# What can we learn from the total width of the Higgs boson?

Qing-Hong Cao<EMAIL_ADDRESS>Department of Physics and State Key
Laboratory of Nuclear Physics and Technology, Peking University, Beijing
100871, China  Collaborative Innovation Center of Quantum Matter, Beijing
100871, China  Center for High Energy Physics, Peking University, Beijing
100871, China  Hao-Lin Li<EMAIL_ADDRESS>CAS Key Laboratory of
Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of
Sciences, Beijing 100190, P. R. China  Ling-Xiao Xu<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics and State Key Laboratory of Nuclear
Physics and Technology, Peking University, Beijing 100871, China  Scuola
Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy  Jiang-Hao Yu
<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, P. R. China
School of Physical Sciences, University of Chinese Academy of Sciences, No.19A
Yuquan Road, Beijing 100049, P.R. China  Center for High Energy Physics,
Peking University, Beijing 100871, China  School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China  International Centre for Theoretical Physics Asia-Pacific,
Beijing/Hangzhou, China

###### Abstract

As one of the key properties of the Higgs boson, the Higgs total width is
sensitive to global profile of the Higgs boson couplings, and thus new physics
would modify the Higgs width. We investigate the total width in various new
physics models, including various scalar extension, composite Higgs models,
and fraternal twin Higgs model. Typically the Higgs width is smaller than the
standard model value due to mixture with other scalar if the Higgs is
elementary, or curved Higgs field space for the composite Higgs. On the other
hand, except the possible invisible decay mode, the enhanced Yukawa coupling
in the two Higgs doublet model or the exotic fermion embeddings in the
composite Higgs, could enhance the Higgs width greatly. The precision
measurement of the Higgs total width at the high-luminosity LHC can be used to
discriminate certain new physics models.

##  I Introduction

After the discovery of the Higgs boson, the next step is to decipher the
particle nature of the Higgs boson, e.g., the mass and width of the Higgs
boson ($h$). The Higgs boson mass $m_{H}$ has been measured very precisely at
the Large Hadron Collider (LHC), and its value sheds lights on the stability
of the vacuum. On the other hand, the width of the Higgs boson ($\Gamma_{h}$)
is not measured yet. We will focus on in this paper the information we can
obtain by precision measurements of the Higgs total width.

At the LHC, information of the total width of the Higgs boson can be extracted
from off-shell Higgs production Caola and Melnikov (2013); Campbell et al.
(2014a, b); Englert et al. (2015); Logan (2015), the global fit results of the
on-shell Higgs signals Barger et al. (2012), $\bar{t}tH$ and four-top
productions Cao et al. (2017). Measurements of the Higgs total width by the
ATLAS and CMS collaboration are found in Refs. Aaboud et al. (2018); Sirunyan
et al. (2019), respectively, assuming standard model (SM)-like Higgs
couplings. More interestingly, it is worthwhile noting that the Higgs width
can also be measured through the Higgs line shape at $\gamma\gamma$ and
$\mu^{+}\mu^{-}$ colliders Gunion and Haber (1993); Barger et al. (1997).

Assuming the SM-like Higgs couplings, the total width of the Higgs boson
$\Gamma$ can be parametrized as

$\displaystyle\Gamma$
$\displaystyle\simeq\Gamma_{\text{SM}}\left(c_{b}^{2}\text{Br}_{bb}+c_{\tau}^{2}\text{Br}_{\tau\tau}+c_{c}^{2}\text{Br}_{cc}\right.$
$\displaystyle\ \ \ \ \ \ +\
c_{g}^{2}\text{Br}_{gg}+c_{W}^{2}\text{Br}_{WW}+c_{Z}^{2}\text{Br}_{ZZ}$
$\displaystyle\left.\ \ \ \ \ \ +\
c_{\gamma}^{2}\text{Br}_{\gamma\gamma}+c_{\gamma Z}^{2}\text{Br}_{\gamma
Z}\right)+\Gamma_{\text{invisible}},$ (1)

where the major decay modes of the SM Higgs boson, and the invisible width,
are included. The numerical values of the SM total width is
$\Gamma_{\text{SM}}\simeq 4.1$ MeV, and the major branching ratios in the SM
are $\text{Br}_{bb}\simeq 0.584,\text{Br}_{\tau\tau}\simeq
0.0627,\text{Br}_{cc}\simeq 0.029,\text{Br}_{gg}\simeq
0.0856,\text{Br}_{WW}\simeq 0.214,\text{Br}_{ZZ}\simeq 0.0262$, respectively
de Florian et al. (2016). The rescaling factors $c_{i}$’s denote the effects
of new physics (NP) in the Higgs boson decay. Since $\text{Br}_{\gamma\gamma}$
and $\text{Br}_{\gamma Z}$ are highly suppressed in the SM, we neglect these
two decay channels when considering the total width of the Higgs boson. The
invisible decay of the Higgs boson takes place in many NP models, therefore,
we also include the invisible width $\Gamma_{\text{invisible}}$ in our study.
We refer to Ref. Brivio et al. (2019) for a recent analysis of the Higgs width
in the framework of SM effective field theory (EFT).

It remains unknown whether the Higgs boson is fundamental or composite, while
the couplings of the Higgs boson to gauge bosons often tend to be smaller than
the SM values regardless of this nature. In the case of a fundamental Higgs
boson, the decrease of the Higgs-gauge-boson couplings is usually caused by
mixing between the Higgs and the new scalar, whereas the exception exist for
the Georgi-Machacek model Georgi and Machacek (1985); Chanowitz and Golden
(1985) where $c_{W,Z}$ can be enhanced. On the other hand, when the Higgs
boson is composite and is the pNGB comes from the global symmetry breaking,
the Higgs boson to gauge boson couplings decrease due to the misalignment
between the gauged direction and the true vacuum direction where the physical
Higgs boson fluctuates around. The above arguments for the decrease of the
Higgs boson to gauge boson couplings sometimes also hold for the the Higgs
boson to fermion couplings, which leads to a decreased Higgs-boson total width
if no invisible Hidden sector decay considered. However, in the two Higgs
doublet models (2HDMs) Branco et al. (2012), the couplings of the Higgs boson
to fermions can be enhanced when deviating from the alignment limit, while in
the minimal composite Higgs boson model, an exotic fermion embedding can also
leads to the enhanced Yukawa couplings of the top quark to the Higgs boson.
Therefore we investigate several popular models that can modify the Higgs-
boson total width significantly.

Given the measurement of the total Higgs width, it is possible to discriminate
certain classes of new physics models. For the models we studied in this
paper, an increased Higgs decay width indicating either an enhanced Yukawa
coupling or the existence of invisible decay in the extended scalar models
when the Higgs is an elementary particles. When the Higgs is an pNGB, an
increased Higgs width usually needs an exotic fermion setup where the SM
fermions are embedded in higher dimensional representations. On the other
hand, If one observes a smaller Higgs width compared to the SM, a heavy scalar
particle that mixes with Higgs is expected to be found for the Higgs as an
elementary particle, such a particle may or may not exist in the pNGB Higgs
scenario depending on whether the UV is strong coupled or not. In Fig. 1, we
present that how different kinds of new physics models are classified based on
the Higgs width modification and nature of the Higgs boson.

Figure 1: Various kinds of new physics models are classified in the flowchart,
based on the nature of the Higgs boson, and future measurement of the total
Higgs width.

The paper is organized as follows. In Sec. II we discuss the simplest case of
the scalar extension, real singlet scalar model Profumo et al. (2007). In Sec.
III we focus on four types of two Higgs doublet models (2HDMs), which
correspond to four different assignments of Yukawa couplings between fermions
and Higgs doublets Branco et al. (2012). In Sec. V we focus on minimal
composite Higgs boson model Agashe et al. (2005), in which the Higgs boson
emerges as a pseudo Nambu-Goldstone boson (PNGB) from the coset $SO(5)/SO(4)$.
In Sec. VI we turn our attention to the twin Higgs boson paradigm Chacko et
al. (2006). In contrast to the original mirror twin Higgs boson model Chacko
et al. (2006), the fraternal twin Higgs-boson model Craig et al. (2015a) is
considered, in which only the third generation of the SM fermions have twin
partners. Finally, we conclude.

##  II Real Singlet Model

The real singlet scalar model is the most simplest extension to the SM scalar
sector that is possible to generate strong first order electroweak phase
transition. It has been extensively studied in the literature O’Connell et al.
(2007); Profumo et al. (2007). In this model, after electroweak symmetry
breaking, the real singlet scalar $S$ mixes with the neutral CP-even component
$h_{0}$ in the Higgs boson doublet such that the physical Higgs boson $h$ we
observed can be expressed as:

$\displaystyle h=h_{0}\cos\theta+S\sin\theta.$ (2)

Therefore the couplings of the Higgs boson $h$ to other SM particles are
scaled by an overall factor $\cos\theta$. If we assume that, in the minimal
setup, there is no hidden sector particles that the Higgs boson $h$ can decays
to and the mass of the other singlet-like scalar is larger than half of the
SM-like Higgs mass, then the Higgs-boson total width is the SM value scaled by
$\cos\theta$. In this case, any confirmation of an enhanced Higgs total width
can help to ruled out this minimal setup. This conclusion could be extended to
extended scalar models, such as the general model setup considered in Ref.
Ramsey-Musolf et al. (2021).

Figure 2 displays the ratio of the Higgs boson width to its SM value in the
plane of the invisible decay branching ratio $B_{inv}$ and the mixing angle
$\sin^{2}\theta$. The chosen ranges of the two axes are within the current LHC
constraints Aad et al. (2019). The pink region denotes the parameter space
where the deviation of the total width is within 5% of the SM value such that
it cannot be discriminated with the SM with the future high luminosity LHC
(HL-LHC). However, if the Higgs boson width is more than the SM value by 25%,
then it is likely that there is an invisible decays of the Higgs boson in the
singlet model, regardless of the mixing angle $\theta$.

Figure 2: Contour plots on the total width of the Higgs boson in the singlet
scalar extended model in the plane of the mixing angle $\sin^{2}\theta$ and
the invisible decay branching ratio $B_{inv}$. The parameter space in the pink
region the SM value cannot be differentiated with the SM with the future HL-
LHC.

##  III Two Higgs Doublet Models

We consider the general CP-conserving 2HDM with soft $Z_{2}$ breaking mass
term Branco et al. (2012). The Higgs potential is

$\displaystyle V(\phi_{1},\phi_{2})$ $\displaystyle=$ $\displaystyle
m_{11}^{2}(\phi_{1}^{\dagger}\phi_{1})+m_{22}^{2}(\phi_{2}^{\dagger}\phi_{2})-\left(m_{12}^{2}(\phi_{1}^{\dagger}\phi_{2})+{\rm
h.c.}\right)$ (3)
$\displaystyle+\frac{\lambda_{1}}{2}(\phi_{1}^{\dagger}\phi_{1})^{2}+\frac{\lambda_{2}}{2}(\phi_{2}^{\dagger}\phi_{2})^{2}+\lambda_{3}(\phi_{1}^{\dagger}\phi_{1})(\phi_{2}^{\dagger}\phi_{2})$
$\displaystyle+\lambda_{4}(\phi_{1}^{\dagger}\phi_{2})(\phi_{2}^{\dagger}\phi_{1})+\frac{1}{2}\left[\lambda_{5}(\phi_{1}^{\dagger}\phi_{2})^{2}+{\rm
h.c.}\right]\ ,$

where all the parameters in the above potential are assumed to be real. Four
types of Yukawa interactions can be introduced with different assignments of
the $Z_{2}$ charge of the fermion fields to prevent the tree-level flavor
changing neutral current (FCNC). In each model, different types of the right
handed fermion fields couple to different Higgs doublets; see Table 1.

Table 1: Models without tree-level FCNC where the $u_{R}^{i}$, $d_{R}^{i}$ and $e_{R}^{i}$ represents the right-handed up-type quark, down-type quark and charged lepton fields, respectively. In different types of models they couple to different Higgs doublets. Model | $u_{R}^{i}$ | $d_{R}^{i}$ | $e_{R}^{i}$ | Model | $u_{R}^{i}$ | $d_{R}^{i}$ | $e_{R}^{i}$
---|---|---|---|---|---|---|---
Type-I | $\phi_{2}$ | $\phi_{2}$ | $\phi_{2}$ | Type-II | $\phi_{2}$ | $\phi_{1}$ | $\phi_{1}$
Type-X | $\phi_{2}$ | $\phi_{2}$ | $\phi_{1}$ | Type-Y | $\phi_{2}$ | $\phi_{1}$ | $\phi_{2}$

People usually trade the parameters in the potential with a set of physical
parameters: $v$, the electroweak vacuum expectation value (vev); $\tan\beta$,
the ratio of the vevs of two Higgs doublets; $\alpha$, the rotation angle
which diagonalizes the mass matrix of the CP-even neutral Higgs sector;
$m_{1(2,3)}$, the mass for the tree neutral Higgs, where $m_{1}=125.1$ GeV is
identified with the mass of the SM-like Higgs boson, $m_{2}$ and $m_{3}$ are
masses of the heavy CP even and CP-odd Higgs boson, respectively;
$m_{H^{\pm}}$, the mass of the charged Higgs boson; $m_{12}^{2}$, the
parameters that is sensitive to several the theoretical bonds of the theory
(perturbative unitarity, stability) Kling et al. (2016). In addition, we
denotes the physical SM-like Higgs boson as $h$ and the charged Higgs boson as
$H^{\pm}$.

The Higgs boson width in the 2HDM is dominantly determined by the parameters
$\alpha$, $\beta$, which fix the Higgs couplings to fermions and gauge bosons.
The dependence on $m_{12}^{2}$, $m_{2}$ and $m_{H^{\pm}}$ appears when taking
into account the partial widths of $h\to\gamma\gamma$ and $h\to Z\gamma$. We
define the rescaling of the SM Higgs coupling and the $h$ to $H^{\pm}$
couplings as follows:

$\displaystyle{\mathcal{L}}_{h}$ $\displaystyle=$
$\displaystyle-\sum_{f=u,d,\ell}\frac{m_{f}}{v}c_{f}{\overline{f}}fh-\sum_{V=Z,W}\frac{(1+\delta_{V})m_{V}^{2}}{v}c_{V}V^{\mu}V_{\mu}h$
(4) $\displaystyle-\frac{g_{c}}{v}hH^{+}H^{-},$

where the rescaling factors of the Yukawa couplings $c_{f}$’s in each models
are summarized in Table 2, and the $c_{V}$ and $g_{c}$ are universal for each
model,

$\displaystyle c_{V}$ $\displaystyle=$ $\displaystyle\sin(\beta-\alpha),$ (5)
$\displaystyle g_{c}$ $\displaystyle=$ $\displaystyle
2\left(m_{1}^{2}-\frac{2m_{12}^{2}}{\sin 2\beta}\right)\cot
2\beta\cos(\beta-\alpha)$ (6) $\displaystyle+$
$\displaystyle\left(m_{1}^{2}+2m_{H^{\pm}}^{2}-\frac{4m_{12}^{2}}{\sin
2\beta}\right)\sin(\beta-\alpha).$

Table 2: The rescaling factor of the Yukawa couplings to the SM value. | $c_{u}$ | $c_{d}$ | $c_{l}$
---|---|---|---
Type-I | $c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$ | $c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$ | $c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$
Type-II | $c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$ | $s_{\beta-\alpha}-c_{\beta-\alpha}t_{\beta}$ | $s_{\beta-\alpha}-c_{\beta-\alpha}t_{\beta}$
Type-X | $c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$ | $c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$ | $s_{\beta-\alpha}-c_{\beta-\alpha}t_{\beta}$
Type-Y | $c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$ | $s_{\beta-\alpha}-c_{\beta-\alpha}t_{\beta}$ | $c_{\beta-\alpha}/t_{\beta}+s_{\beta-\alpha}$

We use the public code 2HDMC-1.7.0 Eriksson et al. (2010) to calculate the
total width of the Higgs boson numerically. Figure 3 plots the contours of the
ratio of the total width to the SM value in four types of models in the plane
of $\cos(\beta-\alpha)$ and $\tan\beta$. Different color lines represent the
contours of the ratio, the gray region is excluded by the global fit of the
single Higgs boson production, which is obtained directly from Fig. 1 in Ref.
Chowdhury and Eberhardt (2018). In each model, we have set mass parameters as:
$m_{H^{\pm}}=m_{2}=m_{3}=1$ TeV, $m_{12}^{2}=m_{2}^{2}\sin\beta\cos\beta$. The
reason that we chose such a $m_{12}^{2}$ is to satisfy the unitarity
constraint in large $\tan\beta$ regions Kling et al. (2016). The mass
difference between the charged Higgs boson and the neutral heavy Higgs boson
should be less than 300 GeV, i.e., $|m_{H^{\pm}}-m_{2,3}|\leq 300~{}{\rm GeV}$
Chowdhury and Eberhardt (2018). The range of the $\tan\beta$ is taken to be
(0.3, 32), this region is well consistent with the region allowed by 95.4%
confidential level when taking into account all the available constraints
Chowdhury and Eberhardt (2018).

Figure 3: Contour of the ratio of the total width of the Higgs boson to the SM
value in various 2HDMs in the $\cos(\beta-\alpha)$ vs $\tan\beta$ plane. We
have set the mass parameters in each types of model as:
$m_{H^{\pm}}=m_{2}=m_{3}=1$ TeV, $m_{12}^{2}=1$ TeV2.

In general, if one neglects the partial width contributions from
$h\to\gamma\gamma$ and $h\to Z\gamma$, then the Higgs-boson total width is
uniquely determined by two parameters: $\cos(\beta-\alpha)$ and $\tan\beta$,
while the information of other parameters enters via the coupling of the
charged Higgs boson to the SM Higgs boson $g_{c}$ when these two decay
channels $h\to\gamma\gamma$ and $h\to Z\gamma$ are taken into account.

Figure 3 shows that ratio of the Higgs-boson total width to the SM value in
the 2HDMs roughly ranges from 0.8 to 1.25, and only the Type-I model can have
the ratio less than 0.85; on the other hand, only the Type-X model can have
the ratio larger than 1.15 . Therefore, all the four types of models are
likely to be excluded if the LHC find a ratio larger than 1.25 or less than
0.8 . If one find the ratio is in the range of (1.15,1.25) ((0.8,0.85)), then
it must be Type-X (Type-I) model provided the CP conserving 2HDM were realized
in the nature.

Figure 3 also demonstrates that in the Type-I model only a positive
$\cos(\beta-\alpha)$ combined with a rather small $\tan\beta$ can give an
enhanced Higgs total width, the reason is simply that all the fermion
couplings are enhanced in this region. However, in the Type-II and Type-X
model, an enhanced Higgs total width can only obtained by a negative
$\cos(\beta-\alpha)$, the main reason is that the couplings to the bottom
quarks are enhanced. In Type-X model, one can obtain an enhanced Higgs total
width with either a positive $\cos(\beta-\alpha)$ combined with a relatively
small $\tan\beta$ or a negative $\cos(\beta-\alpha)$ combined with a
relatively large $\tan\beta$. In the former case, the increase of the Higgs
width is mainly due to enhanced couplings to the top quarks which further
increases the decay width to gluons, while for the later case, the increase of
the Higgs width is mainly caused by the enhanced decay width to bottom quarks.
In addition, in Type-II, Type-X and Type-Y models, one can find a nearly
vertical line corresponding to the ratio of the Higgs width equal to one, the
reason is that the increase of decay widths of some channels due to increased
fermions couplings is exactly compensated by the reduction in decay widths of
some other channels caused by decreased Higgs couplings to other fermions and
gauge bosons.

In the decoupling limit, where we assume the masses of the three Higgs are
equal and much larger than the electroweak scale:
$m_{2}=m_{3}=m_{H^{\pm}}\sim\Lambda\gg v$. In this case, one can work in an
effective field theory derived from the Higgs basis, where
$\cos(\beta-\alpha)$ scales as:

$\displaystyle\cos(\beta-\alpha)=-Z_{6}\frac{v^{2}}{\Lambda^{2}},$ (7)

where $Z_{6}$ is the coefficient of the $|H_{1}|^{2}H_{1}^{\dagger}H_{2}$
which can be either positive or negative and $\Lambda^{2}$ is the coefficient
of $|H_{2}|^{2}$ in the Higgs basis Bélusca-Maïto et al. (2017). Then one can
translate the bond on the Higgs width onto the new physics scale $\Lambda$
with different value of $\tan\beta$ assuming $Z_{6}$ a ${\cal O}(1)$
parameter. We plot the contours of the ratio of the Higgs width to the SM
value in different models in Fig. 4 with $Z_{6}=1$, the gray regions are
excluded by the Higgs global fit of the LHC Run1 and Run2 results as above
Chowdhury and Eberhardt (2018), the pink region represents the parameter space
in which the deviation of the Higgs total width is within 5% of the SM value
so that the new physics effects cannot be probed by the measurement of the
Higgs total width at the HL-LHC.

Figure 4: Contour of the ratio of the total width of the Higgs boson to the SM
value in various 2HDMs with the parameter $Z_{6}=1$. The gray regions are
excluded by the Higgs global fit, the pink regions are the parameter spaces
such that the deviation of the Higgs total width from the SM value are too
small to probe at future HL-LHC.

##  IV Georgi-Machacek Model

In the Georgi-Machacek Model, the scalar sector of the SM is extended with a
real triplet $(\xi^{+},\xi^{0},\xi^{-})$ with hypercharge $Y=0$ and a complex
triplet ($\chi^{+}+,\chi^{+},\chi^{0}$) with hypercharge $Y=2$. A good review
of the model can be found in Ref Hartling et al. (2014). After electroweak
symmetry breaking, the Higgs doublet gets a vev $v_{H}$ and the neutral
components of the real and complex triplet get an equal vev $v_{\chi}$. The
relation between the electroweak vev $v$ defined by the Fermi constant $G_{F}$
and $v_{H}$ and $v_{\chi}$ is given by:

$\displaystyle v^{2}=v_{H}^{2}+8v_{\chi}^{2}.$ (8)

The rescaling factor for the gauge boson and fermions are given by:

$\displaystyle c_{W,Z}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2s_{W}^{2}c_{W}^{2}}\left(\cos\alpha\sin\beta-\frac{2\sqrt{6}}{3}\sin\alpha\cos\beta\right)$
(9) $\displaystyle c_{f}$ $\displaystyle=$
$\displaystyle\frac{\cos\alpha}{\sin\beta},$ (10)

where the angle $\alpha$ related to the diagonalization of the mass matrix of
the neutral scalars, and $\cos\beta$ is defined by the ratio of $v_{H}$ and
$v$. From Eq. 10, one can find that the Higgs boson couplings of fermions and
gauge bosons can be larger than 1, which may lead to an enhanced Higgs boson
width.

To analyze the Higgs boson width in the model, we use GMCALC-1.4.1 Hartling et
al. (2014) to scan the allowed parameter space. The scanning method and the
parameter ranges are given in Ref. Hartling et al. (2014). Each parameter
point in the scan is required to satisfy various constraints: the correct
electroweak vev, the stability of the scalar potential, the tree-level
unitarity, the experimental bounds on the $S$-parameter and $b\to s\gamma$.
Figure 5 shows the results of the parameter scan in the plane of $\sin\beta$
and $\sin\alpha$ (left) and in the plane of $c_{f}$ and $c_{V}$ (right). The
blue dots represents the ratio $\Gamma/\Gamma_{SM}$ within the range
$(0.95,1.05)$ and are not be able to discriminant from the SM value. The red
and green dots denotes the ratio $\Gamma/\Gamma_{SM}$ larger than 1.05 and
smaller 0.95, respectively, which are possible to be probed by the future
experiments. We also find that enhanced fermion couplings strongly correlate
with an enlarged Higgs width, while an enlarged Higgs width does not
necessarily corresponds to enhanced gauge boson couplings. Especially for a
Higgs boson width enlarged by a factor of 1.5 or more, a decreased gauge boson
coupling is observed. We also plot the scanned points in the plane of
$\sin\beta$ and $\sin\alpha$. One can find from the plot that these two angle
parameters almost determine the value of the Higgs boson width, as the
parameter points with different colors are well separated. As one expected,
the SM value corresponds to the point with $\sin\beta=1$ and $\cos\alpha=0$,
and the red dots corresponding to an increased Higgs boson width tend to have
smaller $\sin\beta$’s.

Figure 5: The scattered plot of the scanned points on the $\sin\beta$ vs
$\sin\alpha$ plane (left) and $c_{f}$ vs $c_{V}$ plane (right), each points
satisfies the theoretical bounds and indirect experimental bounds. The blue
dots are those with $\Gamma/\Gamma_{SM}$ within [0.95, 1.05], which cannot be
distinguished by the future HL-LHC experiment. The red and green dots
corresponds to $\Gamma/\Gamma_{SM}>1.05$ and $\Gamma/\Gamma_{SM}<0.95$
respectively and are possible to be probed by the future HL-LHC experiments.

##  V Minimal Composite Higgs Models

If the Higgs boson is a pseudo Nambu-Goldstone boson (PNGB) emerging from
strong dynamics at TeV scale, its coupling to light fermions $b,c,\tau$ and
electroweak gauge bosons $W^{\pm},Z$ would be modified with respect to their
SM values. In this case, the sign and magnitude of Higgs coupling
modifications are mainly dictated by two effects: the Higgs nonlinearity and
the composite resonances Li et al. (2019).

The Higgs nonlinearity denotes the nontrivial curvature of the coset space
from which the PNGB Higgs boson emerges Alonso et al. (2016). To be concrete,
the compact cosets have positive curvatures, while the non-compact ones have
negative curvatures. Among various cosets Bellazzini et al. (2014),
$SO(5)/SO(4)$ receives the most attention, and it is also known as the minimal
coset that accommodates the custodial symmetry. Therefore this is called as
the minimal composite Higgs model Agashe et al. (2005). As a result, the Higgs
coupling to the electroweak gauge bosons $W^{\pm},Z$ are universally shifted
Liu et al. (2018) as

$\displaystyle
c_{W}=c_{Z}=\frac{\kappa_{hWW}}{\kappa^{\text{SM}}_{hWW}}=\frac{\kappa_{hZZ}}{\kappa^{\text{SM}}_{hZZ}}=\sqrt{1-\xi}$
(11)

at the leading order of the chiral expansion, due to Higgs nonlinearity. Here
$\xi\equiv v^{2}/f^{2}$ is defined as the ratio of the electroweak scale and
the decay constant of the PNGB Higgs boson.

In the fermionic sector, modifications of Higgs boson couplings also depend on
fermion embeddings into $SO(5)$ multiplets. In this work, we consider the
cases where the top quark and the light fermions $b,c,\tau$ are all embedded
in the fundamental representation $5$, or the symmetric tensor representation
$14$, of the $SO(5)$ group. Under the paradigm of partial compositeness, there
are composite particles mixing with the elementary $t,b,c,\tau$, which are
usually considered as the source of explicit $SO(5)$ breaking in the fermionic
sector; otherwise the Higgs boson would be a massless exact Goldstone boson
when these mixings (as well as gauge couplings) are turned off. After
integrating out these composite fermions, they would contribute to the low
energy effective couplings of the Higgs boson, through both the chirality-
conserving wave functions and the chirality-flipping Yukawa vertices of the
elementary fermions. Following Ref. Li et al. (2019), the chirality-conserving
wave functions are universally expanded as

$\displaystyle\begin{aligned} \Pi_{f_{L}}&=\Pi_{0f_{L}}+\Pi_{1f_{L}}\
s^{2}_{h}+\Pi_{2f_{L}}\ s^{4}_{h}+\cdots,\ \\\
\Pi_{f_{R}}&=\Pi_{0f_{R}}+\Pi_{1f_{R}}\ s^{2}_{h}+\Pi_{2f_{R}}\
s^{4}_{h}+\cdots;\end{aligned}$ (12)

while the chirality-flipping Yukawa interactions satisfy the expansion

$\displaystyle\Pi_{t_{L}t_{R}}$ $\displaystyle=\Pi_{1t_{L}t_{R}}\
c_{h}s_{h}+\Pi_{2t_{L}t_{R}}\ c_{h}s^{3}_{h}+\cdots\ ,$ (13)

or

$\displaystyle\Pi_{t_{L}t_{R}}$ $\displaystyle=\Pi_{1t_{L}t_{R}}\
s_{h}+\Pi_{2t_{L}t_{R}}\ s^{3}_{h}+\cdots\ ,$ (14)

depending on fermion representations. For the cases considered in this work,
$(t,b,c,\tau)_{L,R}\subset 5$ and $(t,b,c,\tau)_{L,R}\subset 14$ corresponds
to the Eq. 13, while $(t,b,c,\tau)_{L}\subset 5,(t,b,c,\tau)_{R}\subset 1$
corresponds to Eq. 14, as discussed in Ref. Li et al. (2019).

Accordingly, up to the linear order of $\xi$, the fermion couplings are
modified as following:

$\displaystyle c_{f}$
$\displaystyle\equiv\frac{\kappa_{f}}{\kappa_{f}^{\text{SM}}}$
$\displaystyle=1-\frac{3}{2}\xi-\xi\left(\frac{\Pi_{1f_{L}}}{\Pi_{0f_{L}}}+\frac{\Pi_{1f_{R}}}{\Pi_{0f_{R}}}\right)+2\xi\frac{\Pi_{2f_{L}f_{R}}}{\Pi_{1f_{L}f_{R}}}.$
(15)

By naive dimensional analysis,

$\displaystyle\Pi_{1f_{L}}\sim\frac{y_{L}^{2}f^{2}}{m_{*}^{2}},\
\Pi_{1f_{R}}\sim\frac{y_{R}^{2}f^{2}}{m_{*}^{2}},\
\Pi_{0f_{L}}\sim\Pi_{0f_{R}}\sim 1,$ (16)

where the mass scale $m_{*}\sim g_{*}f$ and $y_{L,R}$ are the mixing
parameters, while

$\displaystyle\Pi_{1f_{L}f_{R}}\sim\frac{y_{L}y_{R}f^{2}}{m_{*}},\
\Pi_{2f_{L}f_{R}}\sim\frac{y_{L}y_{R}f^{2}}{m_{*}}.$ (17)

For the light fermions $b,c,\tau$, due to their small masses
$m_{b,c,\tau}\sim\frac{y_{L}y_{R}}{g_{*}}f$, the mixing parameters $y_{L,R}$
are expected to be much smaller than the coupling strength $g_{*}$ if the
decay constant $f$ is at the order of $1$ TeV. As a result, one can neglect
the effect of composite resonances in the wave functions of $b,c,\tau$.
Furthermore, $\Pi_{2f_{L}f_{R}}$ does not vanish only in the case of
$(t,b,c,\tau)_{L,R}\subset 14$.

The effective coupling between the Higgs boson and the gluon is, up to the
linear order of $\xi$,

$\displaystyle c_{g}$
$\displaystyle\equiv\frac{\kappa_{hgg}}{\kappa^{\text{SM}}_{hgg}}$
$\displaystyle=1-\frac{3}{2}\xi+2\xi\frac{\Pi_{2t_{L}t_{R}}}{\Pi_{1t_{L}t_{R}}}+\sum_{f=b,c,\tau}\xi\left(\frac{\Pi_{1f_{L}}}{\Pi_{0f_{L}}}+\frac{\Pi_{1f_{R}}}{\Pi_{0f_{R}}}\right)\
.$ (18)

Note that contribution of the composite resonances in the $b,c,\tau$ sectors
(but not the light fermions $b,c,\tau$) are included in the above equation Liu
et al. (2017).

Therefore, the Higgs effective couplings in the fermionic sectors are
concretely

  * •

$(t,b,c,\tau)_{L,R}\subset 5$:

$\displaystyle c_{b,c,\tau}=c_{g}\simeq 1-\frac{3}{2}\xi\ .$ (19)

  * •

$(t,b,c,\tau)_{L}\subset 5$ and $(t,b,c,\tau)_{R}\subset 1$:

$\displaystyle c_{b,c,\tau}=c_{g}\simeq 1-\frac{1}{2}\xi\ .$ (20)

  * •

$(t,b,c,\tau)_{L,R}\subset 14$:

$\displaystyle c_{b,c,\tau}=c_{g}\simeq 1-\frac{3}{2}\xi+2\ r_{12}\ \xi\ ,$
(21)

where $r_{12}$ is the ratio of $\Pi_{2f_{L}f_{R}}$ to $\Pi_{1f_{L}f_{R}}$.

With the above results, we obtain the total width of the Higgs boson in
minimal composite Higgs models, as shown in Fig. 6. We see explicitly that
$\xi\to 0$ corresponds to the decoupling limit where one cannot distinguish a
composite Higgs boson from an elementary one. For small fermion
representation, the total width of the Higgs boson tends to be smaller than
the value in the SM; while for symmetric tensor representation, the total
width could be larger.

Figure 6: The total width of the Higgs boson in minimal composite Higgs
models, in which the left-handed and right-handed fermions are embedded in
different representations of the $SO(5)$ group.

##  VI Fraternal Twin Higgs Model

It is fascinating to consider models in which the Higgs boson could decay into
invisible particles. Typically invisible decay exist in neutral naturalness
models Chacko et al. (2006); Craig et al. (2015b); Xu et al. (2020). One of
these typical model is the twin Higgs scenario Chacko et al. (2006), where the
naturalness problem is addressed with some particles which are not charged
under the SM gauge groups.

In this section we consider the fraternal twin Higgs model Craig et al.
(2015a). In contrast to the original mirror twin Higgs setup Chacko et al.
(2006), the fraternal model is more minimal and cosmologically safe. The model
ingredients of the twin sector consists of one additional Higgs doublet, twin
fermions (including
$\widetilde{t}_{R},\widetilde{b}_{R},\widetilde{\tau}_{R},\widetilde{Q}_{L},\widetilde{L}_{L}$),
twin weak gauge bosons from the $SU(2)$ gauge group in the twin sector, and
twin gluons $\widetilde{g}$ Craig et al. (2015a). The Higgs boson is
identified as a pseudo Nambu-Goldstone boson from the coset $SU(4)/SU(3)$, or
equivalently $SO(8)/SO(7)$, depending whether the custodial symmetry is
included in the unbroken group. After the global symmetry breaking, three of
the Goldstone bosons would be eaten by the twin weak gauge bosons, which makes
their masses roughly at the order of symmetry-breaking scale $f$. For
reasonable values of $f$, these twin gauge bosons can be sufficiently heavy.
Only the other light degrees of freedom, lighter than $m_{h}/2$, could
contribute to the invisible width of the Higgs boson. On top of this, all the
SM couplings are modified due to the Higgs nonlinearity.

Up to the linear order of $\xi=v^{2}/f^{2}$, all the relevant Higgs couplings
are as follows: i) for the couplings in the SM sector,

$\displaystyle c_{b,c,\tau}$ $\displaystyle=c_{g}\simeq 1-\frac{1}{2}\xi\ ,$
$\displaystyle c_{W}$ $\displaystyle=c_{Z}=\sqrt{1-\xi};$ (22)

ii) for the couplings in the twin sector,

$\displaystyle
c_{\widetilde{b},\widetilde{\tau}}=c_{\widetilde{g}}\equiv\frac{\kappa_{h\widetilde{b}\widetilde{b},h\widetilde{\tau}\widetilde{\tau},h\widetilde{g}\widetilde{g}}}{\kappa^{\text{SM}}_{hbb,h\tau\tau,hgg}}\simeq\sqrt{\xi},$
(23)

assuming the Yukawa couplings are the same, i.e.
$y_{t,b,\tau}=y_{\widetilde{t},\widetilde{b},\widetilde{\tau}}$. All the above
Higgs boson couplings in the twin sector are correlated to their counterparts
in the SM sector, in order to realize the naturalness condition Li et al.
(2019); as one can see that
$c^{2}_{\widetilde{b},\widetilde{\tau},\widetilde{g}}+c^{2}_{b,\tau,g}$ does
not depend on $\xi$. Furthermore, the masses of the twin fermions are also
connected to the partners in the SM sector, e.g.,

$\displaystyle\frac{m_{\widetilde{t},\widetilde{b},\widetilde{\tau}}}{m_{t,b,\tau}}\simeq\frac{\sqrt{1-\xi}}{\sqrt{\xi}}\sim\frac{f}{v},$
(24)

assuming the Yukawa couplings are the same in the SM and the twin sectors. The
twin gluons are massless. They could form glueballs by themself or other bound
states with twin fermions below the scale of twin confinement Craig et al.
(2015a). Note that $m_{\widetilde{b}}$ and $m_{\widetilde{\tau}}$ can be
larger or smaller than $m_{h}/2$ depending on the scale $f$, which would
result in different invisible widths of the Higgs boson.

The particles in the twin sector induce the invisible width as following:

$\displaystyle\Gamma_{\text{invisible}}\simeq\Gamma_{\text{SM}}\xi$
$\displaystyle\Bigg{[}~{}~{}\text{Br}_{gg}$
$\displaystyle+\text{Br}_{\tau\tau}\left(\frac{1-x_{\widetilde{\tau}}}{1-x_{\tau}}\right)^{\frac{3}{2}}\Theta(m_{h}-2m_{\widetilde{\tau}})$
$\displaystyle+\text{Br}_{bb}\left(\frac{1-x_{\widetilde{b}}}{1-x_{b}}\right)^{\frac{3}{2}}\Theta(m_{h}-2m_{\widetilde{b}})\Bigg{]},$
(25)

where $x_{i}\equiv 4m_{i}^{2}/m_{h}^{2}$ for the $i$-particle and
$\text{Br}_{jj}$ denotes the branching ratio of the decay channel of $h\to jj$
in the SM.

Note that the running masses of the fermions $b$ and $\tau$ in the SM model
are roughly $m_{b}(M_{Z})\simeq 2.9$ GeV and $m_{\tau}(M_{Z})\simeq 1.746$
GeV, respectively. We calculate the total width of the Higgs boson in the
fraternal twin Higgs model from Eq. 24 and Eq. 25. Figure 7 plots the ratio of
the total width of the Higgs boson to the SM value as a function of $\xi$. We
notice that the total width is moderately smaller than the SM value,
therefore, it is very hard to distinguish the fraternal twin Higgs model from
the SM with only the information of Higgs total width.

Figure 7: The total width of the Higgs boson in fraternal twin Higgs model.

##  VII Conclusion

The nature of the Higgs boson remains as one unsolved puzzle nowadays.
Motivated from the future precise measurement of the Higgs width in the future
HL-LHC, we investigated that if the Higgs is elementary or composite, how the
model setup could affect the Higgs total width. Typically the Higgs total
width is expected to be smaller than the standard model values because of the
mixture with other scalars or curved Higgs field space for the composite
Higgs. If there is invisible decay channels, the Higgs width could be
enhanced, as expected. We further showed that there are also cases that the
Higgs width is enhanced due to model setup.

Depending on the model setup, the following results are in order:

  * •

For the singlet extended model, and typical scalar extension models other than
the doublet and GM models, the Higgs width is smaller than the SM if no other
invisible decay presents.

  * •

In the general two Higgs doublet models, the Higgs width can be larger or
smaller than the SM value, depending on values of the model parameters. It
depends on the concrete value of the Higgs width measured to determine whether
a given type of the two Higgs doublet models are excluded or not.

  * •

In the Georgi-Machacek model, one will find that an increased Higgs boson
coupling always accompanies an enhance Higgs to fermion couplings.

  * •

For the case of a pseudo-Nambu-Goldstone Higgs boson, including both the
minimal composite Higgs model and the twin Higgs model, the Higgs total width
is strongly preferred to be smaller than the SM prediction except for the case
where the left handed and right handed top quark are both embedded in the 14
representation of SO(5). Therefore, only if the Higgs width is measured to be
smaller than the SM value, the minimal composite Higgs model and the twin
Higgs model are favored.

Overall, if the Higgs total width is measured to be significantly larger or
smaller than the SM value, it is possible for us to falsify several new
physics models considered above, although the caveat still exists, in which
new physics might contain some new hidden sector we never consider and thus
the SM Higgs can decay invisibly.

We note that the Higgs total width is sensitive to almost all kinds of Higgs
couplings. On the other hand, the Higgs production and partial decay width
provide different kinds of information on the Higgs coupling. We expect that
combining the Higgs coupling information and the Higgs total width information
at the HL-LHC, the Higgs sector could be better understood in near future.

###### Acknowledgements.

The work is supported in part by the National Science Foundation of China
under Grant Nos. 11725520, 11675002, 11635001, 12022514, 11875003 and
12047503. JHY is also supported by the National Key Research and Development
Program of China under Grant No. 2020YFC2201501.

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