DESY-22-128

Higgs-boson visible and invisible constraints
on hidden sectors

Thomas<EMAIL_ADDRESS>& Mathias
<EMAIL_ADDRESS>
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

ABSTRACT

We investigate the impact of interactions between hidden sectors and the
discovered Higgs boson $h_{125}$, allowing for additional invisible decay
channels of $h_{125}$. We perform $\chi^{2}$-fits to the measurements of the
Higgs-boson cross sections as a function of the invisible branching ratio and
different combinations of coupling modifiers, where the latter quantify
modifications of the couplings of $h_{125}$ compared to the predictions of the
Standard Model. We present generic results in terms of exclusion limits on the
coupling modifiers and the invisible branching ratio of $h_{125}$.
Additionally, we apply our results to a variety of concrete model realizations
containing a hidden sector: dark matter within Higgs- and singlet-portal
scenarios, models featuring (pseudo) Nambu-Goldstone bosons and two Higgs
doublet extensions. One of the main conclusions of our work is that in a wide
class of models the indirect constraints resulting from the measurements of
the cross sections of $h_{125}$ provide substantially stronger constraints on
the invisible Higgs-boson branching ratio compared to the direct limits
obtained from searches for the invisible decay of $h_{125}$. However, we
demonstrate that the presence of an invisible decay mode of $h_{125}$ can also
open up parameter space regions which otherwise would be excluded as a result
of the indirect constraints. As a byproduct of our analysis, we show that in
light of the new results from the LZ collaboration a fermionic DM candidate
within the simplest Higgs-portal scenario is ruled out under standard
assumptions.

###### Contents

  1. 1 Introduction
  2. 2 Indirect limits on the invisible decay of the Higgs boson $h_{125}$
    1. 2.1 $h_{125}$ with SM couplings and a non-zero $\mathrm{BR}_{\rm inv}$
    2. 2.2 $h_{125}$ with universally modified couplings and a non-zero $\mathrm{BR}_{\rm inv}$
    3. 2.3 $h_{125}$ with non-universally modified couplings and a non-zero $\mathrm{BR}_{\rm inv}$
  3. 3 Application to hidden sector models
    1. 3.1 The Higgs portal dark matter model
    2. 3.2 Singlet portal dark matter
    3. 3.3 (Pseudo) Nambu-Goldstone bosons
    4. 3.4 Two Higgs doublet models
  4. 4 Summary and conclusions
  5. A Singlet portal dark matter: additional content
    1. A.1 Minimization of the potential
    2. A.2 Decay rates

##  1 Introduction

In 2012 a particle was discovered at the Large Hadron Collider (LHC) by both
the ATLAS [1] and CMS collaborations [2] with a mass of about $125\
\mathrm{GeV}$ that within the current experimental uncertainties behaves in
agreement with the predictions for a Higgs boson of the Standard Model (SM) of
particle physics [3, 4]. As a consequence, theories beyond the SM (BSM) in
which the shortcomings of the SM can be addressed have to contain a particle
that plays the role of the discovered Higgs boson $h_{125}$. This is
especially relevant for BSM theories in which $h_{125}$ is coupled to so far
unknown hidden sectors. If the hidden sector contains particles with masses
below $125\ \mathrm{GeV}$, modifications of the properties of $h_{125}$ can be
present as a result of exotic decays into lighter BSM states. Here it is
important to note that, in addition to the fact that the exotic decays can be
searched for at colliders, additional decay modes also suppress the ordinary
decay modes of $h_{125}$ into SM particles. Therefore, the discovered Higgs
boson acts as a probe for new physics beyond the SM (BSM), and the LHC plays a
vital role (and will do so for many more years to come) in order to shed light
on the existence of light hidden sectors that might have escaped discovery
until today.

A remarkable amount of experimental knowledge about the recently discovered
Higgs boson has been gathered at the LHC within the last ten years, where
nature was kind to us by choosing a Higgs-boson mass for which several
production modes and decay channels of $h_{125}$ as predicted by the SM can be
observed at the LHC [5]. The Higgs boson has been discovered in the gluon-
fusion production mode (ggH) with subsequent decay into di-photon pairs and
via its decay into pairs of off-shell vector bosons giving rise to four-lepton
final states [1, 2].333It is remarkabe that the main discovery channel $gg\to
h_{125}\to\gamma\gamma$ involves two loop-induced processes according to the
fact that the Higgs-boson does not carry color- or electric charge. As a
consequence, the Higgs boson is also a valuable probe for indirect effects of
BSM particles that could leave their footprint exclusively via radiative
corrections to such loop-induced couplings of $h_{125}$ (see e.g. Ref. [6]).
This possibility is, however, not further considered in this paper. The fine
mass resolution of the discovery channels permit a measurement of the mass of
$h_{125}$ at the sub-percent level [7, 8]. As of today, in addition to the ggH
production mode, the production via the (tree-level) couplings to vector
bosons in the vector-boson fusion (VBF) mode and via production in association
with a $W$\- or a $Z$-boson (VH) have been observed with a statistical
significance of $5\sigma$ or more [3, 4]. Finally, although indirect
experimental evidence for the coupling of $h_{125}$ to top-quarks was already
present as a consequence of the measurement of ggH production, the presence of
a large Yukawa coupling of $h_{125}$ to top quarks has also been directly
confirmed by means of the observation of Higgs-boson production in association
with a top-quark pair (ttH) [3, 4]. With regards to the decays of $h_{125}$,
in addition to the decay modes into pairs of photons and off-shell vector
bosons as stated above, more recently also the decays of $h_{125}$ into pairs
of bottom quarks and into pairs of tau leptons have been observed with a
statistical significance at the level of $5\sigma$ [3, 4], such that the
presence of couplings to the third-generation quarks and charged leptons has
been established. Finally, there is first experimental evidence, although not
yet statistically significant enough for a discovery, for the presence of rare
decays of $h_{125}$ into pairs of muons [9] and of the decays into a photon
and a $Z$-boson [10, 11]. There is so far no statistically significant
indication for exotic decay modes into hidden sectors, but the current
experimental uncertainties leave room for such decays at the level of a few
percent, as will be the main topic of this paper.

One of the most puzzling open problems of modern physics concerns the presence
of a dark matter (DM) component in our universe, whose relative contribution
to the total energy budget has been measured to a great accuracy by the Planck
collaboration to be $\Omega_{\text{DM}}h^{2}=0.11933\pm 0.00091$ [12].
According to the standard Weakly Interacting Massive Particles (WIMP)
paradigm, such a dark component could have been produced by the well-known
freeze-out mechanism, by decoupling from the SM thermal bath while becoming
non-relativistic. This scenario has been extensively studied in the literature
and strong constraints from dark matter direct detection experiment, such as
LUX [13], Xenon1T [14], PandaX [15] and LUX-ZEPLIN (LZ) [16], have pushed the
most classic models towards corners of allowed parameter space [17, 18, 19,
20, 21, 22, 23]. The Higgs boson plays a central role in many theories
attempting to solve the dark matter problem in this context as any generic
theory containing a fundamental scalar could allow for this state to couple to
the Higgs doublet via a quartic coupling. As the possibilities for a WIMP to
carry a charge with respect to the SM gauge group are very constrained and
limited [24, 25, 26], this coupling is one of the only options for a
renormalizable coupling of SM singlet states to one of the SM fields, while
respecting the SM gauge invariance [27]. This coupling plays a major role in
the context of Higgs portal scenarios where a dark matter candidate interacts
with the SM only via coupling to the Higgs sector [28, 17, 20, 22, 23].

Collaboration |  $\sqrt{s}$ [TeV] | Data $[\mathrm{fb}^{-1}]$ |  $h_{125}$ production | Exp. [%] | Obs. [%]
---|---|---|---|---|---
CMS [29] | $8+13$ | $19.7+140$ | VBF | 10 | $\mathbf{18^{*}}$
CMS [30] | $7+8+13$ | $4.9+19.7+38.2$ | VBF + VH + ggHj | 15 | 19
CMS [31] | 13 | 35.9 | VH + ggHj | 40 | 53
CMS [32] | 13 | 35.9 | ZH | 44 | 45
ATLAS [33] | $7+8+13$ | $4.7+20.3+139$ | VBF + ttH | 11 | $\mathbf{11^{*}}$
ATLAS [34] | $7+8+13$ | $4.7+20.3+36.1$ | VBF + VH | 17 | 26
ATLAS [35] | $13$ | $139$ | VBF | 10.3 | 14.5
ATLAS [36] | 13 | 36.1 | ZH | 39 | 67
ATLAS [37] | 13 | 36.1 | VH | 58 | 83
Table 1: Expected (exp) and observed (obs) experimental upper limits at the
95% confidence level on the invisible branching ratio of $h_{125}$ as reported
by ATLAS and CMS, obtained from direct searches for the invisible decay of
$h_{125}$ at the LHC at $\sqrt{s}=13\ \mathrm{TeV}$. The currently strongest
upper limit reported by each collaboration are marked with a star. The
production modes of $h_{125}$ targeted in the different searches are vector-
boson fusion (VBF), production in association with a $Z$-boson (ZH) or $Z$\-
and $W$-bosons (VH), and gluon-fusion production in association with a jet
(ggHj).

Assuming, as discussed above, the presence of a hidden sector which
exclusively couples to the visible sector via the Higgs portal, Higgs-boson
decays into particles of the hidden sector can be kinematically allowed. Here
it should be noted that one motivation to investigate hidden sectors with BSM
states in this mass range is that the strongest direct detection constraints
can be evaded for dark-matter masses at the level of a few GeV or below. If
the BSM particles are stable or sufficiently long-lived in order to escape the
detector, so-called invisible (inv) decays of $h_{125}$ are present which can
be searched for at the LHC in final states with large missing transverse
energy. Direct searches for the invisible decay of $h_{125}$ have been
performed by both the ATLAS and the CMS collaboration utilizing various
different production modes. We summarize in Tab. 1 the current LHC searches
for the invisible decay of $h_{125}$ that include part or all of the Run 2
dataset at $13\ \mathrm{TeV}$. No significant excesses over the SM background
have been observed, such that upper limits on the branching ratio for the
invisible decay of $h_{125}$, denoted ${\mathrm{BR}_{\rm inv}}$ from hereon,
have been determined (see last column of Tab. 1). Assuming that $h_{125}$
couples to the SM fermions and gauge bosons according to the predictions of
the SM, the currently strongest limit on ${\mathrm{BR}_{\rm inv}}$ was
reported by ATLAS [33],

${\mathrm{BR}_{\rm inv}}<11\%\ \textrm{ at 95\% confidence level (CL)},$ (1.1)

which was obtained by combining the datasets collected at 7, 8 and $13\
\mathrm{TeV}$, and by utilizing the VBF and ttH production modes. In the
following, we will refer to the constraint shown in Eq. (1.1) as the direct
limit on ${\mathrm{BR}_{\rm inv}}$, according to the fact that it is extracted
from directly searching for the invisible decay of $h_{125}$.

In many phenomenological analyses of BSM theories in which a Higgs-boson
invisible decay plays a role, only the above mentioned direct limit on
${\mathrm{BR}_{\rm inv}}$ is applied in order to exclude parameter space
regions of the theory that are in disagreement with the Higgs-boson
measurements at the LHC (see e.g. Refs. [19, 17, 21, 38, 39, 40]). However,
additional, although more model-dependent, constraints on ${\mathrm{BR}_{\rm
inv}}$ arise from the cross-section measurements of $h_{125}$, considering
that additional BSM decay modes give rise to a suppression of the ordinary
decay modes of $h_{125}$. By performing global scans to the LHC Higgs-boson
measurements in terms of ${\mathrm{BR}_{\rm inv}}$ and so-called coupling
modifiers that quantify deviations of the couplings of $h_{125}$ with respect
to the SM predictions, one can set limits on ${\mathrm{BR}_{\rm inv}}$ as a
function of the coupling modifiers (see, e.g. Refs. [41, 42, 43, 44, 45, 46]
for earlier analyses of this kind). Since in this approach the measurements of
the ordinary decay modes of $h_{125}$ are used to constrain ${\mathrm{BR}_{\rm
inv}}$, in contrast to directly searching for the invisible decay of
$h_{125}$, we will refer to the limits on ${\mathrm{BR}_{\rm inv}}$ resulting
from the global analysis of the cross-section measurements as the indirect
constraints in the following.

In this paper our goal is to exploit the complementarity of the direct and the
indirect constraints on ${\mathrm{BR}_{\rm inv}}$. To this end, the first step
of our analysis is to determine the indirect constraints on ${\mathrm{BR}_{\rm
inv}}$ via global fits to the cross-section measurements of $h_{125}$, where
we will make use of the public code HiggsSignals v.3 [47]. We perform such
global scans assuming that the couplings of $h_{125}$ remain unchanged
compared to the couplings of a SM Higgs boson, but also assuming more
complicated Higgs-portal models in which, in addition to the presence of an
invisible decay mode, there are modifications of the couplings of $h_{125}$ to
the fermions and gauge bosons.444Similar analyses have been performed, for
instance, in Refs. [42, 48, 49, 46], and more recently both CMS and ATLAS
presented constraints on ${\mathrm{BR}_{\rm inv}}$ obtained in the so-called
$\kappa$-framework [3, 4]. If coupling modifications are considered, the upper
limit on ${\mathrm{BR}_{\rm inv}}$ is a function of the coupling modifiers.
Having determined the indirect constrains on ${\mathrm{BR}_{\rm inv}}$, we
then compare them to the direct limit on ${\mathrm{BR}_{\rm inv}}$ in order to
investigate which of the two kind of constraints results in a stronger
exclusion in a variety of different BSM scenarios. To give a brief outlook on
the key results that we have found, we emphasize already here that in many of
the Higgs-portal scenarios considered in the literature the indirect
constraints can provide substantially stronger exclusions and should therefore
not be overlooked. However, in models in which the properties of $h_{125}$ are
not as predicted by the SM, we also found parameter space regions that are in
conflict with the LHC Higgs-boson measurements if there is no invisible decay
mode of $h_{125}$, whereas the same parameter regions are well in agreement
with the LHC measurements if a value of ${\mathrm{BR}_{\rm inv}}$ at the level
of a few percent is present.

The outline of the paper is as follow. In Sec. 2 we describe the setup and
approach used to derive the indirect constraints on ${\mathrm{BR}_{\rm inv}}$
as a function of the couplings modifiers, thus accounting for deviations with
respect to the SM expectation of the couplings of $h_{125}$ to the SM fields.
We present generic constraints in terms of the coupling modifiers and the
invisible branching fraction for a different set of assumption regarding the
structure of the coupling modifiers. Going beyond this generic framework, we
analyse in Sec. 3 the constraints in a variety of concrete BSM scenarios:
Higgs- and singlet-mediated dark matter models, constructions featuring
(pseudo) Nambu-Goldstone bosons, and extended Higgs sectors featuring a second
Higgs doublet. In Sec. 4, we summarize our main results and conclude.

##  2 Indirect limits on the invisible decay of the Higgs boson $h_{125}$

As mentioned above, in the first step of our analysis we will remain agnostic
about the precise nature of the hidden sector. Instead of specifying a
concrete model, we perform $\chi^{2}$-fits to the measurements of the Higgs
boson $h_{125}$ as a function of the branching ratio for invisible decay modes
of $h_{125}$ and, in addition, as a function of coupling modifiers that
quantify modifications of the couplings of $h_{125}$ compared to the SM
predictions. These $\chi^{2}$-analyses, for which we make use of the public
code HiggsSignals v.3 [50, 46, 47] (see the discussion below for details),
will be discussed in this section. The $\chi^{2}$-analyses provide us with
$68\%$ and $95\%$ confidence-level (CL) upper limits on $\mathrm{BR}_{\rm
inv}$ as a function of the coupling modifiers. These upper limits can be
compared to the upper limits on the invisible branching ratio resulting from
direct-searches for the decays of $h_{125}$ into invisible final state. At a
later stage of our analysis, discussed in Sec. 3, we will apply the
constraints on the invisible braching ratio $\mathrm{BR}_{\rm inv}$ that we
obtained from the global $\chi^{2}$-fit to a range of commonly studied hidden
sectors possessing dark matter candidates, extended scalar content (2HDM) and
pseudo Nambu-Goldstone bosons, in which the Higgs portal plays a major role.

Before starting the discussion of the results of the $\chi^{2}$-analyses, we
briefly discuss the strategy that is implemented in the public code
HiggsSignals, and which is used here in order to compare the predicted cross
sections and signal rates of $h_{125}$ to the experimental measurements. For a
more detailed description of the code, we refer the reader to Ref. [47] (see
also Ref. [46] for a more detailed discussion on the statistical
interpretation of the $\chi^{2}$-values provided by HiggsSignals). The code
contains a large set of experimental data from LHC measurements at $8\
\mathrm{TeV}$ and (mostly) $13\ \mathrm{TeV}$ center-of-mass energy.555The
data repository of HiggsSignals can be found at
https://gitlab.com/higgsbounds/hsdataset. The measurements are implemented not
only in the form of inclusive cross sections (or signal rates), but also in
the form of the simplified template cross sections [51] which were designed to
allow for a combination of measurements of different decay channels of
$h_{125}$. In total, the HiggsSignals dataset currently comprises 24
independent measurements from the CMS and the ATLAS collaborations. For the
purpose of our paper, as will be discussed in detail below, the most important
measurements are the ones utilizing the di-photon final state due to the
sensitivity of the di-photon branching ratio on the presence of BSM decay
modes of $h_{125}$.

On the theory side, in order to compare the model predictions to the
experimental data, HiggsSignals requires as input either directly the signal
rates (or alternatively the cross sections and branching ratios) of $h_{125}$
in the various different production or decay modes, or the user has the option
to provide effective coupling modifiers from which the cross sections and
branching ratios are derived internally by a re-scaling of the SM predictions.
The coupling modifiers are defined as the couplings of the particle state at
$125\ \mathrm{GeV}$ in the given BSM theory normalized to the couplings of a
SM Higgs boson at the same mass (see below for details). Thus, in order to
check the SM against the experimental data, one would choose all modifiers to
be equal to one, whereas in models that feature modifications of the
properties of $h_{125}$ the couplings modifiers deviate from one. In our
analysis, we will combine both input formats. This is possible since the most
recent update of HiggsSignals, which is now incorporated into the public code
HiggsTools [47]. We make use of the input in terms of the coupling modifiers
in order to obtain the LHC cross sections of $h_{125}$ and in order to
calculate the partial decay widths of all conventional decay modes of
$h_{125}$ into SM particles. Subsequently, in order to set the desired value
of the branching ratio for the additional decay mode into invisible final
states, we give the respective partial decay width $\Gamma_{\rm inv}$ as
input. The partial width $\Gamma_{\rm inv}$ that is required to set a desired
value of $\mathrm{BR}_{\rm inv}$ can be calculated as

$\Gamma_{\rm inv}=\frac{\mathrm{BR}_{\rm inv}}{1-\mathrm{BR}_{\rm
inv}}\sum_{i}\Gamma_{i}^{\rm SM}\,,$ (2.1)

where $\Gamma^{\rm SM}_{i}$ stands for the individual partial widths for
decays of $h_{125}$ into SM final states, which were calculated previously as
a function of the coupling modifiers. Accordingly, the sum runs over all
relevant SM decay modes of $h_{125}$, i.e.
$i=\\{b\bar{b},gg,WW^{*},ZZ^{*},c\bar{c},s\bar{s},\tau^{+}\tau^{-},\gamma\gamma,\mu^{+}\mu^{-},Z\gamma\\}$.
The total width of $h_{125}$ is then given by

$\Gamma_{\rm tot}=\Gamma_{\rm inv}+\sum_{i}\Gamma_{i}^{\rm SM}\ .$ (2.2)

As already discussed in the introduction, the fact that the total width of
$h_{125}$ is modified via the presence of the additional contribution
$\Gamma_{\rm inv}$ gives rise to the fact that $\mathrm{BR}_{\rm inv}$ can be
constrained via the measured cross sections and signal rates of $h_{125}$.

In order to capture the coupling modifications that arise in as many UV-
complete models as possible while maintaining a manageable number of free
parameters, we define four independent coupling modifiers $c_{i}$, with
$i={V,u,d,\ell}$. The coefficients $c_{i}$ modify the couplings of the
discovered Higgs boson $h_{125}$ to the SM mass eigenstates according to

$\mathcal{L}\,=\,\sum_{f=u,d,\ell}c_{f}\left(\dfrac{m_{f}}{v}\right)h_{125}\,\bar{f}f+c_{V}\left(\dfrac{2m_{W}^{2}}{v}\right)h_{125}\,W^{+\mu}W^{-}_{\mu}+c_{V}\left(\dfrac{m_{Z}^{2}}{v}\right)h_{125}\,Z^{\mu}Z_{\mu}+\mathcal{L}_{\text{inv}}\,,$
(2.3)

where $c_{V}$, $c_{u}$, $c_{d}$ and $c_{\ell}$ are, respectively, the coupling
modifier of the massive gauge bosons, to up-type quarks, to down-type quarks
and charged leptons.666Consequently, we do not take into account flavour-
dependent modifications to the Higgs-boson couplings. We also do not consider
possible sources of CP violation, such that $h_{125}$ is assumed to be a
purely CP-even state. $\mathcal{L}_{\text{inv}}$ is the contribution that
contains the portal couplings to the hidden sector responsible for the
presence of the invisible branching ratio $\mathrm{BR}_{\rm inv}$, which we
will further specify when we consider concrete BSM theories in Sec. 3. For the
loop-induced couplings of $h_{125}$ to photons and gluons, we assume that
there are no sizable BSM contributions in additions to the loop diagrams with
SM particles in the loops, reflecting the fact that light hidden-sector
particles can only couple very weakly via the SM gauge interactions in order
to be physically viable. Hence, the respective coupling coefficients
$c_{\gamma\gamma}$ and $c_{gg}$ can be calculated as a function of $c_{V}$,
$c_{u}$ and $c_{d}$.777The contributions from the charged leptons can safely
be neglected due to the suppression from the smaller Yukawa couplings
$Y_{\ell}\ll Y_{t},Y_{b}$. The structure of the coupling modifiers in the
various BSM theories considered in this work are summarized in Tab. 2.

Scenario | $c_{u}$ | $c_{d}$ | $c_{\ell}$ | $c_{V}$ | Discussion
---|---|---|---|---|---
SM-like couplings | 1 | 1 | 1 | 1 | Sec. 2.1
Universal couplings | $c_{\text{uni}}$ | $c_{\text{uni}}$ | $c_{\text{uni}}$ | $c_{\text{uni}}$ | Sec. 2.2
Non-universal couplings | $c_{f}$ | $c_{f}$ | $c_{f}$ | $c_{V}$ | Sec. 2.3
Non-universal couplings | $c_{u}$ | $c_{d}$ | $c_{\ell}=c_{d}$ | $1$ | Sec. 2.3
Higgs portal DM | 1 | 1 | 1 | 1 | Sec. 3.1
Singlet portal DM | $c_{\text{uni}}\leq 1$ | $c_{\text{uni}}\leq 1$ | $c_{\text{uni}}\leq 1$ | $c_{\text{uni}}\leq 1$ | Sec. 3.2
(P)NGB | $c_{\text{uni}}\leq 1$ | $c_{\text{uni}}\leq 1$ | $c_{\text{uni}}\leq 1$ | $c_{\text{uni}}\leq 1$ | Sec. 3.3
2HDM Type I | $c_{f}$ | $c_{f}$ | $c_{f}$ | $c_{V}\leq 1$ | Sec. 3.4
2HDM Type II | $c_{u}$ | $c_{d}$ | $c_{\ell}=c_{d}$ | $c_{V}\leq 1$ | Sec. 3.4
2HDM Type III | $c_{u}$ | $c_{d}=c_{u}$ | $c_{\ell}=c_{u}$ | $c_{V}\leq 1$ | Sec. 3.4
2HDM Type IV | $c_{u}$ | $c_{d}$ | $c_{\ell}=c_{u}$ | $c_{V}\leq 1$ | Sec. 3.4
Table 2: Summary of the coupling modifcications in terms of the modifiers
defined in Eq. (2.3) for the various models considered in this work and
references to the (sub)sections of the manuscript where each specific case is
discussed.

As already mentioned above, in order to derive the indirect limits on
$\mathrm{BR}_{\rm inv}$ from the measurements related to $h_{125}$ we perform
a $\chi^{2}$-analysis utilizing the public code HiggsSignals. We determine in
each scan

$\Delta\chi^{2}\left(c_{i},\mathrm{BR}_{\rm
inv}\right)=\chi^{2}\left(c_{i},\mathrm{BR}_{\rm inv}\right)-\chi^{2}_{\rm
SM}\ ,$ (2.4)

where $\chi^{2}(c_{i},\mathrm{BR}_{\rm inv})$ is the fit result for the
respective BSM scenario given as a function of the coupling modifiers $c_{i}$
and $\mathrm{BR}_{\rm inv}$, and $\chi^{2}_{\rm SM}$ is the fit result
assuming properties of $h_{125}$ as predicted by the SM, i.e.
$\mathrm{BR}_{\rm inv}=0$ and $c_{i}=1$.888Strictly speaking, also in the SM
one finds $\mathrm{BR}_{\rm inv}>0$ due to the decay mode $h_{125}\to
ZZ^{*}\to\nu\nu\bar{\nu}\bar{\nu}$. However, the resulting branching ratio is
of the order of $0.1\%$ [52]. Considering the current experimental precisions
of the signal-rate measurements of $h_{125}$, this value can be approximated
by zero for all practical purposes. In order to set limits on the coupling
modifiers or model parameters, we demand that the BSM scenario is not
disfavoured compared to the SM fit result at the 95% CL or more. In a one-
dimensional parameter estimation, this translates into the condition [53]

$\Delta\chi^{2}\leq 3.84\ .$ (2.5)

In a joint estimation of two free parameters, the respective condition is

$\Delta\chi^{2}\leq 5.99\ .$ (2.6)

In addition to the allowed regions of the coupling modifiers obtained using
the 95% CL conditions, we will show in our plots also allowed regions using
the 68% ($1\sigma$) CL for illustrative reasons, which corresponds to the
conditions $\Delta\chi^{2}\leq 1$ and $\Delta\chi^{2}\leq 2.30$ in one-
dimensional and two-dimensional fits, respectively [53].

By defining the allowed/excluded regions for the coupling modifiers based on
the $\chi^{2}$-definition shown in Eq. (2.4), where we utilize the SM as a
reference model, the obtained limits only correspond to the 95% CL limits if
the SM result $\chi^{2}_{\rm SM}$ is a good approximation of the best-fit
result of the BSM scenario under consideration, i.e. $\chi^{2}_{\rm
min}=\min(\chi^{2})\simeq\chi^{2}_{\rm SM}$. As we will discuss below, we
encountered situations in which $\chi^{2}_{\rm min}$ was considerably smaller
than $\chi^{2}_{\rm SM}$ for certain parameter configurations.999We emphasize
that $\chi^{2}_{\rm min}<\chi^{2}_{\rm SM}$ does not necessarily signify a
global statistical preference of the respective BSM scenario compared to the
SM, as such a conclusion can only be drawn by also taking into account the
different numbers of degrees of freedom of both models. For the context of our
analysis, in which we are interested only in the the constraints on the
coupling modifiers and $\mathrm{BR}_{\rm inv}$ that can be derived within a
certain BSM scenario, the global preference of different BSM scenarios against
the SM (and also against each other) is not relevant. In such a situation, the
more common approach (in a frequentist analysis) would be to construct the
confidence intervals of the free parameters by comparing to the parameter
point that features the best-fit value $\chi^{2}_{\rm min}$ instead of
comparing to $\chi^{2}_{\rm SM}>\chi^{2}_{\rm min}$, i.e. constructing the
limits based on $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}$. This approach
would yield stronger constraints on the parameters and therefore smaller
allowed regions in the $c_{i}$ parameter space as compared to our approach,
such that the latter should be regarded as more conservative. We found that
the experimental measurements responsible for values of $\Delta\chi^{2}<0$ are
mainly the ones related to the ttH production of $h_{125}$, where the signal
extraction is affected by systematic uncertainties regarding the theoretical
predictions for the background estimation [54, 55]. Taking this into account,
it is reasonable to be conservative in the determination of the exclusion
limits, thus sticking to the definition of $\Delta\chi^{2}$ as shown in Eq.
(2.4), even though $\chi^{2}_{\rm min}$ is smaller than $\chi^{2}_{\rm SM}$ in
some scenarios. We also note that in our plots we indicate the
$\Delta\chi^{2}$-distribution together with the value of $\chi^{2}_{\rm min}$
for all parameter points in addition to the 68% CL and 95% CL exclusions, such
that the reader can also apply a different criterion in order to define the
allowed regions. Finally, it should also be noted that the opposite case with
$\chi^{2}_{\rm min}>\chi^{2}_{\rm SM}$ is not possible for the different BSM
scenarios considered in the following, because in the fits of the BSM theories
in terms of the coupling modifiers we always include the parameter space point
$c_{i}=1$ and $\mathrm{BR}_{\rm inv}=0$ in which the particle state $h_{125}$
resembles exactly the properties of a SM Higgs boson.

As was already mentioned above, we start our analysis be performing the
$\chi^{2}$-fits in different BSM scenarios in a generic fashion in terms of
the coupling modifiers $c_{i}$. The different scenarios are discussed in the
following subsections. We summarize in Tab. 2 the different constructions of
coupling modifications that we consider. The applications of the constraints
on the modifiers $c_{i}$ and ultimately on $\mathrm{BR}_{\rm inv}$ obtained in
this way to the parameter space of UV-complete BSM theories, in which both
$c_{i}$ and $\mathrm{BR}_{\rm inv}$ are functions of the model parameters,
will be discussed in Sec. 3.

###  2.1 $h_{125}$ with SM couplings and a non-zero $\mathrm{BR}_{\rm inv}$

In the simplest hidden sector models, the SM is augmented by new fields which
do not give rise to modifications of the couplings of $h_{125}$ compared to a
SM Higgs boson. As a result, the cross sections at colliders for the
production of $h_{125}$ are unchanged compared to the SM predictions. However,
as discussed above, the branching ratios of $h_{125}$ can be modified due to
additional decay modes into the hidden sector if such decay modes are
kinematically allowed. In this case, the modifications of the properties of
$h_{125}$ compared to the SM predictions can be quantified exclusively in
terms of the invisible branching ratio $\mathrm{BR}_{\rm inv}$. In the
following we will discuss how non-zero values of $\mathrm{BR}_{\rm inv}$ give
rise to a suppression of the signal rates in the conventional decay modes of
$h_{125}$. By comparing the predicted signal rates to the experimental
measurements, we will derive an upper limit on $\mathrm{BR}_{\rm inv}$ at the
$95\%$ CL via the condition shown in Eq. (2.5). Finally, we will compare the
indirect limit resulting from the global Higgs-boson measurements to the
direct limit on $\mathrm{BR}_{\rm inv}$ from direct searches for the invisible
decay of $h_{125}$.

Figure 1: Result of the global fit to the Higgs signal-rate measurements
assuming a Higgs boson $h_{125}$ that apart from the presence of an invisible
decay mode behaves according to the predictions of the SM model, i.e. the
couplings of $h_{125}$ to the SM particles are unchanged. The black solid line
indicates the value $\Delta\chi^{2}$ taking into account the full HiggsSignals
data set. The black dashed and the black dotted lines indicate the value of
$\Delta\chi^{2}$ considering only the measurements in the diphoton final state
utilizing the full Run 2 data by CMS [56] and ATLAS [57], respectively. The
gray horizontal lines indicate the 68% and 95% CL, respectively. The orange
vertical line indicates the currently strongest upper limit on
$\mathrm{BR}_{\rm inv}$ resulting from direct searches for
$h_{125}\to\mathrm{inv}$ as published by ATLAS [33].

To this end, we show in Fig. 1 the results of the $\chi^{2}$-analysis
performed with HiggsSignals, where we set $c_{i}=0$, and we varied
$\mathrm{BR}_{\rm inv}$ from 0% to 18%. The black solid line indicates the
value of $\Delta\chi^{2}$, which is obtained including the complete data set
implemented in HiggsSignals. We find the best-fit with $\Delta\chi^{2}=0$ at
$\mathrm{BR}_{\rm inv}=0$, with $\chi^{2}_{\rm min}=\chi^{2}_{\rm SM}=151.67$.
For increasing values of $\mathrm{BR}_{\rm inv}$ the $\chi^{2}$ function is
monotonically increasing. Consequently, under the assumption that the
couplings of $h_{125}$ are unchanged compared to the SM predictions, the
presence of an invisible decay mode of $h_{125}$ with sizable branching ratio
worsens the fit result to the Higgs-boson measurements for all possible values
of ${\mathrm{BR}_{\rm inv}}$. Based on the HiggsSignals analysis, we find an
indirect upper limit of

${\mathrm{BR}_{\rm inv}}<6.22\%\leavevmode\nobreak\ (3.08\%)\qquad\textrm{at
}95\%\leavevmode\nobreak\ (68\%)\leavevmode\nobreak\
\textrm{CL}\qquad\textrm{for }c_{i}=1\ .$ (2.7)

The 95% CL limit is almost a factor of two smaller than the currently
strongest limits from direct searches for the decay mode
$h_{125}\to\mathrm{inv}$ (see Tab. 1). The strongest direct limit
${\mathrm{BR}_{\rm inv}}<11\%$ is indicated by the orange dotted-dashed
vertical line in Fig. 1 [33]. Accordingly, for phenomenological studies of
models with hidden sectors in which the couplings of $h_{125}$ are not
modified compared to the SM predictions, but where the decay of $h_{125}$ into
the hidden sector is kinematically open, one should include the indirect
constraints from the measurements of the signal rates of $h_{125}$, currently
giving rise to the upper limit shown in Eq. (2.7). On the contrary, taking
into account only the limit on ${\mathrm{BR}_{\rm inv}}$ from direct searches
allows also parameter space regions that are already excluded by the
measurements regarding the discovered Higgs boson at a CL of more than
$3\sigma$, i.e. $\Delta\chi^{2}>9$, as is visible in Fig. 1.

In order to shed light on which particular measurements are most relevant for
the increase of $\chi^{2}$ with increasing values of ${\mathrm{BR}_{\rm
inv}}$, we show in Fig. 1 the $\chi^{2}$-values that are obtained only taking
into account the measurement with the largest and the second largest
individual contributions to the total $\chi^{2}$.101010The total
$\chi^{2}$-value including all measurements (black solid line) can be smaller
than the sum of the two individual contributions from the
$h_{125}\to\gamma\gamma$ measurements (black dashed and dotted lines) shown in
Fig. 1, because for the computation of the total $\chi^{2}$-values the
correlations are taken into account. The most important $\chi^{2}$-penalty has
its origin in the CMS measurement of Higgs-boson production and subsequent
decay into di-photon pairs including the full Run 2 data at $13\ \mathrm{TeV}$
center of mass energy [56]. The fit to the corresponding cross-section
measurements gives rise to the $\Delta\chi^{2}$ values indicated by the black
dashed line in Fig. 1. One can see that based on this measurement alone one
obtains a stronger upper limit on ${\mathrm{BR}_{\rm inv}}$ than the one
obtained from direct searches for $h_{125}\to\mathrm{inv}$. Also the second
largest $\chi^{2}$-penalty is caused by signal rates of $h_{125}$ utilizing
the di-photon decay mode, but here as a result of the corresponding ATLAS
measurements [57], indicated by the black dotted line in Fig. 1. The fact that
the two most relevant measurements are both related to the
$h_{125}\to\gamma\gamma$ decay mode can be understood by realizing that, from
the experimental side, it is the most precisely measured decay mode, and that,
from the theory side, the di-photon branching ratio is very sensitive to
modifications of the total width of $h_{125}$.

We finally compare the indirect limit on ${\mathrm{BR}_{\rm inv}}$ to similar
results that have been obtained in the past and to projections for future runs
of the LHC and for the high-luminosity LHC (HL-LHC). An early analysis
including Tevatron and LHC first-year Run 1 data found a limit of
${\mathrm{BR}_{\rm inv}}\leq 23\%$ via a global fit to the Higss-boson data
assuming SM-like couplings [43]. Shortly after, in Ref. [44] a slightly
stronger limit of ${\mathrm{BR}_{\rm inv}}\leq 17\%$ was found making use of
HiggsSignals using the same fit strategy as applied here. Taking into account
the full Run 1 LHC data, an upper limit of ${\mathrm{BR}_{\rm inv}}\leq 12\%$
has been determined in Ref. [49]. Global fits including also 13 TeV LHC data
have been performed in Ref. [45] and Ref. [46], finding upper limits on
${\mathrm{BR}_{\rm inv}}$ under the assumption of SM-like couplings of 5% and
10%, respectively. These numbers are of comparable size as the limit
${\mathrm{BR}_{\rm inv}}\leq 6.2\%$ found in this analysis.

It is interesting to note that the current indirect limits are also comparable
to the HL-LHC projections assuming that 3000 fb-1 will be collected at 14 TeV
[58]. The reason for this is mainly that the projected limits on
${\mathrm{BR}_{\rm inv}}$ were obtained under the assumptions that the
experimental data will be in agreement with the SM predictions, whereas the
current limits on ${\mathrm{BR}_{\rm inv}}$ have been determined from actual
data in which the central values of the various measurements naturally
fluctuate within statistical and systematic uncertainties. If differences
exist between the central values in the current data and the SM predictions
(which are expected at least at the level of statistical fluctuations), the
extracted upper limit on ${\mathrm{BR}_{\rm inv}}$ derived today can be
smaller than the projected HL-LHC limit even though the precision of the
individual cross-section measurements is larger now compared to what is
expected in the future for the HL-LHC. This also means that, even though the
uncertainties of the cross-section measurements of $h_{125}$ will improve in
the future, the indirect limit on ${\mathrm{BR}_{\rm inv}}$ might not
necessarily become substantially stronger, because the limit will depend on
how the measured central values of the cross sections will evolve within the
current uncertainty bands.

###  2.2 $h_{125}$ with universally modified couplings and a non-zero
$\mathrm{BR}_{\rm inv}$

In the previous section we discussed the case in which the Higgs sector
remained unchanged compared to the SM except for the presence of an additional
invisible decay mode into a hidden sector. However, in many BSM scenarios
additional scalar fields can mix with the SM-like Higgs boson at $125\
\mathrm{GeV}$. In this case, the properties of $h_{125}$ can be modified not
only by additional decay modes into invisible final states, but also the
couplings of $h_{125}$ to the fermions and gauge bosons of the SM can be
modified compared to the SM. In this section we will analyze the most simple
possibility, in which the couplings of $h_{125}$ to the SM particles are
modified by a universal coupling modifier $c_{\rm
uni}=c_{V}=c_{u}=c_{d}=c_{\ell}$. The simplest UV-complete model in which this
pattern of coupling modifiers arises is a model in which a gauge singlet
scalar field mixes with $h_{125}$, giving rise to a suppression of the
couplings of $h_{125}$ by a universal factor $c_{\rm uni}<1$ which can be
identified as $c_{\rm uni}=\cos\theta$ where $\theta$ is the singlet mixing
angle with $h_{125}$. Models in which also a coefficient $c_{\rm uni}>1$ can
be realized comprise, for instance, extensions of the SM by a real or complex
SU(2) triplet Higgs field.

Figure 2: Result of the global fit to the Higgs signal-rate measurements
assuming a Higgs boson $h_{125}$ that has an invisible decay mode and whose
couplings to the SM particles are modified w.r.t. the SM predictions by a
universal coupling coefficient $c_{\rm uni}$. The color coding indicates the
value of $\Delta\chi^{2}$ resulting from the HiggsSignals analysis. The black
solid and dashed lines indicate the allowed regions at the $95\%$ and the
$68\%$ confidence level, respectively. The orange star indicates the currently
strongest upper limit on $\mathrm{BR}_{\rm inv.}$ resulting from direct
searches for $h_{125}\to\mathrm{inv}$ obtained according to the assumption
that $h_{125}$ is produced as predicted in the SM, i.e. $c_{\rm uni}=1$ [33].
The orange dashed line indicates an approximate extrapolation of this direct
limit for the case $c_{\rm uni}\neq 1$ (see text for details).

Under the assumption of the universal coupling coefficient $c_{\rm uni}$ and
the presence of an invisible decay mode $h_{125}\to\text{inv}$, we can perform
a $\chi^{2}$-fit to the measurements of the properties of $h_{125}$ in
dependence of the two free parameters $c_{\rm uni}$ and $\mathrm{BR}_{\rm
inv}$ with the help of HiggsSignals. By doing so, we obtain the two-
dimensional $\chi^{2}$-distribution that is shown in Fig. 2. In this plot the
color grading indicates the value of $\Delta\chi^{2}$ in the parameter plane
$\\{c_{\rm uni},{\mathrm{BR}_{\rm inv}}\\}$, and the solid and dashed lines
indicate the $95\%$ and the $68\%$ CL limits, respectively. Also indicated
with an orange star is the 95% CL upper limit ${\mathrm{BR}_{\rm inv}}<11\%$
which was obtained assuming a VBF production cross sections of $h_{125}$
according to the SM prediction [33]. As such, the direct limit can in
principle only be applied for $c_{\rm uni}=1$, in which case the production
cross sections of the state at $125\ \mathrm{GeV}$ considered here agree with
the SM predictions. However, for $c_{\rm uni}\neq 1$ the direct limit on
${\mathrm{BR}_{\rm inv}}$ can still be applied in an approximate form. Taking
into account that the VBF production cross sections is given by $\sigma_{\rm
VBF}=c_{\rm uni}^{2}\times\sigma_{\rm VBF}^{\rm SM}$ at leading order, with
$\sigma_{\rm VBF}^{\rm SM}$ being the SM cross section, the number of signal
events would be enhanced or reduced by the factor $c_{\rm uni}^{2}$ depending
on whether $c_{\rm uni}$ is larger or smaller than one, respectively. Under
the assumption that the kinematical shape of the signal events is not modified
substantially, which can be expected to be the case for small deviations of
$c_{\rm uni}$ from unity, one can therefore apply the upper limit
${\mathrm{BR}_{\rm inv}}<11\%/c_{\rm uni}^{2}$ from direct searches for
$h_{125}\to\mathrm{inv}$ in order to account for the modification of the
production cross section $\sigma_{\rm VBF}$ if $c_{\rm uni}\neq 1$. We will
call this limit an extrapolation of the direct limit on ${\mathrm{BR}_{\rm
inv}}$, and this extrapolated limit is shown as the orange dotted-dashed line
in Fig. 2.

One can see in Fig. 2 that there is a flat direction with
$\Delta\chi^{2}\simeq 0$ in the $\\{{c_{\rm uni}},{\mathrm{BR}_{\rm inv}}\\}$
plane. Along this flat direction, values of ${c_{\rm uni}}>1$ give rise to an
enhancement of the cross sections for the production of $h_{125}$, but at the
same time values of ${\mathrm{BR}_{\rm inv}}>0$ suppress the decay modes of
$h_{125}$ decaying into SM particles (see also Ref. [44]). As a result, for
each value of ${c_{\rm uni}}>1$ there is a value of ${\mathrm{BR}_{\rm
inv}}>0$ for which production cross sections times branching ratios remain
equal to the SM predictions. Accordingly, we find that in the range $c_{\rm
uni}\gtrsim 1.015$ the direct upper limit on ${\mathrm{BR}_{\rm inv}}$ (orange
lines) is stronger than the indirect limit on ${\mathrm{BR}_{\rm inv}}$
resulting from the fit to the signal-rate measurements of $h_{125}$. For
${c_{\rm uni}}\gtrsim 1.085$ the values of ${\mathrm{BR}_{\rm inv}}$ that are
required in order to cancel the enhancement of the production cross sections
of $h_{125}$ are larger than the direct limit, such that the combination of
the indirect constraints from signal-rate measurements and the direct
constraint on ${\mathrm{BR}_{\rm inv}}$ is able to exclude this part of the
parameter plane entirely.

In contrast to the case with ${c_{\rm uni}}>1$, in models in which the
couplings of $h_{125}$ are suppressed compared to the SM, i.e. ${c_{\rm
uni}}<1$, one can observe that the indirect limit on ${\mathrm{BR}_{\rm inv}}$
from the fit to the signal-rate measurements (black solid line) is always
substantially stronger than the direct limit from searches for
$h_{125}\to\mathrm{inv}$ (orange dotted-dashed line). In this region of the
parameter plane, both the values of ${c_{\rm uni}}<1$ and ${\mathrm{BR}_{\rm
inv}}>0$ give rise to a suppression of the signal rates of $h_{125}$. This is
why we observe that, the smaller the value of ${c_{\rm uni}}$, the smaller is
the indirect upper limit on ${\mathrm{BR}_{\rm inv}}$ resulting from the
signal-rate measurements of $h_{125}$. Finally we note that, according to the
discussion of Sec. 2.1, also for ${c_{\rm uni}}=1$ the indirect limit is
stronger than the direct limit, as indicated with the orange star.

An approximate estimate of the allowed parameter space, corresponding to the
region delimited by the two black lines in Fig. 2, is given at 95% CL by the
condition

$0.520\,{\mathrm{BR}_{\rm inv}}+0.960\,<\,{c_{\rm
uni}}\,<\,0.565\,{\mathrm{BR}_{\rm inv}}+1.043\,.$ (2.8)

The fact that for ${c_{\rm uni}}<1$ the indirect constraints provide the
strongest constraints on ${\mathrm{BR}_{\rm inv}}$, whereas the direct limits
are (in the considered scenario) essentially irrelevant, is important for
phenomenological studies in models with extended Higgs sectors featuring a
gauge-singlet scalar. Then the deviations of the universal coupling modifier
$c_{\text{uni}}$ to unity is generated via the mixing of the SM-like Higgs
boson with the singlet scalar, and the coupling coefficient can be written as
$c_{\text{uni}}=\cos\theta\simeq 1-\theta^{2}/2$ for small $\theta$, where
$\theta$ is the mixing angle. In such case, the constraints from Eq. (2.8)
translate in very good approximation into

${\mathrm{BR}_{\rm
inv}}<0.078\left(1-\left(\dfrac{\theta}{0.285}\right)^{2}\right)\ ,$ (2.9)

which implies that in the limit ${\mathrm{BR}_{\rm inv}}\rightarrow 0$ values
$\theta<0.285$ are excluded and reciprocally if $\theta\rightarrow 0$,
${\mathrm{BR}_{\rm inv}}$ is constrained to be ${\mathrm{BR}_{\rm
inv}}<0.078$, which is stronger than all current direct bounds listed in Tab.
1.111111If the invisible decay of $h_{125}$ is absent, and modifications of
the properties of $h_{125}$ arise only by means of a universal coupling
modification, we find in a one-dimensional parameter estimation 95% CL limits
of ${c_{\rm uni}}>0.968$, or equivalently $\theta<0.253$. We will discuss the
parameter constraints resulting from the analysis of this section in two
concrete BSM theories, a hidden-sector composed of a fermionic DM candidate as
well as a singlet scalar in Sec. 3.2, and a construction featuring (pseudo)
Nambu-Goldstone bosons in Sec. 3.3.

###  2.3 $h_{125}$ with non-universally modified couplings and a non-zero
$\mathrm{BR}_{\rm inv}$

In the previous section we investigated the case in which the couplings of the
Higgs boson $h_{125}$ are modified via a universal coupling coefficient
${c_{\rm uni}}$. Although this procedure captures the modifications that arise
in many BSM constructions featuring additional scalars, there is also a wide
range of models in which the modifications to the couplings of $h_{125}$
compared to the SM prediction cannot be captured in terms of a single
coefficient. Since it is not feasible to use independent coupling modifiers
for each of the couplings of $h_{125}$, we have to make assumptions on the
total number of coupling modifiers that we include, and on how the couplings
of $h_{125}$ depend on these modifiers. As already discussed in Sec. 2, for
the case of non-universal coupling modifiers we will consider the four
independent coefficients $c_{V}$, $c_{u}$, $c_{d}$ and $c_{\ell}$ defined in
2.3, and for the loop-induced couplings to gluons and photons we assume that
no additional BSM loop-contributions play a role. Thus, including
$\mathrm{BR}_{\rm inv}$ as an additional free parameter, HiggsSignals can be
used to determine $\Delta\chi^{2}$ as a function of up to five independent
parameters (see also Tab. 2 for a summary of the combinations of coupling
modifiers for each case considered in this work). In order to be able to
present the results in a clearer fashion, we will restrict our analysis in
this section to parameter scans with only two independent coupling modifiers.
The corresponding parameter space can be understood as a subspace of the more
complicated coupling configurations allowed by the effective Lagrangian shown
in 2.3, requiring further relations amongst the four individual coupling
modifiers $c_{V}$, $c_{u}$, $c_{d}$ and $c_{\ell}$. This approach allows us to
show the allowed regions of the two varied parameters in two-dimensional
plots, and we can investigate how these regions change with increasing values
of $\mathrm{BR}_{\rm inv}$.

As a first example, we show constraints in a benchmark model commonly utilized
by the CMS and the ATLAS collaborations (see, for instance, Ref. [59, 60, 61,
3, 4]), in which it is assumed that the couplings of $h_{125}$ to fermions are
modified by a common factor $c_{f}=c_{u}=c_{d}=c_{\ell}$, and, in addition,
the coupling to the massive vector bosons is allowed to vary, i.e. $c_{V}\neq
1$. Such coupling modifications can be present in various different UV-
complete BSM theories. For example, the presence of a gauge-singlet scalar
that mixes with $h_{125}$ gives rise to $c_{V}=c_{f}<1$, as discussed in Sec.
2.2. Other examples comprise the 2HDM with Yukawa structure of type I and
extensions thereof, where $c_{V}\leq 1$, and where $c_{f}$ can be both smaller
and larger than 1. Also extensions of the SM containing SU(2) triplet scalar
fields that mix with the SM Higgs field can give rise to coupling
modifications in terms of $c_{V}$ and $c_{f}$ as discussed above, where in
contrast to 2HDMs also $c_{V}>1$ is possible (at tree level). All these models
can be extended by a hidden sector in order to feature one or more valid DM
particles, and where the Higgs sector can act as the portal between the hidden
and the SM sector (see, for instance, Refs. [62, 63, 64, 65]). Thus, if the
decay of $h_{125}$ into states of the hidden sector is kinematically allowed,
it is interesting to analyze how the experimental limits on the coupling
modifiers are modified for different values of the invisible branching ratio
of $h_{125}$.

Figure 3: Result of the global fit to the Higgs-boson signal-rate measurements
in the plane $\\{c_{f},c_{V}\\}$ for different values of ${\mathrm{BR}_{\rm
inv}}$. The color coding indicates the value of $\Delta\chi^{2}$ resulting
from the HiggsSignals analysis. The black solid and dashed lines indicate the
allowed regions at the $95\%$ and the $68\%$ confidence level, respectively.
The black crosses indicate the point at which the couplings of $h_{125}$ are
identical to the SM predictions, i.e. $c_{f}=c_{V}=1$.

In Fig. 3 we show the values of $\Delta\chi^{2}$ as a function of $c_{V}$ and
$c_{f}$ for different choices of $\mathrm{BR}_{\rm inv}$. The upper left plot
is obtained assuming that there is no invisible decay mode of $h_{125}$, and
the other plots are obtained by increasing $\mathrm{BR}_{\rm inv}$ in steps of
2% up to $\mathrm{BR}_{\rm inv}=10\%$ (lower right plot), covering the allowed
range of ${\mathrm{BR}_{\rm inv}}$ according to the direct limits. The solid
and the dashed contours indicate the 95% and 68% CL exclusion limits based on
the HiggsSignals $\chi^{2}$-fit to the cross-section measurements of
$h_{125}$. One can see that with increasing value of $\mathrm{BR}_{\rm inv}$
the allowed regions of the coupling modifiers move to larger values. This
observation is in line with the discussion in Sec. 2.2, where we demonstrated
that the suppression of the decay modes into SM particles due to the presence
of the invisible decay mode can be compensated by en enhancement of the
production cross sections as a result of coupling modifiers larger than one
(see also Fig. 2). In contrast to the location of the allowed regions, the
shape and the size of the allowed regions are practically unchanged in all
plots. Furthermore, also the values of $\Delta\chi^{2}$ featured by the best-
fit points for the different values of $\mathrm{BR}_{\rm inv}$ are effectively
the same. Based on the results shown in Fig. 3, one can conclude that for BSM
scenarios in which the restriction $c_{V}\leq 1$ applies, the presence of the
invisible decay mode gives rise to a decrease of the allowed ranges of both
$c_{V}$ and $c_{f}$. On the other hand, for models like the Higgs-triplet
extension in which one can find values of $c_{V}>1$, one can see that the
presence of an invisible decay mode of $h_{125}$ with sizable branching ratios
can open up parameter space regions that would be excluded if such a novel
decay mode is not present. It should be noted also that the best-fit points do
not lie at the point at which the couplings of $h_{125}$ are identical to the
SM predictions (black crosses) even if the decay mode $h_{125}\to\mathrm{inv}$
is absent. Instead, for ${\mathrm{BR}_{\rm inv}}=0\%$ the parameter points
with the smallest values of $\chi^{2}$ are found for $c_{V}>1$ and $c_{f}<1$,
featuring values of $\Delta\chi^{2}\approx-6$.121212The values of
$\Delta\chi^{2}<0$ are driven here by the following measurements, where for
simplicity we only quote inclusive signal strength measurements here, and we
remind the reader that $\mu=1$ according to the SM predictions: (i)
$\mu(\mathrm{ttH},H\to b\bar{b})=0.35^{+0.36}_{-0.34}$ reported by ATLAS [55],
where it should be taken into account that the measurement uncertainty is
dominated by systematic uncertainties regarding the background estimation,
(ii) $\mu(\mathrm{ggH},H\to\tau^{+}\tau^{-})=0.59^{+0.28}_{-0.32}$ and
$\mu(\mathrm{VBF},H\to\tau^{+}\tau^{-})=1.39^{+0.56}_{-0.47}$ reported by CMS
[66], (iii) $\mu(\mathrm{ttH})=0.58^{+0.36}_{-0.33}$ reported by ATLAS [54],
where the signal extraction is affected by the fact that the ttW background
was found to be larger than the SM prediction. Hence, if one would define the
exclusion regions based on $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}$,
instead of $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm SM}$ as applied throughout
this paper, one would obtain even stronger exclusion limits in the
$\\{c_{f},c_{V}\\}$ plane (see also the related discussion in Sec. 2). The
location of the best-fit points as a function of ${\mathrm{BR}_{\rm inv}}$
moves to larger values of both coupling modifiers investigated here for
increasing values of ${\mathrm{BR}_{\rm inv}}$.

Figure 4: As Fig. 3, but in the plane $\\{c_{u},c_{d}\\}$ and assuming that
$c_{\ell}=c_{d}$ and $c_{V}=1$.

A second BSM scenario that we consider is the case in which the coupling of
$h_{125}$ to gauge bosons is not modified compared to the SM predictions
($c_{V}=1$), in contrast to all the other scenarios discussed above, but in
which the couplings to the fermions can deviate from the SM
($c_{u},c_{d},c_{\ell}\neq 1$). We furthermore impose the condition
$c_{\ell}=c_{d}$ in order to stick to a parameter scan with two free
parameters for a given value of ${\mathrm{BR}_{\rm inv}}$. UV-complete models
in which coupling modifications of this form arise comprise generically
extensions of the SM containing a second SU(2) doublet scalar field in which
the so-called Yukawa structure of type II (see Sec. 3.4 for details) is
imposed, which is also the Yukawa structure that arises in supersymmetric
extensions of the SM (see e.g. Ref. [67]). We show the $\chi^{2}$-distribution
in the $(c_{u},c_{d})$ for different values of ${\mathrm{BR}_{\rm inv}}$ as
they were obtained with the help of HiggsSignals in Fig. 4, where the color
coding and the definitions of the CL exclusion limits is as in Fig. 3. One can
observe that increasing values of ${\mathrm{BR}_{\rm inv}}$ shifts the allowed
parameter regions to larger values of $c_{u}$ and smaller values of
$c_{d}=c_{\ell}$. The reason for the preference of larger values of $c_{u}$ if
${\mathrm{BR}_{\rm inv}}>0$ lies in the fact that the enhancement of the
coupling of $h_{125}$ to top quarks gives rise to an enhancement of the gluon-
fusion production cross section that partially compensates the suppression of
the branching ratios for the ordinary decay modes of $h_{125}$. However, the
VBF production mode, for instance, is unchanged since $c_{V}=1$, such that the
minimal values of $\chi^{2}$ that are found for each value of
${\mathrm{BR}_{\rm inv}}$ considered increase with ${\mathrm{BR}_{\rm inv}}$,
in contrast to the results depicted in Fig. 3. The trend towards lower values
of $c_{d}$ with increasing values of ${\mathrm{BR}_{\rm inv}}$ has its origin
in the fact that a suppression of the coupling of $h_{125}$ to bottom quarks
suppresses the total width of $h_{125}$. As a consequence, sizable values of
the decay width $\Gamma_{\rm inv}$ (see Eq. (2.1)) can be be in agreement with
the signal-rate measurements of $h_{125}$, because the measured branching
ratios of the ordinary decays of $h_{125}$ remain closer to the SM
predictions. However, since $c_{V}=1$ and $c_{\ell}=c_{d}$ in this scenario,
the suppression of the couplings of $h_{125}$ to tau leptons if $c_{d}<1$, and
the unchanged partial decay with for the decay modes $h_{125}\to W^{+}W^{-}$
and $ZZ$ still result in significant modifications of the branching ratios of
$h_{125}$ compared to the SM predictions, such that overall the fit result
becomes worse with increasing values of ${\mathrm{BR}_{\rm inv}}$. As a
result, even though for all possible values of ${\mathrm{BR}_{\rm inv}}$ one
can find ranges of $c_{u}$ and $c_{d}$ that describe the experimental data
regarding $h_{125}$ as accurate as the SM, the size of the allowed regions of
the coupling modifiers decreases and the fit result deteriorates with
increasing value of ${\mathrm{BR}_{\rm inv}}$.

The two examples discussed above demonstrate that the presence of sizable
values of ${\mathrm{BR}_{\rm inv}}$ can improve or worsen the fit result to
the Higgs-boson measurements, depending on which ranges of the coupling
modifiers are considered, and depending on whether additional relations
between the different modifiers are imposed. Since in both cases viable
parameter ranges of the coupling modifiers survived even for values of
${\mathrm{BR}_{\rm inv}}\simeq 10\%$, it becomes apparent that both the direct
and the indirect constraints on ${\mathrm{BR}_{\rm inv}}$ have to be
considered in order to be in agreement with the measurements regarding
$h_{125}$. In Sec. 3.4 we will analyze the impact of the values of
${\mathrm{BR}_{\rm inv}}$ on the allowed parameter regions in UV-complete
models in which the couplings of the state at $125\ \mathrm{GeV}$ can be
captured by the full set of coupling modifiers considered in our analysis, as
defined in Eq. (2.3). Therein, as an illustrative example of popular BSM
theories for which our results are relevant, we will focus on models
containing, in addition to the hidden sector as the origin of the invisible
decay mode of $h_{125}$, a second Higgs doublet.

##  3 Application to hidden sector models

In this section we will apply the indirect constraints on ${\mathrm{BR}_{\rm
inv}}$ obtained previously to concrete BSM scenarios featuring a hidden
sector. We will start by considering Higgs-portal dark-matter models in Sec.
3.1 in which no coupling modifications of $h_{125}$ are present, i.e.
$c_{i}=1$. As demonstrated in Sec. 2.1, in this case the indirect constraints
on ${\mathrm{BR}_{\rm inv}}$ are stronger than the direct constraints, such
that the latter do not have to be considered. Afterwards we will discuss in
Secs. 3.2–3.4 models featuring a hidden sector and an extended Higgs sector,
in which the couplings of $h_{125}$ to ordinary matter deviate from the SM
predictions. Following the discussions in Sec. 2.2 and Sec. 2.3, here
depending on the values of the coupling modifieres $c_{i}$ predicted in each
model, both the indirect or the direct constraints on ${\mathrm{BR}_{\rm
inv}}$ can be stronger, such that both constraints will have to be considered
and their complementarity can be studied.

###  3.1 The Higgs portal dark matter model

Figure 5: $\sigma_{\rm SI}$ as a function of the DM mass for the Higgs-portal
parameter space in which the relic abundance as observed by Planck is
reproduced (dark grey), for scalar DM (left), Dirac fermion DM (center) and
vector DM (right). Constraints from LZ, projection for Darwin and the neutrino
floor are represented in blue (dashed), green (dotted-dashed) and red (solid)
respectively.

One of the simplest approaches to accommodate a valid DM candidate in a BSM
scenario consists of the so-called Higgs portal scenario. Here it is assumed
that the discovered Higgs boson $h_{125}$ is the only portal to the dark
sector, such that no direct (gauge) interactions with the SM fermions and the
gauge bosons exist. We will consider here three possible kind of DM candidates
coupled to the Higgs boson via
$\mathcal{L}=\mathcal{L}_{\text{SM}}+\mathcal{L}_{\text{DM}}$ where
$\mathcal{L}_{\text{SM}}$ is the SM Lagrangian and
$\mathcal{L}_{\text{DM}}=\mathcal{L}_{S}$ for real scalar DM ($\equiv S$),
$\mathcal{L}_{\text{DM}}=\mathcal{L}_{\chi}$ for Dirac fermion DM
($\equiv\chi$) and $\mathcal{L}_{\text{DM}}=\mathcal{L}_{V}$ for real vector
DM ($\equiv V^{\mu}$) with

$\mathcal{L}_{S}\,=\,-\dfrac{1}{4}\lambda_{S}|H|^{2}S^{2}\,,\quad\mathcal{L}_{\chi}\,=\,-\dfrac{1}{4}\dfrac{\lambda_{\chi}}{\Lambda}|H|^{2}\bar{\chi}\chi\,,\quad\mathcal{L}_{V}\,=\,-\dfrac{1}{4}\lambda_{V}|H|^{2}V^{\mu}V_{\mu}\,,$
(3.1)

where $\lambda_{S,\chi,V}$ and $\Lambda$ are respectively dimensionless and
dimension-one couplings. We parameterize the SM Higgs doublet in unitary gauge
$H=(0,\leavevmode\nobreak\ v+h)^{T}/\sqrt{2}$ with $v\simeq 246$ GeV and
$h=h_{125}$ is the SM Higgs boson. In addition, in each of the three cases we
consider bare mass terms for our DM candidates. We will denote the physical
scalar, fermion and vector dark-matter masses after electroweak symmetry
breaking by $m_{S,\chi,V}$, respectively. The partial widths of the Higgs-
boson decays into a pair of DM particles are given by 131313The expression for
the scalar DM is in agreement with Ref. [20] and Ref. [68]. The expression for
the vector DM is in agreement with Ref. [69].

$\displaystyle\Gamma_{h\rightarrow SS}\,=$
$\displaystyle\,\dfrac{\lambda_{S}^{2}v^{2}}{128\pi
m_{h}}\left(1-\frac{4m_{S}^{2}}{m_{h}^{2}}\right)^{1/2}\,,$
$\displaystyle\Gamma_{h\rightarrow\bar{\chi}\chi}\,=$
$\displaystyle\,\dfrac{\lambda_{\chi}^{2}v^{2}m_{h}}{128\pi\Lambda^{2}}\left(1-\frac{4m_{\chi}^{2}}{m_{h}^{2}}\right)^{3/2}\leavevmode\nobreak\
\,,$ $\displaystyle\Gamma_{h\rightarrow VV}\,=$
$\displaystyle\,\dfrac{\lambda_{V}^{2}v^{2}m_{h}^{3}}{512\pi
m_{V}^{4}}\left(1-4\dfrac{m_{V}^{2}}{m_{h}^{2}}+12\dfrac{m_{V}^{4}}{m_{h}^{4}}\right)\left(1-\frac{4m_{V}^{2}}{m_{h}^{2}}\right)^{1/2}\leavevmode\nobreak\
.$ (3.2)

Higgs-portal models are strongly constrained by the null-results of direct-
detection experiments. Currently, the most stringent experimental constraints
on the DM-nucleon spin-independent scattering cross section
$\sigma_{\text{SI}}$ were reported by the Xenon1T [14], PandaX-4T [15] and the
LUX-ZEPLIN (LZ) experiments [16], whose most recent results exclude
$\sigma_{\text{SI}}\gtrsim 10^{-47}\,\text{cm}^{2}$ for a $50$ GeV DM mass and
up to $\sigma_{\text{SI}}\gtrsim 10^{-45}\,\text{cm}^{2}$ for $10$ TeV DM
mass. The sensitivity of the upcoming Darwin experiment [70] should improve
the current bounds from LZ by more than an order of magnitude, and will almost
reach the so-called neutrino floor [71]. The direct detection cross section
with a nucleon $N$, mediated by the Higgs boson, can be expressed for the
three scenarios considered here as

$\sigma^{S}_{\text{SI}}\,=\,\dfrac{\lambda_{S}^{2}\mu_{SN}^{2}m_{N}^{2}}{16\pi
m_{S}^{2}m_{h}^{4}}f_{N}^{2}\leavevmode\nobreak\
,\quad\sigma^{\chi}_{\text{SI}}\,=\,\dfrac{\lambda_{\chi}^{2}\mu_{\chi
N}^{2}m_{N}^{2}}{16\pi\Lambda^{2}m_{h}^{4}}f_{N}^{2}\leavevmode\nobreak\
,\quad\sigma^{V}_{\text{SI}}\,=\,\dfrac{\lambda_{V}^{2}\mu_{VN}^{2}m_{N}^{2}}{16\pi
m_{V}^{2}m_{h}^{4}}f_{N}^{2}\leavevmode\nobreak\ ,$ (3.3)

with $\mu_{AB}=m_{A}m_{B}/(m_{A}+m_{B})$.141414The expression for scalar DM is
in agreement with Ref. [20, 23, 68], and the one for vector DM is in agreement
with Refs. [69, 23] The nucleon form factor $f_{N}\sim 0.3$ is defined as
$f_{N}\equiv f_{Tu}^{(N)}+f_{Td}^{(N)}+f_{Ts}^{(N)}+(6/27)f_{TG}^{(N)}$ where
$f_{Tq}^{(N)}\equiv\langle N|m_{q}\bar{q}q|N\rangle/m_{N}$ is the contribution
from a quark $q$ to the nucleon mass and $f_{TG}^{(N)}\equiv
1-\sum_{q}f_{Tq}^{(N)}$ is the gluon contribution.151515Numerical values for
these form factor can be found in Ref. [72].

Figure 6: Higgs-portal parameter plane of dimensionless couplings and the DM
mass for scalar DM $S$ (first row), Dirac fermion DM $\chi$ (second row) and
vector DM $V$ (third row). The dark-grey lines indicates the parameter space
reproducing the relic abundance as observed by Planck. Constraints from LZ,
projection for Darwin and the neutrino floor are represented in dashed blue,
dotted-dashed green and solid red respectively. In the plots on the right, the
LZ constraint is rescaled according to the predicted DM abundance via Eq.
(3.4), and the light-grey areas represent regions where the DM relic abundance
exceeds the Planck value. The solid black and dotted-dashed orange lines
represent respectively our constraints derived using HiggsSignal and
constraints from direct searches for the invisible decay of $h_{125}$.

After implementing the models in Feynrules [73], we used micrOMEGAs [74, 75]
to compute numerically both relic abundance and direct detection cross
sections.161616With micrOMEGAs, we obtained numerical results for the direct
detection cross section within a $10\%$ agreement with respect to our
analytical expression of Eq. (3.3). In Fig. 5 we represented with a dark gray
line in the plane $\\{\sigma_{\rm SI},m_{S,\chi,V}\\}$ the parameter points
that satisfy the relic density as determined by the Planck collaboration [12]
within a $3\sigma$ interval around the best-fit value,
$\Omega_{\chi}h^{2}\in[0.11933-3\times 0.00091,0.11933+3\times 0.00091]$ for
the three kind of dark matter candidates of the Higgs portal scenarios
considered in this work. We also indicate the constraints from LZ in dashed
blue, sensitivity prediction for Darwin in dotted-dashed green and the
neutrino floor in solid red. From Fig. 5, one can see that the Higgs portal
models considered in this section are essentially only allowed for bosonic
dark matter candidates and for a very narrow region of DM masses close to the
resonance $m_{\chi}\simeq m_{h}/2\simeq 62.5\leavevmode\nobreak\ \text{GeV}$,
if one assumes that the dark matter abundance saturates the Planck best fit
value. The fermionic dark matter candidate considered in this section is
entirely excluded by the most recent results from the LZ collaboration.

We represented the relic density and direction detection constraints in the
left panels of Fig. 6 in terms of a dimensionless coupling as a function of
the DM mass. In addition, we represented constraints on Higgs physics from
HiggsSignal and direct limits from ATLAS respectively in black and dotted-
dashed orange line. For a given set of parameters, one can rescale the LZ
constraints according to the predicted value for the relic abundance via

$\sigma_{\text{SI}}^{\text{LZ}}\,=\,\sigma_{\text{SI}}^{\text{LZ}}|_{\Omega_{\text{DM}}h^{2}=0.11933}\left(\dfrac{0.11933}{\Omega_{\text{DM}}h^{2}}\right)\,,$
(3.4)

where $\sigma_{\text{SI}}^{\text{LZ}}|_{\Omega_{\text{DM}}h^{2}=0.11933}$ is
the LZ constraint assuming a DM abundance $\Omega_{\text{DM}}h^{2}=0.11933$
and $\Omega_{\text{DM}}h^{2}$ is the DM abundance predicted for a given set of
parameters. The rescaled LZ constraint is represented on the plots in the
right panels of Fig. 6 where the blue areas represent the excluded regions.
One can see from this figure that such constraint weaken precisely around the
peak value for bosonic DM candidates, as a smaller DM abundance
$\Omega_{\text{DM}}h^{2}<0.11933$ is generated in this region. However, the
fermionic dark matter candidate is completely excluded, even if it constitutes
just a fraction of the total dark matter.171717Notice that in this work we
considered a scalar operator $\bar{\chi}\chi$ connected to the Higgs field.
Our statement here depends on the choice of operators and could differ, for
example, for a pseudoscalar operator $\bar{\chi}\gamma_{5}\chi$. It remains to
be investigated whether one can evade the LZ constraints upon inclusion of the
pseudoscalar operator, or whether the presence of new mediators between the
hidden sector and the visible sector at or below the electroweak scale have to
be introduced in order to predict a viable fermionic Higgs-portal DM scenario
[76] (see also Sec. 3.2).

One can relax the condition of achieving the correct relic abundance via a
single Higgs-portal parameter by assuming the presence of additional DM
annihilation channels as typically expected in constructions featuring a more
complex hidden sector. In this case, the correct relic abundance could be
achieved for a smaller value of the Higgs-portal dimensionless coupling. As
can be seen in the left panels of Fig. 6, constraints from Higgs physics would
become the strongest, for fermion and vector DM, in the region of the
parameter space corresponding to masses $\lesssim m_{h}/2$. In addition, if
one assumes a multi-component dark matter setup, direct detection constraints
would have to be rescaled according to the local density of the relevant dark
matter component. The parameter space at small masses $\lesssim m_{h}/2$ could
open up and the constraints derived in this work would become the strongest.
In such cases, for DM masses typically $\lesssim m_{h}/2$, our constraints
derived using HiggsSignal, independent of the local dark matter density, could
be stronger than both direct detection bounds and from an extrapolation of
direct searches for a invisible decay of the Higgs boson.

###  3.2 Singlet portal dark matter

The simplest possibility to extend the Higgs portal DM scenario is to assume
the presence of an additional real singlet scalar field $\Phi$ that can act as
a portal between the dark sector and the visible sector. This can be achieved
by considering a discrete $\mathbb{Z}_{4}$ symmetry acting on
$\Phi\rightarrow-\Phi$ [38] and by introducing components of a fermionic Dirac
DM candidate $\chi_{L,R}$ with opposite chiralities
$\chi\equiv\chi_{L}+\chi_{R}$ that couples to $\Phi$ via

$\mathcal{L}\supset-y_{\chi}\Phi\bar{\chi}_{L}\chi_{R}+\text{h.c.}\,,$ (3.5)

where $y_{\chi}$ is a Yukawa coupling.181818Under the $\mathbb{Z}_{4}$
symmetry, the two chirality components of $\chi$ would transform as
$\chi_{R}\rightarrow i\chi_{R}$ and $\chi_{L}\rightarrow-i\chi_{L}$. The most
general scalar potential respecting the $\mathbb{Z}_{4}$ symmetry is given by

$V=\mu_{H}^{2}H^{\dagger}H+\frac{1}{2}\mu_{\Phi}^{2}\Phi^{2}+\lambda_{H}\left(H^{\dagger}H\right)^{2}+\frac{1}{4}\lambda_{\Phi}\Phi^{4}+\frac{1}{2}\lambda_{\Phi
H}\Phi^{2}H^{\dagger}H\ ,$ (3.6)

where $\mu_{H,\Phi}$ and $\lambda_{H,\Phi,\Phi H}$ are respectively dimension-
one and dimensionless parameters. $H$ is a $SU(2)$ Higgs doublet, following
the same notation as in the previous section. The quartic coupling
$\lambda_{\Phi H}$ in combination with the Yukawa interaction of Eq. (3.5)
allow for the interactions between the DM candidate and the SM. The
$\mathbb{Z}_{4}$ symmetry is spontaneously broken down to a remaining
$\mathbb{Z}_{2}$ symmetry by the vacuum expectation value (vev) of the singlet
field $\langle\Phi\rangle=v_{\phi}$. We parametrize the scalar field as
$\Phi=v_{\phi}+\phi$ where $\phi$ denotes a real scalar degree of freedom. The
singlet vev $v_{\phi}$ generates a mass term $m_{\chi}=y_{\chi}v_{\phi}$ for
the DM fermion, whose stability is ensured by the remaining $\mathbb{Z}_{2}$
symmetry. In the following, we will assume that no bare mass term for the DM
fermion is present (or equivalently that it can be neglected), such that the
physical mass of $\chi$ is given by $m_{\chi}$. In order to obtain the
physical mass eigenstates $h_{1,2}$ an orthogonal field transformation can be
performed, parameterized by an angle $\theta$

$\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}=\begin{pmatrix}c_{\theta}&-s_{\theta}\\\
s_{\theta}&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}-}c_{\theta}\end{pmatrix}\begin{pmatrix}h\\\
\phi\end{pmatrix}\ ,$ (3.7)

with $s_{\theta}\equiv\sin\theta$ and $c_{\theta}\equiv\cos\theta$. The
expression for the mixing angle $\theta$ in terms of the vevs and the quartic
scalar couplings and details about the minimization of the scalar potential
are given in App. A. In the following, the physical state $h_{1}\simeq
h_{125}$ denotes the neutral scalar whose mass is identical to the discovered
Higgs boson, i.e. $m_{h_{1}}\simeq 125$ GeV, but with couplings modified by
factors of $c_{\rm uni}=\cos\theta$ with respect to the SM prediction. In the
following the mixing angle $\theta$ is taken as free parameter in combination
with the physical masses $m_{h_{1}}$ and $m_{h_{2}}$ and the vev of the
singlet $v_{\phi}$. Taking into account that the Yukawa coupling $y_{\chi}$ is
fixed by the DM mass $m_{\chi}=y_{\chi}v_{\phi}$ if $v_{\phi}$ is used as free
parameter, we are left with $(m_{h_{1}},m_{h_{2}},m_{\chi},v_{\phi},\theta)$
as set of independent free parameters.

Both physical scalars mediate DM-nucleon scatterings whose corresponding cross
section can be expressed as

$\sigma_{\text{SI}}\,=\,\dfrac{\mu_{\chi
N}^{2}}{\pi}\left[\dfrac{m_{N}m_{\chi}}{vv_{\phi}}c_{\theta}s_{\theta}\left(\dfrac{1}{m_{h_{2}}^{2}}-\dfrac{1}{m_{h_{1}}^{2}}\right)\right]^{2}f_{N}^{2}\leavevmode\nobreak\
.$ (3.8)

where notations are identical to the ones used in Sec. 3.1. In Fig. 7 we
represented the value of the direct detection cross section as a function of
the dark matter mass for the parameter space allowing to reproduce the correct
relic abundance for selected value of $v_{\phi}$ and $m_{h_{2}}$ as dark-grey
lines.191919The DM-nucleon scattering cross sections obtained with the
analytical expression shown in Eq. (3.8) are in agreement within $10\%$ with
the results obtained using the public code micrOMEGAs [74, 75]. In Fig. 7 the
resonances correspond to DM annihilating via (quasi) on-shell mediators
($h_{1}$ and $h_{2}$) and therefore are peaked around $m_{\chi}\simeq
m_{h_{1}}/2\simeq 62.5\leavevmode\nobreak\ \text{GeV}$ and $m_{\chi}\simeq
m_{h_{2}}/2$. The vertical lines at $m_{\chi}\simeq 30\ \mathrm{GeV}$ in the
left plot and $m_{\chi}\simeq 80\ \mathrm{GeV}$ in the middle plot,
respectively, have their origin in the channel $\bar{\chi}\chi\rightarrow
h_{2}h_{2}$ opening, which is kinematically forbidden at zero temperature for
$m_{\chi}<m_{h_{2}}$ and exponentially suppressed at finite temperature but
yet still efficient enough to yield the correct relic abundance. For this
annihilation channel the DM abundance is set by the value of the Yukawa
coupling between the fermionic DM and the singlet scalar, which also sets the
DM mass resulting in $\theta-$independent generated relic abundance. Larger DM
masses correspond to larger annihilation cross sections and a resulting dark
matter under-abundance.

We also indicate in Fig. 7 the currently strongest upper limit on $\sigma_{\rm
SI}$ from the LZ collaboration with the blue shaded region. One can see that
the LZ constraints exclude most parts of the parameter space that predicts the
experimentally determined DM relic abundance. In the scenario with
$m_{h_{2}}<m_{h_{1}}/2$ (left plot) and $m_{h_{1}}/2<m_{h_{2}}<m_{h_{1}}$
(middle plot) only DM masses of $m_{\chi}\simeq m_{h_{2}}/2$ and
$m_{\chi}\lesssim m_{h_{2}}$ remain viable, whereas the scenario with
$m_{h_{2}}>m_{h_{1}}$ (right plot) is entirely ruled out in the low-mass
regime investigated here. Thus, we will not discuss the latter scenario any
further in the following.

Figure 7: As in Fig. 5 for the singlet portal DM scenario for different values
of the mass of the additional scalar $m_{h_{2}}$ and the vev $v_{\phi}$.

In the singlet portal dark matter model the presence of a second scalar
particle gives rise to additional experimental constraints that have to be
applied. The Higgs boson $h_{1}\simeq h_{125}$ can decay invisibly into a dark
matter pair, but $h_{1}$ can also decay into pairs of light scalars $h_{2}$ if
$m_{h_{2}}<m_{h_{1}}/2$. Depending on whether the singlet-like state $h_{2}$
decays predominantly into SM particles or into DM pairs, the decay mode
$h_{1}\to h_{2}h_{2}$ either gives rise to exotic visible decay modes of
$h_{1}$, mainly resulting in $b\bar{b}b\bar{b}$, $b\bar{b}\tau^{+}\tau^{-}$
and $b\bar{b}\mu^{+}\mu^{-}$ final states, or the decay mode $h_{1}\to
h_{2}h_{2}$ gives rise to an additional invisible decay mode of $h_{1}$. The
decay rates for the decays of $h_{1}$ into BSM states are given in App. A.2.
Accordingly, the additional Higgs boson $h_{2}$ can be searched for via the
decays $h_{1}\to h_{2}h_{2}$ if kinematically allowed in the final states
mentioned above. In addition, $h_{2}$ can be directly searched for at $pp$
colliders via its production in the ggH or VBF production modes and at lepton
colliders via Higgsstrahlung production. Therefore, we here also include
constraints from collider searches for additional Higgs bosons by using the
public code HiggsBounds [77, 78, 79, 80, 47], which are complementary to the
constrains resulting from the signal-rate measurements of $h_{1}$.

We present in Fig. 8 the constraints from the Higgs-boson measurements using
HiggsSignal and from searches for additional Higgs bosons using HiggsBounds.
The top-panels show $\Delta\chi^{2}$ obtained with HiggsSignal, and the
corresponding upper bounds on the mixing angle as a function of the dark
matter mass for selected value of $v_{\phi}$ and $m_{h_{2}}$ at the 68% and
the 95% CL indicated by the black dashed and solid lines, respectively. We
also indicate the region which is excluded by the observed 95% CL cross-
section limits with regards to $h_{2}$ with the red line, where the regions
above the red lines are excluded (further details are given below). Finally,
the orange dotted-dashed lines indicate the regions at which the predicted
invisible branching ratio of the discovered Higgs boson at $125\ \mathrm{GeV}$
are equal to the upper limit obtained from direct searches for the invisible
decay mode, i.e. ${\mathrm{BR}_{\rm inv}}=11\%$. As mentioned above, here
${\mathrm{BR}_{\rm inv}}$ is determined by adding the contributions from the
decay modes $h_{1}\to\bar{\chi}\chi$ and $h_{1}\to
h_{2}h_{2}\to\bar{\chi}\chi\bar{\chi}\chi$ if kinematically allowed. One can
see that in both scenarions the indirect constraints on ${\mathrm{BR}_{\rm
inv}}$ from the cross-section measurements of $h_{125}$ (black lines) are
stronger than the direct limit on ${\mathrm{BR}_{\rm inv}}$ for all DM masses
considered.

Figure 8: In the upper plots we show $\Delta\chi^{2}$ in the
$\\{m_{\chi},\theta\\}$ plane for two different choices of $m_{h_{2}}$ and
$v_{\phi}$. The resulting exclusion limits at 95% and 68% CL are shown with
solid and dashed black lines, respectively, excluding the areas above the
lines. The 95% CL exclusion region from searches for additional scalars is
shown in red, excluding the areas above the lines. The parameter points for
which ${\mathrm{BR}_{\rm inv}}$ is equal to the upper limit from direct
searches for the invisible decay of $h_{125}$ are indicated with the orange
lines. In the lower plots we additionally indicate the parameter space
allowing to reproduce all (dark grey) or part of (light grey) the measured
relic abundance.

The lower panels show the same exclusion lines superimposed to the parameter
space allowed by the LZ bounds, and in which the correct relic abundance (dark
grey) or a lower value (light grey) is achieved. As discussed in Sec. 3.1, the
LZ constraints are rescaled according to Eq. (3.4). According to the previous
discussion, the parameter space allowed by the LZ constraints corresponds to
two different regimes: a narrow region around $m_{\chi}\simeq m_{h_{2}}/2$
where DM annihilation is resonantly enhanced by $s-$channel diagrams and the
region $m_{\chi}\simeq m_{h_{2}}$ where $\bar{\chi}\chi\rightarrow h_{2}h_{2}$
DM annihilation processes are efficient enough in order to not overclose the
universe. In the following, we discuss in more detail the two benchmark
scenarios with $m_{h_{2}}<m_{h_{1}}/2$ (left) and
$m_{h_{1}}/2<m_{h_{2}}<m_{h_{1}}$ (right) that are depicted in Fig. 8.

#### A second Higgs boson below $\mathbf{125/2}$ GeV:

In the scenario depicted in the left panels of Fig. Fig. 8, the decay of the
SM-like Higgs boson $h_{1}$ into a pair of scalars $h_{2}$ is kinematically
allowed, which mainly determines the exclusions obtained from the experimental
data regarding $h_{1}\simeq h_{125}$. For DM masses of
$2m_{\chi}>m_{h_{2}}=30\ \mathrm{GeV}$, the scalar $h_{2}$ decays to visible
final state (mostly $\bar{b}b$). If $2m_{\chi}>m_{h_{1}}\simeq 125\
\mathrm{GeV}$ the HiggsSignals constraint ($\theta<0.073$ at $95\%$ CL) is
then insensitive to the DM mass. For $m_{\chi}\lesssim m_{h_{2}}/2$, the
invisible decay channel $h_{2}\rightarrow\bar{\chi}\chi$ becomes relevant,
such that ${\mathrm{BR}_{\rm inv}}>0$ and one finds slightly stronger
constraints on $\theta$ from the cross-section measurements of $h_{125}\simeq
h_{1}$. One can compare the indirect constraints from the cross-section
measurements of $h_{125}$, indicated by the black line, to the exclusions from
the direct constraint ${\mathrm{BR}_{\rm inv}}<11\%$ obtained from searches
for the invisible decay of $h_{125}$, indicated by the orange line. The direct
limit on ${\mathrm{BR}_{\rm inv}}$ gives rise to two disconnected exclusion
regions. The parameter space above the right parabola-shaped orange line is
excluded because the decay mode $h_{1}\to\chi\bar{\chi}$ has a branching ratio
larger than 11%. The second exclusion region is the parameter space that lies
within the closed orange line at $m_{\chi}<15\ \mathrm{GeV}$, in which
additionally the decay mode $h_{1}\to
h_{2}h_{2}\to\chi\bar{\chi}\chi\bar{\chi}$ contributes to ${\mathrm{BR}_{\rm
inv}}$. However, both exclusion regions resulting from the direct limit on
${\mathrm{BR}_{\rm inv}}$ lie above the black line and are therefore already
excluded based on the signal-rate measurements of $h_{125}$. Finally, the
exclusions as a consequence of the searches for additional Higgs bosons,
indicated by the red line, have their origin in two different collider
searches. For $m_{\chi}>15\ \mathrm{GeV}$ we find an upper limit of
$\theta\lesssim 0.15$ due to constraints from searches for Higgs-boson decays
into a pair of two lighter scalars in the $b\bar{b}\mu^{+}\mu^{-}$ final state
performed by the ATLAS collaboration at $13\ \mathrm{TeV}$ [81]. For
$m_{\chi}<15\ \mathrm{GeV}$ we find a substantially stronger limit of
$\theta\lesssim 0.10$ from searches for invisibly decaying scalars performed
by the OPAL collaboration at the LEP collider at up to $209\ \mathrm{GeV}$
[82]. However, for all values of $m_{\chi}$ considered the constraints from
the searches for additional Higgs bosons are weaker than the constraints
determined with HiggsSignals.

#### A second Higgs boson below $\mathbf{125}$ GeV:

Whereas in the previous case the indirect constraints from the $h_{125}$
measurements were dominantly determined by the presence of the decay mode
$h_{2}\to h_{1}h_{1}$, in the benchmark scenario depicted in the right plot of
Fig. 8 this decay is kinematically not possible. Thus, the constraints depend,
on the one hand, on the presence of the mixing between $h_{1}$ and $h_{2}$,
and, on the other hand, also on the presence of the invisible decay
$h_{1}\to\chi\bar{\chi}$ if $2m_{\chi}<125\ \mathrm{GeV}$. We find that the
HiggsSignal constraints (black line) are weaker at large DM masses
$m_{\chi}>m_{h_{1}}/2$, yielding an upper limit of $\theta<0.29$. This limit
is only set by modifications of the SM prediction of the Higgs-boson universal
couplings and matches our results from Sec. 2.2 of Eq. (2.9). The invisible
decay of the SM-like Higgs boson opens up for $m_{\chi}<m_{h_{1}}/2$, and the
exclusion line from the HiggsSignals analysis drops below the one from
HiggsBounds (red line) at DM masses below about $60\ \mathrm{GeV}$. With
regards to the direct limit on ${\mathrm{BR}_{\rm inv}}$ (orange line), the
island of an excluded region at $m_{\chi}<15\ \mathrm{GeV}$ disappears in the
right plot, since only the invisible decay $h_{1}\rightarrow\bar{\chi}\chi$ is
kinematically allowed, whereas the decay $h_{1}\to h_{2}h_{2}$ is not
possible, leaving only the parabolic-shaped upper bound set by the invisible
decay of the Higgs boson $h_{125}$. As before, the indirect constraints from
the cross-section measurements of $h_{125}$ are always stronger than the
constraints from the direct limit on ${\mathrm{BR}_{\rm inv}}$. In contrast to
the previous case, we find for $m_{h_{2}}>m_{h_{1}}/2$ that the HiggsBounds
analysis excluded parameter regions that otherwise would be allowed. In the
right plot, with $m_{h_{2}}=80\ \mathrm{GeV}$, we find an upper limit of
$\theta\lesssim 0.225$ as a result of the cross section limits from searches
for scalar particles produced via Higgsstrahlung production and decaying into
pairs of bottom quarks performed at LEP [83].202020In addition, the
HiggsBounds analysis provides constraints that are stronger than the ones from
HiggsSignals at $m_{\chi}\lesssim 3\ \mathrm{GeV}$, where however the dark-
matter direct-detection constraints already rule out the relevant part of the
parameter space. The most sensitive search as determined by HiggsBounds here
is the search for invisibly decaying scalars at LEP performed by the L3
collaboration [84]. The exclusion power of this search becomes much weaker for
$m_{\chi}<40\ \mathrm{GeV}$, where the scalar $h_{2}$ is able to decay into
pairs of $\chi$ and the branching ratio for the decay $h_{2}\to b\bar{b}$ is
smaller.

In this section we have considered a discrete $\mathbb{Z}_{4}$ symmetry as a
specific example. However this construction could be extended by considering
instead a global continuous $U(1)$ symmetry, broken spontaneously. In this
case the imaginary part of the complex scalar field responsible for the
symmetry breaking would be a massless Goldstone boson. The analysis performed
in this section would not be drastically affected by the presence of this new
state but would need to be accounted for in the possible DM annihilation final
states. Constraints from Higgs physics would have to be modified as the SM-
like Higgs boson could decay into a pair of Goldstone bosons. This precise
point is discussed in the following subsection.

###  3.3 (Pseudo) Nambu-Goldstone bosons

In this section we consider a model with an additional (approximate) global
$U(1)$ symmetry and a complex scalar field parameterized in the exponential
form by $\Phi=(v_{\phi}+\phi)e^{ia/v_{\phi}}/\sqrt{2}$, singlet under the SM
gauge group but charged under the new global $U(1)$. All SM fields are
singlets under the additional $U(1)$ symmetry. The scalar potential including
all terms that respect this symmetry has the same form as the one of Eq.
(3.6), and we adopt the same notation as in Sec. 3.2. Depending on the
specificity of the model, the global symmetry could be spontaneously broken
($v_{\phi}\neq 0$) or not ($v_{\phi}=0$). If the symmetry is spontaneously
broken, a mixing by an angle $\theta$ between the real part of the complex
field and the Higgs field is generated, while the state $a$ is a Nambu-
Goldstone Boson (NGB) or Pseudo Nambu-Goldstone Boson (PNGB) depending on
whether the broken symmetry was exact or approximate before spontaneous
symmetry breaking. In the latter case, we assume that any explicit symmetry
breaking term is small compared to the electroweak scale such that the mass of
the (P)NGB can be set to zero in the following analysis. Using the exponential
parametrization, the field $a$ disappears from the scalar potential and the
only coupling between the Higgs boson at $125\ \mathrm{GeV}$ and the (P)NGB is
generated via mixing from the kinetic terms of the scalar $\Phi$ after
spontaneous symmetry breaking [85, 86],

$\mathcal{L}\supset|\partial_{\mu}\Phi|^{2}=\dfrac{1}{2}\left(\partial_{\mu}\phi\partial^{\mu}\phi+\partial_{\mu}a\partial^{\mu}a\left(\dfrac{\phi^{2}}{v_{\phi}^{2}}+2\dfrac{\phi}{v_{\phi}}+1\right)\right)\,,$
(3.9)

before performing a rotation to the mass eigenstate basis. We obtain
canonically normalized kinetic terms for the massless pseudoscalar field $a$
in addition to higher dimensional operators involving derivative couplings.
One can perform a rotation to the physical CP-even neutral scalar mass
eigenstate basis $\\{h_{1},h_{2}\\}$ by considering the transformation
described in the previous section (see Eq. (3.7)). By selecting only terms
relevant for the decay of the SM-like Higgs boson into pairs of (P)NGBs, one
obtains

$\mathcal{L}\supset\dfrac{1}{2}\partial_{\mu}a\partial^{\mu}a\Big{(}1-2\dfrac{s_{\theta}h_{1}}{v_{\phi}}\Big{)}\supset-\partial_{\mu}a\partial^{\mu}a\dfrac{s_{\theta}h_{1}}{v_{\phi}}\,.$
(3.10)

This parametrization makes obvious that the decay into pairs of (P)NGBs of the
SM-like Higgs state is triggered by the mixing angle $\theta$. The coupling
term of Eq. (3.10) generates a partial decay width

$\Gamma_{h_{1}\rightarrow
aa}\,=\,\dfrac{s_{\theta}^{2}}{32\pi}\dfrac{m_{h_{1}}^{3}}{v_{\phi}^{2}}\,.$
(3.11)

One can recover this result by choosing the linear parametrization
$\Phi=(v_{\phi}+\phi+ia)/\sqrt{2}$ and by expressing the scalar potential of
Eq. (3.6) in terms of the mass eigenstates, as performed in Ref. [64].

Figure 9: $\Delta\chi^{2}$, indicated by the colour coding, in the
$\\{\cos\theta,v_{\phi}/m_{h_{1}}\\}$ plane for the (Pseudo) Nambu-Goldstone
boson scenario. The resulting exclusion limits at 95% and 68% CL are shown
respectively with solid and dashed black lines, excluding the areas below the
lines. Areas below the orange dashed line are excluded based on the
experimental limit on ${\mathrm{BR}_{\rm inv}}$ from direct searches for
$h_{125}\to\mathrm{inv}$. The areas above the dotted red lines are excluded by
constraints from perturbative unitarity assuming
$m_{h_{2}}=v_{\phi}/2,v_{\phi},2v_{\phi}$ (see discussion in Sec. 3.3 for
details).

As the couplings between the SM field content and the Higgs boson $h_{1}\simeq
h_{125}$ are modified by a universal factor $c_{\mathrm{uni}}=\cos\theta$, we
can derive constraints in the plane $\\{\cos\theta,(v_{\phi}/m_{h_{1}})\\}$,
where we divide the vev $v_{\phi}$ by $m_{h_{1}}=125\ \mathrm{GeV}$ in order
to have a dimensionless quantity. We depict in Fig. 9 with the color coding
the $\Delta\chi^{2}$-values from HiggsSignals in this plane. In addition, we
show with the black dashed and solid lines the resulting exclusion limits at
the 68% and 95% CL, respectively, and we also show exclusion lines based on
perturbative unitarity constraints on the dimensionless coupling of the scalar
potential for a selection of masses $m_{h_{2}}=v_{\phi}/2,v_{\phi},2v_{\phi}$
(explicit expressions for these constraints can be found, for instance, in
Ref. [87]).212121The $m_{h_{2}}$ dependence of these limits enters via the
relations shown in Eqs. (A.5)–(A.6). In agreement with the discussions of Sec.
2.2, we find that the indirect constraints from the cross-section measurements
of $h_{125}$ are considerably stronger than constraints from the experimental
limit on ${\mathrm{BR}_{\rm inv}}$ from direct searches for the invisible
decay of $h_{125}$, where the latter are indicated with the orange dashed-
dotted line in Fig. 9. For instance, for $v_{\phi}/m_{h_{1}}=10$ we find that
the mixing angle is constrained to be $\cos\theta>0.99$ or equivalently
$\theta<0.14$ at $95\%$ CL based on the values of $\Delta\chi^{2}$, whereas
the bounds from direct searches for the decay $h_{125}\to\mathrm{inv}$ allow
for values as small as $\cos\theta>0.98$ or equivalently $\theta<0.20$. As can
be seen in Eq. (3.11), in the limit where the ratio $v_{\phi}/m_{h_{1}}$
becomes large, the partial width for the invisible decay of $h_{1}\simeq
h_{125}$ becomes negligible. Accordingly, in this limit the bound on the
mixing angle saturates at a constant value of $\cos\theta>0.959$ which matches
our results derived in Sec. 2.2 of Eq. (2.8) in the limit ${\mathrm{BR}_{\rm
inv}}\rightarrow 0$. The limits on $\cos\theta$ can be even stronger when one
combines the limits from the cross-section measurements of $h_{125}$ with the
perturbative unitarity constraints. For instance, assuming that
$m_{h_{2}}=v_{\phi}$ imposes $\cos\theta>0.966$, corresponding to
$\theta<0.26$.

It is tempting to demand independent limits on ${\mathrm{BR}_{\rm inv}}$ and
$\theta$ in a phenomenological analysis in order to account in an approximate
form for the experimental constraints, on the one hand, from the invisible
decay mode of $h_{125}$ and, on the other hand, from the mixing between
$h_{125}$ and the additional scalar state. In many studies this amounts to
applying the direct limit on ${\mathrm{BR}_{\rm inv}}$ (orange line) and a
constant lower limit on the coupling modifier $c_{\rm
uni}=\cos\theta(>0.959)$, where the value in the brackets is the limit we
found to be valid in the limit ${\mathrm{BR}_{\rm inv}}\to 0$ according to the
discussion above. The results depicted in Fig. 9 illustrate that this approach
would allow substantial parts of the parameter space that are actually
excluded by the cross-section measurements of $h_{125}$ (black line). This
demonstrates that in order to fully exploit the experimental data with regards
to the discovered Higgs boson in models with light hidden sectors it is vital
to take into account simultaneously the presence of a non-zero
${\mathrm{BR}_{\rm inv}}$ and the modifications of the couplings of $h_{125}$
to SM particles, as was done in our global $\chi^{2}$-analysis using
HiggsSignals.

The results derived in this section go beyond the global symmetry case and can
also be applied to the case of a local $U(1)$ gauge symmetry to some extent.
In the regime where the mass of the new light vector state
$m_{Z^{\prime}}=g_{Z^{\prime}}q_{\phi}v_{\phi}\ll m_{h_{125}}$, with
$g_{Z^{\prime}}$ being the extra $U(1)$ gauge coupling and $q_{\phi}$ the
charge of the complex scalar breaking this symmetry, by virtue of the
Goldstone boson equivalence theorem the decay rate $\Gamma_{h_{125}\rightarrow
aa}$ into a pair of Goldstone bosons $a$ is identical to the rate
$\Gamma_{h_{125}\rightarrow Z^{\prime}Z^{\prime}}$ into a pair of massive
gauge fields $Z^{\prime}$ in the small mass limit. Therefore, our constraints
also apply to this case, provided that the produced $Z^{\prime}$ is
sufficiently weakly coupled in order to escape the detector without
interacting.

###  3.4 Two Higgs doublet models

One of the most prominent example where observables related to the Higgs boson
depart from their SM predicted values are models containing two Higgs doublet
fields $\Phi_{1}$ and $\Phi_{2}$, called 2 Higgs doublet models (2HDM) [88,
89] (see also Ref. [90] for a review). In the CP-conserving 2HDM, the physical
Higgs spectrum consists of two CP-even states $h$ and $H$, where $h$ in the
following plays the role of $h_{125}$, a CP-odd state $A$, and a pair of
charged Higgs bosons $H^{\pm}$. In the decoupling limit of the 2HDM, in which
the BSM particle states have masses considerably larger than the electroweak
scale, the couplings of the state $h=h_{125}$ to the fermions and gauge bosons
are determined by only two parameters: the ratio of the vacuum expectation
values of the neutral CP-even components of the Higgs doublets $v_{1}$ and
$v_{2}$ written in terms of the parameter

$\tan\beta\equiv\dfrac{v_{2}}{v_{1}}\,,\textrm{ where
}v=\sqrt{v_{1}^{2}+v_{2}^{2}}\simeq 246\ \mathrm{GeV}\ ,$ (3.12)

and the rotation angle $\alpha$ that determines the mixing of the two CP-even
states $h$ and $H$. In the so-called alignment limit, defined by the condition
$\cos(\alpha-\beta)=0$, the couplings of $h$ to the SM particles are identical
to the predictions of the SM.

Model | $u_{R}$ | $d_{R}$ | $\ell_{R}$
---|---|---|---
Type I | $\Phi_{2}$ | $\Phi_{2}$ | $\Phi_{2}$
Type II | $\Phi_{2}$ | $\Phi_{1}$ | $\Phi_{1}$
Type III (lepton-specific) | $\Phi_{2}$ | $\Phi_{2}$ | $\Phi_{1}$
Type IV (flipped) | $\Phi_{2}$ | $\Phi_{1}$ | $\Phi_{2}$
Table 3: Summary of which of the two Higgs doublet fields $\Phi_{1,2}$ is
coupled to up-type fermions ($u_{R}$), down-type fermions ($d_{R}$) and
charged leptons ($\ell_{R}$) in the four Yukawa types of the 2HDM.

In order to eliminate sources of flavour-changing neutral currents at the
classical level, one can introduce a softly broken $\mathbb{Z}_{2}$ symmetry,
under which one of the Higgs doublets changes sign, whereas the second Higgs
doublet transforms trivially, and the $\mathbb{Z}_{2}$ charges of the fermions
depend on the so-called Yukawa type that is assumed. In total, there are four
different possibilities to assign the fermion charges. Depending on the
fermion charge, either $\Phi_{1}$ or $\Phi_{2}$ can be coupled to the
corresponding fermion. In Tab. 3 we explicit which Higgs doublet field is
coupled to which kind of fermion in each of the four types of the 2HDM
considered in this work. The resulting structure of the coupling modifiers
that is realized in each Yukawa type is given in Tab. 2. The explicit
dependence of the modifiers $c_{V,u,d,\ell}$ on the parameters $\alpha$ and
$\beta$ can be found, for instance, in Ref. [90].

In the following we perform a $\chi^{2}$-scan in the plane
$\\{\cos(\alpha-\beta),\tan\beta\\}$ for the four types of the 2HDM in the
decoupling limit, and we analyze how the allowed regions of parameter space
are modified by the existence of an invisible decay mode of $h$. The resulting
constraints in terms of exclusion regions can be applied to a variety of 2HDMs
that are extended by a hidden sector. Such models comprise, for instance,
scalar gauge-singlet extensions of the 2HDM such as the N2HDM [91], the S2HDM
[65, 92] or the 2HDM+a [93]. Here it should be taken into account that for a
concrete model realization of a particular configuration of 2HDM-like coupling
modifications also other experimental and theoretical constraints would have
to be considered. We refrain from doing such a model-specific analysis here,
as we want to focus on the constraints that arise from the properties of
$h_{125}$ under the presence of an additional invisible decay mode.

Figure 10: $\Delta\chi^{2}$-distribution in the
$\\{\cos(\alpha-\beta),\tan\beta\\}$ plane for different values of
${\mathrm{BR}_{\rm inv}}$ in the type I (top) and the type II (bottom). The
dashed black and the solid black lines indicate the exclusion limits at the
68% and the 95% confidence level, respectively.

#### Type I

We show in the upper plots of Fig. 10 the $\chi^{2}$-distribution in the
$\\{\cos(\alpha-\beta),\tan\beta\\}$ plane for the Yukawa type I for different
values of ${\mathrm{BR}_{\rm inv}}$ (varied in steps of 2%). The black dashed
and solid lines indicate the excluded regions of the model parameters at the
68% and the 95% CL, respectively. One can see that, as expected, the best
agreement with the experimental data regarding $h_{125}$ is found at or about
the alignment limit $\cos(\alpha-\beta)=0$, in which the state $h$ resembles a
SM Higgs boson. By comparing the plots for different values of
${\mathrm{BR}_{\rm inv}}$ it becomes apparent that the values of
$\Delta\chi^{2}$ increase with increasing values of ${\mathrm{BR}_{\rm inv}}$
in the whole parameter plane. Consequently, in the Yukawa type I the presence
of an invisible decay mode of $h_{125}$ deteriorates the fit result to the
experimental Higgs-boson data independently of the values of the coupling
modifiers. For a value of ${\mathrm{BR}_{\rm inv}}=6\%$ only a small region
with $\tan\beta\lesssim 5$ and $\cos(\alpha-\beta)\simeq 0$ remains allowed at
the $1\sigma$ level, whereas for ${\mathrm{BR}_{\rm inv}}=8\%$ even the
alignment limit is almost excluded at the 95% CL. The type I Yukawa structure
is excluded at more than 95% CL for ${\mathrm{BR}_{\rm inv}}=10\%$, such that
the experimental limits from direct searches for $h_{125}\to\mathrm{inv}$
(currently at the level of ${\mathrm{BR}_{\rm inv}}<11\%$ or larger) do not
provide additional constraints. The observations made here are in agreement
with the results depicted in Fig. 3 for the scan in the benchmark model
featuring the coupling modifiers $c_{V}$ and $c_{f}=c_{u}=c_{d}=c_{\ell}$,
which is also the pattern of coupling modifications that is realized in the
type I, where here the additional restriction $c_{V}\leq 1$ applies.
Accordingly, we also observe here values of $\Delta\chi^{2}$ substantially
smaller than zero. The origin of these values was discussed already in Sec.
2.3.

Figure 11: Same as in Fig. 10, but for the type III (top) and the type IV
(bottom).

#### Type II

The corresponding results for the Yukawa type II are depicted in the bottom
panels of Fig. 10. Regarding the fit result in the vicinity of the alignment
limit of the Yukawa type II, one can observe that with increasing value of
${\mathrm{BR}_{\rm inv}}$ the parameter region that is in agreement with the
measurements of $h_{125}$ moves towards values of $\cos(\alpha-\beta)>0$. This
is related to the fact that one finds $|c_{b}|<1$ and
$|c_{V}|\simeq|c_{u}|\simeq 1$ for $\cos(\alpha-\beta)>0$, such that the
additional contribution $\Gamma_{\rm inv}$ to the total decay width of
$h_{125}$ is compensated by a reduced partial decay width $\Gamma_{b\bar{b}}$,
while at the same time the ggH and the VBF production cross sections of
$h_{125}$ are not significantly modified compared to the SM predictions. On
the other hand, for $\cos(\alpha-\beta)<0$ one finds $|c_{d}|>1$, such that
the branching ratio of the decay $h_{125}\to\gamma\gamma$ is suppressed by
both the non-zero value of $\Gamma_{\rm inv}$ and the enhancement of
$\Gamma_{b\bar{b}}$, and consequently the fit result deteriorates in this case
with increasing values of ${\mathrm{BR}_{\rm inv}}$.

In the type II, in addition to the alignment limit, there is a tight branch in
the parameter plane with values of $\cos(\alpha-\beta)>0.08$ and
$\tan\beta\gtrsim 7$ that predicts a state $h$ that can be in agreement with
the experimental measurements and hence features small values of
$\Delta\chi^{2}$. This branch is a 2HDM realization of the so-called wrong-
sign Yukawa coupling regime, in which the absolute values of the coupling
modifiers are approximately equal to one, i.e. $|c_{V,u,d,\ell}|\simeq 1$, but
in which the couplings to down-type quarks have the opposite sign as compared
to the couplings to gauge bosons and up-type quarks [42, 94] (see also Ref.
[95] for a discussion of this limit in the 2HDM in light of recent LHC
measurements). In the absence of a decay mode $h_{125}\to\mathrm{inv}$ the
wrong-sign Yukawa coupling regime is in tension with the signal-rate
measurements of $h_{125}$, as can be seen in the upper left plot. This tension
is mainly driven by the enhancement of the gluon-fusion production cross
section of $h_{125}$ compared to the SM prediction as a result of the modified
interference effects between the contributions from the top-quark loop and the
bottom-quark loop [95]. We find here that the agreement with the experimental
data can be improved significantly in the wrong-sign Yukawa coupling regime if
a sizable value of ${\mathrm{BR}_{\rm inv}}$ is present. Even for values as
large as ${\mathrm{BR}_{\rm inv}}=10\%$ we find parameter points in this limit
that fit the signal-rate measurements of $h_{125}$ just as well as the SM,
whereas the same parameter points would be excluded at a CL substantially
larger than $2\sigma$ for ${\mathrm{BR}_{\rm inv}}=0$. Here, the additional
partial decay with $\Gamma_{\rm inv}$ leads to a suppression of the ordinary
decay modes of $h_{125}$ which compensates the enhancements of the gluon-
fusion production cross sections. The relative sizes of the signal rates of
$h_{125}$ can then remain close to the SM predictions because the type II
Yukawa structure allows for the individual modifcation of both $c_{u}$ and
$c_{d}$, such that one can find $|c_{u}|\simeq|c_{V}|\simeq 1$ and $|c_{d}|<1$
(compare also to Fig. 4 and the related discussion).

As a consequence of the fact that we find both in the vicinity of the
alignment limit and in the wrong-sign Yukawa coupling regime parameter points
that are in agreement with the indirect constraints from the signal-rate
measurements of $h_{125}$ even for values of ${\mathrm{BR}_{\rm inv}}=10\%$,
whereas the same parameter points would be excluded in the absence of the
invisible decay mode, it becomes clear that in the Yukawa type II the upper
limits on ${\mathrm{BR}_{\rm inv}}$ from the direct searches for
$h_{125}\to\mathrm{inv}$ are (and will be) relevant and have to be taken into
account. Finally, we emphasize that in order to find a valid parameter point
in the allowed parameter regions for ${\mathrm{BR}_{\rm inv}}>0$ in a concrete
BSM theory also other constraints, for instance, from LHC searches for the BSM
Higgs bosons $H$, $A$ and $H^{\pm}$ would have to be checked against. We will
leave such a model-specific analysis for future investigations.

#### Type III

The results for the type III (lepton-specific) Yukwa structure are depicted in
the top panels of Fig. 11. The couplings of $h_{125}$ to vector bosons and to
up- and down-type quarks in type III are identical to the ones in type I.
Thus, the predictions for the production cross sections of $h_{125}$ are very
similar to type I. Accordingly, we find for type III that the agreement with
the indirect constraints becomes worse with increasing value of
${\mathrm{BR}_{\rm inv}}$, similarily to what we observed for type I (see Fig.
10). This is also true for the wrong-sign Yukawa coupling regime, which cannot
be realized in type I, whereas, as can be seen in the top row of plots in Fig.
11, in type III the wrong-sign Yukawa coupling regime is still allowed with
regards to the constraints from the signal-rate measurements of $h_{125}$.
However, in contrast to what we observed in type II, in type III the fit-
result in the wrong-sign Yukawa coupling regime becomes worse with increasing
value of ${\mathrm{BR}_{\rm inv}}$, and it is effectively absent for values of
${\mathrm{BR}_{\rm inv}}>6\%$. We find that for ${\mathrm{BR}_{\rm inv}}=8\%$
only a small region around the alignment limit remains allowed if exclusion
limits at the 95% CL are applied. For ${\mathrm{BR}_{\rm inv}}=10\%$ the whole
parameter plane is excluded in type III. Hence, in this case the upper limits
on ${\mathrm{BR}_{\rm inv}}$ resulting from the direct searches for
$h_{125}\to\mathrm{inv}$ do not provide additional constraints, and one should
instead apply the indirect constraints resulting from the signal-rate
measurements of $h_{125}$.

#### Type IV

The results for the Yukawa type IV (flipped) are depicted in the bottom panels
of Fig. 11. In this type, the coupling modifiers $c_{V}$, $c_{u}$ and $c_{d}$
are identical to the ones of the Yukawa type II. As a result, the cross-
section predictions are also (practically) identical, and also the main
branching ratios of $h_{125}$ remain effectively unchanged compared to type
II, with the exception of the decays of $h_{125}$ into pairs of charged
leptons. Accordingly, we find that the allowed region in the
$\\{\cos(\alpha-\beta),\tan\beta\\}$ plane resemble the allowed regions that
we found for type II (see Fig. 10). Both in the vicinity of the alignment
limit, but also in the wrong-sign Yukawa coupling regime, we find allowed
regions that accommodate the experimental data regarding $h_{125}$ within the
$1\sigma$ level. Hence, as in type II, the direct limit ${\mathrm{BR}_{\rm
inv}}<11\%$ from the searches for the $h_{125}\to\mathrm{inv}$ decay mode have
to be taken into account in type IV, since they exclude parameter space points
with larger values of ${\mathrm{BR}_{\rm inv}}$ that otherwise would be
allowed.

##  4 Summary and conclusions

We explored the possibility of connecting a hidden sector to the discovered
Higgs boson at $125\ \mathrm{GeV}$. We parameterized deviations of the Higgs-
boson couplings to SM states by introducing coupling modifiers $c_{i}$ for
$i={V,u,d,\ell},$ and by introducing a possible invisible branching ratio
${\mathrm{BR}_{\rm inv}}$ as a free parameter. We performed
$\chi^{2}$-analyses taking into account a large set of cross-section and
signal-rate measurements of the discovered Higgs boson by using the public
code HiggsSignals, and we identified the ranges of the coupling modifiers that
are allowed or excluded at $68\%$ and $95\%$ confidence level depending on the
value of ${\mathrm{BR}_{\rm inv}}$. We furthermore derived constraints on
generic constructions of modifications of the properties of $h_{125}$ compared
to the SM, by making general assumptions about the structure of the couplings,
in order for the corresponding constraints to be applicable to a large
category of models. Finally, we derived constraints on the model parameters
for a variety of more specific BSM constructions featuring hidden sectors such
as Higgs-portal dark matter models, scenarios featuring Nambu-Goldstone bosons
and extended Higgs sector in the context of the 2HDM.

The first general conclusion of this paper is that currently in a wide class
of models the LHC measurements of the signal rates of conventional visible
signals of $h_{125}$, in particular in the golden di-photon channel, impose
indirect constraints on the invisible branching ratio of the discovered Higgs
boson that can be substantially stronger than the direct constraints resulting
from searches for the invisible Higgs-boson decay. Hence, for a
phenomenological analysis of a model in which decays of $h_{125}$ into hidden
sectors are kinematically open it is crucial to not only consider the direct
limits on ${\mathrm{BR}_{\rm inv}}$, but to also take into account the
indirect constraints on ${\mathrm{BR}_{\rm inv}}$ in order to not include
parameter regions that are in disagreement with the experimental data from the
LHC.

Secondly, we also found that the presence of a sizable value of
${\mathrm{BR}_{\rm inv}}$ can render scenarios beyond the SM that predict
modifications to the couplings of $h_{125}$ compared to the SM viable which
would otherwise be excluded by the Higgs-boson signal-rate measurements. This
is trivially true in case of a universal enhancement of all Higgs-boson
couplings by a common factor, where the enhancements of the couplings cancel
the suppression of ordinary decay modes of $h_{125}$ as a result of the
additional decay mode $h_{125}\to\mathrm{inv}$. However, also more intricate
coupling modifications can be rendered viable via the presence of an invisible
decay mode of $h_{125}$. The most striking example of our analysis is the
wrong-sign Yukawa coupling regime realized in two Higgs doublet extensions of
the SM with Yukawa type II and IV, which we showed to be in tension with the
cross-section measurements of $h_{125}$ assuming ${\mathrm{BR}_{\rm inv}}=0$,
whereas it is in very good agreement with these measurements assuming values
of ${\mathrm{BR}_{\rm inv}}$ as large as $10\%$.

For the variety of structures of coupling-modifiers and BSM model considered
in this work, our main results are summarized in the following.

#### SM-like couplings $c_{i}=1$ and $\mathbf{{\mathrm{BR}_{\rm inv}}\neq 0}$.

If the Higgs boson possesses couplings to ordinary matter as predicted by the
SM, we found that the invisible Higgs-boson branching ratio is constrained to
be ${\mathrm{BR}_{\rm inv}}<6.2\%$ at $95\%$ CL as a result of the indirect
constraints from $h_{125}$ signal-rate measurements, which is substantially
smaller than the currently strongest limit ${\mathrm{BR}_{\rm inv}}<11\%$
resulting from the direct searches for the invisible decay of $h_{125}$.

#### Universal couplings $c_{i}=c_{\text{uni}}$ and
$\text{BR}_{\text{inv}}\neq 0$.

We found a flat direction with $\Delta\chi^{2}=0$ in the $\\{{c_{\rm
uni}},{\mathrm{BR}_{\rm inv}}\\}$ plane where the universal enhancements of
the Higgs-boson couplings for $c_{\mathrm{uni}}>1$ are compesated by a non-
vanishing ${\mathrm{BR}_{\rm inv}}$. Based on the indirect constraints,
universal couplings are allowed at 95% CL in the interval
$0.520\,{\mathrm{BR}_{\rm inv}}+0.960\,<\,{c_{\rm
uni}}\,<\,0.565\,{\mathrm{BR}_{\rm inv}}+1.043\,$, and are ultimately only
bound by the direct limit on ${\mathrm{BR}_{\rm inv}}$. In the case where the
origin of the coupling modification is the mixing of an additional gauge-
singlet scalar with the Higgs boson $h_{125}$, i.e. ${{c_{\rm
uni}}=\cos\theta}$ with $\theta$ being the mixing angle, we provided indirect
limits on ${\mathrm{BR}_{\rm inv}}$ as a function of $\theta$ in an
approximate form as given in Eq. (2.9).

#### Non-universal couplings and $\text{BR}_{\text{inv}}\neq 0$.

We considered two examples for non-universal coupling modifications of the
discovered Higgs boson. The first example consists in a common coupling
modifier for the couplings to fermions, i.e. $c_{f}=c_{u}=c_{d}=c_{\ell}$, and
a second independent coupling modifier $c_{V}$ for the couplings to the gauge
bosons. We found that if $c_{V}\leq 1$ applies, invisible $h_{125}$ decay
modes gives rise to a decrease of the allowed ranges of both $c_{V}$ and
$c_{f}$. However, if also $c_{V}>1$ is considered, the $\chi^{2}$ values do
not generically degrade as $\mathrm{BR}_{\rm inv}$ is increased to 10%. Hence,
the presence of an invisible decay mode of $h_{125}$ with sizable branching
ratios opens up parameter space regions in the $\\{c_{f},c_{V}\\}$ plane that
would be excluded if such a novel decay mode is not present. The second
example considered is a SM-like Higgs-boson coupling to vector bosons
$c_{V}=1$ but modified couplings to up and down-type quarks with the
additional condition $c_{\ell}=c_{d}$. We found that increasing values of
${\mathrm{BR}_{\rm inv}}$ shifts the allowed parameter space to larger values
of $c_{u}$ and smaller values of $c_{d}=c_{\ell}$, but overall yielding a
degraded fit result.

#### Higgs portal dark matter.

We considered dark matter candidates (scalar, fermion and vector) coupled to
the Higgs boson via a single operator and computed the relic abundance and
direct detection cross section in terms of the dimensionless Higgs-portal
coupling parameter for each case. Direct detection experiments tightly
constrain the parameter space suitable to achieve the measured relic
abundance, leaving for the bosonic candidates only the mass region at the
Higgs-boson resonance, i.e. if the DM mass is $\mathcal{O}(m_{h_{125}}/2)$. We
showed that new results from the LZ collaboration rule out the fermionic DM
candidate within the simplest Higgs-portal scenario under standard
assumptions.222222Given the fact that the LZ limit is close to the parameter
space corresponding to the correct relic abundance, several uncertainties can
affect this statement. To derive bounds on the DM nucleon scattering cross
section, a local DM density of $\rho_{\odot}=0.3\,\text{GeV}\,\text{cm}^{-3}$
and the standard halo model for the distribution of dark matter in the Milky
Way are standard assumptions from direct detection collaborations. Both
quantities suffer from astrophysical uncertainties that could affect
constraints from direct searches by a $\mathcal{O}(1-3)$ factor [96]. Such
uncertainties could allow the parameter space around the resonance to still be
viable but on the edge of being excluded.. Around the resonance, if we relax
the hypothesis of the DM candidates to account for all the dark matter, the
indirect constraints resulting from the cross-section measurements of
$h_{125}$ derived using HiggsSignal are the strongest bounds in the narrow
range of DM masses that is not excluded by direct detection experiments.

#### Singlet portal dark matter.

We studied an extension of the Higgs-portal scenario comprising an additional
singlet-like scalar $h_{2}$ acting as the portal between a dark matter fermion
and the visible sector. As a consequence of the presence of $h_{2}$, we also
took into account constraints from collider searches for additional Higgs
bosons using the public code HiggsBounds. Firstly, we investigated a benchmark
scenario in which both the decays of $h_{125}$ into the dark-matter state and
into a pair of the additional scalar are kinematically allowed, which both can
contribute to ${\mathrm{BR}_{\rm inv}}$ depending on whether $h_{2}$ decays
invisibly. Secondly, we investigated a benchmark scenario in which the second
scalar has a mass of $80\ \mathrm{GeV}$, allowing for a larger mixing between
discovered Higgs boson and $h_{2}$. We identified the regions of the parameter
space that are still allowed by direct detection constraints and in which part
or all of the measured relic abundance of dark matter is predicted. We
demonstrated that in such regions of the parameter space the indirect
constraints from the cross-section measurements of $h_{125}$ provide the
strongest constraints, whereas the direct limit on ${\mathrm{BR}_{\rm inv}}$
from searches for the invisible decay of $h_{125}$ do not give rise to any
additional constraints.

#### (Pseudo) Nambu-Goldstone bosons ((P)NGB).

We considered the possibility of the Higgs boson $h_{125}$ decaying into pairs
of (P)NGBs, where the (P)NGB arises as the angular mode of a new complex
scalar after the spontaneous breaking of an additional (approximate)
continuous symmetry once the radial mode acquires a vev $v_{\phi}$. Using the
cross-section measurements of $h_{125}$, we determined constraints on the
mixing between $h_{125}$ and the radial mode and on the invisible branching
ratio ${\mathrm{BR}_{\rm inv}}$. We highlighted that if one would apply
independent constraints on the mixing angle $\theta$ and the invisible
branching ratio ${\mathrm{BR}_{\rm inv}}$, one would consider parameter
regions as allowed which are actually excluded by the cross-section
measurements of $h_{125}$, as we showed by simultaneously taking into account
both the mixing in the Higgs sector and the presence of a non-vanishing
${\mathrm{BR}_{\rm inv}}$ in a global fit to the experimental Higgs-physics
observables.

#### Two Higgs doublet models.

We derived constraints in the plane $\\{\tan\beta,\cos(\alpha-\beta)\\}$ for
$\tan\beta\in[0,20]$ in the four Yukawa types of the 2HDM. We found that the
$\chi^{2}$-fit result deteriorates in the type I and III as ${\mathrm{BR}_{\rm
inv}}$ is increased, leaving both types as excluded at the 95% CL for values
of ${\mathrm{BR}_{\rm inv}}>8\%$. On the other hand, for type II and IV a
narrow region with small but positive values of $\cos(\alpha-\beta)$ remains
viable even for ${\mathrm{BR}_{\rm inv}}=10\%$. We also found that in these
two types the wrong-sign Yukawa coupling regime, which is barely in agreement
with the measurements of $h_{125}$ for ${\mathrm{BR}_{\rm inv}}=0$, can be in
very good agreement with the experimental data if the presence of an invisible
decay mode of $h_{125}$ with ${\mathrm{BR}_{\rm inv}}>2\%$ is assumed to be
present.

## Acknowledgements

The authors thank Anton Sokolov for useful discussions. The authors
acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany‘s Excellence Strategy – EXC 2121 “Quantum
Universe” – 390833306. This work has been partially funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - 491245950. This
work was made possible by with the support of the Institut Pascal at
Université Paris-Saclay during the Paris-Saclay Astroparticle Symposium 2021,
with the support of the P2IO Laboratory of Excellence (program
“Investissements d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and
ANR-10-LABX-0038), the P2I axis of the Graduate School Physics of Université
Paris-Saclay, as well as IJCLab, CEA, IPhT, APPEC, the IN2P3 master projet
UCMN and EuCAPT ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038.

##  Appendix A Singlet portal dark matter: additional content

###  A.1 Minimization of the potential

The vevs are related to the bilinear mass parameters via the tadpole equations

$\mu_{H}^{2}=-\frac{1}{2}\left(2\lambda_{H}v^{2}+\lambda_{\Phi
H}v_{\phi}^{2}\right)\ ,\quad\mu_{\Phi}^{2}=-\frac{1}{2}\left(\lambda_{\Phi
H}v^{2}+2\lambda_{\Phi}v_{\phi}^{2}\right)\ .$ (A.1)

Using these relations, the mass matrix of the CP-even fields $h$ and $\phi$
can be written as

$\mathcal{M}^{2}=\begin{pmatrix}2\lambda_{H}v^{2}&\lambda_{\Phi H}vv_{\phi}\\\
\lambda_{\Phi H}vv_{\phi}&2\lambda_{\Phi}v_{\phi}^{2}\end{pmatrix}\ .$ (A.2)

This matrix can be diagonalized by a rotation of angle $\theta$, expressed in
terms of the various parameters as

$\tan(2\theta)\,=\,\dfrac{\lambda_{\Phi
H}vv_{\phi}}{\lambda_{\Phi}v_{\phi}^{2}-\lambda_{H}v^{2}}\,.$ (A.3)

The quartic scalar couplings are then dependent parameters that can be
computed via the relations

$\displaystyle\lambda_{H}$
$\displaystyle=\frac{1}{2v^{2}}\left(m_{h_{1}}^{2}\cos^{2}\theta+m_{h_{2}}^{2}\sin^{2}\theta\right)\
,$ (A.4) $\displaystyle\lambda_{\Phi}$
$\displaystyle=\frac{1}{2v_{\phi}^{2}}\left(m_{h_{1}}^{2}\sin^{2}\theta+m_{h_{2}}^{2}\cos^{2}\theta\right)\
,$ (A.5) $\displaystyle\lambda_{\Phi H}$
$\displaystyle=\frac{1}{vv_{\phi}}\left(m_{h_{2}}^{2}-m_{h_{1}}^{2}\right)\cos\theta\sin\theta\
.$ (A.6)

###  A.2 Decay rates

The relevant partial decay width for the SM-like Higgs $h_{1}\simeq h_{125}$
are

$\Gamma_{h_{1}\rightarrow
h_{2}h_{2}}\,=\,\frac{\left(c_{\theta}^{3}\lambda_{\Phi
H}v+2c_{\theta}^{2}s_{\theta}v_{\phi}(\lambda_{\Phi
H}-3\lambda_{S})+2c_{\theta}s_{\theta}^{2}v(3\lambda_{H}-\lambda_{\Phi
H})-\lambda_{\Phi H}s_{\theta}^{3}v_{\phi}\right)^{2}}{32\pi
m_{h_{1}}}\sqrt{1-\dfrac{4m_{h_{2}}^{2}}{m_{h_{1}}^{2}}}\,,$ (A.7)

Decay rates from the physical scalars to a DM pair are given by

$\Gamma_{h_{1}\rightarrow\bar{\chi}\chi}\,=\,\dfrac{y_{\chi}^{2}s_{\theta}^{2}}{8\pi}m_{h_{1}}\left(1-\frac{4m_{\chi}^{2}}{m_{h_{1}}^{2}}\right)^{3/2}\,=\,\frac{s_{\theta}^{2}m_{h_{1}}m_{\chi}^{2}}{8\pi
v_{\phi}^{2}}\left(1-\frac{4m_{\chi}^{2}}{m_{h_{1}}^{2}}\right)^{3/2}\leavevmode\nobreak\
,$ (A.8)

and

$\Gamma_{h_{2}\rightarrow\bar{\chi}\chi}\,=\,\dfrac{y_{\chi}^{2}c_{\theta}^{2}}{8\pi}m_{h_{2}}\left(1-\frac{4m_{\chi}^{2}}{m_{h_{2}}^{2}}\right)^{3/2}\,=\,\frac{c_{\theta}^{2}m_{h_{2}}m_{\chi}^{2}}{8\pi
v_{\phi}^{2}}\left(1-\frac{4m_{\chi}^{2}}{m_{h_{2}}^{2}}\right)^{3/2}\leavevmode\nobreak\
.$ (A.9)

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