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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

cosmic string.

The possible existence of a light pseudo scalar particle

is a very interesting possibility. For example, the axion [1]

- [3], 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 [4], [5].

Related to that, in a series of recent publications [6]

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 [7].

Together with this, the cosmic string solutions which

contain a magnetic flux in their core have been exten-

sively studied [8]. 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 [9].

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-

tremely prominent.

To see this, let us write the Lagrangian describing the

relevant light pseudoscalar coupling to the photon,

L = −1

4FµνFµν+1

−g

2∂µφ∂µφ −1

2m2

aφ2−

8φ?µναβFµνFαβ,

(1)

and, following Ref. [10] (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

independent.

∗Electronic address: guendel@bgu.ac.il

†Electronic address: silon@bgu.ac.il

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

LI= −βφEz,

(2)

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

L2=1

2∂µA∂µA +1

2∂µφ∂µφ −1

2m2

aφ2+ βφ∂tA ,

(3)

where A is the z-polarization of the photon, so that

Ez= −∂tA.

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:

LI=1

2β(φ∂tA − A∂tφ) .

(4)

Defining now the axion-photon complex field, Ψ, as

Ψ =

1

√2(φ + iA) (5)

and plugging this into the Lagrangian results in

L = ∂µΨ∗∂µΨ −i

2β(Ψ∗∂tΨ − Ψ∂tΨ∗) ,

(6)

arXiv:0810.4665v1 [hep-th] 26 Oct 2008

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where Ψ∗is the charge conjugation of Ψ. From this we

obtain the equation of motion for Ψ

∂µ∂µΨ + iβ∂tΨ = 0 .

(7)

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,

(8)

?k2φ(?k) + gB0ωψ(0) = ω2φ(?k) ,

(9)

yields the solution

φ(?k) = −gB0ωψ(0)

?k2− ω2

.

(10)

Following Thorn [12], 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

?

d2k

(2π)2φ(?k) = ψ(0) = −

?

gB0ωψ(0)

?k2− ω2

d2k

(2π)2,

(11)

to obtain an eigenvalue condition on ω2

1 = −gB0ω

4π2

?

d2k

?k2− ω2.

(12)

In order to stop this integral from diverging we intro-

duce a cutoff at |?k| = Λ. Hence,

1 = −gB0ω

4π

ln

?

1 −Λ2

ω2

?

≈ −gB0ω

4π

ln

?

−Λ2

ω2

?

, (13)

where in the last step we assume Λ ? |ω|. By manip-

ulating the latter to the form

2π

B0gωexp

?

2π

B0gω

?

=

2πi

B0gΛ,

(14)

we find that ω is described by Lambert’s W function

[11]

ω =

2π

?

B0g W

2πi

B0gΛ

? ,

(15)

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 [13]. 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 [9] 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.

5.?10?15

1.?10?14

1.5?10?14

Η ?GeV?1?

?2.?1015

?1.?1015

1.?1015

2.?1015

Ω ?GeV?

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

lim

η→∞Re[ω] → 0 ,

Hence, the imaginary part of ω is bounded from above

by −Λ, rendering the instability extremely strong, for

lim

η→∞Im[ω] → −Λ .

(16)

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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

string theory.

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

z.

1 = −gB0

?ω2− k2

z

4π

ln

?

−

Λ2

ω2− k2

z

?

(17)

and we find ω to be

ω =

4π2

(B0g)2W2?

2πi

B0gΛ

? + k2

z

1/2

.

(18)

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

[14]. 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 [15], thus mak-

ing the m = 0 assumption we made even more valid.

Acknowledgment

We would like to thank Doron Chelouche for helpful

discussions on the subject.

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