KCL-PH-TH/2023-41

Fate of oscillating homogeneous $\mathbb{Z}_{2}$-symmetric scalar condensates
in the early Universe

Wen-Yuan Ai∗1††∗<EMAIL_ADDRESS>and Zi-Liang Wang†2†††
<EMAIL_ADDRESS>

1Theoretical Particle Physics and Cosmology, King’s College London,
Strand, London WC2R 2LS, UK

2Department of Physics, School of Science,
Jiangsu University of Science and Technology, Zhenjiang, 212003, China

###### Abstract

Dark matter, if represented by a $\mathbb{Z}_{2}$-symmetric scalar field, can
manifest as both particles and condensates. In this paper, we study the
evolution of an oscillating homogeneous condensate of a
$\mathbb{Z}_{2}$-symmetric scalar field in a thermal plasma in an FLRW
universe. We focus on the perturbative regime where the oscillation amplitude
is sufficiently small so that parametric resonance is inefficient. This
perturbative regime necessarily comprises the late stage of the condensate
decay and determines its fate. The coupled coarse-grained equations of motion
for the condensate, radiation, and spacetime are derived from first principles
using nonequilibrium quantum field theory. We obtain analytical expressions
for the relevant microscopic quantities that enter the equations of motion and
solve the latter numerically. We find that there is always a nonvanishing
relic abundance for a condensate with a $\mathbb{Z}_{2}$ symmetry that is not
spontaneously broken. This is because its decay rate decreases faster than the
Hubble parameter at late times due to either the amplitude dependence or the
temperature dependence in the condensate decay rate. Consequently, accounting
for the condensate contribution to the overall dark matter relic density is
essential for $\mathbb{Z}_{2}$ scalar singlet dark matter.

###### Contents

  1. 1 Introduction
  2. 2 Model and the condensate equation of motion
    1. 2.1 Model
    2. 2.2 Coupled fundamental equations of motion for the condensate and plasma.
    3. 2.3 Condensate equation of motion in the small-field regime
  3. 3 Coupled coarse-grained equations of motion
  4. 4 “Freeze-out” of the $\mathbb{Z}_{2}$ condensate
  5. 5 Numerical results
    1. 5.1 $m_{\phi}>m_{\chi}$
    2. 5.2 $m_{\phi}<m_{\chi}$
  6. 6 Conclusion and discussion
  7. A Closed-Time-Path formalism
  8. B Self-energies
    1. B.1 Proper four-vertex function
      1. B.1.1 Summary of the results
      2. B.1.2 Computation of $I_{a}$
    2. B.2 Sunset self-energy
      1. B.2.1 Computation of $J_{\phi}$
      2. B.2.2 Computation of $J_{\chi}$
  9. C Solving the condensate equation of motion with a time-dependent mass term

##  1 Introduction

Scalar fields appear in many theories beyond the Standard Model (SM) of
particle physics. For instance, in theories with extra dimensions, there are
numerous moduli scalar fields [1, 2]. The inflationary paradigm, which
describes the very early stage of the Universe, commonly assumes the inflaton
to be a scalar field [3, 4, 5, 6, 7]. Scalar fields can also explain the
strong CP problem [8, 9]111See Refs. [10, 11] for a different perspective on
the strong CP problem., be dark matter (DM) [12, 13, 14, 15, 16, 17] and dark
energy [18, 19, 20].

Scalar fields with $\mathbb{Z}_{2}$ symmetry are natural DM candidates [12,
13, 14]. In this scenario, it is typically assumed that DM exists in the form
of particles. However, a scalar field may also form condensate in the early
Universe. For example, the scalar field may be pushed away from its
equilibrium value during inflation due to the dramatic universe expansion [21,
22, 23]. After the inflation and when the Hubble parameter is smaller than its
mass, it starts oscillating coherently, forming an oscillating condensate. It
is known that the dynamics of condensates is very different from that of
particles. The $\mathbb{Z}_{2}$ symmetry cannot prohibit the coherently
oscillating condensate from decaying.

In this paper, we study the evolution of a $\mathbb{Z}_{2}$-symmetric scalar
condensate in the early Universe, taking into account fully the interactions
between the condensate, plasma, and spacetime. We focus on the late stage
where the oscillation amplitude is sufficiently small such that
nonperturbative processes are inefficient. This stage determines the fate of
the condensate. Our analysis is based on the Closed-Time-Path (CTP) formalism
[24, 25]. Earlier studies on the dissipation of scalar backgrounds using the
same formalism can be found in Refs. [26, 27, 28, 29, 30, 31, 32, 33].
Different from these references, we adopt the multiple-scale analysis [34, 35]
to derive the coupled Markovian equations of motion from first principles,
closely following the recent work [36]. We present an explicit computation for
the relevant quantities that enter the coupled equations of motion (EoMs),
allowing us to solve the latter numerically.

From the obtained equations, the decay rate of the oscillating condensate is
shown to be both temperature-dependent and amplitude-dependent. This kind of
time-dependent dissipation rate is usually considered only phenomenologically
[37, 38, 39, 40, 41, 42]. In this work, we go beyond the high-temperature
approximation for the condensate decay rate. This is crucial because it leads
to a conclusion different from that given in Refs. [43, 44]. If one uses the
high-temperature approximation, it is possible to have a decay rate that is
always larger than the Hubble parameter, and one may conclude that the
condensate can transfer all its energy to radiation for some values of the
coupling and mass parameters. However, going beyond the high-temperature
approximation we find that the decay rate always decreases faster than the
Hubble parameter at low temperatures and thus the condensate must “freeze out”
at some point, leading to a nonvanishing relic density provided the
$\mathbb{Z}_{2}$ symmetry is not spontaneously broken.

The coupled EoMs derived in this work could be used for investigating
perturbative production of DM from an oscillating scalar field [45] and
perturbative reheating [46, 47, 48, 49]. However, it is important to note that
our derivation assumes all particles in the plasma are in thermal equilibrium.
In scenarios such as freeze-in [50, 51], where certain particle species remain
out of equilibrium, it becomes necessary to track the evolution of their
particle distribution functions. A derivation of the coupled EoMs from
nonequilibrium quantum field theory in such cases is provided in Ref. [52].

The paper is organised as follows. In Sec. 2 we introduce the model and the
non-local condensate EoM derived from the CTP formalism. The non-local terms
are due to self-energy corrections. In order to describe the condensate’s
evolution in the early Universe, it is sufficient to derive the EoM for its
envelope function. In Sec. 3 we therefore derive the coupled coarse-grained
EoMs for the condensate, radiation, and spacetime. This requires the
Markovianization of the original non-local condensate EoM. In Sec. 4 we
analyse the freeze-out behaviour of the condensate analytically. The obtained
coupled coarse-grained equations are then solved numerically in Sec. 5. Sec. 6
is left for conclusion and discussion. We collect several technical details in
appendices. Appendix A is a brief introduction to the CTP formalism. In
Appendix B we present an explicit computation of the self-energies. In
Appendix C, we solve the non-local condensate EoM with a time-dependent mass
term using multiple-scale analysis.

##  2 Model and the condensate equation of motion

###  2.1 Model

We consider the following model

$\displaystyle
S[\Phi,\chi]=\int\mathrm{d}^{4}x\,\sqrt{-g}\biggl{[}\frac{1}{2}(\partial_{\mu}\Phi)(\partial^{\mu}\Phi)+\frac{1}{2}(\partial_{\mu}\chi)(\partial^{\mu}\chi)-V(\Phi,\chi)\biggr{]}\,,$
(1)

with

$V(\Phi,\chi)=\frac{m_{\phi}^{2}}{2}\Phi^{2}+\frac{m_{\chi}^{2}}{2}\chi^{2}+\frac{\lambda_{\phi}}{4!}\Phi^{4}+\frac{\lambda_{\chi}}{4!}\chi^{4}+\frac{g}{4}\Phi^{2}\chi^{2}\,.$
(2)

Here $\Phi$ and $\chi$ are two real scalars. We assume that $\Phi$ forms
condensate, possessing a nonvanishing expectation value
$\langle\Phi\rangle\equiv\varphi$. The other scalar field $\chi$ may interact
with the SM fields. For example, $\chi$ could be the Higgs but in this case,
one needs to be careful with the electroweak phase transition that may occur
during the decay of the condensate. To avoid additional complications from
spontaneous symmetry breaking, in this work we assume that both $m_{\phi}^{2}$
and $m_{\chi}^{2}$ are positive. For the case that $\chi$ is the Higgs field
and there is the electroweak symmetry breaking, our main conclusions are not
changed. We will comment on this in the Conclusion section. For a spatially
flat FLRW universe,

$\mathrm{d}{s}^{2}=\mathrm{d}{t}^{2}-a^{2}(t)\mathrm{d}{\mathbf{x}}^{2}\,,$
(3)

and $\sqrt{-g}=a^{3}(t)$.

Expanding $\Phi=\varphi+\phi$, one can view $\varphi$ as a background field
and $\phi$, $\chi$ as fluctuations about this background. Particles are then
defined as excitations of these fluctuation fields. Discarding the linear
terms,222The linear terms in fluctuations, e.g., $m^{2}_{\phi}\varphi\phi$,
would not contribute to the perturbative diagrammatic expansion of the
effective action basically due to $\langle\phi\rangle=\langle\chi\rangle=0$
and will not induce any dynamics. For a more detailed explanation see, e.g.,
Ref. [53]. the action $S[\Phi,\chi]$ can be written as
$S_{\varphi}[\varphi]+\hat{S}[\phi,\chi;\varphi]$ where

$\displaystyle
S_{\varphi}[\varphi]=\int\mathrm{d}^{4}x\,a^{3}(t)\left[\frac{1}{2}(\partial_{\mu}\varphi)(\partial^{\mu}\varphi)-\frac{1}{2}m^{2}_{\phi}\varphi^{2}-\frac{\lambda_{\phi}}{4!}\varphi^{4}\right]\,,$
(4a)
$\displaystyle\hat{S}[\phi,\chi;\varphi]=\int\mathrm{d}^{4}x\,a^{3}(t)\left[\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)+\frac{1}{2}(\partial_{\mu}\chi)(\partial^{\mu}\chi)-V(\phi,\chi;\varphi)\right]\,,$
(4b)

with

$\displaystyle V(\phi,\chi;\varphi)$
$\displaystyle=\frac{1}{2}\left(m_{\phi}^{2}+\frac{\lambda_{\phi}}{2}\varphi^{2}\right)\phi^{2}+\frac{1}{2}\left(m^{2}_{\chi}+\frac{g}{2}\varphi^{2}\right)\chi^{2}+\frac{\lambda_{\phi}}{3!}\varphi\phi^{3}+\frac{g}{2}\varphi\phi\chi^{2}+\frac{\lambda_{\phi}}{4!}\phi^{4}$
$\displaystyle\quad+\frac{\lambda_{\chi}}{4!}\chi^{4}+\frac{g}{4}\phi^{2}\chi^{2}\,.$
(5)

The oscillation of $\varphi$ could induce particle production for both $\phi$
and $\chi$, either through the time-dependent mass terms or the interactions
$\varphi\phi^{3}$, $\varphi\phi\chi^{2}$.

###  2.2 Coupled fundamental equations of motion for the condensate and
plasma.

In addition to the condensate $\varphi$, our system also includes a plasma.
The plasma can be described by the two-point functions of the fluctuation
fields, $\langle\phi\phi\rangle\equiv\Delta_{\phi}$,
$\langle\chi\chi\rangle\equiv\Delta_{\chi}$. In the present subsection, for
simplicity, we avoid distinguishing from various types of two-point functions.
For more details, see Appendix A. In principle, we have coupled EoMs for these
one- and two-point functions. Such coupled EoMs can be most generally derived
from the two-particle-irreducible (2PI) effective action [54, 55]. The latter
is a functional of the one- and two-point functions, $\Gamma_{\rm
2PI}[\varphi,\Delta_{\phi},\Delta_{\chi}]$, constructed from the Legendre
transform of the generating functional with local one-point and non-local two-
point sources (Appendix A). The EoMs are given by

$\displaystyle\frac{\delta\Gamma_{\rm
2PI}[\varphi,\Delta_{\phi},\Delta_{\chi}]}{\delta\varphi(x)}$
$\displaystyle=0\,,$ (6a) $\displaystyle\frac{\delta\Gamma_{\rm
2PI}[\varphi,\Delta_{\phi},\Delta_{\chi}]}{\delta\Delta_{\phi}(x,x^{\prime})}$
$\displaystyle=0\,,$ (6b) $\displaystyle\frac{\delta\Gamma_{\rm
2PI}[\varphi,\Delta_{\phi},\Delta_{\chi}]}{\delta\Delta_{\chi}(x,x^{\prime})}$
$\displaystyle=0\,.$ (6c)

The last two equations, typically studied in the absence of a background
field, can be reduced to Boltzmann equations with a certain localization
procedure [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67].333More
specifically, the Boltzmann equation is obtained from the EoM for the Wightman
functions (cf. Eq. (A39), the same for $\chi$). The presence of the condensate
introduces both $\varphi$-dependent masses and vertices.

The 2PI effective action $\Gamma_{\rm 2PI}$ can be expanded in terms of
Feynman diagrams. In such diagrams, the propagators are the full two-point
functions $\Delta_{\phi}$ and $\Delta_{\chi}$. Of particular interest is the
condensate EoM at the one-loop444We emphasise that here “one-loop” is said in
the sense of the 2PI effective action. If we use free propagators, the full
two-point functions $\langle\phi\phi\rangle$ and $\langle\chi\chi\rangle$ can
be expressed as a sum of infinitely many diagrams with arbitrarily higher
loops.

$\displaystyle\left(\Box+m_{\phi}^{2}\right)\varphi(x)+\frac{\lambda_{\phi}}{6}\varphi^{3}(x)+\frac{\lambda_{\phi}}{2}\varphi(x)\Delta_{\phi}(x,x)+\frac{g}{2}\varphi(x)\Delta_{\chi}(x,x)=0\,,$
(7)

which is local. At this level, one can already have damping for the condensate
if the plasma is not in thermal equilibrium, i.e., if

$\displaystyle\Delta_{\phi}\neq\Delta_{\phi}^{\rm
eq}\,,\qquad\Delta_{\chi}\neq\Delta_{\chi}^{\rm eq}\,,$ (8)

where the superscript “eq” indicates thermal equilibrium. Writing
$\Delta_{\phi/\chi}=\Delta_{\phi/\chi}^{\rm eq}+\delta\Delta_{\phi/\chi}$, the
equilibrium parts $\Delta^{\rm eq}_{\phi/\chi}(x,x)$ simply correspond to
thermal corrections to the squared mass of the field $\Phi$ (and hence also of
the condensate). The resulting damping term is proportional to
$\delta\Delta_{\phi/\chi}$, and the latter can be determined by solving the
Boltzmann equations that are derived from the last two equations in (6). At
high temperatures $T\gg M_{\phi}$ where $M_{\phi}$ is the oscillation
frequency, the damping coefficient can be related to the bulk viscosity [68].
Such a map must break down at low temperatures
$T\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;M_{\phi}$
because the bulk viscosity diverges for $T\rightarrow 0$ [69, 70] while the
damping coefficient due to the deviation from equilibrium should decrease to
zero for $T\rightarrow 0$ as in that limit the condensate oscillates simply in
vacuum.

##### The small-field regime.

The plasma can be driven out of equilibrium essentially because of the
$\varphi$-dependent masses (see e.g., [68]). In this paper, we are interested
in the evolution of the condensate at a sufficiently late stage so that

$\displaystyle\frac{\lambda_{\phi}}{2}\varphi^{2}\ll M^{2}_{\phi}\equiv
m_{\phi}^{2}+\frac{\lambda_{\phi}}{2}\varphi^{2}+m^{2}_{\phi,\rm
th}\,,\quad\frac{g}{2}\varphi^{2}\ll M^{2}_{\chi}\equiv
m_{\chi}^{2}+\frac{g}{2}\varphi^{2}+m_{\chi,\rm th}^{2}\,,$ (9)

where $m_{\phi,\rm th}$ and $m_{\chi,\rm th}$ are thermal corrections to the
masses of $\phi$ and $\chi$ particles. The first condition also indicates that
the oscillation is quasi-harmonic. This late stage is still interesting
because it determines the fate of the oscillating condensate, i.e., whether
the condensate can have a nonvanishing relic abundance. The conditions in Eq.
(9) define the so-called small-field regime. In this case, one can perform a
perturbative expansion in $\varphi$ in $\Delta_{\phi,\chi}[\varphi]$ and at
the leading order, the two-point functions are independent of $\varphi$. In
the small-field regime, it is expected that the damping due to the out-of-
equilibrium effects mentioned above becomes inefficient as the particle masses
are not sensitive to the change of $\varphi$ anymore. Furthermore, the small-
field regime is precisely the regime in which nonperturbative particle
production from condensate oscillations can be neglected and the condensate
decays mainly via perturbative processes [71].

In the small-field regime, we have to consider other types of damping for the
condensate oscillations. Going beyond the one-loop of the 2PI effective
action, one has non-local terms in the condensate EoM which describe direct
interactions between the condensate and particles induced by the
$\varphi$-dependent vertices. Such interactions cause damping even if the
plasma is in thermal equilibrium. This is the damping more often studied in
the literature [26, 27, 28, 29, 30, 31, 32, 33, 36, 44] and is the focus of
the present work.

The above observation also applies to bubble wall dynamics in cosmological
first-order phase transitions. The local EoM (7) is usually used for studying
the bubble expansion. In the ultrarelativistic regime, the third and fourth
terms, usually written as

$\displaystyle\sum_{i}\frac{\mathrm{d}m^{2}_{i}(\phi)}{\mathrm{d}\phi}\int\frac{\mathrm{d}^{3}{\bf
p}}{(2\pi)^{3}2E_{i}}\,f_{i}(p,z)\,,$ (10)

give rise to the Bödeker-Moore thermal friction [72, 73]. However, there is
also additional friction caused by particle-production processes induced by
condensate-dependent vertices, see e.g. Refs. [74, 75]. This force is not
captured by the local terms in Eq. (7) but rather is encoded in the non-local
terms when one goes beyond the one-loop of the 2PI effective action [76].

###  2.3 Condensate equation of motion in the small-field regime

We make the approximation that the plasma is in thermal equilibrium during the
evolution of the condensate under study. Therefore, the leading-order two-
point functions $\Delta_{\phi,\chi}[\varphi=0]$ in the small-field expansion
are fixed to be the thermal equilibrium propagators and essentially, one only
needs to solve the condensate EoM [36]. This dramatically simplifies the
problem. The condensate EoM can be derived in the CTP formalism [24, 25] using
the effective action, see e.g., Ref. [36]. To be self-contained, we give a
brief introduction to the CTP formalism in Appendix A.

Properly truncating the effective action and taking into account the
homogeneity of the condensate and plasma, one obtains the condensate EoM as
[36, 44]

$\displaystyle\ddot{\varphi}(t)+M_{\phi}^{2}\varphi(t)+3H\dot{\varphi}(t)+\frac{\lambda_{\phi}\varphi^{3}(t)}{6}+\int_{t_{i}}^{t}\mathrm{d}t^{\prime}\,\pi^{\rm
R}(t-t^{\prime})\varphi(t^{\prime})+\frac{\varphi(t)}{6}\int_{t_{i}}^{t}\mathrm{d}t^{\prime}\,v^{\rm
R}(t-t^{\prime})\varphi^{2}(t^{\prime})=0\,,$ (11)

where $t_{i}$ is the initial time at which initial conditions $\varphi(t_{i})$
and $\dot{\varphi}(t_{i})$ are specified. $\pi^{\rm R}$ and $v^{\rm R}$ are
the retarded self-energy and proper four-vertex function integrated over
space, respectively. The proper four-vertex function is also called self-
energy in a broad sense. They correspond to the following diagrams in the
effective action,

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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (12b)

where a solid/dashed line denotes the free thermal equilibrium propagator for
$\phi/\chi$ (cf. Eq. (B54)), and a solid line ended with a wheel cross
indicates a quantum of the condensate. $\pi^{\rm R}$ and $v^{\rm R}$ are
explicitly defined in Appendix B. The self-energies are discussed in Appendix
B and the expressions for the relevant quantities that enter the coarse-
grained EoMs will be given below. Note that the diagrams of $v^{\rm R}$ are
generated from the one loop diagrams in $\Gamma_{\rm 2PI}$ by the small-field
expansion while $\pi^{\rm R}$ is obtained from two loop diagrams in
$\Gamma_{\rm 2PI}$ [36].

The last two terms in Eq. (11) exhibit non-locality in time and therefore
contain memory effects, making it challenging to obtain a rigorous solution
for the EoM. However, one can reasonably expect the first two terms in Eq.
(11) to dominate. By treating the remaining terms as perturbations, one can
solve the condensate EoM using multiple-scale analysis [34, 35] as described
in Refs. [36, 44]. There, constant temperature and $M_{\phi}$ are assumed. In
the present work, we account for the evolution of temperature which makes
$M_{\phi}$ time dependent. In Appendix C, we extend the analysis of Refs. [36,
44] to the case with time-dependent but adiabatic $M_{\phi}$. The solution
takes the following form:

$\displaystyle\varphi(t)\approx
A(t)\cos\left[\int^{t}M_{\phi}(t^{\prime})\mathrm{d}t^{\prime}+f(t)\right]\,,$
(13)

where $A(t)$ is the envelope function of the oscillation and $f(t)$ represents
corrections to the local frequency $M_{\phi}$. The EoMs for $A(t)$ and $f(t)$
are found to be

$\displaystyle\frac{\mathrm{d}A(t)}{\mathrm{d}t}+\left(\gamma+\frac{3}{2}H(t)+\frac{1}{2M_{\phi}(t)}\frac{\mathrm{d}M_{\phi}(t)}{\mathrm{d}t}\right)A(t)+\frac{\sigma}{2}[A(t)]^{3}$
$\displaystyle=0\,,$ (14a)
$\displaystyle\frac{\mathrm{d}f(t)}{\mathrm{d}t}-\mu-\left(\frac{\lambda_{\phi}}{16M_{\phi}}+\alpha\right)[A(t)]^{2}$
$\displaystyle=0\,,$ (14b)

where555Note that the $\gamma$ defined in Ref. [44] is half of that defined in
Ref. [36] and here we follow the former.

$\displaystyle\mu$ $\displaystyle\equiv\frac{\text{Re}[\widetilde{\pi}^{\rm
R}(M_{\phi})]}{2M_{\phi}}\,,\;\gamma\equiv-\frac{\text{Im}[\widetilde{\pi}^{\rm
R}(M_{\phi})]}{2M_{\phi}}\,,\;\sigma\equiv-\frac{\text{Im}[\widetilde{v}^{\rm
R}(2M_{\phi})]}{24M_{\phi}}\,,$ $\displaystyle\alpha_{1}$
$\displaystyle\equiv\frac{\text{Re}[\widetilde{v}^{\rm R}(0)]}{24M_{\phi}}\,,\
\ \,\alpha_{2}\equiv\frac{\text{Re}[\widetilde{v}^{\rm
R}(2M_{\phi})]}{48M_{\phi}}\,,\ \;\alpha\equiv\alpha_{1}+\alpha_{2}\,,$ (15)

with

$\widetilde{\pi}^{\rm
R}(\omega)=\int_{-\infty}^{\infty}\mathrm{d}t^{\prime}\,\mathrm{e}^{\mathrm{i}\omega(t-t^{\prime})}\pi^{\rm
R}(t-t^{\prime})\,,\qquad\widetilde{v}^{\rm
R}(\omega)=\int_{-\infty}^{\infty}\mathrm{d}t^{\prime}\,\mathrm{e}^{\mathrm{i}\omega(t-t^{\prime})}v^{\rm
R}(t-t^{\prime})\,.$ (16)

Above, one actually has neglected the early-time transient effects related to
the initial conditions in the non-local terms. This is why the EoMs for $A(t)$
and $f(t)$ above depend on the normal Fourier transforms of the retarded self-
energy and proper four-vertex function.

##  3 Coupled coarse-grained equations of motion

In order to understand the energy transfer between the condensate and plasma,
it is sufficient to study the EoM of the envelope function, Eq. (14a). Indeed,
the energy of the condensate can be approximated as

$\displaystyle\rho_{\varphi}=\frac{1}{2}\dot{\varphi}^{2}+\frac{1}{2}M^{2}_{\phi}\varphi^{2}\approx\frac{1}{2}M^{2}_{\phi}A(t)^{2}\,,$
(17)

where we have assumed that

$\displaystyle\frac{\dot{A}}{A}\ll
M_{\phi}\,,\quad\frac{\mathrm{d}f}{\mathrm{d}t}\ll M_{\phi}\,.$ (18)

The above conditions imply that the amplitude and the frequency of
oscillations do not change much during one oscillation. Therefore, the
evolution of $A(t)$ contains information on the dissipation of the condensate
energy. From Eq. (14a) we obtain

$\displaystyle\dot{\rho}_{\varphi}+3H\rho_{\varphi}+\left(2\gamma-\frac{\dot{M}_{\phi}}{M_{\phi}}\right)\rho_{\varphi}+\frac{2\sigma\rho_{\varphi}^{2}}{M_{\phi}^{2}}=0\,.$
(19)

The $\gamma$ and $\sigma$ depend on the temperature and thus are time
dependent. Eq. (19) is accompanied by the EoM for the radiation energy density
and the Friedmann equation

$\displaystyle\dot{\rho}_{\rm R}+4H\rho_{\rm
R}-\left(2\gamma-\frac{\dot{M}_{\phi}}{M_{\phi}}\right)\rho_{\varphi}-\frac{2\sigma\rho_{\varphi}^{2}}{M_{\phi}^{2}}=0\,,$
(20) $\displaystyle H^{2}=\frac{8\pi(\rho_{\varphi}+\rho_{\rm R})}{3M^{2}_{\rm
P}}\,,$ (21)

where $M_{\rm P}$ is the Planck mass. The radiation energy density is related
to the temperature through

$\displaystyle\rho_{\rm R}=\frac{\pi^{2}}{30}g_{*}T^{4}\,,$ (22)

where $g_{*}$ is the effective number of relativistic degrees of freedom. Here
we have assumed that during the dissipation of the condensate, there are no
other forms of energy densities.

If we define the number density of the condensate quanta $n_{\varphi}$ through
$n_{\varphi}=\rho_{\varphi}/M_{\phi}$, then Eq. (19) becomes

$\displaystyle\dot{n}_{\varphi}+3Hn_{\varphi}=-2\gamma
n_{\varphi}-\frac{2\sigma}{M_{\phi}}n_{\varphi}^{2}\,.$ (23)

This implies that the comoving number density of the condensate quanta is
conserved for vanishing $\gamma$ and $\sigma$.

We note that the evolution of the envelope function (and of $\rho_{\varphi}$)
depends on two particular microscopic quantities, $\gamma$ and $\sigma$, the
imaginary parts of the retarded self-energy and retarded proper four-vertex
function evaluated at particular frequencies. These quantities have an
interpretation in terms of particle production via the cutting rules at finite
temperature [77, 78, 79, 80, 81, 82, 83], which explains the origin of the
dissipation. For more details see Ref. [44] and Appendix B. In summary,
nonvanishing $\gamma$ describes the following processes

$\displaystyle\gamma{\rm\
channels:}\quad\varphi\phi\leftrightarrow\phi\phi\,,\quad\varphi\chi\leftrightarrow\phi\chi\,,\quad\varphi\phi\leftrightarrow\chi\chi\,,$
(24)

while $\sigma$ describes

$\displaystyle\sigma\ {\rm
channel:}\quad\quad(\varphi\varphi)\leftrightarrow\chi\chi\quad{\rm for\
}M_{\phi}>M_{\chi}\,.$ (25)

Eq. (24) are scattering processes that are absent at zero temperature and are
called Landau damping. We leave the details of an explicit computation of
$\gamma$ and $\sigma$ in Appendix B.

The imaginary part of the retarded proper four-vertex is known in the
literature [84], and reads

$\displaystyle\sigma=\theta\left(M_{\phi}-M_{\chi}\right)\frac{g^{2}}{256\pi
M_{\phi}}\sqrt{1-\frac{M_{\chi}^{2}}{M_{\phi}^{2}}}\,\left[1+2f_{\rm
B}\left(M_{\phi}\right)\right]\,.$ (26)

In the limit of $T=0$, one has

$\displaystyle\sigma_{0}\equiv\left.\sigma\right|_{T=0}=\theta\left(m_{\phi}-m_{\chi}\right)\frac{g^{2}}{256\pi
m_{\phi}}\sqrt{1-\frac{m_{\chi}^{2}}{m_{\phi}^{2}}}\,.$ (27)

The $\gamma$ has contributions from the self-interactions of $\Phi$ and the
interaction between $\Phi$ and $\chi$:
$\gamma=\gamma_{\phi}+\gamma_{\phi\chi}$. The imaginary part of the self-
energy from the $\Phi$ self-interaction has been obtained in Refs. [69, 85,
86] and reads666Note a sign difference in the definition of the imaginary part
of Ref. [69]. See Eq. (2.35) of that reference.

$\displaystyle\left.{\rm Im}[\widetilde{\pi}^{\rm R}(M_{\phi})]\right|_{\rm
self\ interaction}=-\frac{\lambda_{\phi}^{2}T^{2}}{128\pi^{3}}{\rm
Li}_{2}\left(\mathrm{e}^{-M_{\phi}/T}\right)\,.$ (28)

where ${\rm Li}_{2}(z)$ is the dilogarithm function defined as

$\displaystyle{\rm
Li}_{2}(z)=-\int^{1}_{0}\frac{\mathrm{d}x}{x}\ln(1-z\,x)\,,$ (29)

for $z\in(-\infty,1)$. The dilogarithm function has the following asymptotic
behaviours

$\displaystyle\text{Li}_{2}(\mathrm{e}^{-M_{\phi}/T})=\frac{\pi^{2}}{6}+O\left[\frac{M_{\phi}}{T}\ln(\frac{M_{\phi}}{T})\right]\quad{\rm
for\ }T\gg M_{\phi}\,,$ (30)
$\displaystyle\text{Li}_{2}(\mathrm{e}^{-M_{\phi}/T})=\mathrm{e}^{-M_{\phi}/T}+O\left(\mathrm{e}^{-2M_{\phi}/T}\right)\quad{\rm
for\ }T\ll M_{\phi}\,.$ (31)

Substituting the above into the definition of $\gamma$, we obtain

$\displaystyle\gamma_{\rm\phi}=\frac{\lambda_{\phi}^{2}T^{2}}{256M_{\phi}\pi^{3}}{\rm
Li}_{2}\left(\mathrm{e}^{-M_{\phi}/T}\right)\,.$ (32)

The high-temperature limit of Eq. (32) is consistent with the result given in
Refs. [87, 88]. For $T\rightarrow 0$, $\gamma_{\phi}$ vanishes. This is
reasonable because at zero temperature all the $\gamma$ channels are not
possible.

While we cannot provide a closed form for $\gamma_{\phi\chi}$, we do obtain an
analytical fit for it (cf. Appendix B.2):

$\displaystyle\gamma_{\phi\chi}\sim\gamma_{\phi\chi,\rm fit}$
$\displaystyle\equiv 4.5\times\frac{g^{2}T^{2}}{128M_{\phi}\pi^{5}}{\rm
Li}_{2}\left(\frac{4}{(M_{\chi}/M_{\phi})^{3}+4}\right){\rm
Li}_{2}\left(\mathrm{e}^{-\frac{M_{\chi}+2M_{\phi}}{3T}}\right)$
$\displaystyle+6.8\times\frac{g^{2}T^{2}}{64M_{\phi}\pi^{5}}{\rm
Li}_{2}\left(\frac{1.6}{(M_{\chi}/M_{\phi})+1.6}\right){\rm
Li}_{2}\left(\mathrm{e}^{-\frac{2M_{\chi}+M_{\phi}}{3T}}\right)\,.$ (33)

One can use the above fit to solve the coupled dynamics. For $T\rightarrow 0$,
$\gamma_{\phi\chi,\rm fit}$ vanishes as one would expect.

##  4 “Freeze-out” of the $\mathbb{Z}_{2}$ condensate

Before we present some exact numerical results, let us first estimate the
condition for the “freeze-out” of the $\mathbb{Z}_{2}$ condensate. From Eq.
(19), one can read the condensate “decay rate” as

$\displaystyle\Gamma_{\varphi}=2\gamma-\frac{\dot{M}_{\phi}}{M_{\phi}}+\frac{2\sigma\rho_{\varphi}}{M^{2}_{\phi}}\,.$
(34)

Note that this decay rate is not necessarily equal to the damping rate defined
from the envelop function $A(t)$ [36]. Nevertheless, it can still characterise
how fast the condensate depletes its energy into the plasma. An efficient
decay requires
$\Gamma_{\varphi}\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle>}}{{\sim}}$}\;H$
so that one can use $\Gamma_{\varphi}\sim H$ as an estimate of the “freeze-
out” time of the condensate. The full evolution is complicated because there
could be some reheating effects due to the energy transfer from the condensate
to the plasma. However, the “freeze-out” of the condensate typically occurs
after the reheating. At that stage, one would have $\rho_{\rm
R}\gg\rho_{\varphi}$,777If the initial conditions already satisfy this, then
reheating effects are negligible. and

$\displaystyle H\sim
1.66\times\sqrt{g_{*}}T^{2}\sim\mathcal{O}(10)\,T^{2}\sim\frac{1}{t}\,.$ (35)

#####  $\gamma$ channels.

In the high-temperature limit, $\gamma$ exhibits a decrease proportional to
the temperature $T$ as the Universe cools down. When the temperature drops to
the regime where $M_{\phi}\sim m_{\phi}$, $\gamma$ decreases with a quadratic
dependence on $T$, thus scaling as $T^{2}$. At lower temperatures, $\gamma$
experiences an exponential suppression. As a consequence, when the temperature
drops to a sufficiently low value, $\gamma$ decreases faster than $H$.

#####  $\sigma$ channel.

While $\sigma$ is not vanishing even for $T\rightarrow 0$ provided
$m_{\phi}>m_{\chi}$, the condensate decay rate due to the sigma channel is
proportional to $\rho_{\varphi}$ which decreases fast. For the case of
$\gamma=0$ and $\dot{M}_{\phi}=0$, $\rho_{\varphi}$ drops as $t^{-3/2}$ [44]
while $H$ drops as $1/t$. With nonvanishing $\gamma$ and $\dot{M}_{\phi}$,
$\rho_{\varphi}$ should decrease even faster.

##### Damping due to the change of $M_{\phi}$.

In the high-temperature limit, one has

$\displaystyle\frac{\dot{M}_{\phi}}{M_{\phi}}=\frac{(\lambda_{\phi}+g)T\dot{T}/24}{{m^{2}_{\phi}+(\lambda_{\phi}+g)T^{2}/24}}\sim\frac{\dot{T}}{T}$
(36)

which scales as $H$. Going beyond the high temperature limit,
$\dot{M_{\phi}}/M_{\phi}$ decreases faster and quickly approaches zero when
$(\lambda_{\phi}+g)T^{2}/24<m_{\phi}^{2}$. Actually, in the parameter region
for our numerical calculations below, $\dot{M}_{\phi}/M_{\phi}$ is negligible.

In summary, $\Gamma_{\varphi}$ would decrease faster than $H$ as the Universe
cools down and therefore the condensate must “freeze out” at some point. This
will be confirmed further with our numerical results below. We emphasize that
it is crucial to go beyond the high-temperature approximation for $\gamma$.
Otherwise, one can have, for some chosen parameters, a $\gamma$ that is larger
than $H$ all the time and conclude that the condensate energy can be fully
transferred to radiation [43, 44]. The above analysis does not draw any
conclusion about the relic density of the condensate. The latter strongly
depends on the parameters in the theory as well as the initial conditions.

##  5 Numerical results

In this section, we numerically solve the coupled equations (19), (20), and
(21). The purpose of this section is to numerically demonstrate the “freeze-
out” behaviour of the $\mathbb{Z}_{2}$ condensate analysed in the last
section. Since we do not aim for a rigorous phenomenological study, the mass
and coupling parameters will be to some extent chosen randomly without good
phenomenological motivations. A complete phenomenological study on the
condensate contribution to the $\mathbb{Z}_{2}$ singlet scalar DM will be left
for future work. Above, analytical expressions for $\sigma$ and $\gamma$
(approximated as $\gamma_{\rm fit}\equiv\gamma_{\phi}+\gamma_{\phi\chi,\rm
fit}$) have already been presented. For the thermal corrections to the masses,
we shall use the high-temperature approximation (B58). We assume that the
thermal corrections to the $\chi$ mass will be dominated by the term
$\kappa^{2}T^{2}$ and we take $\kappa^{2}\approx 0.1$. We note however that in
the numerical calculation below, the high-temperature limit may not hold.
However, for perturbatively small couplings, the thermal mass corrections away
from the high-temperature regime are immaterial anyway (cf. Appendix B).
Therefore whether or not one uses the high-temperature approximation for the
thermal mass corrections does not change the results quantitatively. However,
as we emphasised in the Introduction, it is crucial to use the complete
$T$-dependence for the $\gamma$ parameter as its high-temperature
approximation would lead to incorrect conclusions.

We will not be concerned with the formation of the condensate and its early
nonperturbative decay through parametric resonance. Rather, we assume that
these processes produce some initial condition
$\\{T_{i},\bar{\varphi}_{i}\\}$, where $T_{i}$ is the initial temperature and
$\bar{\varphi}_{i}$ is the initial oscillation amplitude, for the perturbative
regime at the late stage. In this work, we randomly choose the initial
condition in performing the numerical calculation, provided that it falls
within the perturbative regime of particle production, satisfying $(i)\
M^{2}_{\phi}\gg\lambda_{\phi}\bar{\varphi}_{i}^{2}/2$ and $(ii)\
M^{2}_{\chi}\gg g\bar{\varphi}_{i}^{2}/2$. We will consider different sets of
$\\{T_{i},\bar{\varphi}_{i}\\}$ to explore the sensitivity of the results to
the initial condition. For our analysis, we set $g_{*}=100$, $m_{\chi}=150{\rm
GeV}$ and consider two cases for $m_{\phi}$, $m_{\phi}=100{\rm GeV}<m_{\chi}$
and $m_{\phi}=500{\rm GeV}>m_{\chi}$. An example of the constraints on the
initial conditions around $g=10^{-6}$ and $\lambda_{\phi}=10^{-6}$ is given in
Fig. 1 where the boundary is obtained with
$M_{\phi}^{2}=10\times\lambda_{\phi}\bar{\varphi}_{i}^{2}/2$,
$M_{\chi}^{2}=10\times g\bar{\varphi}_{i}^{2}/2$. One can also find the
constraints on $\\{T_{i},\bar{\varphi}_{i}\\}$ for other chosen couplings.
When fixing $\lambda_{\phi}$ but changing $g$, the regions are mostly affected
by the condition $(ii)$ (see the upper panel). When fixing $g$ but changing
$\lambda_{\phi}$, the regions are mostly affected by the condition $(i)$ (see
the lower panel). Note that the seemingly vertical lines are not strictly
vertical; they just have a very large slope.

Figure 1: Initial conditions that fall within the perturbative regime with
fixed masses $m_{\phi}=500{\rm GeV}$, $m_{\chi}=150{\rm GeV}$.
$\lambda_{\phi}=10^{-6}$ for the upper panel and $g=10^{-6}$ for the lower
panel.

###  5.1 $m_{\phi}>m_{\chi}$

Let us first consider the case $m_{\phi}=500{\rm GeV}>m_{\chi}=150{\rm GeV}$.

##### Different couplings.

We fix the initial condition as $\bar{\varphi}_{i}=5m_{\phi}$,
$T_{i}=2m_{\phi}$ and consider three sets of couplings:
$\\{g=10^{-7},\lambda_{\phi}=10^{-6}\\}$,
$\\{g=10^{-6},\lambda_{\phi}=10^{-6}\\}$, and
$\\{g=10^{-6},\lambda_{\phi}=10^{-5}\\}$. Fig. 2 shows the relic comoving
energy density $\rho_{\varphi}a^{3}$ for these three cases while Fig. 3 shows
the details about the evolution. For $g=10^{-7}$, $\lambda_{\phi}=10^{-6}$,
the condensate decay is rather incomplete. This is because both $\gamma$ and
$\sigma\rho_{\varphi}/M_{\phi}^{2}$ are smaller than $H$ all the time and no
efficient decay can occur (see the first row of Fig. 3). For the second case,
$\sigma\rho_{\varphi}/M_{\phi}^{2}$ is larger than $H$ initially while
$\gamma$ is still smaller than $H$ all the time. Hence the decay is dominated
by the $\sigma$ channel. However, due to dependence on $\bar{\varphi}^{2}$,
$\sigma\rho_{\varphi}/M_{\phi}^{2}$ decreases quickly and the channel is
efficient only for a short time, leading to again an incomplete decay. In the
last case, $g=10^{-6}$, $\lambda_{\phi}=10^{-5}$, $\gamma$ is larger than $H$
for a period of the universe expansion and the decay is quite complete. In the
last two cases, one clearly sees a “freeze-out” behaviour of the condensate
which is characterised by the crossing between the evolution line of either
$\gamma$ or $\sigma\rho_{\varphi}/M_{\phi}^{2}$ and that of $H$. We note that
even if the relic comoving energy density of the condensate is very small
compared with the radiation comoving energy density, its actual relic energy
density decreases slower than the radiation energy density and can catch up
with the latter after a sufficiently long time of universe expansion. This
should lead to additional phenomenological constraints on such
$\mathbb{Z}_{2}$-symmetric scalar fields. For example, for the chosen initial
temperature $T_{i}=2m_{\phi}=1000{\rm GeV}$, the temperature decreases to
$10{\rm MeV}$ for $a/a_{I}\sim 10^{5}$. Since the Universe must be radiation-
dominated at the Big Bang nucleosynthesis (BBN), the first two cases in Fig. 3
would then be excluded as a phenomenological model.888We thank the anonymous
referee for pointing out this to us. However, our initial conditions are
chosen randomly. To have concrete conclusions, one has to carefully analyse
the nonperturbative process prior to the perturbative condensate decay studied
here. We leave a detailed study on this for future work.

Figure 2: Relic comoving energy densities of the condensate with various
coupling constants. The initial condition is chosen as
$\bar{\varphi}_{i}=5m_{\phi}$, and $T_{i}=2m_{\phi}$.  Figure 3: $\gamma$,
$\sigma\rho_{\varphi}/M_{\phi}^{2}$ and the Hubble parameter $H$ (the first
column) and the energy densities (the second column) for different couplings
but with fixed initial condition $\bar{\varphi}_{i}=5m_{\phi}$, and
$T_{i}=2m_{\phi}$.

##### Different initial conditions.

Fig. 4 and Fig. 5 present the numerical solutions for different initial
conditions, $\\{\bar{\varphi}_{i}=m_{\phi},T_{i}=0.2m_{\phi}\\}$,
$\\{\bar{\varphi}_{i}=5m_{\phi},T_{i}=0.2m_{\phi}\\}$, and
$\\{\bar{\varphi}_{i}=m_{\phi},T_{i}=2m_{\phi}\\}$, but with fixed couplings
$g=10^{-7}$, $\lambda_{\phi}=10^{-5}$. For the first initial condition, the
decay is rather incomplete because both $\gamma$ and
$\sigma\rho_{\varphi}/M_{\phi}^{2}$ are smaller than $H$ all the time. A
larger value of either $\bar{\varphi}_{i}$ or $T_{i}$ can enhance the decay
efficiency. A larger $\bar{\varphi}_{i}$ gives a larger initial decay rate of
the $\sigma$ channel. Particle production from the $\sigma$ channel then leads
to reheating and increases the temperature. As a consequence, the decay rate
from the $\gamma$ channels increases at early times, see the left panel of the
second row in Fig. 5. Increasing $T_{i}$ would directly increase $\gamma$ at
the beginning, see the left panel of the third row in Fig. 5. Fig. 6 presents
the evolution of $\gamma$, $\sigma\rho_{\varphi}/M_{\phi}^{2}$, and $H$ at
early times for all the three cases. Again, the first two cases would lead to
a matter-dominated universe already before BBN.

Figure 4: Comoving energy densities of the condensate for various initial
conditions but with fixed couplings $g=10^{-7}$ and $\lambda_{\phi}=10^{-5}$.
Figure 5: $\gamma$, $\sigma\rho_{\varphi}/M_{\phi}^{2}$ and the Hubble
parameter $H$ (the first column) and the energy densities (the second column)
for different initial conditions but fixed couplings $g=10^{-7}$ and
$\lambda_{\phi}=10^{-5}$. Figure 6: Early evolution of $\gamma$ and
$\sigma\rho_{\varphi}/M_{\phi}^{2}$ with various initial conditions:
$\\{\bar{\varphi}_{i}=m_{\phi},T_{i}=0.2m_{\phi}\\}$ (black),
$\\{\bar{\varphi}_{i}=5m_{\phi},T_{i}=0.2m_{\phi}\\}$ (blue),
$\\{\bar{\varphi}_{i}=m_{\phi},T_{i}=2m_{\phi}\\}$ (red) with fixed couplings
$g=10^{-7}$, $\lambda_{\phi}=10^{-5}$.

###  5.2 $m_{\phi}<m_{\chi}$

Now we consider the case $m_{\phi}=100{\rm GeV}<m_{\chi}=150{\rm GeV}$. In
this case, one typically has $M_{\chi}>M_{\phi}$ unless $\lambda_{\phi}$ is
too large which would then probably lead to unitarity problems. Therefore, for
this case, the $\sigma$ channel is closed.

##### Different couplings.

Again we fix the initial condition to $\bar{\varphi}_{i}=5m_{\phi}$ and
$T_{i}=2m_{\phi}$. Fig. 7 presents the obtained comoving energy densities of
the condensate for $\\{g=10^{-7},\lambda_{\phi}=10^{-6}\\}$,
$\\{g=10^{-6},\lambda_{\phi}=10^{-6}\\}$ and
$\\{g=10^{-6},\lambda_{\phi}=5*10^{-6}\\}$. The “freeze-out” behaviour can be
explained by the evolution of $\gamma$ and $H$ shown in Fig. 8.

Figure 7: Comoving energy densities of the condensate for various coupling
constants but with fixed initial condition $\bar{\varphi}_{i}=5m_{\phi}$, and
$T_{i}=2m_{\phi}$.  Figure 8: $\gamma$ and the Hubble parameter $H$ (the first
column) and the energy densities (the second column) for different couplings
but fixed initial condition $\bar{\varphi}_{i}=5m_{\phi}$, and
$T_{i}=2m_{\phi}$.

##### Different initial conditions.

Figs. 9 and 10 present the numerical solutions for different initial
conditions, $\\{\bar{\varphi}_{i}=m_{\phi},T_{i}=0.2m_{\phi}\\}$,
$\\{\bar{\varphi}_{i}=5m_{\phi},T_{i}=0.2m_{\phi}\\}$, and
$\\{\bar{\varphi}_{i}=m_{\phi},T_{i}=m_{\phi}\\}$, but fixed couplings
$g=10^{-6}$, $\lambda_{\phi}=10^{-5}$. Similar to the case
$m_{\phi}>m_{\chi}$, a larger $T_{i}$ would directly increase $\gamma$ at the
beginning, hence improving the decay efficiency. However, we still observe an
enhancement of the decay rate with a larger initial oscillation amplitude even
though the $\sigma$ channel is closed. This is again due to reheating effects
from particle production but via the $\gamma$ channels now. In the middle
panel of Fig. 10, one can see an early increase of $\gamma$ (reheating), which
is followed by a slower decrease. The evolution of the temperature in three
cases is shown in Fig. 11.

Figure 9: Comoving energy densities of the condensate for various initial
conditions with fixed couplings $g=10^{-6}$ and $\lambda_{\phi}=10^{-5}$.
Figure 10: Numerical solutions for coupled equations (19), (20), and (21). The
first column shows the evolution of $\gamma$ and the Hubble parameter $H$. The
second column shows the evolution of energy densities. Figure 11: Evolution of
radiation temperature for various initial conditions with fixed couplings
$g=10^{-6}$ and $\lambda_{\phi}=10^{-5}$.

##  6 Conclusion and discussion

In this work, we have studied the evolution of homogeneous
$\mathbb{Z}_{2}$-symmetric scalar condensates in the early Universe in the
perturbative regime where the oscillation amplitude is sufficiently small so
that dissipation occurs via perturbative processes. This also indicates that
the oscillation is quasi-harmonic. Otherwise, self-interactions of the scalar
field would lead to significant parametric resonance. This perturbative regime
necessarily comprises the late stage of the condensate evolution and
determines the fate of the condensate. Our analysis is based on the CTP
formalism which naturally accommodates quantum statistical effects of the
plasma as well as the backreaction effects from particle production.

We have derived, in the perturbative regime, the coupled coarse-grained EoMs
for the condensate energy, radiation, and spacetime from the original non-
local condensate EoM obtained from nonequilibrium quantum field theory. This
derivation makes use of multiple-scale analysis that was first introduced in
Ref. [36] to study the dissipation of scalar background fields. In this work,
we have extended the analysis in Ref. [36] to take into account both the
expanding spacetime and the time dependence in the temperature. The coarse-
grained EoMs depend on two microscopic quantities $\gamma$ and $\sigma$,
defined in Eq. (2.3), which have an interpretation in terms of particle
production. We have presented a very detailed computation of these two
quantities. The analytic result for $\gamma$ contributed from the sunset
diagram generated by the $\varphi\phi\chi^{2}$ interaction (the second diagram
from Eq. (12a)) is a new result. The coupled EoMs have been solved
numerically.

The key finding in this work is that a scalar condensate with $\mathbb{Z}_{2}$
symmetry cannot transfer all its energy to the plasma. The condensate decay
rate is given in Eq. (34) and depends on both the temperature and the
oscillation amplitude (through $\rho_{\varphi}$). As a result, the condensate
decay rate always decreases faster than the Hubble parameter at late times
(see the analysis in Sec. 4). Therefore, similar to the freeze-out process for
particles, the condensate comoving energy density approaches a constant once
the condensate decay rate drops below the Hubble parameter. A homogeneous
$\mathbb{Z}_{2}$-symmetric scalar condensate must have a nonvanishing relic
density. This should put additional constraints on $\mathbb{Z}_{2}$-symmetric
scalar fields as DM.

In our model, we did not consider spontaneous symmetry breaking. If
$m_{\chi}^{2}<0$, as it would be when $\chi$ is identified as the Higgs, there
may be spontaneous symmetry breaking during the evolution of the condensate.
If such spontaneous symmetry breaking occurs, we should have a cubic
interaction term $v\Phi^{2}\chi$ where $v$ is the symmetry broken value of the
$\chi$ field. In the presence of the condensate, one has the vertex
$v\varphi\phi\chi$ which leads to the following additional diagram in the
effective action that contributes to the condensate decay,

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This diagram would contribute to $\gamma$ because it induces a linear term in
the EoM of $\varphi$. By the cutting rules [77, 78, 79, 80, 81, 82, 83], this
diagram describes the scattering process $\varphi\phi\leftrightarrow\chi$ [the
other processes ($\varphi\leftrightarrow\phi\chi$,
$\varphi\chi\leftrightarrow\phi$) cannot satisfy the on-shell condition]. The
process $\varphi\phi\leftrightarrow\chi$ again belongs to Landau damping and
is suppressed at low temperatures. Therefore, $\gamma$ would still drop below
$H$ at later times. The key finding discussed above is thus not affected by a
spontaneous symmetry breaking from the $\chi$ field but the actual evolution
of the condensate may be quite different.

We emphasize that in this work, $m_{\phi}^{2}>0$ is assumed, i.e., the
$\mathbb{Z}_{2}$ symmetry of $\Phi$ is not spontaneously broken. Different
from the portal field $\chi$, a negative $m^{2}_{\phi}$ would completely
change our conclusion. This is because such a negative squared-mass of $\Phi$
would induce symmetric broken minima for field $\Phi$, say $v_{\phi}$ and
$-v_{\phi}$. Then the condensate would oscillate about one of them. In this
case, one has to substitute $\Phi=v_{\phi}+\varphi+\phi$ (assuming
oscillations about the minimum $v_{\phi}$ without loss of generality) into the
Lagrangian and now one has an interaction term $v_{\phi}\varphi\chi^{2}$. This
term gives the decay $\varphi\rightarrow\chi\chi$ (provided
$M_{\phi}>2M_{\chi}$) which is not suppressed at low temperatures and
meanwhile would contribute to the non-local term linear in $\varphi$ in the
EoM (11). This means that such a decay channel is suppressed neither by low
temperature nor by small oscillation amplitude at late times. Therefore, it
would lead to a complete decay of the condensate.

At last, since our analysis is based on the one and two-loop diagrams given in
Eqs. (12), one may wonder whether higher loop processes would spoil our
conclusion or not. In particular, it is known that in computing correlation
functions in the zero-momentum, zero-frequency limit, e.g., in studying
hydrodynamic transport coefficients [69] or damping coefficients in the slow-
roll regime or at very high temperatures [68], resummation of an infinity set
of diagrams is required. This is because a near on-shell singularity appears
whenever there is a product of two equal four-momentum propagators. However,
such near on-shell singularity only appears in the zero-momentum, zero-
frequency limit of the correlation functions. Since we do not assume $T\gg
M_{\phi}$ (high temperature) nor $\Gamma_{\varphi}\gg M_{\phi}$ (over-damped
or slow-roll), the self-energies in our problem are always evaluated at finite
frequencies related to $M_{\phi}$ depending on the powers of $\varphi$
associated with the external vertices (e.g., $\gamma$ and $\sigma$ in Eq.
(2.3) are evaluated at $M_{\phi}$ and $2M_{\phi}$, respectively). Therefore,
the near on-shell singularity does not appear in our case. For scalar field
theory at finite temperature, there is the famous problem with infrared
divergences, essentially related to the Matsubara zero mode [89]. Such an
infrared problem is cured if one does a daisy resummation. This has already
been done with the thermal corrections being included in the propagators used
in this work.

## Acknowledgments

WYA is supported by the UK Engineering and Physical Sciences Research Council
(EPSRC), under Research Grant No. EP/V002821/1. ZLW is supported by the
Natural Science Foundation of Jiangsu Province (BK20220642). We are grateful
to John Ellis and Jian Wang for helpful discussions.

##  Appendix A Closed-Time-Path formalism

In this section, we give a brief introduction to the CTP formalism, with the
main purpose of setting up the notation. For a more detailed introduction, we
refer to Refs. [90, 56, 55].999See Ref. [91] for a recent review. The CTP
formalism has been widely applied to the studies of baryogenesis, especially
in the form of leptogenesis (see, e.g., Refs. [58, 60, 92, 93, 94, 95, 62, 96,
97, 98, 64, 65, 99, 100, 101]).

Out-of-equilibrium dynamics is an initial-value problem with the initial
conditions given by a density matrix at a given time, $\rho_{D}(t_{i})$. The
expectation value of an observable $\mathcal{O}$ at time $t$ is given by

$\displaystyle\langle\mathcal{O}(t)\rangle={\rm
Tr}\left\\{\rho_{D}(t_{i})\mathcal{O}(t)\right\\}\,,$ (A38)

where
$\mathcal{O}(t)=\exp(\mathrm{i}H(t-t_{i}))\mathcal{O}(t_{i})\exp(-\mathrm{i}H(t-t_{i}))$.
The operator is sandwiched between the same state. For this reason, the CTP
formalism is also called in-in formalism. Viewed from the Schrödinger picture,
the state evolves first forward in time from $t_{i}$ to $t$ and then backward
in time to $t_{i}$.101010In the conventional zero-temperature quantum field
theory studying scattering amplitudes, $t_{i}=-\infty$ and the backward
evolution from $t$ to $-\infty$ can be reversed to that from $t$ to $\infty$
by inserting a $S$-matrix, $S(\infty,-\infty)$, into the leftmost in the
brackets in Eq. (A38), which induces only a total phase. Expectation values
can thus be obtained by a generating functional formulated on a closed time
contour $\mathcal{C}$, as illustrated in Fig. 12.111111If the density matrix
at the initial time is thermal, it can be encoded with an additional contour
segment of length $1/T$ to the negative imaginary direction of time, glued to
the end point of the lower branch of $\mathcal{C}$ at $t_{i}$. For the
quantity given in Eq. (A38), $t_{f}=t$, but one can also choose any $t_{f}\geq
t$ by inserting an operator
$\exp(\mathrm{i}H(t-t_{f}))\exp(-\mathrm{i}H(t_{f}-t))=\mathbf{1}$ on the left
of $\mathcal{O}(t)$ in Eq. (A38). In particular, one can choose
$t_{f}\rightarrow\infty$ so that the generating functional is independent of
the quantities to be computed.

Figure 12: The Keldysh contour $\mathcal{C}$ for the generating functional in
the CTP formalism. In the plot, the forward and backward time contours are
slightly shifted off the real line only for illustration; both contours should
be understood as lying exactly on the real line.

All the information in a quantum field system is encoded in its correlation
functions. For most purposes, it is sufficient to study the one- and two-point
functions. For the scalar field $\Phi$, there are two independent connected
two-point functions

$\displaystyle\langle\Phi(x_{1})\Phi(x_{2})\rangle_{c}$
$\displaystyle=\langle\phi(x_{1})\phi(x_{2})\rangle\equiv\Delta_{\phi}^{>}(x_{1},x_{2})\,,$
(A39a) $\displaystyle\langle\Phi(x_{2})\Phi(x_{1})\rangle_{c}$
$\displaystyle=\langle\phi(x_{2})\phi(x_{1})\rangle\equiv\Delta_{\phi}^{<}(x_{1},x_{2})\,.$
(A39b)

They are called Wightman functions. Here and in what follows, analogous
expressions also apply for the $\chi$ field and corresponding quantities are
denoted similarly with the subscript $\phi$ replaced by $\chi$. Since the
generating functional is formulated on the closed time contour $\mathcal{C}$,
we would have two-point functions time-ordered on $\mathcal{C}$,

$\displaystyle\Delta^{\mathcal{C}}_{\phi}(x_{1},x_{2})=\theta_{\mathcal{C}}(t_{1},t_{2})\Delta_{\phi}^{>}(x_{1},x_{2})+\theta_{\mathcal{C}}(t_{2},t_{1})\Delta_{\phi}^{<}(x_{1},x_{2})\,.$
(A40)

To distinguish the times on the forward and backward contours in
$\Delta^{\mathcal{C}}_{\phi}$, one can denote the forward time with a
superscript “$+$” and the backward time “$-$”: $x^{+}=\\{t^{+},\mathbf{x}\\}$,
$x^{-}=\\{t^{-},\mathbf{x}\\}$. According to the $\mathcal{C}$-ordering, one
would have

$\displaystyle\Delta^{\mathcal{C}}_{\phi}(x_{1}^{+},x_{2}^{-})$
$\displaystyle=\Delta_{\phi}^{<}(x_{1},x_{2})\,,$ (A41a)
$\displaystyle\Delta^{\mathcal{C}}_{\phi}(x_{1}^{-},x_{2}^{+})$
$\displaystyle=\Delta_{\phi}^{>}(x_{1},x_{2})\,,$ (A41b)
$\displaystyle\Delta^{\mathcal{C}}_{\phi}(x_{1}^{+},x_{2}^{+})$
$\displaystyle=\langle T\phi(x_{1})\phi(x_{2})\rangle\,,$ (A41c)
$\displaystyle\Delta^{\mathcal{C}}_{\phi}(x_{1}^{-},x_{2}^{-})$
$\displaystyle=\langle\overline{T}\phi(x_{1})\phi(x_{2})\rangle\,,$ (A41d)

where $T$ and $\overline{T}$ denote the normal time-ordering and anti-time-
ordering operators, respectively.

Compared to the closed time variable, a more convenient notation is the
“double-field” formulation where one distinguishes the fields on the forward
contour from those on the backward contour while using a common time variable.
For example, one denotes $\Phi(x^{+})$ as $\Phi^{+}(x)$, $\Phi(x^{-})$ as
$\Phi^{-}(x)$, and similarly for $\chi$. Then the action integrated over the
contour $\mathcal{C}$ can be written as

$\displaystyle
S_{\mathcal{C}}[\Phi,\chi]=S[\Phi^{+},\chi^{+}]-S[\Phi^{-},\chi^{-}]\,,$ (A42)

where the minus sign is due to the backward direction in the integral on the
backward contour in $S_{\mathcal{C}}[\Phi,\chi]$. In this notation, Eq. (A41)
can be rewritten as

$\displaystyle\Delta^{+-}_{\phi}(x_{1},x_{2})$
$\displaystyle=\Delta_{\phi}^{<}(x_{1},x_{2})\,,$ (A43a)
$\displaystyle\Delta^{-+}_{\phi}(x_{1},x_{2})$
$\displaystyle=\Delta_{\phi}^{>}(x_{1},x_{2})\,,$ (A43b)
$\displaystyle\Delta^{++}_{\phi}(x_{1},x_{2})$ $\displaystyle=\langle
T\phi(x_{1})\phi(x_{2})\rangle\,,$ (A43c)
$\displaystyle\Delta^{--}_{\phi}(x_{1},x_{2})$
$\displaystyle=\langle\overline{T}\phi(x_{1})\phi(x_{2})\rangle\,.$ (A43d)

The indices “$\pm$” are called the Schwinger-Keldysh polarity indices.

Usually, it is convenient to introduce the spectral and statistical
correlation functions defined as, respectively,

$\displaystyle\Delta_{\phi}^{-}(x_{1},x_{2})$
$\displaystyle=\mathrm{i}\langle[\Phi(x_{1}),\Phi(x_{2})]\rangle_{c}=\mathrm{i}\left(\Delta_{\phi}^{-+}(x_{1},x_{2})-\Delta_{\phi}^{+-}(x_{1},x_{2})\right)\,,$
(A44a) $\displaystyle\Delta_{\phi}^{+}(x_{1},x_{2})$
$\displaystyle=\frac{1}{2}\langle\\{\Phi(x_{1}),\Phi(x_{2})\\}\rangle_{c}=\frac{1}{2}\left(\Delta_{\phi}^{-+}(x_{1},x_{2})+\Delta_{\phi}^{+-}(x_{1},x_{2})\right)\,.$
(A44b)

The spectral correlation function encodes the information about the spectrum
of the theory and the statistical correlation function gives the information
about occupation numbers of different modes for the fluctuations.

The Schwinger-Dyson equations for the one- and two-point functions can be
derived from the one-particle-irreducible (1PI) or two-particle-irreducible
(2PI) effective actions [102, 54].121212The $n$PI effective actions are very
powerful and can also be used to study higher-order corrections to false
vacuum decay [103, 104, 105, 106, 107, 108]. Here we use the 2PI effective
action. If the effective action is not truncated, the 1PI and 2PI give the
same EoMs. For a given order in the loop expansion, the 2PI effective action
includes resummation in the propagators compared to the 1PI effective action.
The definition of the 2PI effective action is based on the generating
functional with local and non-local sources,

$\displaystyle
Z[J_{\Phi},J_{\chi},K_{\Phi},K_{\chi}]=\int\mathcal{D}\Phi^{+}\mathcal{D}\Phi^{-}\mathcal{D}\chi^{+}\mathcal{D}\chi^{-}\,$
$\displaystyle\times\exp\left(\mathrm{i}\left[S[\Phi^{+},\chi^{+}]-S[\Phi^{-},\chi^{-}]+J^{A}_{\Phi,x}\Phi^{A}_{x}+J^{A}_{\chi,x}\chi^{A}_{x}+\frac{1}{2}\Phi^{A}_{x}K^{AB}_{\Phi,xy}\Phi^{B}_{y}+\frac{1}{2}\chi^{A}_{x}K^{AB}_{\chi,xy}\chi^{B}_{y}\right]\right).$
(A45)

where $A,B=\pm$, and we have used the DeWitt notation

$\displaystyle
J^{A}_{\Phi,x}\Phi^{A}_{x}=\int\mathrm{d}^{4}x\,J^{A}_{\Phi}(x)\Phi^{A}(x)\,,$
(A46a)
$\displaystyle\Phi^{A}_{x}K^{AB}_{\Phi,xy}\Phi^{B}_{y}=\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,\Phi^{A}(x)K^{AB}_{\Phi}(x,y)\Phi^{B}(y)\,.$
(A46b)

The Einstein summation convention is understood for repeated indices. One can
easily check that

$\displaystyle\frac{\delta\ln
Z[J_{\Phi},J_{\chi},K_{\Phi},K_{\chi}]}{\mathrm{i}\delta J^{A}_{\Phi}(x)}$
$\displaystyle=\langle\Phi^{A}(x)\rangle\equiv\varphi^{A}(x)\,,$ (A47a)
$\displaystyle\frac{\delta\ln
Z[J_{\Phi},J_{\chi},K_{\Phi},K_{\chi}]}{\mathrm{i}\delta K^{AB}_{\Phi,xy}}$
$\displaystyle=\frac{1}{2}\left(\Delta^{AB}_{\phi}(x_{1},x_{2})+\varphi^{A}(x)\varphi^{B}(y)\right)\,,$
(A47b)

and similarly for quantities involving $\chi$. Note that in the above
equations, the one- and two-point functions on the RHS are defined with
corresponding sources. The 2PI effective action is then defined as the
Legendre transform,

$\displaystyle\Gamma_{\rm
2PI}[\varphi,\bar{\chi},\Delta_{\phi},\Delta_{\chi}]=$
$\displaystyle-\mathrm{i}\,\ln
Z[J_{\Phi},J_{\chi},K_{\Phi},K_{\chi}]-J^{A}_{\Phi,x}\varphi^{A}_{x}-J^{A}_{\chi,x}\bar{\chi}^{A}_{x}\,,$
(A48)
$\displaystyle-\frac{1}{2}K^{AB}_{\Phi,xy}(\varphi^{A}_{x}\varphi^{B}_{y}+\Delta^{AB}_{\phi,xy})-\frac{1}{2}K^{AB}_{\chi,xy}(\bar{\chi}_{x}^{A}\bar{\chi}_{y}^{B}+\Delta_{\chi,xy}^{AB})\,,$
(A49)

where $\bar{\chi}^{A}=\langle\chi^{A}\rangle$. The Schwinger-Dyson equations
for the one- and two-point functions read

$\displaystyle\frac{\delta\Gamma_{\rm
2PI}[\varphi,\bar{\chi},\Delta_{\phi},\Delta_{\chi}]}{\delta\varphi^{A}(x)}=-J^{A}_{\Phi}(x)-K^{AB}_{\Phi,xy}\varphi^{B}_{y}\,,$
(A50a) $\displaystyle\frac{\delta\Gamma_{\rm
2PI}[\varphi,\bar{\chi},\Delta_{\phi},\Delta_{\chi}]}{\delta\Delta^{AB}_{\phi/\chi}(x,y)}=-\frac{1}{2}K^{AB}_{\Phi/\chi}(x,y)\,.$
(A50b)

Taking all the $J$ and $K$ to be vanishing, we obtain the EoMs in the absence
of external sources

$\displaystyle\left.\frac{\delta\Gamma_{\rm
2PI}[\varphi,\bar{\chi},\Delta_{\phi},\Delta_{\chi}]}{\delta\varphi^{+}(x)}\right|_{\varphi^{-}=\varphi^{+}=\varphi}=0\,,$
(A51) $\displaystyle\left.\frac{\delta\Gamma_{\rm
2PI}[\varphi,\bar{\chi},\Delta_{\phi},\Delta_{\chi}]}{\delta\Delta^{AB}_{\phi/\chi}(x,y)}\right|_{\varphi^{-}=\varphi^{+}=\varphi}=0\,.$
(A52)

In the Feynman-diagram representation of the $2$PI effective action, we now
have two types of vertices: the + and - types. Depending on the vertices, one
would have four different types of propagators: $\Delta_{\phi/\chi}^{++}$,
$\Delta_{\phi/\chi}^{+-}$, $\Delta_{\phi/\chi}^{-+}$, and
$\Delta_{\phi/\chi}^{--}$. For example, a propagator connecting two + type
vertices is of type $\Delta^{++}$, etc. Solving first on-shell for the two-
point functions and assuming that $\bar{\chi}=0$, one gets
$\Delta_{\phi}[\varphi]$ and $\Delta_{\chi}[\varphi]$ as functionals of the
condensate $\varphi$. Plugging them back into the 2PI effective action gives
the EoM for the condensate

$\left.\frac{\delta\Gamma_{\text{2PI}}[\varphi,\Delta_{\phi},\Delta_{\chi}]}{\delta\varphi(x)}\right|_{\Delta_{\phi}=\Delta_{\phi}[\varphi],\,\Delta_{\chi}=\Delta_{\chi}[\varphi]}=0\,.$
(A53)

Arriving here, one can follow Section 2 of Ref. [36] to perform the small-
field expansion of the on-shell two-point functions $\Delta_{\phi}[\varphi]$
and $\Delta_{\chi}[\varphi]$. To the lowest order, they are independent of
$\varphi$ and become the free thermal equilibrium propagators. Properly
truncating the effective action, one ends up with Eq. (11).

##  Appendix B Self-energies

The Feynman-diagram rules for the free thermal equilibrium propagators with
thermal corrections to the masses taken into account are

$\leavevmode\leavevmode\hbox to59.87pt{\vbox
to15.57pt{\pgfpicture\makeatletter\hbox{\hskip
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} \pgfsys@invoke{\lxSVG@closescope
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.16335pt}{-8.783pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\hbox{{$\scriptstyle(x^{\prime},B)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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G_{\phi}^{AB}(x-x^{\prime})\,,\qquad\qquad\leavevmode\leavevmode\hbox
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29.55794pt\lower-13.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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G_{\chi}^{AB}(x-x^{\prime})\,,$ (B54)

where

$\displaystyle G_{\phi/\chi}^{++}(x-x^{\prime})=$
$\displaystyle\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\,\mathrm{e}^{-\mathrm{i}k(x-x^{\prime})}\left[\frac{\mathrm{i}}{k^{2}-M_{\phi/\chi}^{2}+\mathrm{i}\varepsilon}+2\pi
f_{\rm B}(|k_{0}|)\delta\left(k^{2}-M_{\phi/\chi}^{2}\right)\right]\,,$ (B55a)
$\displaystyle G_{\phi/\chi}^{+-}(x-x^{\prime})=$
$\displaystyle\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\,\mathrm{e}^{-\mathrm{i}k(x-x^{\prime})}\left[2\pi
f_{\rm B}(k_{0})\rm
sign(k_{0})\,\delta\left(k^{2}-M_{\phi/\chi}^{2}\right)\right]\,,$ (B55b)

with the remaining components being complex conjugates,
$G^{--}_{\phi/\chi}(x-x^{\prime})=\left[G^{++}_{\phi/\chi}(x-x^{\prime})\right]^{*}$,
$G^{-+}_{\phi/\chi}(x-x^{\prime})=\left[G^{+-}_{\phi/\chi}(x-x^{\prime})\right]^{*}$.
Here, the thermally corrected masses are

$\displaystyle M_{\phi}^{2}$
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.75938pt}{-1.75pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle(x,+)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (B56a)
$\displaystyle M_{\chi}^{2}$
$\displaystyle=m_{\chi}^{2}-\frac{\mathrm{i}\lambda_{\chi}}{2}\leavevmode\leavevmode\hbox
to52.4pt{\vbox to28.95pt{\pgfpicture\makeatletter\hbox{\hskip
14.47638pt\lower-14.47638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.5pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.72638pt}{0.0pt}\pgfsys@curveto{15.72638pt}{0.82843pt}{15.05481pt}{1.5pt}{14.22638pt}{1.5pt}\pgfsys@curveto{13.39795pt}{1.5pt}{12.72638pt}{0.82843pt}{12.72638pt}{0.0pt}\pgfsys@curveto{12.72638pt}{-0.82843pt}{13.39795pt}{-1.5pt}{14.22638pt}{-1.5pt}\pgfsys@curveto{15.05481pt}{-1.5pt}{15.72638pt}{-0.82843pt}{15.72638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.75938pt}{-1.75pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle(x,+)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\frac{\mathrm{i}g}{2}\leavevmode\leavevmode\hbox
to52.4pt{\vbox to28.95pt{\pgfpicture\makeatletter\hbox{\hskip
14.47638pt\lower-14.47638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.5pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.72638pt}{0.0pt}\pgfsys@curveto{15.72638pt}{0.82843pt}{15.05481pt}{1.5pt}{14.22638pt}{1.5pt}\pgfsys@curveto{13.39795pt}{1.5pt}{12.72638pt}{0.82843pt}{12.72638pt}{0.0pt}\pgfsys@curveto{12.72638pt}{-0.82843pt}{13.39795pt}{-1.5pt}{14.22638pt}{-1.5pt}\pgfsys@curveto{15.05481pt}{-1.5pt}{15.72638pt}{-0.82843pt}{15.72638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.75938pt}{-1.75pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle(x,+)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+({\rm other\
diagrams\ with\ SM\ fields})\,.$ (B56b)

Writing $M_{\phi/\chi}^{2}=m_{\phi/\chi}^{2}+m_{\phi/\chi,\rm th}^{2}$, we
then have [109]

$\displaystyle m^{2}_{\phi,\rm
th}=\frac{\lambda_{\phi}}{2}\int\frac{\mathrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\frac{f_{\rm
B}(\omega_{\phi,\mathbf{k}})}{\omega_{\phi,\mathbf{k}}}+\frac{g}{2}\int\frac{\mathrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\frac{f_{\rm
B}(\omega_{\chi,\mathbf{k}})}{\omega_{\chi,\mathbf{k}}}\,,$ (B57a)
$\displaystyle m^{2}_{\chi,\rm
th}=\frac{\lambda_{\chi}}{2}\int\frac{\mathrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\frac{f_{\rm
B}(\omega_{\chi,\mathbf{k}})}{\omega_{\chi,\mathbf{k}}}+\frac{g}{2}\int\frac{\mathrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\frac{f_{\rm
B}(\omega_{\phi,\mathbf{k}})}{\omega_{\phi,\mathbf{k}}}+({\rm contributions\
from\ SM\ fields})\,,$ (B57b)

where $\omega_{\phi/\chi,\mathbf{k}}=\sqrt{M^{2}_{\phi/\chi}+\mathbf{k}^{2}}$
and we have subtracted the divergent $T=0$ part. Note that the above equations
are self-consistent equations as $m^{2}_{\phi/\chi,\rm th}$ appear on both
sides. This is due to that resumed propagators are used in the 2PI effective
action. In particular, with the assumed form for the propagators given in Eqs.
(B55), the famous daisy resummation has been taken into account. In the high-
temperature limit $m_{\phi,\chi}\ll T$, we get the familiar results [109]

$\displaystyle m^{2}_{\phi,\rm th}$
$\displaystyle=(\lambda_{\phi}+g)\frac{T^{2}}{24}+\mathcal{O}(\lambda_{\phi}^{3/2})+\mathcal{O}(g^{3/2})\,,$
(B58a) $\displaystyle m_{\chi,\rm th}^{2}$
$\displaystyle=(\lambda_{\chi}+g)\frac{T^{2}}{24}+\mathcal{O}(\lambda_{\chi}^{3/2})+\mathcal{O}(g^{3/2})+\kappa^{2}T^{2}\,,$
(B58b)

where $\kappa^{2}T^{2}$ is due to possible interactions between $\chi$ and the
SM fields. We shall take $\kappa^{2}\approx 0.1$ in our numerical
calculations, which corresponds to roughly the value of the thermal
corrections to the Higgs mass in the SM. Beyond the high-temperature limit
$m_{\phi,\chi}\ll T$, one in principle needs to solve Eqs. (B57) iteratively.
At the leading order, we simply have Eqs. (B57) with the following
replacements on the RHS

$\displaystyle\omega_{\phi,\mathbf{k}}\rightarrow\sqrt{m^{2}_{\phi}+\mathbf{k}^{2}}\,,\qquad\omega_{\chi,\mathbf{k}}\rightarrow\sqrt{m^{2}_{\chi}+\mathbf{k}^{2}}\,,$
(B59)

i.e., by neglecting the thermal corrections on the RHS. Taking the self-
interaction as an example, in Fig. 13 we plot $m_{\phi,\rm
th}/(\sqrt{\lambda_{\phi}}m_{\phi})$ as a function of $T/m_{\phi}$ with and
without the high-temperature approximation. It shows that away from the high-
temperature regime, i.e., for
$T\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;m_{\phi}$,
the thermal corrections to the mass are small anyway compared with the zero-
temperature mass $m_{\phi}$. Therefore, it is not material whether or not one
uses the high-temperature approximation for the thermal corrections to the
mass throughout. Note $m_{\phi,\rm th}/m_{\phi}$ is further suppressed by a
factor of $\sqrt{\lambda_{\phi}}$ compared with what has been plotted in Fig.
13.

Figure 13: $m_{\phi,\rm th}/\sqrt{\lambda_{\phi}}m_{\phi}$ as a function of
$T/m_{\phi}$ for the self-interactions. The red solid curve is obtained from
the analytical expression in the high-temperature approximation, i.e., (B58a)
with vanishing $g$. The black dashed curve is obtained from numerically
integrating (B57a) with the simplification (B59). Note that the ratio
$m_{\phi,\rm th}/m_{\phi}$ is further suppressed by a factor of
$\sqrt{\lambda_{\phi}}$ compared with what has been plotted. This plot shows
that, away from the high-temperature regime, the thermal corrections to the
masses could be simply neglected for perturbatively small couplings.

The various self-energies and proper four-vertex functions read

$\displaystyle\Pi^{AB}(x-x^{\prime})$ $\displaystyle=\
-\,\frac{\mathrm{i}(AB)\lambda_{\phi}^{2}}{6}\,\leavevmode\leavevmode\hbox
to80.31pt{\vbox to28.95pt{\pgfpicture\makeatletter\hbox{\hskip
39.3986pt\lower-14.47638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.5pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.5pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@moveto{-12.72638pt}{0.0pt}\pgfsys@curveto{-12.72638pt}{0.82843pt}{-13.39795pt}{1.5pt}{-14.22638pt}{1.5pt}\pgfsys@curveto{-15.05481pt}{1.5pt}{-15.72638pt}{0.82843pt}{-15.72638pt}{0.0pt}\pgfsys@curveto{-15.72638pt}{-0.82843pt}{-15.05481pt}{-1.5pt}{-14.22638pt}{-1.5pt}\pgfsys@curveto{-13.39795pt}{-1.5pt}{-12.72638pt}{-0.82843pt}{-12.72638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.0656pt}{-1.75pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle(x,A)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.72638pt}{0.0pt}\pgfsys@curveto{15.72638pt}{0.82843pt}{15.05481pt}{1.5pt}{14.22638pt}{1.5pt}\pgfsys@curveto{13.39795pt}{1.5pt}{12.72638pt}{0.82843pt}{12.72638pt}{0.0pt}\pgfsys@curveto{12.72638pt}{-0.82843pt}{13.39795pt}{-1.5pt}{14.22638pt}{-1.5pt}\pgfsys@curveto{15.05481pt}{-1.5pt}{15.72638pt}{-0.82843pt}{15.72638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.75938pt}{-1.75pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle(x^{\prime},B)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ -\
\frac{\mathrm{i}(AB)g^{2}}{2}\,\leavevmode\leavevmode\hbox to80.31pt{\vbox
to28.95pt{\pgfpicture\makeatletter\hbox{\hskip 39.3986pt\lower-14.47638pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.5pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.5pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@moveto{15.72638pt}{0.0pt}\pgfsys@curveto{15.72638pt}{0.82843pt}{15.05481pt}{1.5pt}{14.22638pt}{1.5pt}\pgfsys@curveto{13.39795pt}{1.5pt}{12.72638pt}{0.82843pt}{12.72638pt}{0.0pt}\pgfsys@curveto{12.72638pt}{-0.82843pt}{13.39795pt}{-1.5pt}{14.22638pt}{-1.5pt}\pgfsys@curveto{15.05481pt}{-1.5pt}{15.72638pt}{-0.82843pt}{15.72638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.75938pt}{-1.75pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle(x^{\prime},B)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@moveto{-12.72638pt}{0.0pt}\pgfsys@curveto{-12.72638pt}{0.82843pt}{-13.39795pt}{1.5pt}{-14.22638pt}{1.5pt}\pgfsys@curveto{-15.05481pt}{1.5pt}{-15.72638pt}{0.82843pt}{-15.72638pt}{0.0pt}\pgfsys@curveto{-15.72638pt}{-0.82843pt}{-15.05481pt}{-1.5pt}{-14.22638pt}{-1.5pt}\pgfsys@curveto{-13.39795pt}{-1.5pt}{-12.72638pt}{-0.82843pt}{-12.72638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.0656pt}{-1.75pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle(x,A)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (B60a)
$\displaystyle V^{AB}(x-x^{\prime})$ $\displaystyle=\ -\
\frac{3\mathrm{i}(AB)\lambda_{\phi}^{2}}{2}\,\leavevmode\leavevmode\hbox
to80.31pt{\vbox to28.95pt{\pgfpicture\makeatletter\hbox{\hskip
39.3986pt\lower-14.47638pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.5pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{7.8571pt}{7.8571pt}{14.22638pt}{0.0pt}{14.22638pt}\pgfsys@curveto{-7.8571pt}{14.22638pt}{-14.22638pt}{7.8571pt}{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-7.8571pt}{-7.8571pt}{-14.22638pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{7.8571pt}{-14.22638pt}{14.22638pt}{-7.8571pt}{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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