# Recurrent Axinovae and their Cosmological Constraints

Patrick J. Fox1 Neal Weiner2 Huangyu Xiao3 1Particle Theory Department, Theory
Division, Fermilab, Batavia, IL 60510, USA  2Center for Cosmology and Particle
Physics, Department of Physics, New York University, New York, NY 10003, USA
3 Astrophysics Theory Department, Theory Division, Fermilab, Batavia, IL
60510, USA

###### Abstract

Axion-like dark matter whose symmetry breaking occurs after the end of
inflation predicts enhanced primordial density fluctuations at small scales.
This leads to dense axion minihalos (or miniclusters) forming early in the
history of the Universe. Condensation of axions in the minihalos leads to the
formation and subsequent growth of axion stars at the cores of these halos.
If, like the QCD axion, the axion-like particle has attractive self-
interactions there is a maximal mass for these stars, above which the star
rapidly shrinks and converts an $\mathcal{O}(1)$ fraction of its mass into
unbound relativistic axions. This process would leave a similar (although in
principle distinct) signature in cosmological observables as a decaying dark
matter fraction, and thus is strongly constrained. We place new limits on the
properties of axion-like particles that are independent of their non-
gravitational couplings to the standard model.

††preprint: FERMILAB-PUB-23-029-T

##  I Introduction

The stars in our galaxy are made of $\sim 10^{57}$ fermions bound together by
gravity and protected from collapse by thermal pressure or fermion degeneracy
pressure. In the presence of a light, long-lived boson similar gravitationally
bound states of that boson may exist, but in the absence of nuclear burning
they are instead supported by gradient pressure, which is a result of the
uncertainty principle. Axion stars are one such example of these bosonic
objects.

In the Standard Model, stars convert approximately $0.1\%$ of their mass
energy into radiation over their lifetime. The small energy released (compared
to rest mass) in the pp-chain, for instance, is due to the relatively small
binding energy inside the star. Only very compact objects like neutron stars
are relativistic in nature. Moreover, because of baryon number conservation,
there is a limitation on overall energy release given the (approximately
degenerate) neutrons and protons which must remain in the final state.

However, in the dark sector, there are reasons to expect the overall energy
conversion could be much higher if a similar process were to occur. Complete
conversion of rest mass from e.g. a $3\rightarrow 1$ process is possible
because there is no “baryon-number” conservation for bosonic dark matter. For
example, axion stars will collapse and emit relativistic axions when they
reach a critical mass. We call such processes that drastically convert dark
matter to dark radiation as Axinovae. There is no mechanism to quench the
axionovae if axion stars form ubiquitously in the Universe, as expected in the
post-inflationary scenario where axion miniclusters form after matter-
radiation equality. Therefore, a large formation rate of axion stars that lead
to axinovae is very constraining. We take the formation of axion stars as a
concrete example to study but the result can apply to generic scalars whose
self-interaction is attractive since the properties of axion stars do not
depend on any interactions other than gravity and the axion self-couplings.

A natural cosmic history that can occur generically for these models is, after
matter-radiation equality, these axions stars form, grow, and finally explode
as an axinova, converting a significant fraction of energy into semi-
relativistic axions. After this, the remnant can continue to grow, until it
explodes again. This process of recurrent axinovae can convert a significant
fraction of the dark matter into relativistic energy, which is then
constrained by cosmological observations.

This paper is organized as follows: in Sec. II, we discuss the formation of
enhanced structures at small scales due to the axion perturbations and study
the formation history of axion stars inside those structures. In Sec. III, we
study the constraints on axion parameter space by requiring the decay fraction
of axion dark matter should not exceed an upper bound. In Sec. IV, we present
our conclusions.

##  II Axions, Axion minihalos, and axion stars

The axion is a well-motivated dark matter candidate, which can also leave
unique fingerprint on the matter power spectrum at small scales if the PQ
symmetry breaking occurs _after_ inflation. In such scenarios, different
horizon patches have different matter densities when the axion acquires its
mass, leading to the formation of axion miniclusters or axion minihalos at
matter-radiation equality [1, 2]. More interestingly, coherent objects called
axion stars can form in the center of axion minihalos due to Bose-Einstein
condensation [3], which may eventually accrete into a critical object and emit
relativistic axions. We call such phenomenon axinovae, which can occur with an
attractive axion self-coupling and the formation of axion minihalos at matter-
radiation equality.

Originally proposed to solve the strong CP problem [4, 5, 6], the present-day
landscape of axions and axion-like particles (ALPs) is broad. One common
feature across this landscape is that the axion, $\phi$, is a pseudo-Goldstone
boson of a global $U(1)_{PQ}$ symmetry broken at a scale $f_{a}$. The
$U(1)_{PQ}$ is anomalous under a confining gauge group which means that the
axion’s potential is generated through instanton effects occurring at the
compositeness scale of the gauge group, $\Lambda$, and takes the form

$V(\phi)=\frac{\Lambda^{4}}{c_{ud}}\sqrt{1-4c_{ud}\,\sin^{2}\left(\frac{\phi}{f_{a}}\right)}~{}.$
(1)

In the case of the QCD axion $\Lambda\approx 200\,{\rm MeV}$ and
$c_{ud}\approx m_{u}m_{d}/(m_{u}+m_{d})^{2}\approx 0.2$. In addition to the
self couplings, coupling to gravity, and the anomaly-induced coupling to QCD
(or QCD-like group), the axion may have model-dependent couplings to other SM
gauge bosons and fermions. We focus here on the self and gravitational
couplings only, which can already lead to interesting dynamics such as
axinovae.

###  II.1 Axion Minihalos

In the post-inflationary scenario, the present-day Universe contains a large
number of patches which were causally disconnected at the time of QCD phase
transition. In each causally disconnected patch of the Universe, axion field
values are uncorrelated. Once the axion acquires a mass, and Hubble friction
is small enough, the axion behaves as cold dark matter and isocurvature
fluctuations are present in the matter density. When the Universe becomes
matter dominated this small-scale structure will start to collapse under
gravity, leading to axion minihalos. Furthermore, there may be large
overdensities of axions at even smaller scales arising from the evolution of
the network of axion strings and domain walls [7] set up when the PQ symmetry
breaks. Even for the much studied case of the QCD axion, there is controversy
[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 10, 17, 19, 14, 15, 16, 13, 20, 21,
22] as to what fraction of the relic dark matter axions arise from
misalignment or from the decay of topological defects. Along with those
topological defects, objects called oscillons or axitons that can contribute
to the small scale overdensities will form after the axion accquires its mass
[15, 14]. Those objects can form when the axion self-interaction dominates
over the Hubble expansion term, which is easily satisfied in the early
Universe when the self-interaction is strong due to the high density. As the
axion density drops, the formation of oscillons will be turned off and
oscillons themselves will dissipate via emitting relativistic axions.

It is worth noting that the post-inflationary scenario is not essential for
the axinovae. Any matter power spectrum which is enhanced at small scales can
lead to the formation of axion minihalos around matter-radiation equality, but
the post-inflationary scenario is a minimal realisation. We take a simple
ansatz for the spectrum of initial fluctuations in the axion field, namely
that the spectrum of isocurvature fluctuations in the axion field follow a
white-noise spectrum, cut off at small scales i.e.

$\frac{\delta\rho_{a}}{\rho_{a}}=A_{0}\left(\frac{k}{k_{0}}\right)^{3}\,\Theta(k_{0}-k)~{}.$
(2)

Here $k_{0}\approx a_{\mathrm{osc}}H_{\mathrm{osc}}$is the (comoving)
wavenumber determined by the horizon size at the time the axion starts to
oscillate, i.e. $m_{a}(T_{\mathrm{osc}})\sim 3H_{\mathrm{osc}}$. While here we
consider a pure white noise spectrum we extend this analysis to a more general
power law spectrum in Appendix C. In reality one would expect a softening of
the cutoff in the white noise power spectrum at small scales. The exact
details of how this occurs is related to the dynamics of string network and
axitons, and is unknown. It will not affect our conclusions, see Appendix C
for details. As mentioned above, the contribution of strings and domain walls
to the abundance of non-relativistic axions is uncertain and will impact the
size of the power spectrum. Simulations typically show the density
perturbations have $A_{0}\sim 0.1$ at $k=a_{\mathrm{osc}}H_{\mathrm{osc}}$,
but they also show larger sub-horizon (larger $k$) fluctuations. These sub-
horizon fluctuations can collapse earlier than those at the horizon scale,
leading to high concentration minihalos. These halos are at smaller scales,
$k\geq a_{\mathrm{osc}}H_{\mathrm{osc}}$, and have larger
$\delta\rho_{a}/\rho_{a}$ and lower axion speeds in the mini-halos, resulting
in a faster star growth rate. We take the conservative limit of holding
$k_{0}=H_{\mathrm{osc}}$ as the scale at which $A_{0}=1$. With this white
noise power spectrum (2) the first structures, of mass $M_{0}$, form at
redshift $z_{c}\approx A_{0}^{1/2}z_{\rm eq}$ and the characteristic structure
mass, defined as the peak in the distribution $Mdn/d\log M$, occurs at

$M_{peak}(z)=M_{0}\left(\frac{1+z_{c}}{1+z}\right)^{2}~{}.$ (3)

Where

$M_{0}=6\pi^{2}\overline{\rho}_{0}\left(\frac{1}{k_{0}}\right)^{3}~{},$ (4)

is the co-moving mass in the horizon at the time the axion starts to roll and
$\overline{\rho}_{0}$ is the present-day cosmological axion density. The
minihalos have a distribution of masses but for simplicity we use the
characteristic mass $M_{peak}$ to provide a measure of the overall behavior.

The growth continues till around $z\sim 10-20$ when the minihalos merge into
standard CDM halos and their growth stalls [23].

We take the minihalos to have an NFW [24] density profile, defined by a scale
radius $r_{s}$ and density $\rho_{s}$,

$\rho(r)=\frac{\rho_{s}}{\frac{r}{r_{s}}\left(1+\frac{r}{r_{s}}\right)^{2}}~{}.$
(5)

At the scale radius the circular speed is given by

$v_{s}^{2}=4\pi G_{N}\rho_{s}r_{s}^{2}\left(\log 4-1\right)~{}.$ (6)

This speed will be relevant for the calculation of axion star formation rate,
and in the minihalos that will form axion stars this speed is much smaller
than typical speeds in the Milky Way. Numerical studies have shown that the
dark matter halos at the characteristic mass will first reach a concentration
factor 111The concentration is defined as the ratio $c=r_{200}/r_{s}$ where
$r_{200}$ is the radius at which the halos average density is 200 times the
background DM density. $c\approx 4$ at the time of halo collapse [26]. For
lighter halos that grow more through accretion than mergers, the halo
concentration will grow linearly with the scale factor due to the decreasing
background density. In the next subsections we discuss further structures that
can develop in the core of these minihalos. In addition to the low speeds in
mini-halos, the scale density in the mini-halos that form early is large,

$\rho_{s}=\overline{\rho}_{0}(1+z)^{3}\frac{\Delta_{200}\,c^{3}}{3\left(\log(1+c)-\frac{c}{1+c}\right)}~{},$
(7)

where $\Delta_{200}\approx 200$ in the spherical collapse model. For mini-
halos that collapse at $z_{\mathrm{eq}}$ this density is $\rho_{s}\approx
10^{14}\overline{\rho}_{0}$.

###  II.2 Lifecycle of an Axion Star

We now turn to the question of formation of axion stars at the core of the
minihalos discussed above. The subsequent growth and explosion of axion stars
(axinovae) will also be studied. There are a few timescales we will discuss
that are relevant in the lifecycle of axion stars:

  * •

The condensation timescale from gravitational interactions

  * •

The condensation timescale from axion self-interactions.

  * •

The evaporation timescale of light axion stars.

  * •

The Hubble time when the axion star formation is active.

We will discuss those timescales later in this subsection. For the parameter
space that axinovae can place meaningful constraints on, the axion self-
interaction always dominates over gravity.

As discussed in Appendix A, there are two branches of axion star
configurations: the dilute branch, which, below a certain mass, is stable and
the dense branch which will explode and emit relativistic axions. There is a
critical star mass (23) that separates the two branches, which we denote as
$M_{*}^{\mathrm{m}ax}$. Therefore, if they continue to accrete mass, the
lighter dilute axion stars will eventually become unstable in a minihalo
environment. Axion minihalos are ideal environments for the axion star
formation because they are dense and cold, owing to the high collapse
redshifts and small virial masses. When the star formation rate is
sufficiently large, stars will form in the minihalo center and grow to a
critical mass star if the minihalo is massive enough. The critical star will
contract under self-interaction and gravity, converting a large fraction of
its mass to relativistic axions. Until the axinovae consume most of the
minihalo mass, axion minihalos remain ideal environments for the axion star
formation and axinovae shall occur again within the same timescale. Thus, we
naturally expect the axinovae phenomenon to be recurrent, when the growth
timescale is fast enough. The crucial calculation to determine the fate of
axion stars is the formation rate in the minihalo center and the corresponding
star mass.

Once minihalos exist, gravitational interactions or self-interactions can
subsequently lead to the formation of Bose-Einstein condensed axion stars at
their center. The timescale for this formation, and subsequent growth, in an
environment where the axions being captured have typical number density $n$
and speed $v$ is determined [27, 28, 29, 30] by

$\tau\sim\left(f_{\mathrm{BE}}n\sigma v\right)^{-1}~{}.$ (8)

With $\sigma$ the total scattering cross section. This formation rate is Bose-
enhanced from the naive expectation due to the large phase space density,
$f_{\mathrm{BE}}=6\pi^{2}n(m_{a}v)^{-3}$. The gravitational Rutherford
transport cross section is
$\sigma_{\mathrm{gr}}=8\pi(G_{N}m_{a}v^{-2})^{2}\log(m_{a}vR)$, where the
Coulomb logarithm has been cutoff at a characteristic length scale of the
minihalo, $R$. Attractive self-couplings can also lead to formation and the
scattering cross section is
$\sigma_{\mathrm{self}}=\lambda^{2}m_{a}^{-2}/128\pi$. The total condensation
time, considering both gravity and self-interaction, is

$\tau=\frac{\tau_{\rm self}\tau_{gr}}{\tau_{\mathrm{gr}}+\tau_{\rm self}}.$
(9)

With each individual process having a timescale of

$\tau_{\mathrm{gr}}=\frac{b}{48\pi^{3}}\frac{m_{a}v^{6}}{G_{N}^{2}n^{2}\log\left(m_{a}vR\right)}~{},$
(10)

for gravity, and

$\tau_{\rm self}=\frac{64dm_{a}^{5}v^{2}}{3\pi n^{2}\lambda^{2}}~{}.$ (11)

The parameters $b,d\sim\mathcal{O}(1)$ are numerical coefficients that are
extracted from numerical simulations [31]. Comparing these two timescales,
(10) and (11), we see that the self interactions will determine the axion star
formation rate if $f_{a}\ \raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}\ M_{\mathrm{pl}}v$. Furthermore, if the relevant speed
is determined by gravitational collapse of a minihalo (6) then self
interactions dominate in the limit $f_{a}^{2}\ \raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}\
\rho_{s}r_{s}^{2}$. When determining the gravitational relaxation timescale
for formation of axion stars in minihalos we take, as typical, the densities
and speeds at the scale radius, see Eqs. (5) and (6).

In addition to the timescale for axion star growth there is also a rate for
evaporation of the star. Axions in the halo that are not part of the star can
collide with bound axions causing them to be ejected. The rate for this
process shrinks with axion star mass and is approximately [32]
$\Gamma_{\mathrm{evap}}\sim(m_{a}vR_{*})^{2}\tau^{-1}$. The competition
between growth and evaporation means only axion stars above a certain mass
will gain mass by gathering axions from the halo. As observed in numerical
simulations [27, 32] such stars first appear after time $\tau$ and then
proceed to grow. The growth is initially fast ($d\log M/dt$ is constant) but
once the virial velocity of the minicluster falls below the speed of the
axions in the axion star the rate of growth slows, $d\log M/dt$ becomes
inversely proportional to (a power of) the star mass [32] which results in the
mass growing with time as a power law. The characteristic axion star mass
where this change in behavior occurs is obtained by equating the virial
velocity of the minicluster with that of the axion star [27, 31, 28, 32, 33,
34] is

$\overline{M_{*}}\approx 3\rho_{a}^{1/6}G_{N}^{-1/2}m_{a}^{-1}M_{h}^{1/3}~{},$
(12)

where $M_{h}$ is the halo mass. The behavior of the growth rate once the
axions in the star are moving faster than those in the halo is not
definitively known, and there is evidence that it may continue to evolve with
star mass [32]. This would result in the mass growing as a power law with a
running index. However, to simplify our analysis and to partially account for
the numerical uncertainties, we will use a single power law but consider a
range of possible powers. In particular, we parametrize the power-law mass
growth as $M_{*}=\overline{M_{*}}\,(t/\tau)^{1/\alpha}$ and vary $\alpha$ in
the range of 1 to 5. With initial exponential growth followed by constant
power law growth, the timescale to form an axion star at critical mass
$M_{*}^{\rm max}$ depends in which regime the critical mass falls. Thus,

$t_{\rm crit}=\tau\times\begin{cases}\log\left(\overline{M_{*}}/M_{*}^{\rm
max}\right)+1,\,&\quad M_{*}^{\rm max}\leq\overline{M_{*}}\\\ (M_{*}^{\rm
max}/\overline{M_{*}})^{\alpha},\,&\quad M_{*}^{\rm
max}>\overline{M_{*}}\end{cases}.$ (13)

The numerical simulations discussed above have mostly been carried out
assuming a homogenous gas of axions as the initial background upon which an
axion star forms. For stars that form in minihalos the gas has a density and
velocity profile. In Appendix B we argue that for an NFW profile the
exponential growth is replaced with a power law, and the whole growth becomes
a single power law, with $\alpha=3/2$ when self interactions dominate.

Given that the majority of the dark matter has collapsed into axion minihalos
with a characteristic mass $M_{\mathrm{peak}}(z)$, the total fraction of dark
matter rest mass that has been converted to kinetic energy per unit time can
be calculated as

$\frac{df_{\rm decay}}{dt}=\frac{\kappa\,M_{*}^{\rm
max}}{M_{\mathrm{peak}}(z)\,t_{\rm crit}}~{},$ (14)

where $\kappa$ is the fraction of the axion star’s mass that is converted to
relativistic axions during axinovae. From simulations of these processes [35],
it is seen that approximately $50\%$ of the star’s mass is lost during the
nova and of this about $20\%$ is in the form of relativistic axions, so
$\kappa\approx 0.1$. The time to reach a critical star given in (13) assumes
the star grows from an undistorted minihalo. After the first axinova there is
a remanent of mass $\sim 0.5M_{*}^{\rm max}$ already present and the time for
this to grow to $M_{*}^{\rm max}$ is slightly shorter than for the first star.
For the power law considered here this correction is small and we ignore it,
assuming all subsequent stars take time $t_{\rm crit}$ to explode.

##  III Cosmological Constraints

###  III.1 The decay rate of axion stars

The process of forming axion stars which subsequently become nova converts
non-relativistic dark matter axions into boosted
($\gamma\sim\mathcal{O}(\mathrm{few}))$ axions. The kinetic energy of the
outgoing axions will red-shift away after the scale factor has grown by
$\sim\sqrt{\gamma}$ and thus the dark matter’s contribution to the matter-
energy budget is depleted. Here we study the impact of the cumulative loss of
mass in the dark sector but it is possible that the temporary existence of a
new relativistic species may lead to a measurable effect on large-scale
structure and is worthy of future study.

This process is closely related to the scenario of decaying dark matter, which
is well constrained by recent cosmological data [36, 37, 38]. For dark matter
which decays after recombination, the decrease of the dark matter fraction
will increase the angular diameter distance to the last scattering surface
over time. Furthermore, the amount of CMB lensing is reduced due to a smaller
gravitational potential than expected. This scenario is constrained by a
combination of CMB [39] and, for very long lived dark matter, SDSS [40] data.
If the decay of dark matter occurs well before recombination or even before
matter-radiation equality, the primary effect of the decaying dark matter is
to enhance $N_{\rm eff}$ since the decay products behave as dark radiation. In
the short-lived situation the constraints are primarily from CMB measurements.
We will be interested in the long-lived case, and in particular decays which
occur after matter-radiation equality but are no longer ongoing. The
equivalent bound [38] for decaying dark matter on the fraction of the initial
amount of dark matter that can decay is

$f_{dDM}\equiv\frac{\Omega_{dDM}}{\Omega_{dDM}+\Omega_{DM}}\leq
2.62\%\quad(\mathrm{at}\ 2\sigma)~{}.$ (15)

Although the cosmological evolution of the dark sectors for decaying dark
matter and axinovae are not identical they are similar and since the above
constraint is independent of decaying dark matter lifetime over a wide range
of lifetimes we will use it to constrain axions. We leave a more detailed
numerical analysis, and an investigation of other possible signals, for future
work. Converting (15) to the case of axinova leads to the requirement that

$\int_{z_{c}}^{z=20}\\!\\!\\!dz\,\frac{df_{\rm decay}}{dz}\leq 2.62\%~{}.$
(16)

In the scenario of axinovae, the decay of dark matter occurs when axion
miniclusters start to form, which is always after matter-radiation equality.
To avoid the constraint of (16) requires either that the formation rate of
axion stars is too small to be cosmologically relevant or that the formed
axion star mass is smaller than the critical mass so there are no axinovae.
Note that this bound does not rely upon there being a coupling to any SM
particles e.g. photons, gluons, or SM fermions. However, our constraints do
rely on the assumption that axions do make up the dark matter relic abundance
and that the fluctuations in the axion field are isocurvature in nature and
approximately power law. If axions make up a fraction of the dark matter, this
can be encoded as a reduction of $\kappa$, see (14), and a corresponding
weakening of the bounds.

For normal misalignment production of axions, where
$\langle\theta^{2}\rangle\approx 4$ [41], the typical initial halos that form
have a mass that depends upon the horizon size when the axion starts to
oscillate $m_{a}(T_{\mathrm{osc}})=3H(T_{\mathrm{osc}})/2$. For the QCD axion,
where the temperature dependence of the axion mass is known, this oscillation
time is uniquely determined. However, in more general axion scenarios the
oscillation temperature, and therefore $M_{0}$, is a free parameter. In the
radiation dominated era $H(T)=\pi\left(8\pi
g_{*}(T)/90\right)^{1/2}T^{2}/M_{Pl}$ and the halos form with mass,

$M_{h}=\frac{4\pi}{3}\left(\frac{1}{a(T_{\mathrm{osc}})H(T_{\mathrm{osc}})}\right)^{3}\overline{\rho}_{0}\approx
2\times
10^{8}M_{\odot}\left(\frac{\mathrm{keV}}{T_{\mathrm{osc}}}\right)^{3}~{}.$
(17)

The existence of DM structure down to small scales requires that the axions
behave as dark matter by the time the temperature of the Universe is
$\sim\,{\rm keV}$, i.e. $T_{\mathrm{osc}}\ \raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}\ 1\,{\rm keV}$.
Thus, there is an upper bound on the initial halo mass. More sophisticated
analysis of the constraints on the axion isocurvature power spectrum at small
scales can be found in Ref. [42].

Going forward we will assume that the axion makes up a sizable fraction of the
dark matter abundance and place a bound on its self-coupling, equivalently
$f_{a}$, through recurrent axinova. There are four parameters that determine
the amount of axion dark matter that is converted to dark radiation: the axion
mass $m_{a}$, the axion self coupling $\lambda$ which in simple models is
determined by the decay constant $f_{a}$, the structure mass $M_{0}$ (or
equivalently $M_{peak}(z_{c})$), and the red-shift at which minihalos first
form $z_{c}$. Numerical simulations [14, 15] indicate that the white noise
spectrum has large amplitude at small scales $A_{0}\sim\mathcal{O}(1)$ and
thus minihalos form as early as possible $z_{c}\sim z_{\mathrm{eq}}$, with
mass given by (17).

As times evolves, the characteristic mass grows as $M_{h}\sim(1+z)^{-2}$ as
minihalos merge with each other. Since a characteristic mass halo has
concentration $c\approx 4$ its scale radius and density vary with redshift as
$r_{s}\sim(1+z)^{-5/3}$, $\rho_{s}\sim(1+z)^{3}$ and consequently the speed at
the scale radius depends on redshift as $v_{s}\sim(1+z)^{-1/6}$. From Eqs.
(10) and (11) this implies that the time scales for collapse scale as
$\tau_{gr}\sim(1+z)^{-7}$, $\tau_{self}\sim(1+z)^{-19/3}$. This rapid
lengthening of the axion star formation time as the Universe ages means that
the dominant DM mass loss occurs as soon as the minihalo mass is larger than
the critical star mass, and the earlier that occurs the greater the fraction
lost. More precisely, assuming $t_{\rm crit}$ is in the power law regime, the
decay rate for halos of mass $M_{0}$ which initially form at redshift $z_{c}$
is,

$\begin{split}\frac{df_{\rm decay}}{dz}&\sim
76500\pi^{2/3}\kappa\frac{M_{pl}^{3}\overline{\rho}_{\rm
col}^{2}}{M_{0}f_{a}^{5}m_{a}^{4}}\left(\frac{1+z}{1+z_{c}}\right)^{8}\frac{1}{(1+z)^{5/2}H_{0}}\\\
&\times\left[1+75\pi^{4/3}\left(\frac{f_{a}}{M_{0}^{1/3}\overline{\rho}_{\rm
col}^{1/6}}\right)^{4}\left(\frac{1+z}{1+z_{c}}\right)^{2/3}\right]\\\
&\times\left(\frac{\overline{M_{*}}}{M_{*}^{\rm
max}}\right)^{\alpha-2}\Theta\left(M_{peak}(z)-M_{*}^{\rm
max}\right)~{},\end{split}$ (18)

where we have suppressed the logarithmic corrections to the Rutherford cross
section in (10), taken $b=d=1$, and $\overline{\rho}_{\rm
col}=(1+z_{c})^{3}\overline{\rho}_{0}$ is the background density at the time
of initial collapse.

The from of (18) makes clear that the rate is peaked to early redshift and
this rate is enhanced by decreasing both $m_{a}$ and $f_{a}$. If the timescale
for scattering is set by self interactions, i.e. $f_{a}\ \raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}\
M_{0}^{1/3}\overline{\rho}_{0}^{1/6}$, then along curves where $f_{a}\sim
m_{a}^{-4/5}$ the decay rate is constant. Furthermore, for any choice of
parameters there is a maximal $f_{a}$ above which there is not enough time to
form a critical mass star in a minihalo. This leads to a region, bounded from
below (above), in $m_{a}-f_{a}$ ($m_{a}-f_{a}^{-1}$) space which is
constrained by the cosmological data discussed above (14).

Figure 1: The exclusion region from axinovae for different assumptions for
axion parameters, see text for more details. Existing limits from the black
hole superradiance are shown in grey. The green region is the most
conservative bound using a constant rate of $d{\rm log}M/dt$ at $M_{*}^{\rm
max}\leq\overline{M_{*}}$ and a late $T_{\rm osc}$, with a formation timescale
given in Eq. (13) The red region uses a power-law growth (PL) with $M\propto
t^{2/3}$ over all the mass ranges and it also assumes the lowest oscillation
temperature. The blue region presents the bound after optimizing over
oscillation temperature.

In Fig. 1, we plot the region that is constrained by the axinovae, for various
assumptions. The gray regions are excluded by black hole superradiance
constraints [43, 44, 45, 46]. The most conservative (weakest) constraint,
shown in green, comes from assuming that the oscillation temperature is low
and that the time to reach a critical star is given by (13). Over most of the
green region the critical star mass is low and the growth ($d\log M/dt$) is
still in the constant regime. Given constraints on large scale structure we
take the lowest possible oscillation temperature to $\sim 1$ keV. The later an
axion starts oscillating the larger the mass of the initial axion
miniclusters, which leads to a longer axion star production time $\tau$,
suppressing the resulting appearance of axinova.

In the red region we again assume the lowest possible oscillation but now
assume that the star growth is power law,
$M\sim\rho_{s}r_{s}^{3}(t/\tau)^{2/3}$, for all star masses, as discussed in
Appendix B. At masses below $\rho_{s}r_{s}^{3}$ the power law predicts faster
growth than the constant growth assumed in (13) and the green region. This
makes the bound stronger. For $f_{a}\ \raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}\
10^{15}\mathrm{GeV}$ the axion star critical mass is larger than where
exponential growth transitions to power law in the green and the two
constraints coincide.

Finally, the blue region is the strongest constraint and is found by
optimizing over the oscillation temperature. The maximum possible oscillation
temperature arises when the axion starts oscillating with its zero temperature
mass, $T_{\rm osc}=\sqrt{m_{a}M_{\rm pl}}$. These high temperatures will lead
to the lightest axion miniclusters and the shortest star production times, but
such miniclusters may not be massive enough to contain a critical star. At
each point in the parameter space, we select the highest possible $T_{\rm
osc}$ that leads to a massive enough minicluster. Since, $M_{h}\sim T_{\rm
osc}^{-3}$ this selected temperature is still close to $\sqrt{m_{a}M_{\rm
pl}}$. In Fig. 2, in Appendix D we show the constraint for $M_{*}\sim
t^{1/5}$, when the leading order the decay rate is independent of $T_{\rm
osc}$.

In the excluded regions an $\mathcal{O}(1)$ fraction of all dark matter has
passed through an axionova. This may lead to other observables in axion
experiments or in cosmological observations. Given the high powers that appear
in (18) if the constraints on decaying dark matter are improved in the future
the region of parameter space excluded will not be greatly altered.

##  IV Conclusions

We obtain new bounds on axion dark matter parameters $m_{a},f_{a}$ assuming
the formation of dense axion minihalos, motivated by the post-inflationary
scenario. Axion perturbations in the post-inflationary scenario will lead to
the formation of dense substructures known as axion miniclusters or minihalos
after matter-radiation equality, which can subsequently form coherent objects
known as axion stars at the core of axion minihalos. Low mass dilute axion
stars, supported by gradient pressure, can be cosmologically stable. However,
they will accrete more axions from minihalos and continue to grow in mass
until the axion self-coupling becomes important and the gradient pressure can
no longer stop them from collapsing and emitting relativistic axions, in an
_axinova_. The remnant of an axinova is a less massive star which will again
grow, leading to recurrent axinova.

If the recurrent formation rate is large enough and axinovae are active, they
can convert a significant fraction of dark matter into radiation which can be
constrained by measurements of large scale structure formation. Our constraint
only depends on the axion self-coupling and gravity. The self coupling can be
mapped to axion-photon and axion-neutron couplings in specific models. Those
constraints are obtained by requiring the population of dense axion stars
formed in axion minihalos at high redshifts shall not dominate the mass of
dark matter. If the axion is only a fraction of dark matter or only a few
percent of axion dark matter is decaying, the conversion to dark radiation may
be cosmologically significant in future observations but consistent with the
current data. Alternatively, if the axinova has a branching fraction into
standard model states there may be observables in the region or parameter
space close to our bound. We leave a more detailed study of the cosmological
evolution or possible visible signals to future work.

## Acknowledgements

We thank Asimina Arvanitaki, Masha Baryakhtar, Nikita Blinov, Abhish Dev,
George Fleming, Junwu Huang for helpful discussions. We especially thank Josh
Eby for comments on an early draft. PJF and HX are supported by Fermi Research
Alliance, LLC under Contract DE-AC02-07CH11359 with the U.S. Department of
Energy. N.W. is supported by NSF under award PHY-1915409, by the BSF under
grant 2018140, and by the Simons Foundation. HX thanks NYU for support and
hospitality while a portion of this work was completed. This work was
performed in part at the Aspen Center for Physics, which is supported by the
National Science Foundation grant PHY-1607611.

## Appendix

##  Appendix A Axion Star Configurations

The stable axion-field configuration for the gravitational bound-state of non-
relativistic axions can be found by solving the Gross-Pitaevskii-Poisson
equations, which must be done numerically. For a thorough review, see [47] and
references therein. However, it has been shown that a good approximation of
these solutions is obtained by using a Gaussian ansatz for the field profile
[48, 49, 50]. Doing so gives some insight into the competing effects driving
the physics [51]. Expanding the axion potential (1) to quartic order one finds
an attractive self interaction

$V=\frac{1}{2}m_{a}^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4}~{},$ (19)

with $\lambda=(1-3c_{ud})m_{a}^{2}/f_{a}^{2}$. An axion star of mass $M_{*}$
and radius $R_{*}$ has energy

$E_{*}=-\frac{G_{N}M_{*}^{2}}{R_{*}}+c_{1}\frac{M_{*}}{2\,m_{a}^{2}R_{*}^{2}}-c_{2}\frac{\lambda
M_{*}^{2}}{12\,m_{a}^{4}R_{*}^{3}}~{}.$ (20)

In order, these terms correspond to the gravitational self energy, the
gradient pressure, and the internal energy from self interactions. The
numerical coefficients, $c_{i}$, depend upon the details of the field profile
and are found numerically [52, 53, 51] to be $c_{1}=9.9$, $c_{2}=0.85$. The
mass-radius relation for axion stars, found by minimizing $E_{*}$, has two
solutions

$R_{*}^{\pm}=\frac{c_{1}}{2\,G_{N}M_{*}m_{a}^{2}}\left(1\pm\sqrt{1-\frac{c_{2}}{c_{1}^{2}}\lambda
G_{N}M_{*}^{2}}\right)~{}.$ (21)

The $R_{*}^{+}$ root corresponds to the so-called dilute branch and the axion
field value is small. On this branch gravitational attraction is balanced by
gradient pressure leading to a stable configuration. As is typical for objects
supported by uncertainty pressure the product of the radius and mass of the
star is a constant

$R_{*}^{+}=9.9\frac{M_{\mathrm{pl}}^{2}}{m_{a}^{2}M_{*}}~{}.$ (22)

However, as one moves to larger axion star mass the self interactions cannot
be ignored and if they are attractive (as asummed above) they destabilise the
star. There is a maximal mass, beyond which axion stars are no longer stable

$M_{*}^{\mathrm{m}ax}=\frac{10.7}{\sqrt{\lambda}}M_{\mathrm{pl}}~{}.$ (23)

The two solutions (21) meet at this maximal mass. The second solution is one
where gravity can be ignored and the gradient pressure and the axion’s
attractive self interactions are in unstable equilibrium. On this branch
$R_{*}\sim M_{*}$.

The value of the axion field at the center of the star scales as
$a_{0}^{2}\sim M_{*}/(m_{a}^{2}R_{*}^{3})$ so that at the low mass end of the
$R_{*}^{-}$ branch $a_{0}\sim 1$ and the axion field is not dilute. The axions
can no longer be thought of as non-relativistic and the solution is
approximately constant density ($\rho\sim m_{\pi}^{2}f_{\pi}^{2}$) and thus
$R_{*}\sim M_{*}^{1/3}$. However, it is believed that this field configuration
is also unstable, with a lifetime $\sim 10^{3}m_{a}^{-1}$ [51], although
alterations to the axion potential can make these solutions long lived [54,
55, 56].

The upshot of this is that if a dilute axion star with mass below
$M_{*}^{\mathrm{max}}$ were to form and grow, by accumulation of additional
axions, to the maximal mass it would then shrink in size and become a dense
axion star which would survive for a short period. During this time the dense
axion star goes through several oscillations and a density singularity
develops in the central core and this dense region emits relativistic axions
lowering the density [57, 58, 35]. This process repeats and $\sim 30\%$ of the
initial star mass can be emitted, leaving a dilute remnant which may in turn
grow to the maximal mass and emit more relativistic axions. Thus, maximal mass
stars are an engine to turn substantial amounts of cold dark matter into
radiation.

##  Appendix B The growth of axion stars in a large minihalo

When the axion star mass larger than the characteristic star mass
$\overline{M_{*}}$, the mass growth is found to be well described by a power
law, $M_{*}\propto t^{1/\alpha}$. However, the growth rate at lighter masses
in an axion minihalo is still unknown. One would expect the growth rate is
larger at smaller radius in the minihalo environment due to the larger density
and smaller velocity. If a star is formed within a small radius, the mass
contained in this region is small. Therefore, lighter objects always start to
form with a greater rate. For an NFW profile, the mass contained within $r$ is

$M(r)|_{r\rightarrow 0}=4\pi\rho_{s}r_{s}^{3}\left({\rm
ln}\left(1+\frac{r}{r_{s}}\right)-\frac{r}{r+r_{s}}\right)\approx
2\pi\rho_{s}r_{s}r^{2}.$ (24)

The formation timescale given by self-interactions is $\tau_{self}\propto
v^{2}/\rho^{2}$. At small radius of an NFW halo, the density and velocity
scale as $\rho\propto 1/r$ and $v\propto\sqrt{r}$. Therefore, $M(t)\propto
t^{2/3}$. Similarly, if the gravity dominates the axion star formation,
$\tau_{gr}\propto v^{6}/\rho^{2}\propto r^{5}$ at small radius and we obtain
the mass growth power law $M(t)\propto t^{2/5}$. Since this scaling is active
at short distance scales within the minihalo we consider a scenario where
$\alpha=3/2$ at all axion star masses, see Fig. 1.

##  Appendix C Press-Schechter with White Noise-like Power at Short Distances

We consider the density perturbations,
$\delta\equiv\delta\rho/\overline{\rho}$, to consist of two contributions,
conventional $\Lambda$CDM adiabatic perturbations that are present at all
scales and isocurvature perturbations which are only become important over a
finite range of scales. We take the isocurvature contribution to be a power
low with a cut-off at very small scales, corresponding to a wavenumber
$k_{0}$. For the case of the axion it is believed the short-scale behavior has
a power spectrum that is approximately that of white noise, corresponding to
$n=3$ below. Modes from these two contributions have different growth
behaviors after they enter the horizon, in particular the adiabatic
perturbations have logarithmic growth until matter-radiation equality while
the isocurvature modes do not. At late times, in the matter dominated era,
they have similar growth. Taking into account these different growth behaviors
the two-point function of the density perturbations is

$\langle\delta^{2}\rangle=\frac{2\pi^{2}}{k^{3}}\left(D_{adi}^{2}I_{1}^{2}L^{2}A_{s}\left(\frac{k}{k_{s}}\right)^{n_{s}-1}+D_{iso}^{2}A_{0}\left(\frac{k}{k_{0}}\right)^{n}\Theta(k_{0}-k)\right)~{}.$
(25)

For a $\Lambda$CDM-like power spectrum $A_{s}\approx 2\times 10^{-9}$,
$n_{s}\approx 0.97$, and the pivot scale is $k_{s}=5\times 10^{-3}$ Mpc-1. At
late times $D_{adi}\approx D_{iso}\approx a/a_{eq}=(1+z_{\rm eq})/(1+z)$ and
the exact forms can be found in standard references e.g. [59, 60]. The
constant $I_{1}\approx 9.1$ and $L\approx\log(0.1a_{eq}/a)$.

The Press-Schechter formalism assumes spherical collapse of over-densities and
that the probability for these collapses follows a Gaussian distribution whose
variance, smoothed at some scale $R$, is given by

$\sigma^{2}(z,R)=\int\frac{d^{3}k}{\left(2\pi\right)^{3}}\langle\delta^{2}\rangle\left|\widetilde{W}(kR)\right|^{2}~{},$
(26)

where $\widetilde{W}(kR)$ is the window function and can take various forms.
Here we focus on the so-called sharp $k$-filter where
$\widetilde{W}(z)=\Theta(1-z)$. For this choice of window function there is
not a well defined mass, $M$, associated with the co-moving filter scale $R$,
since the real space form of $\widetilde{W}$ does not have local support [61].
However, we will follow the oft-used relation
$M=6\pi^{2}\overline{\rho}_{0}R^{3}$ [62], where $\overline{\rho}_{0}$ is the
present day cosmological axion density. Note that for (26) to be well defined
we have to introduce an IR cut-off $k_{IR}$ and we define
$M_{0}=6\pi^{2}\overline{\rho}_{0}k_{0}^{-3}$. We are typically interested in
halo masses and formation redshifts where the adiabatic perturbations are
subdominant to the isocurvature perturbations, $A_{s}\ll A_{0}$. In this
regime, once structures can form i.e. $z<z_{\rm eq}$, the variance has the
simple form

$\sigma^{2}(z,M)\sim\left(\frac{1+z_{\rm
eq}}{1+z}\right)^{2}\frac{A_{0}}{n}\times\begin{cases}1~{}&M\leq M_{0}\\\
\left(\frac{M_{0}}{M}\right)^{n/3}~{}&M>M_{0}\end{cases}~{}.$ (27)

In the Press-Schechter approach the halo mass function is related to the
probability to find $\delta>\delta_{c}\approx 1.686$, with the fraction of
matter in objects of mass $M$ given by

$\frac{df}{dM}=\sqrt{\frac{2}{\pi}}\frac{\delta_{c}}{M\sigma}\left|\frac{d\log\sigma}{d\log
M}\right|e^{-\delta_{c}^{2}/\sigma^{2}}~{}.$ (28)

The exponential suppression means that the most massive objects, with mass
$M_{peak}$, to have formed are those for which
$\sigma(z,M_{peak})=\delta_{c}$. If the isocurvature perturbations were large
enough, $A_{0}>n\delta_{c}$, these objects would form at $z_{\rm eq}$.
Instead, for more typical isocurvature perturbations of $A_{0}\approx 0.1$,
the first halos to form are of mass $M_{0}$ and they form at

$z_{c}\approx\sqrt{\frac{A_{0}}{n}}\frac{z_{\rm eq}}{\delta_{c}}~{},$ (29)

and subsequently grow, with the peak mass of the halo mass function being

$M_{peak}=M_{0}\left(\frac{1+z_{c}}{1+z}\right)^{6/n}=M_{0}\left(\frac{A_{0}}{\delta_{c}^{2}n}\right)^{3/n}\left(\frac{1+z_{\rm
eq}}{1+z}\right)^{6/n}~{}.$ (30)

##  Appendix D Axion Relic Abundance from Misalignment

We consider the relic abundance from the misalignment mechanism for an axion
coupled to a dark confining gauge group “DarkQCD”, which is taken to be
$SU(N_{C})$ with $N_{F}$ vector-like quarks. The temperature dependence of the
mass is understood in two limits. At low temperature the axion mass is
independent of temperature and at high temperature the dilute instanton gas
approximation is valid, leading to a power law dependence. In between there
could be a first or second order transition or a smooth cross over depending
on $N_{F},N_{C}$ [63, 64]. For simplicity we take the temperature dependence
mass to have the form

$m_{a}(T)=\begin{cases}m_{0}&T<T_{c}\\\
m_{0}\left(\frac{T_{c}}{T}\right)^{b}&T\geq T_{c}\end{cases}~{}.$ (31)

Here we take the critical temperature to be the same as the confinement scale
of DarkQCD, $T_{c}=\Lambda=\sqrt{m_{0}f_{a}}$. The dilute instanton gas
approximation gives $b=(11N_{C}+N_{F}-12)/6$. Taking $b$ large for
temperatures in the vicinity of $T_{c}$ also approximates the form of a first
order phase transition. After PQ symmetry breaking, and before the instantons
generate a potential for the axion, the misalignment angle $\theta=a/f$ has a
flat potential and is free to take on any initial value in each causal patch.
The equation of motion for this angle is

$\ddot{\theta}+3H\dot{\theta}+m^{2}(T)\theta=0~{}.$ (32)

Assuming the cosmology is governed by a fluid with equation of state
$p=\omega\rho$ (RD is $\omega=1/3$) then the scale factor $a\sim
t^{\frac{2}{3(1+\omega)}}$ and $H=\frac{2}{3(1+\omega)t}$. Combining this with
the fact that temperature redshifts with the scale factor, $T\sim a^{-1}$,
(32) becomes

$\ddot{\theta}+\frac{2}{(1+\omega)t}\dot{\theta}+m_{0}^{2}\left(\frac{\Lambda}{T_{i}}\right)^{2b}\left(\frac{t}{t_{i}}\right)^{\frac{4b}{3(1+\omega)}}\theta=0~{}.$
(33)

This equation can be solved exactly by noting that $y=x^{\alpha}J_{n}(\beta
x^{\gamma})$ with $J_{n}$ the $n$-th Bessel function, satisfies the equation

$\frac{d^{2}y}{dx^{2}}-\frac{2\alpha-1}{x}\frac{dy}{dx}+\left(\beta^{2}\gamma^{2}x^{2(\gamma-1)}+\frac{\alpha^{2}-n^{2}\gamma^{2}}{x^{2}}\right)y=0~{}.$
(34)

Thus, the solution to (33) takes the form

$\left(\frac{t_{i}}{t}\right)^{\frac{1-\omega}{2(1+\omega)}}J_{\frac{3(\omega-1)}{2(2b+3(1+\omega))}}\left(m_{0}\,t\left(\frac{\Lambda}{T_{i}}\right)^{b}\frac{3(1+\omega)}{2b+3(1+\omega)}\left(\frac{t}{t_{i}}\right)^{\frac{2b}{3(1+\omega)}}\right)~{}.$
(35)

Requiring that the argument of the Bessel function changes by an
$\mathcal{O}(1)$ amount before oscillation is deemed to have set in, and
identifying various powers of $t$ with $H$ and $m_{a}(T)$, the oscillation
temperature is implicitly defined by

$m_{osc}\sim\frac{3+3\omega+2b}{2}H_{osc}~{}.$ (36)

Notice that for large $b$, an axion mass that rapidly changes from zero to
$m_{0}$ as can arise in a first order phase transition, this is different from
the usual $m\sim 3H/2$ requirement since the rapid evolution of the axion mass
provides its own “friction”. From now on we consider the case of RD and thus
$m_{osc}\sim(2+b)H_{osc}$. We also consider the possibility that the dark
sector and the SM are at different temperatures. Assuming there are no
thermalizing interactions between them, and ignoring the complication of
different thresholds in the two sectors we take the ratio of temperatures to
be a constant, $T_{D}=\xi T_{SM}\equiv\xi T$. Thus, the oscillation
temperature and mass are found by solving

$\left(\frac{8\pi^{3}g_{*}(T)}{90}\right)^{1/2}\frac{T^{2}}{M_{pl}}=\begin{cases}\frac{2m_{0}}{3}&\xi
T<\Lambda\\\ \frac{m_{0}}{2+b}\left(\frac{\Lambda}{\xi T}\right)^{b}&\xi
T\geq\Lambda\end{cases}~{}.$ (37)

If the oscillation begins while the mass is temperature dependent then

$T_{osc}=\Lambda\left(c(T_{osc})\frac{M_{pl}}{f_{a}}\frac{\xi^{-b}}{2+b}\right)^{\frac{1}{2+b}}\xrightarrow{b\rightarrow\infty}\frac{\Lambda}{\xi}~{},$
(38)

where $c(T)=\sqrt{90/8\pi^{3}g_{*}(T)}$ and, assuming the SM dominates the
energy density of the Universe, $0.06\ \raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}\ c(T)\ \raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}\ 0.33$. This
solution is only consistent if $T_{D}>\Lambda$ which places the restriction
$b\ \raise 1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}\
\xi^{2}M_{pl}/f_{a}$ 222Note also that since confinement only occurs for a
negative beta function $b\geq 5/3$.. For $b,\xi$ in violation of this bound
the oscillation starts after the axion has attained its zero-temperature mass
and
$T_{osc}\sim\sqrt{2c(T_{osc})m_{0}M_{pl}/3}=\Lambda\sqrt{2c(T_{osc})M_{pl}/3f_{a}}$.

Once the oscillation temperature is known, and using the fact that ratio of
axion number density to entropy density is constant, the present day axion
mass fraction can be determined:

$\displaystyle\Omega_{a}$ $\displaystyle=$
$\displaystyle\frac{m_{0}m_{osc}f_{a}^{2}\langle\theta^{2}\rangle}{2\rho_{crit}}\frac{g_{*}(T_{0})T_{0}^{3}}{g_{*}(T_{osc})T_{osc}^{3}}=\sqrt{\frac{8\pi^{3}}{90g_{*}(T_{osc})}}\frac{m_{0}f_{a}^{2}\langle\theta^{2}\rangle}{\rho_{crit}M_{pl}}\frac{g_{*}(T_{0})T_{0}^{3}}{T_{osc}}\begin{cases}\frac{3}{4}&\xi
T_{osc}<\Lambda\\\ \frac{2+b}{2}&\xi T_{osc}\geq\Lambda\end{cases}$ (39)
$\displaystyle=$
$\displaystyle\frac{g_{*}(T_{0})T_{0}^{3}}{\rho_{crit}M_{pl}}m_{0}^{1/2}f_{a}^{3/2}\langle\theta^{2}\rangle\begin{cases}\frac{3}{4}\left(\frac{8\pi^{3}}{90}\right)^{3/4}\left(\frac{3f_{a}}{2M_{pl}}\right)^{1/2}g^{-1/4}_{*}(T_{osc})&\xi
T_{osc}<\Lambda\\\
\frac{2+b}{2}\left(\frac{8\pi^{3}}{90}\right)^{\frac{3+b}{2(2+b)}}\left(\frac{(2+b)\xi^{b}f_{a}}{M_{pl}}\right)^{\frac{1}{2+b}}g^{-\frac{b+1}{2(2+b)}}_{*}(T_{osc})&\xi
T_{osc}\geq\Lambda\end{cases}~{}.$ (40)

If the dark sector has roughly the same temperature as the standard model
sector ($\xi\sim 1$), the confinement scale corresponds to a Hubble of $H\sim
m_{a}f_{a}/M_{\rm pl}$, which is always smaller than $m_{a}$ because we
require $f_{a}<M_{\rm pl}$ and the axion self-coupling is stronger than
gravity. The axion mass will not be turned on until the dark confinement
occurs. Therefore, $T_{\rm osc}$ is greatly delayed, which enhances the relic
abundance since it is less diluted. The blue dashed curve in Fig. 2 shows the
axion parameters that give the dark matter relic abundance assuming a slightly
colder dark sector ($T_{\rm DS}=0.5T_{\rm SM}$) and axion mass to be turned on
as $m_{a}\propto T^{-b}$. A large dark gauge group or a first-order phase
transition in the dark sector will be needed for a large $b$. We also
presented the $T_{\rm osc}$ independent constraint in Fig. 2 which assumes
axion star mass grows like $M\propto t^{0.2}$, corresponding to $\alpha=5$.
For this value of $\alpha$ the decay rate (18) is independent of $T_{\rm osc}$
in the region of parameter space dominated by self interactions.

Figure 2: The exclusion plot of axion parameters from axinovae. The colored
region represents the exclusion region assuming the axion star growth
$M\propto t^{0.2}$, where the exclusion is independent of $T_{\rm osc}$. The
gray regions are the existing limits from the black hole superradiance. Dashed
curves are the axion parameters giving the correct relic abundance assuming
axion mass behaves as $m_{a}(T)=m_{a}(\Lambda/T)^{b}$, where
$\Lambda=\sqrt{m_{a}f_{a}}$ is the dark confinement scale. A large $b$ can
naturally come from a first-order dark QCD phase transition.

While we have been focusing on a QCD-like axion model to study the relic
abundance, there are other models that can enhance the self-coupling of axions
while giving the correct relic abundance, such as a clockwork axion [66]
(discussed in Appendix E), friendship axion [67], axions from dilute domain
walls [68, 69],and kinetic misalignment mechanism [70, 71]. In a clockwork
axion scenario, a large field range is naturally produced for the axion field
in the low-energy theory. The axion potential can have two confinement scales
and two effective decay constants which can give the relic abundance that is
needed while keeping the self-coupling strong. The friendship axion can
resonantly convert the energy density in the axion sector with a larger decay
constant to that with a lower decay constant if the mass ratio of two axions
is close to 1. Therefore the relic density of axions with a low decay constant
is greatly enhanced. Axion relic density can be greatly enhanced if the Peccei
Quinn symmetry is followed by a period of inflation such that axion string
networks are inflated away but will eventually reenter the horizon [68, 69].
In this scenario, the decay of diluted domain walls occurs very late,
enhancing the relic density of axions. In kinetic misalignment [70, 71] the
axion field does not start at rest but instead has a nonzero initial velocity.
The process of the axion settling into a minimum of the periodic potential,
and generating an axion number density, is delayed since it can only occur
after its initial kinetic energy has red-shifted away. The initial velocity,
$\dot{\theta}_{i}$, for the field is proportional to the net PQ charge and its
generation requires an explicit breaking of the PQ symmetry at some scale.
This breaking should not be present at later times when the axion potential
should be determined solely by instanton effects as can occur, for instance,
if the breaking is from higher dimensional operators or arises from another
scalar field acquiring a VEV. The kinetic energy of the field becomes
comparable to the potential energy when $\dot{\theta}_{i}(a_{i}/a)^{3}\approx
m_{a}(T)$, so large initial velocity and late generation both delay the onset
of oscillations and increase the relic abundance. Kinetic misalignment tends
to produce denser minihalos than conventional misalignment [72, 73] due to a
parametric resonance that enhances fragmentation [74]. If the fragmentation is
not complete the power spectrum of axion density perturbations has features at
many scales and our power law ansatz will not be a good approximation.
However, if the fragmentation completes before the kinetic motion is depleted
the power spectrum is well approximated by white noise [72]. In both cases the
late-time halo mass function is peaked such that most of the mass is in mini-
halos of mass $M_{peak}$. While there have been many models that can enhance
either the axion relic abundance or the self-coupling, diluting the relic
abundance is also possible in scenarios such as nonstandard thermal histories
that lead to entropy production [75].

##  Appendix E Enhanced Axion Self-Coupling

The axion self-coupling is given by $|\lambda|\sim
m_{a}^{2}/f_{a}^{2}\sim\Lambda^{4}/f_{a}^{4}$, assuming a cosine instanton
potential. To obtain the right relic abundance for axion dark matter, $f_{a}$
is usually large since the relic abundance of axions is proportional to
$f_{a}^{2}$. However, axion self-couplings can be enhanced without affecting
the standard misalignment mechanism or the formation of axion miniclusters. If
the axion couples to two confining sectors, which can be naturally achieved
with clockwork mechanism [66], the axion potential is

$V(a)=V_{1}(a)+V_{2}(a)=\Lambda_{1}^{4}\left(1-{\rm
cos}\frac{a}{f_{1}}\right)+\Lambda_{2}^{4}\left(1-{\rm
cos}\frac{a}{f_{2}}\right)~{}.$ (41)

Here $\Lambda_{1},\Lambda_{2}$ are the confinement scales of the two strongly
coupled sectors and $f_{1},f_{2}$ are the corresponding decay constants.

We consider the situation where the vacuum misalignment mechanism is mostly
set by $V_{1}(a)$ and so we require $V_{1}^{\prime}(a)\gg V_{2}^{\prime}(a)$
and $V_{1}^{\prime\prime}(a)\gg V_{2}^{\prime\prime}(a)$ which corresponds to
the requirements

$\frac{\Lambda_{1}^{4}}{f_{1}}\gg\frac{\Lambda_{2}^{4}}{f_{2}}~{},\,~{}\frac{\Lambda_{1}^{4}}{f_{1}^{2}}\gg\frac{\Lambda_{2}^{4}}{f_{2}^{2}}~{}.$
(42)

Satisfying these constraints will guarantee that the misalignment mechanism
and the axion mass term and the rolling of axion field are solely determined
by the strong sector with a confinement scale of $\Lambda_{1}$ and breaking
scale $f_{1}$, which will be responsible for the relic abundance of the axion
particles. However, this does not fully determine the axion self-couplings. If
$f_{1}\gg f_{2}$, the self-coupling can be dominated by the other strong
sector, as long as the following condition is satisfied

$\frac{\Lambda_{1}^{4}}{f_{1}^{4}}\ll\frac{\Lambda_{2}^{4}}{f_{2}^{4}}~{}.$
(43)

The conditions (42) and (43) can be consistent with each other provided
$f_{1}\gg f_{2}$. For instance, if $\Lambda_{2}/\Lambda_{1}\equiv\epsilon\ll
1$ then $f_{2}/f_{1}\sim\epsilon^{\zeta}$, with $1<\zeta<2$, will satisfy the
conditions. Assuming the strong coupling sectors satisfy these requirements
then $m_{a}^{2}\sim\Lambda_{1}^{4}/f_{1}^{2}$ and
$|\lambda|\sim\Lambda_{2}^{4}/f_{2}^{4}$ and the effective decay constant that
labels the self-coupling strength is

$\tilde{f_{a}}=\frac{m_{a}}{\sqrt{|\lambda|}}=f_{1}\left(\frac{f_{2}\Lambda_{1}}{f_{1}\Lambda_{2}}\right)^{2}\ll
f_{1}.$ (44)

Therefore, the effective decay constant of an axion model that gives the self-
coupling strength can be much smaller than the decay constant that is
responsible for the relic abundance. They can be considered as two independent
parameters.

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# Chiral Condensate in Two Dimensional Models

V.G. Ksenzov  State Scientific Center Institute for Theoretical and
Experimental Physics, Moscow, 117218, Russia

###### Abstract

We investigate two different models. In one of them massive fermions interact
with a massive scalar field and in the other the fermion field is in an
electrical field (QED2). The chiral condensates are calculated in one-loop
approximation. We found that the chiral condensate in the case of the Yukawa
interaction the fermions and scalar field does not vanish if the mass of the
fermion field tends to zero. The chiral condensate disappears in QED2, if the
fermion mass is zero.

##  I Introduction

Dynamical breaking of the symmetry is described by a quantity known as the
order parameter. In models with a broken chiral symmetry the chiral condensate
is the order parameter that disappears, if the chiral symmetry is restored.
The investigation of the chiral condensate plays a crucial role in attempts to
describe phase transitions related to the dynamical chiral symmetry breaking.

In QCD chiral condensate is investigated via lattice simulations. This
approach is usually employed in studies of the chiral condensate in the
presence of external factors such as temperature, chemical potential, magnetic
fields and magnetic monopoles, etc (see Refs. BCL –RS  and references
therein).

Apart from numerical studies there are a lot of models employed for the
investigation of the chiral condensate in various physical systems BBM –JLK ,
NJL –KR2 . In the first paper on this subject Nambu and Jona-Lasinio (NJL)
analyzed a specific field model in four dimensions NJL . Later, the chiral
condensate was studied by Gross and Neveu (GN) in two dimensions spacetime in
the limit of a large number of fermion flavors $N$ GN . These two models are
similar, but in contrast to NJL-model, GN-model is a renormalizable and
asymptotically free theory. Due to these properties, GN-model is used for
qualitative modeling of QCD. The relative simplicity of both models is a
consequence of the quartic fermion interaction.

In our previous papers we investigated a system of a self-interacting massive
scalar and massless fermion fields with the Yukawa interaction in a
$(1+1)$-dimensional spacetime. In the limit of a large mass of the scalar
field, the model equivalent to the GN-model. The chiral condensate was
obtained by the path integral using the method of the stationary phase KR –KR2
.

In this paper we present a study of two models. One of them is the model with
massive fermions and massive scalars with the Yukawa interaction between these
two fields. It is worth to note, that the scalars is not a self-interacting
fields. The other model is QED in two-dimensional space-time.

The purpose of this paper is to obtain the chiral condensate in QED2-model. To
do this we must obtain an effective action of the model. Evaluating the one-
loop effective potential is equivalent to the summation of an infinite class
of Feynman diagrams, therefore we are unable to calculate a fermionic
determinant with an arbitrary vector potential $A_{\mu}(x)$. For this reason
we begin studying of the first model, that we use to investigate the effective
potential and the chiral condensate, if discrete chiral symmetry was
explicitly broken. This model exhibits the essential feature of the
techniques, that we use to construct the effective action of QED2. In
particular we can see, how the chiral symmetry breaking manifests itself in
the effective potential. As a result we found the chiral condensate in the
electrical field.

##  II Fermions in a scalar field

The model, that will be discussed in this section involves $N$ massive fermion
fields and a massive scalar field with the Yukawa interaction between those
two fields in $(1+1)$-dimensional spacetime.

Lagrangian of the model is

$L=L_{b}+L_{f}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}\mu^{2}\phi^{2}(x)+i\bar{\psi}^{a}\\!\\!\\!\not\partial\psi^{a}-g\left(\phi(x)-\frac{m}{g}\right)\bar{\psi}^{a}\psi^{a},$
(1)

here $\phi(x)$ is a real scalar field, $\psi^{a}(x)$ is a fermion fields,
index $a$ runs from 1 to $N\gg 1$ and $m$ is a mass of the fermions.

The model with the massless fermions was investigated in our previous papers
KR –KR2 . In the case, if $m=0$, the Lagrangian is invariant under a discrete
symmetry

$\psi^{a}\to\gamma_{5}\psi^{a},\;\bar{\psi}^{a}\to-\bar{\psi}^{a}\gamma_{5},\;\phi\to-\phi,$
(2)

which is broken by the chiral condensate. If $m\neq 0$, the discrete symmetry
(2) disappears, and chiral condensate is not a qualitative criterion.

As before in our papers we will determine the chiral condensate by an
effective potential. For our purpose we formally define the chiral correlator,
using a functional integral in Minkowski space

$\left\langle 0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle=\frac{1}{Z}\int D\phi
D\bar{\psi}^{a}D\psi^{a}g\bar{\psi}^{a}\psi^{a}\exp\left(i\int
d^{2}xL(x)\right),$ (3)

here $Z$ is a normalization constant. The chiral correlator (3) is rewritten
as

$\left\langle 0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle=\frac{1}{Z}\int
D\phi\exp\left({i\int d^{2}xL_{b}(x)}\right)i\frac{\delta}{\delta\phi}\int
D\bar{\psi}^{a}D\psi^{a}\exp\left(i\int d^{2}xL_{f}(x)\right).$ (4)

The fermionic Lagrangian is quadratic in the field and we can integrate over
them, getting

$\left\langle 0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle=\frac{1}{Z}\int
D\phi\frac{g^{2}N\left(\phi(x)-\frac{m}{g}\right)}{2\pi}\ln\frac{g^{2}\left(\phi(x)-\frac{m}{g}\right)^{2}}{\Lambda^{2}}\times$
$\times\exp\left(i\int
d^{2}x\left(\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{\mu^{2}}{2}\phi^{2}-\frac{Ng^{2}\left(\phi(x)-\frac{m}{g}\right)^{2}}{4\pi}\left(\ln\frac{g^{2}\left(\phi(x)-\frac{m}{g}\right)^{2}}{\Lambda^{2}}-1\right)\right)\right),$
(5)

here $\Lambda$ is the ultraviolet cutoff.

We want to obtain the chiral condensate in the framework of one-loop
approximation. Therefore we calculate (5) using the method of the stationary
phase. A minimum of the effective action of the system is reached if the
effective potential and kinetic energy are minimal on its own:

$\partial_{\mu}\phi=0\text{ and }U_{\text{eff}}(\phi)=\min.$ (6)

Let the constant scalar field $\phi_{m}$ satisfies the condition (6). The
factor in front of the exponent in (5) is fixed at the point $\phi=\phi_{m}$
and we obtain

$\left\langle
0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle=\frac{Ng^{2}\left(\phi_{m}-\frac{m}{g}\right)}{2\pi}\ln\frac{g^{2}\left(\phi_{m}-\frac{m}{g}\right)^{2}}{\Lambda^{2}}.$
(7)

It is worth to note, that the correlator $\left\langle
0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle$ is a chiral condensate only in the
minimum of the effective potential $\phi_{m}$. The effective potential and the
chiral condensate require renormalization. We renormalize effective potential
following Coleman and Weinberg CW  and Gross and Neveu GN  by demanding that

$\left.\frac{d^{2}U_{\text{eff}}}{d\phi_{m}^{2}}\right|_{\phi_{m}=M^{2}}=\mu_{R}^{2}.$
(8)

Then the renormalized chiral condensate is written as

$\left\langle
0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle_{R}=\frac{Ng^{2}\left(\phi_{m}-\frac{m}{g}\right)}{2\pi}\ln\frac{\left(\phi_{m}-\frac{m}{g}\right)^{2}}{M^{2}}.$
(9)

$\phi_{m}$ is determined by mean the renormalized effective potential
$U^{R}_{\text{eff}}$, which is written as

$U^{R}_{\text{eff}}=\frac{1}{2}\mu_{R}^{2}\phi^{2}+\frac{g^{2}N}{4\pi}\left(\phi-\frac{m}{g}\right)^{2}\left(\ln\frac{\left(\phi-\frac{m}{g}\right)^{2}}{M^{2}}-3\right),$
(10)

Figure 1: a – The value $U_{\text{eff}}(\phi)$; b – Chiral condensate
$R=\left\langle 0|m\bar{\psi}^{a}\psi^{a}|0\right\rangle_{R}$ at
$g^{2}N=1,69\text{ MeV}^{2}$, $\mu^{2}_{R}=0,5\text{ MeV}^{2}$, $m/g=2,17$,
$M^{2}=100$.

We determine the stationary point $\phi_{m}$ by numerical solution of equation

$\left.\frac{dU_{\text{eff}}}{d\phi}\right|_{\phi=\phi_{m}}=\mu^{2}_{R}\phi_{m}+\frac{g^{2}N}{2\pi}\left(\phi_{m}-\frac{m}{g}\right)\left(\ln\frac{\left(\phi_{m}-\frac{m}{g}\right)^{2}}{M^{2}}-2\right)=0.$
(11)

One can see, that there are two different solutions of (11), and for this
reason appear two different vacuums. One of them is a global, and the other is
a local minima (see Fig. 1a).

Using (9) and (11) we get

$\left\langle
0|m\bar{\psi}^{a}\psi^{a}|0\right\rangle_{R}=\frac{m}{g}\frac{N}{\pi}(g^{2}-g^{2}_{\text{cr}})\phi_{m}-\frac{m^{2}N}{\pi},$
(12)

here $g^{2}_{\text{cr}}=\frac{m^{2}_{R}\pi}{N}$.

It is worth to note, that the chiral condensate does not disappear, if the
fermions mass tends to zero. In such a case $(m=0)$ the effective potential
has the minimum at the point $\phi^{2}_{m}$, which is given in an explicit
form

$\phi^{2}_{m}=M^{2}\exp 2\left(1-\frac{\pi\mu^{2}_{R}}{g^{2}N}\right),$ (13)

and the vacuum energy is

$\mathcal{E}_{V}=-\frac{g^{2}N}{4\pi}\phi^{2}_{m}.$ (14)

If $g^{2}N=\pi\mu^{2}_{R}$, then $\phi^{2}_{m}=M^{2}$ and the vacuum energy is
completely perturbative one. It is known, that the vacuum energy is determined
by the vacuum condensate of the trace of the energy-momentum tensor
$\theta_{\mu\mu}$ MS , SH , SVZ  and SVNZ  as

$\frac{1}{d}\left\langle 0|\theta_{\mu\mu}|0\right\rangle=\mathcal{E}_{V},$
(15)

here $d$ is the dimension of the spacetime.

The diagrams shown in Fig. 2 determine the quantum correction to the trace of
the energy-momentum tensor in the case of the massive fermions. Then the
quantum correction to the trace of the energy-momentum tensor is written as

$\theta_{\mu\mu}=\frac{N}{2\pi}(g^{2}\phi^{2}-2mg\phi+m^{2}),$ (16)

here the fist term defines quantum anomaly if mass fermions are zero, the
second term defines the discrete symmetry breaking Fig. 2b, and the term
$\propto m^{2}$ defines the vacuum energy noninteracting fermions Fig. 2c.

It will be noted, that although the discrete symmetry is broken in the model
with the free massive fermions, but there is no violation of this symmetry in
the effective potential of the theory. This is due to the fact, that it is
impossible to construct a combination of the energy dimension, that violates
this symmetry.

The diagrams shown in Fig. 2 determine the quantum correction to the trace of
the energy-momentum tensor in the case of massive fermions.

Figure 2: The dashed line denotes the fermion loops, solid line denotes the
scalar field $\phi$.

##  III Fermions in an electrical field

The model that will be discussed here is QED2. A Lagrangian of the model is
well-known

$L=L_{b}+L_{f}=-\frac{1}{4}F^{2}_{\mu\nu}+i\bar{\psi}^{a}\\!\\!\\!\not\partial\psi^{a}-m\bar{\psi}^{a}\psi^{a}-A_{\mu}\bar{\psi}^{a}\gamma_{\mu}\psi^{a},$
(17)

If $m=0$ then the Lagrangian is invariant under a discrete chiral simmetry

$\psi^{a}\to\gamma_{5}\psi^{a},\;\bar{\psi}^{a}\to-\bar{\psi}^{a}\gamma_{5}.$
(18)

We formally define the chiral correlator as

$\begin{split}&\int d^{2}x\left\langle
0|m\bar{\psi}^{a}\psi^{a}|0\right\rangle=\\\ =\frac{1}{Z}\int
DA_{\mu}\exp\left(i\int
d^{2}x\right.&\left.L_{b}(x)\\!\\!\\!\\!\\!\\!\phantom{\int}\right)im\frac{d}{dm}\int
D\bar{\psi}^{a}D\psi^{a}\exp\left(i\int d^{2}xL_{f}(x)\right).\end{split}$
(19)

It is hard enough to calculate a fermionic determinant with an arbitrary
vector potential $A_{\mu}(x)$ therefore we will use an alternative method of
getting an effective potential.

It is known that the expression of the effective potential was derived within
gluodynamics by Migdal and Shifman MS  and SH . In the paper VK  the method
was used to obtain the vacuum condensate of the trace of the energy-momentum
tensor in massless theories in various spacetime dimensions. In this paper the
method will be used for construction the effective Lagrangian in QED2. The
expression for the effective Lagrangian was obtained from the requirement that
the anomaly be reproduced under a scale transformation. The effective
potential has the form

$V_{\text{eff}}=\frac{1}{d}\sigma\left(\ln\frac{\sigma}{\upsilon}-1\right),$
(20)

here $\sigma=\theta_{\mu\mu}$ and $\upsilon$ emerges as a constant of
integration in solving the respective differential equation MS , SH . We
should find $\theta_{\mu\mu}$ and $\upsilon$ for our model.

Figure 3: The dashed line denotes the fermion loops, solid line denotes the
vector potential $A_{\mu}$.

Above we see that the mass term is fundamentally important for calculating of
the chiral condensate (19) therefore we must construct $\theta_{\mu\mu}$
taking into account the mass term. The diagrams shown at Fig. 3 determine
$\theta_{\mu\mu}$ in QED2. The chiral symmetry breaking at the expense of the
mass term is given by the diagram Fig. 3b. A result of calculating the
diagrams is given

$\theta_{\mu\mu}=-e^{2}_{0}\frac{E^{2}}{6\pi}-e_{0}m\frac{E}{3\pi}+\frac{m^{2}}{2\pi},$
(21)

here we introduced a dimensionless coupling constant
$e^{2}_{0}=\frac{e^{2}}{m^{2}}$ and an electrical field
$E=\varepsilon_{\mu\nu}\partial_{\mu}A_{\nu}$.

To find $\upsilon$ we assume that the coupling constant $e_{0}$ is zero at the
time then $V_{\text{eff}}(\sigma)$ coincides with the effective potential for
free fermions and we get $\upsilon=\frac{1}{2\pi}M_{0}^{2}$, here $M_{0}$ is
an arbitrary substraction parameter.

Now the effective Lagrangian can be expressed as

$L_{\text{eff}}=\frac{E^{2}}{2}+\frac{\sigma}{2}\left(\ln\frac{2\pi\sigma}{M^{2}_{0}}-1\right).$
(22)

A minimum of the effective Lagrangian of the system is reached if

$\frac{dL_{\text{eff}}}{dE}=E+\frac{1}{2}\frac{d\sigma}{dE}\ln\frac{2\pi\sigma}{M^{2}_{0}}=0,$
(23)

here

$\frac{d\sigma}{dE}=-e^{2}_{0}\frac{E}{3\pi}-e_{0}\frac{m}{3\pi}$ (24)

The chiral correlation is

$\int d^{2}x\left\langle 0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle=\int
d^{2}x\frac{m}{2}\frac{d\sigma}{dm}\ln\frac{2\pi\sigma}{M_{0}},$ (25)

here

$m\frac{d\sigma}{dm}=-e_{0}\frac{mE}{3\pi}+\frac{m^{2}}{\pi}.$ (26)

Let’s $E_{m}$ is the solution of the equation (23) then using (23) and (25)
and accounting that $E_{m}$ and $\left\langle
0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle$ are constants we get

$\left\langle
0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle=-mE_{m}\frac{d\sigma}{dm}\left(\frac{d\sigma}{dE_{m}}\right)^{-1}$
(27)

or

$\left\langle
0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle=\left(3m^{2}-e_{0}mE_{m}\right)\frac{E_{m}}{e_{0}m+e^{2}_{0}E_{m}}$
(28)

In generally there are no analytical solutions (23) but we can find a few of
them in special cases if

  1. 1.

$m=0$, then $E_{m}=0$ and the chiral condensate disappears,

  2. 2.

$e_{0}\ll 1$, then
$E_{M}\simeq\cfrac{e_{0}m}{6\pi}\ln\cfrac{m^{2}}{M^{2}_{0}}$ and

$\left\langle
0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle=\frac{m^{2}}{2\pi}\ln\frac{m^{2}}{M^{2}_{0}}\left(1-\frac{e_{0}m}{18\pi}\ln\frac{m^{2}}{M^{2}_{0}}\right),$

here $\left|\cfrac{e_{0}m}{6\pi}\ln\cfrac{m^{2}}{M^{2}_{0}}\right|\leqslant
1.$

It is worth to note that $E$ may be a constant quantity. Really, fixing a
gauge $A_{1}=0$ and using the fact that the Coulomb potential is a leaner one
$A_{0}=-ax$,we get $E=a$.

In Fig. 4 the effective Lagrangian and the chiral codensate are shown as the
function of $E$.

Figure 4: a – The value $L_{\text{eff}}(\phi)$; b – Chiral condensate
$R=\left\langle 0|g\bar{\psi}^{a}\psi^{a}|0\right\rangle_{R}$ at $e_{0}=1$,
$m=0,5$ MeV, $M_{0}^{2}=100\text{ MeV}^{2}$.

##  IV Conclusions

The models analyzed in this paper, formulated in two dimensional spacetime,
are unrealistic. However, we believe that obtained results are showing the
method calculation of the chiral condensate in more realistic models.

It was shown that the massive fermions in the massive scalar field have the
effective potential with two different vacuums. One of them is global the
other is local minima. The chiral condensate obtained in the model does not
disappear if the fermion mass tends to zero.

The trace of the energy-momentum tensor taking into account the mass of
fermions demonstrates a clear violation of discrete symmetry. This fact
allowed to construct the effective Lagrangian in QED2. The chiral condensate
was obtained in the model. If the fermion mass vanishes, then the chiral
condensate disappears.

Technically, the central point is the construction of the effective Lagrangian
in QED2.

Acknowledgment: I am grateful to O.V. Kancheli for useful discussions.

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