=V_{\text{tree}}+\Delta V_{\text{CW}}+\Delta V_{T}~{}+C.$ (14)

$C$ 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

$\displaystyle\Delta V_{\text{CW}}$
$\displaystyle=\frac{1}{64\pi^{2}}\sum_{i}(\pm)n_{i}m_{i}^{4}\bigg{[}\log{\frac{m_{i}^{2}}{v^{2}}}-c_{i}\bigg{]}~{},$
(15)

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

$\displaystyle m_{W}^{2}$
$\displaystyle=\frac{1}{4}g^{2}(h^{2}+4\phi_{0}^{2})~{},$ (16) $\displaystyle
m_{Z}^{2}$ $\displaystyle=\frac{1}{4}(g^{2}+g^{\prime 2})h^{2}~{},$ (17)
$\displaystyle m_{G^{0,\pm}}^{2}$
$\displaystyle=-\mu_{H}^{2}+\lambda_{H}h^{2}+\frac{1}{2}\lambda_{H\Phi}\phi_{0}^{2}~{},$
(18) $\displaystyle m_{\phi^{\pm}}^{2}$
$\displaystyle=\mu_{\Phi}^{2}+\frac{1}{2}\lambda_{H\Phi}h^{2}+\lambda_{\Phi}\phi_{0}^{2}~{},$
(19) $\displaystyle m_{t}^{2}$ $\displaystyle=\frac{1}{2}y_{t}^{2}h^{2}~{}.$
(20)

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

$\displaystyle\mathcal{M}^{2}=$ (21)
$\displaystyle\bigg{[}\begin{array}[]{cc}-\mu_{H}^{2}+3\lambda_{H}h^{2}+\frac{1}{2}\lambda_{H\Phi}\phi_{0}^{2}{}&\lambda_{H\Phi}h\phi_{0}\\\
\lambda_{H\Phi}h\phi_{0}&\mu_{\Phi}^{2}+3\lambda_{\Phi}\phi_{0}^{2}+\frac{1}{2}\lambda_{H\Phi}h^{2}\end{array}\bigg{]}$
(24)

Finally, the finite temperature corrections read

$\displaystyle\Delta V_{T}=$
$\displaystyle\frac{T^{4}}{2\pi^{2}}\sum_{i}(\pm)n_{i}\int_{0}^{\infty}y^{2}\log{\bigg{[}1\mp
e^{-\sqrt{\frac{m_{i}^{2}}{T^{2}}+y^{2}}}\bigg{]}}~{}.$ (25)

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:

$\displaystyle\frac{\partial V}{\partial
h}=0~{},\quad\frac{\partial^{2}V}{\partial
h^{2}}=m_{h}^{2}~{},\quad\frac{\partial^{2}V}{\partial\phi_{0}^{2}}=m_{\Phi}^{2}~{}.$
(26)

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.

Figure -1077: Evolution of the VEV with the temperature for the parameter
space point $m_{\Phi}=120$ GeV, $\lambda_{h\Phi}=1.1$, $\lambda_{\Phi}=1.1$.
At high temperatures, the EW symmetry is restored. It is spontaneously broken
to $(\langle h\rangle,\langle\phi_{0}\rangle)\sim(0,10)$ GeV at $T\sim 160$
GeV, evolving until $T_{n}\sim 85$ GeV at which the step
$(0,120)\rightarrow(220,0)$ GeV takes place. This latter transition is clearly
strong; $\langle h\rangle/T_{n}>1$. The shape of the potential at $T_{n}$ is
also shown. Figure -1076: Same as Fig. -1077 but for the parameter space point
$m_{\Phi}=460$ GeV, $\lambda_{H\Phi}=4.9$, $\lambda_{\Phi}=0.1$. The EWPT
proceeds in one step at $T_{n}=115$ GeV in this case. $\langle\phi_{0}\rangle$
vanishes for all values of $T$.

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. -1077. In this case, the EWPT proceeds in two steps.
An example of a first-order EWPT in one step is shown in Fig. -1076. 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. -1075, $v_{n}/T_{n}>1$; 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.

Figure -1075: The region in the plane $(m_{\Phi},\lambda_{h\Phi})$ where the
EWPT is strongly first order is enclosed by the two dashed green lines, each
one labeling the structure of the phase transition (1- or 2-step). The sub-
regions above the dotted lines can be tested at LISA. The area enclosed by the
solid (dashed) orange line is (could be) excluded at the $95$ % CL by current
(future) measurements of the Higgs to photon width.

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

$\alpha\sim\frac{\epsilon(T_{n})}{35T_{n}^{4}}~{};$ (27)

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

$\frac{\beta}{H}\sim T_{n}\frac{d}{dT}\frac{S_{3}}{T}\sim
T_{n}\frac{\Delta(S_{3}/T)}{\Delta T}~{}.$ (28)

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. -1075. 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 .)

$m_{\Phi}$ | $\lambda_{H\Phi}$ | $\lambda_{\Phi}$ | $(\mu_{H}^{2})^{0}$ | $(\mu_{\Phi}^{2})^{0}$ | $\lambda_{H}^{0}$ | $\mu_{H}^{2}$ | $\mu_{\Phi}^{2}$ | $\lambda_{H}$
---|---|---|---|---|---|---|---|---
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 | $-0.024$
Table 1: Comparison of tree-level obtained parameters (0) versus the ones
computed at one loop for different values of the three inputs. Figure -1074:
Main contribution to the electron (left) and neutron (right) EDMs in the
scalar triplet extension of the SM with a low cutoff. Figure -1073:
Representative diagrams of the main $\Phi$ production mechanisms at $pp$
colliders. Left) Pair production via EW currents. Middle-left) Single
production via gluon fusion. Middle-right) Production in association with
$t\overline{b}$ ($\overline{t}b$). Right) Production from the decay of a top
quark.

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

$\displaystyle\Gamma(h\rightarrow\gamma\gamma)=\frac{\alpha^{2}m_{h}^{3}}{1024\pi^{3}}\bigg{\\{}$
$\displaystyle\frac{2}{v}\bigg{[}A_{1}(\tau_{W})+\frac{4}{3}A_{1/2}(\tau_{t})\bigg{]}$
(29)
$\displaystyle+\lambda_{H\Phi}\frac{v}{m_{\Phi}^{2}}A_{0}(\tau_{\Phi})\bigg{\\}}^{2}~{},$

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. -1075.
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. -1077) 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. -1074. It
gives Choi:2016hro

$\bigg{|}\frac{d_{n}}{e}\bigg{|}\sim
20~{}\text{MeV}\frac{g_{3}^{3}}{(4\pi)^{4}}\frac{y_{t}^{2}v^{2}}{m_{t}^{2}m_{\Phi}^{2}}\frac{c\kappa}{f}h(m_{t}^{2}/m_{h}^{2})~{},$
(30)

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

$\displaystyle h(z)$ $\displaystyle=$ (31)
$\displaystyle=z^{2}\int_{0}^{1}dx\int_{0}^{1}dy\frac{x^{3}y^{3}(1-x)}{[zx(1-xy)+(1-x)(1-y)]^{2}}~{}.$

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. -1074. Using the expressions of Ref.
Keus:2017ioh , we find that the value of the electron EDM in our case reads

$\frac{d}{e}\sim\frac{\alpha v^{2}c}{6\pi^{3}m_{t}f}y_{t}\frac{\kappa
y_{e}}{m_{\Phi}^{2}}\bigg{[}f(m_{t}^{2}/m_{h}^{2})+g(m_{t}^{2}/m_{h}^{2})\bigg{]}~{},$
(32)

with

$\displaystyle f(z)$
$\displaystyle=\frac{1}{2}z\int_{0}^{1}\frac{1-2x(1-x)}{x(1-x)-z}\log{\frac{x(1-x)}{z}}dx~{},$
(33) $\displaystyle g(z)$
$\displaystyle=\frac{1}{2}z\int_{0}^{1}\frac{1}{x(1-x)-z}\log{\frac{x(1-x)}{z}}dx~{}.$
(34)

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

Figure -1072: Cross section of the different production modes for $\Phi$ at
$pp$ colliders of different c.m.e. The coupling $c/f$ is set to
$1\rm{TeV}^{-1}$. The production cross section of $\phi_{0}$ via gluon fusion
was rescaled from Ref. Primulando:2016eod , at $\sqrt{s}=13$ TeV. To obtain
the cross sections for other c.m.e., we have computed the corresponding ratio
in MadGraph, by using the Higgs EFT of the ggHFullLoop model. This ratio turns
out to be a good approximation of the ratio in the full theory; see Ref.
Li:2010fu .

The scalar triplet can be produced at $pp$ colliders in a variety of ways; see
Fig. -1073. The corresponding cross sections at $\sqrt{s}=8,13$ TeV are given
in Fig. -1072. For completeness, we also provide numbers for $27$ and $100$
TeV center-of-mass energy.

Figure -1071: Current (solid) and future (dashed) bounds on the parameter
space $(m_{\Phi},c/f)$. The red region to the left comes from LHC searches for
rare top decays, $t\rightarrow b\phi^{\pm},\phi^{\pm}\rightarrow$ jets; see
Ref. Khachatryan:2015uua . The upper blue region comes from LHC searches for
single production of $\phi^{0}\rightarrow b\overline{b}$ via gluon fusion; see
Ref. Sirunyan:2018ikr . The green region to the right comes from LHC searches
for single production of $\phi^{\pm}$ in association with a top and a bottom
quark; see Ref. Aaboud:2018cwk . The middle vertical orange band is the region
that could be tested using our dedicated analysis.

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. -1071. 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. -1071. 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. -1071. 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. -1071, 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:

  1. 1.

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,

  1. 3.

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. -1070. In Fig.
-1069, 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,

  1. 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. -1070 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.

Figure -1070: Normalized distribution of $m_{\phi_{0},\mathrm{rec}}$ in the
main background (dashed black) and in the signal for $m_{\Phi}=185$ GeV (solid
red) and $m_{\Phi}=310$ GeV (solid blue). Figure -1069: Same as Fig. -1070 but
for $p_{T}$ of the reconstructed $\phi^{0}$.

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 .

Figure -1068: Current (solid) and futute (dashed) bounds on the plane
$(m_{\Phi},c)$ for $\lambda_{H\Phi}=0.1c$ (left), $\lambda_{H\Phi}=c$ (center)
and $\lambda_{H\Phi}=10c$ (right) and $f=1$ TeV.

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
$3~{}b$-tags | 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

Table 2: 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 HL-LHC
is also shown.

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. -1071.

##  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. -1068 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

$\displaystyle A_{0}(x)$
$\displaystyle=-x^{2}\bigg{[}x^{-1}-f(x^{-1})\bigg{]}~{},$ (35) $\displaystyle
A_{1/2}(x)$
$\displaystyle=2x^{2}\bigg{[}x^{-1}+(x^{-1}-1)f(x^{-1})\bigg{]}~{},$
$\displaystyle A_{1}(x)$
$\displaystyle=-x^{2}\bigg{[}2x^{-2}+3x^{-1}+3(2x^{-1}-1)f(x^{-1})\bigg{]}~{}$

with

$f(x)=\arcsin^{2}{\sqrt{x}}~{}.$ (36)

## References

  * (1) J. R. Espinosa and M. Quiros, _Higgs triplets in the supersymmetric standard model_ , _Nucl. Phys._ B384 (1992) 113–146.
  * (2) S. Di Chiara and K. Hsieh, _Triplet Extended Supersymmetric Standard Model_ , _Phys. Rev._ D78 (2008) 055016, [0805.2623].
  * (3) L. Vecchi, _The Natural Composite Higgs_ , 1304.4579.
  * (4) G. Ferretti, _Gauge theories of Partial Compositeness: Scenarios for Run-II of the LHC_ , _JHEP_ 06 (2016) 107, [1604.06467].
  * (5) M. Chala, _$h\rightarrow\gamma\gamma$ excess and Dark Matter from Composite Higgs Models_, _JHEP_ 01 (2013) 122, [1210.6208].
  * (6) M. Cirelli, N. Fornengo and A. Strumia, _Minimal dark matter_ , _Nucl. Phys._ B753 (2006) 178–194, [hep-ph/0512090].
  * (7) P. Fileviez Perez, H. H. Patel, M. Ramsey-Musolf and K. Wang, _Triplet Scalars and Dark Matter at the LHC_ , _Phys. Rev._ D79 (2009) 055024, [0811.3957].
  * (8) A. Carmona and M. Chala, _Composite Dark Sectors_ , _JHEP_ 06 (2015) 105, [1504.00332].
  * (9) G. Panico and A. Wulzer, _The Composite Nambu-Goldstone Higgs_ , _Lect. Notes Phys._ 913 (2016) pp.1–316, [1506.01961].
  * (10) N. Blinov, J. Kozaczuk, D. E. Morrissey and C. Tamarit, _Electroweak Baryogenesis from Exotic Electroweak Symmetry Breaking_ , _Phys. Rev._ D92 (2015) 035012, [1504.05195].
  * (11) S. Inoue, G. Ovanesyan and M. J. Ramsey-Musolf, _Two-Step Electroweak Baryogenesis_ , _Phys. Rev._ D93 (2016) 015013, [1508.05404].
  * (12) Particle Data Group collaboration, M. Tanabashi et al., _Review of Particle Physics_ , _Phys. Rev._ D98 (2018) 030001.
  * (13) A. Delgado, G. Nardini and M. Quiros, _A Light Supersymmetric Higgs Sector Hidden by a Standard Model-like Higgs_ , _JHEP_ 07 (2013) 054, [1303.0800].
  * (14) G. Ballesteros, A. Carmona and M. Chala, _Exceptional Composite Dark Matter_ , _Eur. Phys. J._ C77 (2017) 468, [1704.07388].
  * (15) D. Egana-Ugrinovic, M. Low and J. T. Ruderman, _Charged Fermions Below 100 GeV_ , _JHEP_ 05 (2018) 012, [1801.05432].
  * (16) M. Quiros, _Finite temperature field theory and phase transitions_ , in _Proceedings, Summer School in High-energy physics and cosmology: Trieste, Italy, June 29-July 17, 1998_ , pp. 187–259, 1999.  hep-ph/9901312.
  * (17) M. Laine and A. Vuorinen, _Basics of Thermal Field Theory_ , _Lect. Notes Phys._ 925 (2016) pp.1–281, [1701.01554].
  * (18) J. Choi and R. R. Volkas, _Real Higgs singlet and the electroweak phase transition in the Standard Model_ , _Phys. Lett._ B317 (1993) 385–391, [hep-ph/9308234].
  * (19) A. Ashoorioon and T. Konstandin, _Strong electroweak phase transitions without collider traces_ , _JHEP_ 07 (2009) 086, [0904.0353].
  * (20) K. Enqvist, S. Nurmi, T. Tenkanen and K. Tuominen, _Standard Model with a real singlet scalar and inflation_ , _JCAP_ 1408 (2014) 035, [1407.0659].
  * (21) C. Caprini et al., _Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions_ , _JCAP_ 1604 (2016) 001, [1512.06239].
  * (22) M. Kakizaki, S. Kanemura and T. Matsui, _Gravitational waves as a probe of extended scalar sectors with the first order electroweak phase transition_ , _Phys. Rev._ D92 (2015) 115007, [1509.08394].
  * (23) P. Schwaller, _Gravitational Waves from a Dark Phase Transition_ , _Phys. Rev. Lett._ 115 (2015) 181101, [1504.07263].
  * (24) J. Jaeckel, V. V. Khoze and M. Spannowsky, _Hearing the signal of dark sectors with gravitational wave detectors_ , _Phys. Rev._ D94 (2016) 103519, [1602.03901].
  * (25) F. P. Huang, Y. Wan, D.-G. Wang, Y.-F. Cai and X. Zhang, _Hearing the echoes of electroweak baryogenesis with gravitational wave detectors_ , _Phys. Rev._ D94 (2016) 041702, [1601.01640].
  * (26) K. Hashino, M. Kakizaki, S. Kanemura and T. Matsui, _Synergy between measurements of gravitational waves and the triple-Higgs coupling in probing the first-order electroweak phase transition_ , _Phys. Rev._ D94 (2016) 015005, [1604.02069].
  * (27) T. Tenkanen, K. Tuominen and V. Vaskonen, _A Strong Electroweak Phase Transition from the Inflaton Field_ , _JCAP_ 1609 (2016) 037, [1606.06063].
  * (28) P. Huang, A. J. Long and L.-T. Wang, _Probing the Electroweak Phase Transition with Higgs Factories and Gravitational Waves_ , _Phys. Rev._ D94 (2016) 075008, [1608.06619].
  * (29) M. Artymowski, M. Lewicki and J. D. Wells, _Gravitational wave and collider implications of electroweak baryogenesis aided by non-standard cosmology_ , _JHEP_ 03 (2017) 066, [1609.07143].
  * (30) F. P. Huang and X. Zhang, _Probing the gauge symmetry breaking of the early universe in 3-3-1 models and beyond by gravitational waves_ , _Phys. Lett._ B788 (2019) 288–294, [1701.04338].
  * (31) K. Hashino, M. Kakizaki, S. Kanemura, P. Ko and T. Matsui, _Gravitational waves and Higgs boson couplings for exploring first order phase transition in the model with a singlet scalar field_ , _Phys. Lett._ B766 (2017) 49–54, [1609.00297].
  * (32) V. Vaskonen, _Electroweak baryogenesis and gravitational waves from a real scalar singlet_ , _Phys. Rev._ D95 (2017) 123515, [1611.02073].
  * (33) W. Chao, H.-K. Guo and J. Shu, _Gravitational Wave Signals of Electroweak Phase Transition Triggered by Dark Matter_ , _JCAP_ 1709 (2017) 009, [1702.02698].
  * (34) M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, _Shape of the acoustic gravitational wave power spectrum from a first order phase transition_ , _Phys. Rev._ D96 (2017) 103520, [1704.05871].
  * (35) A. Kobakhidze, A. Manning and J. Yue, _Gravitational waves from the phase transition of a nonlinearly realized electroweak gauge symmetry_ , _Int. J. Mod. Phys._ D26 (2017) 1750114, [1607.00883].
  * (36) A. Beniwal, M. Lewicki, J. D. Wells, M. White and A. G. Williams, _Gravitational wave, collider and dark matter signals from a scalar singlet electroweak baryogenesis_ , _JHEP_ 08 (2017) 108, [1702.06124].
  * (37) A. Kobakhidze, C. Lagger, A. Manning and J. Yue, _Gravitational waves from a supercooled electroweak phase transition and their detection with pulsar timing arrays_ , _Eur. Phys. J._ C77 (2017) 570, [1703.06552].
  * (38) R.-G. Cai, M. Sasaki and S.-J. Wang, _The gravitational waves from the first-order phase transition with a dimension-six operator_ , _JCAP_ 1708 (2017) 004, [1707.03001].
  * (39) L. Bian, H.-K. Guo and J. Shu, _Gravitational Waves, baryon asymmetry of the universe and electric dipole moment in the CP-violating NMSSM_ , _Chin. Phys._ C42 (2018) 093106, [1704.02488].
  * (40) F. P. Huang and C. S. Li, _Probing the baryogenesis and dark matter relaxed in phase transition by gravitational waves and colliders_ , _Phys. Rev._ D96 (2017) 095028, [1709.09691].
  * (41) D. Croon, V. Sanz and G. White, _Model Discrimination in Gravitational Wave spectra from Dark Phase Transitions_ , _JHEP_ 08 (2018) 203, [1806.02332].
  * (42) A. Mazumdar and G. White, _Cosmic phase transitions: their applications and experimental signatures_ , 1811.01948.
  * (43) K. Hashino, R. Jinno, M. Kakizaki, S. Kanemura, T. Takahashi and M. Takimoto, _Fingerprinting models of first-order phase transitions by the synergy between collider and gravitational-wave experiments_ , 1809.04994.
  * (44) A. Ahriche, K. Hashino, S. Kanemura and S. Nasri, _Gravitational Waves from Phase Transitions in Models with Charged Singlets_ , 1809.09883.
  * (45) A. Beniwal, M. Lewicki, M. White and A. G. Williams, _Gravitational waves and electroweak baryogenesis in a global study of the extended scalar singlet model_ , 1810.02380.
  * (46) F. P. Huang and J.-H. Yu, _Exploring inert dark matter blind spots with gravitational wave signatures_ , _Phys. Rev._ D98 (2018) 095022, [1704.04201].
  * (47) M. Breitbach, J. Kopp, E. Madge, T. Opferkuch and P. Schwaller, _Dark, Cold, and Noisy: Constraining Secluded Hidden Sectors with Gravitational Waves_ , 1811.11175.
  * (48) C. L. Wainwright, _CosmoTransitions: Computing Cosmological Phase Transition Temperatures and Bubble Profiles with Multiple Fields_ , _Comput. Phys. Commun._ 183 (2012) 2006–2013, [1109.4189].
  * (49) J. Ellis, M. Lewicki and J. M. No, _On the Maximal Strength of a First-Order Electroweak Phase Transition and its Gravitational Wave Signal_ , _Submitted to: JCAP_ (2018) , [1809.08242].
  * (50) CMS collaboration, A. M. Sirunyan et al., _Combined measurements of Higgs boson couplings in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, _Submitted to: Eur. Phys. J._ (2018) , [1809.10733].
  * (51) _Projections for measurements of Higgs boson cross sections, branching ratios and coupling parameters with the ATLAS detector at a HL-LHC_ , Tech. Rep. ATL-PHYS-PUB-2013-014, CERN, Geneva, Oct, 2013.
  * (52) V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, _On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe_ , _Phys. Lett._ 155B (1985) 36.
  * (53) A. G. Cohen, D. B. Kaplan and A. E. Nelson, _Progress in electroweak baryogenesis_ , _Ann. Rev. Nucl. Part. Sci._ 43 (1993) 27–70, [hep-ph/9302210].
  * (54) A. Riotto and M. Trodden, _Recent progress in baryogenesis_ , _Ann. Rev. Nucl. Part. Sci._ 49 (1999) 35–75, [hep-ph/9901362].
  * (55) D. E. Morrissey and M. J. Ramsey-Musolf, _Electroweak baryogenesis_ , _New J. Phys._ 14 (2012) 125003, [1206.2942].
  * (56) J. R. Espinosa, B. Gripaios, T. Konstandin and F. Riva, _Electroweak Baryogenesis in Non-minimal Composite Higgs Models_ , _JCAP_ 1201 (2012) 012, [1110.2876].
  * (57) F. P. Huang, Z. Qian and M. Zhang, _Exploring dynamical CP violation induced baryogenesis by gravitational waves and colliders_ , _Phys. Rev._ D98 (2018) 015014, [1804.06813].
  * (58) B. Grzadkowski and D. Huang, _Spontaneous $CP$-Violating Electroweak Baryogenesis and Dark Matter from a Complex Singlet Scalar_, _JHEP_ 08 (2018) 135, [1807.06987].
  * (59) K. Choi, S. H. Im, H. Kim and D. Y. Mo, _750 GeV diphoton resonance and electric dipole moments_ , _Phys. Lett._ B760 (2016) 666–673, [1605.00206].
  * (60) C. A. Baker et al., _An Improved experimental limit on the electric dipole moment of the neutron_ , _Phys. Rev. Lett._ 97 (2006) 131801, [hep-ex/0602020].
  * (61) V. Keus, N. Koivunen and K. Tuominen, _Singlet scalar and 2HDM extensions of the Standard Model: CP-violation and constraints from $(g-2)_{\mu}$ and $e$EDM_, _JHEP_ 09 (2018) 059, [1712.09613].
  * (62) ACME collaboration, V. Andreev et al., _Improved limit on the electric dipole moment of the electron_ , _Nature_ 562 (2018) 355–360.
  * (63) R. Primulando and P. Uttayarat, _Probing Lepton Flavor Violation at the 13 TeV LHC_ , _JHEP_ 05 (2017) 055, [1612.01644].
  * (64) Q. Li, M. Spira, J. Gao and C. S. Li, _Higgs Boson Production via Gluon Fusion in the Standard Model with four Generations_ , _Phys. Rev._ D83 (2011) 094018, [1011.4484].
  * (65) CMS collaboration, V. Khachatryan et al., _Search for a light charged Higgs boson decaying to $\mathrm{c}\overline{\mathrm{s}}$ in pp collisions at $\sqrt{s}=8$ TeV_, _JHEP_ 12 (2015) 178, [1510.04252].
  * (66) CMS collaboration, A. M. Sirunyan et al., _Search for low-mass resonances decaying into bottom quark-antiquark pairs in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, 1810.11822.
  * (67) ATLAS collaboration, M. Aaboud et al., _Search for charged Higgs bosons decaying into top and bottom quarks at $\sqrt{s}$ = 13 TeV with the ATLAS detector_, _Submitted to: JHEP_ (2018) , [1808.03599].
  * (68) CMS collaboration, A. M. Sirunyan et al., _Search for pair-produced resonances decaying to quark pairs in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, 1808.03124.
  * (69) CMS collaboration, C. Collaboration, _Search for pair production of resonances decaying to a top quark plus a jet in final states with two leptons_ , .
  * (70) G. D. Kribs, A. Martin, B. Ostdiek and T. Tong, _Dark Mesons at the LHC_ , 1809.10184.
  * (71) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., _The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations_ , _JHEP_ 07 (2014) 079, [1405.0301].
  * (72) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten et al., _An Introduction to PYTHIA 8.2_ , _Comput. Phys. Commun._ 191 (2015) 159–177, [1410.3012].
  * (73) D. E. Soper and M. Spannowsky, _Finding physics signals with shower deconstruction_ , _Phys. Rev._ D84 (2011) 074002, [1102.3480].
  * (74) D. E. Soper and M. Spannowsky, _Finding top quarks with shower deconstruction_ , _Phys. Rev._ D87 (2013) 054012, [1211.3140].
  * (75) D. E. Soper and M. Spannowsky, _Finding physics signals with event deconstruction_ , _Phys. Rev._ D89 (2014) 094005, [1402.1189].
  * (76) A. Abdesselam et al., _Boosted objects: A Probe of beyond the Standard Model physics_ , _Eur. Phys. J._ C71 (2011) 1661, [1012.5412].
  * (77) G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, _Minimal flavor violation: An Effective field theory approach_ , _Nucl. Phys._ B645 (2002) 155–187, [hep-ph/0207036].
  * (78) A. G. Akeroyd, S. Moretti and M. Song, _Light charged Higgs boson with dominant decay to quarks and its search at the LHC and future colliders_ , _Phys. Rev._ D98 (2018) 115024, [1810.05403].
  * (79) R. Harnik, J. Kopp and J. Zupan, _Flavor Violating Higgs Decays_ , _JHEP_ 03 (2013) 026, [1209.1397].
  * (80) S. Banerjee, M. Chala and M. Spannowsky, _Top quark FCNCs in extended Higgs sectors_ , _Eur. Phys. J._ C78 (2018) 683, [1806.02836].