Instability of Axions and Photons In The Presence of Cosmic Strings
Eduardo I. Guendelman∗and Idan Shilon†
Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
We report that axions and photons exhibit instability in the presence of cosmic strings that are
carrying magnetic flux in their core. The strength of the instability is determined by the symmetry
breaking scale of the cosmic string theory. This result would be evident in gamma ray bursts and
axions emanating from the cosmic string. These effects will eventually lead to evaporation of the
The possible existence of a light pseudo scalar particle
is a very interesting possibility. For example, the axion 
- , which was introduced in order to solve the strong
CP problem, has since then also been postulated as a
candidate for the dark matter. A great number of ideas
and experiments for the search this of particle have been
proposed , .
Related to that, in a series of recent publications 
one of us showed that an axion-photon system displays
a continuous axion - photon duality symmetry when an
external magnetic field is present and when the axion
mass is neglected. This allows one to analyze the behav-
ior of axions and photons in external magnetic fields in
terms of an axion-photon complex particle. For example,
the deflection of light from magnestars has been recently
studied using these techniques .
Together with this, the cosmic string solutions which
contain a magnetic flux in their core have been exten-
sively studied . In particular, we concentrate here on
cosmic strings that are generated by breaking of a local
U(1) symmetry in the abelian Higgs model. This admits
string solutions in the form of the Nielsen and Olesen
vortex lines .
In this letter we show that the coupling of axion-photon
complex particles to the magnetic flux of the cosmic
string renders the cosmic string unstable. This results
in strong gamma ray bursts away from the cosmic string,
which would make the existence of cosmic strings ex-
To see this, let us write the Lagrangian describing the
relevant light pseudoscalar coupling to the photon,
L = −1
and, following Ref.  (and references therein), spe-
cialize to the case where we consider an electromagnetic
field with propagation along the x and y directions and
a strong magnetic field pointing in the z-direction to be
present. The magnetic field may have an arbitrary space
dependence in x and y, but it is assumed to be time
∗Electronic address: firstname.lastname@example.org
†Electronic address: email@example.com
For the small perturbations, we consider only small
quadratic terms in the Lagrangian for the axion and the
electromagnetic fields, but now considering a static mag-
netic field pointing in the z direction having an arbitrary
x and y dependence and specializing to x and y depen-
dent electromagnetic field perturbations and axion fields.
This means that the interaction between the background
field and the axion and photon fields reduces to
where β = gB(x,y). Choosing the temporal gauge
for the photon excitations and considering only the z-
polarization for the electromagnetic waves (since only
this polarization couples to the axion) we get the fol-
lowing 2+1 dimensional effective Lagrangian
aφ2+ βφ∂tA ,
where A is the z-polarization of the photon, so that
Without assuming any particular x and y-dependence
for β, but still insisting that it will be static, we see that
in the ma= 0 case (the validity of this assumption will be
discussed at the end of this report), we discover a contin-
uous axion photon duality symmetry. This is due to a ro-
tational O(2) symmetry in the axion-photon field space,
allowed by the axion and photon kinetic terms, and also
since the interaction term, after dropping a total time
derivative, can also be expressed in an O(2) symmetric
way as follows:
2β(φ∂tA − A∂tφ) .
Defining now the axion-photon complex field, Ψ, as
√2(φ + iA) (5)
and plugging this into the Lagrangian results in
L = ∂µΨ∗∂µΨ −i
2β(Ψ∗∂tΨ − Ψ∂tΨ∗) ,
arXiv:0810.4665v1 [hep-th] 26 Oct 2008
where Ψ∗is the charge conjugation of Ψ. From this we
obtain the equation of motion for Ψ
∂µ∂µΨ + iβ∂tΨ = 0 .
Writing separately the time and space dependence of
axion-photon field as Ψ = e−iωtψ(? x) and considering the
magnetic field of an infinitely thin cosmic string lying
along the z axis, reduces Eq. (7) to
[−?∇2+ gB0ωδ(x)δ(y)]ψ(? x) = ω2ψ(? x) ,
where B0is magnetic flux of the cosmic string. Trans-
forming to Fourier space,
?k2φ(?k) + gB0ωψ(0) = ω2φ(?k) ,
yields the solution
φ(?k) = −gB0ωψ(0)
Following Thorn , who solved a similar problem of
a non relativistic Schr¨ odinger equation with a two dimen-
sional delta function, we integrate the latter over?k
(2π)2φ(?k) = ψ(0) = −
to obtain an eigenvalue condition on ω2
1 = −gB0ω
In order to stop this integral from diverging we intro-
duce a cutoff at |?k| = Λ. Hence,
1 = −gB0ω
where in the last step we assume Λ ? |ω|. By manip-
ulating the latter to the form
we find that ω is described by Lambert’s W function
where W(z) satisfies z = W(z)eW(z). Since the W
function has an imaginary argument, ω must be a com-
plex function. Therefore, the axion-photon complex par-
ticles will excess a (time) instability which will result in
axion and photon bursts away from the cosmic string,
thus making the string unstable.
Turning now to estimate the strength of the instabil-
ity, we denote the term B0g/2π by η and evaluate the
magnitudes of η and Λ.
Recent results from the CAST collaboration, that
searches for solar produced axions, has set an upper limit
on the magnitude of the axion-photon coupling constant
of g < 8.8 × 10−11GeV−1for an axion mass of ma ?
0.02 eV . Along with this, in Planck units the mag-
netic flux of a cosmic string is given by B0= 2πn/√α,
where n is an integer, so called the string winding num-
ber. Therefore, η = gn/√α ? n × 10−9GeV−1.
To evaluate the order of the cutoff, Λ, we understand
from dimensional analysis that it is inversely propor-
tional to the only length scale of the system, which is
the cosmic string radius. Studying the structure of vor-
tex lines, Nielsen and Olesen  showed that the string
width is inversely proportional to the symmetry breaking
scale of the theory. Thus, for GUT scale strings, we take
Λ ∼ 1015GeV.
The real and imaginary parts of ω as functions of η are
depicted, for Λ = 1015GeV, in Fig. 1.
FIG. 1: The real and imaginary parts of ω, for Λ = 1015GeV.
The real part is described by the solid line, while the dashed
line represents the imaginary part of ω.
Im[ω] goes to −Λ already at the scale η ∼ 1/Λ, since
Im[1/W(1.3i)] ≈ −1. On the other hand, it would require
a very large value of η in order to make the real part of ω
negligible (which could be misleading from the graph) since
only for η ∼ 10 GeV−1, with Λ ∼ 1015GeV, do we get
Re[ω] ∼ 10−3GeV. For the largest allowed value of η (with
n = 1), Re[ω] ≈ 8.65 × 107GeV.
One notes that
Taking the limit η → ∞, the asymptotic values of the
real and imaginary parts of ω are given by
η→∞Re[ω] → 0 ,
Hence, the imaginary part of ω is bounded from above
by −Λ, rendering the instability extremely strong, for
η→∞Im[ω] → −Λ .
a cutoff of the GUT scale. In particular, this analysis
shows that the instability strength is independent of the
axion-photon coupling constant magnitude, but is rather
determined by the symmetry breaking scale of the cosmic
The effective energy of the axion-photon particles is
given by ω2. From the properties of ω, we see that the
particles effective energy will be complex as well, with
a very large and negative real part (in the order −1 ×
1015GeV2) and a negligible imaginary part (in the order
10−8GeV2). Hence, as long as the axion-photon particles
are localized around the string they will be in a bound
state. However, the perturbation will exponentially grow
in time eventually causing the cosmic string to evaporate.
The results we have obtained so far are valid for a par-
ticle moving on a plane perpendicular to the string axis.
We now claim that the same conclusions will be obtained
for the more general case of a wave function ψ(? x) with a
general momentum as well. To address the general situa-
tion, we write Eq. (13) in an invariant form with respect
to boosts in the z direction. The electromagnetic field
is unchanged by this transformation since the boost is
pointing along the direction of the magnetic field. The
quantity that will transform under the boost is ω. How-
ever, there is an invariant quantity in the form of ω2−k2
Thus, Eq. (13) in an arbitrary frame becomes
1 = −gB0
and we find ω to be
? + k2
Therefore, ω is complex for any kz and our previous
results appear in all modes.
Lastly, we turn to discuss the validity of the ma= 0 ap-
proximation we made at the beginning of this letter and
show that our conclusions are valid for massive axions as
well. In order to verify the massless axion approxima-
tion we compare the Compton wavelength of the axion,
∼ 1/ma, with the localization length of the axion-photon
particles wave function. The axion mass is known to be
restricted to the region 3 × 10−3eV > ma ? 10−6eV
. Along with this, as one can see from the explicit so-
lution, the wave funcion will be localized around the cos-
mic string with a size |1/√−ω2|, which is much smaller
than the compton wavelength of the axion and therefore
making the axion mass irrelevant to our problem.
In conclusion, we have shown that axionic and electro-
magnetic excitations will be extremely unstable in the
presence of a magnetic flux carrying cosmic string. The
axions and the photons will be trapped in a bound state
as long as they are localized in the vicinity of the string,
but as the perturbation becomes significant extremely
rapidly, strong axion and gamma ray bursts will quickly
emanate from the string, taking their energy from it and
thus bringing its existence to an early end. We also note
that our conclusions are true for any value of the axion-
photon coupling constant and are determined solely by
the symmetry breaking scale of the cosmic string theory.
The same conclusions are of course valid as well for
axion like particles, which have no lower bound on their
mass and are usually considered very light , thus mak-
ing the m = 0 assumption we made even more valid.
We would like to thank Doron Chelouche for helpful
discussions on the subject.
 R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38, 1440
 S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
 F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
 For a very early proposal see: J.T. Goldman and C.M.
Hoffman, Phys. Rev. Lett. 40, 220 (1978).
 For a review see: G.G. Raffelt, arXiv: hep-ph/0611118.
 E.I. Guendelman, Mod. Phys. Lett. A 23 191 (2008),
arXiv: 0711.3685 [hep-th]; E.I. Guendelman, Phys. Lett.
B 662 227 (2008), arXiv: 0801.0503 [hep-th]; E.I. Guen-
delman, Phys. Lett. B 662 445 (2008), arXiv: 0802.0311
 D. Chelouche and E.I. Guendelman, arXiv: 0810.3002
 A. Vilenkin and E.P.S. Shellard, Cosmic Strings and
Other Topological Defects, (Cambridge University Press,
 H.B. Nielsen and P. Olesen, Nucl. Phys. B 61 45 (1973).
 E.I. Guendelman and I. Shilon, arXiv: 0808.2572 [hep-th]
 R.M. Corless, G.H. Gonnet, D.E.G Hare, D.J. Jeffrey
and D.E. Knuth, Adv. in Comp. Math. 5 1 (1996).
 C. Thorn, Phys. Rev. D 19 639 (1979).
 S. Andriamonje et al., J. Cosmol. Astropart. Phys. 04
010 (2007), arXiv: hep-ex/0702006.
 P .Sikivie, Nucl. Phys. Proc. Suppl. 87 41 (2000), arXiv:
 A. De Angelis, O. Mansutti and M. Roncandelli, Phys.
Rev. D 76 121301 (2007), arXiv:0707.4312[astro-ph].