More Doublets and Singlets

Abstract

I give an overview on models that extend the Standard Model scalar sector by additional gauge singlets or multiplets. I discuss current constraints on such models, as well as possible signatures and discovery prospects at current and future colliders.

Full text

RBI-ThPhys-2022-16
MORE DOUBLETS AND SINGLETS
T. ROBENS
Ruder Boskovic Institute, Bijenicka cesta 54, 10000 Zagreb, Croatia
I give an overview on models that extend the Standard Model scalar sector by additional
gauge singlets or multiplets. I discuss current constraints on such models, as well as possible
signatures and discovery prospects at current and future colliders.
1 Introduction
After the discovery of a particle that complies with the properties of the Standard Model (SM)
Higgs boson, particle physics has entered an exciting era. One important question is to inves-
tigate whether the scalar sector realized in nature corresponds indeed to the SM, or whether it
is enhanced by additional scalars, which can be singlets, doublets, or any other multiplets un-
der the electroweak gauge group. Such extensions then typically come with additional particle
content, i.e. additional neutral or charged scalars which can also differ by their CP properties.
Naturally, such models then need to obey current constraints from both theory and experiment,
as e.g. stabilization conditions for the vacuum, positivity, perturbativity, and constraints from
direct searches, signal strength, or electroweak precision observables. In turn, in the allowed
regions novel signatures can appear that could be of interest for current and future collider
searches.
In the following, I discuss several such extensions. For various of these, I present work done
by myself and collaborators; for these, we made use of private codes as well as publicly available
tools such as HiggsBounds 1, HiggsSignals 2, 2HDMC 3, micrOMEGAs 4,5, and ScannerS 6,7,8,9.
Predictions for production cross sections shown here have been obtained using Madgraph5 10.
2 Real singlet extension
We first turn to a simple example, where the SM scalar sector has been enhanced by a real
singlet field obeying a Z2symmetry11,12,13,14,15,16. The Z2symmetry is softly broken by a vacuum
expectation value (vev) of the singlet field, inducing mixing between the gauge-eigenstates which
introduces a mixing angle α. The model has in total 5 free parameters. Two of these are fixed
by the measurement of the 125GeV resonance mass and electroweak precision observables. We
then have
sin α, m2,tan β≡v
vs
(1)
as free parameters of the model, where v(vs) are the doublet and singlet vevs, respectively. We
concentrate on the case where m2≥125 GeV, where SM decoupling corresponds to sin α→0.
The model is subject to a number of theoretical and experimental constraints. Exemplary
limits are shown in figure 1, which correspond to an update to results presented in 17 , including
a comparison of the currently maximal available rate of H→h125h125 with the combination
arXiv:2205.06295v1 [hep-ph] 12 May 2022

limits from ATLAS 18 as well as novel results for b¯
bb¯
b19 and b¯
bγγ 20 final states. The most
constraining direct search bounds are in general dominated by searches for diboson final states
21,22,23,24. In some regions, the Run 1 Higgs combination 25 is also important. Especially 23,24
currently correspond to the best probes of the models parameter space
a.
0.1
0.2
0.3
0.4
0.5
200 300 400 500 600 700 800 900 1000
| sinα | (upper limit)
mH [GeV]
W boson mass
λ1 perturbativity
LHC SM Higgs searches
Higgs signal rates
300 400 500 600 700 800 900 1000
m2[GeV]
10−3
10−2
10−1
1
σ(pp →h2)×BR(h2→h1h1) [pb]
ATLAS comb.(1906.02025)
obs.(95% CL)
obs.(95% CL),bbbb(new)
obs.(95% CL),bbaa(new)
exp.(95% CL)
±1σ
±2σ
Maximal pp →h2→h1h1rate (13 TeV)
all constraints EW scale constraints
Figure 1 – Current constraints on the Higgs singlet extension with a Z2symmetry. Constraints have been updated
to reflect results prior to Moriond 2022. Left: Various constraints on the mixing angle as a function of the second
scalar mass, for fixed tan β= 0.1. Right: Current constraints for H→h125h125 searches. Comparison with
ATLAS Run I combination as well as novel results (see text for details).
3 Two Higgs Doublet Models
Two Higgs doublet models (2HDMs) constitute another example of new physics models intro-
ducing additional scalar states. A general discussion of such models is e.g. given in 26 and will
not be repeated here. In these models, the SM scalar sector is augmented by a second scalar
doublet which also acquires a vev; electroweak symmetry breaking then involves both fields.
Several structures of couplings in the Yukawa sector are possible and distinguish the different
models. The particle content in the scalar sector containa, besides the SM candidate, two ad-
ditional neutral scalars which differ in CP properties as well as a charged scalar, denoted by
h, H, A, H±, where one of the two CP-even neutral scalars h, H needs to be identified with
the 125 GeV resonance discovered at the LHC. Besides the masses of the scalar particles, the
scalar sector is also characterized by different mixing angles, which are typically parametrized
in terms of cos (β−α) and tan β.
Exemplarily, in figure 2 we show constraints on two different types of 2HDMs from various
searches, taken from 27, for the degenerate case where all heavy scalar masses are set equal. In
that work, the authors considered various searches at the 8 TeV and early 13 TeV LHC runs.
We refer to the original work for a complete list of all channels.
Another important result is that in 2HDMs, the parameter space for the mixing angle cos (β−α)
is stronlgy constrained from signal strength measurements, where the exact allowed region again
depends on the Yukawa structure of the model. In figure 3, we display the results as obtained by
the ATLAS collaboration in a fit combination using full Run 2 data 28. The actual constraints
differ for each model, but the absolute value of the mixing angle does not exceed 0.25.
4 Two real scalar extension at hadron colliders
We now turn to the two-real-singlet extension (TRSM) 29 , a model that extends the SM scalar
sector by two real scalars that are singlets under the SM gauge group. This model allows for
aWe include searches currently available via HiggsBounds.

Figure 2 – Constraints on various types of 2HDM models, from various search channels at the LHC, status 2020.
See text for reference.
Figure 3 – Signal strength fit constraints and allowed regions for cos(β−α) for various types of 2HDMs; prelim-
inary ATLAS combination results. See text for reference.
interesting new final states, as e.g. scalar-to-scalar decays. Single scalar production following by
asymmetric scalar decays or symmetric decays, where all scalar masses differ from the SM-like
scalar mass at 125 GeV, have not yet been fully explored. Simple counting reveals that for such
scenarios at least three physical scalar states need to be present in the model, out of which one

takes the role of the state already discovered by the LHC experiments.
The potential in the scalar sector is given by
V(Φ, S, X) = µ2
ΦΦ†Φ + λΦ(Φ†Φ)2+µ2
SS2+λSS4+µ2
XX2+λXX2
+λΦSΦ†ΦS2+λΦXΦ†ΦX2+λSX S2X2.(2)
where Φ denotes the doublet also present in the SM potential and X, S are the two additional
real scalars. The model obeys an additional Z2⊗Z20symmetry Z2S:S→ −S , Z2X:X→ −X,
while all other fields transform evenly under the respective Z2symmetry. All three scalars acquire
a vev and mix. This leads to three physical states with all possible scalar-scalar interactions.
In the following, we will use the convention that
M1≤M2≤M3(3)
and denote the corresponding physical mass eigenstates by hi. 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 κi, such that ghiAB
i=κighiAB,SM
ifor any
hiAB coupling, where A, B denote SM particles. Orthogonality of the mixing matrix implies
Piκ2
i= 1.
The model allows for interesting scalar-to-scalar decays
pp →ha(+X)→hbhb(+X),(4)
pp →h3(+X)→h1h2(+X),(5)
where a, b ∈ {1,2,3}. In 29 , six benchmark planes (BPs) were suggested for the decay chains
above: three involving asymmetric decays (5) and three symmetric decays (4), where in the
latter case we concentrated on scenarios where Ma,b 6= 125 GeV. Depending on the specific
benchmark point, production rates can reach O(60 pb) for the above production modes. The
benchmark scenario which considers h3→h2h2, where M2>250 GeV, can furthermore lead
to interesting h125h125h125 and h125 h125h125 h125 final states. Largest rates for these processes,
including further decays to SM final states, are in the O(fb) range.
Figure 4 shows predicted production rates for several of these benchmark planes: BPs 1/3
for the asymmetric decay, where h125 ≡h3/h1, as well as two symmetric scenarios BPs 4/ 5
where a non-SM like scalar decays into light scalars with M1≤125 GeV. Depending on the
benchmark point, dominant final states typically contain multiple b-jets b¯
bb¯
bb¯
b. While some
of these are already under investigation by the LHC experiments, I encourage the collaborations
to explore all possible channels and collider signatures. These can lead to sizeable rates at the
13 TeV LHC, while still complying with all current theoretical and experimental bounds.
5 Inert Doublet Model
We now turn to a new physics model that contains a dark matter candidate. The Inert Doublet
Model is a two Higgs doublet model with an exact discrete Z2symmetry 30,31,32. The model
contains four additional scalar states H, A, H±, and has in total 7 free parameters prior to
electroweak symmetry breaking: v, mh, mH, mA, mH±
| {z }
second doublet
, λ2, λ345 ≡λ3+λ4+λ5. Here, the
λis denote standard couplings appearing in the 2HDM potential. Two parameters (mhand v)
are fixed by current measurements. We have provided a detailed investigation of the models
parameter space in33,15,34,35,36. One important observation is the existance of a relatively strong
degeneracy between the additional masses of the second doublet, as well as a minimal mass scale
for the dark matter candidate Hresulting from a combination of relic density and signal strength
measurement constraints (see 33,35 for a detailed discussion). These features are displayed in
figure 5.

Figure 4 – Benchmark planes for the TRSM, for asymmetric (top) and symmetric (bottom) final states, as defined
in the text. Top left: BP1, where h3≡h125 ; displayed is the branching ratio h125 →h1h2in the two-dimensional
mass plane. Top right: BP3; Bottom left: BP5; Bottom right: BP6. For the last three, production cross sections
at a 13 TeV pp collider are shown. Exclusion bounds on these planes from various constraints are also given.
0
50
100
150
200
250
300
350
400
0 50 100 150 200 250 300 350
MA-MH
MH+-MH
benchmark points
Z mass
W mass
-0.1
-0.05
0
0.05
0.1
30 40 50 60 70 80 90 100
λ345
MH [GeV]
excluded from XENON
excluded from relic density
surviving points
exact relic density
Figure 5 – Left: allowed range in the (MH±−MH, MA−MH) plane, including benchmark points discussed in
the text. The lines discriminate on- off-shellness of the decay products of H±and Arespectively. Right: allowed
and excluded regions in the MH, λ345 parameter space. Red points are still allowed after all constraints, golden
render exact relic density.
In 35, we have performed a sensitivity comparison for selected benchmark points 34,37,35 ,
relying on a simple counting criteria: a benchmark point is considered reachable if at least 1000
signal events are produced using nominal luminosity of the respective collider. The corresponding
results in terms of mass scales of the pair-produced particles are displayed in table 1, with
accompagnying plots in figure 6, taken from 35 . We here have used Madgraph5 10 with a UFO
input file from 38 for cross-section predictions. In contrast, results for CLIC have been obtained
using a detailed study of signal and background 37,39 .

collider all others AA AA +VBF
HE-LHC 2 TeV 400-1400 GeV 800-1400 GeV
FCC-hh 2 TeV 600-2000 GeV 1600-2000 GeV
CLIC, 3 TeV 2 TeV - 300-600 GeV
µµ, 10 TeV 2 TeV - 400-1400 GeV
µµ, 30 TeV 2 TeV - 1800-2000 GeV
Table 1: Sensitivity of different collider options, using the sensitivity criterium of 1000 generated events in the
specific channel. x−ydenotes minimal/ maximal mass scales that are reachable.
0 500 1000 1500 2000
mass sum [GeV]
3−
10
2−
10
1−
10
1
10
2
10
3
10
cross section [fb]
= 13 TeVs
+
H H
+
A H
H A
-
H
+
H
-
H H
-
A H
A A
0 500 1000 1500 2000
mass sum [GeV]
2−
10
1−
10
1
10
2
10
3
10
4
10
cross section [fb]
+
H H 13 TeV
H A 27 TeV
-
H H 100 TeV
0 500 1000 1500 2000
mass sum [GeV]
3−
10
2−
10
1−
10
1
10
2
10
3
10
4
10
cross section [fb]
A A 13 TeV
-
H
+
H 27 TeV
100 TeV
0 500 1000 1500 2000
mass sum [GeV]
3−
10
2−
10
1−
10
1
10
cross section [fb]
A A 10 TeV
-
H
+
H 30 TeV
Figure 6 – Predictions for production cross sections for various processes and collider options. Top left: Predictions
for various pair-production cross sections for a pp collider at 13 TeV, as a function of the mass sum of the produced
particles. Top right: Same for various center-of-mass energies. Bottom left: VBF-type production of AA and
H+H−at various center-of-mass energies for pp colliders. Bottom right: Same for µ+µ−colliders. The lines
correspond to the cross-sections required to produce at least 1000 events using the respective design luminosity.
6 Conclusions
I have presented results for various models that extend the scalar sector of the SM by additional
gauge singlets or doublets. These models are subject to additional theoretical and experimental
constraints limiting the parameter space. On the other hand, all of these allow for additional
signatures that have not fully been explored yet at current colliders, and therefore provide
encouragement for the experimental collaborations to investigate these at future hadron or
lepton machines. Despite the discovery of a particle that complies with the properties of a SM
Higgs, the structure of the electroweak potential is not completely understood yet. The models
discussed here will allow to further investigate the electroweak structure realized in nature.

Acknowledgements
The author wants to thank the organizers for the invitation, a very pleasant atmosphere, as well
as financial support. I furthermore thank all collaborators who helped to achieve the results
presented here.
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