#  Snowmass 2021  KIAS–P22014
Global fit of 2HDM with future collider results

Ankit Beniwal  Theoretical Particle Physics and Cosmology Group, Department of
Physics, King’s College London, Strand, London, WC2R 2LS, UK Filip Rajec  ARC
Centre of Excellence for Dark Matter Particle Physics, Department of Physics,
University of Adelaide, South Australia 5005, Australia  Markus Tobias Prim
Physikalisches Institut der Rheinischen Friedrich-Wilhelms-Universität Bonn,
53115 Bonn, Germany  Pat Scott  School of Mathematics and Physics, The
University of Queensland, St. Lucia, Brisbane, QLD 4072, Australia  Department
of Physics, Imperial College London, Blackett Laboratory, Prince Consort Road,
London SW7 2AZ, UK  Wei Su  Korea Institute for Advanced Study, Seoul 02455,
Korea  Martin White  ARC Centre of Excellence for Dark Matter Particle
Physics, Department of Physics, University of Adelaide, South Australia 5005,
Australia  Anthony G. Williams  ARC Centre of Excellence for Dark Matter
Particle Physics, Department of Physics, University of Adelaide, South
Australia 5005, Australia  Alex Woodcock  ARC Centre of Excellence for Dark
Matter Particle Physics, Department of Physics, University of Adelaide, South
Australia 5005, Australia

###### Abstract

In this work, we summarize a global fit study of Type-II two Higgs doublet
models (2HDM), and explore the impact of future SM-like Higgs and $Z$-pole
precision measurements on the allowed parameter space. The work is based on
the study results of a global fit of 2HDMs with the tool GAMBIT, utilising
various current constraints including theoretical constraints (unitarity,
perturbativity and vacuum stability), Higgs searches at colliders, electroweak
physics and flavour constraints. We further investigate the ability of future
facilities, such as the HL-LHC, CEPC, ILC and FCC-ee to explore the 2HDM
parameter space.

11footnotetext: Email address<EMAIL_ADDRESS>
Thematic Areas:

$\blacksquare$ (EF01) EW Physics: Higgs Boson properties and couplings
$\blacksquare$ (EF02) EW Physics: Higgs Boson as a portal to new physics
$\blacksquare$ (EF04) EW Precision Physics and constraining new physics
$\blacksquare$ (EF08) BSM: Model specific explorations
$\blacksquare$ (TF07) Collider phenomenology

##  1 Introduction

The discovery of a Standard Model (SM)-like Higgs boson at the Large Hadron
Collider (LHC) set a milestone for high energy physics, confirming the self-
consistency of the SM. At the same time there are also various unsolved
mysteries, such as the source of dark matter, the origin of the baryon
asymmetry of the universe, and the muon $g$-2 anomaly. Models with extended
Higgs sectors provide promising solutions to these problems.

As one of the simplest such frameworks, the Two Higgs Doublet Model (2HDM) is
embedded in various models with extended Higgs sectors, such as the Minimal
Supersymmetric Standard Model, and gauge extensions (such as the Left-Right
symmetric model). After electroweak symmetry breaking (EWSB), the general CP-
conserving 2HDM can generate five physical eigenstates: the observed 125 GeV
CP-even neutral scalar $h$, an additional CP-even neutral scalar $H$, one CP-
odd Higgs boson $A$, and a pair of charged Higgs bosons $H^{\pm}$ [1].
Exploring the properties of 2HDMs with various experimental constraints can
help us understand the new physics potential of a broad class of BSM
scenarios.

In this paper, we present preliminary results of a forthcoming global fit of
the $Z_{2}$-Yukawa symmetric, Type-II 2HDM [2]. This analysis is carried out
using the open-source tool GAMBIT [3] (Global and Modular beyond-Standard
Model Inference Tool) GAMBIT is compatible with both the Bayesian and
frequentist statistical frameworks, and we here focus on frequentist results
obtained with the Diver [4] implementation of the differential evolution
algorithm. We investigate the effect of theoretical constraints (unitarity,
perturbativity and vacuum stability), Higgs searches at colliders, electroweak
physics and flavour constraints individually, as well as displaying the final
results with all constraints. We also investigate the impact on the allowed
parameter space of a series of future collider measurements, by reweighting
the likelihoods of the GAMBIT samples outside of the GAMBIT framework

Our paper is organised as follows. Section 2 provides the details of our
assumed future collider measurements. Section 3 summarises the general 2HDM
and our results are presented in Section 4.

##  2 Higgs precision measurements at future lepton colliders

At future lepton colliders, the dominant channel to measure the properties of
the Higgs boson is the Higgsstrahlung process, $e^{+}e^{-}\rightarrow hZ$, at
center of mass energies $(\sqrt{s})$ of around 240$-$250 GeV. Due to the
nature of lepton colliders, both the inclusive cross section, $\sigma(hZ)$,
and the exclusive $\sigma(hZ)\times{\rm BR}$ values for different Higgs decay
modes, can be measured with remarkable precision. The invisible decay width of
the Higgs can also be very well constrained. In addition, the cross section
for a Higgs production via the $WW$ fusion process grows with energy. While it
cannot be measured very well at 240$-$250 GeV, at higher center of mass
energies (in particular, at linear colliders), such a fusion process becomes
significantly more important and can provide crucial complementary
information. For $\sqrt{s}>500$ GeV, $tth$ production can also be
investigated.

collider | CEPC | FCC-ee | ILC
---|---|---|---
$\sqrt{s}$ | 240 GeV | 240 GeV | 365 GeV | 250 GeV | 350 GeV | 500 GeV
$\int{\mathcal{L}}dt$ | $\text{5.6 ab}^{-1}$ | $\text{5 ab}^{-1}$ | $\text{1.5 ab}^{-1}$ | $\text{2 ab}^{-1}$ | $\text{200 fb}^{-1}$ | $\text{4 ab}^{-1}$
production | $Zh$ | $Zh$ | $Zh$ | $\nu\bar{\nu}h$ | $Zh$ | $Zh$ | $\nu\bar{\nu}h$ | $Zh$ | $\nu\bar{\nu}h$
$\Delta\sigma/\sigma$ | 0.5% | 0.5% | 0.9% | $-$ | 0.71% | 2.0% | $-$ | 1.05 | $-$
decay | $\Delta(\sigma\cdot BR)/(\sigma\cdot BR)$
$h\to b\bar{b}$ | 0.27% | 0.3% | 0.5% | 0.9% | 0.46% | 1.7% | 2.0% | 0.63% | 0.23%
$h\to c\bar{c}$ | 3.3% | 2.2% | 6.5% | 10% | 2.9% | 12.3% | 21.2% | 4.5% | 2.2%
$h\to gg$ | 1.3% | 1.9% | 3.5% | 4.5% | 2.5% | 9.4% | 8.6% | 3.8% | 1.5%
$h\to WW^{*}$ | 1.0% | 1.2% | 2.6% | 3.0% | 1.6% | 6.3% | 6.4% | 1.9% | 0.85%
$h\to\tau^{+}\tau^{-}$ | 0.8% | 0.9% | 1.8% | 8.0% | 1.1% | 4.5% | 17.9% | 1.5% | 2.5%
$h\to ZZ^{*}$ | 5.1% | 4.4% | 12% | 10% | 6.4% | 28.0% | 22.4% | 8.8% | 3.0%
$h\to\gamma\gamma$ | 6.8% | 9.0% | 18% | 22% | 12.0% | 43.6% | 50.3% | 12.0% | 6.8%
$h\to\mu^{+}\mu^{-}$ | 17% | 19% | 40% | $-$ | 25.5% | 97.3% | 178.9% | 30.0% | 25.0%
$(\nu\bar{\nu})h\to b\bar{b}$ | 2.8% | 3.1% | $-$ | $-$ | 3.7% | $-$ | $-$ | $-$ | $-$
Table 1: Estimated statistical precisions for Higgs measurements obtained at
the proposed CEPC program with 5.6 ab-1 integrated luminosity [5, 6], FCC-ee
program with 5 ab-1 integrated luminosity [7, 8], and ILC with various center-
of-mass energies [9].

When investigating the impact of future facilities, our study makes use of the
following scenarios of various machines (in terms of the center of mass energy
and the corresponding integrated luminosity), as well as the estimated
precision of relevant Higgs measurements:

  * •

CEPC  According to the preCDR [10], CEPC plans to collect 5 ab-1 data points
at 240 GeV. The estimated precision of measurements for the Higgsstrahlung
process $e^{+}e^{-}\rightarrow hZ$ with various final states, as well as the
$WW$ fusion process with Higgs decaying to bottom pairs
($e^{+}e^{-}\rightarrow\nu\bar{\nu}h$, $h\to b\bar{b}$) are summarized in
Table 1. As systematic uncertainties of the Higgs measurements can be reduced
using $Z$-pole calibration, they are assumed to be much smaller than the
statistical uncertainties and are therefore neglected.

  * •

FCC-ee  The FCC-ee CDR is finished in 2018 [11, 7, 8].At the current moment,
the white paper proposes total luminosities of $\text{5 ab}^{-1}$ at 240 GeV
and $1.5\text{ab}^{-1}$ at 350 GeV. The estimated precision of
$e^{+}e^{-}\rightarrow hZ$ measurements at 240 GeV, as well as $h\rightarrow
b\bar{b}$ channel in $WW$ fusion are listed in Table 1. In addition, the cross
sections of vector boson fusion processes for the Higgs production ($WW,ZZ\to
h$) grow with the center of mass energy logarithmically. While their rates are
still rather small at 240-250 GeV, at higher energies such as 350 GeV, such
fusion processes become significantly more important and can provide crucial
complementary information.

  * •

ILC  The proposed run scenarios in the ILC TDR [12] have been updated in
recent documents [13, 14], which suggested that the ILC could collect
$2\text{ab}^{-1}$ data points at 250 GeV, $200\text{fb}^{-1}$ at 350 GeV, and
$4\text{ab}^{-1}$ at 500 GeV. However, the estimation of signal strengths, as
summarized in Ref. [14], are only available for smaller benchmark luminosities
for which the full detector studies are performed. We take these estimations
and scale them up to the current run scenarios, assuming statistical
uncertainties dominate [15]; these are summarized in Table 1. Such scaling
provides a reasonable approximation as long as the luminosities are not
excessively large and the systematic uncertainties are under control.

With large center of mass energies up to 3 TeV, CLIC is also able to measure
the Higgs properties very well through the $WW$ fusion process [16, 17]. On
the other hand, with its extensive coverage of energy scales, the primary goal
of CLIC is to directly search for new particles, in particular, the ones
coupled to SM particles only through electroweak interactions. A comprehensive
study of the CLIC physics potential including both the direct and indirect
searches of new physics is beyond the scope of this paper.

In our global fit to the Higgs measurements, we only include the rate
information for the Higgsstrahlung as well as the $WW$ fusion process.
Electroweak (EW) precision measurements at the $Z$-pole also impose strong
constraints on new physics [18, 19]. The current constraints from the Large
Electron Positron (LEP) collider can be significantly improved by a $Z$-pole
run at any of the future lepton colliders. While these constraints are not
explicitly considered in our study, we do restrict ourselves to models with
suppressed EW precision corrections (e.g., by imposing custodial symmetries)
such that these constraints are automatically satisfied.

It is also important to study the reach of the future High Luminosity LHC (HL-
LHC) [20, 21, 22]. Current LHC Higgs measurements are included via the GAMBIT
interfaces to HiggsBounds-5.3.2 [23, 24] and HiggsSignals-2.2.3 [25]. For the
future HL-LHC, we take their designed precision measurements to construct
likelihood, which will be introduced in detail later.

Based on these analyses, we propose a global fit study of the 2HDM with
hypothetical data from Higgs and Z pole precision measurements at future HL-
LHC and Higgs factories.

##  3 2HDM and study strategy

The general 2HDM has two ${\rm SU}(2)_{L}$ scalar doublets $\Phi_{i}\ (i=1,2)$
with hyper-charge $Y=+1/2$,

$\Phi_{i}=\begin{pmatrix}\phi_{i}^{+}\\\
(v_{i}+\phi^{0}_{i}+iG_{i})/\sqrt{2}\end{pmatrix}\,,$ (1)

where $v_{i}\ (i=1,\,2)$ are the vacuum expectation values (VEVs) of the two
doublets after EWSB with $v_{1}^{2}+v_{2}^{2}=v^{2}=(246\ {\rm GeV})^{2}$ and
$\tan\beta=v_{2}/v_{1}$.

The 2HDM Lagrangian for the Higgs sector can be written as

$\mathcal{L}=\sum_{i}|D_{\mu}\Phi_{i}|^{2}-V(\Phi_{1},\Phi_{2})+\mathcal{L}_{\rm
Yuk}\,,$ (2)

with a Higgs potential of

$\displaystyle V(\Phi_{1},\Phi_{2})$
$\displaystyle=m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-m_{12}^{2}(\Phi_{1}^{\dagger}\Phi_{2}+h.c.)+\frac{\lambda_{1}}{2}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\frac{\lambda_{2}}{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}$
$\displaystyle\hskip
14.22636pt+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\frac{\lambda_{5}}{2}\Big{[}(\Phi_{1}^{\dagger}\Phi_{2})^{2}+h.c.\Big{]}\,,$
(3)

where we have assumed $CP$ conservation and a soft $\mathbb{Z}_{2}$ symmetry
breaking term $m_{12}^{2}$. The physical degrees of freedom after EWSB are a
pair of singly-charged Higgs bosons $H^{\pm}$, a CP-odd Higgs boson $A$ and
two CP-even Higgs bosons $h$ and $H$. Here we take $h$ as the observed Higgs
boson with a mass of 125 GeV. Our current study foucs on the Type-II 2HDM,
where one Higgs doublet couples to up-type quarks, and the other Higgs doublet
couples to down-type quarks and leptons.

For the study of current constraints, we refer to the forthcoming work of the
GAMBIT collaboration [2], which includes the latest theoretical constraints
(unitarity, perturbativity and vacuum stability), Higgs searches at colliders,
electroweak physics and flavour constraints. To further investigate the
precise measurement constrains of future colliders, such as HL-LHC, CEPC, ILC
and FCC-ee, we define new likelihoods for the proposed Higgs factories as

$-2\ln\mathcal{L}_{\rm{Future}}=\frac{(m_{h}-m_{h}^{\rm
obs})^{2}}{\delta_{m_{h}}^{2}}+\sum_{i}\frac{(\mu_{i}-\mu_{i}^{\rm
obs})^{2}}{\sigma_{\mu_{i}}^{2}}+\sum_{ij}(X_{i}-\hat{X}_{i})(\sigma^{2})_{ij}^{-1}(X_{j}-\hat{X}_{j})\,,$
(4)

Here $\mu_{i}^{\rm BSM}\equiv(\sigma\cdot\rm{BR})_{\rm
BSM}/(\sigma\cdot\rm{BR})_{\rm SM}$ for various Higgs search channels,
$\sigma_{\mu_{i}}$ is the experimental precision on a particular channel, and
the index $i$ runs over all the Higgs search channels in Table 1. For the
future $\mu_{i}^{\rm\rm obs}$, we take them to 1. For the future
$\delta_{m_{h}}$, since the present experimental uncertainty,
$\sigma_{m_{h}}^{\rm exp}=0.17$ GeV [26], we suppose that $\delta_{m_{h}}$
will be dominated by the theoretical uncertainty, to be 1 GeV.***Our study
results show a small difference between $\delta_{m_{h}}$ = 1 GeV and 3 GeV.
For the $Z$-pole observables, $X_{i}=(\Delta S\,,\Delta T\,,\Delta U)_{\rm
2HDM}$ are the 2HDM predicted values, and $\hat{X}_{i}=(\Delta S\,,\Delta
T\,,\Delta U)$ are the best-fit central values for current measurements (or
zero for future measurements)†††Here we only consider effects of $S,T$ and $U$
parameters. There are other paramters discussed such $U,V,W$ [27] are not
included in our study.. The $\sigma_{ij}$ are the components of the error
matrix, $\sigma_{ij}^{2}\equiv\sigma_{i}\rho_{ij}\sigma_{j}$ where
$\sigma_{i}$ and the correlation matrix components $\rho_{ij}$ are given in
Table 2.

|  Current ($1.7\times 10^{7}\ Z$’s) |  CEPC ($10^{10}Z$’s) |  FCC-ee ($7\times 10^{11}Z$’s) |  ILC ($10^{9}Z$’s)
---|---|---|---|---
| $\sigma$ | correlation | $\sigma$ | correlation | $\sigma$ | correlation | $\sigma$ | correlation
$S$ | $T$ | $U$ |  ($10^{-2}$) | $S$ | $T$ | $U$ |  ($10^{-2}$) | $S$ | $T$ | $U$ |  ($10^{-2}$) | $S$ | $T$ | $U$
$S$ | $0.04\pm 0.11$ | 1 | 0.92 | $-0.68$ | $2.46$ | 1 | 0.862 | $-0.373$ | $0.67$ | 1 | 0.812 | 0.001 | $3.53$ | 1 | 0.988 | $-0.879$
$T$ | $0.09\pm 0.14$ | $-$ | 1 | $-0.87$ | $2.55$ | $-$ | 1 | $-0.735$ | $0.53$ | $-$ | 1 | $-0.097$ | $4.89$ | $-$ | 1 | $-0.909$
$U$ | $-0.02\pm 0.11$ | $-$ | $-$ | 1 | $2.08$ | $-$ | $-$ | 1 | $2.40$ | $-$ | $-$ | 1 | $3.76$ | $-$ | $-$ | 1

Table 2: Estimated $S$, $T$, and $U$ parameter ranges and correlation matrices
$\rho_{ij}$ from $Z$-pole precision measurements, mostly from LEP-I [28] and
future lepton colliders such as CEPC [10], FCC-ee [29] and ILC [15]. Gfitter
package [30] is used to obtain these constraints.

##  4 Study results

| $\lambda_{1}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{5}$
---|---|---|---|---|---
Region | $(0,\,5.5)$ | $(0,\,1)$ | $(-1.8,\,1.5)$ | $(-2,\,2)$ | $(-1.4,\,1.2)$
Table 3: Allowed region in the generic basis $\lambda_{1-5}$ at the 95%
confidence level.

$\lambda_{1}$ |  |  |  |  |
---|---|---|---|---|---
$\lambda_{2}$ |  |  |  |  |
$\lambda_{3}$ |  |  |  |  |
$\lambda_{4}$ |  |  |  |  |
$\lambda_{5}$ |  |  |  |  |
|  $\lambda_{1}$ |  $\lambda_{2}$ |  $\lambda_{3}$ |  $\lambda_{4}$ |  $\lambda_{5}$

Figure 1: Combined 1D and 2D profile likelihood distributions for the
couplings in the generic basis, with marked boundaries for the 1$\sigma$,
2$\sigma$ and 3$\sigma$ confidence regions. The global fit includes
theoretical constraints and current collider Higgs, electroweak precision and
flavour constraints.

In Fig. 1, we show the 1D and 2D profile likelihood distributions for the
couplings in the generic basis, with marked boundaries for the 1$\sigma$,
2$\sigma$ and 3$\sigma$ confidence regions. These results include all of the
latest relevant constraints, including theoretical constraints (unitarity,
perturbativity and vacuum stability), Higgs searches at colliders based on
latest version of HiggsBounds-5.3.2 [23, 24] and HiggsSignals-2.2.3 [25],
electroweak physics and flavour constraints. A more detailed description will
be provided in the forthcoming GAMBIT paper [2]. Generally speaking, there are
still large allowed regions in the general basis as summarized in Table 3, and
the regions change little after including hypothetical future results.

In Fig. 2, we show the global fit results in the $m_{H}$-$m_{A}$ plane (top),
$m_{H}$-$m_{H^{\pm}}$ plane (middle) and $m_{H}$-$\tan\beta$ plane (bottom).
The left panels show the global fit results with current data and theoretical
constraints. Generally speaking, there are lower limits on the heavy scalar
masses of approximately 400 GeV, which mainly comes from the $Z$-pole
precision measurements and flavour physics. There is also a lower limit on
$\tan\beta$ that arises mainly from flavor physics. The right panels compare
the current 2$\sigma$ confidence regions (black) with those arising from the
inclusion of future precision measurements, including those from the HL-LHC
(orange), HL-LHC + CEPC (red), HL-LHC + ILC (blue), and HL-LHC + FCC-ee
(green). The inclusion of HL-LHC precision measurements pushes the lower limit
on the scalar mass up to 500 GeV, and these rise further up to 700 GeV after
including constraints from future lepton colliders.

Finally in Fig. 3, we show the 1$\sigma$ and 2$\sigma$ regions allowed by
current measurements and theoretical constraints in the plane of $\Delta
m_{A}=m_{A}-m_{H}$ vs $\Delta m_{C}=m_{H^{\pm}}-m_{H}$ (left panel). The right
panel compares the 2$\sigma$ region (black) with those arising from the
inclusion of future precision measurements including those from the HL-LHC
(orange), HL-LHC + CEPC (red), HL-LHC + ILC (blue), and HL-LHC + FCC-ee
(green). At present, $\Delta m_{A}$ and $\Delta m_{C}$ are limited to the
range $(-200,150)$ GeV and $(-200,250)$ GeV respectively, both of which will
be reduced to $(-200,100)$ GeV in the future.

We also notice that the expected limit from the FCC-ee proposal is a little
stronger than those arising from the ILC and CEPC proposals. This is mainly
because the currently proposed FCC-ee will have a larger luminosity around 250
GeV, which is about 2 times that of CEPC. At the same time, it will also
produce more $Z$ bosons than the other two lepton colliders.

To summarise, after comparing global fit results with current and hypothetical
future data, we find that:

  1. 1.

In the generic coupling basis, there is not much ability to constrain
$\lambda_{1-5}$ from future Higgs mass, coupling, and $Z$-pole precision
measurements.

  2. 2.

The same is not true when considering the physical masses of the heavy
scalars. Currently, the masses $m_{H,A,H^{\pm}}$ have a lower limit of
approximately 400 GeV, which is increased to 500 and 700 GeV after including
hypothetical results from the HL-LHC and HL-LHC + lepton colliders.

  3. 3.

There are strong constraints on the mass splittings $\Delta m_{A}=m_{A}-m_{H}$
and $\Delta m_{C}=m_{H^{\pm}}-m_{H}$ from Higgs and $Z$-pole precision
measurements, which are currently limited to the ranges $(-200,150)$ GeV and
$(-200,250)$ GeV. Future precision measurements will shrink the allowed range
on $\Delta m_{C}$ to $(-200,100)$ GeV.

Figure 2: Left panel: Global fit results showing the 1$\sigma$ and 2$\sigma$
regions in the $m_{H}$-$m_{A}$ plane (top), $m_{H}$-$m_{H^{\pm}}$ plane
(middle) and $m_{H}$-$\tan\beta$ plane (bottom) based on current measurements
and theoretical constraints. Right panel: Comparison of current $2\sigma$
constraints (black) with those arising from the inclusion of future precision
measurements, including those from the HL-LHC (orange), HL-LHC + CEPC (red),
HL-LHC + ILC (blue), and HL-LHC + FCC-ee (green).

Figure 3: Left panel: The 1$\sigma$ and 2$\sigma$ allowed regions in $\Delta
m_{A}-\Delta m_{C}$ plane, based on current measurements and theoretical
constraints. Right panel: Comparison of the $2\sigma$ region allowed by
current constraints (black) with those arising from the inclusion of future
precision measurements, including those from the HL-LHC (orange), HL-LHC+CEPC
(red), HL-LHC + ILC(blue), and HL-LHC + FCC-ee (green).

##  5 Acknowledgement

We acknowledge helpful discussions with the GAMBIT community.

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