CERN-TH-2022-075

KEK-TH-2422

Charge transfer between
rotating complex scalar fields

Valerie Domckea,b, Keisuke Harigayaa,c, Kyohei Mukaidad,e

a | _Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland_
---|---
b | _Laboratory for Particle Physics and Cosmology, Institute of Physics,_
| _School of Basic Sciences, EPFL, 1015 Lausanne, Switzerland_
c | _Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan_
d | _Theory Center, IPNS, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan_
e | _Graduate University for Advanced Studies (Sokendai),_
| _1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan_

We consider the transfer of a $U(1)$ charge density between Bose-Einstein
condensates of complex scalar fields coupled to a thermal bath, focusing on
the case of a homogeneous Affleck-Dine field transmitting the charge stored in
its angular motion to an axion field. We demonstrate that in the absence of
additional symmetries this charge transfer, aided by cosmic expansion as well
as the thermal effective potential of the Affleck-Dine field, can be very
efficient. The charge redistribution between the scalar fields becomes
possible if the interactions with the thermal bath break the original
$U(1)\times U(1)$ symmetry down to a single $U(1)$ symmetry; the charge
distribution between the two fields is then determined by minimizing the free
energy. We discuss implications for cosmological setups involving complex
scalars, with applications to axion dark matter, baryogenesis, kination
domination, and gravitational wave production.

###### Contents

  1. 1 Introduction
  2. 2 Charge transfer in the zero-temperature potential
    1. 2.1 Charge transfer through an anomalous coupling to a gauge field
    2. 2.2 Charge transfer including charged fermions
  3. 3 Thermal potential and charge transfer by thermal fluctuations
    1. 3.1 Effective potential for the AD field
    2. 3.2 Effective potential including the axion
    3. 3.3 Phase transition including the thermal corrections
  4. 4 Phenomenological applications
    1. 4.1 Axion dark matter by kinetic misalignment
    2. 4.2 Baryogenesis
    3. 4.3 Kination domination by axion rotation
    4. 4.4 Gravitational waves
  5. 5 Summary and Discussion
  6. A Boltzmann equations for charge transfer
  7. B Thermodynamics of rotating scalar fields
    1. B.1 Preliminary
    2. B.2 Effective potential

##  1 Introduction

Dynamical complex scalar fields may play an important role in the early
Universe. Here we consider (composite) complex scalar fields subject to an
approximate $U(1)$ symmetry. For a homogeneous field configuration, the
corresponding approximately conserved charge corresponds to a rotation of the
complex scalar in field space, i.e., a non-vanishing velocity for the angular
degree of freedom. The angular motion may be a source of dark matter
generation [1]. Moreover, this motion spontaneously breaks $CP$ and can thus
contribute to the generation of a matter antimatter asymmetry [2]. Concrete
realizations of this are the Affleck-Dine (AD) mechanism [3], spontaneous
baryogenesis [4, 5], or axiogenesis [6].

In AD baryogenesis, the role of an rotating complex AD field is naturally
played by a condensate of scalar particles aligned along a flat direction of
the scalar potential of the minimal supersymmetric standard model (MSSM). See
[7] for the list of the MSSM flat directions and [8] for a review on their
dynamics. The rotation is induced by higher-dimensional baryon or lepton
number-violating operators and the charge associated with this rotation is
transferred to thermal bath through the decay of the AD field into Standard
Model (SM) fermions. In the more recently proposed axiogenesis scenario, the
rotating complex field is instead a fundamental scalar, namely the complex
Peccei-Quinn (PQ) field introduced to explain the absence of any observed $CP$
violation in QCD [9, 10]. The angular degree of freedom is referred to as the
axion [11, 12], which simultaneously provides a promising dark matter
candidate [13, 14, 15]. The charge transfer between the rotating axion and the
SM thermal bath occurs via sphaleron interactions. The kinetic energy of the
axion field arising from the rotation of the PQ field is transferred into an
axion dark matter density through the kinetic misalignment mechanism [1].

In this paper, we study the charge transfer between Bose-Einstein condensates
(BECs) of different complex scalar fields in the presence of a thermal bath.
For two or more scalar fields coupled through efficient interactions with a
thermal bath, the $U(1)$ charge will be redistributed among the fields and the
thermal bath to minimize the free energy. If after coupling to the thermal
bath, the system has more than one relevant conserved charge, this additional
symmetry implies that a charge transfer between the scalar fields would be
accompanied by a large chemical potential in the thermal bath so that the free
energy condition disfavours a charge transfer. On the other hand, in the
absence of additional symmetries we find that the charge transfer can be very
efficient, and must be taken into account in cosmologies involving rotating
complex scalar fields. Our work is related to earlier works that discuss the
generation of a non-zero velocity of an axion field by asymmetries of other
particles or fields, such as a quark chiral asymmetry [16], helical magnetic
fields [17], and a baryon asymmetry [18].

Concretely, we study a system of two complex scalar fields coupled to a
thermal bath via sphaleron, Yukawa, and gauge interactions. We consider one of
the fields, the ‘AD field’, to have a large initial charge encoded in an
angular motion. The second field, the ‘PQ field’, is initially at rest at the
minimum of its zero-temperature Mexican-hat scalar potential with the PQ
symmetry spontaneously broken. Minimizing the free energy of this system, we
demonstrate that in the ground state of the system, the charge has largely
been transferred to the axion field (i.e., the angular component of the PQ
field) unless additional conserved symmetries (such as a chiral symmetry for
the fermions of the thermal bath) require that the transfer involve a large
fermion asymmetry. This charge transfer occurs once the transfer rates
governing this process are efficient compared to the Hubble expansion rate and
once the charge associated with the AD field rotation has been red-shifted
close to a critical value. At earlier times, the system is trapped in a state
with the charge largely stored in the rotation of the AD field. For
sufficiently large initial values for the AD field, a phase of AD field
domination with a subsequent kination era by the axion field is possible.

Thermal contributions to the effective potential governing the dynamics of the
AD field and the axion can further facilitate the charge transfer. We compute
the relevant thermodynamic quantities in the presence of rotating scalar
fields to obtain the one-loop thermal effective potential of the coupled
system of the AD field, axion, and thermal bath. If the thermal bath contains
particles which obtain zero-temperature masses from a finite AD field value
(e.g., due to a Yukawa coupling or due to the spontaneous breaking of a gauge
symmetry by the AD field), the resulting contribution to the effective
potential prefers a vanishing AD field value. When the total charge is
sufficiently small (but is still above the critical value), this creates an
absolute minimum at the origin, while the state with a large AD field value
remains a local minimum. This is analogous to the shape of the potential that
appears in thermal inflation models [19, 20]. As Hubble expansion dilutes the
charge of the coupled system, the latter vacuum is eventually destabilized and
all the remaining charge is transferred to the axion field. This process may
occur via nucleation of bubbles of true vacuum in which the axion field is
rotating [21, 22, 23] or through a more gradual phase mixing driven by sub-
critical bubble formation [24, 25, 26, 27]. The former will generate
gravitational waves. In both cases, fluctuations of the axion field are
created, which contribute to axion dark matter. Also, the formation of Q-balls
[28, 29] is possible, and the fluctuations associated with the Q-ball
formation and decay may lead to an additional source of axion dark matter.

The remainder of this paper is organized as follows. In Sec. 2, we discuss
charge transfer neglecting the thermal corrections to the potential of the AD
field and clarify the conditions for the transfer to occur by computing the
effective potential for a given finite charge density. We demonstrate how
additional conserved charges can prevent the charge transfer. In Sec. 3, we
include the thermal correction to the AD field potential and show how the
shape of the potential is modified. Phenomenological implications of the
charge transfer are discussed in Sec. 4. Finally, we give a summary and
discussion in Sec. 5. Technical details can be found in the appendices. App. A
provides numerical solution for the Boltzmann equations of Sec. 2. A
derivation of the effective potential of the Affleck Dine field, key to the
discussion in Sec. 3, is given in App. B.

##  2 Charge transfer in the zero-temperature potential

In this section, we discuss the charge transfer between a homogeneous complex
scalar field $\phi$ and an axion field $a=f\theta_{a}$ and illustrate the
conditions for this charge transfer to occur efficiently. Throughout this
paper, we assume that the radial direction of the complex PQ symmetry breaking
field $P$ is fixed at its VEV$\sim f$, so that we can integrate it out and use
the effective field theory of the axion field without the radial direction,

$\displaystyle P=\frac{1}{\sqrt{2}}fe^{i\theta_{a}}\,.$ (2.1)

In particular, we discuss the dynamics of the axion at $T\ll f,|\phi|$, where
both complex scalar fields can be described as coherently rotating in field
space. This does however not exclude a restoration of the PQ symmetry at
higher temperatures. The era with $|\phi|<T$ is discussed in detail in Sec. 3.

We assume that the complex scalar field $\phi$ initially rotates with a large
radius. The rotation can be initiated by the Affleck-Dine (AD) mechanism [3],
and we call the rotating scalar field $\phi$ the AD field. We refer to the
radial and angular directions of $\phi$ as $r$ and $\theta_{\phi}$,
respectively,

$\displaystyle\phi=\frac{1}{\sqrt{2}}re^{i\theta_{\phi}}.$ (2.2)

To be concrete, we assume that the zero temperature potential of the AD field
is nearly quadratic, $m^{2}|\phi|^{2}$, which is indeed the case when the AD
field is a flat direction in the MSSM. See [7] for the list of MSSM flat
directions. Throughout this paper we will assume $m\ll f$, motivated e.g., by
the QCD axion with an axion decay constant constrained to $f>10^{8}$ GeV and
low scale supersymmetry with a supersymmetry breaking scale of $m\sim$ TeV. In
realistic cases, the zero temperature potential is not exactly quadratic.
Indeed, the MSSM flat directions receive logarithmic corrections by
renormalization group effects,

$\displaystyle V(\phi)=m^{2}|\phi|^{2}\quantity(1+K\,{\rm
ln}\frac{\phi^{2}}{M^{2}}),$ (2.3)

where $K$ is a constant of $O(10^{-3}-10^{-1})$ and $M$ is a mass scale. We
assume $K>0$ so that the rotation is stable against fragmentation into Q-balls
[30]. The potential of the AD field moreover receives thermal corrections,
which we will discuss in Sec. 3.

We introduce interactions that can transfer the charge of the AD field to the
axion field, inducing rotations of the axion field with almost all of the
charges eventually transferred to the axion. For simplicity, we will take the
axion field to be at rest initially, $\dot{\theta}_{a}=0$. We discuss several
models to illustrate the conditions for the efficient transfer to occur. In
particular, Sec. 2.1 computes the charge transfer rate in a situation where
the charge transfer can occur and Sec. 2.2 shows how additional (approximate)
symmetries can prevent the charge transfer.

###  2.1 Charge transfer through an anomalous coupling to a gauge field

#### Setup and conserved charge.

We first consider a simple example that will illustrate the condition on the
free energy for the transfer to occur. We introduce a non-Abelian gauge field
$G$ that couples to the AD and axion fields through a quantum anomaly,

$\displaystyle\left(\theta_{\phi}+\theta_{a}\right)G^{\mu\nu}\tilde{G}_{\mu\nu}.$
(2.4)

Such an interaction of the AD field with a gauge field can be generated
through a fermion which is charged under the non-Abelian gauge group and which
obtains a mass from the coupling with the AD field, see App. B for a concrete
realization. We may neglect the fermion dynamics and simply use the
interaction in Eq. (2.4) as long as the AD field value, and hence the fermion
mass, is much larger than the temperature. This assumption will break down
towards the end of the transfer; we will comment on the relevant fermion
dynamics below. The interaction (2.4) breaks the axion shift symmetry and the
$U(1)$ symmetry of the AD field down to the linear combination,

$\displaystyle\frac{\differential}{\differential
t}\left[R^{3}\left(\dot{\theta}_{\phi}r^{2}-\dot{\theta}_{a}f^{2}\right)\right]=0\,,$
(2.5)

allowing the charge of the AD field to be transferred to the axion field. Here
$R(t)$ denotes the scale factor of the Friedmann Robertson Walker metric. The
total comoving charge $R^{3}q$ remains constant, with the physical charge

$\displaystyle
q=q_{\phi}-q_{a}=\dot{\theta}_{\phi}r^{2}-\dot{\theta}_{a}f^{2}=mr^{2}-\dot{\theta}_{a}f^{2}$
(2.6)

decreasing as $q\propto R^{-3}$ due to cosmic expansion, where in the last
equality we have used
$\dot{\theta}_{\phi}=m$.\natural1\natural1$\natural$1This equality follows
from the AD equation of motion which permits a stationary solution $\dot{r}=0$
for $\dot{\theta}_{\phi}=m$ with equal kinetic and potential energy according
to the virial theorem.

Figure 1: Free energy (2.7) for different values of the total charge $q$,
leading to a finite vev for the AD field for $q$ above the critical value
$q_{c}$. For thermal corrections, see Sec. 3.

#### Free energy.

Assuming that the charge transfer is efficient, whether the charge is
dominantly stored in the AD or axion fields is determined so that the
thermodynamic free energy density of the system is minimized. The contribution
of a coherent rotation to the free energy density is simply the energy
density.\natural2\natural2$\natural$2 The change in (Helmholtz) free energy
corresponds to the maximal amount of work a thermodynamic system can perform
at a constant temperature. It can be obtained from the canonical partition
function, which assumes a constant temperature and the number of particles. In
practice, it is simpler to start from the grand canonical partition function,
which assumes a constant temperature and chemical potential, and then obtain
the Helmholtz free energy through a Legendre transformation. See App. B for
details.  The AD field should have a vanishing ellipticity to minimize the
energy density. Using the charge conservation, the (free) energy density can
then be expressed as a function of the axion velocity $\dot{\theta}_{a}$ or as
a function of the AD field radius $r$,

$\displaystyle\rho=m^{2}r^{2}+\frac{1}{2}\dot{\theta}_{a}^{2}f^{2}$
$\displaystyle=mq+m\dot{\theta}_{a}f^{2}+\frac{1}{2}\dot{\theta}_{a}^{2}f^{2}\,,$
$\displaystyle=m^{2}f^{2}\left[\frac{q^{2}}{2m^{2}f^{4}}+\left(1-\frac{q}{mf^{2}}\right)\frac{r^{2}}{f^{2}}+\frac{r^{4}}{2f^{4}}\right]\,,$
(2.7)

which is minimized at $\dot{\theta}_{a}=-m$. The radius of the AD field at
this equilibrium point is

$\displaystyle r_{\rm eq}^{2}=r_{0}^{2}-f^{2},$ (2.8)

where $r_{0}=\sqrt{q/m}$ is the radius of the AD field when all of the charge
$q$ is stored in the AD field. If $q\gg mf^{2}$ ($r_{0}\gg f$), the charge
remains dominantly stored in the AD field. Since cosmic expansion implies
$r_{0}\propto R^{-3/2}$, even if $r_{0}\gg f$ initially, cosmic expansion will
dilute the charge in the AD field, reducing $r_{0}$ to any critical value
$r_{c}$, triggering the charge transfer to the axion field. Once $q$ ($r_{0}$)
is close to $mf^{2}$ ($f$), the larger fraction of the charge is transferred
to the axion field, and for $q\rightarrow mf^{2}\equiv q_{c}$
($r_{0}\rightarrow f\equiv r_{c}$), almost all charge is ends up in the axion
field as $r_{\text{eq}}\rightarrow 0$, see Fig. 1.

For $q<q_{c}$, $r_{\rm eq}^{2}$ formally becomes negative, signaling the
breakdown of our computation. In fact, as $q$ approaches $q_{c}$, the radius
of the AD field decreases. The mass of the fermion charged under $G$ and
coupling to the AD field, responsible for the quantum anomaly in Eq. (2.4), is
now much smaller than the temperature and hence this fermion can no longer be
integrated out. The chiral asymmetry of the fermions should hence be also
taken into account in the conservation law. Also, the charge asymmetry of the
AD field should be no longer interpreted as a coherent rotation, but is rather
given by particle-antiparticle asymmetry. The asymmetries of the fermion and
the AD field are, however, of the order of $\dot{\theta}_{a}T^{2}$, which is
much smaller than the charge in the axion rotation. With this, we can estimate
the velocity of the axion field from charge conservation,

$\displaystyle\dot{\theta}_{a}=-\frac{q}{f^{2}}=-m\frac{r_{0}^{2}}{f^{2}}.$
(2.9)

See Sec. 3 for a discussion of the transition region $q\sim q_{c}$ accounting
for thermal corrections.

In the above discussion, we implicitly assumed that the thermal bath is
“large”, namely, that the back-reaction of the AD and axion fields to the
thermal bath is negligible. In this picture, the equilibrium state minimizes
the combined free energy of the AD and axion fields. One the other hand, when
the energy of the AD field is comparable to the radiation energy, we must take
into account the backreaction and treat the thermal bath together with the AD
and axion fields as an isolated system. In this picture, we should rather
maximise the entropy of the system, but the equilibrium state is still given
by minimizing the energy of the AD and axion fields, since that is the state
with the maximal temperature of the thermal bath and hence with the maximal
entropy.

#### Transfer rate.

We have so far implicitly assumed that the charge transfer rate is larger than
the Hubble expansion rate. Let us now estimate this transfer rate, see also
App. A for numerical results. The Boltzmann equations governing the transfer
are

$\displaystyle\frac{\differential}{\differential
t}\left(\dot{\theta}_{\phi}r^{2}\right)=-\gamma_{s}T^{2}\left(\dot{\theta}_{\phi}+\dot{\theta}_{a}\right)-3H\dot{\theta}_{\phi}r^{2},$
$\displaystyle\frac{\differential}{\differential
t}\left(\dot{\theta}_{a}f^{2}\right)=-\gamma_{s}T^{2}\left(\dot{\theta}_{\phi}+\dot{\theta}_{a}\right)-3H\dot{\theta}_{a}f^{2}.$
(2.10)

Here the first terms encode the charge transfer due to sphaleron processes,
where $\gamma_{s}=c_{\text{sph}}\,\alpha_{G}^{4}T$ is the sphaleron transition
rate with e.g. $c_{\text{sph}}\simeq 100$ for SU(3) sphalerons in the SM and
$\alpha_{G}=g^{2}/(4\pi)$ denoting the gauge coupling constant, and the second
terms account for the cosmic expansion. At the equilibrium,
$\tfrac{\differential}{\differential t}(R^{3}\dot{\theta}_{a}f^{2})=0$, this
yields $\dot{\theta}_{a}=-\dot{\theta}_{\phi}=-m$, confirming the computation
based on the free energy.

For $r_{0}>f$, starting from the initial condition $\dot{\theta}_{a}=0$, the
velocity of the axion field approaches the equilibrium value
$-\dot{\theta}_{\phi}$ sourced by the term
$\gamma_{s}T^{2}\dot{\theta}_{\phi}/f^{2}$. The rate of the transfer of
comoving charge from the AD to axion fields is therefore given by

$\displaystyle\Gamma(r_{0}>f)=-\frac{\frac{\differential}{\differential
t}(q_{a}R^{3})}{(q_{a}-q_{a}^{\text{eq}})R^{3}}=\frac{\gamma_{s}T^{2}(\dot{\theta}_{\phi}+\dot{\theta}_{a})}{f^{2}(\dot{\theta}_{a}+m)}=\frac{T^{2}}{f^{2}}\gamma_{s}\qquad\text{with}\quad
q_{a}^{\text{eq}}=-mf^{2},$ (2.11)

which is suppressed by a factor $T^{2}/f^{2}$ in comparison with the sphaleron
transition rate $\gamma_{s}$. Note that in this regime, only a very small
fraction of the initial AD charge is transferred to the axion,
$\dot{q}_{\phi}/q_{\phi}\ll\dot{q}_{a}/q_{a}$.

For $r_{0}<f$, charge conservation implies the equilibrium value
$\dot{\theta}_{a}=-mr_{0}^{2}/f^{2}$. The rate is therefore

$\displaystyle\Gamma(r_{0}<f)=\frac{\frac{\differential}{\differential
t}(q_{a}R^{3})}{q_{a}^{\text{eq}}R^{3}}\simeq\frac{-\gamma_{s}T^{2}\dot{\theta}_{\phi}}{-mr_{0}^{2}}=\frac{T^{2}}{r_{0}^{2}}\gamma_{s}\qquad\text{with}\quad
q_{a}^{\text{eq}}=-mr_{0}^{2},$ (2.12)

which is enhanced compared to Eq. (2.11) by a factor of $f^{2}/r_{0}^{2}$. In
this regime, the charge density in the AD field is rapidly decreasing,
$\dot{q}_{\phi}/q_{\phi}\sim\dot{r}/r\sim\gamma_{s}T^{2}/r_{0}^{2}$. Note that
growth rate of the axion charge in Eq. (2.12) is the same as this depletion
rate for the AD field. This can be understood from the observation that once
the AD field is depleted, the charge conservation requires that the depleted
charge go to the axion field. In this picture, $r_{0}<f$ is crucial, so that
at the equilibrium the AD field loses almost all of its charge density. Note
that we have assumed $r_{0}>T$. Otherwise, the coherent picture of the AD
field is not applicable and the transfer rate is simply given by $\gamma_{s}$.

To determine if the charge transfer is efficient, we compare these comoving
charge transfer rates with the Hubble expansion rate. For $r_{0}>f$, the
transfer rate decreases in proportion to $T^{3}$, faster than the Hubble
expansion rate does. If the equilibrium was not reached initially, it is never
reached when $r_{0}>f$. Even if the equilibrium is reached initially, the
decoupling may occur later. For these cases, the axion velocity at a given
temperature is given by

$\displaystyle\dot{\theta}_{a}\simeq-m\times\text{min}\left[1,\frac{\Gamma(r_{0}>f)}{3H}\right]=-\frac{\gamma_{s}T^{2}m}{3f^{2}H}\propto
T\quad\text{for $\Gamma\ll H$}.$ (2.13)

Note that the axion velocity decreases only linearly with $T$, slower than
what would be caused by the Hubble friction ($\propto T^{3}$), so the axion
velocity at a given temperature is dominated by the charge transfer at that
temperature and rather insensitive to previous charge transfer at higher
temperatures. Once $r_{0}$ drops below $f$, the transfer rate remains constant
(recall $r_{0}^{2}\propto q\propto T^{3}$) and thus eventually dominates over
the Hubble expansion rate. Even if the axion field has not yet reached the
equilibrium value when $r_{0}\sim f$ (for which the AD field would collapse
toward the origin given an efficient charge transfer), the equilibrium is
eventually reached, and almost all charge is transferred into the axion field.
In summary, cosmic expansion acts towards removing the metastable minimum of
vacuum potential of the AD field at finite field value $r_{\text{eq}}$ and
eventually ensures an efficient transfer of the AD charge to the axion field.

#### Grand canonical ensemble.

Before closing this subsection, we compute the equilibrium value of $r$ using
the grand canonical ensemble with the chemical potential $\mu$ associated with
the total charge $q$ and obtaining the free energy density by a Legendre
transformation. This framework is reviewed in Appendix B and used in the
remainder of this paper. The effective potential from the rotation of the AD
field and axion fields under a fixed chemical potential $\mu$
is\natural3\natural3$\natural$3 Here the subscript $\mu$ is a reminder that
this effective potential is obtained under a fixed $\mu$.

$\displaystyle
V_{\mu,\text{eff}}(r,\mu)=\frac{1}{2}m^{2}r^{2}-\frac{1}{2}\mu^{2}(r^{2}+f^{2}),$
(2.14)

which is related to the thermodynamic pressure $p$ after extremizing it with
respect to $r$ (see Appendix B). The relation between $q$ and $\mu$ is

$\displaystyle q=-\frac{\partial
V_{\mu,\text{eff}}(r,\mu)}{\partial\mu}=\mu(r^{2}+f^{2}).$ (2.15)

In our context, we would like to minimize the free energy for a fixed $q$
rather than a fixed $\mu$ since we do not have the bath for the charge $q$.
Hence, the effective potential in our context is obtained through the Legendre
transformation,

$\displaystyle V_{\rm eff}(r,q)=V_{\mu,\text{eff}}(r,\mu)+\mu
q=\frac{1}{2}m^{2}r^{2}+\frac{q^{2}}{2(r^{2}+f^{2})}.$ (2.16)

This is minimized at $r^{2}=q/m-f^{2}$, implying $\mu=m$. Using the relation
(2.15) and $q=\dot{\theta}_{\phi}r^{2}-\dot{\theta}_{a}f^{2}$, we obtain the
equilibrium value $\dot{\theta}_{\phi}=-\dot{\theta}_{a}=m$.

###  2.2 Charge transfer including charged fermions

#### Conserved charges and free energy.

We next consider an example that will illustrate an important condition on the
symmetries of the system for the transfer to occur in the presence of
additional (approximate) symmetries. To the system discussed in the previous
subsection, let us add a gauge charged Dirac fermion pair $\psi$ and
$\overline{\psi}$. If the Dirac fermion is massless, there are three conserved
charges (up to cosmic expansion),

$\displaystyle\dot{\theta}_{\phi}r^{2}-\dot{\theta}_{a}f^{2}\equiv q(\neq
0)\,,\quad\frac{1}{2}\left(q_{\psi}+q_{\bar{\psi}}\right)-\dot{\theta}_{a}f^{2}\equiv
q_{A}(=0)\,,\quad q_{\psi}-q_{\bar{\psi}}\equiv q_{B}(=0),$ (2.17)

where $q_{\psi}$ and $q_{\bar{\psi}}$ are particle-antiparticle asymmetry of
$\psi$ and $\bar{\psi}$, respectively, and we have chosen the initial
condition so that the latter two conserved charges vanish. The associated
chemical potentials are $\mu$, $\mu_{A}$, and $\mu_{B}$, respectively. The
chemical potential-dependent part of the effective potential is (see App. B)

$\displaystyle-
V_{\mu,\text{eff}}(r,T,\bm{\mu})\supset\frac{1}{2}\mu^{2}r^{2}+\frac{1}{2}\quantity(\mu+\mu_{A})^{2}f^{2}+\frac{d_{\psi}}{6}\quantity(\frac{1}{4}\mu_{A}^{2}+\mu_{B}^{2})T^{2},$
(2.18)

where $\bm{\mu}$ collectively denotes the chemical potentials
$\bm{\mu}=(\mu,\mu_{A},\mu_{B})$, and $d_{\psi}$ is the dimension of the gauge
representation of $\psi$, i.e., $d_{\psi}=1$ for a gauge-singlet Dirac
fermion. The relation between the conserved charges and chemical potentials is

$\displaystyle q=-\frac{V_{\mu,\text{eff}}(r,T,\bm{\mu})}{\partial\mu}$
$\displaystyle=\quantity(\mu+\mu_{A})f^{2}+\mu r^{2},$ $\displaystyle
q_{A}=-\frac{V_{\mu,\text{eff}}(r,T,\bm{\mu})}{\partial\mu_{A}}$
$\displaystyle=\quantity(\mu+\mu_{A})f^{2}+\frac{d_{\psi}}{12}\mu_{A}T^{2},$
$\displaystyle
q_{B}=-\frac{V_{\mu,\text{eff}}(r,T,\bm{\mu})}{\partial\mu_{B}}$
$\displaystyle=\frac{d_{\psi}}{3}\mu_{B}T^{2}.$ (2.19)

Using $q_{A}=q_{B}=0$, we obtain

$\displaystyle\mu$
$\displaystyle=\frac{q}{r^{2}}\frac{1+d_{\psi}T^{2}/(12f^{2})}{1+d_{\psi}T^{2}/(12r^{2})+d_{\psi}T^{2}/(12f^{2})},$
$\displaystyle\mu_{A}$
$\displaystyle=-\frac{q}{r^{2}}\frac{1}{1+d_{\psi}T^{2}/(12r^{2})+d_{\psi}T^{2}/(12f^{2})},$
$\displaystyle\mu_{B}$ $\displaystyle=0.$ (2.20)

The fixed-$q$ effective potential from the AD and axion rotations and the
asymmetry of $\psi$ is obtained as

$\displaystyle V_{\rm eff}(r,T,q)$
$\displaystyle=V_{\mu,\text{eff}}(r,T,\bm{\mu})_{q_{A,B}=0}+\mu q$
$\displaystyle=\frac{1}{2}m^{2}r^{2}+\frac{q^{2}}{2r^{2}}\frac{1+d_{\psi}T^{2}/(12f^{2})}{1+d_{\psi}T^{2}/(12r^{2})+d_{\psi}T^{2}/(12f^{2})}\simeq\frac{1}{2}m^{2}r^{2}+\frac{q^{2}}{2r^{2}},$
(2.21)

where we assume $r,f\gg T$ in the last inequality. The effective potential is
minimized at $r^{2}\simeq q/m=r_{0}^{2}$. At the equilibrium, the charge of
the axion rotation is

$\displaystyle\quantity(\mu+\mu_{A})f^{2}=\frac{d_{\psi}}{12}mT^{2}\frac{1}{1+d_{\psi}T^{2}/(12f^{2})}\simeq\frac{d_{\psi}}{12}mT^{2},$
(2.22)

where we used $f\gg T$ in the second equality.

In contrast to the situation discussed in Sec. 2.1, the axion obtains a
subdominant fraction of the charge as long as $q>mT^{2}$ (i.e., $r_{0}>T$)
even if $r_{0}\lesssim f$. This is due to the second conservation law in Eq.
(2.17); for the axion rotation to obtain a large charge, the charge asymmetry
of the fermion must be also large, leading to a large free energy. If the
Dirac fermion has a non-zero Dirac mass, the conservation law is violated and
the axion field can obtain a large charge. In general, in order for the axion
field to obtain most of the charge, all symmetries that would require large
charge asymmetry of fermions must be violated. For example, any chiral
symmetry with $G$ anomaly must be explicitly broken.

#### Transfer rate.

Let us estimate the transfer rate including the mass term
$m_{\psi}\psi\overline{\psi}$. The Boltzmann equation is given by

$\displaystyle\frac{\differential}{\differential
t}\left(\dot{\theta}_{\phi}r^{2}\right)=$
$\displaystyle-\gamma_{s}\left(\dot{\theta}_{\phi}T^{2}+\dot{\theta}_{a}T^{2}+q_{\chi}\right)-3H\dot{\theta}_{\phi}r^{2},$
$\displaystyle\frac{\differential}{\differential
t}\left(\dot{\theta}_{a}f^{2}\right)=$
$\displaystyle-\gamma_{s}\left(\dot{\theta}_{\phi}T^{2}+\dot{\theta}_{a}T^{2}+q_{\chi}\right)-3H\dot{\theta}_{a}f^{2},$
$\displaystyle\frac{\differential}{\differential t}q_{\chi}=$
$\displaystyle-2\gamma_{s}\left(\dot{\theta}_{\phi}T^{2}+\dot{\theta}_{a}T^{2}+q_{\chi}\right)-\gamma_{\chi}q_{\chi}-3Hq_{\chi},$
(2.23)

where $\gamma_{\chi}$ is the chiral symmetry breaking rate
$\sim\alpha_{G}m_{\psi}^{2}/T$ and $q_{\chi}=q_{\psi}+q_{\overline{\psi}}$.
Starting from the initial condition $\dot{\theta}_{a}T^{2}=q_{\chi}=0$ and
$\dot{\theta}_{\phi}=m$, the fermion charge $q_{\chi}$ first reaches the
quasi-equilibrium value determined by
$\differential(q_{\chi}R^{3})/\differential t\simeq\differential
q_{\chi}/\differential t=0$,

$\displaystyle q_{A,{\rm
eq}}=-\frac{2\gamma_{s}}{\gamma_{\chi}+2\gamma_{s}}\left(\dot{\theta}_{\phi}+\dot{\theta}_{a}\right)T^{2},$
(2.24)

with a rate

$\displaystyle\frac{\frac{\differential}{\differential
t}(R^{3}q_{\chi})}{R^{3}q_{\chi}^{\text{eq}}}\simeq-\frac{2\gamma_{s}\quantity(\dot{\theta}_{\phi}+\dot{\theta}_{a})T^{2}}{q_{A,{\rm
eq}}}=\gamma_{\chi}+2\gamma_{s},$ (2.25)

for $q_{\chi}\ll q_{\chi}^{\text{eq}}$. As we will see shortly, the system
reaches the true equilibrium state with a rate suppressed by $T^{2}/{\rm
min}(r_{0}^{2},f^{2})$ and the assumption of $q_{\chi}$ reaching the quasi-
equilibrium value is consistent. Putting $q_{A,{\rm eq}}$ to Eq. (2.24), we
obtain the Boltzmann equation of $\dot{\theta}_{\phi}r^{2}$ and
$\dot{\theta}_{a}f^{2}$ at this quasi-equilibrium,

$\displaystyle\frac{\differential}{\differential
t}\left({R^{3}}\dot{\theta}_{\phi}r^{2}\right)=\frac{\differential}{\differential
t}\left({R^{3}}\dot{\theta}_{a}f^{2}\right)=-\frac{\gamma_{s}\gamma_{\chi}}{\gamma_{\chi}+2\gamma_{s}}\quantity(\dot{\theta}_{\phi}+\dot{\theta}_{a})T^{2}{R^{3}}.$
(2.26)

For $\gamma_{\chi}\gg\gamma_{s}$, we recover Eq. (2.10), i.e., an efficient
charge between the AD field and the axion.

Applying the same discussion as Sec. 2.1, we obtain the transfer rate from the
AD to axion field

$\displaystyle\Gamma=\frac{\gamma_{s}\gamma_{\chi}}{\gamma_{\chi}+2\gamma_{s}}\frac{T^{2}}{f^{2}+r_{0}^{2}}\simeq{\rm
min}\quantity(\gamma_{s},\gamma_{\chi})\times\frac{T^{2}}{{\rm
min}\quantity(f^{2},r_{0}^{2})}.$ (2.27)

As anticipated, the rate is much smaller than the rate in Eq. (2.25) and the
assumption of $q_{\chi}$ first reaching the quasi-equilibrium value is
justified. For $\gamma_{\chi}<\gamma_{s}$, the transfer rate $\Gamma$
decreases slower than $H$ does even if $r_{0}>f$. Thus, even if the transfer
is not effective at high temperatures, it can be effective at low
temperatures.

##  3 Thermal potential and charge transfer by thermal fluctuations

In the previous section, we have shown how charge can be transferred from an
AD field to an axion field assuming a zero-temperature quadratic potential for
the AD field. In realistic setups, the AD field couples to the thermal bath
and may obtain a non-negligible thermal potential. In this section, we discuss
the effect of thermal corrections to the potential and show that the charge
transfer may involve thermal tunneling.

###  3.1 Effective potential for the AD field

#### Setup.

Let us first discuss the dynamics of the AD field taking into account thermal
corrections, but for now ignoring the charge transfer to the axion field. We
consider the following exemplary setup:

  * •

The AD field spontaneously breaks a gauge symmetry, giving a mass $gr$ to the
gauge bosons.

  * •

The AD field moreover couples to fermions $\psi$ in a thermal bath (whose mass
does not depend on the AD field), such that the charge of the AD field can be
transferred into a particle-antiparticle asymmetry of these particles.

The conserved charge is

$\displaystyle q=q_{\phi}+q_{\psi}$ (3.1)

with the associated chemical potential $\mu$. Without loss of generality, we
take $q>0$. In realistic setups, there typically also exist fermions which
contribute to the conserved charge $q$ and obtain masses from the AD field. We
analyse such a case in App. B and find that the qualitative behavior of the
potential does not change.

#### Local minima of the effective potential.

Following the formulation described in Appendix B, parts of the effective
potential depending on the AD field radius $r$ and the total charge $q$ are
given by

$\displaystyle V_{\rm eff}(r,T,q)=$ $\displaystyle
V_{0}(r)+V_{g}(r,T)+V_{q}(r,T,q),$ $\displaystyle V_{0}(r)=$
$\displaystyle\frac{1}{2}m^{2}r^{2},~{}~{}\quad
V_{g}(r,T)=2N_{g}\frac{T^{4}}{2\pi^{2}}\int\differential xx^{2}{\rm
log}\left(1-e^{-\sqrt{x^{2}+g^{2}r^{2}/T^{2}}}\right),$ $\displaystyle
V_{q}(r,T,q)=$
$\displaystyle\frac{1}{12}\mu(r,T,q)^{2}\quantity(T^{2}+6r^{2})+\frac{1}{8\pi^{2}}\mu(r,T,q)^{4},$
(3.2) $\displaystyle q=$
$\displaystyle\frac{1}{6}\mu(r,T,q)\quantity(T^{2}+6r^{2})+\frac{1}{6\pi^{2}}\mu(r,T,q)^{3}.$
(3.3)

Here $V_{0}$ is the vacuum potential of the AD field and $V_{g}$ is the free
energy of $N_{g}$ gauge fields which obtain a mass from the AD field. (To be
precise, a resummation to include the thermal mass of the gauge boson is
necessary, but we find that this does not change the qualitative behavior.)
$V_{q}$ is the $q$-dependent part of the free-energy contribution associated
with the asymmetry of $\psi$ and the rotation of the AD field. The chemical
potential $\mu$ should be determined as a function of $(r,T,q)$ according to
Eq. (3.3). The fluctuations of the AD field also contribute to the free
energy, but their inclusion does not change the qualitative behaviors of the
free energy; see App. B.

In Fig. 2, we show the effective potential as a function of $r$ for several
values of $q$. Here we take $m=10^{-4}T$. At large $q>T^{3}$, the free-energy
density has a unique minimum at $r\simeq\sqrt{q/m}=r_{0}$. As $q$ approaches
$T^{3}$, $r_{0}$ becomes a local minimum and a global minimum appears at a
smaller $r$. For $q\ll T^{3}$, $r_{0}$ is no longer a local minimum and the
unique minimum is at $r=0$. This behaviour is due to several competing effects
as we explain below. Large values of the AD charge imply a centrifugal force
which creates an a local minimum for the radial component of the AD field at
finite $r$. This is completely analogous to our observation in Sec. 2.1 before
the charge transfer to the axion becomes efficient. On the other hand, thermal
masses for the gauge bosons drive a restoration of the gauge symmetry at high
$T$, thus pushing the order parameter $r$ of the spontaneous gauge symmetry
breaking to zero. Consequently we expect a global minimum at $r=0$ to develop
once $q/T^{3}\lesssim 1$ [29].

Figure 2: Effective potential including thermal contributions for the case
without a charge transfer to the axion as a function of the radius of the AD
field $r$ for several total charges $q$. The left panel shows a global
picture, while the right panel focuses on the potential around the minimum at
$r_{0}$.

Let us understand this behavior analytically. The first and second derivatives
of $V_{\rm eff}$ are

$\displaystyle\frac{\partial V_{\rm eff}}{\partial r}=$
$\displaystyle\frac{\partial\quantity(V_{0}+V_{g})}{\partial r}-\mu^{2}r,$
$\displaystyle\frac{\partial^{2}V_{\rm eff}}{\partial r^{2}}=$
$\displaystyle\frac{\partial^{2}\quantity(V_{0}+V_{g})}{\partial
r^{2}}-\mu^{2}-2\mu r\frac{\partial\mu}{\partial r}.$ (3.4)

Thus, $V_{\rm eff}$ has extrema at $r=0$ and at non-zero $r$ which satisfy
$m^{2}+(\partial V_{g}/\partial r)/r=\mu(r,T,q)^{2}$. When $\partial
V_{g}/\partial r$ is negligible, in the limit $m\ll T$, the latter has a
solution $r^{2}\simeq q/m-T^{2}/6\equiv r_{\rm eq}^{2}$. So when $q\gg
mT^{2}$, there is an extrema at $r=\sqrt{q/m}=r_{0}$.

The second derivative at $r=0$ is

$\displaystyle\left(\frac{\partial^{2}V_{\rm eff}}{\partial
r^{2}}\right)_{r=0}=m^{2}+\frac{g^{2}N_{g}}{6}T^{2}-\mu^{2}.$ (3.5)

By using the relation between $q$ and $\mu$ at $r=0$, one can show that if
$q\gtrsim 0.07g\sqrt{N_{g}}T^{3}$, the third term dominates and the second
derivative is negative; $r=0$ is a maximum. If $q\lesssim
0.07g\sqrt{N_{g}}T^{3}$, $r=0$ is an minimum. This can be intuitively
understood in the following way. Having $r=0$ requires that all charge be
stored in the chiral asymmetry of the fermions and hence the chemical
potential is fixed. For large $q$, the chemical potential is so large that the
thermal mass from the gauge bosons cannot keep the AD field at the origin and
hence a BEC forms.

To obtain the second derivative at $r=r_{\rm eq}$, we need
$\partial\mu/\partial r$. This can be obtained by differentiating Eq. (3.3)
with respect to $r$. This yields

$\displaystyle\left(\frac{\partial^{2}V_{\rm eff}}{\partial
r^{2}}\right)_{r=r_{\rm
eq}}=\left[V_{0}^{\prime\prime}+V_{g}^{\prime\prime}-\frac{V_{0}^{\prime}+V_{g}^{\prime}}{r}+\frac{4r\quantity(V_{0}^{\prime}+V_{g}^{\prime})}{r^{2}+\frac{1}{6}T^{2}+\frac{1}{24\pi^{2}r}\quantity(V_{0}^{\prime}+V_{g}^{\prime})}\right]_{r=r_{\rm
eq}},$ (3.6)

where the primes denote the derivative with respect to $r$. For $q\gg mT^{2}$,
$r_{\rm eq}^{2}\simeq q/m\gg T^{2}$. For such large $r$, $V_{g}^{\prime}$ is
exponentially suppressed and

$\displaystyle\left(\frac{\partial^{2}V_{\rm eff}}{\partial
r^{2}}\right)_{r=r_{\rm eq}}\simeq 4m^{2}>0,$ (3.7)

so $r_{\rm eq}$ is a minimum. As $q$ approaches $mT^{2}$, $r_{\rm eq}$
approaches $T$ and $V_{g}$ is not negligible, and the minimum at $r_{\rm eq}$
eventually disappears. The non-zero value of $r$ for large enough $q$ can be
understood as a result of the centrifugal force provided by the rotation.

For $mT^{2}\ll q\lesssim 0.07g\sqrt{N_{g}}T^{3}$, there are two minima at
$r=0$ and $r_{\rm eq}$. The free energies at these two minima are

$\displaystyle V_{\rm eff}\quantity(0,T,q)\simeq$ $\displaystyle 3q^{4/3}\,,$
$\displaystyle V_{\rm eff}\quantity(r_{0},T,q)\simeq$ $\displaystyle
mq+\frac{\pi^{2}N_{g}}{15}T^{4}\simeq\frac{\pi^{2}N_{g}}{15}T^{4}\,.$ (3.8)

Therefore, for $q\lesssim 0.3N_{g}^{3/4}T^{3}$, the minimum at $r=0$ is the
global minimum and that at $r_{\rm eq}$ is a local minimum. The inequality is
indeed satisfied for $q\lesssim 0.07g\sqrt{N_{g}}T^{3}$ with perturbative $g$.

To sum up, if $q\gtrsim 0.07g\sqrt{N_{g}}T^{3}$, there is a unique minimum at
$r^{2}=q/m$ and if $mT^{2}\ll q\lesssim 0.07g\sqrt{N_{g}}T^{3}$, there is a
global minimum at $r=0$ and a local minimum at $r^{2}=q/m$. The local minimum
disappears as $q$ approaches $mT^{2}$.

#### Additional contributions to the thermal potential.

Before closing this subsection, we point out another possible contribution to
thermal potential of $r$ that appears in realistic setups and can destabilize
the local minimum at $r^{2}=q/m$. After integrating out the heavy degrees of
freedom that obtain masses from the AD field, some of the couplings of the low
energy effective theory may depend on the AD field value. Consequently, the
corrections to the free energy arising from these couplings depend on the AD
field value, generating additional thermal potential of the AD field. Let us
consider an exemplary situation where the gauge coupling constant depends on
the AD field and a thermal potential $\sim\alpha^{2}T^{4}{\rm
ln}(r^{2}/T^{2})$ is generated [31]. The potential gradient from this term
exceeds the gradient arising from $V_{0}+V_{q}$ at $r^{2}=q/m$ if
$q<\alpha^{2}T^{4}/m$ (assuming $T<q/T^{3}(m/\alpha^{2})$), leading to a
destabilization of the local minimum at $r^{2}=q/m$.

###  3.2 Effective potential including the axion

We now include the axion field. As we will see, in this case the origin $r=0$
becomes the absolute minimum more easily than in the case discussed above,
which neglected the charge transfer to the axion field. This is in line with
the discussion in Sec. 2.1, where we saw that the charge transfer to the axion
drives the local minimum for the AD field to zero. The total conserved charge
is now

$\displaystyle q=q_{\phi}+q_{\psi}-q_{a}$ (3.9)

with the associated chemical potential $\mu$.

#### Local minima of the effective potential.

The effective potential is given by (see App. B)

$\displaystyle V_{\rm eff}\quantity(r,T,q)=$ $\displaystyle
V_{0}(r)+V_{g}(r,T)+V_{q,a}(r,T,q),$ $\displaystyle V_{q,a}\quantity(r,T,q)=$
$\displaystyle\frac{1}{12}\mu(r,T,q)^{2}\quantity(T^{2}+6r^{2}+6f^{2})+\frac{1}{8\pi^{2}}\mu(r,T,q)^{4},$
(3.10) $\displaystyle q=$
$\displaystyle\frac{1}{6}\mu(r,T,q)\quantity(T^{2}+6r^{2}+6f^{2})+\frac{1}{6\pi^{2}}\mu(r,T,q)^{3}.$
(3.11)

The expression is greatly simplified when $q\ll Tf^{2}$, for which the
$\mu^{4}$ term in $V_{q,a}$ and the $\mu^{3}$ term in $q$ are always
negligible. Then $V_{q,a}$ is given by

$\displaystyle
V_{q,a}\quantity(r,T,q)\simeq\frac{3q^{2}}{6\quantity(r^{2}+f^{2})+T^{2}}.$
(3.12)

In the upper panel of Fig. 3, we show the effective potential as a function of
$r$ for several representative values of $q$. When $q$ is very large, there is
a unique minimum at $r=r_{0}$. As $q$ decreases, even if $q>T^{3}$, $r_{0}$
becomes only a local minimum and the global minimum is at $r=0$. As $q$
approaches $q_{c}$, the local minimum approaches $0$ and eventually
disappears. The lower panel shows the evolution of the effective potential for
a fixed $m$, $f$, and $q/T^{3}$ around the time when the transfer completes.
Here $T_{c}$ is the temperature at which $q=mf^{2}$. The minimum at $r_{0}$
becomes an local one as $T$ approaches $T_{c}$, and disappears for $T<T_{c}$.

Figure 3: Effective potential including thermal contributions for the case
with charge transfer to the axion as a function of the radius of the AD field
$r$ for several total charges $q$.

Let us understand this behavior analytically. The first and the second
derivatives of $V_{\rm eff}$ are given by Eq. (3.1), and the second derivative
at $r=0$ is given by Eq. (3.5). When $q\gg f^{2}T$, Eq. (3.11) gives
$\mu\quantity(0,T,q)\simeq q/f^{2}\gg T$. Then the second derivative of
$V_{\rm eff}$ at $r=0$ is negative, so the origin cannot be a minimum.
Instead, there is a unique minimum at $r$ such that $\mu\quantity(r,q,T)\simeq
m$, namely, $r^{2}\simeq q/m(\gg f^{2})$. When $q<f^{2}T$,
$\mu\quantity(0,T,q)\simeq q/f^{2}<T^{2}$, so the origin is a minimum. For
$q>mf^{2}$, $r_{\rm eq}$ is also a minimum. The effective potentials at $r=0$
and $r_{\rm eq}$ are

$\displaystyle V_{\rm eff}\quantity(0,T,q)\simeq$
$\displaystyle\frac{q^{2}}{2f^{2}},$ $\displaystyle V_{\rm
eff}\quantity(r_{\rm eq},T,q)\simeq$ $\displaystyle
mq+\frac{\pi^{2}N_{g}}{15}T^{4}.$ (3.13)

When $q>(2\pi^{2}N_{g}/15)^{1/2}fT^{2}$, $r_{\rm eq}$ has a smaller effective
potential and is the global minimum, while for
$q<(2\pi^{2}N_{g}/15)^{1/2}fT^{2}$, $r=0$ is the global minimum. As $q$
approaches $mf^{2}$, $r_{\rm eq}=\sqrt{q/m-f^{2}}$ approaches $T$, and $V_{g}$
eventually destabilizes the local minimum at $r_{\rm eq}$.

If $q/T^{3}>(2\pi^{2}N_{g}/15)^{3/4}(f/m)^{1/2}$, $q$ drops below $mf^{2}$
before it drops below $(2\pi^{2}N_{g}/15)^{1/2}fT^{2}$. In this case, $r_{\rm
eq}$ continues to be the global minimum until $q$ becomes very close to
$mf^{2}$. For such a large $q/T^{3}$, the AD field dominates the energy
density of the universe before $q$ reaches the critical value and kination
domination occurs after the charge is transferred into the axion field, see
Sec. 4.

###  3.3 Phase transition including the thermal corrections

With the effective potentials we have shown above, we now discuss the
evolution of the AD field and the axion field towards the end of the charge
transfer around $q=q_{c}$. We assume $q/T^{3}\gg 1$ and $T\gg m$.

When $q>(2\pi^{2}N_{g}/15)^{1/2}fT^{2}$, the state with almost all charge in
the rotation of the AD field has the least free-energy, and the rotation with
$r^{2}=q/m-f^{2}\simeq q/m$ is absolutely stable. As the temperature
decreases, $q$ drops below $(2\pi^{2}N_{g}/15)^{1/2}fT^{2}$ and $r^{2}\simeq
q/m$ becomes a local minimum, while the minimal free energy is achieved by the
state where almost all of charges are in the axion rotation and the AD field
is trapped around the origin. A thermal transition from the local minimum to
the absolute minimum can occur, but since $r^{2}\simeq q/m\gg T^{2}$, we
expect that the transition is suppressed and does not occur within the
cosmological time scale. As $q$ approaches $q_{c}=mf^{2}$, $r^{2}=q/m-f^{2}$
approaches $T^{2}$, and the transition to the absolute minimum may become
efficient. Note that at this point $q-q_{c}$ is much smaller than $q_{c}$, so
almost all charges have been already transferred into the axion field before
the thermal transition becomes effective. The thermal transition may occur as
a first order phase transition, which proceeds via nucleation of bubbles [21,
22, 23], or as phase mixing, where the two minima are populated via creation
of subcritical bubbles [24, 25, 26, 27].

#### Uncertainties in the phase transition dynamics.

A full investigation of the transition dynamics is beyond the scope of the
present paper. Instead we highlight only some differences from the usual phase
transition problem: First, the fields relevant for the phase transition rotate
rather than just moving in the radial direction. The effect of the angular
motion in field space and associated conserved charges must hence be taken
into account. Second, the charge can be transferred from the AD field both to
the axion field and the thermal bath. Whereas the transfer rate for the former
is suppressed by $T^{2}/f^{2}$, the latter is only suppressed only by
$T^{2}/r^{2}$, which is much larger than $T^{2}/f^{2}$ when $q$ is close to
$q_{c}$. Also, the transfer involving the axion may require additional
processes, further suppressing the transfer rate. For example, in the toy
model discussed in Sec. 2.2, the transfer rate into the axion involves the
chiral symmetry breaking by the fermion mass. This implies that the charge
transfer during the tunneling process may involve only the AD field and the
thermal bath, or could also involve the axion field. In the former case, we
expect bubbles of true vacuum to form with $r=0$ and $\dot{\theta}_{a}=m$ in a
false vacuum with $r=r_{\text{eq}}$ and $\dot{\theta}_{a}=m$. In the latter
case, bubbles are formed with $r=0$ and $\dot{\theta}_{a}>m$, with the charge
of the AD field transmitted to the rotation of the PQ field inside the bubble.
Once these bubbles expand, more charge is converted from the AD field into the
axion motion and/or the thermal bath. Again, a comparison between the two
competing transfer rates is required to understand the dynamics of the bubble
expansion. Finally, unlike the AD field, the axion field does not experience
any phase transition during the tunneling; only the axion velocity changes,
which is not an order parameter. This means that sub-critical bubbles can be
more important than the usual case; inside the sub-critical bubble, the AD
field is at the origin while the axion field has a larger charge density than
outside. As the sub-critical bubble collapses, the AD field will revert to
large field values, but the excess of the axion charge inside the bubble can
spread out as density waves.

#### Inhomogeneities in the axion field.

The first order phase transition or the phase mixing at the end of the
transfer could in fact be phenomenologically interesting. After the first
order phase transition or the phase mixing, the axion field will have large
fluctuations, which can contribute to axion dark matter if they are not
thermalized.\natural4\natural4$\natural$4One may worry that the fluctuations
may produce stable domain walls. We expect this is not the case, since the
fluctuations are produced in sub-horizon modes. Whether or not thermalization
occurs will depend on the length scale of the fluctuations, since the
interaction of the axion with the thermal bath is suppressed for long-
wavelength modes. We also note that Q-balls may form when $q$ is close to
$q_{c}$. For $q\gtrsim q_{c}$, $r$ is not much above $T$, for which the
thermal potential becomes comparable to the vacuum potential. Since the
thermal potential is flatter than the quadratic one, fluctuations around the
rotating background grow and Q-balls are formed. This occurs only after the
majority of the charges have been transferred into the axion field, so the
basic picture of the transfer is not altered. However, the fluctuations
associated with the Q-ball formation may have some phenomenological
implications, such as the production of axion dark matter. These observations
motivate further studies on the dynamics of the transition, which are beyond
the scope of the present paper. We will instead present a lower bound on the
axion dark matter production based on the homogeneous axion component in Sec.
4.1.

##  4 Phenomenological applications

In this section, we discuss phenomenological applications of the charge
transfer from the AD field to the axion field.

###  4.1 Axion dark matter by kinetic misalignment

If the axion field receives a large enough charge, the axion field continues
to rotate even after the Hubble expansion rate drops below the axion mass
$m_{a}$ and the axion would begin to oscillate around the minimum. In this
case, axions are produced by the kinetic misalignment mechanism [1], where the
kinetic energy of the axion rotation is transferred into axion dark matter
density. The resulting number density of the axion $n_{a}$ is as large as the
charge density $q_{a}$, so

$\displaystyle\frac{\rho_{a}}{s}\simeq m_{a}Y_{a}=0.4~{}{\rm
eV}\frac{m_{a}}{\rm meV}\frac{Y_{a}}{400}=0.4~{}{\rm eV}\frac{10^{9}~{}{\rm
GeV}}{f}\frac{Y_{a}}{70},$ (4.1)

where $Y_{a}=q_{a}/s$. In addition, the axion abundance can receive
contributions sourced by the inhomogeneous axion component generated in a
tunneling or phase-mixing process completing the charge transfer between the
AD field and the axion field. Since this process occurs only after most of the
charge has already been transferred to the axion field, we will focus only on
the homogeneous axion component and the production via kinetic misalignment
here. This gives a lower bound for the total axion abundance produced. In the
following, we demonstrate how a large enough $Y_{a}$ can be obtained to
explain the observed dark matter abundance via kinetic misalignment.

#### Efficient transfer.

Let us first assume an efficient transfer from the AD field to the axion
field. The transfer to the axion field completes when $r_{0}\simeq f$ and
$q_{a}\simeq mf^{2}$ at this point. If the entropy production around the
completion of the charge transfer is negligible, $Y_{a}$ is the same as the
initial $Y_{\rm\phi}$, which can be easily large enough to produce axion dark
matter by kinetic misalignment.

As discussed in Sec. 3.3, the charge transfer may be completed by a tunneling
process in the thermal potential. From the local minimum at
$r_{\text{eq}}<r_{0}\simeq f$ the AD field can tunnel to the true minimum at
$r=0$. Let us parameterize the AD field value when the tunneling occurs by
$r=XT$. The parameter $X$ ($>1$ and $\ll f/T$) can be computed once the
tunneling rate has been determined. The energy density of the thermal bath is
then bounded from below by $X^{2}m^{2}T^{2}$. From this, we obtain

$\displaystyle Y_{a}\lesssim 6\times 10^{5}\left(\frac{f}{10^{9}~{}{\rm
GeV}}\frac{100~{}{\rm
TeV}}{m}\right)^{2}\left(\frac{g_{*}}{200}\right)^{1/2}\left(\frac{10}{X}\right)^{3}$
(4.2)

where $g_{*}$ is the number of effective degrees of freedom of the thermal
bath. The bound is saturated when the entropy of the universe is dominantly
created by the tunneling process. Requiring that the maximal possible $Y_{a}$
can explain the axion dark matter by kinetic misalignment, we obtain an upper
bound on the mass of the AD field,

$\displaystyle m\lesssim 10^{4}~{}{\rm TeV}\left(\frac{f}{10^{9}~{}{\rm
GeV}}\right)^{1/2}\left(\frac{g_{*}}{200}\right)^{1/4}\left(\frac{10}{X}\right)^{3/2}.$
(4.3)

Interpreting the AD field as a supersymmetric flat direction, this upper bound
does not require low scale supersymmetry.

#### Inefficient transfer.

The transfer may not be efficient when $r_{0}$ reaches $f$. In this case, the
transfer completes for $r_{0}<f$. The value of such $r_{0}$ depends on the
model, but let us consider a representative case where a process with
$\gamma=\epsilon T$ is the bottleneck process. Since the transfer occurs by an
interaction coming into equilibrium, the transfer is a non-thermal process and
entropy is produced. Using the following relations at the completion of the
transfer,

$\displaystyle
m^{2}r_{0}^{2}<\frac{\pi^{2}g_{*}}{30}T^{4},~{}\epsilon\frac{T^{3}}{r_{0}^{2}}=H(T),$
(4.4)

we obtain

$\displaystyle Y_{a}\lesssim 1000\left(\frac{10~{}{\rm
TeV}}{m}\right)^{1/3}\left(\frac{200}{g_{*}}\right)^{1/2}\left(\frac{\epsilon}{10^{-2}}\right)^{1/3},$
(4.5)

where we use as reference value the strong sphaleron process,
$\epsilon=100\alpha_{3}\simeq 10^{-2}$. The upper bound on $m$ from successful
DM production from kinetic misalignment is

$\displaystyle m\lesssim 10^{5}~{}{\rm TeV}\left(\frac{10^{9}~{}{\rm
GeV}}{f}\right)^{3}\left(\frac{200}{g_{*}}\right)^{3/2}\frac{\epsilon}{10^{-2}}.$
(4.6)

For $f<10^{11}$ GeV, the upper bound does not require low scale supersymmetry
if the strong sphaleron process is indeed the bottleneck process. The bound
can be strong enough to have implications for low scale supersymmetry if a
less efficient process is the bottleneck process. For example, explicit $R$
symmetry breaking provided by the gaugino mass $m_{\lambda}$ may be required
for the transfer to be free-energetically favored. In this case, $\gamma\sim
0.1m_{\lambda}^{2}/T$, and we obtain a bound

$\displaystyle\frac{m^{3}}{m_{\lambda}^{2}}\lesssim 40{\rm
TeV}\left(\frac{10^{9}~{}{\rm
GeV}}{f}\right)^{5}\left(\frac{200}{g_{*}}\right)^{3/2}.$ (4.7)

The compatibility with $m,~{}m_{\lambda}>$ TeV requires $f<$ few $10^{9}$ GeV.
In the context of the MSSM (or its extension), the transfer rate can be
computed for different flat directions. The rate may also depend on the
sfermion mixing, since this can enhance chiral symmetry breaking rate [32]. We
leave the detailed estimation of the transfer rate for future work.

###  4.2 Baryogenesis

A rotating axion field generically induces a non-zero baryon number as long as
the baryon number violation from the electroweak sphaleron process is
efficient [6, 33, 34]. The baryon number is frozen after the electroweak phase
transition.

For the QCD axion, however, the axion velocity that can explain the observed
baryon asymmetry produces too much axion dark matter via the kinetic
misalignment mechanism [6]. This problem can be avoided by introducing
additional BSM physics which raises temperature of the electroweak phase
transition in comparison with the standard model prediction.

Alternatively, the mass of an axion-like particles can be lighter than that of
the QCD axion, so that the overproduction of dark matter is avoided, see Eq.
(4.1). However, in this case it turns out to be non-trivial to transfer the
charge of the AD field to the axion-like particle. The transfer requires the
violation of the shift symmetry of the axion-like particle, see Sec. 2. If
this is induced by the QCD anomaly, this also gives a mass to the axion-like
particle and dark matter is overproduced by kinetic misalignment. One can
additionally introduce the QCD axion which couples to QCD more strongly than
the axion-like particle does, so that the axion-like particle does not obtain
a mass. However, when the QCD anomaly induces the rotation of the axion-like
particle, the QCD axion also rotates, and the overproduction occurs. If the
violation of the shift symmetry is instead induced by the weak anomaly, the
axion-like particle does not obtain too large a mass. However, within the
standard model, electroweak sphalerons are the only interaction which violates
$B+L$ through the weak anomaly, so the transfer into the axion-like particle
requires large asymmetry of $B+L$ fermions and thus does not minimize the free
energy. This can be avoided if the initial rotation of the AD field carries
$B+L$ charge and can thus absorb the change in the total $B+L$ charge without
invoking large fermion asymmetries or if there are additional $B+L$ violating
interactions. In the former case, the $B-L$ charge of the AD field should be
zero to avoid the overproduction of baryon asymmetry. In the latter case, the
extra $B+L$ violation can directly create a baryon asymmetry if $B-L$ is
simultaneously violated [33, 35] (see also [32, 36, 37, 38]), so the
production from the axion motion through the electroweak sphaleron process may
be subdominant. We conclude that if the axion-like particle couples to the
weak anomaly, there is explicit $B+L$ violation in the system (which initiates
the rotation of the AD field or washes-out $B+L$ charges) and $B-L$ is
(approximately) preserved, then the motion of this axion-like particle at the
electroweak phase transition could be responsible for the observed matter-
antimatter asymmetry of the universe without overproducing dark matter.

###  4.3 Kination domination by axion rotation

Since the energy density of the rotation of the AD field decreases as
$R^{-3}$, if the initial field value of the AD field is sufficiently large,
the rotation energy of the AD field can come to dominate over the energy of
the thermal bath. We show that in this case the universe can enter a kination-
dominated era driven by the axion rotation.

#### From an AD field era to a kination era.

Let us start from the initial state where the universe is dominated by the AD
field and almost all of the charge $q>q_{c}$ is in the AD field, with the
initial charge in the axion much below the equilibrium value $\sim mf^{2}$.
Once the charge transfer becomes efficient and the axion rotation reaches
equilibrium, the total energy density of the AD and axion fields decreases by
$m^{2}f^{2}/2$. This amount of energy should go to the thermal bath and
entropy is created. If the thermal bath has an energy density smaller than
this before this entropy production, the temperature of the universe when the
axion and AD field begin to be in chemical equilibrium is determined by
$T^{4}\sim m^{2}f^{2}$. Otherwise, the temperature of the bath remains
approximately constant up to cosmic expansion.

As long as the charge transfer between the axion and AD fields continues to be
efficient, the system evolves adiabatically. In particular, the axion and AD
fields follow the equilibrium values $\dot{\theta_{a}}=-m$ and
$r^{2}=r_{0}^{2}-f^{2}$, and the entropy and energy density of the thermal
bath decrease as $R^{-3}$ and $R^{-4}$, respectively. Thus, the universe
continues to be dominated by the AD field.

The energy density of the axion and AD fields is given by

$\displaystyle\rho=\rho_{a}+\rho_{\phi}=\frac{1}{2}\dot{\theta}_{a}^{2}f^{2}+m^{2}r^{2}=mq-\frac{1}{2}m^{2}f^{2},$
(4.8)

and decreases following

$\displaystyle\frac{{\rm dln}\rho}{{\rm
dln}R}=\frac{-3q}{q-mf^{2}/2}\rightarrow\begin{cases}-3&:q\gg q_{c}=mf^{2}\\\
-6&:q\rightarrow q_{c}\end{cases},$ (4.9)

where we used the charge conservation $q\propto R^{-3}$. For $q\gg q_{c}$, the
equation of state is that of matter, while as $q$ approaches $q_{c}$, it
approaches that of kination. The charge transfer completes at $q=q_{c}$, after
which the universe is dominated by the axion rotation with
$\dot{\theta}_{a}\propto R^{-3}$, and the universe is kination-dominated.

#### Implications.

A kination-dominated era due to a rotating axion, called axion kination, is
realized in [6] starting from the rotation of a PQ symmetry breaking field
that has a nearly quadratic potential. Our realization is applicable to more
generic potentials of the PQ symmetry breaking field since the rotation is
initiated in another sector. The duration of the kination-dominated era
depends on the amount of the radiation energy density at the completion of the
charge transfer, and thus on the preceding cosmological history, e.g., the
initial field value of the AD field, the reheating temperature, and when the
AD field is thermalized. Also, as we discussed in Sec. 3, the very end of the
charge transfer can involve tunneling process that creates small amount of
entropy, which will also limit the duration of the kination-dominated era. We
leave an investigation of the duration of the kination era to future work.

In the above discussion, we assume $m\ll T$ to argue that the energy density
of the bath decreases as $R^{-4}$. If $m$ is larger than $T$, the chemical
potential $\mu\sim m>T$ ensures that the energy density remains constant $\sim
m^{4}$. This is smaller than the axion energy density, $m^{2}f^{2}$, as long
as $m\ll f$ and hence the axion kination era still occurs. Note that in this
limit, scalar fields do not receive thermal masses larger than $\mu\sim m$. If
there exist an additional scalar field that can receive charges from the AD
field and has a smaller energy per charge, the scalar field necessarily is
destabilized by the negative mass from the chemical potential; see also Sec.
5.

###  4.4 Gravitational waves

The charge transfer between an AD field and an axion field offers two
interesting possibilities for gravitational wave observations. First, any
gravitational waves produced prior to a kination era (such as those produced
by inflation or cosmic strings) are enhanced [39, 40]. As in the setup in [6],
the kination-dominated era in our setup is preceded by a matter-dominated era.
This leaves peculiar signatures in the spectrum of primordial gravitational
waves [41, 42, 43]. Second, if the charge transfer occurs via a first order
phase transition (see discussion in Sec. 3.3), the associated bubble dynamics
source gravitational waves (see [44] for a recent review.) The magnitude of
the gravitational waves will depend on the detail of the phase transition,
such as the latent heat and the duration of the phase transition, motivating
the detailed investigation of the dynamics.

##  5 Summary and Discussion

In this paper, we discussed charge transfer between complex scalar fields. A
rotating complex field $\phi$ has a non-zero $U(1)_{\phi}$ charge. This $U(1)$
charge can be transferred into a $U(1)_{P}$ charge of another complex scalar
field $P$ so that $P$ begins rotation if $U(1)_{\phi}\times U(1)_{P}$ is
explicitly broken with only one linear combination of the original charges
conserved. We considered the charge transfer through couplings to a thermal
bath and focused on the case where $\phi$ has a nearly quadratic potential
with a mass $m$, the angular direction of $P$ is an axion field, and the
radial direction of $P$ is strongly fixed at a constant value $f$. Whether or
not the $U(1)$ charge is dominantly in $\phi$ or $P$ is determined by
minimizing the free energy. We found that almost all of the $U(1)$ charge is
transferred into the axion field once the total charge has been redshifted to
a critical value of about $mf^{2}$ as long as no conservation law requires the
state with large $U(1)$ charge in $P$ to have a large fermion asymmetry. The
latter condition requires that all linear combinations of $U(1)_{\phi}$ and
chiral symmetry as well as those of $U(1)_{P}$ and chiral symmetry be
explicitly broken. We moreover find that the transfer rate is suppressed by
the ratio between the temperature and the smaller of the radii of $\phi$ and
$P$ as well as by explicit symmetry breaking rates.

The scenario has an immediate phenomenological application. The kinetic energy
of the induced axion rotation, if large enough, contributes to the axion dark
matter density through so-called the kinetic misalignment mechanism. Unlike
the original realization in [1], our case does not require a flat potential of
the radial direction of the PQ symmetry breaking field, and thus is compatible
with a wider class of PQ symmetry breaking models including dynamical symmetry
breaking models [45].\natural5\natural5$\natural$5In supersymmetric models
[46], the PQ charged scalar degree of freedom may have a flat potential at
field values much above the dynamical scale. However, the potential may be
flatter than a quadratic one and the dynamics of the PQ field be complicated
by Q-ball formation. This comes with the advantage that in dynamical PQ
symmetry breaking models, the smallness of the PQ breaking scale compared to
the Planck scale is understood by dimensional transmutation and the PQ
symmetry is more easily understood as an accidental symmetry than the models
with a fundamental PQ scalar [47]. Also, the axion rotation can be induced
even when the PQ breaking occurs after cosmic inflation, leading to cosmic
string configurations rotating in field space. See [48] for the existence of
rotating cosmic string solutions. This may impact the axion dark matter
abundance produced from the string-domain wall network.

Further phenomenological implications are tied to the detailed dynamics of the
charge transfer. Contributions to the effective potential of $\phi$ induced by
particles coupling to $\phi$ favour $\phi=0$, and hence the state with most
charge in $P$. Once the total charge density becomes sufficiently small
through cosmic expansion, the state with most charge in $\phi$ becomes a
metastable state and the state with most charge in $P$ becomes the absolute
minimum. The charge transfer can then involve a first order phase transition
or phase mixing. The former will source gravitational waves. Both will create
fluctuations of the axion field, which can contribute to axion dark matter. We
hope that these observations can serve as a motivation for a more detailed
investigation of the dynamics of this charge transfer.

Our work can be immediately generalized to charge transfer process between
generic complex fields. Transfer is possible as long as it is favored by the
free energy and the transfer rate exceeds the Hubble expansion rate. For
example, charge transfer can occur from a MSSM flat direction to another flat
direction, if no conservation law requires a large fermion asymmetry and if
the latter flat direction has a smaller energy per charge. It is also possible
that the final state is composed of rotations of several flat directions. The
charge transfer may complete by first order phase transition or phase mixing,
depending on the thermal potentials of the fields involved in the charge
transfer. We note that the axion case, i.e., a recipient field for the charge
with a fixed value for the radial component, is special in the sense that the
energy per charge of the axion rotation $\sim q/f^{2}$ is always smaller than
that of a complex scalar in a quadratic potential for a sufficiently small
total $U(1)$ charge.

## Acknowledgement

K.M. was supported in part by MEXT Leading Initiative for Excellent Young
Researchers Grant No. JPMXS0320200430.

##  Appendix A Boltzmann equations for charge transfer

In this appendix we show numerical results obtained by solving the Boltzmann
equations (2.10) for the model discussed in Sec. 2.1. We fix $f=10^{9}$ GeV,
$m=10^{3}$ GeV, $\alpha_{G}=0.1$ and consider a radiation-dominated
background, $H=T^{2}/M_{*}$ with $M_{*}=7\cdot 10^{17}$ GeV. We initialize the
AD field at $r=r_{0}=10f$, treating the temperature $T_{0}$ at this time as a
free parameter. We denote the temperature at which the AD field would collapse
assuming an efficient charge transfer ($q=q_{c}$) as $T_{\text{c}}$. The
results are shown in Fig. 4 as a function of the temperature of the thermal
bath $T$.

Figure 4: Evolution of charges stored in the AD field and the axion, governed
by the AD field radius $r$ and the axion velocity $\dot{\theta}_{a}$ for
different initial temperatures $T_{0}$. The black curve depicts their
equilibrium values, obtained if the charge transfer rate is significantly
faster than the Hubble expansion. Here cosmic expansion is accounted for but
thermal contributions to the effective potential are not included.

The left panel of Fig. 4 shows the evolution of the radial component of the AD
field (recall that the angular component is fixed to $\theta_{\phi}=m$). For
low temperatures, the charge transfer is not efficient, and we observe that
the field value decreasing as $r\simeq r_{0}\propto R^{-3/2}$, in agreement
with the conserved charge $R^{3}r_{0}^{2}m$ in the absence of charge transfer.
At larger temperatures the charge transfer becomes efficient, tracing the
equilibrium value for $r$ for $\gamma_{s}(T/f)^{2}>H$. Once we reach $r\sim f$
the AD field collapses if the charge transfer is efficient, otherwise the
collapse is delayed until the Hubble rate falls below the charge transfer
rate.

The right panel shows the evolution of the axion velocity, starting from
$\dot{\theta}_{a}=0$. The equilibrium value $\dot{\theta}_{a}=-m$ is reached
if the transfer is efficient at $T=T_{0}$. More generally the axion velocity
is well described by Eq. (2.13), as indicated by the dashed lines. The curves
for larger values of $T_{0}$ end once the AD field collapses, and all charge
is transferred to the axion, see left panel. After this occurs,
$\dot{\theta}_{a}$ should decrease in proportion to $R^{-3}$.

In the model discussed in Sec. 2.2, the charge density for the fermions
efficiently tracks the equilibrium value given in Eq. (2.24) as long as at
least one of the two interactions rates $\gamma_{s}$ and $\gamma_{\chi}$ is
faster than the Hubble expansion. In this case, after replacing the $q_{\chi}$
by its equilibrium value in Eq. (2.23), the dynamics of the axion and AD field
are analogous to the the model of Sec. 2.1. For the impact of thermal
corrections, see Sec. 3.

##  Appendix B Thermodynamics of rotating scalar fields

The main purpose of this section is to demonstrate how to obtain thermodynamic
quantities in the presence of rotating scalar fields by performing explicit
calculations in a simplified model. Unlike the simplified setup in Sec. 3, we
include an AD-field dependent mass term for the fermion which participates in
the charge transfer between the AD and the axion field. We moreover include
the fluctuations of the AD field. Nevertheless, we find a qualitatively
similar result as in Sec. 3.

###  B.1 Preliminary

#### Model.

To make our discussion concrete, we consider the following toy model,

$\displaystyle\mathcal{L}=$
$\displaystyle\absolutevalue{\partial\phi}^{2}-m^{2}\absolutevalue{\phi}^{2}+\overline{\psi}i\not{D}\psi+\frac{1}{2}f^{2}\quantity(\partial\theta_{a})^{2}-\frac{1}{2}\Tr
G_{\mu\nu}G^{\mu\nu}$
$\displaystyle-\quantity(y\phi\overline{\psi}P_{L}\psi+\text{H.c.})-\frac{g^{2}\theta_{a}}{16\pi^{2}}\Tr
G_{\mu\nu}\tilde{G}^{\mu\nu}.$ (B.1)

Here $\phi$ is a complex scalar field whose charge is transferred to the axion
$\theta_{a}$ in the end, and $\psi$ is a Dirac fermion charged under a gauge
group with field strength tensor $G_{\mu\nu}$. The chiral symmetry of the
fermion is explicitly broken by the chiral anomaly. Through the Yukawa
coupling $y$, the fermion obtains a $\phi$-dependent mass. In a supersymmetric
theory, where a nearly quadratic potential of $\phi$ can be achieved, $y$ is
as large as the gauge coupling constant, so we assume $y={\cal O}(1)$ in the
following. The relevant current equations are

$\displaystyle\partial\cdot J_{\phi}+\frac{1}{2}\partial\cdot
J_{5}=-\frac{g^{2}}{16\pi^{2}}\Tr
G_{\mu\nu}\tilde{G}^{\mu\nu},\qquad\partial\cdot
J_{a}=-\frac{g^{2}}{16\pi^{2}}\Tr G_{\mu\nu}\tilde{G}^{\mu\nu},$ (B.2)

where each current is defined by

$\displaystyle
J_{\phi}^{\mu}=\phi^{\dagger}i\overleftrightarrow{\partial}\phi,\qquad
J_{5}^{\mu}=\overline{\psi}\gamma^{\mu}\gamma_{5}\psi,\qquad
J_{a}^{\mu}=f^{2}\partial^{\mu}\theta_{a}.$ (B.3)

This Lagrangian has a U$(1)$ symmetry, whose conserved charge is given by

$\displaystyle Q=Q_{\phi}+\frac{1}{2}Q_{5}-Q_{a},$ (B.4)

with $Q_{\bullet}=\int_{\bm{x}}J_{\bullet}^{0}$. We also define charge
densities for later convenience as $q_{\bullet}=\langle
Q_{\bullet}\rangle/\mathbb{V}$ with $\mathbb{V}$ being a spatial volume. On
top of Eq. (B.1), we introduce a higher dimensional operator for $\phi$ that
explicitly breaks this U$(1)$ symmetry. When the field value of $\phi$ is
large in the early Universe, this breaking term induces the rotation of $\phi$
via the Affleck-Dine mechanism. After some time, the breaking becomes
inefficient as the field value of $\phi$ decreases due to the cosmic
expansion, and then $Q$ becomes conserved.

When the field value of $\phi$ is large enough, one may integrate out the
Dirac fermion $\psi$. The Lagrangian in this case reads

$\displaystyle\mathcal{L}=$ $\displaystyle\frac{1}{2}\quantity(\partial
r)^{2}-\frac{1}{2}\quantity(m^{2}-\quantity(\partial\theta_{\phi})^{2})r^{2}+\frac{1}{2}f^{2}\quantity(\partial\theta_{a})^{2}-\frac{1}{2}\Tr
G_{\mu\nu}G^{\mu\nu}-\frac{g^{2}\quantity(\theta_{a}+\theta_{\phi})}{16\pi^{2}}\Tr
G_{\mu\nu}\tilde{G}^{\mu\nu},$ (B.5)

and the current equations are

$\displaystyle\partial\cdot J_{\phi}=-\frac{g^{2}}{16\pi^{2}}\Tr
G_{\mu\nu}\tilde{G}^{\mu\nu},\qquad\partial\cdot
J_{a}=-\frac{g^{2}}{16\pi^{2}}\Tr G_{\mu\nu}\tilde{G}^{\mu\nu}.$ (B.6)

We can see that the model (B.1) is a simple realization of the toy model
discussed in Sec. 2.1.

#### Review of thermal field theory.

Before going into our model calculations, we briefly summarize some basic
facts of thermal field theory [49, 50]. Starting from the Hamiltonian
$\mathcal{H}$, thermodynamic quantities with a conserved charge $Q$ are
obtained from the grand canonical ensemble of

$\displaystyle\rho_{\text{GC}}=\frac{1}{Z}e^{-\frac{1}{T}\quantity(\mathcal{H}-\mu
Q)},\qquad Z=\Tr\rho_{\text{GC}},$ (B.7)

which is related to the thermodynamic pressure

$\displaystyle p(T,\mu)=\frac{T}{\mathbb{V}}\ln Z,$ (B.8)

with $\mathbb{V}$ being a spatial volume. The charge density is obtained from
\natural6\natural6$\natural$6 In the main text, we often simply refer to $q$
as ‘charge’ for brevity.

$\displaystyle q=\frac{\partial p\quantity(T,\mu)}{\partial\mu}.$ (B.9)

These quantities are useful when considering a bath of fixed temperature $T$
and chemical potential $\mu$. However, we are rather interested in the case of
a fixed charge $\langle Q\rangle$. The Helmholtz free energy, $f(T,q)$, is
more useful in this case, which is obtained from the Legendre transformation

$\displaystyle f\quantity(T,q)=\mu q-p.$ (B.10)

In the context of field theory, it is more convenient to express these
quantities by means of an effective potential, i.e., as a function of $\phi$.
Thermodynamic quantities are obtained by evaluating it at its extrema,

$\displaystyle
p\quantity(T,\mu)=-\left.V_{\mu,\text{eff}}\quantity(\phi,T,\mu)\right|_{\differential
V_{\mu,\text{eff}}/\differential\phi=0},\qquad
f\quantity(T,q)=\left.V_{\text{eff}}\quantity(\phi,T,q)\right|_{\differential
V_{\text{eff}}/\differential\phi=0}.$ (B.11)

with the subscript indicating the thermodynamic quantity taken to be constant.
These effective potentials are related through the Legendre transform of

$\displaystyle V_{\text{eff}}\quantity(\phi,T,q)=\mu
q+V_{\mu,\text{eff}}\quantity(\phi,T,\mu),\qquad q=-\frac{\partial
V_{\mu,\text{eff}}\quantity(\phi,T,\mu)}{\partial\mu},$ (B.12)

Note that we assume that the background $\phi$ is homogeneous. The grand
canonical partition function can be expressed as an Euclidean path integral

$\displaystyle
Z=e^{\frac{\mathbb{V}}{T}p\quantity(T,\mu)}=\int_{\text{b.c.}}\quantity[\differential\mu]\,\exp\quantity[-\int_{0}^{\frac{1}{T}}\differential\tau\differential^{3}x\,\mathcal{L}_{\text{E}}\quantity(\partial_{\tau}\to\partial_{\tau}-c_{\phi}\mu)],$
(B.13)

with $c_{\phi}$ denoting the charge of $\phi$ with respect to the conserved
charge $Q$. The boundary condition (b.c.) is taken so that the bosonic fields
are periodic while fermionic fields are anti-periodic. The effective potential
$V_{\mu}$ is obtained by performing this path integral on a background of
$\phi$. Then we obtain the second effective potential $V_{q}$ via the Legendre
transform.

###  B.2 Effective potential

#### Without axion.

We first consider the case where the conversion of charge between $\phi$ and
$\theta_{a}$ mediated by the sphaleron is negligible. The axion is sequestered
and hence we omit it hereafter, as in Sec. 3.1. The relevant conserved charge
is then

$\displaystyle Q_{\text{AD}}=Q_{\phi}+\frac{Q_{5}}{2}.$ (B.14)

The Euclidean Lagrangian reads

$\displaystyle\mathcal{L}_{\text{E}}=(\partial_{\tau}\phi^{\dagger}+\mu\phi^{\dagger})(\partial_{\tau}\phi-\mu\phi)+\absolutevalue{\nabla\phi}^{2}+m^{2}\absolutevalue{\phi}^{2}+\overline{\psi}\quantity(\gamma^{0}\partial_{\tau}-i\gamma^{i}\partial_{i}+\frac{\mu}{2}\gamma^{0}\gamma_{5})\psi+\quantity(y\phi\overline{\psi}P_{L}\psi+\text{H.c.}).$
(B.15)

Assuming a homogeneous background of $\phi=re^{i\theta_{\phi}}/\sqrt{2}$, one
may compute its effective potential

$\displaystyle
V_{\mu,\text{eff}}\quantity(r,T,\mu)=\quantity(m^{2}-\mu^{2})\frac{r^{2}}{2}-d_{\psi}\ln\det\quantity(i\not{\partial}+\frac{yr}{\sqrt{2}}+\frac{\mu}{2}\gamma^{0}\gamma_{5})+\ln\det\quantity(\Box+m^{2}-\mu^{2}+\mu
i\overleftrightarrow{\partial_{t}})$ (B.16)

at the one-loop level. Here $d_{\psi}$ counts the number of Dirac fermions
(e.g., $d_{\psi}=1/2$ for a Weyl fermion, $3$ for a Dirac color triplet, …).
The determinants are evaluated as

$\displaystyle\ln\det\quantity(i\not{\partial}+\frac{yr}{\sqrt{2}}+\frac{\mu}{2}\gamma^{0}\gamma_{5})=$
$\displaystyle
2\int\frac{\differential^{3}p}{(2\pi)^{3}}\left[\frac{\omega^{\prime}}{2}+T\ln\left(1+e^{-\omega^{\prime}/T}\right)\right]_{\omega^{\prime}=\sqrt{y^{2}r^{2}/2+(p-\mu/2)^{2}}}+\left(\mu\rightarrow-\mu\right),$
$\displaystyle=$
$\displaystyle\int\frac{\differential^{3}p}{(2\pi)^{3}}\left[\frac{\omega^{\prime}}{2}+T\ln\left(1-e^{-\omega^{\prime}/T}\right)\right]_{\omega^{\prime}=\sqrt{m^{2}+p^{2}}-\mu}+\left(\mu\rightarrow-\mu\right).$
(B.17)

for the fermionic and bosonic contribution, respectively. The first terms in
the integrands lead to UV divergence, which are the usual zero temperature
divergent parts and can be dropped.\natural7\natural7$\natural$7The divergence
in the fermion contribution includes the logarithmic divergence of the
coefficient of $\mu^{2}r^{2}$. One may worry about this divergence at finite
densities, but this contribution is cancelled by the wave-function
renormalization of $\phi$, owing to the non-renormalization of the charge.
From (B.17), the approximate form of the fermion contribution is

$\displaystyle
V_{\mu,f}/d_{\psi}\simeq\begin{cases}-4\frac{7}{8}\frac{\pi^{2}T^{4}}{90}-\frac{\mu^{2}T^{2}}{24}-\frac{\mu^{4}}{192\pi^{2}}+\frac{y^{2}T^{2}r^{2}}{24}&yr\ll
T,\\\
-\frac{\sqrt{2}}{\pi^{3/2}}e^{-\frac{yr}{\sqrt{2}T}}\left(\frac{yr}{\sqrt{2}}\right)^{3/2}T^{5/2}-\frac{3}{4\sqrt{2}\pi^{2}}e^{-\frac{yr}{\sqrt{2}T}}\sqrt{\frac{yr}{\sqrt{2}}T^{3}}\mu^{2}-\frac{\mu^{4}}{192\pi^{2}}&yr\gg
T.\end{cases}$ (B.18)

For the fluctuations of the AD field, (B.17) yields

$\displaystyle V_{\mu,{\rm
AD}}\simeq-\frac{\pi^{2}}{45}T^{4}-\frac{1}{6}\mu^{2}T^{2},$ (B.19)

where we have assumed $m,\mu<T$ and have dropped higher-order terms in $\mu$.

Around $r=0$, assuming $\mu<T$ (dropping $O(\mu^{4})$ terms),

$\displaystyle
V_{\mu,\text{eff}}(r,T,\mu)\simeq\quantity(m^{2}-\mu^{2})\frac{r^{2}}{2}-d_{\psi}\quantity(4\frac{7}{8}\frac{\pi^{2}T^{4}}{90}+\frac{\mu^{2}T^{2}}{24}+\frac{\mu^{4}}{192\pi^{2}}-\frac{y^{2}T^{2}r^{2}}{24})-\quantity(\frac{\pi^{2}T^{4}}{45}+\frac{\mu^{2}T^{2}}{6}).$
(B.20)

Using the Legendre transform, the effective potential for a fixed charge
density $q$ is

$\displaystyle
V_{\text{eff}}(r,T,q)=-\frac{\pi^{2}}{45}\quantity(1+\frac{7}{4}d_{\psi})T^{4}+\frac{1}{2}\quantity(m^{2}+\frac{d_{\psi}}{12}y^{2}T^{2})r^{2}+\frac{6q^{2}}{12r^{2}+(4+d_{\psi})T^{2}}.$
(B.21)

This has an extremum at $r=0$. The second derivative of $V_{q}$ with respect
to $r$ at $r=0$ is

$\displaystyle\left(\frac{\partial^{2}V_{\text{eff}}(r,T,q)}{\partial
r^{2}}\right)_{r=0}=m^{2}+\frac{d_{\psi}}{12}y^{2}T^{2}-\frac{144q^{2}}{(4+d_{\psi})^{2}T^{4}}.$
(B.22)

This is positive and the origin is a (local) minimum if the charge $q$ is
small enough,

$\displaystyle
q^{2}<\frac{(4+d_{\psi})^{2}d_{\psi}}{1728}y^{2}T^{6}+\frac{(4+d_{\psi})^{2}}{144}m^{2}T^{4}$
(B.23)

This condition is qualitatively the same as the one derived below Eq. (3.8).
Note that $q$ satisfying this condition implies $\mu<T$, and hence our
assumption of $\mu<T$ is self-consistent. When $q$ significantly violates this
condition, $\mu^{2}>m^{2}+d_{\psi}y^{2}T^{2}/12$. The integration in Eq.
(B.17), even after the resummation to include the thermal mass $\sim yT$,
results in an imaginary part, signaling the instability of $r=0$ and leading
to the formation of the BEC of the AD field.

For $yr\gg T$, assuming $\mu<T$,

$\displaystyle
V_{\mu,\text{eff}}(r,T,\mu)\simeq\quantity(m^{2}-\mu^{2})\frac{r^{2}}{2}-\quantity(\frac{\pi^{2}T^{4}}{45}+\frac{\mu^{2}T^{2}}{6}).$
(B.24)

The effective potential for a fixed charge density $q$ is

$\displaystyle
V_{\text{eff}}(r,T,q)=-\frac{\pi^{2}}{45}T^{4}+\frac{1}{2}m^{2}r^{2}+\frac{3q^{2}}{6r^{2}+2T^{2}}.$
(B.25)

This has an extremum at $r^{2}=|q|/m\equiv r_{\rm eq}^{2}$. The second
derivative at $r_{\rm eq}$ is $4m^{2}$. The assumption of $yr_{\rm eq}\gg T$
is valid if $q\gg mT^{2}/y^{2}$. At $r_{\rm eq}$, $|\mu|=m$, so the
integration in Eq. (B.17) does not yield an imaginary part and $V_{q}$ can be
interpreted as the effective potential as usual. Moreover, for $m<T$, the
assumption $\mu=m<T$ is again self-consistent. For $m>T$, Eq. (B.16) is simply
dominated by the first term, and thermal corrections are irrelevant to the
dynamics of the phase transition.

There are two minima of the effective potential for $mT^{2}/y^{2}<q<yT^{3}$.
As in Sec. 3.1, one can compare the effective potential at these two minima
and determine which is the absolute minimum, obtaining qualitatively the same
result.

We are interested in the possible transition from $r=r_{\rm eq}$ to $r=0$ by
bubble nucleation. To compute the nucleation rate, one needs the effective
potential for intermediate field values, $0<r<r_{\rm eq}$. When $|q|>mT^{2}$
and $r<r_{\rm eq}$, $|\mu|$ is larger than $m$. Also, for $yr>T$, the thermal
mass squared of the AD field given by the coupling $y$ is negative. Therefore,
for $T/y<r<r_{\rm eq}$, the integration in Eq.(B.17) yields an imaginary part,
and the effective potential inside the bubble wall formally contains an
imaginary part. This may cast a doubt on the validity of the computation, but
we argue that the appearance of the imaginary part is an artifact of the
inclusion of long-wave length modes despite the finiteness of the width of the
bubble wall [51]. In computing the effective potential, the field value $r$ is
approximated to be homogeneous, which is justified only for the excitations
with wavelength shorter than the wall width. The effective potential contains
a term $-\mu^{2}(q,r)r^{2}/2$, so the width of the bubble-wall is at most
$O(\mu^{-1})$. We should, therefore, use a prescription where $p<\mu$ is
excluded from the integration in Eq. (B.17). This regularization prescription
introduces some uncertainty in the coefficient of $\mu^{2}T^{2}$ terms; they
depend on the choice the IR cut off. However, since we are interested in
$r>T/y>T$, for which the $\mu^{2}r^{2}$ term dominates over $\mu^{2}T^{2}$
terms, this IR dependence does not introduce any relevant uncertainty in the
computation of the bubble nucleation rate. An explicit computation of this
nucleation rate is beyond the scope of this paper.

#### With axion.

Finally, we turn on the sphaleron processes to consider the charge transfer to
the axion through a Chern-Simons coupling, as in Sec. 3.2. In this case, the
axion and AD charges are no longer separately conserved. Rather their linear
combination is the only conserved quantity of the system, i.e.,

$\displaystyle Q=Q_{\text{AD}}-Q_{a}=Q_{\phi}+\frac{Q_{5}}{2}-Q_{a}.$ (B.26)

Assuming a homogeneous background of $\phi=re^{i\theta_{\phi}}/\sqrt{2}$
again, the one-loop effective potential acquires an additional contribution
from the axion

$\displaystyle
V_{\mu,\text{eff}}(r,T,\mu)\simeq\left.V_{\mu,\text{eff}}(r,T,\mu)\right|_{\text{Eq.~{}\eqref{eq:effmu-
AD}}}-\frac{\mu^{2}f^{2}}{2}+\frac{1}{2}\ln\det\Box.$ (B.27)

Here we neglect the potential from the coupling with $G\tilde{G}$ and non-
perturbative gauge dynamics, which makes a particular linear combination of
$\theta_{\phi}-\theta_{a}$ massive. The last term comes from the axion
fluctuations, which does not lead to a $\mu$-dependent effective potential and
hence one may omit it for our purpose.

Using Eq. (B.27), we may compute $V_{q}$ and check the minimum of the
effective potential. The result is identical to the discussion in Sec. 3.2 up
to the modification of $O(1)$ factors.

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