DESY-21-034
IFT-UAM/CSIC-21-018
Fate of electroweak symmetry in the early Universe:
Non-restoration and trapped vacua in the N2HDM
Thomas Biekötter
1∗
, Sven Heinemeyer
2,3,4†
, José Miguel No
2,5‡
,
María Olalla Olea
1§
and Georg Weiglein
1,6¶
1
Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany
2
Instituto de Física Teórica UAM-CSIC, Cantoblanco, 28049, Madrid, Spain
3
Campus of International Excellence UAM+CSIC, Cantoblanco, 28049, Madrid, Spain
4
Instituto de Física de Cantabria (CSIC-UC), 39005, Santander, Spain
5
Departamento de Física Teórica, Universidad Autónoma de Madrid (UAM),
Campus de Cantoblanco, 28049 Madrid, Spain
6
II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149,
D-22761 Hamburg, Germany
Abstract
Extensions of the Higgs sector of the Standard Model allow for a rich cosmological
history around the electroweak scale. We show that besides the possibility of strong first-
order phase transitions, which have been thoroughly studied in the literature, also other
important phenomena can occur, like the non-restoration of the electroweak symmetry
or the existence of vacua in which the Universe becomes trapped, preventing a transition
to the electroweak minimum. Focusing on the next-to-minimal two-Higgs-doublet model
(N2HDM) of type II and taking into account the existing theoretical and experimental
constraints, we identify the scenarios of electroweak symmetry non-restoration, vacuum
trapping and first-order phase transition in the thermal history of the Universe. We
analyze these phenomena and in particular their relation to each other, and discuss their
connection to the predicted phenomenology of the N2HDM at the LHC. Our analysis
demonstrates that the presence of a global electroweak minimum of the scalar potential
at zero temperature does not guarantee that the corresponding N2HDM parameter space
will be physically viable: the existence of a critical temperature at which the electroweak
phase becomes the deepest minimum is not sufficient for a transition to take place,
necessitating an analysis of the tunnelling probability to the electroweak minimum for a
reliable prediction of the thermal history of the Universe.
∗
thomas.biekoetter@desy.de
†
Sven.Heinemeyer@cern.ch
‡
josemiguel.no@uam.es
§
maria.olalla.olea.romacho@desy.de
¶
georg.weiglein@desy.de
arXiv:2103.12707v1 [hep-ph] 23 Mar 2021
Contents
1 Introduction 1
2 The next-to-minimal two Higgs doublet model 3
2.1 Model definition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Effective potential and renormalization . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Scale dependence and perturbativity . . . . . . . . . . . . . . . . . . . . . . . 7
3 Constraints on the N2HDM 8
3.1 Theoretical constraints: Vacuum stability and unitarity . . . . . . . . . . . . . 8
3.2 Flavor-physics observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Properties of the observed Higgs boson at 125 GeV . . . . . . . . . . . . . . . 9
3.4 Direct searches for additional Higgs bosons . . . . . . . . . . . . . . . . . . . . 10
3.5 Electroweak precision observables . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 The N2HDM at finite temperature 10
4.1 Finite-T effective potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 N2HDM thermal history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Symmetry non-restoration at high T 14
5.1 Analytical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 The EW phase transition and symmetry non-restoration . . . . . . . . . . . . 25
6 Trapped metastable singlet vacua 28
6.1 Case 1: Singlet admixture in H . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 Case 2: Singlet admixture in h
125
. . . . . . . . . . . . . . . . . . . . . . . . . 32
7 Conclusions 36
1 Introduction
The discovery of a Higgs boson with a mass of about 125 GeV at the Large Hadron Collider
(LHC) [1, 2] was a milestone in our understanding of the laws of nature. Within the current
experimental and theoretical uncertainties, the properties of the detected particle agree with
the predictions of the Standard Model (SM) [3–5]. However, they are also compatible with
a wide variety of extensions of the SM that are motivated in view of several shortcomings
of the SM which require new physics beyond the SM (BSM). Among others, the ingredients
of the SM are not sufficient to generate the observed matter-antimatter asymmetry of the
Universe [6–8], and the SM lacks a particle candidate to explain the observed cosmological
abundance of dark matter [9].
Extensions of the SM scalar sector, e.g. by gauge singlets or SU(2) doublets, provide scope
to address the above shortcomings. For instance, adding further Higgs doublets to the SM [10–
12] allows the generation of the observed matter-antimatter of the Universe via electroweak
1
(EW) baryogenesis [13–16] (see [17–19] for general reviews on EW baryogenesis). A necessary
ingredient in such a case is a (strongly) first order EW phase transition (FOEWPT) to provide
the required out-of-equilibrium conditions for baryogenesis in the early Universe [20]. Scenarios
featuring a FOEWPT have also re-gained attention in recent years since they could lead to
a stochastic gravitational wave background detectable with future space-based gravitational
wave interferometers [21, 22].
Higgs sector extensions like the two-Higgs-doublet model (2HDM) (see [23] for a review) or
the Next-to-2HDM (N2HDM) [24–27], which extends the 2HDM by a real scalar singlet field,
give rise to a rich collider phenomenology that has an important interplay with the physics of
the early Universe, see e.g. [28–32]. In this work we explore this interplay within the N2HDM,
focusing on type II. We study the thermal history, i.e. the evolution of the Higgs fields in the
early Universe, and demonstrate that over large parts of the parameter space of the N2HDM
the thermal history in the N2HDM differs very significantly from the commonly expected
scenario of EW symmetry breaking around an early Universe temperature T of O(100 GeV).
The two phenomena of EW symmetry non-restoration and “vacuum trapping” play a key role.
Concerning the feature of EW symmetry non-restoration, it is well-known that the EW
symmetry can be broken already at temperatures much larger than the EW scale, resulting in
EW symmetry non-restoration [33–37] up to these (possibly very high) temperatures, or even
in no restoration at all. In our study of the N2HDM, we find that the presence of this non-
restoration behavior is related to the effect of the resummation of infrared divergent modes in
the scalar potential within the high-temperature expansion. We provide compact analytical
expressions for the quantities that determine the EW symmetry restoration or non-restoration
behavior and supplement our analytical analysis with a detailed numerical investigation.
Vacuum trapping occurs if the Universe remains trapped in an EW symmetric phase down
to T → 0, even though a global EW symmetry breaking minimum of the potential exists at zero
temperature. We analyze this feature, which has also recently been discussed in the context of
the NMSSM [38], in detail for the N2HDM. Since parameter regions where vacuum trapping
occurs are unphysical, we demonstrate that the incorporation of this constraint, which up to
now has not been taken into account for the N2HDM, has very important consequences for
the allowed parameter space of the model.
As a further aspect of our investigation of the N2HDM thermal history we study FOEWPT
scenarios in the type II N2HDM and discuss the interplay between EW symmetry non-
restoration, the occurrence of a FOEWPT and vacuum trapping, as well as the connection of
such early Universe processes to possible signatures of the N2HDM at the LHC. Our results
illustrate the rich variety of thermal histories that can be realized in extended Higgs sectors, as
well as the phenomenological impact of these different histories. In particular, we demonstrate
that the results for the thermal history of the early Universe can rule out large parts of the
otherwise unconstrained N2HDM parameter space.
Our paper is organized as follows: In section 2 we introduce the N2HDM, paying particular
attention to the inclusion of radiative corrections to the scalar potential in section 2.2, and
to the renormalization group evolution of the scalar couplings in section 2.3. In section 3
we describe the theoretical and experimental constraints that we take into account for the
(zero-temperature) analysis of the N2HDM parameter space. Then, in section 4 we discuss
the inclusion of finite-temperature corrections to the scalar potential and provide a qualitative
2
discussion of the N2HDM thermal history, which we investigate in detail in the following
two sections. We study the phenomenon of EW symmetry non-restoration in section 5, both
via an analytical and a numerical approach, and analyze its interplay with the occurrence of
a FOEWPT in the N2HDM. In section 6 we investigate the possible occurrence of vacuum
trapping, together with the connection between the thermal history of the N2HDM and its
LHC phenomenology. We conclude in section 7.
2 The next-to-minimal two Higgs doublet model
In order to specify our notation and conventions, we briefly review below the extension of
the CP-conserving (real) 2HDM with a softly broken Z
2
symmetry by a real scalar field, the
so-called next-to-minimal 2HDM (N2HDM). Afterwards, we describe the precise form of the
one-loop zero-temperature effective potential and the renormalization group running of scalar
couplings.
2.1 Model definition and notation
The tree-level scalar potential of the two SU(2)
L
Higgs doublets Φ
1
and Φ
2
and the real singlet
field Φ
S
is given by [27]
V
tree
= m
2
11
|Φ
1
|
2
+ m
2
22
|Φ
2
|
2
− m
2
12
Φ
†
1
Φ
2
+ h.c.
+
λ
1
2
Φ
†
1
Φ
1
2
+
λ
2
2
Φ
†
2
Φ
2
2
+ λ
3
Φ
†
1
Φ
1
Φ
†
2
Φ
2
+ λ
4
Φ
†
1
Φ
2
Φ
†
2
Φ
1
+
λ
5
2
Φ
†
1
Φ
2
2
+ h.c.
+
1
2
m
2
S
Φ
2
S
+
λ
6
8
Φ
4
S
+
λ
7
2
Φ
†
1
Φ
1
Φ
2
S
+
λ
8
2
Φ
†
2
Φ
2
Φ
2
S
. (1)
The Z
2
symmetry of the 2HDM potential in (1), Φ
1
→ Φ
1
, Φ
2
→ −Φ
2
, whose extension to the
Yukawa sector prevents the occurrence of flavor-changing neutral currents (FCNCs) at lowest
order, is softly broken by the m
2
12
term. The third line of the tree-level potential (1) includes
the contribution of the singlet field. Here an extra discrete Z
0
2
symmetry is imposed,
Φ
1
→ Φ
1
, Φ
2
→ Φ
2
, Φ
S
→ −Φ
S
, (2)
which is not explicitly broken. The original motivation to introduce this symmetry for the
N2HDM was the fact that, when not spontaneously broken, it will give rise to a dark matter
(DM) candidate after EWSB (see e.g. [24–26, 39–43]). In this work we do not restrict to such
a scenario but study the case where Φ
S
does acquire a vacuum expectation value (vev). We
expand the fields around the EW minimum as follows,
Φ
1
=
φ
+
1
1
√
2
(v
1
+ ρ
1
+ iη
1
)
!
, Φ
2
=
φ
+
2
1
√
2
(v
2
+ ρ
2
+ iη
2
)
!
, Φ
S
= v
S
+ ρ
3
, (3)
where v
1
, v
2
and v
S
are the field vevs for the Higgs doublets and the singlet field, respectively,
at zero temperature. The doublet vevs v
1
and v
2
define the EW scale v =
q
v
2
1
+ v
2
2
≈ 246 GeV.
3
The minimization (or tadpole) equations for v
1
, v
2
and v
S
read
v
2
v
1
m
2
12
− m
2
11
=
1
2
v
2
1
λ
1
+ v
2
2
λ
345
+ v
2
S
λ
7
, (4)
v
1
v
2
m
2
12
− m
2
22
=
1
2
v
2
1
λ
345
+ v
2
2
λ
2
+ v
2
S
λ
8
, (5)
−m
2
S
=
1
2
v
2
1
λ
7
+ v
2
2
λ
8
+ v
2
S
λ
6
, (6)
with λ
345
≡ λ
3
+ λ
4
+ λ
5
.
Since the CP symmetry and the electric charge are conserved, the (squared-)mass matrix
for the fields φ
±
1,2
, η
1,2
, ρ
1,2,3
can be split into three blocks: a 3 ×3 matrix M
2
ρ
for the CP-even
states ρ
1,2,3
, a 2 × 2 matrix M
2
η
for the CP-odd states η
1,2
and a 2 × 2 matrix M
2
C
for the
charged scalars φ
±
1,2
. The matrices M
2
η
and M
2
C
correspond to the ones obtained in the 2HDM,
i.e.,
M
2
η,C
=
m
2
A,H
±
v
2
v
2
2
−v
1
v
2
−v
1
v
2
v
2
1
!
, (7)
with m
2
A
= v
2
(m
2
12
/(v
1
v
2
) − λ
5
) and m
2
H
±
= m
2
A
+ v
2
(λ
5
− λ
4
) /2. They can be diagonalized
via the rotation matrix
R
β
=
c
β
s
β
−s
β
c
β
!
, (8)
with the abbreviations s
x
≡ sin x and c
x
≡ cos x, and the angle β is defined by
t
β
≡ tan β ≡ v
2
/v
1
. After diagonalization we are left with the charged and neutral massless
Goldstone bosons G
±
and G
0
and the charged and neutral CP-odd physical mass eigenstates
H
±
and A with masses m
H
±
and m
A
.
The neutral CP-even sector of the N2HDM is modified with respect to that of the 2HDM
by the presence of the singlet ρ
3
. The mass matrix M
2
ρ
in the basis ρ
1,2,3
can be expressed as
M
2
ρ
=
v
2
λ
1
c
2
β
+ m
2
12
t
β
v
2
λ
345
c
β
s
β
− m
2
12
v v
S
λ
7
c
β
v
2
λ
345
c
β
s
β
− m
2
12
v
2
λ
2
s
2
β
+ m
2
12
/t
β
v v
S
λ
8
s
β
v v
S
λ
7
c
β
v v
S
λ
8
s
β
v
2
S
λ
6
. (9)
In the physical basis h
1,2,3
, the mass matrix M
2
ρ
is diagonal. The rotation matrix R between
the h
1,2,3
and ρ
1,2,3
bases satisfies R M
2
ρ
R
T
= diag
m
2
h
1
, m
2
h
2
, m
2
h
3
, with m
2
h
i
the squared
tree-level mass for h
i
. The matrix R can be parametrized in terms of the angles α
1,2,3
R =
c
α
1
c
α
2
s
α
1
c
α
2
s
α
2
−(c
α
1
s
α
2
s
α
3
+ s
α
1
c
α
3
) c
α
1
c
α
3
− s
α
1
s
α
2
s
α
3
c
α
2
s
α
3
−c
α
1
s
α
2
c
α
3
+ s
α
1
s
α
3
−(c
α
1
s
α
3
+ s
α
1
s
α
2
c
α
3
) c
α
2
c
α
3
. (10)
Without loss of generality, the angles α
1,2,3
are defined in the range −π/2 ≤ α
i
< π/2,
and we choose the convention that the mass eigenstates are ordered by ascending mass as
m
h
1
< m
h
2
< m
h
3
. The singlet composition of the mass eigenstates h
i
will be denoted by
Σ
h
i
= R
2
i3
.
4
