# The Higgs program and open questions in particle physics and cosmology

Beate Heinemann1,2a, and Yosef Nir3b 1Deutsches Elektronen-Synchrotron, 22607
Hamburg, Germany
2Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, 79104 Freiburg,
Germany
3Department of Particle Physics and Astrophysics, Weizmann Institute of
Science, Rehovot, Israel 7610001<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Higgs program is relevant to many of the open fundamental questions in
particle physics and in cosmology. Thus, when discussing future collider
experiments, one way of comparing them is by assessing their potential
contributions to progress on these questions. We discuss in detail the
capabilities of the various proposed experiments in searching for singlet
scalars, which are relevant to several of the open questions, and in measuring
Higgs decays to fermion pairs, which are relevant to the flavor puzzles. On
other interesting questions, we list the most relevant observables within the
Higgs program.

##  I Introduction

The jewel in the crown of the achievements of the LHC experiments to date is
the Higgs boson discovery in July 2012 Aad:2012tfa ; Chatrchyan:2012xdj . The
discovery of a Higgs boson is, however, not just an end of a story - a quest
that began with the theoretical predictions of Brout, Englert Englert:1964et
and Higgs Higgs:1964pj  \- but also a beginning of one. While the presence of
the Higgs boson provides a solution to the question of how fundamental
particles can acquire mass, it does not actually explain the mass values
themselves, and it also raises new questions. “The Higgs program” is a major
research project at the LHC, and at all proposed future collider experiments.
From the experimental side, the Higgs program refers to a large set of
measurements aimed to learn the detailed properties of this unique particle.
From the theoretical side, this is also a very exciting program, as it touches
upon several open questions and puzzles in particle physics and particle
cosmology.

In this article, we explore the relationship of the properties of the Higgs
boson to big questions in the field of particle physics and cosmology. This
work was initiated by discussions in the Scientific Policy Committee of CERN
in the context of discussions on future accelerators. All proposed future
accelerators consider measurements of the Higgs boson a major part of their
scientific program, and this short article is designed to summarize concisely
the conclusions from the currently available literature that relates Higgs
precision measurements to fundamental open questions about our Universe. We
also summarize the precision these future colliders estimate, and try to
contrast that with the precision required to answer a given question. It is
worth noting that for many of the questions there are also important
observables unrelated to the Higgs boson measurements (at both collider and
non-collider experiments), but discussing those is beyond the scope of this
article.

Here is a list of seven open questions at the forefront of particle physics
and cosmology, to which the Higgs program may provide answers:

  1. 1.

Is $h$ the only scalar degree of freedom?

  2. 2.

Is $h$ elementary?

  3. 3.

What keeps $m_{h}^{2}\ll m_{\rm Pl}^{2}$?

  4. 4.

Was the electroweak phase transition first order?

  5. 5.

Did CP violating $h$ interactions generate the baryon asymmetry?

  6. 6.

Are there light SM-singlet degrees of freedom (in particular, related to Dark
Matter)?

  7. 7.

What is the solution of the flavor puzzle(s)?

In what follows, we focus on two topics. In Section III, we describe the
search for singlet scalars. We choose this topic for four reasons:

  * •

It is relevant to a large number (four) of the above questions;

  * •

The Higgs program is likely to be the unique portal to such particles;

  * •

The search for singlet scalars by itself constitutes a rich and broad
experimental program;

  * •

For the question of the electroweak phase transition, there is a clear target
for the required accuracy that will give a definite answer on this question.

In Section IV, we discuss the measurements that are relevant to the flavor
puzzles. Again, there are several reasons for this choice of topic:

  * •

The flavor puzzles are long standing, and the Higgs program provides an
opportunity to measure new flavor parameters and make progress;

  * •

The main ingredient in the flavor-related Higgs program – the measurement of
Yukawa couplings – is also relevant to other questions.

  * •

The flavor related measurements by themselves constitute a rich and broad
experimental program;

  * •

In contrast to the question of electroweak phase transition, for the flavor
measurements there is no lower bound on the size of new physics effects.
Instead, the better accuracy, the higher the scale of new physics to which
there is sensitivity.

All other topics are discussed briefly in Section V, where we describe the
most relevant experimental measurements to each physics question, but do not
compare the capabilities of the various experiments (which is still work in
progress). In Section VI we present three important conclusions about the
scientific significance of the Higgs program.

##  II Experiments

For the purpose of discussing the expected accuracy of future measurements,
the following future collider parameters (integrated luminosity ${\cal L}$,
center of mass energy $\sqrt{s}$, and, where relevant, polarization) are
assumed:

  * •

HL-LHC: $pp$ collider, ${\cal L}=3-4$ ab-1 at $\sqrt{s}=14$ TeV (2026 to late
2030s: $\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}10$ years) Bordry:2018gri .

  * •

ILC: $e^{+}e^{-}$ collider, with $e^{-}$ 80% polarized and $e^{+}$ 30%
polarized, to be operated in two stages (22 years total including shutdowns)
Bambade:2019fyw :

    1. 1.

ILC250: ${\cal L}=2$ ab-1 at $\sqrt{s}=250$ GeV (11 years),

    2. 2.

ILC500: ${\cal L}=4$ ab-1 at $\sqrt{s}=500$ GeV (10 years). This includes also
a 1-year run at $\sqrt{s}=365$ GeV.

  * •

CLIC: $e^{+}e^{-}$ collider, with $e^{-}$ 80% polarized and $e^{+}$ not
polarized, to be operated in three stages (26 years including shutdowns)
Bordry:2018gri :

    1. 1.

CLIC380: ${\cal L}=1.0$ ab-1 at $\sqrt{s}=380$ GeV (8 years),

    2. 2.

CLIC1500: ${\cal L}=2.5$ ab-1 at $\sqrt{s}=1400$ GeV (7 years),

    3. 3.

CLIC3000: ${\cal L}=5.0$ ab-1 at $\sqrt{s}=3000$ GeV (7 years).

  * •

CEPC: $e^{+}e^{-}$ collider, without polarization, ${\cal L}=5.6$ ab-1 at
$\sqrt{s}=240$ GeV (7 years) CEPCStudyGroup:2018ghi . It is assumed that there
are two experiments collecting the data, and the datasets are combined.

  * •

FCC-ee: $e^{+}e^{-}$ collider, without polarization, to be operated in two
stages Bordry:2018gri :

    1. 1.

FCC240: ${\cal L}=5.0$ ab-1 at $\sqrt{s}=240$ GeV (3 years).

    2. 2.

FCC365: ${\cal L}=1.5$ ab-1 at $\sqrt{s}\approx 365$ GeV (5 years).

In both cases it is assumed that there are two experiments collecting the
data, and the datasets are combined.

  * •

LHeC: $ep$ collider, colliding the 7 TeV HL-LHC protons with 60 GeV electrons,
aiming to deliver ${\cal L}=1$ ab-1 within 12 years. For the last 4 years the
HL-LHC is assumed not to operate concurrently for $pp$ collisions
Bordry:2018gri .

  * •

HE-LHC: $pp$ collider, ${\cal L}=10$ ab-1 at $\sqrt{s}=27$ TeV (20 years)
Bordry:2018gri . For the numbers presented in Table 7, ${\cal L}=15$ ab-1 is
considered, corresponding to two collider experiments combining at least 75%
of the data Mangano:2018mur .

  * •

FCC-hh: $pp$ collider, ${\cal L}=20$ ab-1 at $\sqrt{s}=100$ TeV (25 years)
Bordry:2018gri . For the numbers presented in Table 7, ${\cal L}=30$ ab-1 is
considered, corresponding to two collider experiments combining at least 75%
of the data Mangano:2018mur .

  * •

Muon collider: a muon collider is also a very interesting option to collide
leptons at very high energies. This is, however, not considered in this report
as the studies related to this are currently less advanced.

##  III Singlet scalars

One of the experimentally most challenging extensions of the Standard Model
(SM), yet one that is relevant to many interesting Higgs-related questions, is
that of a singlet scalar. In fact, it appears with relation to four of the
seven open questions that we posed:

  1. 1.

Is $h$ alone?

  2. 2.

What keeps $m_{h}^{2}\ll m_{\rm Pl}^{2}$?

  3. 3.

Was the electroweak phase transition first order?

  4. 4.

Are there light SM-singlet degrees of freedom?

The simplest extension is to add to the SM a real scalar field $S(x)$, in the
$(1,1)_{0}$ (gauge-singlet) representation. The most general renormalizable
Lagrangian is Huang:2016cjm ; Craig:2013xia ; Curtin:2014jma  (we use the
notations of Huang:2016cjm )

${\cal L}={\cal L}_{\rm
SM}+\frac{1}{2}(\partial_{\mu}S)(\partial^{\mu}S)-\frac{m_{s}^{2}}{2}S^{2}-\frac{a_{s}}{3}S^{3}-\frac{\lambda_{s}}{4}S^{4}-\lambda_{hs}\Phi^{\dagger}\Phi
S^{2}-2a_{hs}\Phi^{\dagger}\Phi S,$ (1)

where $\Phi$ is the Higgs doublet in the $(1,2)_{+1/2}$ representation. In the
general case, both $\Phi$ and $S$ can acquire VEVs:

$\langle\Phi\rangle=(0,v/\sqrt{2}),\ \ \ \langle S\rangle=v_{s},$ (2)

and the Higgs and singlet fields mix:

$\sin 2\theta=\frac{4v(a_{hs}+\lambda_{hs}v_{s})}{M_{h}^{2}-M_{s}^{2}},$ (3)

where $M_{h}$ and $M_{s}$ are the mass eigenvalues of the two scalar mass
eigenstates ($M_{h}\simeq 125$ GeV).

Deviations from the SM predictions for various couplings are often
parameterized by “$\kappa$ parameters”:

$\kappa_{x}\equiv\frac{g_{hxx}}{g_{hxx}^{\rm SM}}.$ (4)

Within the SM extended by singlet scalars, the couplings of the Higgs boson to
all fermion and vector boson pairs are modified by a universal factor. (For a
discussion, see Section I of Ref. Craig:2013xia .) Defining $\delta g_{h}$ via

$\kappa_{x}\equiv 1+\delta g_{h},$ (5)

we have Huang:2016cjm

$\delta
g_{h}\approx(\cos\theta-1)-\frac{|a_{hs}+\lambda_{hs}v_{s}|^{2}}{8\pi^{2}}I_{B}(M_{h}^{2};M_{h}^{2},M_{s}^{2})-\frac{|\lambda_{hs}|^{2}v^{2}}{16\pi^{2}}I_{B}(M_{h}^{2};M_{s}^{2},M_{s}^{2}),$
(6)

where Fan:2014axa

$I_{B}(p^{2};m_{1}^{2},m_{2}^{2})=\int_{0}^{1}dx\frac{x(1-x)}{x(1-x)p^{2}-xm_{1}^{2}-(1-x)m_{2}^{2}}.$
(7)

As concerns Eq. (6), we emphasize the following points:

  * •

The first term on the right hand side (RHS) is a consequence of the mixing
(see Eq. 3).

  * •

The second and third terms are a consequence of the wave function
renormalization of the Higgs at one loop.

Usually, the first term dominates. This is not the case, however, in models
where a $Z_{2}$ symmetry, under which $S$ ($\Phi$) is odd (even), is imposed.
This scenario is considered the most difficult one for finding experimental
evidence for first order electroweak phase transition Curtin:2014jma . In this
model, we have $a_{hs}=v_{s}=0$ and consequently $\theta=0$, so that the first
two terms of Eq. (6) vanish. It follows that Fan:2014axa

$\delta
g_{h}^{(Z_{2})}=-\frac{|\lambda_{hs}|^{2}v^{2}}{16\pi^{2}}I_{B}(M_{h}^{2};M_{s}^{2},M_{s}^{2})=-\frac{|\lambda_{hs}|^{2}v^{2}}{16\pi^{2}M_{h}^{2}}\left[1-\frac{4M_{s}^{2}{\rm
arctan}\left(\sqrt{\frac{M_{h}^{2}}{4M_{s}^{2}-M_{h}^{2}}}\right)}{\sqrt{M_{h}^{2}(4M_{s}^{2}-M_{h}^{2})}}\right].$
(8)

In the limit that $M_{s}^{2}\gg M_{h}^{2}$ Fan:2014axa ,

$\delta
g_{h}^{(Z_{2})}\to+\frac{|\lambda_{hs}|^{2}v^{2}}{96\pi^{2}M_{s}^{2}}.$ (9)

###  III.1 $M_{s}<M_{h}/2$

If $M_{s}<M_{h}/2$, then a new decay channel for the Higgs boson opens up,
$h\to SS$, with

$\Gamma(h\to SS)=\frac{\lambda_{hSS}^{2}}{8\pi
M_{h}}\sqrt{1-\frac{4M_{s}^{2}}{M_{h}^{2}}},$ (10)

and

$\lambda_{hSS}=+c_{\theta}^{3}\lambda_{hs}v+c_{\theta}^{2}s_{\theta}(a_{s}+3\lambda_{s}v_{s}+2\lambda_{hs}v_{s}-2a_{hs})+c_{\theta}s_{\theta}^{2}(3\lambda_{h}v-2\lambda_{hs}v)+s_{\theta}^{3}(\lambda_{hs}v_{s}+a_{hs}).$
(11)

In the $Z_{2}$ limit

$\lambda_{hSS}=\lambda_{hs}v,$ (12)

and, furthermore, $S$ is stable, and therefore, the $h\to SS$ mode contributes
to $h\to{\rm invisible}$. Ref. Frugiuele:2018coc  collected the upper bounds
and projections on BR($h\to$invisible). We quote updated results and future
projections in Table 1.

The BR($h\to$invisible) is determined directly by analyses searching for a
Higgs boson produced in association with a $Z$ boson (at lepton and hadron
colliders) or with forward jets (in VBF process at hadron colliders), and the
Higgs boson decaying invisibly which is observed as missing transverse
momentum in the experiments. Several $e^{+}e^{-}$ colliders are able to probe
this with a precision of 0.3%. The best sensitivity is achieved by FCChh with
0.025%.

Table 1: Current upper bounds and projections on ${\rm BR}_{\rm inv}\equiv{\rm BR}(h\to{\rm invisible})$ and ${\rm BR}_{\rm und}\equiv{\rm BR}(h\to{\rm undetected})$. All upper bounds are given at 95% CL unless explicitly stated. The LHC2 line updates the most recent bounds from CMS Sirunyan:2018owy  and ATLAS Aaboud:2018sfi . Projections for future colliders are also shown: CLIC Robson:2018zje , CEPC CEPCStudyGroup:2018ghi , ILC Barklow:2017suo  , FCCee and FCChh Mangano:2018mur . For FCChh the SM background from $H\to 4\nu$, corresponding to a BR of $0.113\%$, has been subtracted and the limit refers to additional invisible new physics contributions. For the LHC and HL-LHC BRund is inferred assuming $\kappa_{Z}\leq 1$. The symbol ”$-$” means that no value is available in the literature.  Collider |  $\sqrt{s}$ [TeV] |  ${\cal L}$ [ab-1] | ${\rm BR}_{\rm inv}[\%]$ | ${\rm BR}_{\rm und}[\%]$
---|---|---|---|---
LHC1 | $7,8$ | $0.022$ | $37$ | 20
LHC2 | $13$ | $0.036$ | $26$ | $-$
LHC3 | $13$ | $0.300$ | $8.8\ (68\%)$ | $7.6\ (68\%)$
HL-LHC | $14$ | $3$ | $2.6$ | $2.5$
CLIC | $0.38$ | $1$ | $0.69(90\%)$ | $1.6$
CEPC | $0.25$ | $5.6$ | $0.3$ | $1.0$
ILC | $0.25$ | $2$ | $0.3$ | $1.5$
FCCee | $0.24$ | $5$ | $0.3$ | $0.87$
FCChh | $100$ | $20$ | $0.025$ | $-$

In the case that no $Z_{2}$ symmetry is imposed and, in particular,
$\sin\theta\neq 0$, the mass eigenstate $S$ inherits the Higgs couplings,
suppressed by $\sin\theta$:

$\lambda_{Sf\bar{f}}=\sin\theta\ y_{f}^{\rm SM},$ (13)

where $y_{f}^{\rm SM}$ is the corresponding SM Yukawa coupling. The $h\to SS$
decay will be followed by $S\to f\bar{f}$ where $m_{f}<M_{s}/2$ with coupling
given by Eq. (13). There are two complementary ways of searching for these
decays. One is the direct search for the final four fermion state, which we
discuss further below. The second is the indirect search, via a global fit to
the Higgs couplings, which can reveal (or constrain) a contribution to the
Higgs width that is not accounted for by the final states that are included in
the fit Bechtle:2014ewa ; Flacke:2016szy ; Frugiuele:2018coc , the so-called

$h\to{\rm undetected}.$ (14)

Upper bounds on ${\rm BR}(h\to{\rm undetected})$ are given in Table 1, and put
an upper bound on $\sin^{2}\theta$ for a given $M_{s}$. The branching ratios
to “undetected” final state are derived from the observed cross sections and
branching ratios using certain constraints. For the LHC and HL-LHC this can
only be done if $\kappa_{Z}\leq 1.0$ is assumed, which holds for singlets with
$M_{s}<M_{h}/2$ if the dominant effect is due to $\sin\theta\neq 0$ (and in
general only holds for a fraction of new physics models). For the $e^{+}e^{-}$
colliders it is possible to use a much more direct and very model-independent
constraint Frugiuele:2018coc  ${\rm BR}(h\to{\rm undetected})\leq
1-(1-2\delta_{\kappa})^{2}$, where $\delta_{\kappa}$ is the precision on the
measurement of $\kappa_{Z}$ (CLIC380: 0.4%, CEPC: 0.25%, ILC: 0.38%, FCCee:
0.22%).

Examples of other relevant experimental searches are collected in Table 2. For
the $\mu\mu bb$ final state it is expected that the HL-LHC Atlas:2019qfx  will
have a sensitivity down to ${\rm BR}(h\to\mu^{+}\mu^{-}b\bar{b})\sim 5\times
10^{-5}$. For the other decays no projections are available.

Table 2: Upper bounds on $(\sigma_{h}/\sigma_{h}^{\rm SM})\times{\rm BR}(h\to SS\to\overline{f_{1}}f_{1}\overline{f_{2}}f_{2})$. $\overline{f_{1}}f_{1}\overline{f_{2}}f_{2}$ |  $M_{s}$ [GeV] | Upper bound | Ref.
---|---|---|---
$\tau\tau bb$ | $15-60$ | $0.03-0.12$ | CMS Sirunyan:2018pzn
$\mu\mu bb$ | $25-63$ | $(1-8)\times 10^{-4}$ | CMS Khachatryan:2017mnf , ATLAS Aaboud:2018esj
$\tau\tau\tau\tau$ | $9-15$ | $0.2-0.3$ | CMS Khachatryan:2017mnf
$\mu\mu\tau\tau$ | $25-63$ | $(1-5)\times 10^{-4}$ | CMS Khachatryan:2017mnf ; Sirunyan:2018mbx
$\gamma\gamma jj$ | $20-60$ | $0.034-0.173$ | ATLAS Aaboud:2018gmx

The total width of $h$ is, on one hand, suppressed by the suppression of the
$h$-couplings to pairs of SM particles but, on the other hand, enhanced by the
addition of the $h\to SS$ channel. It thus provides an interesting observable
to probe this scenario.

The width of the $h$ boson can be inferred at the LHC using three methods:

  * •

The cross section of off-shell $h$-boson production is proportional to
$[(s-m_{h}^{2})^{2}+m_{h}^{2}\Gamma_{H}^{2}]^{-1}$. Measuring it at different
values of $s$ allows an extraction of the width under the assumption that the
couplings of $h$ to the particles involved are independent of $s$ (this is not
a good assumption for several BSM models). The most precise extraction
Sirunyan:2019twz  has been done in the $ZZ$ decay mode by the CMS
collaboration using $\sim 80$ fb-1 of run-2 data, and gives
$3.2^{+2.8}_{-2.2}$ MeV. It is foreseen that, with the HL-LHC and improvements
in the theoretical calculations, $\Gamma_{H}$ will be extracted with a
precision of 20% using this method Cepeda:2019klc .

  * •

Using the diphoton decay mode it is also possible to determine the width
through interference effects. This is complementary, as well as theoretically
robust, and will be able to probe values of $(8-22)\times\Gamma_{h}$
Cepeda:2019klc .

  * •

Using the coupling measurements and assuming that $\kappa_{Z}\leq 1$ (which
holds if the dominant effect is due to $\sin\theta\neq 0$), $\Gamma_{h}$ can
be extracted with a precision of 5% with the HL-LHC. This determination of the
width is only possible if one assumes that there are no undetected final
states. We can either fix the width and determine the BR to undetected
particles or we can set the latter to zero and determine the width.

In $e^{+}e^{-}$ colliders, $\Gamma_{h}$ can be measured based on
$\sigma(e^{+}e^{-}\to Zh)$ and ${\rm BR}(h\to ZZ^{*})$:

$\frac{\sigma(e^{+}e^{-}\to Zh)}{{\rm BR}(h\to
ZZ^{*})}=\frac{\sigma(e^{+}e^{-}\to Zh)}{\Gamma(h\to
ZZ^{*})/\Gamma_{h}}=\left[\frac{\sigma(e^{+}e^{-}\to Zh)}{\Gamma(h\to
ZZ^{*})}\right]_{\rm SM}\times\Gamma_{h},$ (15)

where we used the fact that, within the framework discussed here, both
$\sigma(e^{+}e^{-}\to Zh)$ and $\Gamma(h\to ZZ^{*})$ are modified from their
SM values by the same factor, $\kappa_{Z}^{2}$. The precision of this
determination is then limited by the statistics of $h\to ZZ^{*}$ decays. In
practice, for most colliders the $\kappa$-framework is used to extract the
width parameter which benefits from additional measurements that are also
sensitive to the width via similar reasoning. In particular, both the
statistical precision and the model dependence can be reduced significantly if
also $\sigma(e^{+}e^{-}\to h\nu_{e}\nu_{e})$ and BR($h\to WW^{*}$) as is
possible for FCCee at 365 GeV, CLIC (at all proposed energies) and ILC at 500
GeV. This explains the substantial improvement observed when higher energy
data are included. The estimated sensitivities of future $e^{+}e^{-}$
colliders to the total width are given in Table 3. It is much superior to the
first and second determinations of the width at the LHC discussed above. A
word of caution is, however, in place here. While the assumption that both the
$Zh$ cross section and the $h\to ZZ$ decay width are modified by one simple
factor, $\kappa_{Z}^{2}$, holds in the framework discussed in this subsection
(extending the SM with a single light real singlet scalar), it is a model-
dependent assumption, and cannot be assumed in general Barklow:2017suo . Thus,
for ILC, the width has been extracted using an EFT fit with use of
polarization and angular information instead. This is the reason that we quote
also the method used in each experiment in Table 3.

Table 3: Precision on $\Gamma_{h}$ for the following $e^{+}e^{-}$ colliders: CLIC Robson:2018zje , CEPC CEPCStudyGroup:2018ghi , ILC Bambade:2019fyw  and FCCee Mangano:2018mur . Also shown is the method used to determine the width. Collider |  $\sqrt{s}$ [TeV] |  ${\cal L}$ [ab-1] |  $\delta\Gamma_{h}/\Gamma_{h}$ [%] | method
---|---|---|---|---
CLIC 380 | $0.38$ | $1.0$ | $4.7$ | $\kappa$
CLIC 1.5 | $0.38+1.5$ | $2.5$ | $2.6$ | $\kappa$
CLIC 3.0 | $0.38+1.5+3$ | $5$ | $2.5$ | $\kappa$
ILC 250 | $0.25$ | $2.0$ | $2.4$ | EFT
ILC 500 | $0.25+0.5$ | $2.0+4.0$ | $1.6$ | EFT
CEPC | $0.25$ | $5.6$ | 2.8 | $\kappa$
FCCee 240 | $0.24$ | $5.0$ | $2.7$ | $\kappa$
FCCee |  $0.24$+0.365 | $6.5$ | $1.3$ | $\kappa$

###  III.2 $M_{s}>M_{h}/2$

The case of $M_{s}>M_{h}/2$ is more challenging. In this case, there is a
universal modification of all $hxx$ couplings, see Eq. (6). The partial width
into any SM final state, $\Gamma(h\to f)$, as well as the total Higgs width,
$\Gamma_{h}$, change by the same factor, $(1+\delta g_{h})^{2}$. Consequently,
the branching ratios into the various final states are unchanged from the SM,
but the production rates are modified:

$\mu_{i}^{f}\equiv\frac{\sigma_{i}(pp\to h)\times{\rm BR}(h\to
f)}{\left[\sigma_{i}(pp\to h)\times{\rm BR}(h\to f)\right]_{\rm SM}}=(1+\delta
g_{h})^{2}.$ (16)

Here $i$ is the Higgs production mode (ggF, VBF, etc.). If there is no $Z_{2}$
symmetry, then doublet-singlet mixing is allowed, i.e. $\sin 2\theta\neq 0$
(see Eq. (3)), and the dominant contribution to $\delta g_{h}$ in Eq. (6) is
likely to come from the $(\cos\theta-1)$ term, and consequently

$\mu_{i}^{f}\approx\cos^{2}\theta.$ (17)

If there is an unbroken $Z_{2}$ symmetry,

$(\mu_{i}^{f})^{(Z_{2})}=1-\frac{|\lambda_{hs}|^{2}v^{2}}{8\pi^{2}}I_{B}(M_{h}^{2};M_{s}^{2},M_{s}^{2}).$
(18)

For $M_{s}^{2}\gg M_{h}^{2}$, we have

$(\mu_{i}^{f})^{(Z_{2})}(M_{s}^{2}\gg M_{h}^{2})\approx
1+\frac{|\lambda_{hs}|^{2}}{48\pi^{2}}\frac{v^{2}}{M_{s}^{2}}.$ (19)

Under the assumption that the values of the signal strengths $\mu_{i}^{f}$ are
the same for all production processes $i$ and decay channels $f$, a fit to the
ATLAS and CMS data at $\sqrt{s}=7$ and 8 TeV, with $\mu$ as the parameter of
interest, results in the best fit value Khachatryan:2016vau :

$\mu=1.09^{+0.11}_{-0.10}.$ (20)

With the full HL-LHC dataset, the LHC now projects, for the two best measured
production rates, $1.6\%$ for the ggF channel and $3.1\%$ for the VBF channel.
There is still a question of theory uncertainties on the ggF and VBF cross
sections. One can argue that the theory precision on VBF will be
$\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}1\%$ while for ggF this is harder to estimate. In principle
the above numbers already fold in acceptance uncertainties due to theory but
not yet overall normalization uncertainties.

As concerns the ILC, Ref. Barklow:2017suo  (Table 6) estimates the accuracy on
$\sigma_{Zh}$ to be $0.7\%$, while for CEPC and FCCee a precision of 0.5% is
expected. The accuracy on the measurement of $\sigma_{Zh}$ in various proposed
$e^{+}e^{-}$ colliders is presented in Table 4.

Table 4: Precision on the dominant Higgs production cross sections for the following $e^{+}e^{-}$ colliders: CLIC Robson:2018zje , CEPC CEPCStudyGroup:2018ghi , ILC Barklow:2017suo  and FCCee. Collider |  $\sqrt{s}$ [TeV] |  ${\cal L}$ [ab-1] |  $\delta\sigma_{Zh}/\sigma_{Zh}$ [%]
---|---|---|---
CLIC | $0.38$ | $1.0$ | $1.3$
ILC | $0.25$ | $2.0$ | $0.7$
CEPC | $0.25$ | $5.6$ | $0.5$
FCCee | $0.24$ | $5.0$ | $0.5$

A particularly interesting issue in which the addition of a singlet scalar to
the SM is relevant is the possibility that its coupling to the Higgs field
makes the EWPT first order. Ref. Katz:2014bha  obtains that the lower bound on
$|\lambda_{hs}|^{2}/M_{s}^{2}$ is such that

$\mu_{i}^{f}-1\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.6\%.$ (21)

We conclude that the ILC, CEPC and FCCee may be able to lend support or
exclude this scenario as they reach a precision of 0.5-0.7%. However, the
sensitivity may only be at the level of one standard deviation for all three
colliders if the deviation is near its lower bound.

At CLIC, this scenario has been studied in detail clicbsm  and it was
estimated that, based on the Higgs coupling measurements, a constraint of

$\mu_{i}^{f}-1\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.24\%\ \mbox{at 95\% C.L.}$ (22)

can be derived when including all stages, making it sensitive to probing the
order of the EWPT.

The width measurements presented in Table 3 are also sensitive to this
scenario but the precision is inferior than that of the total cross sections.

##  IV What is the solution of the flavor puzzle(s)?

There are several reasons that make the study of flavor physics via Higgs
physics well motivated. First, flavor physics raises three puzzles Nir:2013maa
:

  * •

The Standard Model flavor puzzle: Why is there structure (smallness and
hierarchy) in the charged fermion masses and the CKM mixing angles?

  * •

The neutrino flavor puzzle: Why, in contrast to the charged fermions, there
seems to be no structure (neither hierarchy nor degeneracy nor smallness) in
the neutrino-related flavor parameters?

  * •

The new physics flavor puzzle: If there is new physics at the TeV scale, what
is the mechanism (alignment and/or degeneracy) that prevents it from
significantly modifying the SM predictions for flavor changing neutral current
processes?

Various models have been suggested to answer one or more of these questions.
The best hope to make further progress is by measuring new flavor parameters
(beyond the matrix elements of the CKM matrix). Measurements of the Yukawa
couplings of $h$ provide such an opportunity.

Second, within the Standard Model (SM), flavor changing neutral current (FCNC)
processes are suppressed by three factors:

  * •

Loop suppression;

  * •

CKM suppression;

  * •

GIM suppression.

This unique situation allows the measurements of FCNC processes to probe new
physics at very high scales. The search for Higgs-related flavor violating
processes, such as $t\to ch$ or $h\to\tau\mu$, provides a new arena for FCNC
measurements.

Within the SM, there is a clear prediction for the tree-level values of the
Yukawa couplings:

$Y_{F}=(\sqrt{2}/v)M_{F}\ \ \ (F=U,D,E).$ (23)

This relation between the Yukawa matrices and the mass matrices encompasses,
in fact, three features:

  * •

Proportionality: The diagonal Yukawa couplings ($y_{f}\equiv Y_{ff}$) are
proportional to the corresponding masses. For example,

$y_{\mu}/y_{\tau}=m_{\mu}/m_{\tau}.$ (24)

  * •

Factor of proportionality: The factor that connects the mass to the diagonal
Yukawa coupling is $\sqrt{2}/v$. For example,

$y_{\tau}=(\sqrt{2}/v)m_{\tau}.$ (25)

  * •

Diagonality: The off-diagonal Yukawa couplings vanish. For example,

$Y_{\mu\tau}=0.$ (26)

The three examples in Eqs. (24), (25) and (26) demonstrate that, by measuring
(or constraining) the rates of $h\to\tau\tau$, $h\to\mu\mu$ and $h\to\tau\mu$,
we can test each of the three features predicted by the SM separately. In case
that deviations from the SM predictions are established, the pattern of
deviations will allow us to distinguish between various frameworks that aim to
solve the flavor puzzles. This is demonstrated in Table 5, where the
predictions of various flavor models for the Yukawa couplings are given
(adapted from Dery:2013rta ). In this Table, the scale $\Lambda$ is the scale
that suppresses dimension-six terms in the SM effective-field-theory (SM-EFT)
of the form

$\frac{z_{ij}}{\Lambda^{2}}(\Phi^{\dagger}\Phi)\overline{L_{i}}\tilde{\Phi}E_{j},$
(27)

where $L_{i}$ are the lepton doublet fields in the $(1,2)_{-1/2}$
representation, and $E_{j}$ are the lepton singlet fields in the $(1,1)_{-1}$
representation.

Table 5: Predictions for di-lepton Higgs decays in various flavor models: SM, natural flavor conservation (NFC), minimal supersymmetric SM (MSSM), minimal flavor violation (MFV), Froggatt-Nielsen (FN) models and Giudice-Lebedev (GL). (See Dery:2013rta  for details.) Model |  $\frac{y_{\tau}}{\sqrt{2}m_{\tau}/v}$ | $\frac{y_{\mu}/y_{\tau}}{m_{\mu}/m_{\tau}}$ |  $\frac{{\rm BR}(h\to\mu\tau)}{{\rm BR}(h\to\tau\tau)}$
---|---|---|---
SM | $1$ | $1$ | $0$
NFC | $V_{h\ell}^{*}v/v_{\ell}$ | $1$ | $0$
MSSM | $\sin\alpha/\cos\beta$ | $1$ | $0$
MFV | $1+{\cal O}(v^{2}/\Lambda^{2})$ | $1+{\cal O}(m_{\tau}^{2}/\Lambda^{2})$ | $0$
FN | $1+{\cal O}(v^{2}/\Lambda^{2})$ | $1+{\cal O}(v^{2}/\Lambda^{2})$ | ${\cal O}(|U_{23}|^{2}v^{4}/\Lambda^{4})$
GL | $3$ | $5/3$ | ${\cal O}[(25/9)(m_{\mu}^{2}/m_{\tau}^{2})]$

In Table 6, we present the predictions in various theoretical models for the
flavor changing $t\to hq$ decays. The numerical estimates for the SM, 2HDM,
MSSM and RS frameworks are taken from Ref. Agashe:2013hma  (Table 1-7). The
parametric suppression factors in the SM-EFT subject to minimal flavor
violation (MFV) or to Froggatt-Nielsen (FN) selection rules are taken from
Refs. Dery:2013aba  and Dery:2014kxa , respectively.

Table 6: Predictions BR($t\to hq$) ($q=u,c$) in various models. (See Agashe:2013hma ; Dery:2013aba ; Dery:2014kxa  for details.) | SM | 2HDM | MSSM | RS | SM-EFT(MFV) | SM-EFT(FN)
---|---|---|---|---|---|---
$t\to hc$ | $3\times 10^{-15}$ | $2\times 10^{-3}$ | $\leq 10^{-5}$ | $\leq 10^{-4}$ | $y_{b}^{4}(V_{cb}V_{tb})^{2}(v/\Lambda)^{4}$ | $|V_{cb}|^{2}(v/\Lambda)^{4}$
$t\to hu$ | $2\times 10^{-17}$ | $6\times 10^{-6}$ | $\leq 10^{-5}$ | $-$ | $y_{b}^{4}(V_{ub}V_{tb})^{2}(v/\Lambda)^{4}$ | $|V_{ub}|^{2}(v/\Lambda)^{4}$

In the following, the current measurements and expected accuracy of future
measurements for the parameters relevant to flavor are discussed. For this
purpose, we assume the future collider parameters presented in Section II.

The present status of the measured $y_{f}$ values is presented in Table 7, and
its compatibility with the SM prediction of proportionality is presented in
Fig. 1. For the various colliders, different methods are used to extract the
couplings, leading to some model-dependence at the level of $(0.5-1.0)$%. For
the third generation fermions the current LHC precision is about $(10-25)$%.
With HL-LHC it will be improved to $(2-4)$%. The other future machines will
improve the precision by another factor of $\sim 3$.

Table 7: Experimental status and future projections of the diagonal Yukawa couplings $y_{f}$, and the accuracy estimated for future experiments in %. The accuracy quoted for future experiments is based on combining those data with the HL-LHC, except for CLIC and FCC240 where only the accuracy of the future collider is stated. Upper bounds are given at 95% confidence level. The first line in each row shows the CMS result and the second line the ATLAS result. In both cases, a BSM contribution is allowed, the $\gamma\gamma$ and $gg$ loop processes are treated with effective couplings, and the $Z\gamma$ process is resolved. For the $e^{+}e^{-}$ colliders, the same assumptions on the loops are made when using the so-called $\kappa$-framework. For LHC and HL-LHC Cepeda:2019klc  , BR${}_{\textrm{BSM}}=0$ is assumed. For ILC an EFT fit is used to extract the values and the values are combined with HL-LHC Bambade:2019fyw . For all other colliders the $\kappa$-fit results are quoted from Refs, clic ; CEPCStudyGroup:2018ghi ; Mangano:2018mur ; lhec  and no combination with HL-LHC is made. All upper limits are given at 95% C.L.. For the FCC-ee, an upper limit of 1.6 can be set on $y_{e}/y_{e}^{\rm SM}$ if one year of running at $\sqrt{s}\approx m_{h}$ is performed. When no value is available in the literature, a $-$ is shown.  Observable |  Current range | HL-LHC | ILC250 | ILC250+500 | CLIC380 | CLIC3000 | CEPC | FCC240 | FCC365 | LHeC
---|---|---|---|---|---|---|---|---|---|---
|  |  $\delta y/y$ (%)
$y_{t}/y_{t}^{\rm SM}$ |  |  $1.02^{+0.19}_{-0.15}$ CMSrun2hcomb
---
$1.05^{+0.14}_{-0.13}$ ATLASrun2hcomb
3.4 | $-$ | $6.3$ | $-$ | $2.9$ | $-$ | $-$ | $-$ | $-$
$y_{b}/y_{b}^{\rm SM}$ |  |  $0.91^{+0.17}_{-0.16}$ CMSrun2hcomb
---
$0.85^{+0.13}_{-0.14}$ ATLASrun2hcomb
3.7 | $1.0$ | $0.60$ | $1.3$ | $0.2$ | 1.0 | 1.4 | 0.67 | 1.1
$y_{\tau}/y_{\tau}^{\rm SM}$ |  |  $0.93\pm 0.13$ CMSrun2hcomb
---
$0.95\pm 0.13$ ATLASrun2hcomb
1.9 | $1.2$ | $0.77$ | $2.7$ | $0.9$ | 1.2 | 1.4 | 0.78 | 1.3
$y_{c}/y_{c}^{\rm SM}$ |  $<6.2$ Aaboud:2018fhh ; PerezCharm  | $<220$ | $1.8$ | $1.2$ | $4.1$ | $1.3$ | 1.9 | 1.8 | 1.2 | 3.6
$y_{\mu}/y_{\mu}^{\rm SM}$ |  |  $0.72^{+0.50}_{-0.72}$CMSrun2hcomb
---
$<1.63$ ATLASrun2hcomb
4.3 | $4.0$ | $3.8$ | $-$ | $5.6$ | 5.0 | 9.6 | 3.4 | $-$
$y_{e}/y_{e}^{\rm SM}$ |  $<611$ Khachatryan:2014aep  | $-$ | $-$ |  | $-$ | $-$ | $-$ | $-$ | $<1.6^{(+)}$ | $-$

For the second (and first) generation fermions currently only upper limits are
available. As concerns $y_{\mu}$, the HL-LHC will measure it to better than
5%, and future $e^{+}e^{-}$ colliders are not expected to improve this further
significantly. FCChh will be able to measure the ratio ${\rm
BR}(h\to\mu\mu)/BR(h\to 4\mu)$ with a precision of $1.3$%. As concerns
$y_{c}$, the HL-LHC expects to be sensitive to values 6-21 times larger than
the SM value. Future colliders will probe it in earnest with a precision of
(1-2)%. At hadron colliders it is expected that very large anomalous values of
the strange quark coupling can be probed on $\phi\gamma$ decays Cepeda:2019klc
: for HL-LHC (FCC) the sensitivity is $-35<y_{s}/y_{b}^{\rm SM}<57$
($-2<y_{s}/y_{b}^{\rm SM}<24$). At $e^{+}e^{-}$ colliders values of ${\cal
O}(10)\times$SM rates might be in reach Duarte-Campderros:2018ouv . As
concerns the first generation, the best hope for probing the SM value of
$y_{e}$ directly is to run one of the circular $e^{+}e^{-}$ colliders at
$\sqrt{s}\approx m_{h}$ for several years. For the up and down quarks the
projected sensitivity of the HL-LHC is more than 1000 times the SM value, and
is only obtained indirectly.

The GL model (see Table 5) is already excluded by the current data presented
in Table 7 (by both the $y_{\tau}$ measurement and the $y_{\mu}$ upper limit).
For the FN model, the current constraints imply a lower bound on the scale of
new physics, $\Lambda/\sqrt{z_{\tau\tau}}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}600$ GeV. The
expected precision of HL-LHC will increase the reach to
$\Lambda/\sqrt{z_{\tau\tau}}\sim 2$ TeV, while some of the future facilities
can reach up to about $\Lambda/\sqrt{z_{\tau\tau}}\sim 4$ TeV.

Figure 1: The currently allowed experimental ranges for the Higgs couplings.
For the Yukawa couplings, we present $\kappa_{f}m_{f}/v$. For the coupling to
the electroweak vector bosons, we present $\sqrt{\kappa_{V}}m_{V}/v$. The SM
prediction is presented by the diagonal solid line. (Keith Ellis, private
communication.)

The present status of measurements and the future projections for off-diagonal
branching ratios are given in Table 8. At present most upper limits are
between 0.1% and 1%. With HL-LHC they will be improved by an order of
magnitude to about $1\times 10^{-4}$. For the $t\to qh$ decays, it can be
expected that the sensitivity of future $e^{+}e^{-}$ colliders is to branching
ratios of order $10^{-5}$, similar to other rare decay modes of the top quark
Mangano:2018mur , but no dedicated study is available. With FCC-hh an
improvement by another 1-2 orders of magnitude is expected, depending on the
systematic uncertainties that will be obtained. These values are somewhat
interesting in view of theories beyond the SM, see Table 6.

Table 8: Experimental status of measurements that depend on off-diagonal Yukawa couplings (assuming SM production rates). The LHC $t\to qh$ measurements are based on 36.1 fb-1. For FCChh the range quoted for $t\to qh$ reflects the size of the systematic uncertainty on the background of (0-5)%. Observable |  Upper bound | HL-LHC | FCC-hh
---|---|---|---
${\rm BR}(t\to ch)$ |  $1.1\times 10^{-3}$ Aaboud:2018oqm  |  $10^{-4}$ Azzi:2019yne  |  $(4.9-16)\times 10^{-6}$ Mangano:2018mur
${\rm BR}(t\to uh)$ |  $1.2\times 10^{-3}$ Aaboud:2018oqm  |  $10^{-5}$ Azzi:2019yne  |  $(2.5-28)\times 10^{-6}$ Mangano:2018mur
${\rm BR}(h\to\tau\mu)$ |  $2.5\times 10^{-3}$ Aad:2016blu ; Sirunyan:2017xzt  |  $5\times 10^{-4}$ Cepeda:2019klc  | $-$
${\rm BR}(h\to\tau e)$ |  $6.1\times 10^{-3}$ Aad:2016blu ; Sirunyan:2017xzt  |  $5\times 10^{-4}$ Atlas:2019qfx  | $-$
${\rm BR}(h\to\mu e)$ |  $3.4\times 10^{-4}$ Khachatryan:2016rke  | $-$ | $-$

Finally, let us mention that, while the third generation couplings are already
constrained to be within about 10% of the SM values, large deviations might
still arise in measurements of the Yukawa couplings of the first two
generations and of flavor changing decays. The experimental effort to search
for these decays has thus the potential to make surprising discoveries, even
if the sensitivity is far from the SM predictions.

##  V Additional Questions

###  V.1 Is $h$ alone?

Within the SM, there is a single real scalar field, the Higgs boson $h_{\rm
SM}$. It originates from a field $\Phi(1,2)_{1/2}$ that is a doublet of the
$SU(2)$ gauge symmetry. In the unitary gauge, the Higgs field is given by

$\Phi=\left(\begin{matrix}0\cr\frac{1}{\sqrt{2}}(v+h_{\rm
SM})\cr\end{matrix}\right).$ (28)

The scalar $h$ discovered in the ATLAS and CMS experiments Aad:2012tfa ;
Chatrchyan:2012xdj  has measured properties that are all consistent, within
present experimental accuracy, with the SM predictions for the Higgs boson
Khachatryan:2016vau . The question that is being asked is whether the observed
Higgs boson is the SM Higgs boson. In fact, one can separate this question to
two:

  1. 1.

Does $h$ originate from purely a doublet?

  2. 2.

If there is more than a single doublet, is $h$ in exactly the direction of
$v$?

As concerns the first question, an example of a scenario where $h$ does not
originate from purely a doublet is that of doublet-singlet mixing, discussed
in Section III. As concerns the second question, we briefly discuss in this
subsection the scenario of two Higgs doublet models (2HDM).

In a 2HDM with Natural Flavor Conservation (NFC), there are three well defined
bases:

  1. 1.

The NFC basis $(\Phi_{1},\Phi_{2})$, where each fermionic sector couples to
either $\Phi_{1}$ or $\Phi_{2}$, but not to both.

  2. 2.

The Higgs basis $(\Phi_{v},\Phi_{A})$, where
$\langle\Phi_{v}\rangle=v/\sqrt{2}$ and $\langle\Phi_{A}\rangle=0$.

  3. 3.

The mass basis $(\Phi_{h},\Phi_{H})$, which is the mass basis for the CP-even
neutral scalars $h$ and $H$.

The rotation angle from the $(\Phi_{1},\Phi_{2})$ to the $(\Phi_{v},\Phi_{A})$
[$(\Phi_{h},\Phi_{H})$] basis is denoted $\beta$ [$\alpha$]. We define
$\kappa_{x}$ via

$\kappa_{x}\equiv\frac{g_{hxx}}{g_{hxx}^{\rm SM}}.$ (29)

In all 2HDMs we have ($V=W,Z$)

$\displaystyle\kappa_{V}$ $\displaystyle=$
$\displaystyle\sin(\beta-\alpha)\simeq 1-\frac{1}{2}\cos^{2}(\beta-\alpha),$
(30) $\displaystyle\kappa_{h}$ $\displaystyle\simeq$ $\displaystyle
1-\frac{2m_{A}^{2}}{m_{h}^{2}}\cos^{2}(\beta-\alpha).$

Note that for $m_{A}^{2}\gg m_{h}^{2}$, we have
$(m_{A}^{2}/m_{h}^{2})\cos^{2}(\beta-\alpha)\simeq\cos(\beta-\alpha)$. In all
NFC models, where by definition $\Phi_{1}$ is the doublet that does not couple
to the up sector,

$\displaystyle\kappa_{t}$ $\displaystyle=$
$\displaystyle\sin(\beta-\alpha)+\frac{\cos(\beta-\alpha)}{\tan\beta},$ (31)
$\displaystyle\kappa_{b}$ $\displaystyle=$
$\displaystyle\sin(\beta-\alpha)-\cos(\beta-\alpha)\tan\beta\ \ {\rm or}\ \
\kappa_{t},$ $\displaystyle\kappa_{\tau}$ $\displaystyle=$
$\displaystyle\sin(\beta-\alpha)-\cos(\beta-\alpha)\tan\beta\ \ {\rm or}\ \
\kappa_{t},$

where the two options on the RHS of $\kappa_{b}$ correspond to the cases where
$b$ couples to $\Phi_{1}$ or $\Phi_{2}$, respectively, and similarly for
$\kappa_{\tau}$. The largest deviations in these models arise in the down or
charged lepton sector, if either or both couple to the doublet to which the up
sector does not couple. If all fermions couple to $\Phi_{2}$, then the largest
deviation is expected in $\lambda_{h}$ Efrati:2014uta . At tree level, the
results for the MSSM are the same as those of a type II 2HDM.

We conclude that the most promising precision measurements of Higgs decays for
obtaining hints of 2HDM are those of $h\to b\bar{b}$ and
$h\to\tau^{+}\tau^{-}$. The future projections of measuring $y_{b}$ and
$y_{\tau}$ in future experiments can be read off Table 7. An even more
powerful probe of 2HDM is the combination of these two couplings with a
measurement of $\kappa_{V}$, see Table 4. With $\kappa_{Z}=\sin(\beta-\alpha)$
measured very accurately, an accuracy of ${\cal O}(1\%)$ on $y_{\tau}$ will
essentially constrain $\cos(\beta-\alpha)\tan\beta$ at that level in NFC
models where the $\tau$ couples to $\Phi_{1}$ only.

###  V.2 Is $h$ elementary?

The question of Higgs compositeness is interesting in its own sake, but also
in the context of mechanisms to explain the smallness of $m_{h}^{2}$ compared
to $m_{\rm Pl}^{2}$. Various models have been proposed where the Higgs is a
pseudo-Goldstone boson which emerges from a strongly interesting sector
ArkaniHamed:2002qy ; Contino:2003ve ; Agashe:2004rs . An effective field
theory framework for the strongly interacting light Higgs (SILH) incorporates
the basic features of a large class of these models Giudice:2007fh . The low
energy effective SILH Lagrangian depends on essentially two parameters:
$m_{\rho}$, the mass scales of new resonances emerging from the strongly
interacting sector, and $g_{\rho}$, their coupling. It is useful to define the
dimensionless combination

$\xi=g_{\rho}^{2}v^{2}/m_{\rho}^{2}.$ (32)

The terms in the SILH Lagrangian that are relevant to our purposes are given
by

${\cal L}_{\rm
SILH}=\frac{c_{H}\xi}{2v^{2}}\partial^{\mu}(\phi^{\dagger}\phi)\partial_{\mu}(\phi^{\dagger}\phi)+\frac{c_{y}\xi
y_{f}}{v^{2}}\phi^{\dagger}\phi\overline{f_{L}}\phi f_{R}+{\rm h.c.}.$ (33)

Naive dimensional analysis (NDA) suggests that $c_{H}$ and $c_{y}$ are ${\cal
O}(1)$. The theoretical framework only allows the parameters to be in the
range

$1\leq g_{\rho}\leq 4\pi,\ \ \ \xi\leq 1.$ (34)

Electroweak precision measurements give stronger constraints. The new physics
contribution to $\hat{S}$ and $\hat{T}$ can be parameterized as follows
Thamm:2015zwa :

$\displaystyle\Delta\hat{S}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{96\pi^{2}}\xi\left[\log\left(\frac{\Lambda}{m_{h}}\right)+6\alpha\right]+\frac{m_{W}^{2}}{m_{\rho}^{2}},$
$\displaystyle\Delta\hat{T}$ $\displaystyle=$ $\displaystyle-\frac{g^{\prime
2}}{32\pi^{2}}\xi\left[\log\left(\frac{\Lambda}{m_{h}}\right)-6\beta\frac{y_{t}^{2}}{g^{\prime
2}}\right],$ (35)

where $\Lambda=4\pi m_{\rho}/g_{\rho}$ and the coefficients
$\alpha,\beta={\cal O}(1)$ and could have either sign. The resulting
constraints read Thamm:2015zwa

$\xi\leq 0.15,\ \ \ m_{\rho}\geq 3\ TeV$ (36)

when marginalizing over $\alpha$ and $\beta$ (or $\xi\leq 0.025$ and
$m_{\rho}\geq 4\ TeV$ for $\alpha=\beta=0$).

The modifications of the Higgs couplings are given by

$\displaystyle\Delta g_{V}/g_{V}^{\rm SM}$ $\displaystyle=$ $\displaystyle-
c_{H}\xi/2,$ $\displaystyle\Delta g_{f}/g_{f}^{\rm SM}$ $\displaystyle=$
$\displaystyle-c_{H}\xi/2-c_{y}\xi,$ $\displaystyle\Delta g_{g}/g_{g}^{\rm
SM}$ $\displaystyle=$ $\displaystyle-c_{H}\xi/2-c_{y}\xi,$
$\displaystyle\Delta g_{\gamma}/g_{\gamma}^{\rm SM}$ $\displaystyle=$
$\displaystyle-c_{H}\xi/2+0.3c_{y}\xi.$ (37)

Thus, the maximal possible deviations are Gupta:2012mi

$\displaystyle\Delta g_{V}/g_{V}^{\rm SM}$ $\displaystyle\approx$
$\displaystyle-0.08c_{H},$ $\displaystyle\Delta g_{f}/g_{f}^{\rm SM}$
$\displaystyle=$ $\displaystyle\Delta g_{g}/g_{g}^{\rm
SM}\approx-0.08-0.15\frac{c_{y}}{c_{H}}$ $\displaystyle\Delta
g_{\gamma}/g_{\gamma}^{\rm SM}$ $\displaystyle\approx$
$\displaystyle-0.08+0.05\frac{c_{y}}{c_{H}}.$ (38)

E.g. for $c_{H}=c_{y}=1$ the maximal deviations are $-0.08$, $-0.20$ and
$-0.03$ for the $g_{V}$, $g_{f,g}$ and $g_{\gamma}$ couplings, respectively.
The estimates of the reach on $c_{H}\xi$ of Higgs-related measurements for
various colliders are collected in Table 9. Comparing this to the maximal
possible deviations given above, it is clear that future colliders will probe
this aspect in a very interesting regime. Currently the precision obtained by
ATLAS and CMS on $\kappa_{Z}$ and $\kappa_{W}$ is about 10% each CMSrun2hcomb
; ATLASrun2hcomb , resulting in a sensitivity of about 20% on $c_{H}\xi$. Ref.
Liu:2018qtb  employs a more general framework, based on nonlinear shift
symmetries acting on $h$ and assuming custodial symmetry in the strong sector,
and quotes the presently allowed 68% CL range as

$\xi=-0.041^{+0.090}_{-0.094}.$ (39)

Table 9: Summary of the reach on $\xi$ for various collider options, based on Ref. Thamm:2015zwa  but updated using the recent projections on the $\kappa_{V}$ sensitivity. See discussion in Section V.2. Collider | Energy | Luminosity |  $\Delta g_{V}/g_{V}$ (%) | $(c_{H}\xi)_{max}\ [1\sigma]$
---|---|---|---|---
HL-LHC | 14 TeV | 3 ab-1 | 1.3 | $0.026$
ILC | 250 GeV | 2 ab-1 | 0.56 | 0.011
| +500 GeV | 4 ab-1 | 0.38 | 0.008
CLIC | 380 GeV | 1 ab-1 | 0.6 | $0.012$
| +1.4 TeV | 2.5 ab-1 | 0.6 | $0.012$
| +3.0 TeV | 5 ab-1 | 0.6 | $0.012$
FCCee | 240 GeV |  $2\times 5.0$ ab-1 | 0.2 | 0.004
| +350 GeV | 2.6 ab-1 | 0.17 | $0.003$
CEPC | 240 GeV |  $2\times 5.6$ ab-1 | 0.25 | 0.005

Ref. Thamm:2015zwa  compared the sensitivity of the Higgs related measurements
and the direct resonance searches at (and beyond) the LHC within the framework
of composite Higgs models. They reach the conclusion that the Higgs related
measurements will be more sensitive for large $g_{\rho}$. Specifically, the
indirect measurements explore novel territory for $g_{\rho}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}4.5$.

###  V.3 What keeps $m_{h}^{2}\ll m_{\rm Pl}^{2}$?

If the SM is only a low energy effective theory, and at some high scale
$\Lambda_{\rm NP}$ there are new degrees of freedom which couple to the Higgs
boson, then the mass-squared of the Higgs boson get corrections of ${\cal
O}(\Lambda_{\rm NP}^{2})$, and a fine-tuned cancelation with the bare
$\mu^{2}$ term is needed to keep the Higgs mass light. Possible solutions to
this fine-tuning problem can be classified to symmetry-related mechanisms, and
dynamical scenarios.

The two most intensively studied frameworks where the solution is symmetry
related are composite Higgs and Supersymmetry. We discussed the composite
Higgs framework in Section V.2.

As concerns the Higgs couplings in the minimal supersymmetric SM (MSSM), the
situation is similar to the 2HDM discussed in Section V.1 with NFC type II,
and with some additional constraints arising from relations to processes
involving the supersymmetric partners of the SM degrees of freedom. Ref.
Gupta:2012mi  obtained the maximal possible deviations of the $h$ couplings,
assuming that the scalar degrees of freedom of the second Higgs doublet are
too heavy to be directly observed at the LHC. They obtain

$\displaystyle 1-\kappa_{V}$ $\displaystyle\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$ $\displaystyle
0.01,$ $\displaystyle 1-\kappa_{t}$ $\displaystyle\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$
$\displaystyle 0.03$ $\displaystyle 1-\kappa_{b}$ $\displaystyle\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}$
$\displaystyle 0.1-1.$ (40)

A dynamical solution to the fine tuning problem is provided by relaxion models
Graham:2015cka . The Higgs mass depends on a time-dependent VEV of a scalar
field which rolls until it stops at a value much smaller than the cut-off
scale of the theory. Unlike models of composite Higgs or Supersymmetry, where
many new degrees of freedom are predicted at a scale not much higher than the
electroweak symmetry breaking scale, in this framework there is a single new
scalar, SM-singlet, degree of freedom, the relaxion $\phi_{R}$. Much of the
relevant phenomenology comes from the generic situation where the relaxion and
the Higgs mix Flacke:2016szy . Thus, the relaxion inherits the Higgs
couplings, suppressed by the small $\phi_{R}-h$ mixing angle. This scenario
was discussed in Section III, particularly in the case of $M_{s}<M_{h}/2$.

###  V.4 Was the electroweak phase transition (strongly) first order?

At a temperature of order 100 GeV, the Universe went through a transition from
a high temperature symmetric phase ($\langle\Phi\rangle=0$) to a state with
broken electroweak symmetry ($\langle\Phi\rangle=v/\sqrt{2}\neq 0$). If the SM
is a good low energy effective theory up to the TeV scale, then the
electroweak phase transition (EWPT) is a crossover transition. If, on the
other hand, the EWPT was first-order, then electroweak baryogenesis was
possible. In fact, successful EWBG requires that the phase transition is
strongly first order,

$\frac{v(T_{c})}{T_{c}}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}1,$ (41)

where $T_{c}$ is the temperature at which the phase transition takes place.
This requirement usually implies new degrees of freedom that are not much
heavier than the EW scale and which couple to the Higgs field with a coupling
that is not small.

The new physics that can make a first-order EWPT can be broadly classified to
four classes Chung:2012vg , distinguished by the physics that is responsible
for generating the barrier between the EW-symmetric and EW-breaking minima:

  * •

Thermally driven. New bosonic degrees of freedom generate a finite-temperature
term in the effective potential of the form $-T(h^{2})^{3/2}$.

  * •

Tree-level renormalizable operators. Additional scalars, typically
$SU(2)_{L}$-singlets or doublets, mix with the Higgs boson and generate an
effective, temperature-independent, $\phi^{3}$ term.

  * •

Tree level non-renormalizable operators. The most intensively studied example
is a dimension-six term $(\Phi^{\dagger}\Phi)^{3}$ (with a negative $\lambda$
for the $\lambda(\Phi^{\dagger}\Phi)^{2}$ term).

  * •

Loop corrections. The most intensively studied example is an effective term of
the form $\kappa h^{4}\ln(h^{2}/M^{2})$ in the Higgs potential.

Loops involving scalar particles induce a cubic term in the effective
potential at high temperature that can modify the EWPT into a first-order one
(see, for example, Ref. Katz:2014bha ). Consider then a new scalar $S$, with
coupling to the Higgs field $\Phi$:

$V_{S\Phi}=\kappa|S|^{2}|\Phi|^{2}.$ (42)

To make the phase transition first-order, we need $\kappa={\cal O}(1)$ and
$m_{S}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}400$ GeV. This term modifies the following Higgs observables:

  1. 1.

If $S$ is colored, the $hgg$ coupling is modified. Thus, at hadron colliders,
a measurement of $\mu_{\rm ggF}$ is best suited to probe this scenario, which
is expected to have a precision of about 2% at HL-LHC. At lepton colliders,
the best observable is BR($h\to gg$) which can be probed with a precision of
about 1.5-2.5%.

  2. 2.

If $S$ is electromagnetically charged, the $h\gamma\gamma$ coupling is
modified. Thus, a measurement of BR($h\to{\gamma\gamma}$) is best suited to
probe this scenario. At HL-LHC a precision of about 2.6% is expected, and
lepton colliders are not expected to improve this significantly. A substantial
improvement is expected from FCChh with about 0.5%.

  3. 3.

Independently of the quantum numbers of $S$, the $hZZ$ coupling will be
modified, since $S$ renormalizes the Higgs wavefunction Englert:2013tya ;
Craig:2013xia . The case where $S$ is neither colored nor electromagnetically
charged was discussed in Section III.

Models of strongly first-order EWPT due to renormalizable tree-level operators
will be further tested by Higgs measurements of the invisible Higgs decays or
by the bounds on the mixing with singlet scalars, discussed in Section III,
and by various other measurements (see e.g. Profumo:2014opa ; Kotwal:2016tex
). For models of strongly first-order EWPT due to non-renormalizable tree-
level operators,

$V=\mu^{2}(\Phi^{\dagger}\Phi)+\lambda(\Phi^{\dagger}\Phi)^{2}+\frac{\rho}{\Lambda^{2}}(\Phi^{\dagger}\Phi)^{3},$
(43)

the main consequence is a modification of the relation between the Higgs mass,
the VEV and the trilinear Higgs self-coupling:

$\lambda_{hhh}=3\frac{m_{h}^{2}}{v}+6\rho\frac{v^{3}}{\Lambda^{2}}.$ (44)

The required size of $\rho v^{2}/\Lambda^{2}$ is such that deviation of ${\cal
O}(1)$ in $\lambda_{hhh}$ are predicted. Table 10 summarizes the expected
sensitivity of various future colliders.

Table 10: Summary of the reach on $\lambda_{hhh}$ for various collider options. Given are the collider, the center-of-mass energy, the integrated luminosity and the precision that is expected to be achieved on $\lambda_{hhh}$. For all colliders, except FCCee, it is based on di-Higgs measurements. For FCCee it is based on the $\sqrt{s}$ dependence of higher order corrections to the $Zh$ cross section. In all cases, the luminosity quoted is that of the sum of the experiments. The values correspond to the luminosities given in Sec. II Collider | $\sqrt{s}$ |  $\frac{\Delta\lambda_{hhh}}{\lambda_{hhh}}$ (%)
---|---|---
HL-LHC | 14 TeV | $\pm 50$
ILC | 500 GeV | $\pm 17$
CLIC | 3.0 TeV | ${}^{+11}_{-7}$
FCCee | 240+365 GeV | $\pm 40$
HE-LHC | 27 TeV |  $\pm$(10-20)
FCChh | 100 TeV | $\pm 5$

###  V.5 Do CP violating $h$ interactions generate the baryon asymmetry?

Complex Yukawa couplings could play a CP violating role in electroweak
baryogenesis, see e.g. Refs. Shu:2013uua ; Kobakhidze:2015xlz ; Chiang:2016vgf
; Guo:2016ixx . Various suggestions of how to measure such CP violating Higgs
interactions have been made, in particular for $h\tau\tau$ and $htt$
couplings, see e.g. Refs. Harnik:2013aja ; Demartin:2014fia ; Buckley:2015vsa
; AmorDosSantos:2017ayi ; Huang:2015izx ; Hayreter:2016kyv ; Goncalves:2018agy
; Bernlochner:2018opw .

The ATLAS collaboration atlas:cpv  has recently estimated the accuracy of
measuring the CP violating mixing angle $\phi_{\tau}$ defined via

${\cal
L}_{h\tau\tau}=g_{h\tau\tau}[\cos(\phi_{\tau})\bar{\tau}\tau+\sin(\phi_{\tau})\bar{\tau}i\gamma_{5}\tau]h.$
(45)

With 3 ab-1 at $\sqrt{s}=14$ TeV, $\phi_{\tau}$ can be measured with a
statistical precision between $\pm 18^{o}$ and $\pm 33^{o}$, depending on the
energy resolution of the $\pi^{0}$ achieved in the reconstruction at high
pile-up.

##  VI Conclusions

A list of interesting theoretical questions, and a partial list of observables
that are most relevant to making progress on these questions, are summarized
in Table 11.

Table 11: A list of interesting theoretical questions, and a partial list of observables that are most relevant to making progress on these questions. $\kappa_{3}\ (\kappa_{\ell})$ stands for third (first or second) generation couplings. $\mu_{4f}$ stands for the processes $h\to SS\to\overline{f_{1}}f_{1}\overline{f_{2}}f_{2}$. For more details, see the text. Question | $\kappa_{V}$ | $\kappa_{3}$ | $\kappa_{g}$ | $\kappa_{\gamma}$ | $\lambda_{hhh}$ | $\sigma_{hZ}$ | BRinv | BRund | $\kappa_{\ell}$ | $\mu_{4f}$ | BRτμ | $\Gamma_{h}$
---|---|---|---|---|---|---|---|---|---|---|---|---
Is $h$ Alone? | $+$ | $+$ |  |  | $+$ | $+$ |  |  |  | $+$ |  | $+$
Is $h$ elementary? | $+$ | $+$ | $+$ | $+$ |  | $+$ |  |  |  |  |  |
Why $m_{h}^{2}\ll m_{\rm Pl}^{2}$? | $+$ | $+$ |  |  |  |  | $+$ | $+$ |  | $+$ |  | $+$
1st order EWPT? |  |  | $+$ | $+$ | $+$ | $+$ |  |  |  | $+$ |  |
CPV? |  |  $+$(CP) |  |  |  |  |  |  |  |  |  |
Light singlets? |  |  |  |  |  |  | $+$ | $+$ | $+$ | $+$ |  | $+$
Flavor puzzles? |  | $+$ |  |  |  |  |  |  | $+$ |  | $+$ |

Our review of what can be learned from Higgs precision measurements led us to
three important conclusions, independent of the particular experiments that
will perform these measurements:

  * •

The Higgs program, which is a guaranteed program of any future collider,
touches some of the most significant open questions in particle physics and
cosmology. In particular, this program may lead to progress in two puzzles of
particle cosmology: dark matter and the baryon asymmetry.

  * •

No Higgs related measurement will go unnoticed. Each of the production modes
and each of the decay modes carries in it information that is relevant to
important questions. Higgs decays to two bosons, two fermions, invisible
modes, four fermions, as well as Higgs production by gluon-gluon fusion,
$t\bar{t}h$, and di-Higgs production, all of these (and other measurements)
are highly valuable for better understanding of particle physics and
cosmology.

  * •

For many of the interesting observables, the proposed future colliders
significantly extend the sensitivity beyond that reachable by the HL-LHC
alone.

### Acknowledgements

We thank the members of the Scientific Policy Committee (SPC) of CERN for
initiating discussions that led to this work, and for comments on the
manuscript. We thank the participants of the workshop “Voyages Beyond the
Standard Model II,” and Gilad Perez and Georg Weiglein for useful discussions
and comments on the manuscript. BH acknowledges support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC 2121 ”Quantum Universe” – 390833306. YN is the Amos
de-Shalit chair of theoretical physics, and is supported by grants from the
Israel Science Foundation (grant number 394/16), the United States-Israel
Binational Science Foundation (BSF), Jerusalem, Israel (grant number 2014230),
and the I-CORE program of the Planning and Budgeting Committee and the Israel
Science Foundation (grant number 1937/12).

## References

  * (1) G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716, 1 (2012) [arXiv:1207.7214 [hep-ex]].
  * (2) S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716, 30 (2012) [arXiv:1207.7235 [hep-ex]].
  * (3) F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964).
  * (4) P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964).
  * (5) F. Bordry, M. Benedikt, O. Brüning, J. Jowett, L. Rossi, D. Schulte, S. Stapnes and F. Zimmermann, arXiv:1810.13022 [physics.acc-ph].
  * (6) P. Bambade et al., arXiv:1903.01629 [hep-ex].
  * (7) CEPC Study Group, arXiv:1811.10545 [hep-ex].
  * (8) M. Mangano et al., CERN-ACC-2018-0056.
  * (9) P. Huang, A. J. Long and L. T. Wang, Phys. Rev. D 94, no. 7, 075008 (2016) [arXiv:1608.06619 [hep-ph]].
  * (10) N. Craig, C. Englert and M. McCullough, Phys. Rev. Lett. 111, no. 12, 121803 (2013) [arXiv:1305.5251 [hep-ph]].
  * (11) D. Curtin, P. Meade and C. T. Yu, JHEP 1411, 127 (2014) [arXiv:1409.0005 [hep-ph]].
  * (12) J. Fan, M. Reece and L. T. Wang, JHEP 1508, 152 (2015) [arXiv:1412.3107 [hep-ph]].
  * (13) C. Frugiuele, E. Fuchs, G. Perez and M. Schlaffer, JHEP 1810, 151 (2018) [arXiv:1807.10842 [hep-ph]].
  * (14) A. M. Sirunyan et al. [CMS Collaboration], [arXiv:1809.05937 [hep-ex]].
  * (15) M. Aaboud et al. [ATLAS Collaboration], [arXiv:1809.06682 [hep-ex]].
  * (16) A. Robson and P. Roloff, arXiv:1812.01644 [hep-ex].
  * (17) T. Barklow, K. Fujii, S. Jung, R. Karl, J. List, T. Ogawa, M. E. Peskin and J. Tian, Phys. Rev. D 97, no. 5, 053003 (2018) [arXiv:1708.08912 [hep-ph]].
  * (18) P. Bechtle, S. Heinemeyer, O. Stål, T. Stefaniak and G. Weiglein, JHEP 1411, 039 (2014) [arXiv:1403.1582 [hep-ph]].
  * (19) T. Flacke, C. Frugiuele, E. Fuchs, R. S. Gupta and G. Perez, JHEP 1706, 050 (2017) [arXiv:1610.02025 [hep-ph]].
  * (20) ATLAS and CMS Collaborations, arXiv:1902.10229 [hep-ex].
  * (21) CMS Collaboration, Phys. Lett. B 785, 462 (2018) [arXiv:1805.10191 [hep-ex]].
  * (22) CMS Collaboration, JHEP 1710, 076 (2017) [arXiv:1701.02032 [hep-ex]].
  * (23) M. Aaboud et al. [ATLAS Collaboration], Phys. Lett. B 790, 1 (2019) [arXiv:1807.00539 [hep-ex]].
  * (24) CMS Collaboration, JHEP 1811, 018 (2018) [arXiv:1805.04865 [hep-ex]].
  * (25) ATLAS Collaboration, Phys. Lett. B 782, 750 (2018) [arXiv:1803.11145 [hep-ex]].
  * (26) CMS Collaboration, arXiv:1901.00174 [hep-ex].
  * (27) M. Cepeda et al. [Physics of the HL-LHC Working Group], arXiv:1902.00134 [hep-ph].
  * (28) ATLAS and CMS Collaborations, JHEP 1608, 045 (2016) [arXiv:1606.02266 [hep-ex]].
  * (29) A. Katz and M. Perelstein, JHEP 1407, 108 (2014) [arXiv:1401.1827 [hep-ph]].
  * (30) Y. Nir, Phys. Scripta T 158, 014005 (2013).
  * (31) A. Dery, A. Efrati, Y. Hochberg and Y. Nir, JHEP 1305, 039 (2013) [arXiv:1302.3229 [hep-ph]].
  * (32) K. Agashe et al. [Top Quark Working Group], arXiv:1311.2028 [hep-ph].
  * (33) A. Dery, A. Efrati, G. Hiller, Y. Hochberg and Y. Nir, JHEP 1308, 006 (2013) [arXiv:1304.6727 [hep-ph]].
  * (34) A. Dery, A. Efrati, Y. Nir, Y. Soreq and V. Susič, Phys. Rev. D 90, 115022 (2014) [arXiv:1408.1371 [hep-ph]].
  * (35) CMS Collaboration, arXiv:1809.10733 [hep-ex].
  * (36) ATLAS Collaboration, ATLAS note ATLAS-CONF-2018-031.
  * (37) T. K. Charles et al. [CLICdp and CLIC Collaborations], arXiv:1812.06018 [physics.acc-ph].
  * (38) J. Duarte-Campderros, G. Perez, M. Schlaffer and A. Soffer, arXiv:1811.09636 [hep-ph].
  * (39) Uta Klein, FCC-eh as a Higgs Facility. Talk at FCC Week, Amsterdam, 2018.
  * (40) ATLAS Collaboration, Phys. Rev. Lett. 120, 211802 (2018) [arXiv:1802.04329 [hep-ex]].
  * (41) G. Perez, Y. Soreq, E. Stamou, and K. Tobioka, Phys. Rev. D 92 no. 3, 033016 (2015), [arXiv:1503.00290]
  * (42) CMS Collaboration, Phys. Lett. B 744, 184 (2015) [arXiv:1410.6679 [hep-ex]].
  * (43) M. Aaboud et al. [ATLAS Collaboration], [arXiv:1812.11568 [hep-ex]].
  * (44) P. Azzi et al. [HL-LHC Collaboration and HE-LHC Working Group], arXiv:1902.04070 [hep-ph].
  * (45) ATLAS Collaboration, Eur. Phys. J. C 77, no. 2, 70 (2017) [arXiv:1604.07730 [hep-ex]].
  * (46) A. M. Sirunyan et al. [CMS Collaboration], JHEP 1806, 001 (2018) [arXiv:1712.07173 [hep-ex]].
  * (47) CMS Collaboration, Phys. Lett. B 763, 472 (2016) [arXiv:1607.03561 [hep-ex]].
  * (48) A. Efrati and Y. Nir, arXiv:1401.0935 [hep-ph].
  * (49) N. Arkani-Hamed, A. G. Cohen, E. Katz and A. E. Nelson, JHEP 0207, 034 (2002) [hep-ph/0206021].
  * (50) R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B 671, 148 (2003) [hep-ph/0306259].
  * (51) K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B 719, 165 (2005) [hep-ph/0412089].
  * (52) G. F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, JHEP 0706, 045 (2007) [hep-ph/0703164].
  * (53) A. Thamm, R. Torre and A. Wulzer, JHEP 1507, 100 (2015) [arXiv:1502.01701 [hep-ph]].
  * (54) R. S. Gupta, H. Rzehak and J. D. Wells, Phys. Rev. D 86, 095001 (2012) [arXiv:1206.3560 [hep-ph]].
  * (55) D. Liu, I. Low and Z. Yin, arXiv:1809.09126 [hep-ph].
  * (56) P. W. Graham, D. E. Kaplan and S. Rajendran, Phys. Rev. Lett. 115, no. 22, 221801 (2015) [arXiv:1504.07551 [hep-ph]].
  * (57) D. J. H. Chung, A. J. Long and L. T. Wang, Phys. Rev. D 87, no. 2, 023509 (2013) [arXiv:1209.1819 [hep-ph]].
  * (58) C. Englert and M. McCullough, JHEP 1307, 168 (2013) [arXiv:1303.1526 [hep-ph]].
  * (59) S. Profumo, M. J. Ramsey-Musolf, C. L. Wainwright and P. Winslow, Phys. Rev. D 91, no. 3, 035018 (2015) [arXiv:1407.5342 [hep-ph]].
  * (60) A. V. Kotwal, M. J. Ramsey-Musolf, J. M. No and P. Winslow, Phys. Rev. D 94, no. 3, 035022 (2016) doi:10.1103/PhysRevD.94.035022 [arXiv:1605.06123 [hep-ph]].
  * (61) J. Shu and Y. Zhang, Phys. Rev. Lett. 111, no. 9, 091801 (2013) [arXiv:1304.0773 [hep-ph]].
  * (62) A. Kobakhidze, L. Wu and J. Yue, JHEP 1604, 011 (2016) [arXiv:1512.08922 [hep-ph]].
  * (63) C. W. Chiang, K. Fuyuto and E. Senaha, Phys. Lett. B 762, 315 (2016) [arXiv:1607.07316 [hep-ph]].
  * (64) H. K. Guo, Y. Y. Li, T. Liu, M. Ramsey-Musolf and J. Shu, Phys. Rev. D 96, no. 11, 115034 (2017) [arXiv:1609.09849 [hep-ph]].
  * (65) R. Harnik, A. Martin, T. Okui, R. Primulando and F. Yu, Phys. Rev. D 88, no. 7, 076009 (2013) [arXiv:1308.1094 [hep-ph]].
  * (66) F. Demartin, F. Maltoni, K. Mawatari, B. Page and M. Zaro, Eur. Phys. J. C 74, no. 9, 3065 (2014) [arXiv:1407.5089 [hep-ph]].
  * (67) M. R. Buckley and D. Goncalves, Phys. Rev. Lett. 116, no. 9, 091801 (2016) [arXiv:1507.07926 [hep-ph]].
  * (68) S. Amor Dos Santos et al., Phys. Rev. D 96, 013004 (2017) [arXiv:1704.03565 [hep-ph]].
  * (69) F. P. Huang, P. H. Gu, P. F. Yin, Z. H. Yu and X. Zhang, Phys. Rev. D 93, no. 10, 103515 (2016) [arXiv:1511.03969 [hep-ph]].
  * (70) A. Hayreter, X. G. He and G. Valencia, Phys. Lett. B 760, 175 (2016) [arXiv:1603.06326 [hep-ph]]; Phys. Rev. D 94, 075002 (2016) [arXiv:1606.00951 [hep-ph]].
  * (71) D. Goncalves, K. Kong and J. H. Kim, JHEP 1806, 079 (2018) [arXiv:1804.05874 [hep-ph]].
  * (72) F. U. Bernlochner, C. Englert, C. Hays, K. Lohwasser, H. Mildner, A. Pilkington, D. D. Price and M. Spannowsky, Phys. Lett. B 790, 372 (2019) [arXiv:1808.06577 [hep-ph]].
  * (73) J. de Blas, R. Franceschini, F. Riva, P. Roloff, U. Schnoor, M. Spannowsky, J. D. Wells, A. Wulzer and J. Zupan CERN Yellow Reports: Monographs, Vol. 3/2018, CERN-2018-009-M (CERN, Geneva, 2018). [arXiv:1812.02093]
  * (74) The ATLAS collaboration, ATLAS-PHYS-PUB-2019-008.