#### QCD axion star collapse with the chiral potential

Received: February
QCD axion star collapse with the chiral potential
Joshua Eby 0 1 2 3
Madelyn Leembruggen 0 2 3
Peter Suranyi 0 2 3
L.C.R. Wijewardhana 0 2 3
Open Access 0 3
c The Authors. 0 3
0 2600 Clifton Ave , Cincinnati, OH, 45221 U.S.A
1 Fermi National Accelerator Laboratory
2 Department of Physics, University of Cincinnati
3 [16] E. Braaten, A. Mohapatra and H. Zhang , Dense Axion Stars
In a previous work, we analyzed collapsing axion stars using the low-energy instanton potential, showing that the total energy is always bounded and that collapsing axion stars do not form black holes. In this paper, we provide a proof that the conclusions are unchanged when using instead the more general chiral potential for QCD axions.
Classical Theories of Gravity; Cosmology of Theories beyond the SM
1 Introduction 2 3 4
Review of the variational method
The instanton potential
The chiral potential
3.1 Integral representation Boundedness of the energy Conclusions 1
Introduction
Axion stars [1–5], bound states of condensed axion particles [6–13], have recently received
increased interest in the literature [14–16]. Such states can have important effects in the
context of cosmology and astrophysics, as well as fundamental particle physics [17–27].
QCD axions, which are connected with the strong nuclear force, were originally proposed
to solve the Strong CP problem; such theories have a well-defined parameter space with
a single free parameter, the axion mass. The masses of dilute QCD axion stars have an
upper bound such that they are much lighter than an ordinary star [5, 14, 15], though
alternative axion theories can exceed this bound.
The possibility of a dense state for axion stars was first developed in [16]. If they exist,
dense states can lead to novel astrophysical consequences, including new sources of photon
emission through, for example, accretion of hydrogen into dense configurations [28]. The
existence of these states depends on both attractive and repulsive interaction terms present
in the axion potential. The well-studied dilute state, which is a metastable minimum of
the energy, exists only for particle number N < Nc, a critical value. However, at any
particle number N in the condensate, the global minimum of the energy is at a radius
many orders of magnitude smaller than that of the dilute state (though still larger than
the corresponding Schwarzschild radius).
Lately, a significant amount of work has been done analyzing the stability properties
of dilute axion stars, both in the context of gravitational collapse and decay through
relativistic axion emission [29–35]. In a previous work [32], we analyzed the collapse of an
axion star using a variational method for finding energetically stable bound states. This
analysis built on the formalism of [31] (itself built on the previous work of [5, 36, 37]),
who argued that boson stars with attractive self-interactions collapse to black holes. Our
work [32], which included both attractive and repulsive terms in the interaction potential
for the axion field A [38, 39],1
VI(A) = m2 f 2 1 − cos
concluded that axion stars do not collapse to form black holes. This was due to the existence
of a dense state which is larger than the corresponding Schwarzschild radius. Further,
number changing interactions inside the axion star [30, 35] may cause the collapsing axion
star to emit a significant fraction of its energy in the form of relativistic free axions [32–34].
Our analysis in [32] made use of a common approximate form of the axion potential,
sometimes called the “instanton potential”, eq. (1.1) [38, 39]. It is known, however, that
the system is more precisely described by the so-called “chiral potential” [39, 40]
1 + z −
1 + z2 + 2 z cos
The latter form takes into account the effect of nonzero light quark masses through the
expansion, it is possible that our conclusions (the boundedness of the energy, the existence
of a dense global energy minimum) would have been different if we had used the more
precise form VC. We thus revisit the analysis with the improved potential here.
There is a second potentially limiting assumption in the analysis of [32]. We used a
variational method to minimize the energy, which gives good agreement with other, more
precise analyses [4, 14, 16, 41] in analyzing bound states of axion stars. This method
non-relativistic expansion [15, 42]
A(t, r) = √
e−i m t ψ(r) + ei m t ψ∗(r) .
In that work, we analyzed in detail a Gaussian ansatz,
and also presented the analogous result for a cos2 ansatz,
r ≤ R
We presented both to provide evidence that our results were not an artifact of some
particular choice of ansatz. Indeed, these two general classes, those functions which extend to
1m and f are the mass and decay constant, respectively, of the underlying axion field theory. For QCD
The total axion star energy, in the non-relativistic and weak-binding limits, can be written
E =
d3r |∇ψ|2 + W (ψ) +
2 Vgrav |ψ|2
non-relativistic limit of eq. (1.1) or eq. (1.2)), and
x − x|
or have compact support on a finite radius R.
We will use the generic scaling of macroscopic parameters derived in [14]
port (as does cos2), are those well-behaved enough to be considered reasonable choices for
a ground-state axion star wavefunction. Nonetheless, it could be asked whether anything
changes upon assuming only a general, well-behaved ansatz in one of these two classes. We
thus revisit this question here as well.
This paper is organized as follows. We review the variational method in section 2,
and use it to expand the chiral potential in a tractable form in section 3. We examine the
boundedness of the energy in section 4, and conclude in section 5.
Review of the variational method
2 C2 ρ2 − 2 C22 ρ − ρ3
6 Z
R =
m √
N =
1019 GeV. Then a generic ansatz can be written as
Then the total energy can be written as
is a rescaled self-interaction energy, and where we defined the functions
For a given ansatz, the functions B4, Ck, and D2 are numerical constants.
The instanton potential
In [32], we used the instanton potential of eq. (1.1) to investigate the bound states in the
axion star energy. In that case the rescaled functional v was calculated in closed form,
both as an integral and also as an infinite series:
= X
− 2 C2
[(k + 2)!]2
We used these simplified expressions to show that the total axion star energy was always
bounded from below, and that the global minimum of the energy in eq. (2.5) was at a
radius much larger than the corresponding Schwarzschild radius.
A similar analysis, using the chiral potential of eq. (1.2), is somewhat more challenging.
However, making use of some convenient analytic properties of the self-interaction energy
makes the problem tractable. We present such an analysis in the next section.
The chiral potential
The potential we wish to consider has the form [39, 40]
m2 f 2 ≡ V =
1 + z −
1 + z2 + 2 z cos
Using the particle data group values of
we find that
To recover the form of the instanton potential in eq. (1.1), one merely needs to take the
z → 0 limit, which gives V = 1 − cos(A/f ).
mu = 2.15 ± 0.15 MeV
md = 4.70 ± 0.20 MeV
z =
≃ 0.457.
+ + A−, where A
+ (A−) creates
(annihilates) one axion in the ground state wavefunction. Expanding eq. (3.1) in a power
series of the cosine, we have
V =
1 + z − p1 + z2 X
Now noting that in leading order of N , A
− commute,
Expand the jth power of the cosine,
ei A+/f ei A−/f + e−i A+/f e−i A−/f .
ei (j−2 k)A+/f ei (j−2 k)A−/f
J0 (j − 2 k)
X (j − 2 k)2 m(−1)m
+ − !#
+ −
where in the second step, we expanded the exponentials and kept only those terms with
equal numbers of A
interaction energy density as follows:
V =
1 + z − p1 + z2 X
J0 (j − 2 k)
+ − !#
Using the correspondence with the non-relativistic Gross-Pit¨aevskii approach [32], as
defined by the expansion in eq. (1.3),
A A
+ −
= √
Integral representation
Our aim in this section is to perform the summations over j and k in eq. (3.6) at the price
of introducing integrals, since the latter is easier to estimate. Using the ansatz of eq. (2.3),
the rescaled self-interaction energy, defined in eq. (2.6), is
so that the expression in the energy functional evaluates to
= −
1 − J0 h(j − 2k)√2 ζ F (ξ)i −
1 − J0 h(j − 2 k) √2 ζ F (ξ)i − (j − 2 k)2
First, we will perform the summation over k by using the standard integral representation
J0[(j − 2 k) u] =
ei (j−2 k) u sin tdt
for J0. Substituting (3.10) into (3.9) we can perform the summation over k, yielding
1 Z π
[cos(√2 ζ F (ξ) sin t)]j dt − j ζ2 F (ξ)2
Now the summation over j can also be performed, to yield the following expression for the
energy functional
I[√2 ζ F (ξ)] ξ2 dξ + 1 ,
I(u) =
h1 + z − p1 + z2 + 2 z cos(u sin t)i dt.
it is monotonic, being minimized at z → 0. But the z → 0 limit of I(u) is nothing but the
rescaled instanton potential
zli→m0 I(u) = 1 − J0(u).
The boundedness of the instanton potential therefore implies the boundedness of the chiral
potential. It thus suffices to show that the instanton self-interaction potential is bounded
from below. We showed exactly this in the appendix of [32], but only for the Gaussian
ansatz. For completeness, we prove the general case in the next section.
In [32], the proof that the axion star energy had a dense global minimum depended on
the boundedness of the axion self-interaction potential, which we were able to show in the
specific case of a Gaussian wavefunction. This proof can be generalized to any wavefunction
2 We also assume that F (ξ)
is monotonically decreasing, though even that condition could be relaxed. This could be
relevant, for example, for rotating axion stars, whose wavefunctions have nodes at the
In this section we prove, under general conditions on the ansatz, that the
self-interac
The general form of ansa¨tze for the wave function, which we and other authors use
is of the form in eq. (2.3). As shown in eq. (2.5), such an ansatz leads to the following
equation for the self-interaction energy
Then the integral we evaluate is
The last term of K
is always finite, since we must have3
K =
1 Z
Z ∞
K =
convergent for a normalizable wavefunction. Consequently, we can restrict ourselves to
1 Z
dξ ξ2 n1 − J0[√2 ζ F (ξ)]o .
value taken at √
have the inequality
1 − J0[ 2 ζ F (ξ)] ≤ B ≡ 1 − J0(j1,1) = 1.40276.
2To our knowledge, all relevant ansa¨tze used by other authors fall into one of these categories as well.
Let us consider ansa¨tze in the first class (a). In that case, using the bound B in
eq. (4.6), we have
K =
1 Z 1
dξ ξ2 n1 − J0[√2 ζ F (ξ)]o
Now consider ansa¨tze in class (b). We break up integral K′ of eq. (4.5) such that in
certainly have
Then it is easy to see that there is an a > 0 such that
Now consider integral
K2′ =
1 Z ∞
dξ ξ2 n1 − J0[√2 ζ F (ξ)]o .
As √
2 ζ F (ξ) ≪ 1 at all ξ. Then we can safely
expand the Bessel function and keep terms only up to the second order, to get
1 Z ∞
bounded. Then K
′ = K1 + K2 is also bounded.
We conclude that, for any generic,
monotonically decreasing ansatz for the axion star wavefunction, the instanton potential
self-energy is bounded below. Because the chiral potential, as a function of z, is bounded
below by the instanton potential, we conclude that the chiral potential is also bounded.
Conclusions
In this work, we have extended the analysis of [32] to two important cases, namely, to the
more precise chiral potential for axions, and to a generic ansatz for the wavefunction in
the variational method. In both of these cases, we have found that the general conclusions
of [32] hold: collapsing axion stars do not form black holes, but rather they are stabilized
in a dense state which is a global minimum of the axion star energy.
The analysis in this work, and the majority of [32], are performed in the non-relativistic
limit; we have neglected relativistic effects which lead to stimulated emission of axions
through number changing operators [30, 33, 34, 43, 44]. The possibility exists that these
dense states are unstable to emission of relativistic axions; we will return to this topic in a
future publication.
In previous work, Kolb and Tkachev [45] analyzed the formation of very dense
configurations, which they termed “axitons”, employing the instanton potential; similar
configurations were found by Levkov, Panin, and Tkachev [33] for the chiral potential. The stable,
dense solutions we find in our nonrelativistic analysis could well correspond to these same
been observed in [45] for the instanton potential, and more recently in [33] for the chiral
Acknowledgments
We thank P. Argyres, P. Fox, R. Gass, R. Harnik, A. Kagan, and M. Ma for conversations.
The work of J.E. was supported by the U.S. Department of Energy, Office of Science, Office
of Workforce Development for Teachers and Scientists, Office of Science Graduate Student
Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge
Institute for Science and Education for the DOE under contract number DE-SC0014664.
J.E. also thanks the Fermilab Theory Group for their hospitality.
Open Access.
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