RBI-ThPhys-2022-21

# TRSM Benchmark Planes - Snowmass White Paper

Tania Robens<EMAIL_ADDRESS>Ruder Boskovic Institute, Bijenicka cesta 54,
10000 Zagreb, Croatia

######  Abstract

In this whitepaper, I briefly review the Benchmark Planes in the Two-Real-
Singlet Model (TRSM), a model that enhances the Standard Model (SM) scalar
sector by two real singlets that obey a
$\mathbb{Z}_{2}\,\otimes\,\mathbb{Z}_{2}^{\prime}$ symmetry. In this model,
all fields acquire a vacuum expectation value, such that the model contains in
total 3 CP-even neutral scalars that can interact with each other. All
interactions with SM-like particles are inherited from the SM-like doublet via
mixing. I remind the readers of the previously proposed benchmark planes, and
briefly discuss possible production at future Higgs factories.

##  I Introduction and Model

The model discussed here has been proposed in Robens:2019kga , and I refer the
reader to that reference for a detailed discussion of model setup and
constraints. I just briefly repeat the generic features for completeness.

The potential in the scalar sector is given by

$\displaystyle V\left(\Phi,\,S,\,X\right)$
$\displaystyle=\mu_{\Phi}^{2}\Phi^{\dagger}\Phi+\lambda_{\Phi}{(\Phi^{\dagger}\Phi)}^{2}+\mu_{S}^{2}S^{2}+\lambda_{S}S^{4}+\mu_{X}^{2}X^{2}+\lambda_{X}X^{2}$
(1) $\displaystyle\quad+\lambda_{\Phi S}\Phi^{\dagger}\Phi S^{2}+\lambda_{\Phi
X}\Phi^{\dagger}\Phi X^{2}+\lambda_{SX}S^{2}X^{2}\,.$

Here, $\Phi$ denotes the SM-like doublet, while $X,\,S$ are two additional
real scalar fields. The model obeys an additional
${\mathbb{Z}_{2}}\,\otimes\,{\mathbb{Z}_{2}}^{\prime}$ symmetry
${\mathbb{Z}_{2}}^{S}:\,S\to-S\,,{\mathbb{Z}_{2}}^{X}:\,X\to-X,$ while all
other fields transform evenly under the respective ${\mathbb{Z}_{2}}$
symmetry. All three scalars acquire a vacuum expectation value (vev) and
therefore mix. This leads to three physical states with all possible scalar-
scalar interactions.

Among the important constraints are e.g. the Higgs signal strength
measurements by the LHC experiments, perturbative unitarity as well as the
requirement for the potential to be bounded from below, and current collider
searches. Results have been obtained using the ScannerS Coimbra:2013qq ;
Ferreira:2014dya ; Costa:2015llh ; Muhlleitner:2016mzt ; Muhlleitner:2020wwk
framework. Experimental results from past and current collider experiments
have been implemented using the publicly available tools HiggsBounds
Bechtle:2008jh ; Bechtle:2011sb ; Bechtle:2013gu ; Bechtle:2013wla ;
Bechtle:2015pma ; Bechtle:2020pkv  and HiggsSignals Stal:2013hwa ;
Bechtle:2013xfa ; Bechtle:2014ewa ; Bechtle:2020uwn .

In the following, we will use the convention that

$M_{1}\,\leq\,M_{2}\,\leq\,M_{3}$ (2)

and denote the corresponding physical mass eigenstates by $h_{i}$. Gauge and
mass eigenstates are related via a mixing matrix. Interactions with SM
particles are then inherited from the scalar excitation of the doublet via
rescaling factors $\kappa_{i}$, such that
$g_{i}^{h_{i}AB}\,=\,\kappa_{i}\,g_{i}^{h_{i}AB,\text{SM}}$ for any $h_{i}AB$
coupling, where $A,\,B$ denote SM particles. Orthogonality of the mixing
matrix implies $\sum_{i}\kappa_{i}^{2}\,=\,1$. Furthermore, signal strength
measurements require $|\kappa_{125}|\gtrsim\,0.96$ Robens:2019kga  for the SM-
like scalar $h_{125}$, which can be $h_{1},\,h_{2}$ or $h_{3}$ depending on
the specific parameter choice.

For a certain production process (e.g. gluon gluon fusion) the cross section,
$\sigma$, for $h_{a}$ with mass $M_{a}$ can be obtained from the corresponding
SM Higgs production cross section, $\sigma_{\text{SM}}$, by simply rescaling

$\sigma(M_{a})=\kappa_{a}^{2}\cdot\sigma_{\text{SM}}(M_{a})\,.$ (3)

Since $\kappa_{a}$ rescales all Higgs couplings to SM particles, Eq. 3 is
exact up to genuine electroweak corrections involving Higgs self-interactions,
and in particular holds to all orders in QCD.

The scaling factor $\kappa_{a}$ plays the same role in universally rescaling
the partial widths of $h_{a}$ decays into SM particles, leading to

$\Gamma(h_{a}\to\text{SM};M_{a})=\kappa_{a}^{2}\cdot\Gamma_{\text{tot}}(h_{\text{SM}};M_{a}),$
(4)

where $\Gamma(h_{a}\to\text{SM};M_{a})$ denotes the sum of all partial widths
of $h_{a}$ into SM particle final states. In addition, the branching ratios
(BRs) of $h_{a}$ decays to other scalar bosons, $h_{a}\to h_{b}h_{c}$, are
given by:

$\text{BR}(h_{a}\to h_{b}h_{c})=\frac{\Gamma_{a\to
bc}}{\kappa_{a}^{2}~{}\Gamma_{\text{tot}}(h_{\text{SM}})+\sum_{xy}\Gamma_{a\to
xy}}\,.$ (5)

where the denominator now denotes the total width of the scalar $h_{a}$. In
the absence of BSM decay modes — which is always the case for the lightest
Higgs bosons $h_{1}$ — $h_{a}$ has BRs identical to a SM-like Higgs boson of
the same mass.

##  II Benchmark planes

In Robens:2019kga , several benchmark planes were proposed which were meant to
capture mainly features that by the time of that publication were not yet
adressed by searches at the LHC:

  * •

asymmetric production and decay, in the form of

$p,\,p\,\rightarrow\,h_{3}\,\rightarrow\,h_{1}\,h_{2},$

where, depending on the kinematics, $h_{2}\,\rightarrow\,h_{1}\,h_{1}$ decays
are also possible;

  * •

symmetric decays in the form of

$p\,p\,\rightarrow\,h_{i}\,\rightarrow\,h_{j}\,h_{j},$

where none of the scalars corresponds to the 125 GeV resonance. Note that this
in principle allows for further decays $h_{j}\,\rightarrow\,h_{k}\,h_{k}$,
again depending on the specific benchmark plane kinematics.

We list the definition of these benchmark planes in tables 1 and 2,
respectively.

benchmark scenario |  $h_{125}$ candidate |  target signature |  possible successive decays
---|---|---|---
BP1 |  $h_{3}$ |  $h_{125}\to h_{1}h_{2}$ |  $h_{2}\to h_{1}h_{1}$ if $M_{2}>2M_{1}$
BP2 |  $h_{2}$ |  $h_{3}\to h_{1}h_{125}$ |  -
BP3 |  $h_{1}$ |  $h_{3}\to h_{125}h_{2}$ |  $h_{2}\to h_{125}h_{125}$ if $M_{2}>$250\text{\,}\mathrm{GeV}$$
BP4 |  $h_{3}$ |  $h_{2}\to h_{1}h_{1}$ |  -
BP5 |  $h_{2}$ |  $h_{3}\to h_{1}h_{1}$ |  -
BP6 |  $h_{1}$ |  $h_{3}\to h_{2}h_{2}$ |  $h_{2}\to h_{125}h_{125}$ if $M_{2}>$250\text{\,}\mathrm{GeV}$$
Table 1: Overview of the benchmark scenarios: The second column denotes the Higgs mass eigenstate that we identify with the observed Higgs boson, $h_{125}$, the third column names the targeted decay mode of the resonantly produced Higgs state, and the fourth column lists possible relevant successive decays of the resulting Higgs states. Parameter | Benchmark scenario
---|---
|  BP1 |  BP2 |  BP3 |  BP4 |  BP5 |  BP6
$M_{1}~{}[$\text{\,}\mathrm{GeV}$]$ |  $[1,62]$ |  $[1,124]$ |  $125.09$ |  $[1,62]$ |  $[1,124]$ |  $125.09$
$M_{2}~{}[$\text{\,}\mathrm{GeV}$]$ |  $[1,124]$ |  $125.09$ |  $[126,500]$ |  $[1,124]$ |  $125.09$ |  $[126,500]$
$M_{3}~{}[$\text{\,}\mathrm{GeV}$]$ |  $125.09$ |  $[126,500]$ |  $[255,650]$ |  $125.09$ |  $[126,500]$ |  $[255,1000]$
$\theta_{hs}$ |  $1.435$ |  $1.352$ |  $-0.129$ |  $-1.284$ |  $-1.498$ |  $0.207$
$\theta_{hx}$ |  $-0.908$ |  $1.175$ |  $0.226$ |  $1.309$ |  $0.251$ |  $0.146$
$\theta_{sx}$ |  $-1.456$ |  $-0.407$ |  $-0.899$ |  $-1.519$ |  $0.271$ |  $0.782$
$v_{s}~{}[$\text{\,}\mathrm{GeV}$]$ |  $630$ |  $120$ |  $140$ |  $990$ |  $50$ |  $220$
$v_{x}~{}[$\text{\,}\mathrm{GeV}$]$ |  $700$ |  $890$ |  $100$ |  $310$ |  $720$ |  $150$
$\kappa_{1}$ |  $0.083$ |  $0.084$ |  $0.966$ |  $0.073$ |  $0.070$ |  $0.968$
$\kappa_{2}$ |  $0.007$ |  $0.976$ |  $0.094$ |  $0.223$ |  $-0.966$ |  $0.045$
$\kappa_{3}$ |  $-0.997$ |  $-0.203$ |  $0.239$ |  $0.972$ |  $-0.250$ |  $0.246$
Table 2: Input parameter values and coupling scale factors, $\kappa_{a}$
($a=1,2,3$), for the six defined benchmark scenarios. The doublet vev is set
to $v=$246\text{\,}\mathrm{GeV}$$ for all scenarios.

For this whitepaper, I rescanned all benchmark planes with the newest
HiggsBounds and HiggsSignals versions: HiggsBounds-5.10.2 and
HiggsSignals-2.6.2. For nearly all parameter points, these new versions did
not introduce additional constraints on the parameter space, and I therefore
show the benchmark planes from the original publication. One exception is BP5
which has a slightly more constrained parameter space taking additional
searches into account. I also comment on a possible recast on this plane and
give a list of current experimental searches partially relying on our model.

###  II.1 Asymmetric decays

In this subsection, I discuss the asymmetric decay modes
$h_{3}\,\rightarrow\,h_{1}\,h_{2}$, where successively one of the three
scalars is identified with the 125 GeV resonance. I display the corresponding
benchmark planes in figure 1.

Benchmark Plane 1

Benchmark Plane 2

Benchmark Plane 3

Figure 1:  Benchmark planes for asymmetric production and decay,
$p\,p\,\rightarrow\,h_{3}\,\rightarrow\,h_{1}\,h_{2}$, for various assigments
of the 125 GeV resonance. Top row: BP1, where $h_{3}\,\equiv\,h_{125}$.
Production cross sections are close to the SM production here, of around
$\sim\,4\,{\rm pb}$ at 13 TeV. Shown is the branching ratio to $h_{1}\,h_{2}$
in the two-dimensional mass plane. Middle and bottom rows: BPs 2 and 3, where
$h_{2,1}\,\equiv\,h_{125}$, respectively. Left: Production cross sections at a
13 TeV LHC. Right:  Branching ratios of the $h_{1}\,h_{2}$ state as a function
of the free light scalar mass. Partially taken from Robens:2019kga .

Depending on the benchmark plane, maximal production cross sections are given
by $\sim\,3-4\,{\rm pb},\,\sim\,0.6\,{\rm pb},$ and $0.3\,{\rm pb}$ for
$h_{1}\,h_{2}$ production for BPs 1/2/3, respectively. In BP3, the
$h_{1}h_{1}h_{1}$ final state reaches cross sections up to $\sim\,140\,{\rm
fb}$. Note that as soon as the kinematic threshold for
$h_{2}\,\rightarrow\,h_{125}h_{125}$ is reached, in fact decays from that
state become dominant.

###  II.2 Symmetric decays

Symmetric decays are given by BPs 4/5/6, with again a differing assignment for
$h_{3/2/1}\,\equiv\,h_{125}$, respectively. The corresponding production and
decay modes are displayed in figure 2.

Benchmark Plane 4

Benchmark Plane 5 [color coding:
$\log_{10}\left(\sigma_{h_{3}\,\rightarrow\,h_{1}\,h_{1}}/[{\rm pb}]\right)$]

Benchmark Plane 6

Figure 2:  Benchmark planes for symmetric production and decay,
$p\,p\,\rightarrow\,h_{i}\,\rightarrow\,h_{j}\,h_{j}$, for various assigments
of the 125 GeV resonance. Top /middle/ bottom rows: BPs 4/5/6, where
$h_{3/2/1}\,\equiv\,h_{125}$. Left: Production cross sections at a 13 TeV LHC.
Right:  Branching ratios of the $h_{j}\,h_{j}$ state as a function of the
lighter free scalar mass. Branching ratios for BP4 and 5 are identical,
therefore only one plot is displayed here. Partially taken from Robens:2019kga
.

Depending on the benchmark plane, pair-production cross sections can reach up
to 60/ 2.5/ 0.5 pb for BPs 4/5/6, respectively. For the latter the
$h_{125}h_{125}h_{125}h_{125}$ final state can reach rates up to 14 fb. Also
note that the allowed parameter space in BP5 has slightly shrunk, mainly due
to the implementation of an additional search ATLAS:2018rnh  into HiggsBounds
after the performance of the original scan. For BP6, 6 particle final states
as e.g. $W^{+}W^{-}b\bar{b}b\bar{b}$ can reach branching ratios up to
$\sim\,10\%$, depending on $M_{2}$.

##  III Further investigation of this model

After the original appearance of the paper proposing the TRSM, several
theoretical and experimental works have been performed which at least
partially build on the benchmark planes proposed here. We briefly list some of
these here.

###  III.1 Investigation of the $h_{125}h_{125}h_{125}$ final state

In BP3, for $M_{2}\,\rightarrow\,250\,{\rm GeV}$, the decay
$h_{2}\,\rightarrow\,h_{1}\,h_{1}$ becomes dominant, leading to a
$h_{125}h_{125}h_{125}$ final state. For subsequent decays into $b\,\bar{b}$,
this BP has been investigated in Papaefstathiou:2020lyp . We found that,
depending on the parameter point and integrated luminosity, significances
between 3 and $\sim\,10$ can be achieved. We display the results in table 3

Label | $(M_{2},M_{3})$ | $\varepsilon_{\rm Sig.}$ | $\rm{S}\bigl{|}_{300\rm{fb}^{-1}}$ | $\varepsilon_{\rm Bkg.}$ | $\rm{B}\bigl{|}_{300\rm{fb}^{-1}}$ | $\text{sig}|_{300\rm{fb}^{-1}}$ | $\text{sig}|_{3000\rm{fb}^{-1}}$
---|---|---|---|---|---|---|---
| [GeV] |  |  |  |  | (syst.) | (syst.)
A | $(255,504)$ | $0.025$ | $14.12$ | $8.50\times 10^{-4}$ | $19.16$ | $2.92~{}(2.63)$ | $9.23~{}(5.07)$
B | $(263,455)$ | $0.019$ | $17.03$ | $3.60\times 10^{-5}$ | ${8.12}$ | $4.78~{}(4.50)$ | $15.10~{}(10.14)$
C | $(287,502)$ | $0.030$ | $20.71$ | $9.13\times 10^{-5}$ | $20.60$ | $4.01~{}(3.56)$ | $12.68~{}(6.67)$
D | $(290,454)$ | $0.044$ | $37.32$ | $1.96\times 10^{-4}$ | $44.19$ | $5.02~{}(4.03)$ | $15.86~{}(6.25)$
E | $(320,503)$ | $0.051$ | ${31.74}$ | $2.73\times 10^{-4}$ | $61.55$ | $3.76~{}({2.87})$ | $11.88~{}(4.18)$
F | $(264,504)$ | $0.028$ | $18.18$ | $9.13\times 10^{-5}$ | $20.60$ | $3.56~{}(3.18)$ | $11.27~{}(5.98)$
G | $(280,455)$ | $0.044$ | $38.70$ | $1.96\times 10^{-4}$ | $44.19$ | $5.18~{}(4.16)$ | $16.39~{}(6.45)$
H | $(300,475)$ | $0.054$ | $41.27$ | $2.95\times 10^{-4}$ | $66.46$ | $4.64~{}(3.47)$ | $14.68~{}({4.94})$
I | $(310,500)$ | $0.063$ | $41.43$ | $3.97\times 10^{-4}$ | $89.59$ | $4.09~{}(2.88)$ | ${12.94~{}(3.87)}$
J | $(280,500)$ | $0.029$ | $20.67$ | $9.14\times 10^{-5}$ | $20.60$ | $4.00~{}(3.56)$ | $12.65~{}(6.66)$
Table 3:  The resulting selection efficiencies, $\varepsilon_{\rm Sig.}$ and
$\varepsilon_{\rm Bkg.}$, number of events, $S$ and $B$ for the signal and
background, respectively, and statistical significances. A $b$-tagging
efficiency of $0.7$ has been assumed. The number of signal and background
events are provided at an integrated luminosity of $300~{}\rm{fb}^{-1}$.
Results for $3000~{}\rm{fb}^{-1}$ are obtained via simple extrapolation. The
significance is given at both values of the integrated luminosity excluding
(including) systematic errors in the background. Taken from
Papaefstathiou:2020lyp .

Note we also compared how different channels, e.g. direct decays of the
heavier scalars into $VV$ or $h_{125}h_{125}$ final states, would perform at a
HL-LHC. The results are displayed in figure 3.

Figure 3:  The expected exclusion region for the full integrated luminosity of
the HL-LHC, $3000~{}\rm{fb}^{-1}$, through final states other than
$pp\rightarrow h_{1}h_{1}h_{1}$ as explained in the main text. Points with
green circles are expected to be excluded by $ZZ$ final states, with red
circles by $h_{1}h_{1}$ and with blue circles by $W^{+}W^{-}$. The
$W^{+}W^{-}$ analysis excludes only very few points on the parameter space and
therefore appears infrequently in the figure. The points A–I that we have
considered in our analysis of $pp\rightarrow h_{1}h_{1}h_{1}$ are shown in
black circles overlayed on top of the circles indicating the exclusion. The
two cut-out white regions near $M_{2}\sim 130$ GeV and $M_{2}\sim 170$ GeV
will remain viable at the end of the HL-LHC. Taken from Papaefstathiou:2020lyp
.

We note that all benchmark points that were investigated can additionally be
probed by other production and decay mechanisms. Note, however, that these
test different regions of the parameter space, as they depend on different
parameters in the potential. These searches can therefore be considered to be
complementary.

###  III.2 Recasting current LHC searches

It is also interesting to investigate whether current searches can be
reinterpreted and recasted in such a way that they allow to exclude regions in
the models parameter space that were not directly scrutinized in the
experimental search, or for which no interpretation was presented in the
original publication. In Barducci:2019xkq , the authors have reinterpreted a
CMS search for
$p\,p\,\rightarrow\,H\,\rightarrow\,h_{125}h_{125}\,\rightarrow\,4\,b$
CMS:2018qmt , which corresponds to di-Higgs production via a heavy resonance
and subsequent decays into $b\,\bar{b}$ final states, and extended the mass
ranges for the scalars in the decay chain. I have applied these results to the
TRSM, in particular to BP5. I display the corresponding results in figure
4111I thank the authors of Barducci:2019xkq  for providing us with the
corresponding exclusion limits.. We see that the sensitive region of parameter
space is significantly extended, and therefore, an actual experimental
analysis also in this parameter region is greatly encouraged.

Figure 4:  Reinterpretation of a $36\,{\rm fb}^{-1}$ CMS search for di-Higgs
production via a heavy resonance using the 4 b final state. The exclusion line
uses the results obtained in Barducci:2019xkq . Points to the right and above
the red contour are excluded. Taken from Robens:2022mvi .

###  III.3 Experimental searches with TRSM interpretations

At least one experimental search actually has made use of the predictions
obtained within the TRSM to interpret regions in parameter space that are
excluded: a CMS search for asymmetric production and subsequent decay into
$b\bar{b}b\bar{b}$ final states CMS:2022suh . For this, maximal production
cross sections were provided in the parameter space, allowing all additional
new physics parameter to float; the respective values have been tabulated in
reptr . Figure 5 shows the expected and observed limits in this search for the
TRSM and NMSSM Ellwanger:2022jtd .

Figure 5:  Expected (left) and observed (right) $95\%$ confidence limits for
the $p\,p\,\rightarrow\,h_{3}\,\rightarrow\,h_{2}\,h_{1}$ search, with
subsequent decays into $b\bar{b}b\bar{b}$. For both models, maximal mass
regions up to $m_{3}\,\sim\,1.4{\rm TeV},\;m_{2}\,\sim\,140\,{\rm GeV}$ can be
excluded. Figure taken from CMS:2022suh .

In addition, several searches also investigate decay chains that can in
principle also be realized within the TRSM, as e.g. other searches for the
same final states CMS:2021erw ; CMS:2021fjj ; CMS:2022qww  or
$b\,\bar{b}\mu^{+}\mu^{-}$ ATLAS:2021ypo ; ATLAS:2021hbr  final states.

##  IV Signatures at Higgs factories

The investigation of light scalars has recently gained again more interest,
after the recommendation of the European Strategy Report
EuropeanStrategyforParticlePhysicsPreparatoryGroup:2019qin ; European:2720129
to concentrate on $e^{+}e^{-}$ machines with $\sqrt{s}\,\sim\,240-250\,{\rm
GeV}$. A short review about the current state of the art for such searches and
models which allow for low scalars can e.g. be found in Robens:2022zgk . In
this model, the only feasible production is $Zh$ radiation of the lighter
scalar, with production cross sections given in figure 6. Cross sections have
been derived using Madgraph5 Alwall:2011uj .

Figure 6:  Leading order production cross sections for $Z\,h$ and
$h\,\nu_{\ell}\,\bar{\nu}_{\ell}$ production at an $e^{+}\,e^{-}$ collider
with a com energy of 240 GeV(left) and 250 GeV (right) using Madgraph5 for an
SM-like scalar h. Shown is also the contribution of $Z\,h$ to
$\nu_{\ell}\,\bar{\nu}_{\ell}\,h$ using a factorized approach for the Z decay.
Taken from Robens:2022zgk .

We can now investigate what would be production cross sections for scalar
particles with masses $\lesssim\,160\,{\rm GeV}$ at Higgs factories.

###  IV.1 Production of 125 GeV resonance and subsequent decays

We first turn to the easy case of the production of the 125 GeV resonance in
various benchmark scenarios. Of interest are cases where decays
$h_{125}\,\rightarrow\,h_{i}\,h_{j}$ are kinematically allowed. Note that our
benchmark points were not set up in particular for the scenario where
$i\,=\,j$, so for this rates might be relatively small by construction.

From table 2 we see that for all scenarios the rescaling for the 125 GeV
resonance is $\gtrsim\,0.966$, leading to production cross sections of about
$\sim\,0.2\,{\rm pb}$, close to the SM value. In general, due to constraints
from the invisible branching ratio ATLAS-CONF-2020-052  as well as signal
strength fits, the production cross section for $h_{i}\,h_{j}$ final states
has to be lower by at least an order of magnitude, leading to cross sections
$\mathcal{O}\left(10\,{\rm fb}\right)$. In fact, in the benchmark planes
presented here the largest rate for $Z\,h_{125}$ production and subsequent
scalar decays can be found in BP1, where the rates are given by multiplying
the BRs from figure 1 with the production of $Z\,h_{125}$, giving maximal
cross sections of around 18 fb.

###  IV.2 Additional scalar production

We now turn to the Higgs-Strahlung production of new physics scalars. This
process is in principle possible in all BPs discussed here. However, if we
require production rates of $Z\,h_{i}$ to be larger than $\sim\,10\,{\rm fb}$,
only BPs 4 and 5 render sufficiently large rates for the production of $h_{2}$
and $h_{3}$, respectively. Production rates are independent of the other
scalars, and we therefore depict them for both BPs in figure 7

Figure 7:  Production cross sections for $Zh_{2/3}$ in BPs 4 and 5,
respectively, at a 250 GeVHiggs factory.

BP4 is constructed in such a way that as soon as the corresponding parameter
space opens up, the $h_{1}\,h_{1}$ decay becomes dominant; final states are
therefore mainly $Z\,b\bar{b}b\bar{b}$ if $M_{2}\,\gtrsim\,2\,M_{1}$. Below
that threshold, dominant decays are into a $b\,\bar{b}$ pair, which means that
standard searches as e.g. presented in Drechsel:2018mgd ; Wang:2020lkq  should
be able to cover the parameter space.

Similarly, in BP5 the $h_{3}\,\rightarrow\,h_{1}\,h_{1}$ decay is also
favoured as soon as it is kinematically allowed. Therefore, in this parameter
space again $Zb\bar{b}b\bar{b}$ final states become dominant. Otherwise
$Z\,b\bar{b}$ and $ZW^{+}W^{-}$ final states prevail, with a cross over for
the respective final states at around $M_{3}\,\sim\,135\,{\rm GeV}$. Branching
ratios for these final states are in the $40-50\%$ regime.

##  V Summary

In this whitepaper, I gave a short summary of the status of collider
signatures and searches in the TRSM introduced in Robens:2019kga . I gave a
summary on current state of the art and investigation, including further
detailed collider studies, recasts, as well as current searches that use or
are motivated in this model. I also gave a brief overview on channels within
this model that might be testable at future $e^{+}e^{-}$ machines, with a
focus in Higgs factories with $\sqrt{s}\,\sim\,250\,{\rm GeV}$.

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