# Probing the Electroweak Phase Transition with Exotic Higgs Decays

Marcela Carena  Theoretical Physics Department, Fermi National Accelerator
Laboratory, Batavia, Illinois, 60510, USA  Enrico Fermi Institute, University
of Chicago, Chicago, Illinois, 60637, USA  Kavli Institute for Cosmological
Physics, University of Chicago, Chicago, Illinois, 60637, USA  Jonathan
Kozaczuk  Department of Physics, University of California, San Diego, La
Jolla, CA 92093, USA  Zhen Liu  School of Physics and Astronomy, University of
Minnesota, Minneapolis, MN 55455, USA  Tong Ou  Enrico Fermi Institute,
University of Chicago, Chicago, Illinois, 60637, USA  Michael J. Ramsey-Musolf
Tsung-Dao Lee Institute and School of Physics and Astronomy, Shanghai Jiao
Tong University, 800 Dongchuan Road, Shanghai, 200240 China  Amherst Center
for Fundamental Interactions, Department of Physics, University of
Massachusetts, Amherst, MA 01003, USA  Kellogg Radiation Laboratory,
California Institute of Technology, Pasadena, CA 91125 USA  Jessie Shelton
Illinois Center for the Advanced Study of the Universe, Department of Physics,
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA  Yikun Wang
Walter Burke Institute for Theoretical Physics, California Institute of
Technology, Pasadena, CA 91125, USA  Ke-Pan Xie  Department of Physics and
Astronomy, University of Nebraska, Lincoln, NE 68588, USA

###### Abstract

An essential goal of the Higgs physics program at the LHC and beyond is to
explore the nature of the Higgs potential and shed light on the mechanism of
electroweak symmetry breaking. An important class of models defining the
strength and order of the electroweak phase transition is driven by the Higgs
boson coupling to a light new state. This Snowmass white paper points out the
existence of a region of parameter space where a strongly first order
electroweak phase transition is compatible with exotic decays of the SM-like
Higgs boson. A dedicated search for exotic Higgs decays can actively explore
this framework at the Large Hadron Collider (LHC), while future exotic Higgs
decay searches at the high-luminosity LHC and future Higgs factories will be
vital to conclusively probe the scenario.

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

##  I Motivation

An important goal of the Higgs physics program at the LHC and proposed future
colliders is to explore the nature of the Higgs potential and shed light on
the mechanism of electroweak symmetry breaking (EWSB). Given our current
understanding of the Higgs boson, we do not yet know how EWSB occurred in the
early Universe. The Standard Model (SM) predicts EWSB as a smooth
thermodynamic crossover. However, it may well be that new physics beyond the
SM (BSM) alters this picture, enabling the possibility of a first-order
electroweak phase transition (EWPT). A first-order EWPT could have supplied
one of the necessary ingredients for generating the observed baryon asymmetry
of the Universe through the mechanism of electroweak baryogenesis
Morrissey:2012db . In addition, such a process produces a stochastic
gravitational-wave (GW) signature that could be observed in future GW probes
Caprini:2015zlo ; Caprini:2019egz . A first-order EWPT could have affected the
abundance of primordial relics such as dark matter with masses at or above the
electroweak scale Wainwright:2009mq . Beyond these important cosmological
implications, mapping out the phase diagram of the electroweak sector is an
important undertaking in its own right, analogous to determining the phase
diagram of QCD. Understanding the history of the EWPT can be game changing in
our understanding of nature, and present and future collider experiments have
a crucial role to play in such exploration.

Many BSM scenarios predicting a first-order electroweak phase transition are
being tested at the LHC. An important class of such models is that in which
the EWPT is driven first-order by the Higgs coupling to a new light particle
Profumo:2007wc . As argued in Ref. Kozaczuk:2019pet , if the new degree of
freedom is below $\sim m_{Z}$, it must be a singlet-like scalar denoted as
$s$. The mixing angle ($\sin\theta$) of this singlet with the SM Higgs boson
($h$) control the direct production of such scalars at colliders, and can be
small or absent entirely if additional symmetries are present. Therefore,
these light new particles can be difficult to detect directly. Refs.
Kozaczuk:2019pet ; Carena:2019une  recently emphasized that such scalars can
still have a dramatic impact on the EWPT, provided there is a sizable scalar
coupling to the SM-like Higgs field. This coupling also controls the
$h\rightarrow ss$ branching ratio when $m_{s}<m_{h}/2$. Therefore, these
scalars provide a compelling target for exotic Higgs decay searches, both at
the LHC and future colliders.

Exotic Higgs decays are a cornerstone of the discovery program at both current
and future colliders Curtin:2013fra ; Abada:2019lih ; CEPCStudyGroup:2018ghi ;
Benedikt:2018csr ; Abada:2019zxq ; Cepeda:2021rql . At the HL-LHC, detector
upgrades, new trigger and analysis strategies, and increased datasets will
steadily increase the sensitivity to small branching ratios, especially in
subdominant but cleaner final states (notably $h\to ss\to bb\tau\tau$, $h\to$
invisible) deFlorian:2016spz ; Cepeda:2019klc . Meanwhile, proposed
$e^{+}e^{-}$ colliders offer lower integrated luminosities but substantially
lower backgrounds, resulting in excellent sensitivity to challenging all-
hadronic modes, notably $h\to ss\to bbbb$ Liu:2016zki . This white paper aims
to establish how searches for exotic Higgs decays can inform our understanding
of the early Universe. It will establish parameter space portions compatible
with exotic Higgs decays and first-order phase transitions and provide clear
and achievable targets for current and future colliders.

##  II Higgs exotic decay target from EWPT considerations

For strongly first-order phase transitions driven by light degrees of freedom,
the most exciting possibility is a singlet scalar $s$, which can affect the
Higgs potential at tree-level. The existence of experimentally viable
parameter space where a light singlet-like scalar can drive the EWPT to be
strongly first-order was recently demonstrated in Refs. Carena:2019une ;
Kozaczuk:2019pet . It is useful to classify the possibilities according to
whether or not the light singlet scalar respects the $Z_{2}$ symmetry taking
$s\to-s$. Theoretically, this $Z_{2}$ symmetry, whether or not it is
spontaneously broken, can be a useful proxy for a bigger (gauge or global)
symmetry, as can be the case when $s$ is part of a hidden sector, as
motivated, e.g., by dark matter model-building. Practically, imposing a
$Z_{2}$ symmetry reduces the number of free parameters. Most consequentially,
in models with an unbroken $Z_{2}$, $s$ cannot mix with the SM Higgs. At the
same time, if the $Z_{2}$ is spontaneously broken, the $s$-$h$ mixing angle
$\sin\theta$ is related to the exotic branching fraction Br($h\to ss$). In the
absence of the $Z_{2}$ symmetry, $\sin\theta$ and Br($h\to ss$) are
independent parameters. Ref. Carena:2019une  studies the $Z_{2}$
spontaneously-broken scenario, while Ref. Kozaczuk:2019pet  studies the
$Z_{2}$ unbroken and non-symmetric scenarios.

In the viable light scalar scenarios that yield strongly first-order EWPTs,
symmetry breaking in the thermal history generically proceeds through a two-
step transition, while a one-step transition is feasible in limited regions of
parameter space. The step where the electroweak symmetry is first broken is
required to be strongly first-order. The possibilities are illustrated in Fig.
1 111In the general non-symmetric scenario, one is free to shift the singlet
by a constant without changing the physical predictions of the theory. This
shift is often used to either remove the tadpole term in the potential or set
the singlet vacuum expectation value to zero. We choose the latter.. Since the
SM phase transition is a crossover, if the singlet $s$ is to catalyze a
strongly first-order phase transition, we expect its couplings to the Higgs
cannot be too small in order to induce a sufficiently large deformation to the
scalar potential in the early universe. Thus we generally expect that the
exotic Higgs branching ratio into new scalars compatible with first-order
phase transitions is bounded from below. The specific implications for the
surviving light scalar parameter space depend on the (non-)realization of the
$Z_{2}$ symmetry, as we now discuss.

In the $Z_{2}$-preserving case, a sizeable quartic coupling $s^{2}|H|^{2}$ is
crucial in generating a two-step phase transition (with the second step a
first-order EWPT). In the general $Z_{2}$-explicit breaking scenario, the
singlet cubic term $s^{3}$ can also be important in providing the tree-level
barrier for a first-order EWPT. However, in the spontaneously-broken $Z_{2}$
case, at tree-level, the singlet $s^{3}$ term is absent, while the only
allowed portal coupling, i.e., the quartic term $s^{2}|H|^{2}$, is related to
the scalar-Higgs mixing angle and thus stringently constrained by Higgs
precision tests.

(a) $Z_{2}$ spontaneous breaking scenarios with symmetry restoration.

(b) $Z_{2}$ spontaneous breaking scenarios with symmetry non-restoration.

(c) $Z_{2}$ preserving or explicit breaking scenarios.

Figure 1:  Schematics of EWPT patterns in the singlet extended SM in different
scenarios relevant for Higgs exotic decays Kozaczuk:2019pet ; Carena:2019une .
Solid lines correspond to patterns where EWPT can be strongly first-order,
while dashed lines correspond to patterns where EWPT is weakly first-order,
second order or a crossover. Transition step in which the electroweak symmetry
is first broken, thus relevant for baryogenesis, is labeled as “EWPT”.

However Ref. Carena:2019une  demonstrated that the joint scalar sector with a
spontanously-broken $Z_{2}$ can indeed realize a strong first-order EWPT while
satisfying the current constraints on Higgs properties. This analysis includes
the full one-loop radiative corrections from zero-temperature and thermal
contributions, as well as the daisy resummation. The surviving parameter space
exhibits a sharp correlation between the first-order EWPT and the Higgs exotic
decay. The region that realizes a strongly first-order EWPT parameter space
while remaining compatible with Higgs properties features $m_{s}\lesssim 25$
GeV and gives exotic decay branching ratios ${\rm Br}(h\to ss)\sim
10^{-6}-10^{-1}$. We extract from Ref. Carena:2019une  the motivated region in
such a scenario that is also compatible with the current precision measurement
constraints, as shown by the brown shadowed region in Fig. 4. A more intense
scan might reveal more tuned parameters, beyond the large scans done in Ref.
Carena:2019une , and cover further regions of smaller Higgs mass and lower
branching ratio.

In the general non-symmetric scenario, Ref. Kozaczuk:2019pet  shows that
requiring a strong first-order EWPT together with a small amount of mixing
between the Higgs and the new scalar implies a lower bound on the magnitude of
the $hss$ coupling, and therefore a lower bound on the Higgs branching ratio
for $h\to ss$. Ref. Kozaczuk:2019pet  considers the leading contributions to
the finite-temperature effective potential, keeping terms up to
$\mathcal{O}(g^{3})$ in Landau gauge, and performs numerical scans for fixed
values of $\cos\theta$ that are well below current experimental bounds. The
region of parameter space compatible with first-order phase transitions at
fixed $\sin\theta=0.01$ is shown as a blue shaded region in Fig. 4. In
particular, global fits to Higgs properties require $m_{s}\lesssim 28$ GeV.
Semi-analytical arguments in the small mixing limit, based on an expansion of
the finite temperature effective potential that retains terms up to
$\mathcal{O}(g^{2})$, can be combined to obtain a simple expression for a
lower bound on the value of Br($h\to ss$), as a function of $m_{s}$, that is
compatible with first-order phase transitions.

Unlike Ref. Carena:2019une , Ref. Kozaczuk:2019pet  restricts attention to the
region of parameter space with $m_{s}\gtrsim 5$ GeV. This restriction is
technical: the region of parameter space realizing first-order phase
transitions certainly extends to smaller scalar masses, but the power-counting
arguments that justify the approximations made in the finite-temperature
potential begin to break down at low scalar masses where smaller values of the
mixed quartic coupling must be considered.

Ref. Kozaczuk:2019pet  also performed a similar analysis for the
$Z_{2}$-symmetric scenario. In this scenario, purely analytic arguments (again
based on an $\mathcal{O}(g^{2})$ approximation to the finite-temperature
effective potential) can provide a lower bound on the invisible Higgs
branching ratio that is in excellent agreement with the results of numerical
scans. The identified surviving parameter space of interest is shown in Fig. 4
in Ref. Kozaczuk:2019pet . In this case direct LHC searches for $h\to$
invisible requires $m_{s}\lesssim 20$ GeV.

In all three scenarios, the parameter space compatible with both exotic Higgs
decays and a first-order EWPT predicts exotic branching ratios Br($h\to ss$)
that are bounded from below. The branching ratios of interest are in the range
$10^{-6}-10^{-1}$, with smaller branching ratios possible at smaller values of
$m_{s}$. The resulting parameter space poses an attractive target for both
visible and invisible exotic Higgs decay searches at current and future
colliders, as discussed in the next section.

In summary, both studies suggest that searches for exotic Higgs decays at the
LHC and future colliders can play a vital role in probing the nature of the
EWPT in models with light scalars. Next-generation experiments are likely to
either unearth evidence for or concretely rule out this class of scenarios.

##  III Current status and recent results in exotic decays

The studies discussed in the previous section point us toward an intriguing,
accessible signal in exotic Higgs decays. The 125 GeV SM-like Higgs boson can
decay to pairs of new particles via the portal couplings. As the SM Higgs
boson has a very narrow decay width $\Gamma_{h}=4.07$ MeV, even very tiny BSM
couplings can have appreciable impacts on the decay branching ratios, making
exotic Higgs decays a potent probe of beyond-the-SM interactions
Curtin:2013fra ; Cepeda:2021rql . This section summarizes the current status
and future prospects for Higgs exotic decays that are relevant for the EWPT-
motivated $h\to ss$ decays.

Figure 2: Scalar singlet $s$ branching fractions mediated through mixing with
the Higgs boson, taken from Ref. Gershtein:2020mwi .

Current global fits constrain the Higgs exotic decay branching ratio to be
$\leq 16\%$ at 95% C.L. ATLAS:2021vrm . The $h\to ss$ decay mentioned in the
last section can lead to various final states according to the subsequent
decay channels of the light scalar state $s$. In these SM plus singlet models,
the $s$ decay is controlled by the $s$-$h$ mixing and inherits the Higgs-like
hierarchical branching fractions following the corresponding fermion masses.
The final state is dominated by $h\to ss\to bbbb$ for $m_{s}>2m_{b}\sim 10$
GeV, and by $jjjj,jj\tau\tau$, and $\tau\tau\tau\tau$ for $m_{s}<10$ GeV. In
Fig. 2 we show the singlet decay branching fractions to various final states
from Ref. Gershtein:2020mwi , building on the work of Refs. Spira:1997dg ;
Winkler:2018qyg . In general, the final state arising from $h\to ss$ can be
written as $XXYY$, where $X$ and $Y$ represent (possibly) different particles.
Beyond these visible decays, if $s$ decays dominantly to dark particles or is
stable on collider time scales, the signal would be invisible Higgs decay. For
instance, the $Z_{2}$ symmetry in the $s\to-s$ could be exact, and then the
scalar $s$ could be stable and hence appear as missing energy (prospects for
this case were surveyed in Kozaczuk:2019pet ). Other generalization of $Z_{2}$
symmetric SM-singlet extensions could further alter the cosmological history,
even achieving electroweak non-restoration, and leading to Higgs invisible
decay phenomenology Carena:2021onl . Moreover, supersymmetric extensions of
the Higgs potential, as for example in the next-to-minimal-supersymmetric-
standard-model (NMSSM) can give raise to strongly first order phase transition
and induce interesting phenomenology that deserves a detailed study
Baum:2020vfl .

Final state | Production mode |  $m_{s}$ range [GeV] |  $\mathcal{L}$ [fb-1] | Collaboration
---|---|---|---|---
$\mu\mu\mu\mu$ | $gg$ fusion | $[0.25,8.5]$ | 35.9 |  CMS CMS:2018jid
$[4,8]\cup[11.5,60]$ | 137 |  CMS CMS:2021pcy
$[1.2,2]\cup[4.4,8]\cup[12,60]$ | 139 |  ATLAS ATLAS:2021ldb
$\mu\mu\tau\tau$ | $gg$ fusion | $[3.6,21]$ | 35.9 |  CMS CMS:2020ffa
$[15,62.5]$ | 35.9 |  CMS CMS:2018qvj
$\mu\mu\tau\tau$ |  $gg$ fusion | $[4,15]$ | 35.9 |  CMS CMS:2019spf
$bb\mu\mu$ | $gg$ fusion | $[18,60]$ | 139 |  ATLAS ATLAS:2021hbr
$[20,62.5]$ | 35.9 |  CMS CMS:2018nsh
$bb\tau\tau$ |  $gg$ fusion | $[15,60]$ | 35.9 |  CMS CMS:2018zvv
$bbbb$ | $Zh$ | $[15,30]$ | 36.1 |  ATLAS ATLAS:2018pvw
$Wh/Zh$ | $[20,60]$ | 36.1 |  ATLAS ATLAS:2020ahi
$\gamma\gamma\gamma\gamma$ |  $gg$ fusion | $[15,60]$ | 132 |  CMS CMS:2021bvh
$\gamma\gamma jj$ | VBF | $[20,60]$ | 36.7 |  ATLAS ATLAS:2018jnf
Table 1: The existing experimental searches for exotic Higgs decays $h\to
ss\to XXYY$ at the 13 TeV LHC. Modified from Table 1 of Ref. Cepeda:2021rql .

We summarize existing searches at the LHC for $h\to ss\to XXYY$ final states
at the 13 TeV LHC in Table 1. This table lists the final state targeted by a
given analysis, together with the utilized production mode and the
corresponding integrated luminosity, as well as the intermediate scalar $s$
mass range considered in each search. Except for the $bbbb$ final states,
searches for all the other final states at the LHC involve at least a pair of
non-hadronic final states, such as photons, muons, or tau leptons.

Figure 3: Current bounds (left panel) on exotic Higgs decays $h\to ss\to XXYY$
and corresponding projections (right panel) at the HL-LHC. The horizontal
dotted line is the current and future projection of upper limit for the exotic
Higgs branching ratio from global fits to Higgs properties (16% and 4%
respectively).

In Fig. 3 we show current LHC constraints and future HL-LHC projections on the
branching fraction of $H\to XXYY$ final states as a function of the
intermediate scalar mass. The HL-LHC projections are derived using the simple
assumption that all uncertainties can be taken to scale as $1/\sqrt{L}$.
Searches in these individual final states exclude regions above the lines. We
can see that the $\mu\mu\mu\mu$ channel provides a strong limit on Br($h\to
ss\to XXYY$) to around $10^{-6}$-$10^{-5}$ across the scalar mass. The
$\gamma\gamma\gamma\gamma$ channel also makes a stringent $\sim 10^{-5}$
bound. The constraints from $bb\mu\mu$ and $\mu\mu\tau\tau$ channels are a bit
weaker, around $10^{-4}-10^{-3}$, but still stronger than the $bb\tau\tau$,
$\tau\tau\tau\tau$ and $\gamma\gamma jj$ bounds which are around
$10^{-2}-10^{-1}$. The current $bbbb$ bounds are typically higher than the
allowed maximal exotic branching ratio (16%), but the HL-LHC projections can
reach a few percent. On the other hand, the $\mu\mu\mu\mu$ channel can touch
$10^{-7}$ at the HL-LHC.

The bounds on Br($h\to ss$) can be derived from those on Br($h\to ss\to XXYY$)
once the $s\to XX/YY$ branching ratios are given. Assuming the $s$ decay
branching ratios are dominated by the $h$-$s$ mixing (see Fig. 2), the bounds
on Br($h\to ss$) are given in Fig. 4. We can see that the hierarchies of
various channels are significantly affected compared to those in Fig. 3. For
$m_{s}\lesssim 10$ GeV, the strongest bounds are still from the
$\mu\mu$-relevant channels, e.g. $\mu\mu\mu\mu$ for $m_{s}\lesssim 3.5$ GeV
and $\mu\mu\tau\tau$ for $3.5~{}{\rm GeV}\lesssim m_{s}\lesssim 10$ GeV,
respectively. For $m_{s}\gtrsim 10$ GeV, $bb$ is the main decay channel of
$s$, making $bb$-relevant channels most sensitive. As a result, the most
stringent bounds for $10~{}{\rm GeV}\lesssim m_{s}<62.5$ GeV is $bb\mu\mu$ and
$bb\tau\tau$.

In Fig. 4 we show the projected reach of the $\sim 240$ GeV $e^{+}e^{-}$
colliders with an integrated luminosity of $5~{}{\rm ab}^{-1}$ for the
$\tau\tau\tau\tau$ Shelton:2021xwo  and $bbbb$ Liu:2016zki  channels. The
projections for $qqqq/gggg$ and $cccc$ channels can be found in Ref.
Liu:2016zki , which we do not show here. A ILC-based simulation (250 GeV, 0.9
ab-1) for the $bbbb$ channel is done by Ref. Kato:2021ILC4b  and find similar
projections. There are room for further improvement, e.g., usings Machine
Learning to deal with complex signals and backgrounds for Higgs exotic decays
Jung:2021tym .

Figure 4: The current bounds on Higgs exotic decay $h\to ss$ and the
projections at the HL-LHC, assuming the $s$ decays to SM particles are
mediated by the mixing, and the corresponding branching ratios are taken from
Ref. Gershtein:2020mwi . The upper and lower horizontal dotted lines are the
expected upper limit for Higgs exotic decay branching ratio at the HL-LHC (4%
deBlas:2019rxi ) and statistical limit of $10^{6}$ Higgs at future lepton
colliders, respectively. The brown and light blue shadowed regions are the
strong first-order EWPT regions from Refs. Kozaczuk:2019pet ; Carena:2019une ,
see text for details. Projections of the reach of future lepton colliders are
shown in dashed lines.

The strongly first-order EWPT parameter space for the spontaneous $Z_{2}$
breaking model Carena:2019une  and general singlet scalar extension of the SM
with mixing angle $\sin\theta=0.01$ Kozaczuk:2019pet  are indicated by the
brown and light blue shaded regions in the Br($h\to ss$)-$m_{s}$ plane in Fig.
4, respectively. Combining with the current bounds at the LHC, we see that the
exotic Higgs decay searches have already probed a visible fraction of the EWPT
parameter space, especially for the low mass region $m_{s}\lesssim 10$ GeV.
For the high mass region $m_{s}>10$ GeV, the direct constraints from the
$bb$-relevant channels are slightly weaker than the indirect bound (16% ATLAS-
CONF-2021-053 ) from the exotic Higgs decay. At the HL-LHC, the Br($h\to ss$)
reach in both the low and high mass regions are significantly improved, while
the expected reach from direct searches at high masses is still comparable
with the expected indirect bounds (4% deBlas:2019rxi ). Future Higgs factories
can greatly improve the coverage of EWPT parameter space, as shown in the
dashed lines in the right panel of Fig. 4. Combining the $\tau\tau\tau\tau$
and $bbbb$ channels, future Higgs factories will be able to cover almost all
the EWPT parameter space for the general singlet scalar extension SM with
mixing angle $\sin\theta=0.01$ Kozaczuk:2019pet . For tiny mixing angles
$\sin\theta$, that render the singlet scalar long-lived, recent studies
Feng:2022inv  consider a class of dark Higgs searches at various intensity
frontier machines that can provide useful probes. Note that the statistical
limit of a typical Higgs factory with $10^{6}$ Higgs is $3\times 10^{-6}$.
Given the clean environment with low background at lepton colliders, we
conclude that there is ample scope for future studies in exotic decays.

##  IV Summary and Outlook

Determining the true nature of the electroweak phase transition is a
fundamental question in particle physics, and connects searches at energy
frontier facilities to the history of the early universe. While much study has
been devoted to discovery prospects for theories where heavy ($m>m_{h}$) new
physics drives the electroweak phase transition strongly first-order, we
emphasize that there exists an experiemntally viable region of parameter space
where light new states can accomplish the same goal. Following the studies in
Kozaczuk:2019pet ; Carena:2019une , we describe the frameworks where the
simple singlet scalar extension of the SM can realize a strongly first-order
EWPT along with allowing for exotic Higgs decays, $h\to ss$. Such new physics
tragets are an interesting and accessible target for both the LHC and future
Higgs factories.

At the LHC we find that direct searches in the $bbbb$, $bb\mu\mu$ and
$\mu\mu\mu\mu$ final states lead the sensitivities that can cover new singlet
masses below 10–15 GeV with Higgs exotic branching ratios as low as 10-2, as
shown in the left panel of Fig. 4. The full run of the HL-LHC and future
lepton collider Higgs factories can access most of the relevant parameter
space for singlet SM extension models that enable a strongly first order EWPT,
with the current projections shown in the right panel of Fig. 4. The $bbbb$
and $\tau\tau\tau\tau$ channels have the best sensitivities at future lepton
colliders. The sensitivity projections shown are based on studies of the
signal $e^{+}e^{-}\to hZ$ in various exclusive final states and assuming
leptonic decays of the $Z$. There is ample room for improvement by including
more final states for the Higgs decay, especially in the low mass regions, and
by considering hadronic $Z$ decays. Altogether a combination of future lepton
and hadron colliders offer excellent prospects for full coverage of the
strongly first order EWPT-motivated region of relevance for electroweak
baryogenesis models.

###### Acknowledgements.

This manuscript has been authored by Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of
Science, Office of High Energy Physics. Z.L. is supported in part by the U.S.
Department of Energy (DOE) under grant No. DE-SC0022345. J.S. is supported in
part by U.S. DOE CAREER grant DE-SC0017840. K.-P.X. is supported by the
University of Nebraska-Lincoln. MJRM was supported in part under U.S. DOE
Contract No. DE-SC0011095 and National Natural Science Foundation of China
Grant No. 19Z103010239. Y.W. is supported by the Walter Burke Institute for
Theoretical Physics. T.O. is supported by the Visiting Scholars Program of
Universities Research Association.

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