aainstitutetext: Dipartimento di Fisica, Università di Genova and INFN,
Sezione di Genova,
Via Dodecaneso 33, 16146, Italybbinstitutetext: Department of Physics and
Astronomy, University of Sussex,
BN1 9QH Brighton, UKccinstitutetext: Instituto de Física Corpuscular (IFIC),
Universidad de Valencia-CSIC,
E-46980, Valencia, Spain

# On the impact of the LHC Run2 data on general Composite Higgs scenarios

Charanjit K. Khosa b,c and Veronica Sanz<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the the impact of Run2 LHC data on general Composite Higgs scenarios,
where non-linear effects, mixing with additional scalars and new fermionic
degrees of freedom could simultaneously contribute to the modification of
Higgs properties. We obtain new experimental limits on the scale of
compositeness, the mixing with singlets and doublets with the Higgs, and the
mass and mixing angle of top-partners. We also show that for scenarios where
new fermionic degrees of freedom are involved in electroweak symmetry
breaking, there is an interesting interplay among Higgs coupling measurements,
boosted Higgs properties, SMEFT global analyses, and direct searches for
single- and double-production of vector-like quarks.

##  1 Introduction

The true origin of the Higgs mechanism is still an open question in Particle
Physics, despite the discovery of its key element, the Higgs particle
atlasHiggs ; cmsHiggs , and the observation of the SM-like nature of its
couplings to massive particles Higgscouplings .

The main reason to doubt a purely SM Higgs sector can be traced back its
quantum behaviour, a problem often expressed in the context of naturalness:
how can a fundamental scalar be so light, yet so sensitive to UV effects. And
this is not the only suspicious aspect of the SM Higgs. The hierarchy among
its Yukawa couplings, or its inability to produce enough baryon asymmetry
during the electroweak phase transition also add to the unsatisfactory aspects
of the SM Higgs sector.

However, these shortcomings also open opportunities for new physics. The
Higgs, due to its scalar nature, could connect to other sectors via mixing
with scalars, participate in phase transitions and inflation. Among the
theories bringing the Higgs into a new framework, one particularly appealing
proposal is the concept of compositeness, leading to Composite Higgs Models
(CHMs). In CHMs, the Higgs is not a truly fundamental particle but a bound
state of other, more fundamental, particles.

Primitive proposals for a Composite Higgs CHMoriginal  were replaced by more
realistic realisations based on the idea that a Higgs is not just any
composite state from strong dynamics, but a pseudo-Goldstone boson HiggspGBori
. Pseudo-Goldstone bosons appear when approximate global symmetries are
spontaneously broken. The size of the symmetry and what it breaks down to
determines the amount of scalar degrees of freedom one will have in CHMs. The
partial gauging of some of these global symmetries sets what kinds of quantum
numbers the scalars have. The minimal framework to achieve successful
Electroweak Symmetry Breaking (EWSB), i.e. leading to a Higgs doublet of
$SU(2)_{L}$ with a $m_{h}\sim v$, was found in Ref. CHMoriginal . In this
setup, the Higgs originated from the breaking $SO(5)\to SO(4)$, and its
radiative potential had to be supplemented with a new vector-like fermion with
the same quantum numbers as the top, called top-partner and denoted by $T$.
This minimal set-up and its phenomenological consequences has been thoroughly
explored in the literature, see e.g. Ref. Furlan  for a review.

Typically one would assume the main manifestation of the composite nature of
the Higgs would be the non-linear origin of its couplings, leading to very
specific types of Higgs coupling deviations. Yet Composite Higgs scenarios
could exhibit a richer phenomenology, such as the presence of new scalars or
fermions which would also modify the Higgs properties. The focus of this paper
is the study of the rich patterns which arise in general Composite Higgs
scenarios and what the Run2 LHC data can tell us about them. The data we will
use for this analysis is summarised in Table 1.

Measurement | Reference
---|---
Higgs Measurements |
Run 1 ATLAS and CMS combined Higgs signal strength measurements | Khachatryan:2016vau
Run 2 ATLAS Higgs signal strength measurements | atlasrecent ; atlasvhww
Run 2 CMS Higgs signal strength measurements | cmscombo ; cmshtoggnew ; cmshtobb ; cmshtotautau ; cmstthml ; cmsmumu
ATLAS Higgs+jet differential measurement | ATLASdiphotondiff
LHC direct searches for VLQ |
ATLAS T $\rightarrow$ Wb channel | singleVLQsatlaswb
ATLAS T $\rightarrow$ top+MET channel | AtlastopMET
CMS T $\rightarrow$ Wb channel | CMSsingleVLQbw
CMS T $\rightarrow$ Zt final state | CMSsingleVLQZt
Table 1: Experimental measurements considered in this work.

The paper is organised as follows. In section 2, we describe the different
patterns arising in Composite Higgs scenarios: non-linear effects (Sec. 2.1),
extended scalar sectors (Sec. 2.2) and new fermionic degrees of freedom (Sec.
2.3). In these section we examine the impact that Higgs measurements and
direct top-partner searches have on these patterns and in combinations of
them, e.g. how non-linearities and mixing with new scalars do contribute on
the same direction to reduce Higgs couplings to massive particles. Hence, as
one switches on more than one of these effects, each individual limit becomes
stronger. The last Section 3 is devoted to conclusions.

##  2 Patterns of Composite Higgs scenarios

Figure 1: Different realisations of composite Higgs scenario and the relevant
experimental measurements.

The idea of the Higgs as a composite state could be realised in many ways.
Considering theoretical constraints and precision measurements substantially
reduces the possible scenarios, yet there are still many possibilities to
contemplate. Nevertheless, one can describe modern Composite Higgs scenarios
as exhibiting one or more of the following patterns:

  1. 1.

A Composite Higgs sector would be able to trigger Electroweak Symmetry in a
non-linear fashion. This leads to new possibilities for the light Higgs boson.
Its couplings, and even quantum numbers, could be different from the SM
expectation. Indeed, in the SM the Higgs is a $SU(2)_{L}$-doublet, and the way
it induces EWSB completely determines how it couples to fermions and bosons,
including couplings to more than one Higgs. On the other hand, in CHMs these
couplings would be more general, even opening the possibility of a $SU(2)_{L}$
singlet Higgs singletH . This additional freedom leads to the expectation that
the SM-like Higgs, the 125 GeV state observed by the CERN collaborations,
should exhibit non-standard interactions. We will discuss this first Composite
Higgs pattern in Sec. 2.1.

  2. 2.

The second set of patterns found in CHMs refers to the origin of the Composite
Higgs in the spontaneous breaking of an approximate global symmetry. This
breaking could lead to just the right amount of degrees of freedom to match
one single light Higgs particle and the three missing polarisations of the
$W^{\pm}$ and $Z$ bosons. But more generically, this breaking would lead to
new scalar degrees of freedom which, after partially gauging the global
symmetry, could be identified as $SU(2)_{L}$ singlets, doublets, or even
higher representations. These non-minimal coset embeddings would lead to
additional degrees of freedom which could also develop their own potential,
participate in EWSB, and typically mix with the SM-like Higgs. These
additional scalars would be typically required to be heavier than the Higgs,
and their effects could appear at leading-order at tree-level in the effective
theory via e.g., mixing with the SM-like Higgs. The study of the Run2 limits
on these extensions of the SM scalar sector can be found on Sec . 2.2.

  3. 3.

The third set of patterns found in CHMs refers to the presence of new,
fermionic degrees of freedom which would assist the SM-like Higgs in its job
of triggering EWSB. These fermions, often called top-partners, should be
relatively light to efficiently modify the Higgs potential. They can be
searched for directly, via their production at the LHC through couplings to
the strong or electroweak sector, or indirectly, as they modify the Higgs
properties. The discussion on direct and indirect searches for top-partners
will be developed in Sec 2.3.

  4. 4.

In each section we also discuss situations where more than one dominant
pattern could be at play in the LHC measurements. We will explain how in this
case, their effect in the Higgs couplings would go in the same direction,
ruling out cancellations which could invalidate the limits from the previous
sections.

These patterns are summarised in Fig. 1.

###  2.1 CHM pattern I: Non-linear effects

In CHM one assumes the existence of a global symmetry ${\cal G}$,
spontaneously broken to a smaller subgroup ${\cal H}$. The Higgs particle and
the would-be Goldstone bosons for the $W^{\pm}$ and $Z$ originate from this
breaking. Part of ${\cal H}$ is then weakly gauged to provide the Higgs and
massive gauge bosons with the correct properties under the SM.

To go beyond the SM structure of EWSB, it is useful to follow the Callan,
Coleman, Wess and Zumino prescription Callan:1969sn . Within this
prescription, one aims to build objects with definite transformation
properties under $SU(2)\times U(1)$. In CHMs the object that contains the
Goldstones from the global breaking ${\cal G}\to{\cal H}$ is
$\Sigma=\exp{i\phi^{a}X^{a}}/f$, where $\phi^{a}$ are the Goldstones, $X^{a}$
generators in the coset ${\cal G}/{\cal H}$ and $f$ is the scale associated
with the spontaneous breaking. At low energies, below $f$, one can write an
effective Lagrangian involving $\Sigma$

$\mathcal{L}_{\mathit{kinetic}}^{eff}=\frac{f^{2}}{4}\operatorname{Tr}[D_{\mu}\Sigma^{\dagger}D^{\mu}\Sigma],$
(1)

which, after making a choice for weakly gauging ${\cal H}$, determines the
Higgs and vector boson interactions. Assuming the Higgs is a doublet of
$SU(2)_{L}$, similarly as in the SM, the structure that arises from this
kinetic term is

$\displaystyle g^{2}\,f^{2}\,\sin^{2}{\left(\frac{h}{f}\right)}\ .$ (2)

One could also allow the Higgs to be a singlet under $SU(2)_{L}$, but that
would lead to the need to impose a tuning in the effective Lagrangian to
explain the specific relations between the $Z$ and $W$ masses Contino10 ;
Belenetal . We will not follow this path in this paper, and embed the CHM
global symmetry structure in a way that preserves the custodial symmetry
present in the SM.

Expanding around the physical Higgs boson vacuum expectation value (VEV), one
finds clear predictions of the structure of the Higgs-vector boson couplings

$\kappa_{V}=\frac{g^{CH}_{VVh}}{g^{SM}_{VVh}}=\sqrt{1-\xi}\approx
1-\frac{1}{2}\xi\ ,$ (3)

where $\xi=v^{2}/f^{2}$.

Turning now to the Higgs-fermion couplings, we note that the predictions are
not unique. In particular, the fermion mass generation mechanism in CHMs
usually relies on the concept of partial compositeness Contino06 ; Contino10 ,
i.e. the idea that SM fermions feel EWSB via mixing with fermionic bound
states. To build structures leading to fermion couplings to the Higgs, one
needs to specify the embedding of the fermionic degrees of freedom in the
global symmetry structure ${\cal G}$. As an example, in the minimal composite
Higgs models, based on $SO(5)/SO(4)$ coset group, $\Sigma$ will transform as
5-plet of $SO(5)$ which could form a $SO(5)$ invariant either with two 5-plets
or pair of 5 and 10-plets.

More general CHMs lead to very different patterns of breaking and types of
embeddings for the fermion content of the model. Yes, despite spanning many
model building options, it was noted in Ref. Sanz:2017tco  that the Yukawa
couplings usually fall into two choices,

$\kappa_{F}^{A}=\sqrt{1-\xi}\approx 1-\frac{1}{2}\xi,$ (4)

and

$\kappa_{F}^{B}=\frac{1-2\xi}{\sqrt{1-\xi}}\approx 1-\frac{3}{2}\xi.$ (5)

For example, the models based on coset groups $SO(5)/SO(4)$ Agashe:2004rs ;
Carena:2014ria , $SO(6)/SO(4)\times SO(2)$ DeCurtis:2016tsm ; DeCurtis:2016scv
; Mrazek:2011iu , $SU(5)/SU(4)$ Bertuzzo:2012ya , $SO(8)/SO(7)$ Low:2015nqa ;
Barbieri:2015lqa  have fermions-Higgs couplings modified by $\kappa_{F}^{A}$.
On the other hand, $\kappa_{F}^{B}$-type couplings could exist in
$SO(5)/SO(4)$ Contino:2006tv ; Carena:2014ria ; Carmona:2014iwa ;
Csaki:2017cep , $SU(4)/Sp(4)$ Gripaios09 , $SU(5)/SO(5)$ Ferretti14 ,
$SO(6)/SO(4)\times SO(2)$ DeCurtis:2016tsm ; DeCurtis:2016scv ; Mrazek:2011iu
groups based models. In all these models the Higgs doublet lies within the
unbroken subgroup, but in some of those there could be an extra singlet or
Higgs doublet.

A Composite Higgs, with non-standard couplings to massive fermions and bosons,
would also exhibit non-standard loop-level couplings to gluons and photons
($\kappa_{g,\gamma}$) Khachatryan:2016vau ; Gillioz:2012se

$\displaystyle\kappa_{g}^{2}$ $\displaystyle=$ $\displaystyle
1.06\kappa_{t}^{2}+0.01\kappa_{b}^{2}-0.07\kappa_{b}\kappa_{t}$ (6)
$\displaystyle\kappa_{\gamma}^{2}$ $\displaystyle=$ $\displaystyle
1.59\kappa_{V}^{2}+0.07\kappa_{t}^{2}-0.66\kappa_{V}\kappa_{t}\ ,$ (7)

as well as deviations on the Higgs width ($\kappa_{H}$)

$\kappa_{H}^{2}\approx
0.57\kappa_{b}^{2}+0.25\kappa_{V}^{2}+0.09\kappa_{g}^{2}.$ (8)

Note that $\kappa_{t}$, $\kappa_{b}$ do not have to be equal (see Eqs. 4 and
5) depending on how fermions are embedded within the global symmetry group.

Putting all this together, we are ready to compare measurements of the Higgs
properties with expectations from a non-linear realisation of EWSB via a
Composite Higgs. The experimental inputs, described in Appendix A, are Higgs
signal strengths for different production and decay channels. We compare these
measurements with theoretical predictions as a function of the $\kappa$
modifiers $\kappa_{f}$ , $\kappa_{V}$, $\kappa_{g}$, $\kappa_{\gamma}$, and
$\kappa_{H}$. For example, for the gluon fusion $ggH$
($H\rightarrow\gamma\gamma$) channel, it adopts the form:
$\mu^{CH}=\frac{\kappa_{g}^{2}\kappa_{\gamma}^{2}}{\kappa_{H}^{2}}$. To
compare the CHM predictions with Higgs signal strength measurements from CMS
and ATLAS experiments, we evaluate the $\chi^{2}$ statistic test

$\chi^{2}(f)=\sum_{i}^{Run1,Run2}\left(\frac{\mu_{i}(\kappa(f))^{CH}-\mu_{i}^{Exp}}{\Delta\mu_{i}}\right)^{2}\
.$ (9)

Here $\mu(\kappa(f))^{CH}$, $\mu^{Exp}$ and $\Delta\mu$ denote the model
prediction of the signal strength, experimental measurement, and error for the
experimental measurement, respectively. The index $i$ runs over all the
measurements from Run 1 and Run 2. Note that the correlations among the
different experimental measurements are not considered as they are subdominant
respect to the model uncertainties introduced by the choices of $\kappa_{F}$.

Figure 2: $\Delta\chi^{2}=\chi^{2}(f)-\chi^{2}_{min}$ for the combination of
Run 1 and 2 LHC data (signal strength measurements). The different grey lines
correspond to different choices of fermion couplings $\kappa_{F}^{A,B}$ for
$(\kappa_{t},\kappa_{b},\kappa_{\tau})$. The vertical line is the lowest
allowed value (95 $\%$ C.L.) of the compositeness scale ($f_{min}$) within a
model where fermion couplings are of the ABB type i.e. top quark have
$\kappa_{A}$, bottom and tau have $\kappa_{B}$ factor with the Higgs
couplings. Green and yellow are the 1$\sigma$ and 2$\sigma$ regions,
respectively.

In this section we only focus on non-linear effects, hence the rates
$\mu(\kappa(f))^{CH}$ are sensitive to one parameter, the $\xi=v^{2}/f^{2}$
ratio, and choices for the fermion couplings from Eqs. 4 and 5.

In Figure 2, we show the $\chi^{2}$ fit of different choices for $\kappa_{F}$
to the combined LHC Run 1 and Run 2 data. Run 1 data is taken from the
combined CMS and ATLAS analysis Khachatryan:2016vau . For Run 2 data,
individual measurements from CMS cmscombo ; cmshtoggnew ; cmshtobb ;
cmshtotautau ; cmstthml ; cmsmumu  (see table 3) and ATLAS atlasrecent ;
atlasvhww  (see table 4) are considered. For $H\rightarrow b\bar{b}$ decay
channel, both CMS and ATLAS have combined Run 1 and Run 2 data measurements so
we have not considered the single Run 1 measurements.

In Figure 2 we show the 1- and 2-sigma limits on $\chi^{2}-\chi^{2}_{min}$ in
green and yellow, respectively. The values of $\Delta\chi^{2}$ depend on
whether the top, bottom and tau $\kappa$ modifiers are of type $A$ or $B$.
Following Ref. Sanz:2017tco , we plot all the lines with all combinations and
denote which combination is less constrained (ABB) and which one is
constrained more strongly by LHC data, BAA. Comparing with Ref. Sanz:2017tco ,
we obtain a stronger lower bound on the scale of compositeness of $f_{min}=$
780 GeV. The different curves highlight the fact that there is roughly 500 GeV
variation in the $f_{min}$ scale, reaching $f_{min}\sim 1.3$ TeV for the most
tightly bound scenario.

###  2.2 CHM pattern II: Extended Higgs sectors

When the group ${\cal G}$ breaks down to ${\cal H}$, light degrees of freedom
are generated. Only in specific configurations one would expect exactly four
Goldstones, to match the SM needs. Hence generic Composite Higgs scenarios
would typically exhibit a pattern of Extended Higgs Sectors, with more light
scalars involved in EWSB . In the simplest of such extensions, one would
consider an additional singlet, see e.g. singlet , which could mix with the
SM-like Higgs doublet. The next simplest iteration would introduce new
doublets, as is the case in e.g. the coset group $SO(6)/SO(4)\times SO(2)$
Mrazek:2011iu , where the light degrees of freedom organise as a Two-Higgs
Doublet Model (2HDMs) 2hdm .

Doublets and singlets are not the only model-building possibilities which
Composite Higgs models offer, e.g. on the $SU(6)/SO(6)$ coset one has two
Higgs doublets but also a custodial bi-triplet DMCHM . Generically speaking, a
Composite Higgs scenario leading to one single doublet at low-energies is
minimal but not typical, and one should consider the effect of more scalars
involved in EWSB.

Moreover, this pattern opens new, interesting possibilities for model building
beyond EWSB. These extra singlet or doublet pseudo-Goldstones could play a
role in Inflation DjunaCHM  or act as Dark Matter candidates DMCHM . In
Composite Higgs scenarios, these new scalars could help on enhancing the
strength of the electroweak phase transition EWPTCH  by introducing new
sources of CP violation and more interesting phase diagram structures, and
even play some role in the QCD CP problem Gupta:2020vxb .

In this section, we explore what the Run2 LHC data can tell us about these
extensions. We will discuss the modification of the gauge boson and fermions
couplings to the SM-like Higgs boson when the additional scalars mix with the
SM-like Higgs. Their effect could also be felt at loop-level, even in the
absence of a mixing, but these contributions would be suppressed by loop
factors respect to mixing. Both possibilities, mixing and loop effects, were
computed and explored in Ref. Gorbahn:2015gxa  and here we make use of these
theoretical calculations and update the limits in the context of CHMs and the
possible interplay between mixing and non-linearities.

####  2.2.1 Singlet scalar

The presence of an extra singlet modifies the $\kappa_{F/V}$ by a factor of
$\cos\theta_{S}$, where $\theta_{S}$ denotes the mixing angle between the
neutral scalar $h$ and the extra singlet scalar field. This effect is simply
due to linear mixing terms when both the singlet and the SM-like Higgs get
their VEVs. Non-linearities due to the origin of the SM-like Higgs as a
Composite Higgs would still be present, in exactly the same way we discussed
in Sec. 2.1 111One could consider situations where the additional singlets or
doublets do participate directly in the mechanism for EWSB, as shown in the
see-saw Composite Higgs Sanz:2015sua , where this assumption would fail..
Therefore, the Higgs couplings to vector bosons would be doubly modified as

$\kappa_{V}=\cos\theta_{S}\sqrt{1-\xi}\approx
1-\frac{1}{2}\xi-\frac{1}{2}\theta_{S}^{2}\ ,$ (10)

where we have expanded for small values of non-linearities and mixing to show
how these two effects work in the same direction, i.e. to reduce the coupling
value from the SM expectation.

The non-linear part of the modification of the fermion couplings depends on
the fermion embedding in representations of the global symmetry. As discussed
before, we find two main choices for $\kappa_{F}$, namely

$\kappa_{F}^{A}=\cos\theta_{S}\sqrt{1-\xi}\approx
1-\frac{1}{2}\xi-\frac{1}{2}\theta_{S}^{2}$ (11)

or

$\kappa_{F}^{B}=\cos\theta_{S}\frac{1-2\xi}{\sqrt{1-\xi}}\approx
1-\frac{3}{2}\xi-\frac{1}{2}\theta_{S}^{2}\ ,$ (12)

where again we expand for small modifications $\xi$ and $\theta_{S}$ to show
explicitly the cooperative effort of both effects to lower the coupling.

Figure 3: The bound on the $\xi_{max}$ as a function of $\cos\theta_{S}$ in
case of singlet mixing scenario for the ABB type model.

We perform the fit to the Higgs data as in Sec. 2.1 but now including the
effect of mixing, $\chi^{2}(f,\theta_{S})$. As shown in Figure 2, we vary
options for the fermion couplings and find the $f_{min}$. We perform this fit
for different values of the mixing angle, always finding the minimum $f$
corresponding to a set of options for $\kappa_{F}$. The result of this
procedure is shown in Figure 3, where we plot $\xi_{max}=v^{2}/f^{2}_{min}$
scale as a function of the singlet mixing. On the right side of the plot we
see the effect of small mixing, where we recover the result from the previous
section. As we move towards the left in the plot, the mixing becomes more
important and at some point $\xi_{max}\to 0$, we obtain the pure mixing limit
of $|\cos{\theta_{S}}|\simeq$0.96.

Note that the mixing angle is determined by singlet mass ($m_{S}$), singlet
vev ($v_{S}$) and quadratic coupling of light Higgs doublet and singlet scalar
($\lambda$), as given below

$\cos\theta_{S}=\cos\left(\frac{v^{2}}{(m_{S}^{eff})^{2}}\right)\quad;\quad
m_{S}^{eff}=\frac{m_{S}}{\sqrt{\lambda(v_{S}/v)}}$ (13)

where $v=246$ GeV. Therefore, the limit $|\cos{\theta_{S}}|\simeq$0.96 would
be equivalent to $m^{eff}_{S}\gtrsim$ 450 GeV.

In between these two asymptotic limits, the constraints on the scales $f$ and
$m^{eff}_{S}$ become stronger, as both mixing and non-linearities work
together to reduce the Higgs couplings.

####  2.2.2 Two Higgs Doublet Models (2HDM)

Model | $\kappa_{V}$ | $\kappa_{F}$
---|---|---
$\mathrm{Type}-\mathrm{I}$ | $s_{\beta-\alpha}$ | $\kappa_{u}=\kappa_{d}=\kappa_{\ell}=c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$
$\mathrm{Type-II}$ | $s_{\beta-\alpha}$ | $\kappa_{u}=c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$
|  | $\kappa_{d}=\kappa_{\ell}=s_{\beta-\alpha}-t_{\beta}c_{\beta-\alpha}$
$\ell-\mathrm{Specific}$ | $s_{\beta-\alpha}$ | $\kappa_{u}=\kappa_{d}=c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$
|  | $\kappa_{\ell}=s_{\beta-\alpha}-t_{\beta}c_{\beta-\alpha}$
$\mathrm{Flipped}$ | $s_{\beta-\alpha}$ | $\kappa_{u}=\kappa_{\ell}=c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$
|  | $\kappa_{d}=s_{\beta-\alpha}-t_{\beta}c_{\beta-\alpha}$
Table 2: The $\kappa_{V}$ and $\kappa_{F}$ expressions in Two Higgs doublet
models. Here $s_{x}/c_{x}/t_{x}=\sin x/\cos x/\tan x$.

As discussed in the previous section, the mixing effect due the additional
Higgses in a Composite Higgs model does not typically couple with the non-
linear effects at leading order. Hence, in composite two Higgs doublet models
the vector and femion couplings to the Higgs, $\kappa_{V/f}$, would still
adopt this factorisable form $\kappa^{2HDM}_{V/F}\times\kappa^{CH}_{V/F}$,
where $\kappa_{V/F}^{CH}$ has been discussed in Sec. 2.1 and
$\kappa^{2HDM}_{V/F}$ correspond to the modifications of the SM-like Higgs
couplings in 2HDM models.

The explicit form of $\kappa^{2HDM}_{V/F}$ is given in Table 2 for various
types of 2HDM models. In these scenarios, vector and fermion modifiers are
determined by two parameters, the ratio of symmetry breaking VEVs of Higgs
doublets ($\tan\beta$) and neutral Higgs mixing angle ($\alpha$).

Figure 4: The 68 $\%$(green), 95 $\%$(yellow) and 98 $\%$(red) C.L. limits in
the $\cos(\beta-\alpha)-\tan\beta$ plane for 2HDM Type I (left plot) and Type
II (right plot) with $\xi=0$.

In our analysis, we only consider the main scenarios, Type I and Type II. The
$\chi^{2}_{2CHM}$ statistical test for a Composite Higgs with possible mixing
with another doublet becomes a function of three parameters:
$f,\cos({\beta-\alpha}),$ and $\tan\beta$.

Figure 5: Type I 2HDM results. Left plot: The 95 $\%$ C.L. limit including
non-linear effects. The black, red and green lines correspond to choices of
the parameter $\xi=0$, 0.05 and 0.07, respectively. Right plot:  The 68
$\%$(green), 95 $\%$(yellow) and 98 $\%$(red) C.L. limits in the
$\xi$-$c_{\beta-\alpha}$ plane for $\tan\beta=6$.

First we discuss the information that Run2 LHC data provides on type I and II
2HDMs, without considering the non-linear effects. This information is shown
in Figure 4, where the coloured regions green, yellow and red represent 68
$\%$, 95 $\%$ and 98 $\%$ C.L. limits in the $\cos(\beta-\alpha)-\tan\beta$
plane, respectively. As expected, the SM-like Higgs prefers a region with
$\cos{\beta-\alpha}\simeq$ 0, the alignment limit haberailignment .

The alignment limit can be understood in terms of the potential parameters in
the 2HDM, couplings between the two doublets $\tilde{\lambda}_{i}$ and masses
$\tilde{\mu}_{1,2}$, see Ref. Gorbahn:2015gxa  for more details. In terms of
these parameters, the deviation from the alignment limit can be parametrised
as $\cos{\beta-\alpha}\sim\lambda_{6}v^{2}/\tilde{\mu}_{2}^{2}$.

Next we introduce the additional non-linear effects in the left panel in
Figure 5, where we show how the 95 $\%$ C.L. regions are modified with non-
linear effects. The non-linear effects, as in the case of the singlet add up
to the doublet mixing. Larger non-linearities restrict further the parameter
space of 2HDMs.

Finally, as the $\tan\beta$ dependence is rather mild for $\tan\beta\gtrsim$
2, we fix the value of $\tan\beta=$6 to explore the interplay between
$\cos{\beta-\alpha}$ and $\xi$. In the right panel of Figure 5, we plot the
one-, two- and three-$\sigma$ contours in this plane. In the alignment limit,
$\cos{\beta-\alpha}\simeq$ 0, we recover the pure non-linear limit $\xi\simeq
0.08$ at 95%C.L.. As we move from the alignment limit, and allow a larger
amount of doublet mixing, the limit on $\xi$ becomes milder, leading to a
stronger bound on the compositeness scale $f$. On the left side of the plot
where $\xi\to$ 0, we find the limit on non-decoupling effects
$\cos{\beta-\alpha}\simeq$ 0.25 at 95%C.L., which would correspond to the
scale for the second doublet $\tilde{\mu}_{2}\sim$ 500 GeV for
$\tilde{\lambda}_{6}\sim{\cal O}(1)$.

###  2.3 CHM pattern III: Extended fermionic sector

We finish this section on patterns, analysing a typical building block in
Composite Higgs scenarios: the presence of new fermionic degrees of freedom,
not too far from the electroweak scale. These fermionic degrees of freedom are
usually coming along as top partners, new vector-like fermions which mix with
the SM top quark and modify its couplings with the Higgs.

We first focus on the mixing, and present results for the $SU(2)_{L}$-singlet
top partner scenario $T^{0}_{L,R}$. We calculate the indirect bound on these
states from measurements on the differential distributions of the the boosted
Higgs in association with an energetic jet. The methodology used here was
developed in Ref. BanfiSanzHandj  and further expanded in Refs. andreaHj  and
BanfiBondSanzMartin . We will then evaluate other sources of experimental
constraints, including direct searches for the new state.

To describe the effect of the top-partner in the Higgs behaviour we need to
specify the mixing parameters between the top-partner and the SM top quark.
For the singlet case, the mass matrix of the SM top quark and Dirac fermion
top partner $T=(T_{L},T_{R})$ can be written as

$\begin{pmatrix}\bar{t}_{L}&\bar{T}_{L}\end{pmatrix}\begin{pmatrix}\frac{y_{t}h}{\sqrt{2}}&\Delta\cr
0&M\end{pmatrix}\begin{pmatrix}t_{R}\cr T_{R}\end{pmatrix}\ ,$ (14)

where $\Delta$ describes the mixing between $t$ and $T$, and $M$ is the Dirac
mass of the top partner. After diagonalising this matrix by a bi-unitary
transformation, we can calculate the mass eigenstates $M_{T}$ and $m_{t}$ and
the mixing angle

$\theta_{R}=\frac{1}{2}\sin^{-1}\left(\frac{2m_{t}M_{T}\Delta}{(M_{T}^{2}-m_{t}^{2})M}\right)\
.$ (15)

Note that $\theta_{R}$ and $\theta_{L}$ are related in a simple way,

$\tan\theta_{L}=\frac{M_{T}}{m_{t}}\tan\theta_{R}.$ (16)

####  2.3.1 Top-partner indirect searches

Contrary to the previous sections, total Higgs rates and their respective
limits on vector and fermion couplings $\kappa_{V/F}$ would not be sensitive
to the presence of the top partner. On the other hand, high-$p_{T}$ probes
would access the top partner mixing and mass scale indirectly BanfiSanzHandj ;
andreaHj .

To analyse the effect of $T$ in the Higgs production in association with
radiation, it is useful to define a quantity which depends on $M_{T}$ and
$\theta_{R}$ based on semi-differential measurements,

$\sigma(p_{T}>p_{T}^{cut})=\int_{p_{T}^{cut}}^{\infty}dp_{T}\frac{d\sigma}{dp_{T}}\
.$ (17)

In particular, we define a quantity based on these differential measurements,

$\delta(p_{T}^{cut},M_{t},\sin\theta_{R})=\frac{\sigma_{t+T}(p_{T}^{cut},M_{T},\sin\theta_{R})-\sigma_{t}(p_{T}^{cut})}{\sigma_{t}(p_{T}^{cut})}\
,$ (18)

which exhibits good properties from the point of view of systematic and
statistical fluctuations, see Ref. BanfiSanzHandj  for more details.

Figure 6: Percentage enhancement in the Higgs$+$ jet cross-section by the top
partner contribution as a function of $p_{T}^{cut}$ for different values of
mixing angle and $M_{T}$.

With this observable as an indirect probe for fermionic top-partners, we
calculated $\sigma_{t}$ and $\sigma_{t+T}$ for $\sqrt{s}=$13 TeV by varying
$M_{T}$ and $\sin\theta_{R}^{2}$. The results can be seen in Figure 6, where
we plot the relative change in the Higgs+jet cross-section as a function of
$p_{T}^{cut}$ for different combinations of the mixing angle and $M_{T}$. For
this plot, $M_{T}$ is varied from 500 GeV to 1500 GeV, and one can see that
$\delta$ has a weak dependence on $M_{T}$. We can then use the experimental
measurements in this channel to put a bound on the mixing angle. In
particular, we used the ATLAS Higgs differential cross-section analysis
ATLASdiphotondiff , which reports an 8$\%$ deviation at 95 $\%$ C.L. in the
250-350 GeV (diphoton) $p_{T}$ bin from the SM cross-section. We can then look
back at our Fig. 6 and note the variation of $\delta$ as a function of
$\sin\theta_{R}$ for $p_{T}^{cut}$=250 GeV.

In Fig. 7 we show the variation of $\delta$ as a function of $\sin\theta_{R}$
for a fixed $p_{T}^{cut}=250$ GeV. The green band corresponds to varying
$M_{T}$ from 500 GeV to 1500 GeV. The 8% limit is noted with a hashed black
line, which allows us to set a limit on $\sin^{2}\theta_{R}\lesssim 0.25$.

Figure 7: Percentage enhancement in the Higgs+jet cross-section by the top
partner contribution ($\delta$) as a function of the top-partner mixing angle
for $p_{T}^{cut}$=250 GeV. The horizontal hashed line corresponds to the
experimental bound ATLASdiphotondiff , and the band width to the variation in
$M_{T}\in$[500,1500] GeV.

Another source of constraints from a singlet VLQ top-partner would come from a
SM Effective Field Theory (SMEFT) global analysis of Run2 LHC and LEP
measurements. The singlet top-partner would produce a SMEFT pattern
characterised by relations among some SMEFT operators, whereas all the other
operators would be zero at tree-level, see Ref. GranadaSMEFT  for a dictionary
between many extensions of the SM and their SMEFT matching. See also the
discussion in Ref. CHeftrecent  on the top-partners and the SMEFT framework.
Using this dictionary and a global SMEFT analysis, in Ref. Ellis:2020unq  the
parameters of the singlet top-partner were bound to
$\sin{\theta_{L}}^{2}\lesssim 0.05$ for $M_{T}\simeq$ 1 TeV.

####  2.3.2 Top-partner direct searches

Boosted Higgs measurements provide one handle on top-partners. The LHC New
Physics search programme includes very mature analyses looking for single and
pair production of Vector-Like Quarks (VLQ). Pair production of VLQs is a
dominantly strong interaction process, whereas single production relies on
electroweak couplings.

A singlet up-type VLQ (T) is excluded up to 1.31 TeV by the ATLAS
$\sqrt{s}=13$ TeV (36.1 fb-1) search for pair produced VLQs AtlaspairprodVLQs
after considering all decay modes. This limit is a combined limit from several
searches for the pair production of VLQs.

From the CMS side, there are separate limits from each analysis of pair
production of VLQs. The exclusion limit on singlet T from dilepton final state
of VLQs decaying to Z boson is 1280 GeV CMSVLQpairDilepton . The CMS analysis
for all leptonic final states excludes $M_{T}$ in the range 1140-1300 GeV
corresponding to different branching fractions CMSVLQpairAllleptonfinalstates
, and the analysis of the $bW\bar{b}W$ final state excludes singlet $T$ up to
1295 GeV CMSVLQpairbwbw . The analysis targeting fully hadronic final states
provides a bound up to 1.3 TeV for a specific combination of the branching
fractions of VLQs CMSVLQpairfullthad .

Figure 8: Direct searches bounds on singlet top-partners from single and
double production of VLQs, and indirect bounds from the Higgs+jet final state.
Blue, orange, green and violet curves correspond to the ATLAS single VLQ bound
from $Wb$ channel singleVLQsatlaswb , ATLAS bound from top+MET final state
AtlastopMET , CMS $Wb$ final state CMSsingleVLQbw  and CMS $Zt$ final state
CMSsingleVLQZt , respectively. The vertical line corresponds to the ATLAS pair
production bound AtlaspairprodVLQs . The SMEFT bound is taken from Ref.
Ellis:2020unq  and the red line indicates the limit from Higgs+jet
differential distributions discussed in Sec. 2.3.1.

The single VLQs production process is mediated by its coupling with SM
particles, hence it provides a bound on the mixing as a function of the mass
of VLQs singleVLQsatlaswb ; AtlastopMET ; CMSsingleVLQbw . Both the production
cross-section and decay branching ratios are sensitive to mixing effects. The
communication of the single-production results is a bit more cumbersome than
for double-production. In some analyses, this mixing is directly parametrised
by the mixing angle ($\sin\theta_{L,R}$) and sometimes by the coupling
$c^{bw}$ where $c^{bw}_{L/R}=\sqrt{2}|\sin\theta_{L/R}|$ HandbookVLQs  is just
a re-scaling of the mixing angle. ATLAS analysis cross-section limits are
interpreted in terms of either $|\sin\theta_{L}|$, or both
$C_{W}^{2}=(c^{bw}_{L})^{2}+(c^{bw}_{R})^{2}$ and $|\sin\theta_{L}|$. On the
other hand, CMS exclusion limits on production cross-section times branching
fraction are provided only as a function of $m_{VLQ}$. Additionally, the
theoretical cross-section is provided for the fixed value of $BR$ and
$c^{bw}=0.5$. We use this additional information to calculate the approximate
bound on $c^{bw}$ as a function of $m_{VLQ}$.

We translate CMS bound in terms of mixing angle $|\sin\theta_{L}|$ by
parametrizing the production cross-section as $\sigma(T)\times BR(T\rightarrow
bW)=\tilde{\sigma}(T){c^{bw}}^{2}\times BR(T\rightarrow bW)$, where
$\tilde{\sigma}(T)$ is calculated from the theoretical cross-section with
fixed $c^{bw}$. Then using the bound on cross-section (including BR), we
calculate the limit on $c^{bw}$ which we further translate to
$|\sin\theta_{L}|$ using the relation
$c^{bw}_{L/R}=\sqrt{2}|\sin\theta_{L/R}|$. Note that we have not considered
the effect of $C_{W}$ variation on the width of $T$. We also consider other
decay channels of the singlet vector-like top to $ht$ and $Zt$. Note that CMS
analyses for $Wb$ CMSsingleVLQbw  and $Zt$ CMSsingleVLQZt  final states do
assume 100 $\%$ branching ratio, and the ATLAS collaboration has considered
$Wb$ and top +MET final states AtlastopMET  assuming BR($T\rightarrow
Zt)=0.25$ from the singlet model.

In Figure 8 we show all these direct and indirect limits together in the plane
of top-partner mass versus mixing angle. Double production of VLQs (dark-green
vertical line) sets a limit on the mass $M_{T}\gtrsim$ 1.3 TeV. The SMEFT
limit (light blue horizontal line) is very strong, and independent of the top-
partner mass as long as $M_{T}\geqslant v$. The current sensitivity of the
other indirect probe for top-partners, the $H$+jet channel, is below the SMEFT
fit, although it is worth noticing that our $H$+jet analysis is not at the
same level of sophistication as the SMEFT’s Ellis:2020unq . Note that for the
ATLAS top+MET analysis (black line) the $|\sin\theta_{L}|$ values covered by
the closed contour are excluded.

##  3 Conclusions

In this work we have studied the impact of the LHC Run 2 measurements on
general Composite Higgs scenarios.

The dataset we used includes Higgs signal strength measurements from CMS and
ATLAS, differential properties of boosted Higgs production, several searches
for single- and double-production of vector-like quarks, and a specific result
from a global fit to SMEFT properties obtained in Ref. Ellis:2020unq .

We have described the most significant deviations expected from Composite
Higgs scenarios, going from the simplest effects, non-linear couplings, to the
most complex interplay between direct and indirect searches for additional
fermionic degrees of freedom.

We have shown how in general Composite Higgs scenarios one should expect
simultaneous effects from more than one source. Typically, the composite
nature of the Higgs is tied to the non-linear realisation of EWSB and the
scale of compositeness $f$. But generic Composite Higgs scenarios exhibit a
richer phenomenology, in particular new light scalars or fermions are also
typical predictions of these scenarios.

These new degrees of freedom, if close enough to the electroweak scale, will
also modify the Higgs couplings. For example, mixing of additional scalars
participating in EWSB will add to non-linearities to reduce even further the
couplings of the Higgs to massive particles, leading then to even stronger
limits on the scale of compositeness $f$. Note that this scale is tied to the
mass of new vector resonances like $W^{\prime}$ and $Z^{\prime}$, and to the
degree of tuning of Higgs potential. The Run2 LHC data has pushed this scale
into the TeV range, and we have shown that more complex scenarios would push
this tuning even further. Hence Run3 and future LHC runs should have a good
handle at natural Composite Higgs scenarios.

###### Acknowledgements.

The work of V.S. by the Science Technology and Facilities Council (STFC) under
grant number ST/P000819/1. We want to thank Stephan Huber and Jack Setford for
many fruitful discussions during the early stages of this work. We thank
Andrea Banfi for the help regarding running the Higgs+jet process in his code
implementation.

##  Appendix A Data inputs

Production | Decay Channel | Lumi[Reference] | Signal Strength
---|---|---|---
ggH | $H\to ZZ$ | 137.1 fb-1cmscombo  | $0.98^{+0.12}_{-0.11}$
VBF |  |  | $0.57^{+0.46}_{-0.36}$
VH |  |  | $1.10^{+0.96}_{-0.74}$
$t\bar{t}H,tH$ |  |  | $0.25^{+1.03}_{-0.25}$
ggH | $H\to\gamma\gamma$ | 137 fb-1cmshtoggnew  | $0.98^{+0.13}_{-0.10}$
VBF |  |  | $1.15^{+0.36}_{-0.31}$
VH |  |  | $0.71^{+0.31}_{-0.28}$
$t\bar{t}H$ |  |  | $1.40^{+0.33}_{-0.27}$
ggF | $H\to WW$ | 35.9 fb-1cmscombo  | $1.28^{+0.20}_{-0.19}$
VBF |  |  | $0.63^{+0.65}_{-0.61}$
WH |  |  | $2.85^{+2.11}_{-1.87}$
ZH |  |  | $0.90^{+1.77}_{-1.43}$
ttH |  |  | $0.93^{+0.48}_{-0.45}$
ggH | $H\to b\bar{b}$ | upto 77.4 fb-1cmscombo  | $2.45^{+2.53}_{-2.35}$
WH |  |  | $1.27^{+0.42}_{-0.40}$
ZH |  |  | $0.93^{+0.33}_{-0.31}$
$t\bar{t}H$ |  |  | $1.13^{+0.33}_{-0.30}$
VBF | $H\to b\bar{b}$ | upto 77.2fb-1cmshtobb  | $2.53\pm{1.53}$
ggH | $H\to\tau\bar{\tau}$ | 137 fb-1cmshtotautau  | $0.98^{+0.20}_{-0.19}$
VBF+V(qq)H |  |  | $40.67^{+0.23}_{-0.22}$
$t\bar{t}H$ | $H\to ML$ | 137 fb-1 cmstthml  | $0.92^{+0.26}_{-0.23}$
ggH | $H\to\mu\mu$ | 137 fb-1 cmsmumu  | $0.63^{+0.65}_{-0.64}$
VBF |  |  | $1.36^{+0.69}_{-0.61}$
VH |  |  | $5.48^{+3.10}_{-2.83}$
$t\bar{t}H$ |  |  | $2.32^{+2.27}_{-1.95}$
Table 3: Summary of CMS Run 2 Higgs signal strength measurements. Production | Decay Channel | Lumi[Reference] | Signal Strength
---|---|---|---
ggF | $H\to ZZ$ | 139 fb-1atlasrecent  | $0.94^{+0.11}_{-0.10}$
VBF |  |  | $1.25^{+0.50}_{-0.41}$
VH |  |  | $1.53^{+1.13}_{-0.92}$
ggF | $H\to\gamma\gamma$ | 139 fb-1 atlasrecent  | $1.03\pm{0.11}$
VBF |  |  | $1.31^{+0.26}_{-0.23}$
VH |  |  | $1.32^{+0.33}_{-0.30}$
$ttH+tH$ |  |  | $0.90^{+0.27}_{-0.24}$
ggF | $H\to WW$ | 36.1 fb-1atlasrecent  | $1.08^{+0.19}_{-0.18}$
VBF |  | 36.1 fb-1 atlasrecent  | $0.60^{+0.36}_{-0.34}$
VH |  | 36.1 fb-1atlasvhww  | $2.5^{+0.9}_{-0.8}$
VBF | $H\to bb$ | 24.5-30.6 fb-1atlasrecent  | $3.03^{+1.67}_{-1.62}$
VH |  | 139 fb-1atlasrecent  | $1.02^{+0.18}_{-0.17}$
$ttH+tH$ |  | 36.1 fb-1 atlasrecent  | $0.79^{+0.60}_{-0.59}$
ggF | $H\to\tau\tau$ | 36.1 fb-1atlasrecent  | $1.02^{+0.60}_{-0.55}$
VBF |  |  | $1.15^{+0.57}_{-0.53}$
$ttH+tH$ |  |  | $1.20^{+1.07}_{-0.93}$
$ttH+tH$ | $H\to VV$ | 36.1 fb-1atlasrecent  | $1.72^{+0.56}_{-0.53}$
Table 4: Summary of ATLAS Run 2 (24.5-139 fb-1(13 TeV)) Higgs signal strength
measurements atlasrecent . The measurement of VH,$H\rightarrow WW$ is taken
from atlasvhww .

## References

  * (1) G. Aad et al. [ATLAS], Phys. Lett. B 716 (2012), 1-29, [arXiv:1207.7214 [hep-ex]].
  * (2) S. Chatrchyan et al. [CMS], Phys. Lett. B 716 (2012), 30-61, [arXiv:1207.7235 [hep-ex]].
  * (3) G. Aad et al. [ATLAS and CMS], JHEP 08 (2016), 045, [arXiv:1606.02266 [hep-ex]].
  * (4) D. B. Kaplan, H. Georgi and S. Dimopoulos, Phys. Lett. B 136 (1984), 187-190, M. J. Dugan, H. Georgi and D. B. Kaplan, Nucl. Phys. B 254 (1985), 299-326,
  * (5) R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B 671 (2003), 148-174, [arXiv:hep-ph/0306259 [hep-ph]]. K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B 719 (2005), 165-187, [arXiv:hep-ph/0412089 [hep-ph]].
  * (6) E. Furlan, doi:10.3929/ethz-a-006361307
  * (7) G. Aad et al. [ATLAS and CMS Collaborations], JHEP 1608 (2016) 045, [arXiv:1606.02266 [hep-ex]].
  * (8) The ATLAS collaboration [ATLAS Collaboration], ATLAS-CONF-2020-027.
  * (9) G. Aad et al. [ATLAS Collaboration], arXiv:1903.10052 [hep-ex].
  * (10) [CMS], CMS-PAS-HIG-19-005.
  * (11) [CMS], CMS-PAS-HIG-19-015.
  * (12) A. M. Sirunyan et al. [CMS Collaboration], Phys. Rev. Lett. 121 (2018) no.12, 121801, [arXiv:1808.08242 [hep-ex]].
  * (13) [CMS], CMS-PAS-HIG-19-010.
  * (14) [CMS], CMS-PAS-HIG-19-008.
  * (15) [CMS], CMS-PAS-HIG-19-006.
  * (16) The ATLAS collaboration [ATLAS Collaboration], ATLAS-CONF-2019-029.
  * (17) M. Aaboud et al. [ATLAS Collaboration], [arXiv:1812.07343 [hep-ex]].
  * (18) M. Aaboud et al. [ATLAS Collaboration], [arXiv:1812.09743 [hep-ex]].
  * (19) A. M. Sirunyan et al. [CMS Collaboration], Phys. Lett. B 772 (2017) 634, [arXiv:1701.08328 [hep-ex]].
  * (20) A. M. Sirunyan et al. [CMS Collaboration], Phys. Lett. B 781 (2018) 574, [arXiv:1708.01062 [hep-ex]].
  * (21) F. Feruglio, Int. J. Mod. Phys. A 8 (1993), 4937-4972, [arXiv:hep-ph/9301281 [hep-ph]]. B. Grinstein and M. Trott, Phys. Rev. D 76 (2007), 073002, [arXiv:0704.1505 [hep-ph]].
  * (22) C. G. Callan, Jr., S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247.
  * (23) R. Contino, arXiv:1005.4269 [hep-ph].
  * (24) I. Brivio, T. Corbett, O. J. P. Éboli, M. B. Gavela, J. Gonzalez-Fraile, M. C. Gonzalez-Garcia, L. Merlo and S. Rigolin, JHEP 03 (2014), 024 doi:10.1007/JHEP03(2014)024 [arXiv:1311.1823 [hep-ph]].
  * (25) R. Contino, T. Kramer, M. Son and R. Sundrum, JHEP 0705 (2007) 074, [hep-ph/0612180].
  * (26) V. Sanz and J. Setford, Adv. High Energy Phys. 2018 (2018), 7168480, [arXiv:1703.10190 [hep-ph]].
  * (27) K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B 719 (2005) 165, [hep-ph/0412089].
  * (28) M. Carena, L. Da Rold and E. Pontón, JHEP 1406 (2014) 159, [arXiv:1402.2987 [hep-ph]].
  * (29) S. De Curtis, S. Moretti, K. Yagyu and E. Yildirim, Eur. Phys. J. C 77 (2017) no.8, 513, [arXiv:1610.02687 [hep-ph]].
  * (30) S. De Curtis, S. Moretti, K. Yagyu and E. Yildirim, Phys. Rev. D 94 (2016) no.5, 055017, [arXiv:1602.06437 [hep-ph]].
  * (31) J. Mrazek, A. Pomarol, R. Rattazzi, M. Redi, J. Serra and A. Wulzer, Nucl. Phys. B 853 (2011) 1, [arXiv:1105.5403 [hep-ph]].
  * (32) E. Bertuzzo, T. S. Ray, H. de Sandes and C. A. Savoy, JHEP 1305 (2013) 153, [arXiv:1206.2623 [hep-ph]].
  * (33) M. Low, A. Tesi and L. T. Wang, Phys. Rev. D 91 (2015) 095012 doi:10.1103/PhysRevD.91.095012 [arXiv:1501.07890 [hep-ph]].
  * (34) R. Barbieri, D. Greco, R. Rattazzi and A. Wulzer, JHEP 1508 (2015) 161, [arXiv:1501.07803 [hep-ph]].
  * (35) R. Contino, hep-ph/0609148.
  * (36) A. Carmona and F. Goertz, JHEP 1505 (2015) 002, [arXiv:1410.8555 [hep-ph]].
  * (37) C. Csaki, T. Ma and J. Shu, Phys. Rev. Lett. 119 (2017) no.13, 131803, [arXiv:1702.00405 [hep-ph]].
  * (38) B. Gripaios, A. Pomarol, F. Riva and J. Serra, JHEP 0904 (2009) 070, [arXiv:0902.1483 [hep-ph]].
  * (39) G. Ferretti, JHEP 1406 (2014) 142, [arXiv:1404.7137 [hep-ph]].
  * (40) M. Gillioz, R. Grober, C. Grojean, M. Muhlleitner and E. Salvioni, JHEP 1210 (2012) 004, [arXiv:1206.7120 [hep-ph]].
  * (41) See e.g., V. Barger, P. Langacker, M. McCaskey, M. J. Ramsey-Musolf and G. Shaughnessy, Phys. Rev. D 77 (2008), 035005, [arXiv:0706.4311 [hep-ph]].
  * (42) See e.g., G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept. 516 (2012), 1-102, [arXiv:1106.0034 [hep-ph]].
  * (43) G. Cacciapaglia, H. Cai, A. Deandrea and A. Kushwaha, JHEP 10 (2019), 035, [arXiv:1904.09301 [hep-ph]].
  * (44) D. Croon, V. Sanz and J. Setford, JHEP 10 (2015), 020, [arXiv:1503.08097 [hep-ph]]. D. Croon, V. Sanz and E. R. M. Tarrant, Phys. Rev. D 94 (2016) no.4, 045010, [arXiv:1507.04653 [hep-ph]].
  * (45) J. R. Espinosa, B. Gripaios, T. Konstandin and F. Riva, JCAP 01 (2012), 012, [arXiv:1110.2876 [hep-ph]].
  * (46) R. S. Gupta, V. V. Khoze and M. Spannowsky, [arXiv:2012.00017 [hep-ph]].
  * (47) M. Gorbahn, J. M. No and V. Sanz, JHEP 1510 (2015) 036, [arXiv:1502.07352 [hep-ph]].
  * (48) V. Sanz and J. Setford, JHEP 1512 (2015) 154, [arXiv:1508.06133 [hep-ph]].
  * (49) J. F. Gunion and H. E. Haber, Phys. Rev. D 67 (2003), 075019, [arXiv:hep-ph/0207010 [hep-ph]].
  * (50) A. Banfi, A. Martin and V. Sanz, JHEP 1408 (2014) 053, [arXiv:1308.4771 [hep-ph]].
  * (51) A. Banfi, B. M. Dillon, W. Ketaiam and S. Kvedaraitė, arXiv:1905.12747 [hep-ph].
  * (52) A. Banfi, A. Bond, A. Martin and V. Sanz, JHEP 11 (2018), 171, [arXiv:1806.05598 [hep-ph]].
  * (53) J. de Blas, J. C. Criado, M. Perez-Victoria and J. Santiago, JHEP 03 (2018), 109, [arXiv:1711.10391 [hep-ph]].
  * (54) H. L. Li, L. X. Xu, J. H. Yu and S. H. Zhu, arXiv:1904.05359 [hep-ph].
  * (55) J. Ellis, M. Madigan, K. Mimasu, V. Sanz and T. You, [arXiv:2012.02779 [hep-ph]].
  * (56) M. Aaboud et al. [ATLAS Collaboration], Phys. Rev. Lett. 121 (2018) no.21, 211801, [arXiv:1808.02343 [hep-ex]].
  * (57) A. M. Sirunyan et al. [CMS Collaboration], Eur. Phys. J. C 79 (2019) no.4, 364, [arXiv:1812.09768 [hep-ex]].
  * (58) A. M. Sirunyan et al. [CMS Collaboration], JHEP 1808 (2018) 177, [arXiv:1805.04758 [hep-ex]].
  * (59) A. M. Sirunyan et al. [CMS Collaboration], Phys. Lett. B 779 (2018) 82, [arXiv:1710.01539 [hep-ex]].
  * (60) CMS Collaboration [CMS Collaboration], CMS-PAS-B2G-18-005.
  * (61) J. A. Aguilar-Saavedra, R. Benbrik, S. Heinemeyer and M. Pérez-Victoria, Phys. Rev. D 88 (2013) no.9, 094010, [arXiv:1306.0572 [hep-ph]].