DESY-22-157

# Electroweak Asymmetric Early Universe via a Scalar Condensate

Jae Hyeok Chang<EMAIL_ADDRESS>Department of Physics and Astronomy, Johns
Hopkins University, Baltimore, MD 21218, USA  Maryland Center for Fundamental
Physics, Department of Physics, University of Maryland, College Park, MD
20742, USA  María Olalla Olea-Romacho<EMAIL_ADDRESS>Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL,
CNRS, Sorbonne Université, Université Paris Cité, F-75005 Paris, France
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
Erwin H. Tanin<EMAIL_ADDRESS>Department of Physics and Astronomy, Johns
Hopkins University, Baltimore, MD 21218, USA

###### Abstract

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 vacuum that breaks the
electroweak (EW) symmetry. At high temperatures, the higgs field acquires
positive thermal 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 standard 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 literature. 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.111The possibility of high-
temperature symmetry non-restoration, not specific to the EW sector, was first
pointed out in Weinberg (1974). Indeed, counterexamples to the conventional
picture presented 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 Bajc  _et al._ (1998); Yang  _et al._ (2015);
Freese  _et al._ (2018); Passaglia  _et al._ (2021); Meade and Ramani (2019);
Baldes and Servant (2018); Glioti  _et al._ (2019); Espinosa  _et al._ (2005);
Ramsey-Musolf  _et al._ (2018); Matsedonskyi (2021); Matsedonskyi and Servant
(2020); Bai  _et al._ (2021); Biekötter  _et al._ (2021); Carena  _et al._
(2021); Matsedonskyi  _et al._ (2021). The electroweak phase transition may
not have occurred or instead took place at a much higher temperature as
compared to the SM prediction. Sphaleron processes would remain suppressed at
temperatures 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
appropriately 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
possibility 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,222These quartic couplings are defined as
negative in the potential and positive in the Lagrangian. the 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 Meade and Ramani (2019);
Baldes and Servant (2018); Glioti  _et al._ (2019). 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 symmetry broken at
temperatures well above the electroweak scale. Further studies revealed a
variety of models that display electroweak symmetry non-restoration
Matsedonskyi and Servant (2020); Biekötter  _et al._ (2021); Carena  _et al._
(2021); Matsedonskyi  _et al._ (2021); Bai  _et al._ (2021); Biekötter  _et
al._ (2022), and yet the presence of a large number of new degrees of freedom
remains to be a common feature of existing models for this phenomenon. In some
of these models Matsedonskyi  _et al._ (2021); Biekötter  _et al._ (2021), EW
SNR can technically be realized with an $O(1)$ number of new fields at the
cost of limiting the highest temperature to which EW SNR can be reliably
sustained, which in both studies did not go beyond $\sim
100\mathinner{\mathrm{TeV}}$.

While increasing the multiplicity of the fields coupled to the higgs field
helps to alleviate various constraints Meade and Ramani (2019); Baldes and
Servant (2018); Glioti  _et al._ (2019), in particular those related to the
stability 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 addition 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) Haber and Weldon (1982); Mangano  _et al._
(2001) 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 background
charges of certain kinds. While the baryon asymmetry of the universe has been
observed to be tiny, $n_{B}/s\sim 10^{-10}$ Cyburt  _et al._ (2016); Aghanim
_et al._ (2020), up to $O(1)$ total lepton asymmetry Barenboim and Park (2017)
is still allowed. Charge asymmetries may also reside in the dark sector Kaplan
_et al._ (2009) at an unconstrained level. Global symmetries are expected to
be broken at high energies by higher dimensional operators Kamionkowski and
March-Russell (1992); Holman  _et al._ (1992). 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 Affleck and Dine (1985). Furthermore, if some form of
entropy production Randall  _et al._ (2016); Bramante and Unwin (2017); Evans
_et al._ (2020) or charge washout Buckley and Profumo (2012); Cirelli  _et
al._ (2012); Tulin  _et al._ (2012) took place, these charge asymmetries could
be much greater in the early universe and have stronger impacts 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 $\mathcal{H}$ can be expanded in the unitary gauge as

$\displaystyle\mathcal{H}(x)=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
H+h(x)\end{pmatrix}.$ (1)

where only the real part of the neutral component has a constant background
value $H$ and the physical higgs boson is denoted by $h$. At the minimum
adopted by the universe we have $H=v_{H}$, where $v_{H}$ is the higgs vev. The
EW symmetry is broken in the early universe if the scalar effective potential
has no minimum in which $H$ vanishes. A sufficient condition for EW SNR is the
effective mass squared $m_{H}^{2}(T)$ of the higgs field being negative at the
field space points where $H=0$, i.e.

$m_{H}^{2}(T)=\left.\frac{\partial^{2}V(T,H)}{\partial H^{2}}\right|_{H=0}<0.$
(2)

Here $V(T,H)$ denotes the finite-temperature effective potential evaluated
using the traditional background field method Jackiw (1974), which could also
be a function of additional background fields. At finite temperatures,
$m_{H}^{2}(T)$ acquires large positive contributions from the SM fields
coupled to it, leading to the usual expectation of EW symmmetry restoration
within the SM. All these SM particles contribute positively to $m_{H}^{2}(T)$
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 sectors. On the other hand, new scalar fields can couple with
negative couplings to the higgs field and yield large negative contributions
to $m_{H}^{2}(T)$.

Consider, for instance, the simplest case where a real scalar field $S$ is
coupled to the higgs doublet field $\mathcal{H}$ through a negative higgs-
portal coupling $-\lambda_{HS}|\mathcal{H}|^{2}S^{2}/2$, which also couples
$S$ to the SM thermal bath. Through this coupling, the thermal fluctuations of
$S$ contribute $\sim-\lambda_{HS}T^{2}$ to $m_{H}^{2}(T)$, 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 contributions while keeping the tree-level
scalar potential $V(H,S)$ bounded from below, the quartic self-coupling
$\lambda_{S}$ of the $S$ field would need to be non-perturbatively large Meade
and Ramani (2019); Baldes and Servant (2018); Glioti  _et al._ (2019). This
led to the introduction of $O(100)$ scalars in Refs. Meade and Ramani (2019);
Baldes and Servant (2018); Glioti  _et al._ (2019) 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 $E_{k}\lesssim T$ having $O(1)$ occupation
numbers. In the following we will consider more general momentum
distributions. Schematically, the contribution to $m_{H}^{2}(T)$ from a scalar
with arbitrary occupation numbers $f_{k}$ is proportional to $\int
d^{3}k\;f_{k}/E_{k}$. This contribution is maximized for a given energy
density $\rho\sim\int d^{3}k\;f_{k}E_{k}$ when the occupation number $f_{k}$
is concentrated in the infrared momentum modes, where the particle energy
$E_{k}$ is minimized. Given its low entropy, it appears that a strongly-
coupled field with an IR-concentrated momentum distribution 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 associated
with some conservation law. In fact, when the chemical potential is close to a
critical value, the configuration favoured by thermal equilibrium involves a
large occupation of the ground state, i.e. a BEC. In that case, the
contribution to $m_{H}^{2}(T)$ 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 $S$
with a negative higgs-portal coupling example, now allowing it to develop a
vev $v_{S}$. The effective mass squared of the higgs field at the origin is
then given by

$\frac{m_{H}^{2}(T)}{T^{2}}\approx\kappa_{\rm SM}-\kappa_{S}$ (3)

where $\kappa_{\rm SM}$ and $\kappa_{S}=\lambda_{HS}v_{S}^{2}/2T^{2}$ are,
respectively, the contributions from the SM thermal bath and the vev of $S$.
At temperatures above the electroweak scale, the dominant contributions to
$\kappa_{\rm SM}$ are Meade and Ramani (2019)

$\kappa_{\rm SM}=\frac{y_{t}^{2}}{4}+\frac{3g^{2}+g^{\prime
2}}{16}+\frac{\lambda_{H}}{2}\approx 0.4$ (4)

where $y_{t}$, $g$, $g^{\prime}$, and $\lambda_{H}$ are the top-Yukawa,
$SU(2)_{L}$, $U(1)_{Y}$, and higgs self-quartic coupling, respectively. A
sufficient condition for EW SNR is

$\frac{v_{S}}{T}\gtrsim 0.9\lambda_{HS}^{-1/2}$ (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 temperatures.

##  III Minimal Scalar Condensate Model

We extend the SM with a complex scalar singlet $\phi$, playing the role of the
$S$ field in the previous section. We assume that the tree-level scalar
potential is invariant under a global $U(1)_{\phi}$ symmetry and include all
allowed renormalizable terms

$\displaystyle V_{\rm tree}(H,\phi)=$
$\displaystyle-\mu_{H}^{2}|\mathcal{H}|^{2}+\lambda_{H}|\mathcal{H}|^{4}-\lambda_{H\phi}|\mathcal{H}|^{2}|\phi|^{2}$
$\displaystyle+m_{\phi}^{2}|\phi|^{2}+\lambda_{\phi}|\phi|^{4},$ (6)

where $\lambda_{H\phi}$, $\lambda_{\phi}$, and $m_{\phi}^{2}$ are 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. Lebedev (2021)). Unlike these
previous studies, we assume that the universe has a pre-established large net
conserved charge density $n_{Q}$ associated to the global $U(1)_{\phi}$
symmetry under which $\phi$ transforms. Such a charge density implies that
there is an asymmetry $n_{\phi}-n_{\phi^{\dagger}}=n_{Q}$ in the number
density of $\phi$ particle and antiparticle, denoted as $n_{\phi}$ and
$n_{\phi^{\dagger}}$ respectively. We do not specify the origin of such a
large charge asymmetry, but concrete mechanisms have been proposed Affleck and
Dine (1985); Delepine  _et al._ (2007); Hertzberg and Karouby (2014). Since
both $n_{Q}$ and entropy density $s$ scale with scale factor $a$ as $a^{-3}$,
it is convenient to take their expansion-invariant ratio as a free parameter

$\eta_{Q}=\frac{n_{Q}}{s}$ (7)

###  III.1 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
momentum excitations of the $\phi$ field, $n_{Q}^{k\neq 0}$, 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 $\mu$

$n_{Q}^{k\neq
0}=\int\frac{d^{3}k}{(2\pi)^{3}}\left(\frac{1}{e^{(E_{k}-\mu)/T}-1}-\frac{1}{e^{(E_{k}+\mu)/T}-1}\right).$
(8)

For definiteness, we take $\mu$ and hence the charge density to be positive. A
larger chemical potential $\mu$ corresponds to larger $n_{Q}^{k\neq 0}$ at a
given temperature $T$. In order to keep the $\phi$ particle occupation number
$[e^{(E_{k}-\mu)/T}-1]^{-1}$ positive, the chemical potential $\mu$ must not
exceed the effective mass of $\phi$, $m_{\phi}^{\rm eff}$. The upper limit of
the chemical potential, namely $m_{\phi}^{\rm eff}$, corresponds to the
maximum charge asymmetry that can be accommodated in particle and antiparticle
excitations at a given temperature $T$ Mangano  _et al._ (2001); Fukuyama  _et
al._ (2008)

$\displaystyle\eta_{Q}^{\rm crit}$ $\displaystyle=\left.\frac{n_{Q}^{k\neq
0}}{s}\right|_{\mu\rightarrow m_{\phi}^{\rm eff}}$
$\displaystyle\sim\begin{cases}\displaystyle\frac{15}{2\pi^{2}g_{*}}\left(\frac{m_{\phi}^{\rm
eff}}{T}\right),&T\gtrsim m_{\phi}^{\rm eff}\\\
\displaystyle\frac{45\zeta(3/2)}{\pi^{2}g_{*}}\left(\frac{m_{\phi}^{\rm
eff}}{2\pi T}\right)^{3/2},&T\lesssim m_{\phi}^{\rm eff}\end{cases}$ (9)

where $g_{*}$ is the effective number of relativistic degrees of freedom.

When the charge asymmetry $\eta_{Q}$ is larger than $\eta_{Q}^{\rm crit}$ the
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 $n_{Q}$ can be decomposed into two
parts: the charge density stored in the condensate of $k=0$ quanta,
$n_{Q}^{\rm BEC}$, and the charge density carried by the $k\neq 0$ particle
excitations, $n_{Q}^{k\neq 0}$.

The large occupation of the ground-state quanta in the condensate makes it
possible to treat $\phi$ as a homogeneous classical field $\left<\phi\right>$,
which can be written in terms of its radial component $r$ and phase $\theta$
as $\left<\phi\right>=re^{i\theta}/\sqrt{2}$. In this viewpoint, the charged
BEC corresponds to $\phi$ having a non-zero radial component $r$ coherently
rotating in the field space with an angular velocity $\dot{\theta}$ Co and
Harigaya (2020). The conserved charge density $n_{Q}$ of the spinning $\phi$
is given by its field-space angular momentum, $n_{Q}=\dot{\theta}r^{2}$. The
equations of motion for $r$ read

$\displaystyle\ddot{r}+(3H+\Gamma)\dot{r}-\dot{\theta}^{2}r+\partial_{r}V=0,$
(10)

Here, $H$ is the Hubble rate and $\Gamma$ accounts for dissipative effects
from the interaction of $\phi$ with the near-thermal SM plasma.

The $\phi$ and SM sector can thermalize if the higgs-mediated
$\phi$-$\phi^{\dagger}$ pair annihilation, i.e. the slowest reaction, is in
equilibrium. Assuming that the kinetic-equilibration of the $\phi$ field
occurs much more quickly, this process is efficient if

$\frac{\Gamma_{\rm
th}}{H}\sim\frac{\lambda_{H\phi}^{2}n_{\phi^{\dagger}}^{k\neq
0}/T^{2}}{T^{2}/M_{\rm P}}\sim\frac{\lambda_{H\phi}^{2}M_{\rm P}}{T}\gg 1$
(11)

which is always satisfied at temperatures above the electroweak scale in the
parameter space of our interest.333It is also possible that the early stages
of the relaxation toward thermal equilibrium involve processes whose
timescales are different from and possibly longer than the one considered
here. For example, the early dissipation of the scalar $\phi$ may proceed
through non-perturbative effects Mukaida and Nakayama (2013); Mukaida  _et
al._ (2013); Moroi  _et al._ (2013); Tanin and Stewart (2017) if its initial
radial oscillation amplitude is sufficiently large. However, this is highly
dependent on the reheating scenario and we do not consider it here. We have
assumed in this estimate that all the particles involved are relativistic, in
which case $n_{\phi^{\dagger}}^{k\neq 0}\sim T^{3}$. Once thermal equilibrium
is established, the $\phi$ field moves in the field space in a circular orbit
determined by the balance between the centrifugal force and potential
gradient, $\dot{\theta}^{2}r=\partial_{r}V$. This configuration is nothing but
the previously discussed thermal-equilibrium BEC.

###  III.2 Electroweak asymmetric early universe

We now describe the cosmology of our model at temperatures above the
electroweak scale. In this high-temperature regime, we assume both the higgs
field and the complex singlet $\phi$ acquire vevs. We write the fluctuations
around the constant background values as

$\displaystyle\mathcal{H}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
H+h(x)\end{pmatrix},\quad\phi=\frac{1}{\sqrt{2}}\left(r+\varphi(x)\right)e^{i\theta(x)},$
(12)

where we reserve the notation $H=v_{H}(T)$ and $r=v_{r}(T)$ for the physical
vevs adopted by the universe in our model at a temperature $T$.

To get a sense of the general evolution of the universe, we start by
discussing some estimates where we include only tree-level effects and thermal
masses. In this approximation, the equilibrium radial expectation value $r$ of
the complex scalar $\phi$ is determined by the balance between the
“centrifugal force” $\dot{\theta}^{2}r$ and the potential gradient of the
tree-level $\phi$ potential $\partial_{r}V_{\rm tree}$, which for sufficiently
large $\eta_{Q}$ is dominated by the contribution from the quartic term
$V_{\rm tree}\supset\lambda_{\phi}r^{4}/4$. Setting $\dot{r}=0$ and
$\ddot{r}=0$ in Eq. (10), and using $V_{\rm tree}\approx\lambda_{\phi}r^{4}/4$
and $n_{Q}=\dot{\theta}r^{2}=(2\pi^{2}/45)\eta_{Q}g_{*}T^{3}$, we find

$\frac{v_{r}}{T}\approx 0.76\eta_{Q}^{1/3}g_{*}^{1/3}\lambda_{\phi}^{-1/6}$
(13)

Comparing Eq. (13) with Eq. (5), we find a rough requirement for EW SNR

$\eta_{Q}\gtrsim 1.6g_{*}^{-1}\lambda_{H\phi}^{-3/2}\lambda_{\phi}^{1/2}.$
(14)

It can be easily checked that in the regime where EW SNR occurs, as defined by
Eq. (14), the quartic self-interaction indeed dominates the potential of
$\phi$,444We found that in the parameter space where EW SNR occurs, the $\phi$
condensate can dominate the energy density of the universe by a factor of
$\sim\lambda_{\phi}v_{r}^{4}/g_{*}T^{4}\sim\eta_{Q}^{4/3}\lambda_{\phi}^{1/3}g_{*}^{1/3}$.
Since the energy density of $\phi$ is dominated by its quartic potential,
which scales with the scale factor like $a^{-4}$, the universe would still be
effectively radiation-dominated.  the BEC formation condition Eq. (9) is
temperature-independent and easily satisfied, and we can consider all the
charge to be stored in the BEC, i.e. $n_{Q}\approx n_{Q}^{\rm BEC}\gg
n_{Q}^{\rm k\neq 0}$. Furthermore, the higgs vev is non-zero at high
temperatures

$v_{H}(T)^{2}\approx v_{H}(0)^{2}+\left(\frac{\kappa_{\phi}-\kappa_{\rm
SM}}{\lambda_{H}}\right)T^{2}$ (15)

where $v_{H}(0)=246\mathinner{\mathrm{GeV}}$, $\lambda_{H}=0.13$,
$\kappa_{\phi}\approx\lambda_{H\phi}(v_{r}/T)^{2}/2$ with $v_{r}/T$ shown in
Eq. (13), $\kappa_{\rm SM}\approx 0.4$ as found in Eq. (4), and here
$\kappa_{\phi}>\kappa_{\rm SM}$. In the parameter space of our interest, from
Eq. (15) we observe that $v_{H}/T$ is at most $O(1)$ and mildly temperature
dependent. The latter implies that the higgs vev scales linearly with
temperature, meaning that the SM particles coupled to the higgs can be
considerable heavier in the early universe, though this change is at most
$O(1)$ of their thermal masses.

To account for a more complete description of the phenomenon of EW SNR, the
resummation of a certain set of higher order diagrams, the so-called daisy
diagrams, should be included Gross  _et al._ (1981); Parwani (1992); Arnold
and Espinosa (1993). Furthermore, below the electroweak scale, the change in
$g_{*}$, the zero-temperature masses of the SM particles and of the scalar
field $\phi$ must be taken into account, and moreover the $\phi$ scalar
potential is no longer dominated by the quartic self-interaction. Finally, the
Coleman-Weinberg potential Coleman and Weinberg (1973), which accounts for the
one-loop radiative corrections at zero-temperature should also be considered.
In the next section, we explain how we include all these effects in our
numerical analysis and in our plots, and show that they do not affect our
conclusions qualitatively.

###  III.3 Viable parameter space and numerical analysis

The finite-temperature effective potential for the background fields $H$ and
$r$, as defined in Eq. (12), at one-loop, is given by

$V(H,r,T)=V_{\rm tree}+V_{Q}+V_{\rm CW}+V_{\rm CT}+V_{T}+V_{\rm daisy},$ (16)

where the tree-level scalar potential $V_{\rm tree}$ reads

$\displaystyle V_{\rm tree}=$
$\displaystyle-\frac{1}{2}\mu_{H}^{2}H^{2}+\frac{\lambda_{H}}{4}H^{4}-\frac{\lambda_{H\phi}}{4}H^{2}r^{2}$
$\displaystyle+\frac{1}{2}m_{\phi}^{2}r^{2}+\frac{\lambda_{\phi}}{4}r^{4}$
(17)

$V_{Q}$ is the angular part of the kinetic energy of $\phi$,
$\dot{\theta}^{2}r^{2}/2$, which owing to the conservation of charge
$n_{Q}=\dot{\theta}r^{2}$, acts like an effective potential for $r$

$V_{Q}=\frac{n_{Q}^{2}}{2r^{2}}$ (18)

Note that $V_{Q}$ ensures that $r>0$. $V_{\rm CW}$ represents the one-loop
zero-temperature radiative corrections to the scalar potential in the form of
the Coleman-Weinberg potential, $V_{\rm CT}$ contains UV-finite counterterm
contributions, $V_{T}$ includes the one-loop temperature-dependent corrections
to the scalar potential, and finally $V_{\rm daisy}$ accounts for the
resummation of the daisy diagrams. The expressions for $V_{\rm CW}$, $V_{\rm
CT}$, $V_{T}$, and $V_{\rm daisy}$ can be found in Appendix A.

In our numerical analysis, we choose a set of input parameters which include
the higgs portal coupling $-\lambda_{H\phi}$, the singlet self-coupling
$\lambda_{\phi}$, the singlet charge asymmetry per unit entropy $\eta_{Q}$,
and the present-epoch physical mass of the extra scalar $m_{\phi,0}$,

$\lambda_{H\phi}\,,\lambda_{\phi}\,,m_{\phi,0}\,,\eta_{Q}.$ (19)

To ensure perturbativity and the boundedness of $V_{\rm tree}$, we require
Costa  _et al._ (2015)

$\begin{gathered}\lambda_{H}<4\pi,~{}\lambda_{\phi}<4\pi,~{}\lambda_{H\phi}<8\pi,\\\
\Lambda_{\pm}\equiv
6\lambda_{H}+4\lambda_{\phi}\pm\sqrt{(4\lambda_{\phi})^{2}+(6\lambda_{H}+4\lambda_{\phi})^{2}}<16\pi,\\\
4\lambda_{H}\lambda_{\phi}>\lambda_{H\phi}^{2},\\\
\lambda_{H},\lambda_{\phi}>0.\end{gathered}$ (20)

We look for the extrema of the effective potential where $H=0$. In order to
asses whether a field-space extremum is a minimum, we calculated the principal
minors of the Hessian matrix of the scalar potential shown in Eq. (16). The EW
symmetry could be restored if both minors are positive, otherwise the
phenomenon of EW SNR would take place.

Our results are summarized in Fig. 1, which shows the parameter space where EW
SNR is achieved at $T=1~{}\mathinner{\mathrm{TeV}}$. To illustrate the viable
parameter space, we assume representative values of the charge asymmetry
$\eta_{Q}$ and the scalar singlet mass $m_{\phi,0}$, and scan over the
couplings $\lambda_{H\phi}$ and $\lambda_{\phi}$. Charge asymmetries as large
as $\eta_{Q}\sim 1$ are a natural outcome of the Affleck-Dine mechanism
Affleck and Dine (1985); Linde (1985). Therefore, we fixed
$\eta_{Q}=0.1,1,~{}\textrm{and}~{}10$ in our parameter scan. As per Eq. (9),
the $\phi$ condensate automatically disappears at low temperatures $T\ll
m_{\phi}$ where $\eta_{Q}^{\rm crit}\gg 1$ and so we assume
$\left<\phi\right>=0$ when considering collider constraints. We set the
singlet mass at $m_{\phi,0}=100\mathinner{\mathrm{GeV}}$, given that a singlet
mass $m_{\phi,0}$ larger than $m_{h}/2$ ensures agreement with the measured
properties of the detected SM-like higgs boson at
$125\mathinner{\mathrm{GeV}}$ CMS (2022); ATL (2022), since the decay of the
higgs boson into a pair of additional scalars is forbidden. The blue regions
in Fig. 1 demonstrate the zones where the EW symmetry is not restored at
$T=1\mathinner{\mathrm{TeV}}$ for the three different values of $\eta_{Q}$,
the yellow region indicates where the EW symmetry could be restored at that
temperature, and the grey region features a tree-level potential not bounded
below. The dashed lines depict the cutoff energy scales $\mu_{\rm cutoff}$
above which one of the conditions in Eq. (20) is not satisfied when the
running couplings are inserted therein. The running couplings were obtained by
solving the one-loop renormalization group equations (RGE) in the
$\overline{\mathrm{MS}}$ renormalization scheme shown in Appendix B. On one
hand, the dashed lines parallel to the boundary between the grey and the blue
regions indicate the cutoff energy scales above which tree-level boundedness
from below with inserted one-loop running couplings is not fulfilled anymore.
On the other hand, the dashed horizontal lines show the values of $\mu_{\rm
cutoff}$ for which perturbativity breaks down.

The running of the quartic couplings for two benchmark scenarios is shown in
Fig. 2, whose parameter points are marked in Fig. 1 with stars. The upper plot
in Fig. 2 corresponds to a representative case where $\mu_{\rm cutoff}$ is set
by the tree-level boundedness from below requirements, while the lower plot
exemplifies a situation where $\mu_{\rm cutoff}$ is given by the breakdown of
perturbativity. In most cases, the absolute value of $\lambda_{H\phi}$
increases with the energy scale due to the contribution from the top quark to
its beta function. Furthermore, $\lambda_{H}$ decreases with the
renormalization energy scale. According to equation (15), these facts make it
easier to achieve EW SNR for temperatures much higher than
$1\mathinner{\mathrm{TeV}}$ when one considers the RG improved effective
potential, which is needed to minimize the renormalization scale dependence.
In this paper, we demonstrated the realization of EW SNR at
$T=1\mathinner{\mathrm{TeV}}$, where the condition to achieve this phenomenon
is expected to be stronger. At $T=1\mathinner{\mathrm{TeV}}$, the effective
potential is expected to be only mildly renormalization-scale dependent.555We
numerically checked that at $T=1\mathinner{\mathrm{TeV}}$, $v_{r}$ is of order
$O(1\mathinner{\mathrm{TeV}})$ at most. Therefore, we did not consider the RGE
improved effective potential in our analysis.

Figure 1: $(\lambda_{H\phi},\lambda_{\phi})$ parameter space for
$\eta_{Q}=0.1,~{}1,~{}\textrm{and}~{}10$. The blue regions are the parameter
space zones in which the EW symmetry is not restored at
$T=1~{}\mathinner{\mathrm{TeV}}$ for different values of $\eta_{Q}$, the
yellow region features the possible restoration of the EW symmetry at the same
temperature, and the grey region is where the tree-level potential is not
bounded from below. The dark blue lines show the analytic approximation from
Eq. (14) for the corresponding values of $\eta_{Q}$. The dashed lines are the
cutoff energy scales derived from either perturbativity or boundedness
constraints Eq. (20). The RGE running of couplings for the parameter points
marked with stars are shown in Fig. 2.

Figure 2: The RG flow of couplings for $(\lambda_{H\phi},\lambda_{\phi})$ at
the reference scale $\mu_{0}=246~{}\mathinner{\mathrm{GeV}}$ equal to
$(10^{-2},10^{-3})$ and $(0.1,10^{-0.1})$, which are marked in Fig. 1 with
stars. The cutoff scale $\mu_{\textrm{cutoff}}$ is given by the lowest energy
scale at which any of the curves exceeds 1. $\mu_{\textrm{cutoff}}$ is given
by the breakdown of tree-level boundedness (perturbativity) for the upper
(lower) plot.

###  III.4 Late-time charge washout

The low-temperature ($T\lesssim 100\mathinner{\mathrm{GeV}}$) part of the
cosmological history can be told in any number of ways without affecting the
previous, high-temperature ($T\gtrsim 100\mathinner{\mathrm{GeV}}$)
discussion. Since we are mainly concerned with achieving EW SNR at high
temperatures, we keep the discussion here mostly qualitative. In order not to
affect Big Bang nucleosynthesis (BBN), the universe must be dominated by SM
radiation by $T\sim 10\mathinner{\mathrm{MeV}}$ Cyburt  _et al._ (2016). This
requires a process that depletes the energy density of the $\phi$ field taking
place in the temperature range $10\mathinner{\mathrm{MeV}}\lesssim T\lesssim
100\mathinner{\mathrm{GeV}}$. Here, we give one concrete example of such a
process that is consistent with current experimental and observational data
without introducing new constraints to the high-temperature cosmological
evolution.

For illustration purposes, we set

$\eta_{Q}\sim 0.1,\quad\lambda_{H\phi}\sim\lambda_{\phi}\sim 1,\quad
m_{\phi,0}=100\mathinner{\mathrm{GeV}}$ (21)

From Eq. (9), we see that the BEC evaporates away at $T\sim
10\mathinner{\mathrm{GeV}}$. Below that temperature it suffices to describe
$\phi$ as a particle-antiparticle gas. By that point, $\phi$ is non-
relativistic, the universe is $\phi$ dominated (i.e. matter dominated), and
the antiparticles $\phi^{\dagger}$ have mostly annihilated away due to
Boltzmann suppression. Note that $\eta_{Q}$ and $g_{*}$ always appear in
conjunction in our calculations in this section and $\eta_{Q}g_{*}\sim 1$ at
$T\lesssim 10\mathinner{\mathrm{GeV}}$. For simplicity, we will thus omit
them, together with $\lambda_{H\phi}\sim\lambda_{\phi}\sim 1$, in the
estimates below.

In order to deplete the charge asymmetry, and hence the energy density of the
$\phi$ field, we introduce the following $U(1)_{\phi}$ breaking terms

$V=\frac{1}{2}\Delta^{2}(\phi^{2}+\phi^{\dagger 2})$ (22)

where $\Delta$ can be made real without loss of generality. For a sufficiently
small $\Delta$, these terms become important only at late times, and then they
cause oscillations between the $+1$ and $-1$ charge eigenstates, namely $\phi$
and $\phi^{\dagger}$, respectively. Such oscillations repopulate the otherwise
Boltzmann-suppressed number density of $\phi^{\dagger}$ particles, thus
enabling the depletion of $\phi$ energy density through
$\phi$-$\phi^{\dagger}$ annihilations Buckley and Profumo (2012); Cirelli  _et
al._ (2012); Tulin  _et al._ (2012).

Before we describe the depletion process and derive the relic $\phi$
abundance, let us first outline the key processes involved. A freely
propagating $\phi$ particle would oscillate into $\phi^{\dagger}$ with a
probability
$P_{\phi\rightarrow\phi^{\dagger}}\approx\sin^{2}(\Delta^{2}t/m_{\phi})$ over
a time $t$. However, such oscillations are interrupted because the abundant
$\phi$ particles are constantly colliding off one another,
$\phi\phi\rightarrow\phi\phi$, with the rate $\Gamma_{\rm scat}\sim
n_{\phi}T/m_{\phi}^{3}$. Since each of these collisions projects the
participating particles into one of the charge eigenstates, the oscillations
occur effectively only in the mean free time $\Gamma_{\rm scat}^{-1}$ between
two collisions. Hence, whenever two quanta collide, there is a probability
$P_{\phi\rightarrow\phi^{\dagger}}\sim\left(\Delta^{2}/m_{\phi}\Gamma_{\rm
scat}\right)^{2}$ that one of them is projected into $\phi^{\dagger}$. The
rate at which $n_{\phi}$ is converted to $n_{\phi^{\dagger}}$ through
oscillations is thus given by

$\dot{n}_{\rm osc}\sim n_{\phi}\Gamma_{\rm
scat}P_{\phi\rightarrow\phi^{\dagger}}\sim\Delta^{4}\frac{m_{\phi}}{T}$ (23)

which incidentally depends on neither $n_{\phi}$ nor $n_{\phi^{\dagger}}$.
Furthermore, a $\phi$-$\phi^{\dagger}$ pair can annihilate into SM states via
higgs-mediated processes with the rate per unit volume $\sim\left<\sigma
v\right>_{\rm ann}n_{\phi}n_{\phi^{\dagger}}$. Finally, both the particle and
antiparticle number density are diluted by the expansion of the universe.
These effects are summed up by the following schematic Boltzmann equations

$\displaystyle\dot{n}_{\phi}$ $\displaystyle\sim-\dot{n}_{\rm
osc}-\left<\sigma v\right>_{\rm ann}n_{\phi}n_{\phi^{\dagger}}-3Hn_{\phi}$
(24) $\displaystyle\dot{n}_{\phi^{\dagger}}$ $\displaystyle\sim+\dot{n}_{\rm
osc}-\left<\sigma v\right>_{\rm
ann}n_{\phi}n_{\phi^{\dagger}}-3Hn_{\phi^{\dagger}}$ (25)

where $H$ is the Hubble rate, and we assumed $n_{\phi}\gtrsim
n_{\phi^{\dagger}}$ and correspondingly neglected the
$\phi^{\dagger}\rightarrow\phi$ oscillation. For our choice of parameters the
annihilation cross section is $\left<\sigma v\right>_{\rm ann}\approx
2.0\times 10^{-9}\mathinner{\mathrm{GeV}}^{-2}$.666This annihilation cross
section $\left<\sigma v\right>_{\rm ann}\approx 2.0\times
10^{-9}\mathinner{\mathrm{GeV}}^{-2}$ was computed with the analytical formula
$\left<\sigma_{\rm
ann}v\right>=[\lambda_{H\phi}^{2}v_{H}^{2}/2m_{\phi}(m_{h}^{2}-4m_{\phi}^{2})^{2}]\left.\Gamma_{h}\right|_{m_{h}\rightarrow
2m_{\phi}}$ from Burgess  _et al._ (2001) using the higgs decay width
$\Gamma_{h}$ from Dittmaier  _et al._ (2011).

Our choice of parameters is such that initially oscillations and annihilations
dictate the evolution of $n_{\phi^{\dagger}}$, while $n_{\phi}$ is barely
affected by the oscillations and simply dilutes as the universe expands. In a
timescale $\sim(\left<\sigma v\right>_{\rm ann}n_{\phi})^{-1}$ much less than
a Hubble time, during which $n_{\phi}$ is approximately constant,
$n_{\phi^{\dagger}}$ settles into a quasi-equilibrium where the oscillation
and annihilation rates per unit volume balance, $\dot{n}_{\rm
osc}\sim\left<\sigma v\right>_{\rm ann}n_{\phi}n_{\phi^{\dagger}}$. In several
Hubble times that follow, this balance is maintained while $\dot{n}_{\rm
osc}\propto T^{-1}$ increases and $3Hn_{\phi}$ decreases. Eventually, the two
meet $\dot{n}_{\rm osc}\sim 3Hn_{\phi}$ when the Hubble rate is

$H_{\rm ann}\sim\left(\frac{\Delta^{12}m_{\phi}^{7}}{M_{\rm
P}^{8}}\right)^{1/11}$ (26)

where we have assumed that the universe is $\phi$-dominated when this happens,
$H^{2}\sim T^{3}m_{\phi}/M_{\rm P}^{2}$. At that point, the number density
$n_{\phi}$ begins to decrease considerably. The depletion of $n_{\phi}$
continues approximately linearly with the $n_{\phi}$-independent rate
$\dot{n}_{\rm osc}$, while the quasi-equilibrium antiparticle density scales
as $n_{\phi^{\dagger}}\propto n_{\phi}^{-1}$, i.e. increases rapidly. A few
Hubble times $\sim H_{\rm ann}^{-1}$ later, $n_{\phi}\sim n_{\phi^{\dagger}}$
and the $\phi$ field is approximately symmetrized.

Henceforth, the annihilation rate per unit volume is given by the usual
symmetric thermal freeze out formula $\sim n_{\phi}^{2}\left<\sigma
v\right>_{\rm ann}$. The relic abundance of $\phi$ is thus given by
$n_{\phi}^{\rm relic}\sim H_{\rm ann}/\left<\sigma v\right>_{\rm
ann}$.777While the universe is initially dominated by $\phi$ (i.e. matter
dominated), its energy density is quickly transferred to the SM sector as it
annihilates away. Hence, the later part of the depletion process that
determines the relic abundance of $\phi$ takes place during SM radiation
domination, similar to the standard thermal freeze out. In the standard
symmetric freeze out scenario, the freeze out happens when $m_{\phi}/T\sim 25$
when the Hubble rate is $H_{\rm WIMP}\sim 10^{-3}m_{\phi}^{2}/M_{\rm P}$ and
in that case the WIMP cross-section $\left<\sigma v\right>_{\rm WIMP}\approx
2.6\times 10^{-9}\mathinner{\mathrm{GeV}}^{-2}$ can yield the observed DM
relic abundance for a wide range of masses. In our case, the Hubble rate
$H_{\rm ann}$ at which freeze out takes place is in general not given by
$H_{\rm WIMP}$, but instead is controlled by the free parameter $\Delta$. By
comparing the cross-section and Hubble rate in our case to that of the
Standard WIMP freeze out, we obtain that the $\phi$ relic abundance is

$\displaystyle\frac{\Omega_{\phi}^{\rm relic}}{\Omega_{\rm DM}}$
$\displaystyle\sim\frac{\left<\sigma v\right>_{\rm WIMP}H_{\rm
ann}}{\left<\sigma v\right>_{\rm ann}H_{\rm
WIMP}}\sim\left(\frac{\Delta}{10\mathinner{\mathrm{keV}}}\right)^{\frac{12}{11}},$
(27)

where in the last expression we substituted the previously-mentioned choice of
parameters. Hence, we can make $\phi$ the (WIMP) DM or a subdominant part of
it by tuning $\Delta$ appropriately. Crudely, having $\Delta\lesssim
10\mathinner{\mathrm{keV}}$ ensures that the DM is not overproduced.888Since
the $\Delta^{2}$ Lagrangian terms are the only ones that break the global
$U(1)_{\phi}$ symmetry at low energies, loop corrections to $\Delta^{2}$ are
proportional to itself, thus making its small value technically natural Cui
_et al._ (2009).

In the standard thermal freeze out, the dark matter abundance is depleted by
the increasingly strong Boltzmann suppression as the temperature drops until
the annihilation rates stop being efficient. The timing of the freeze out is
thus completely determined by the mass of the dark matter which sets the
Boltzmann suppression. The freeze out scenario presented here is similar,
except that the chemical potential of $\phi$ associated with its charge
asymmetry blocks the Boltzmann suppression until charge washout becomes
efficient. Hence, the timing of the freeze out is not tied to the mass of
$\phi$ and can be separately controlled. We have made a number of simplifying
assumptions in order to illustrate how the depletion process of $\phi$ might
proceed. Nevertheless, the main consequence of this mechanism, that the relic
abundance of $\phi$ can be controlled by the strength of the $U(1)$ breaking,
should be robust under further dedicated studies of the process.

The low-temperature phenomenology of our $\phi$ field has some resemblance
with that of the minimal singlet scalar dark matter (complex or real) coupled
to the SM via the higgs portal Burgess  _et al._ (2001); Cline  _et al._
(2013); Lebedev (2021). It is known that such models are essentially ruled out
if the scalars are populated via symmetric thermal freeze out, even when they
are allowed to be a small part of the dark matter. The combined constraints
from collider experiments, agreement with the measured dark matter relic
abundance Aghanim  _et al._ (2020), bounds from direct detection of dark
matter, and perturbativity of $\lambda_{H\phi}$ leave no viable parameter
space apart from a narrow mass regime around half of the higgs boson mass
where the annihilation rate is resonantly enhanced Escudero  _et al._ (2016);
Lebedev (2021); Biekötter and Pierre (2022). However due to the mechanism
described above, the relic abundance of $\phi$ in our scenario can be
significantly lower than that predicted in the standard thermal freeze out for
the same mass, and this opens up viable parameter space for our model. For
instance, in the simplified discussion above we found that the relic number
density of $\phi$ particles can be suppressed relative to that of the standard
thermal freeze out by a factor $\sim H_{\rm ann}/H_{\rm WIMP}$ which is
controlled by the $U(1)$ breaking parameter $\Delta$. This lower-than-expected
relic number density $n_{\phi}^{\rm relic}\lesssim\bar{n}_{\phi}^{\rm relic}$,
where $\bar{n}_{\phi}^{\rm relic}$ is the relic abundance of $\phi$ in the
standard freeze out scenario, simultaneously avoids dark matter overproduction
and relaxes the constraints from direct detection, whose rate scales as
$R_{\rm DD}\propto n_{\phi}^{\rm relic}/\bar{n}_{\phi}^{\rm relic}\propto
H_{\rm ann}/H_{\rm WIMP}$ Craig  _et al._ (2016). Furthermore, since the
singlet vev vanishes ($v_{r}(0)=0$) when the BEC evaporates below the
electroweak scale, as far as collider searches are concerned there is no
mixing between the higgs field $h$ and the scalar $\phi$, leaving the
couplings of the higgs boson to SM particles unchanged with respect to the SM.
Thus, collider bounds on our model are currently essentially non-existent
Curtin  _et al._ (2014).

##  IV Discussion and Conclusion

Current observations do not preclude the possibility that the electroweak
symmetry remains broken at temperatures above the electroweak scale in the
early universe. In such electroweak symmetry non-restoration (EW SNR)
scenarios, the higgs vev remains non-zero and typically grows with temperature
in the early universe. This can lead to early universe cosmological histories
radically different from what is commonly assumed. Given its wide-ranging
phenomenological consequences Meade and Ramani (2019), the possibility of EW
SNR is an interesting scenario.

EW SNR requires negative higgs mass squared contributions at high temperatures
to give the higgs field a vev. In most versions of EW SNR proposed so far,
these arise from the thermal fluctuations of new fields coupled to the higgs
field. The difficulties associated with ensuring the boundedness of the scalar
potential and the perturbativity of quartic couplings in such scenarios are
usually overcome by increasing the number of new fields. In this paper, we
argue that EW SNR can be minimally realized if the higgs field is repelled
from the origin by the large vev of a single field, instead of the thermal
fluctuations of many fields.

To demonstrate this, we extend the higgs sector with a complex scalar singlet
with a quartic self-interaction $\lambda_{\phi}|\phi|^{4}$ and introduce a
negative higgs-portal coupling $-\lambda_{H\phi}|\mathcal{H}|^{2}|\phi|^{2}$
to yield EW SNR. If the singlet is in thermal equilibrium with a pre-
established charge asymmetry $\eta_{Q}$ above a critical value, it acquires a
large vev which then drives the higgs field away from the origin through the
higgs-portal coupling, thus realizing EW SNR. Important constraints on
$\eta_{Q}$, $\lambda_{H\phi}$, and $\lambda_{\phi}$ to achieve a viable
scenario where EW SNR is realized come from tree-level perturbative unitarity
and boundedness of the scalar potential from below. The renormalization-group
running of quartic couplings triggers the breakdown of these two requirements
at higher temperatures, but in general EW SNR can be realized up to a upper
cutoff-temperature many orders of magnitude above the electroweak scale.

The high-temperature EW SNR scenario we are proposing is independent of any
specific scenario below the electroweak scale. While constraints arising from
low-temperature observables such as those from BBN, dark matter abundance,
direct and indirect detection experiments, and colliders measurements may
impose further constraints on the main model parameters $\eta_{Q}$,
$\lambda_{H\phi}$, and $\lambda_{\phi}$ that are relevant at high
temperatures, these constraints depend on the cosmological evolution at
temperatures below the EW scale. We found a simple low-temperature scenario
where no new constraint is added to the main model parameters. In this
scenario, we add a small term in the scalar potential, controlled by a mass
scale $\Delta\lesssim 10\mathinner{\mathrm{keV}}$, which softly breaks the
$U(1)_{\phi}$ symmetry, leading to the depletion of the energy density of the
singlet at some point below the electroweak scale and before BBN. The
symmetry-breaking parameter $\Delta$ and the bare mass $m_{\phi,0}$ of the
singlet can be appropriately tuned to meet various constraints without
affecting the high-temperature realization of EW SNR.

The most promising way to probe the model under consideration at present
colliders would be through the searches for invisible decays of the higgs
boson, for a set-up in which the singlet field is lighter than half of the
higgs boson mass. The case in which the singlet state is heavier than half of
the higgs boson mass could only be accessible to future colliders, such as a
$100\mathinner{\mathrm{TeV}}$ hadron collider or a $1\mathinner{\mathrm{TeV}}$
electron-positron collider, by means of precision measurements of the triple
higgs boson self-coupling and the $Zh$ production cross section, and only for
the largest values of $\lambda_{H\phi}$ explored in our analysis Curtin  _et
al._ (2014). The specific depletion scenario we considered predict WIMP-like
relics which can be tuned to make up the entirety of the dark matter or a sub-
component of it. If the relic abundance of the $\phi$ field makes up a
significant part of the dark matter relic abundance, we could probe this
scenario in present and future direct and indirect detection experiments. The
novel freeze out mechanism proposed here could deserve special attention from
the viewpoint of dark matter model building. We leave the detailed study of
this scenario for future work

It would be interesting to see if EW SNR could be realized in other models
involving scalar condensates. The $U(1)$-symmetric scalar condensate model
considered here can be generalized to fields that respect wider global
symmetries Li  _et al._ (2002); Moore (2016). The required chemical potential
to support a Bose-Einstein condensate may also arise from the departure from
thermal equilibrium, in which case the IR-dominated distribution function may
be achieved as an initial condition through the decay a non-relativistic
field, e.g. the inflaton, and kept from cascading toward the UV by suppressing
the thermalization rates with the Standard Model plasma Tenkanen and Vaskonen
(2016). Other ways to generate a non-zero scalar expectation value in the
early universe include coupling the scalar field to fermions Batell and
Ghalsasi (2021) and introducing a non-minimal coupling to gravity Cosme  _et
al._ (2018).

## Acknowledgments

We thank Iason Baldes, Thomas Biekötter, Raymond Co, Pierre Fayet, Sven
Heinemeyer, Zhen Liu, Oleksii Matsedonskyi, Kalliopi Petraki, Harikrishnan
Ramani, Géraldine Servant, and Yikun Wang for useful discussions. JHC is
supported by the NSF grant PHY-1914731, the Maryland Center for Fundamental
Physics, and the JHU Joint Postdoc Fund. MOOR is supported by the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No 101002846, ERC CoG “CosmoChart”. The work of MOOR was also supported by the
Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC2121
“Quantum Universe” - 390833306. This research was supported in part by
Perimeter Institute for Theoretical Physics. Research at Perimeter Institute
is supported by the Government of Canada through the Department of Innovation,
Science and Economic Development and by the Province of Ontario through the
Ministry of Research and Innovation.

##  Appendix A Scalar Effective Potential at One-loop

In the following we present the different pieces that constitute the finite-
temperature effective potential $V$ (see Eq. (16)). $V_{\rm CW}$ is given in
the $\overline{\mathrm{MS}}$ renormalization scheme and in the Landau gauge
by999The Coleman-Weinberg potential introduces gauge dependencies, however it
appears only in subleading terms Patel and Ramsey-Musolf (2011); Garny and
Konstandin (2012).

$V_{\rm
CW}=\sum_{j}\frac{n_{j}}{64\pi^{2}}(-1)^{2s_{i}}\,m_{j}^{4}(H,r)\left[\ln\left(\frac{|m_{j}(H,r)^{2}|}{\mu_{0}^{2}}\right)-c_{j}\right],$
(28)

where $m_{j}(H,r)$ are the field-dependent tree-level masses of all the
particle species, $s_{j}$ the particle spins, $n_{j}$ the corresponding
numbers of degrees of freedom, and $\mu_{0}$ is the reference renormalization
scale. The constants $c_{j}$ arise from the $\overline{\mathrm{MS}}$
renormalization prescription, with $c_{j}=5/6$ for gauge bosons and
$c_{j}=3/2$ for scalars and fermions. The sum in Eq. (28) runs over the two
scalar mass eigenstates, the three goldstone bosons $G_{i}$, the longitudinal
and transversal gauge bosons, $V_{L}=\\{Z_{L},\,W_{L}^{+},\,W_{L}^{-}\\}$ and
$V_{T}=\\{Z_{T},\,W_{T}^{+},\,W_{T}^{-}\\}$ and the SM quarks $q$, and leptons
$\ell$. The degrees of freedom $n_{j}$ for the species of each type are
$n_{h}=1$, $n_{\phi}=2$, $n_{V_{T}}=2$, $n_{G_{i}}=1$, $n_{V_{L}}=1$,
$n_{q}=12$ and $n_{\ell}=4$. The mass matrix for the scalar degrees of freedom
$\\{h(x),\varphi(x)\\}$ obtained from $V_{\rm tree}+V_{Q}$ is

$M^{2}=\left(\begin{array}[]{cc}-\mu_{H}^{2}+3\lambda_{H}H^{2}-\frac{\lambda_{H\phi}}{2}r^{2}&-\lambda_{H\phi}Hr\\\
-\lambda_{H\phi}Hr&\mu_{\phi}^{2}+6\lambda_{\phi}r^{2}-\frac{\lambda_{H\phi}}{2}H^{2}\end{array}\right).$
(29)

Moreover, the three goldstone bosons $G_{i}$ ($i=1,2,3$) tree-level field-
dependent masses read

$m_{G_{i}}^{2}=-\mu_{H}^{2}+\lambda_{H}H^{2}-\frac{1}{2}\lambda_{H\phi}r^{2}.$
(30)

Due to the singlet nature of the additional scalar $\phi$, the field-dependent
masses of the gauge bosons and fermions depend only on $H$, and they are given
by $m_{W}^{2}(H)=(g^{2}/4)H^{2}$, $m_{Z}^{2}(H)=(g^{2}+g^{\prime 2})H^{2}/4$
and $m_{f}^{2}(H)=(y_{f}^{2}/2)H^{2}$, where $y_{f}$ is the higgs Yukawa
coupling to the fermion $f$.

Additionally, we require the zero-temperature loop-corrected vacuum
expectation values and scalar masses to be equal to their tree-level values,
and we refer to this prescription as the “on-shell” (OS) renormalization. To
achieve this, we add a set of UV-finite counterterms $V_{\rm CT}$ to the
effective potential, whose computation was done by following the methods
described in Biekötter  _et al._ (2021); Basler  _et al._ (2017) and with the
help of the public code BSMPT Basler and Mühlleitner (2019); Basler  _et al._
(2020). At finite temperature $T$, the one-loop effective potential receives
thermal corrections $V_{T}$, given by Dolan and Jackiw (1974); Quiros (1999)

$V_{T}=\sum_{j}\,\frac{n_{j}\,T^{4}}{2\pi^{2}}\,J_{\pm}\left(\frac{m_{j}^{2}(H,r)}{T^{2}}\right),$
(31)

where the thermal integrals $J_{-}$ for bosons and $J_{+}$ for fermions are
defined by

$J_{\pm}(y^{2})=\mp\int_{0}^{\infty}dx\,x^{2}\,\log\left[1\pm\exp\left(-\sqrt{x^{2}+y^{2}}\right)\right]\,.$
(32)

Beside the degrees of freedom considered in Eq. (28), the sum in Eq. (31)
includes the photon. We used the implementation of $V_{T}$ of the public code
CosmoTransitions Wainwright (2012). Furthermore, we resummed the daisy
diagrams through the Arnold-Espinosa method Parwani (1992); Arnold and
Espinosa (1993), which amounts to adding the following additional contribution
to the one-loop effective potential at finite temperature101010We also
included in the sum an overall factor $e^{-m_{k}(\varphi,r)/T}$ to avoid
spurious contributions to the effective potential coming from $V_{\rm daisy}$.

$V_{\text{daisy}}=-\sum_{k}\frac{T}{12\pi}\left\\{\left[m_{k}^{2}(H,r,T)\right]^{\frac{3}{2}}-\left[m_{k}^{2}(H,r,0)\right]^{\frac{3}{2}}\right\\}\
$ (33)

where the sum in $k$ runs over the bosonic degrees of freedom, and
$m_{k}^{2}(H,r,T)$ and $m_{k}^{2}(H,r,0)$ denote, respectively, their physical
masses at finite temperature $T$ and at zero temperature Carrington (1992).
The gauge boson thermal masses can be found in Curtin  _et al._ (2014). The
physical masses at finite temperature $m_{k}^{2}(H,r,T)$ for the scalar
degrees of freedom are obtained through the diagonalization of
$M^{2}+\text{diag}(\kappa_{\rm SM},c_{\phi})\,T^{2}$, where $\kappa_{\rm SM}$
is defined in Eq. (4) and

$c_{\phi}=\frac{3n_{Q}^{2}}{T^{2}}\frac{1}{r^{4}}-\frac{\lambda_{H\phi}}{6}+\frac{\lambda_{\phi}}{4}.$
(34)

##  Appendix B Beta functions

We computed the RGEs by using the public code SARAH Schienbein  _et al._
(2019); Staub (2014) to be

$\displaystyle 16\pi^{2}\beta_{\lambda_{H}}=$ $\displaystyle
24\lambda_{H}^{2}+\lambda_{H\phi}^{2}-6y_{t}^{4}+\frac{27}{200}g_{1}^{4}+\frac{9}{20}g_{1}^{2}g_{2}^{2}$
$\displaystyle+\frac{9}{8}g_{2}^{4}+\lambda_{H}(-\frac{9}{5}g_{1}^{2}-9g_{2}^{2}+12y_{t}^{2})$
(35) $\displaystyle 16\pi^{2}\beta_{\lambda_{H\phi}}=$
$\displaystyle-\frac{9}{10}g_{1}^{2}\lambda_{H\phi}-\frac{9}{2}g_{2}^{2}\lambda_{H\phi}+12\lambda_{H}\lambda_{H\phi}$
(36)
$\displaystyle-4\lambda_{H\phi}^{2}+8\lambda_{H\phi}\lambda_{\phi}+6\lambda_{H\phi}y_{t}^{2}$
$\displaystyle 16\pi^{2}\beta_{\lambda_{\phi}}=$ $\displaystyle
2\lambda_{H\phi}^{2}+20\lambda_{\phi}^{2}$ (37) $\displaystyle
16\pi^{2}\beta_{\mu_{H}^{2}}=$
$\displaystyle-\frac{9}{10}g_{1}^{2}\mu_{H}^{2}-\frac{9}{2}g_{2}^{2}\mu_{H}^{2}+12\lambda_{H}\mu_{H}^{2}$
(38) $\displaystyle+2\lambda_{H\phi}m_{\phi}^{2}+6\mu_{H}^{2}y_{t}^{2}$
$\displaystyle 16\pi^{2}\beta_{m^{2}_{\phi}}=$ $\displaystyle
4\lambda_{H\phi}\mu_{H}^{2}+8\lambda_{\phi}m^{2}_{\phi}$ (39)

where $y_{t}(m_{t})=0.93$, $g(m_{t})=0.65=g_{2}(m_{t})$,
$g^{\prime}=0.36=\sqrt{3/5}g_{1}(m_{t})$, $g_{3}(m_{t})=1.17$,
$\lambda_{H}(m_{t})=0.13$,
$\mu_{h}^{2}=(88.4\mathinner{\mathrm{GeV}})^{2}$Glioti  _et al._ (2019);
Martin and Robertson (2019). We found agreement with the RGEs in Costa  _et
al._ (2015). For solving these equations, the values of the parameters at the
reference renormalization scale in the $\overline{\mathrm{MS}}$
renormalization scheme were needed. These values were obtained by performing a
shift of the parameters defined at the reference renormalization scale in the
OS renormalization scheme following Biekötter  _et al._ (2021).

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#  MITP/23-011

Perspective on the Little Group

Jens Erler  PRISMA+ Cluster of Excellence & Institute for Nuclear Physics
Johannes Gutenberg University, 55099 Mainz, Germany

###### Abstract

I very briefly review both the historical and constructive approaches to
relativistic quantum mechanics and relativistic quantum field theory including
remarks on the possibility of a non-vanishing photon mass, as well as a
foolhardy speculation regarding dark matter.

##  1 Relativistic Quantum Mechanics

Historically111Most of the remarks on the historical development of the
relativistic quantum theory that follow are inspired by Ref. [1].,
relativistic quantum mechanics (“first quantization”) was simply understood as
the relativistic generalization of the Schrödinger equation [2]. This is
amusing as Schrödinger apparently conceived of the relativistic wave equation
for a free and spinless particle,

$(\partial_{\mu}\partial^{\mu}+m^{2})\phi=0,$ (1)

before Klein [3] and Gordon [4] did so, whose names are associated with it
today. Including the electromagnetic interaction into Eq. (1) fails to produce
the correct fine structure of the hydrogen atom, so that Schrödinger worked in
the non-relativistic limit until after Refs. [3, 4] had been published. Dirac
was dissatisfied with Eq. (1) as it seemed to lead to negative probabilities.
His celebrated relativistic wave equation [5] for a free particle of spin 1/2,

$(i/\partial-m)\psi=0,$ (2)

supplemented with the electromagnetic interaction, was very successful in
correctly predicting the magnetic moment of the electron, the existence of the
positron and the fine structure of hydrogen.

The Dirac theory had several drawbacks. It interpreted positrons as holes in
an otherwise filled infinite sea of negative energy electrons, invoking the
Pauli exclusion principle which is not generalizable to bosons. This means,
e.g., that the treatments of charged pions or the $W^{\pm}$ bosons remain
obscure. Furthermore, Eq. (2) cannot be applied to neutral scalar bosons like
the helium nucleus or the Higgs boson. It should also be noted that nothing
prevents one to introduce a so-called Pauli term [7] with an arbitrary
adjustable coefficient, which has the exact form of a magnetic moment and
would render the prediction from the Dirac theory an accident.

In any case, Dirac constructed his theory for the wrong reason, thinking that
the alleged negative probabilities of the Klein-Gordon theory were the deeper
reason why the prominent matter fields of the time, electrons and protons, had
to have spin 1/2, as otherwise a consistent quantum theory seemed impossible.
Today it is understood that Eq. (2) represents the correct relativistic
equation for spin 1/2 particles, but other values for the spin are perfectly
possible. The negative probabilities enter only when $\phi$ in Eq. (1) is
interpreted as a probability amplitude rather than a quantum operator acting
on a Hilbert space endowed with positive definite probabilities by
construction.

The modern approach seeks to construct a theory uniting the foundations of
quantum mechanics [8] with the principle of special relativity [9] from the
start. The first step is to classify the unitary representations [10] of the
Poincaré (inhomogeneous Lorentz) group [11] resulting in four distinct types:

Massive particles

The one-particle unitary irreducible representations (irreps) with $m^{2}>0$
mediate short-range interactions. To find additional quantum numbers one
chooses the rest frame of the particle, where the momentum four-vector has the
form $P^{\mu}=(m,0,0,0)$. The subgroup of Lorentz transformations which leave
$P^{\mu}$ unchanged (the little group [10]) is the group $SO(3)$ of three-
dimensional spatial rotations so that the familiar integer and half-integer
spin representations of non-relativistic quantum mechanics are recovered.

Massless particles

The irreps with $m^{2}=0$ mediate long-range interactions. To find additional
quantum numbers one may choose a reference frame in which the momentum four-
vector takes the form $P^{\mu}=(k,0,0,k)$. One can see that the group $SO(2)$
of two-dimensional spatial rotations (in the plane orthogonal to the
$z$-direction) leaves the reference vector $P^{\mu}$ unchanged. This gives
rise to the concept of helicity $h$ (the eigenvalue of the third component of
the angular momentum operator) as an additional quantum number222The CPT
theorem [12, 13] implies that helicity states come in pairs, i.e. if the
helicity $h$ of a particle is present, then so is a state with helicity $-h$
and equal but opposite quantum numbers. For example, the single degree of
freedom represented by a right-handed circularly polarised photon ($h=+1$) is
by itself an irrep, as no Lorentz transformation can change the polarisation
of a massless particle. Nevertheless, in a consistent relativistic quantum
theory left-handed circularly polarised photons ($h=-1$) need to be included,
as well. Likewise a massless left-handed neutrino ($h=-1/2$) will always be
accompanied by a right-handed anti-neutrino ($h=+1/2$). However, unlike in the
case of $SO(3)$, there is no algebraic obstruction to allow arbitrary (real)
values for $h$, and one needs to study the topological structure of the
Poincaré group [1]. This involves the computation of its first homotopy group
and one finds that only integer or half-integer values of $h$ are allowed. On
the other hand, in more than four spacetime dimensions it suffices to consider
infinitesimal Lorentz transformations, i.e., to classify the irreps of the
underlying Lie algebra of symmetry generators, where one finds again integer
and half-integer spin representations.

Continuous spin particles

The complete little group for $m^{2}=0$ states is the Euclidian group $ISO(2)$
extending the afore-mentioned $SO(2)$ to include two-dimensional translations.
The latter are those combinations of Lorentz boosts and three-dimensional
rotations that leave the reference momentum four-vector unchanged. However,
the group $ISO(2)$ is neither compact nor semi-simple and any representation
for which the translations act non-trivially, will possess a continuous spin
variable. Such representations have not been observed in Nature333In condensed
matter systems, they may be understood as massless generalisations of anyons
[14]. and not much is known about them, but there are relatively recent
attempts to construct interacting field theories for them [15]. The only
property that is certain is that continuous spin particles would necessarily
interact gravitationally. Since quite some time I have been wondering whether
they may in fact play a role within the dark matter sector, or conversely,
whether there is a way to exclude this possibility. However, as continuous
spin particles are massless, they themselves could constitute dark matter
candidates only if they can form massive bound states, loosely analogous to
glueballs.

Tachyons

The little group for irreps with $m^{2}<0$ is $SO(1,2)$ which as a simple but
non-compact group permits only the trivial and infinite-dimensional
representations. The appearance of tachyonic representations usually signals
the presence of instabilities in a field theory, and may trigger the process
of tachyon condensation, a phenomenon closely related to a second-order phase
transition. It may be possible to remove the tachyons by field re-definitions
as exemplified by the Higgs mechanism [16, 17].

##  2 Quantum Fields

The first relativistic quantum field theories were constructed by quantizing a
given classical field theory (“second quantization”), specifically the Maxwell
equations [18]. This was a complicated and confusing process, in particular
due to the presence of ultra-violet divergencies [19, 20]. The approach was
ultimately successful, but only after the protagonists [21, 22] had almost
given up or were exploring alternatives [23].

Now, the Wigner classification [10] summarised in Section 1 was only a
necessary step to construct a relativistic quantum theory from first
principles. One also needs a unitary (probability conserving) and Lorentz-
invariant (relativistic) scattering matrix ($S$-matrix) for the interacting
theory. The more recent and very successful developments keep Lorentz
covariance manifest, computing helicity amplitudes at the tree [24, 25] and
loop [26] levels. Unitarity which is not manifest in this approach is used as
a tool instead.

Conversely, the traditional formulation of quantum field theory keeps
unitarity manifest. One builds Hamiltonian densities $\cal H$ out of sums of
products of annihilation ($a$) and creation ($a^{\dagger}$) operators for
particles of specific momentum $\vec{p}$, spin (or helicity) and type444This
represents no loss of generality, as any operator may be expressed in this
way.. On the other hand, Lorentz covariance of the $S$-matrix is implemented
by imposing a (sufficient) causality condition, usually in the form,

$[{\cal H}(x),{\pazocal H}(y)]=0\hskip 60.0pt(x-y)^{2}\leq 0.$ (3)

Equation (3) is a condition in configuration space while the one-particle
states and the corresponding $a^{\dagger}$ that create them from the vacuum
have definite momentum. Thus, to construct $\cal H$ we need to introduce
quantum fields as the Fourier transforms of the $a$ and $a^{\dagger}$
operators. Since the latter have definite Lorentz transformation properties,
one can calculate the Lorentz transformation rules of the fields, in contrast
to classical field theories where the Lorentz transformation rule is
postulated. For example, spin 0 particles are represented by Klein-Gordon
fields, spin 1/2 particles by Dirac fields, and massive spin 1 particles by
Proca fields [27]. These fields turn out to have covariant Lorentz
transformations, so that Lorentz-invariant theories can be constructed from
them straightforwardly.

But if $a$ and $a^{\dagger}$ are the annihilation and creation operators for a
massless particle with helicities $h=\pm 1$, then gauge fields like the photon
would be of the form,

$A_{\mu}(x)=\int\frac{d^{3}p}{(2\pi)^{3}2E_{\vec{p}}}\sum_{h=\pm
1}\left[\epsilon_{\mu}(\vec{p},h)e^{-ipx}a(\vec{p},h)+\epsilon^{*}_{\mu}(\vec{p},h)e^{ipx}a^{\dagger}(\vec{p},h)\right],$
(4)

where $\epsilon_{\mu}$ and $\epsilon^{*}_{\mu}$ are polarisation vectors and
$E_{\vec{p}}$ denotes the energy. The transformation behaviour of $A_{\mu}$
turns out to be,

$U(\Lambda)A_{\mu}(x)U(\Lambda)^{-1}={\Lambda^{\nu}}_{\mu}A_{\nu}(\Lambda
x)+\partial_{\mu}\Omega(\Lambda,x),$ (5)

where ${\Lambda^{\nu}}_{\mu}$ is the Lorentz transformation matrix and
$\Omega$ a linear combination of $a$ and $a^{\dagger}$. The last term shows
that $A_{\mu}$ does not transform as a Lorentz vector. One could utilize a
different kind of field, such as an antisymmetric two-index tensor field
$B_{\mu\nu}$, but this will not give rise to long-range interactions. The way
out is to construct $\cal H$ in such a way that it is invariant under a gauge
transformation,

$A_{\mu}(x)\to A_{\mu}(x)+\partial_{\mu}\Omega(x),$ (6)

so that $\cal H$ is invariant under a Lorentz transformation followed by a
gauge transformation. This implies that breaking gauge invariance is
tantamount to breaking Lorentz invariance.

A similar line of arguments shows that the equivalence principle arises when
one attempts to construct a theory for a massless particle of helicity $h=\pm
2$ with long-range interactions. Thus, neither the gauge nor the equivalence
principle need to be postulated in the quantum context (only the very
existence of the massless $h=\pm 1$ or $h=\pm 2$ particles with long-range
interactions).

##  3 Stückelberg Mechanism

In a gauge invariant theory the gauge fields can only enter through gauge
covariant derivatives or the field strength tensor, whereas an explicit
appearance would be incompatible with Eq. (6). In particular, mass terms
violate gauge invariance, explaining why gluons are massless. But for an
Abelian theory like QED there is a loophole [28, 29] (it is absent for
theories based on non-Abelian gauge groups like QCD or general relativity; see
e.g. Ref. [30]). Consider the Lagrangian,

${\cal
L}=-\frac{1}{4}({\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}})^{2}+\frac{1}{2}(mA_{\mu}-\partial_{\mu}B)^{2}-\frac{1}{2}(\partial_{\mu}A^{\mu}+mB)^{2},$
(7)

where $B(x)$ is a real scalar field. The first two terms in ${\cal L}$ are
invariant under the gauge transformation [7],

$A_{\mu}(x)\to A_{\mu}(x)+\partial_{\mu}\Omega(x)\hskip 60.0ptB(x)\to
B(x)+m\Omega(x)$ (8)

while the third term is invariant provided that the gauge parameter
$\Omega(x)$ satisfies the Klein-Gordon equation,

$(\partial_{\mu}\partial^{\mu}+m^{2})\Omega(x)=0.$ (9)

A restriction like (9) is absent in QED, but the important point is that it
does not change the number of degrees of freedom. The second term in ${\cal
L}$ gives rise to a mass term for $A_{\mu}$ and the kinetic term for $B$.

If one now defines the gauge-invariant combination [7],

$V_{\mu}(x)\equiv A_{\mu}(x)-\frac{1}{m}\partial_{\mu}B(x),$ (10)

one obtains,

${\cal
L}=-\frac{1}{4}({\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}})^{2}+\frac{m^{2}}{2}V_{\mu}V^{\mu}-\frac{1}{2}(\partial_{\mu}A^{\mu}+mB)^{2},$
(11)

so that one is left with a massive vector boson $V_{\mu}(x)$ in a theory with
the additional constraint that matrix elements of $\partial_{\mu}A^{\mu}+mB$
taken between physical states must vanish. This is reminiscent of QED where
one needs to demand the vanishing of the matrix elements of
$\partial_{\mu}A^{\mu}$ taken between physical states. Effectively, we have
traded the degree of freedom represented by $B(x)$ for the longitudinal
polarization (the helicity $h=0$ state) of a massive vector boson
$V_{\mu}(x)$, since the first two terms in the Eq. (11) coincide with the
Lagrangian of the Proca field [27] for a massive spin 1 particle. The
difference is that there one has $\partial_{\mu}V^{\mu}=0$ following directly
from the Euler-Lagrange equations.

The fact that the Stückelberg mass derives from a gauge invariant Lagrangian
may be used to show that this theory is perturbatively renormalizable. Indeed,
the last term in Eq. (11) can be interpreted as a gauge-fixing term,
corresponding in the case $m=0$ (QED) to Feynman gauge. For a clear
discussion, see Ref. [31].

##  4 Conclusion

Following the line of thought reviewed in this note, we conclude that there
are three different ways to obtain a mass for the photon.

  * •

One can speculate that Lorentz invariance may not be an exact symmetry of
Nature [32]. Due to the intimate connection between Lorentz and gauge
invariance, one may expect that Lorentz invariance will protect the photon
from acquiring a mass only up to (large) length scales at which it is broken.

  * •

The Stückelberg mechanism adds an additional degree of freedom to the theory
that in a particular gauge may be interpretable as the longitudinal
polarisation (in addition to the two transversal ones) of a massive vector
boson, as in the case of a Proca field. There is an interesting argument [33]
(resting on conjectures) that in the context of gravity the very small photon
masses ($m_{\gamma}\lesssim 10^{-18}$ eV [34]) consistent with
observations555For a review on photon and graviton mass limits, see Ref. [35].
The effect of a finite photon mass on galactic rotation curves through the
modification of plasma electrodynamics has been considered in Ref. [36]. would
imply an ultraviolet cutoff that is too low. A Stückelberg photon mass would
then be ruled out.

  * •

One can also employ the Higgs mechanism [16, 17] in which case the
longitudinal degree of freedom of the photon is provided together with an
additional physical scalar degree of freedom in complete analogy with the
masses of the $W$ and $Z$ bosons in the Standard Model [37]. But one would
face an aggravated hierarchy problem, as the small photon mass would generally
not be protected to be driven to very large masses by radiative corrections.

The most likely way the photon may be tied to the dark matter sector would be
through its mixing with a dark massive photon [38]. Indeed, kinetic mixing
would occur at the renormalizable level, i.e., in principle unsuppressed by
large new physics scales. A variant of this scenario adds mass mixing of the
dark photon with the Standard Model $Z$ boson [39]. Unlike a dark photon, such
a dark $Z$ boson could induce observable effects in atomic parity violation
experiments [40] and parity violating electron scattering [41].

It will also be interesting to study whether the hypothetical continuous spin
particles could exist in Nature, and if so how they would couple to the
Standard Model sector other than gravitationally. And one would need to find
out whether they would play a r$\hat{\rm o}$le in dark sector physics.

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