#  Addressing the Gravitational Wave – Collider Inverse Problem

Leon S. Friedrich<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Shanghai Key Laboratory for Particle Physics and Cosmology, Key Laboratory for
Particle Astrophysics and Cosmology (MOE), Shanghai Jiao Tong University,
Shanghai 200240, China  Michael J. Ramsey-Musolf<EMAIL_ADDRESS><EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Shanghai Key
Laboratory for Particle Physics and Cosmology, Key Laboratory for Particle
Astrophysics and Cosmology (MOE), Shanghai Jiao Tong University, 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  Tuomas V. I. Tenkanen<EMAIL_ADDRESS>Nordita,
KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden  Tsung-Dao Lee Institute &
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  Shanghai Key Laboratory for Particle Physics and Cosmology, Key
Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Jiao Tong
University, Shanghai 200240, China  Van Que Tran<EMAIL_ADDRESS><EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Faculty
of Fundamental Sciences, PHENIKAA University, Yen Nghia, Ha Dong, Hanoi 12116,
Vietnam

###### Abstract

We provide a roadmap for analyzing the interplay between hypothetical future
collider observations and the detection of a gravitational wave signal
produced by a strong first order electroweak phase transition in beyond the
Standard Model theories. We rely on a combination of a dimensionally reduced,
three-dimensional effective field theory and results of both perturbation
theory and non-perturbative lattice simulations. We apply these state-of-the-
art methods to the real scalar triplet extension of the Standard Model, which
admits a possible two-step electroweak symmetry-breaking thermal history. We
find that (1) a first order transition during the second step could generate a
signal accessible to LISA generation detectors and (2) the gravitational wave
signal displays a strong sensitivity to the portal coupling between the new
scalar and the Higgs boson. We illustrate how a combination of direct and
indirect measurements of the new scalar properties, in combination with the
presence or absence of a gravitational wave detection, could test the model
and identify the values of the model parameters.

††preprint: NORDITA 2022-010

In the Standard Model (SM) of particle physics electroweak symmetry is
smoothly broken through a crossover transition as the temperature is lowered
below the electroweak scale [1, 2]. However, the nature of the transition can
be modified in the presence of physics beyond the Standard Model (BSM).
Extended Higgs sectors in particular can lead to a first order electroweak
phase transition (EWPT) with several associated theoretical and
phenomenological implications. Such a phase transition could provide the
necessary preconditions for the generation of the observed baryon asymmetry
via electroweak baryogenesis [3, 4, 5]. Experimental signatures include
deviations from SM predictions for various high energy collider observables
[6]– at both present day and planned future next-generation detectors – as
well as production of primordial gravitational waves (GW) that may be
accessible by next-generation experiments [7, 8]. The latter include
LISA/eLISA [9], DECi-hertz Interferometer GW Observatory (DECIGO) [10, 11],
Big Bang Observer (BBO) [12], TAIJI [13] and TIANQIN [14]. These experiments
would provide a looking glass into a cosmic era that significantly pre-dates
the epoch of recombination. Thus, the combination of collider and GW probes
hold the prospect of revealing in detail the BSM physics of the hot plasma
consisting of particles and interactions at the time of electroweak symmetry
breaking.

Many BSM theories can accommodate a first order EWPT – which proceeds via
bubble nucleation – when the new degrees of freedom lie close to the
electroweak scale and couple with sufficient strength to the SM Higgs boson.
The EWPT, thus, provides a clear target for collider and GW experiments [6,
8]. One then encounters two questions:

1\. The “GW – collider inverse problem”: can a combination of collider and GW
observations be used to determine the BSM scenario responsible for the
observed signals, and can these GW observables be used to measure relevant
model parameters?

2\. Theoretical robustness: how reliable are the computations that attempt to
address the first question?

Several studies addressing the GW-collider inverse problem have been published
in the past decade (c.f. [15, 8] and e.g. [16, 17, 18]). Most have solely
relied on the use of perturbation theory to analyze the EWPT thermodynamics
and nucleation dynamics. However, enhanced thermal contributions from bosonic
infrared modes render perturbation theory suspect in this context [19].

For quantitative and, in some cases, even qualitative reliability, it is
critical to resort to non-perturbative approaches. Such non-perturbative
studies are provided by lattice simulations [20, 21, 22, 23, 24, 25] of
dimensionally reduced [26, 27], three-dimensional effective field theories (3d
EFT) [28, 29]. In this framework, recent work [30] yields the model-
independent conclusion that a GW signal cannot be accessible to LISA
generation experiments, if the new scalar is sufficiently heavy that
transition dynamics are described by the SM-like 3d EFT. Evidently, a GW
signal detectable by LISA generation experiments is likely only if new scalar
is light enough to be dynamical and actively present in simulations. Such
simulations were recently performed in [24] for the real triplet extension of
the SM ($\Sigma$SM), which non-perturbatively confirmed the possibility of
two-step electroweak symmetry-breaking in the early universe that had been
proposed by previous perturbative studies [31, 32, 33].

While a fully non-perturbative treatment of the $\Sigma$SM bubble dynamics (as
in [34, 35]) remains to be implemented, it is nevertheless of interest to
address the GW-collider inverse problem (c.f. [36]) by drawing on the cutting-
edge perturbative treatment for thermodynamics and results from non-
perturbative (lattice) simulations. Doing so is the goal of this letter. We
emphasize that use of perturbation theory is essential in order to draw
conclusions regarding the full parameter space of a BSM theory; due to the
large computational demand, non-perturbative studies are limited to selected
benchmark points. Our objective is to provide a template for future studies
that may be performed for other models and, ultimately, rely on further
advances in the theory of bubble dynamics. In addition to reliance upon the
combination of lattice and EFT computations for a dynamical BSM scalar, the
novel features of this work include:

  * •

We map the relevant portion of the model phase diagram into the plane of the
key inputs for GW signals, $(\alpha,\beta/H_{*})$ [37, 8], and determine in
detail how these inputs evolve with the parameters that characterise the phase
diagram. To our knowledge, such a mapping and determination of this evolution
has not yet been carried out in the full parameter space for any
phenomenologically-viable BSM scenario while drawing upon state-of-the-art
thermodynamics discussed above. Importantly, our analysis combines multiple
developments in computation of the thermodynamics by utilizing the 3d EFTs,
including non-perturbative information from lattice studies (c.f. [38, 30, 24,
39, 40, 41]).

  * •

We explore the relation between future collider physics phenomenology and GW
probes. In doing so, the specific numbers we assume for hypothetical collider
measurements are less important than the larger lesson that we illustrate: how
the combination of results can address the GW-collider inverse problem.

The choice of the $\Sigma$SM is particularly well-suited for the illustration
of this theoretical “template”. It entails the minimum number of new scalar
degrees of freedom that are charged under the SU(2$)_{L}$ electroweak gauge
group; if a $Z_{2}$-symmetry is imposed, only a relatively small number of new
parameters are introduced – the triplet scalar mass ($m_{\Sigma}$), self-
coupling ($b_{4}$), and Higgs portal coupling ($a_{2}$) – thereby enabling
predictivity; and it allows for the non-trivial thermal history of multi-step
electroweak symmetry-breaking. The first transition, when it is first order,
arises via a radiatively induced barrier. The second comes from a tree-level
barrier in the presence of the thermal loop contributions to the potential.
Even for a dynamical triplet scalar, a LISA-generation experimental signal is
likely possible only from the second step in the two-step scenario. We learn
that the value of the Higgs portal coupling is decisive for this
accessibility. Moreover, strength and duration of the transition depend
strongly on $a_{2}$; $\mathcal{O}(10\%)$ changes in its value can lead to
order of magnitude changes in the strength and duration. Thus, the GW search
provides a particularly powerful probe of the Higgs portal coupling. In the
collider arena, the Higgs diphoton decay rate, as well as the neutral triplet
diboson decay rate, have an analogous sensitivity to $a_{2}$ as well as
sensitivities to the new scalar mass. Thus, a combination of GW and collider
searches provide a means of identifying the relevant region of parameter space
for a two-step transition that is unlikely to be obtainable by either probe
alone.

We consider the most general renormalizable scalar potential for the
$\Sigma$SM [42, 31, 38, 43, 24, 44]

$\displaystyle V(H,\Sigma)\ =$ $\displaystyle\ -\mu_{H}^{2}H^{\dagger}H\ -\
\frac{1}{2}\mu_{\Sigma}^{2}\mathrm{Tr}(\Sigma^{2})\ $ $\displaystyle+\
\lambda_{H}(H^{\dagger}H)^{2}\ +\
\frac{1}{4}b_{4}[\mathrm{Tr}(\Sigma^{2})]^{2}\ $ $\displaystyle+\
\frac{1}{\sqrt{2}}a_{1}H^{\dagger}\Sigma H\ +\
\frac{1}{2}a_{2}\mathrm{Tr}(\Sigma^{2})H^{\dagger}H\,,$ (1)

where $H$ is the SM Higgs doublet and the real scalar

$\Sigma=\begin{pmatrix}\Sigma^{0}/\sqrt{2}&\Sigma^{+}\\\
\Sigma^{-}&-\Sigma^{0}/\sqrt{2}\end{pmatrix}$ (2)

transforms as $(1,3,0)$ under the SM $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}$ gauge group and is comprised of new neutral and charged particles
$\Sigma^{0}$ and $\Sigma^{\pm}$, respectively. The triplets are primarily pair
produced at colliders via charged- and neutral-current Drell-Yan processes
$q\bar{q}\to\Sigma^{+}\Sigma^{-}$ and $q\bar{q}\to\Sigma^{\pm}\Sigma^{0}$
[42]. For the non-$Z_{2}$ symmetric case, the neutral triplet has several
interesting decay channels, such as decay into gauge boson pairs. Here, we
focus on the decay $\Sigma^{0}\to ZZ$ and additionally on $\Sigma^{\pm}$ loop-
induced modifications of the Higgs to diphoton decay rate. Selected formulae
for partial widths, branching fractions and decay rates are collected in the
Supplementary material. These observables may be probed with future data at
the LHC as well as prospective future colliders, including the Future Circular
Collier (FCC) [45], Circular Electron-Positron Collider (CEPC) [46],
International Linear Collider (ILC) [47], Compact Linear Collider (CLIC) [48,
49], or a high energy muon collider [50, 51].

We employ thermal quantum field theory methods to obtain the quantities that
govern the GW spectrum: $T_{n}$, the bubble nucleation temperature; $\alpha$
at $T_{n}$, which describes the strength of the transition in terms of the
trace anomaly, related to entropy density and pressure between the phases; and
$\beta/H_{*}$, inverse duration of the transition, evaluated at $T_{n}$. For
exact definitions, see [8, 39, 36]. The thermodynamics of the EWPT in
$\Sigma$SM were first studied in [31] using 4d perturbation theory and the
$\hbar$-expansion to ensure gauge invariance [52]. As observed in that work,
and subsequently verified in Refs. [31, 32, 33, 24], the model allows for a
“two-step” EWPT. During the first transition, wherein a thermal loop-induced
barrier separates the symmetric and broken phases, the triplet acquires a non-
zero vacuum expectation value (VEV). The second transition to the Higgs phase,
which entails traversing a tree-level barrier, can be a sufficiently strong
first order EWPT to provide for an observable GW signal. Here, we focus solely
on these strong second transitions.

In our computation we work in 3d EFT with next-to-leading order dimensional
reduction [28, 38], and compute the effective potential [53, 24] and the
bubble nucleation rate in analogy to analysis in [39, 40] (c.f. [41]), to
obtain $T_{n},\alpha,\beta/H_{*}$, described above. These parameters can be
used in PTplot tool [8] to obtain GW power spectrum $\Omega(f)$ as function of
frequency, and LISA signal-to-noise ratio (SNR). For relativistic hydrodynamic
simulations of GW production from first order phase transitions, see [54, 55,
56, 57, 58, 59, 60].

The regions of the two-step viable parameter space have been identified using
a combination of the 3d EFT perturbation theory and lattice simulations
involving a dynamical triplet in Ref. [24]. In analogy to [40], we determine
bubble nucleation rate, from which we obtain $T_{n}$ and $\beta/H_{*}$, only
at leading order in 3d EFT, which acts as most limiting theoretical
uncertainty in our computation [39, 40]. For consistency, we also use leading
order result for $\alpha$. We treat the wall velocity ($v_{w}$) – the
remaining important input parameter for GW production, as an unknown input
parameter, setting its value as $v_{w}=0.95$. For recent development on
computing $v_{w}$ from first principles, see [61].

Figure 1:  (a) The real-triplet model phase structure spanned in
($m_{\Sigma},a_{2}$) plane, with fixed $b_{4}=1.0$. The grey, pink, green and
light blue regions present regions of the metastable electroweak minimum, two-
step, one-step first order phase transition and crossover transition regions,
respectively. Inside the dashed black contour within two-step region, bubble
nucleation completes. (b) Results for the completed two-step transitions
spanned in ($\alpha,\beta/H_{*}$) plane, presented as a scatter plot of light
pink points. The solid red and blue lines present results with fixed
$m_{\Sigma}=150$ GeV and $200$ GeV, respectively, which are also visualised in
(a); arrows indicate the increase of the coupling $a_{2}$. Furthermore, the
dashed green, magenta and dot-dashed black lines indicate sensitivities from
the DECIGO, eLISA and LISA, respectively. In bottom row, we illustrate the
interplay between the collider and the GW searches. (c) The $2\sigma$ favoured
regions from hypothetical collider measurements BMA (BMA′) as blue (light
blue) region and the sensitivity from LISA with $\text{SNR}\geq 5$ as black
region, projected on the phase structure. (d) The $2\sigma$ favoured regions
of BMA/BMA′ spanned in ($\alpha,\beta/H_{*}$) plane.

Our numerical analysis consists of two parts:

1\. We scan through the whole model parameter space $(m_{\Sigma},a_{2},b_{4})$
to identify the two-step phase transition regions [24].111 In the non-$Z_{2}$
symmetric case, there is also non-vanishing $a_{1}$ parameter. However, it is
constrained to be small and has negligible effect for thermodynamics [42, 31].
For each point therein we compute the GW parameters, $\Omega(f)$, and LISA SNR
value. Assuming an observation occurs for $\text{SNR}\geq 5$ [62], we project
the result back onto the ($a_{2}$, $m_{\Sigma}$) plane.

2\. We provide a template for future collider diagnostics, by selecting a
benchmark parameter point inside the two-step region and by envisioning
hypothetical measurements. We then identify the hypothetical collider-allowed
region in the ($a_{2}$, $m_{\Sigma}$) and ($\alpha$, $\beta/H_{\ast}$) planes.

We start from 1: Fig. 1 (a) shows the model phase structure projected in the
($m_{\Sigma},a_{2}$) plane with fixed coupling $b_{4}=1.0$. The red area
indicates the parameters accommodating a two-step EWPT history. Therein, the
nucleation process does not complete for parameters outside the black dashed
contour. The green and light blue areas correspond to a one-step first order
EWPT and crossover transition, respectively. We use non-perturbative
information from lattice simulations of [21] along lines of [63, 64, 38, 30]
in order to delineate the boundary between these two regions. Varying $b_{4}$
leads to mild changes in the EWPT-viable regions, c.f. Supplementary material.

Fig. 1 (b) gives a scatter plot of the values of $\alpha$ and $\beta/H_{*}$
corresponding to the interior of the black dashed curve in Fig. 1 (a). The
dashed green and magenta lines indicate the experimental sensitivities of
DECIGO with the “Correlation” design [11] and eLISA with the “$C_{1}$”
configuration shown in Table 1 in [15], respectively; the dot-dashed black
line represents $\text{SNR}=5$ sensitivity for LISA [8].222 For illustration
of this line, we have fixed $T_{*}=100$ GeV, while for all scatter points we
have specific $T_{n}$ determined from thermodynamics individually for each
point.  It is particularly interesting to determine the dependence of the GW
parameters on $a_{2}$ and $m_{\Sigma}$. To that end, for visualisation we
choose two values of the latter, $m_{\Sigma}=$ 150 and 200 GeV, corresponding
to the red and blue lines, respectively, in Fig. 1 (a). Arrows indicate
increasing $a_{2}$. We recast these lines in Fig. 1 (b), thereby depicting the
corresponding trends in $\alpha$ and $\beta/H_{\ast}$. It is evident that the
GW parameters carry a pronounced sensitivity to the Higgs portal coupling and
a milder dependence on the triplet scalar mass. Importantly, a larger $a_{2}$
results in a stronger phase transition and a larger transition duration time.
For example, for the case $m_{\Sigma}=200$ GeV, at $a_{2}=1.54$ the values of
$\alpha\sim 2\times 10^{-3}$ and $\beta/H_{*}\sim 10^{4}$, but increasing
$a_{2}$ by $\sim 10\%$ results in $\alpha$ ($\beta/H_{*}$) increasing
(decreasing) by one order of magnitude.

To illustrate the interplay with collider phenomenology and provide a template
for future analysis (point 2 above) we envision a hypothetical set of
measurements:

$\displaystyle\delta_{\gamma\gamma}$ $\displaystyle=-0.132\pm 0.015,$ (3)
$\displaystyle m_{\Sigma}$ $\displaystyle=(200\pm 5)\;\text{GeV}$ (4)
$\displaystyle\text{BR}(\Sigma^{0}\to ZZ)$ $\displaystyle=0.01\pm 0.002\ \ \
,$ (5)

where $\delta_{\gamma\gamma}$ gives the change of the Higgs diphoton decay
rate relative to the SM prediction and where the decay $\Sigma^{0}\to ZZ$
occurs only for the non-$Z_{2}$ symmetric version of the model
($a_{1}\not=0$). For these two quantities, the central values correspond to
$a_{2}=1.665$ and $b_{4}=1.0$. The uncertainty on the diphoton rate
measurement is chosen to correspond to the sensitivity of future lepton
collider FCC-ee [65]. For a similar discussion in case of a HL-LHC, see
Supplementary material. For uncertainty in the triplet scalar mass, we pick an
arbitrary, yet conservative estimate. The mass can be identified via a
reconstruction of the 4 leptons (or $ZZ$) invariant mass if an excess of $ZZ$
pairs is observed. Note that if the triplet scalar is stable (due to $Z_{2}$
symmetry), its mass can be measured using the disappearing charged tracks
search at the LHC [66, 67]. The uncertainty in the $\text{BR}(\Sigma^{0}\to
ZZ)$ measurement is taken to be $20\%$ of the central value. We note that this
choice for uncertainty is arbitrary, and used here solely for the purpose of
illustration, indicating roughly the level of accuracy required to decisively
indicate two-step phase transition. We denote benchmarks without (with)
$\text{BR}(\Sigma^{0}\to ZZ)$ measurement as BMA (BMA′). In the Supplementary
material we discuss in more detail how $\text{BR}(\Sigma^{0}\to ZZ)$ and
$\delta_{\gamma\gamma}$ depend on $m_{\Sigma}$ and $a_{2}$, and present
another illustrative choices for benchmark points.

The 2$\sigma$ favored region from the hypothetical collider measurements is
given by the blue regions in the ($m_{\Sigma},a_{2}$) plane of Fig. 1 (c). We
see that BMA is consistent with either a first order transition or an unstable
electroweak vacuum. For the cosmologically viable region, however, this
combination of collider measurements is not by itself sufficient to
discriminate between a one-step or two-step transition scenario. For the
non-$Z_{2}$ symmetric version of the model, additional collider information
from $\text{BR}(\Sigma^{0}\to ZZ)$, as in BMA′, could indicate that the
$2\sigma$ region lies solely within two-step parameter space.

A GW signal would provide a more powerful indicator, independent of the
presence or absence of $Z_{2}$ symmetry. To illustrate, we recast the LISA
sensitivity region from Fig. 1 (b) to Fig. 1 (c) by narrow black region. In
this example, the combination of GW observation and collider significantly
narrows down the parameter region of BMA, as well as BMA′. This feature
results, in part, from the strong sensitivity of the GW signal to the Higgs
portal coupling $a_{2}$. It is also worth noting that the combination of
collider and GW results provide an important consistency test of the model.
For example, if the BMA′ region resulting from addition of the $\Sigma^{0}\to
ZZ$ decay were not to overlap with the LISA band, then one would conclude that
the GW signal results from a different source.

It is also interesting to ask how the collider discovery of a given model and
constraints on its parameters might interface with a LISA null result. To
illustrate this possible scenario, we project the BMA/BMA′ allowed parameter
space into the ($\alpha$, $\beta/H_{\ast}$) plane as in Fig. 1 (d). A LISA
null result would then disfavor the portion of the blue bands inside the
$\text{SNR}\geq 5$ region. The remaining portion of the collider-allowed two-
step region could, nevertheless, accommodate a future generation GW probe,
such as DECIGO. Even in this case, however, development of even more sensitive
GW probes would be needed to cover the entire collider-favored, two-step
parameter space. Only in that case would one be able to definitively conclude
that the electroweak symmetry-breaking transition in this model was not first
order (two-step).

We conclude that strong two-step transitions in the $\Sigma$SM are highly
sensitive to the triplet-Higgs portal coupling. Therefore, a detection of a GW
background signal could lead to a precise determination of the portal
coupling, providing a complementary probe to future high energy colliders. We
expect this feature persists in the case of other scenarios. More generally,
the foregoing discussion provides a roadmap for future studies of the GW-
Collider inverse problem, whether in the $\Sigma$SM or other models. Key
ingredients include (1) several “upgrades”in the use of perturbation theory
treatment of EWPT thermodynamics and nucleation dynamics:333 Our cutting-edge
perturbative computation of thermodynamics could still be improved. This is to
go even further beyond by adding several higher order corrections in analogy
to individual state-of-art computations presented in [24, 39, 68, 69, 70, 71].
we include several higher order thermal resummations [38, 24]; our
perturbative expansion is consistent in powers of couplings and includes
renormalization group improvement related to hard thermal scale [72, 73, 40,
74]; our analysis is properly gauge invariant [52, 39]; and we perform self-
consistent computation of the bubble nucleation rate [39, 40, 41]. (2) The
presence of the two-step EWPT-viable region of parameter space and the
character of the each transition has been determined by lattice simulations in
[24]. Findings therein support perturbative computations here. Future
refinements should include a non-perturbative determination of the bubble
nucleation rate [34, 35]. Perturbative 3d EFT computations here pave the way
for such future lattice simulations, that would push the current state-of-the-
art even further.

###### Acknowledgements.

We thank Daniel Cutting, Andreas Ekstedt, Oliver Gould, Chloe Gowling, Mark
Hindmarsh, Joonas Hirvonen, Johan Löfgren, Lauri Niemi, Philipp Schicho and
Jorinde van de Vis for illuminating discussions. Specifically we thank Oliver
Gould for sharing a code for bubble nucleation in 3d EFT [40]; and David J.
Weir for permission to use PTplot [8]. TT thanks Aleksi Vuorinen and Helsinki
Institute of Physics – and VQT would like to thank the Institute of Physics,
Academia Sinica, Taiwan – for their hospitality during completion of part of
this work. This work has been supported in part under National Science
Foundation of China grant no. 19Z103010239.

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Supplementary material

Phase transition thermodynamics and possible resulting gravitational wave
signatures depend sensitively on the scalar couplings, but these couplings are
notoriously difficult to measure experimentally at colliders. This is in
particular the case for the triplet self-coupling $b_{4}$. On the other hand,
triplet-Higgs portal coupling $a_{2}$ has reasonable prospects of being
experimentally measured. This could be achieved by precisely measuring the
triplet loop contribution to the Higgs diphoton decay width, or by precisely
measuring the branching fractions of the neutral triplet. We emphasize, that
at one-loop level, both of these probes are independent of $b_{4}$.

The dominant contributions to the Higgs diphoton decay width come from one-
loop diagrams featuring the top quark and $W^{\pm}$ bosons. With the addition
of the charged triplet scalar, the Higgs diphoton decay width [75, 33] reads

$\displaystyle\Gamma^{\mathrm{\Sigma SM}}_{h\rightarrow\gamma\gamma}=$
$\displaystyle\frac{\alpha^{2}g_{2}^{2}}{1024\pi^{3}}\frac{m_{h}^{3}}{m_{W}^{2}}\left\lvert\
\frac{4}{3}F_{1/2}\left(x_{t}\right)+F_{1}\left(x_{W}\right)+a_{2}\frac{v_{H}^{2}}{2m_{\Sigma}^{2}}F_{0}\left(x_{\Sigma}\right)\right\rvert^{2}\,,$
(6)

where $x_{i}$ = $4\frac{m_{i}^{2}}{m_{h}^{2}}$ and the $F_{0,1/2,1}$ are loop
functions,

$\displaystyle F_{0}(x)$ $\displaystyle=x\Big{(}1-xf(x)\Big{)},$ (7a)
$\displaystyle F_{1/2}(x)$ $\displaystyle=-2x\Big{(}1+(1-x)f(x)\Big{)},$ (7b)
$\displaystyle F_{1}(x)$ $\displaystyle=2+3x\Big{(}1+(2-x)f(x)\Big{)}$ (7c)
$\displaystyle f(x)$ $\displaystyle=\begin{cases}\arcsin^{2}(1/\sqrt{x})&x\geq
1\\\
-\frac{1}{4}\left(\ln\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}-i\pi\right)^{2}&x<1\\\
\end{cases}\,.$ (7d)

The SM result follows by taking $a_{2}\rightarrow 0$ in Eq. (6). Measuring the
deviation of the Higgs diphoton decay rate from the SM prediction will
directly constrain $a_{2}$ and $m_{\Sigma}$. We quantify this deviation using,

$\delta_{\gamma\gamma}\equiv\frac{\Gamma^{\mathrm{\Sigma
SM}}_{h\rightarrow\gamma\gamma}}{\Gamma^{\mathrm{SM}}_{h\rightarrow\gamma\gamma}}-1.$
(8)

We show this quantity as a function of $(m_{\Sigma},a_{2})$ in the left panel
of Fig. 2.

---
Figure 2:  Contours for phenomenological quantities of interest in
$(m_{\Sigma},a_{2})$-plane for $\delta_{\gamma\gamma}$ (left) and
$\mathrm{BR}(\Sigma^{0}\rightarrow ZZ)$ (right), on top of thermodynamic phase
structure of Fig. 1 (a). For illustration, simplified expressions for both
quantities are given in the main text. In both plots, we include small, non-
zero triplet VEV $v_{\Sigma}=1$ GeV. This generates mixing between Higgs and
triplet and leads to singular behaviour for $\delta_{\gamma\gamma}$ around
$m_{\Sigma}\sim m_{H}$, as well as makes neutral triplet unstable, which
allows its decay to $ZZ$ and other channels.

For simplicity, in Eq (6) we have neglected possible mixing between the Higgs
and triplet, that would be present in a case of non-zero triplet VEV
$v_{\Sigma}$. Such mixing only plays a significant role when $m_{\Sigma}\sim
m_{H}$, and we demonstrate that in the aforementioned figure that is based on
more general computation.

Figure 3:  Same as Fig. 1 (c), but the uncertainty on $\delta_{\gamma\gamma}$
is taken to be $\sigma_{\delta_{\gamma\gamma}}=0.04$ which can be archived at
the HL-LHC [76].  Figure 4:  Same as Fig. 1 (a) but for $b_{4}=0.25$. Relative
to Fig. 1 (a) two-step region is significantly narrower.

Another means to determine $a_{2}$ (in non-$Z_{2}$ symmetric case) is by
measuring the neutral triplet’s branching ratios. For low triplet mass, the
triplet will decay primarily into pairs of fermions via its mixing with the
Higgs boson. For large triplet mass, $\gtrsim 120$ GeV, the neutral triplet
will decay primarily into pairs of bosons ($W^{\pm}W^{\mp}$, $ZZ$, $hh$).
These decays depend sensitively on $a_{2}$, and in particular when [43]

$\mu_{\Sigma}^{2}=v^{2}_{\Sigma}b_{4},$ (9)

the off-diagonal terms in the neutral scalar mass matrix disappear, resulting
in

$\Gamma_{\Sigma^{0}\rightarrow ZZ}\ =\ \Gamma_{\Sigma^{0}\rightarrow hh}\ =\
0,\quad\Gamma_{\Sigma^{0}\rightarrow WW}\ \approx\ 1.$ (10)

The region of parameter space where this cancellation occurs is close to the
region where the two-step phase transition takes place. For this reason, the
branching ratios are sensitive to the value of $a_{2}$ in the two-step region,
providing a mean to measure it. Since, the $\Sigma^{0}\rightarrow hh^{*}$
branching ratio strongly suppressed when $m_{\Sigma}<2m_{h}$, we will focus on
the $ZZ$ decay. When $m_{\Sigma}>2m_{Z}$, the branching ratio for this decay
can be approximated using the tree level two-body partial widths, resulting

$\displaystyle\mathrm{BR}(\Sigma^{0}\rightarrow ZZ)$
$\displaystyle\approx\frac{\Gamma_{\Sigma^{0}\rightarrow
ZZ}}{\Gamma_{\Sigma^{0}\rightarrow
WW}}=\frac{\sqrt{m_{\Sigma}^{2}-4m_{W}^{2}}\left(m_{\Sigma}^{4}-4m_{\Sigma}^{2}m_{W}^{2}+12m_{W}^{4}\right)}{\frac{1}{2}\sqrt{m_{\Sigma}^{2}-4m_{Z}^{2}}\left(m_{\Sigma}^{4}-4m_{\Sigma}^{2}m_{Z}^{2}+12m_{Z}^{4}\right)}\left(\frac{4v_{\Sigma}}{v_{H}}\cot\theta+1\right)^{2}.$
(11)

This expression is defined in the mass eigenstate basis

$\displaystyle\begin{pmatrix}h_{1}\\\ h_{2}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\cos\theta&\ -\sin\theta\\\ \sin\theta&\
\cos\theta\end{pmatrix}\begin{pmatrix}H^{0}\\\ \Sigma^{0}\end{pmatrix},$ (12)

where mixing angle $\theta$ is defined to diagonalise neutral scalar sector

$\displaystyle\frac{1}{2}\begin{pmatrix}H^{0}&\Sigma^{0}\end{pmatrix}\mathcal{M}_{N}\begin{pmatrix}H^{0}\\\
\Sigma^{0}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}h_{1}&h_{2}\end{pmatrix}\begin{pmatrix}m_{h}^{2}&0\\\
0&m_{\Sigma}^{2}\end{pmatrix}\begin{pmatrix}h_{1}\\\ h_{2}\end{pmatrix},$ (13)

where $m_{h}=125.1$ GeV is the observed Higgs boson mass. The neutral scalar
mass matrix reads

$\displaystyle\mathcal{M}_{N}$
$\displaystyle=\begin{pmatrix}2\lambda_{H}v_{H}^{2}\ &\
a_{2}v_{H}v_{\Sigma}-\frac{1}{2}a_{1}v_{H}\\\
a_{2}v_{H}v_{\Sigma}-\frac{1}{2}a_{1}v_{H}\ &\
-\mu^{2}_{\Sigma}+\frac{1}{2}a_{2}v_{H}^{2}+3b_{4}v_{\Sigma}^{2}\end{pmatrix}.$
(14)

The $a_{1}$ term in the potential and the triplet’s non-zero VEV are
responsible of mixing between the neutral component of the triplet and SM
Higgs. In right panel of Fig. 2 we visualise contours of
$\mathrm{BR}(\Sigma^{0}\rightarrow ZZ)$ near two-step region. A non-vanishing
$v_{\Sigma}$ ensures that the neutral component of the triplet unstable, which
is also required to avoid dark matter direct detection constraints [43, 77].
However, the VEV has to be sufficiently small to satisfy constraints from
electroweak precision observables, which require [78]

$\displaystyle\rho_{0}=1+\frac{4v_{\Sigma}^{2}}{v_{H}^{2}}=1.0004\pm
0.0002\,,$ (15)

which implies $v_{\Sigma}\lesssim 3.5$ GeV at $2\sigma$ accuracy.

To obtain 2$\sigma$ favored regions from the collider measurements BMA′ in the
main body, we define

$\chi^{2}=\frac{\left({\rm BR}{(\Sigma^{0}\to ZZ)}^{\rm th}-{\rm
BR}{(\Sigma^{0}\to ZZ)}^{\rm obs}\right)^{2}}{\sigma_{{\rm BR}({\Sigma^{0}\to
ZZ})}^{2}}+\frac{\left(\delta_{\gamma\gamma}^{\rm
th}-\delta_{\gamma\gamma}^{\rm
obs}\right)^{2}}{\sigma_{\delta_{\gamma\gamma}}^{2}}+\frac{(m_{\Sigma}^{\rm
th}-m_{\Sigma}^{\rm obs})^{2}}{\sigma_{m_{\Sigma}}^{2}},$ (16)

where $\sigma$ indicates the uncertainty, and superscripts $th$ and $obs$
represent theory and observed quantities respectively. We note that, in this
analysis, the collider measurements of $m_{\Sigma}$, ${\rm BR}({\Sigma^{0}\to
ZZ})$ and $\delta_{\gamma\gamma}$ are hypothetical measurements. We then
demand $\chi^{2}<2.71$ to obtain 2$\sigma$ favoured region.

In Fig. 3 we present a plot in analogy to Fig. 1 (c). We take the uncertainty
on $\delta_{\gamma\gamma}$ from HL-LHC [76],

$\displaystyle\delta_{\gamma\gamma}=-0.132\pm 0.04,$ (17)

while the triplet mass and ${\rm BR}{(\Sigma^{0}\to ZZ)}$ measurements are
kept the same as in BMA. We denote this benchmark without (with)
$\text{BR}(\Sigma^{0}\to ZZ)$ measurement as BMB (BMB′).

Compared to BMA, the worse accuracy reflects to much wider $2\sigma$ region
(blue) in Fig. 3, making such measurement alone indecisive for character of
the phase transition, as blue region spans from two-step to one-step to
crossover. When accompanied with measurement of
$\mathrm{BR}(\Sigma^{0}\rightarrow ZZ)$ (BMB′) corresponding $2\sigma$ region
overlaps with two-step region and LISA sensitivity, but note that there is
another possible, disconnected region in crossover regime, implying that
neither BMB′ alone can be decisive to determine character of the EWPT.

While $\delta_{\gamma\gamma}$ and $\mathrm{BR}(\Sigma^{0}\rightarrow ZZ)$ are
independent on $b_{4}$ at leading order, thermodynamic phase structure depends
on this parameter. To illustrate this, in Fig. 4 we show a similar plot as
Fig. 1 (a) but fixing $b_{4}=0.25$. Comparing with Fig. 1 (a), one observes
that two-step region is significantly narrower for a smaller value of $b_{4}$;
requiring a degree of fine-tuning for a model to live in this region. Similar
observation has been made e.g. in [79] in case of the real singlet extension
model. We note that Fig. 4 reproduces Fig. 1 in [24], except that here we are
unable to delineate the boundary between the first-order and the crossover
regions in two step transitions.

Figure 5:  Same as Fig. 1 but in ($b_{4},a_{2}$)-plane with fixed
$m_{\Sigma}=250$ GeV, instead of $(m_{\Sigma},a_{2})$-plane. Purple regions in
bottom row correspond to BMC, and black region in (c) illustrates DECIGO
sensitivity.

For further illustration, we present yet another choice for a benchmark point
of the main body. We define a benchmark set BMC:

$\displaystyle\text{BR}(\Sigma^{0}\to ZZ)=0.0012\pm 0.00024,$ (18)

with the value for $\delta_{\gamma\gamma}$ taken to be the same as in BMA. The
central value of the $\text{BR}(\Sigma^{0}\to ZZ)$ corresponds to
$m_{\Sigma}=250$ GeV and $a_{2}=2.24$, and the uncertainty is taken to be
$20\%$ of the central value. In Fig. 5 we present an analysis similar to Fig.
1, with following differences: instead of $(m_{\Sigma},a_{2})$-plane we
illustrate ($b_{4},a_{2}$)-plane, with fixed $m_{\Sigma}=250$ GeV. Note that
for illustration we treat here $m_{\Sigma}$ as known input. From Fig. 5 (a) we
learn how thermodynamic phase structure behaves as function of $b_{4}$, and
from Fig. 5 (b) observe, similar to Fig. 1 (b), how sensitively thermal
parameters depend on $a_{2}$. The 2$\sigma$ favoured region for BMC is
depicted by purple colour in Fig. 5 (c) and (d), and instead of LISA
sensitivity, black region in Fig. 5 (c) illustrates the DECIGO sensitivity.
The lesson we illustrate is, that if 1) the triplet mass is known from
collider experiments, 2) DECIGO discovers GW background, and 3) we assume that
such a background is a remnant of a strong first order phase transition in the
triplet model, we get relatively stringent constraint on $a_{2}$ (as well as
lower bound – and relatively less tight constraint – on $b_{4}$) that
translates roughly to collider measurements of BMB. On the other hand, a null
result of DECIGO is still viable together with measurement BMB; this puts on
upper bound on $b_{4}$ (as function of $a_{2}$).