DESY-22-157 Electroweak Asymmetric Early Universe via a Scalar Condensate Jae Hyeok Chang1 2 Mar a Olalla Olea-Romacho3 4 and Erwin H. Tanin1

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DESY-22-157

Electroweak Asymmetric Early Universe via a Scalar Condensate

Jae Hyeok Chang,1, 2, ∗Mar´ıa Olalla Olea-Romacho,3, 4, †and Erwin H. Tanin1, ‡

1Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA

2Maryland Center for Fundamental Physics, Department of Physics,

University of Maryland, College Park, MD 20742, USA

3Laboratoire de Physique de l’ ´

Ecole Normale Sup´erieure, ENS, Universit´e PSL,

CNRS, Sorbonne Universit´e, Universit´e Paris Cit´e, F-75005 Paris, France

4Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

Finite temperature eﬀects in the Standard Model tend to restore the electroweak symmetry in

the early universe, but new ﬁelds coupled to the higgs ﬁeld may as well reverse this tendency,

leading to the so-called electroweak symmetry non-restoration (EW SNR) scenario. Previous works

on EW SNR often assume that the reversal is due to the thermal ﬂuctuations of new ﬁelds with

negative quartic couplings to the higgs, and they tend to ﬁnd that a large number of new ﬁelds

are required. We observe that EW SNR can be minimally realized if the ﬁeld(s) coupled to the

higgs ﬁeld develop(s) a stable condensate. We show that one complex scalar ﬁeld with a suﬃciently

large global-charge asymmetry can develop a condensate as an outcome of thermalization and keep

the electroweak symmetry broken up to temperatures well above the electroweak scale. In addition

to providing a minimal benchmark model, our work hints on a class of models involving scalar

condensates that yield electroweak symmetry non-restoration in the early universe.

I. INTRODUCTION

It is an empirical fact that we live at present in a vac-

uum that breaks the electroweak (EW) symmetry. At

high temperatures, the higgs ﬁeld acquires positive ther-

mal mass squared contributions from the fermions and

gauge bosons coupled to it. This thermal mass tends to

conﬁne the higgs ﬁeld at the origin, leading to the stan-

dard theoretical expectation that the EW symmetry is

restored in the early universe. The latter scenario is true

within the Standard Model (SM) and assumed in most

beyond the Standard Model explorations in the litera-

ture. However, thus far there has been no evidence for

a period of restored EW symmetry in the early universe

and current observational limits permit a wide variety of

extensions to the SM which might reverse the tendency

to restore the EW symmetry at high-temperatures.1In-

deed, counterexamples to the conventional picture pre-

sented above do exist. Such alternative scenarios where

the EW symmetry remains broken at temperatures above

the EW scale feature the phenomenon commonly referred

to as electroweak symmetry non-restoration (EW SNR).

The possibility of the higgs ﬁeld acquiring a nonzero

vacuum expectation value (vev) in the early universe has

wide reaching consequences, some of which have been

explored in [9–22]. The electroweak phase transition

may not have occurred or instead took place at a much

∗jaechang@umd.edu

†mariaolalla.olearomacho@phys.ens.fr

‡etanin1@jhu.edu

1The possibility of high-temperature symmetry non-restoration, not

speciﬁc to the EW sector, was ﬁrst studied in [1–8].

higher temperature as compared to the SM prediction.

Sphaleron processes would remain suppressed at tem-

peratures well above the electroweak scale, thus making

e.g. high-temperature electroweak baryogenesis viable.

Early universe calculations that rely on the properties

of the primordial SM plasma would have to be appropri-

ately modiﬁed. The impacts of these modiﬁcations may

be imprinted in relics such as gravitational waves, dark

matter, and dark radiation that decoupled early. It is

therefore important to consider the less explored possi-

bility that the broken electroweak phase persists above

the electroweak scale.

One way to modify the thermal evolution of the higgs

vev is to couple the higgs ﬁeld to new scalar degrees

of freedom via higgs-portal couplings. If these quartic

couplings are negative,2the higgs ﬁeld would acquire a

negative thermal mass from the thermal ﬂuctuations of

the new scalars. This fact was utilized to achieve EW

SNR in [12–14]. There it was shown that the presence of

at least O(100) thermalized scalars with negative quartic

couplings to the higgs ﬁeld can keep the electroweak sym-

metry broken at temperatures well above the electroweak

scale. Further studies revealed a variety of models that

display electroweak symmetry non-restoration [18–23],

and yet the presence of a large number of new degrees of

freedom remains to be a common feature of existing mod-

els for this phenomenon. In some of these models [20,22],

EW SNR can technically be realized with an O(1) num-

ber of new ﬁelds at the cost of limiting the highest tem-

perature to which EW SNR can be reliably sustained,

2These quartic couplings are deﬁned as negative in the potential and

positive in the Lagrangian.

arXiv:2210.05680v2 [hep-ph] 11 Dec 2022

which in both studies did not go beyond ∼100 TeV.

While increasing the multiplicity of the ﬁelds cou-

pled to the higgs ﬁeld helps to alleviate various con-

straints [12–14], in particular those related to the stabil-

ity of the scalar potential and perturbativity, this feature

is not a proximate cause of the high-temperature EW

SNR phenomenon. In this paper, we show that the addi-

tion of one scalar ﬁeld coupled to the higgs is suﬃcient to

achieve EW SNR if the scalar develops a suﬃciently large

vev at high temperatures in the early universe. Note that

increasing the scalar vev neither destabilizes the scalar

potential nor exacerbates the running of couplings. We

demonstrate this idea of realizing EW SNR via a vev in

a simple model of a complex scalar singlet coupled to

the EW sector through the higgs portal with negative

coupling. In the presence of a suﬃciently large chemical

potential, the thermal equilibrium state of the new scalar

includes a Bose-Einstein condensate (BEC) [24,25] and

this condensate yields the requisite large negative higgs

mass squared for EW SNR.

Chemical potentials in the universe naturally arise in

the presence of net background charges associated with

some global symmetries. In fact, current observations

are consistent with the universe possessing large back-

ground charges of certain kinds. While the baryon asym-

metry of the universe has been observed to be tiny,

nB/s ∼10−10 [26,27], up to O(1) total lepton asym-

metry [28] is still allowed. Charge asymmetries may also

reside in the dark sector [29] at an unconstrained level.

Global symmetries are expected to be broken at high en-

ergies by higher dimensional operators [30,31]. Thus,

a ﬁeld whose Lagrangian respects a global symmetry at

low energies could carry a net charge as an after eﬀect of

its high-energy dynamics. A concrete example of this is

the Aﬄeck-Dine mechanism [32]. Furthermore, if some

form of entropy production [33–35] or charge washout

[36–38] took place, these charge asymmetries could be

much greater in the early universe and have stronger im-

pacts then.

In this paper we show that EW SNR can be minimally

realized by coupling the higgs to a scalar that develops a

vev in the early universe. We elaborate this point further

in section II. In section III, we present a simple example

model (with a new complex scalar that forms a BEC)

that demonstrates this idea, analyze the viable parameter

space for achieving EW SNR, and describe its cosmology.

Finally, we conclude in section IV.

II. HIGH-TEMPERATURE ELECTROWEAK

SYMMETRY NON-RESTORATION WITH A

SCALAR CONDENSATE

The higgs doublet Hcan be expanded in the unitary

gauge as

H(x) = 1

√20

H+h(x).(1)

where only the real part of the neutral component has

a constant background value Hand the physical higgs

boson is denoted by h. At the minimum adopted by the

universe we have H=vH, where vHis the higgs vev. The

EW symmetry is broken in the early universe if the scalar

eﬀective potential has no minimum in which Hvanishes.

A suﬃcient condition for EW SNR is the eﬀective mass

squared m2

H(T) of the higgs ﬁeld being negative at the

ﬁeld space points where H= 0, i.e.

H(T) = ∂2V(T, H)

∂H2H=0

<0.(2)

Here V(T, H) denotes the ﬁnite-temperature eﬀective po-

tential evaluated using the traditional background ﬁeld

method [39], which could also be a function of additional

background ﬁelds. At ﬁnite temperatures, m2

H(T) ac-

quires large positive contributions from the SM ﬁelds cou-

pled to it, leading to the usual expectation of EW symm-

metry restoration within the SM. All these SM particles

contribute positively to m2

H(T) because the fermion and

gauge-boson contributions are quadratic in their Yukawa

and gauge couplings to the higgs, respectively. For the

same reason, the EW symmetry remains to be restored

in many early universe models with extended EW sec-

tors. On the other hand, new scalar ﬁelds can couple

with negative couplings to the higgs ﬁeld and yield large

negative contributions to m2

H(T).

Consider, for instance, the simplest case where a real

scalar ﬁeld Sis coupled to the higgs doublet ﬁeld H

through a negative higgs-portal coupling −λH S |H|2S2/2,

which also couples Sto the SM thermal bath. Through

this coupling, the thermal ﬂuctuations of Scontribute

∼ −λHS T2to m2

H(T), which tend to push the higgs ﬁeld

away from the origin (i.e. the ﬁeld space points where the

higgs background ﬁeld vanish H= 0). In order for one

such scalar contribution to overcome the SM contribu-

tions while keeping the tree-level scalar potential V(H, S)

bounded from below, the quartic self-coupling λSof the

Sﬁeld would need to be non-perturbatively large [12–

14]. This led to the introduction of O(100) scalars in

Refs. [12–14] in order to realize EW SNR in the early

universe, while allowing for a perturbative treatment of

the theory and keeping the tree-level potential bounded

from below.

The preceding discussion assumes that the new scalar

ﬁelds have no appreciable chemical potential, in which

case the Bose-Einstein distribution corresponds to modes

with energy Ek.Thaving O(1) occupation numbers. In

the following we will consider more general momentum

distributions. Schematically, the contribution to m2

H(T)

from a scalar with arbitrary occupation numbers fkis

proportional to Rd3k fk/Ek. This contribution is max-

imized for a given energy density ρ∼Rd3k fkEkwhen

the occupation number fkis concentrated in the infrared

momentum modes, where the particle energy Ekis mini-

mized. Given its low entropy, it appears that a strongly-

coupled ﬁeld with an IR-concentrated momentum distri-

bution would not last for a long time. However, such

a conﬁguration can be the favoured thermal-equilibrium

state if there exists a non-zero chemical potential asso-

ciated with some conservation law. In fact, when the

chemical potential is close to a critical value, the conﬁg-

uration favoured by thermal equilibrium involves a large

occupation of the ground state, i.e. a BEC. In that case,

the contribution to m2

H(T) is maximized and, as we will

show, this allows for a minimal realization of EW SNR

with a single new scalar ﬁeld.

In light of the above observation, we return to the real

scalar singlet Swith a negative higgs-portal coupling ex-

ample, now allowing it to develop a vev vS. The eﬀective

mass squared of the higgs ﬁeld at the origin is then given

H(T)

T2≈κSM −κS(3)

where κSM and κS=λHS v2

S/(2T2) are, respectively, the

contributions from the SM thermal bath and the vev of S.

At temperatures above the electroweak scale, the domi-

nant contributions to κSM are [12]

κSM =y2

4+3g2+g02

16 +λH

2≈0.4 (4)

where yt,g,g0, and λHare the top-Yukawa, SU(2)L,

U(1)Y, and higgs self-quartic coupling, respectively. A

suﬃcient condition for EW SNR is

T&0.9λ−1/2

HS (5)

Thus, a single new scalar ﬁeld that has a negative higgs-

portal coupling and acquires a vev satisfying the above

can reverse the EW symmetry restoring eﬀect of the SM

thermal bath. In the next section, we discuss a simple

mechanism for sustaining such a large vev at high tem-

peratures.

III. MINIMAL SCALAR CONDENSATE MODEL

We extend the SM with a complex scalar singlet φ,

playing the role of the Sﬁeld in the previous section.

We assume that the tree-level scalar potential is invariant

under a global U(1)φsymmetry and include all allowed

renormalizable terms

Vtree(H, φ) = −µ2

H|H|2+λH|H|4−λHφ|H|2|φ|2

+µ2

φ|φ|2+λφ|φ|4,(6)

where λHφ,λφ, and µ2

φare all positive. This model has

been studied extensively in connection to dark matter,

baryogenesis, and gravitational wave production through

a strong ﬁrst order phase transition (for a review, see e.g.

[40]). Unlike these previous studies, we assume that the

universe has a pre-established large net conserved charge

density nQassociated to the global U(1)φsymmetry un-

der which φtransforms. Such a charge density implies

that there is an asymmetry nφ−nφ†=nQin the number

density of φparticle and antiparticle, denoted as nφand

nφ†respectively. We do not specify the origin of such a

large charge asymmetry, but concrete mechanisms have

been proposed [32,41,42]. Since both nQand entropy

density sscale with scale factor aas a−3, it is convenient

to take their expansion-invariant ratio as a free parame-

ter

ηQ=nQ

s(7)

A. Thermal Bose-Einstein condensate

We begin by specifying the conditions under which a

BEC develops as a thermal-equilibrium state. Any net

charge density that is carried by the nonzero momen-

tum excitations of the φﬁeld, nk6=0

Q, manifests itself as

an asymmetry in the Bose-Einstein distributions for the

particles and antiparticles due to the existence of a non-

vanishing chemical potential µ

nk6=0

Q=Zd3k

(2π)31

e(Ek−µ)/T −1−1

e(Ek+µ)/T −1.

(8)

For deﬁniteness, we take µand hence the charge density

to be positive. A larger chemical potential µcorresponds

to larger nk6=0

Qat a given temperature T. In order to

keep the φparticle occupation number [e(Ek−µ)/T −1]−1

positive, the chemical potential µmust not exceed the

eﬀective mass of φ,meﬀ

φ. The upper limit of the chemi-

cal potential, namely meﬀ

φ, corresponds to the maximum

charge asymmetry that can be accommodated in parti-

cle and antiparticle excitations at a given temperature

T[25,43]

ηcrit

Q=nk6=0

sµ→meff

∼









2π2g∗ meﬀ

T!, T &meﬀ

45ζ(3/2)

π2g∗ meﬀ

2πT !3/2

, T .meﬀ

(9)

where g∗is the eﬀective number of relativistic degrees of

freedom.

When the charge asymmetry ηQis larger than ηcrit

Qthe

excess charge must be stored in the ground state, which

leads to the development of a high occupation number

ground state, i.e. a BEC. In the presence of a BEC, the

total change density nQcan be decomposed into two

parts: the charge density stored in the condensate of

k= 0 quanta, nBEC

Q, and the charge density carried by

the k6= 0 particle excitations, nk6=0

The large occupation of the ground-state quanta in the

condensate makes it possible to treat φas a homogeneous

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DESY-22-157ElectroweakAsymmetricEarlyUniverseviaaScalarCondensateJaeHyeokChang,1,2,*MaraOlallaOlea-Romacho,3,4,andErwinH.Tanin1,1DepartmentofPhysicsandAstronomy,JohnsHopkinsUniversity,Baltimore,MD21218,USA2MarylandCenterforFundamentalPhysics,DepartmentofPhysics,UniversityofMaryland,CollegePark,M...

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DESY-22-157 Electroweak Asymmetric Early Universe via a Scalar Condensate Jae Hyeok Chang1 2 Mar a Olalla Olea-Romacho3 4 and Erwin H. Tanin1

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