Instability of Axions and Photons In The Presence of Cosmic Strings
IInstability 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 arecarrying magnetic flux in their core. The strength of the instability is determined by the symmetrybreaking scale of the cosmic string theory. This result would be evident in gamma ray bursts andaxions emanating from the cosmic string. These effects will eventually lead to evaporation of thecosmic string.
The possible existence of a light pseudo scalar particleis a very interesting possibility. For example, the axion [1]- [3], which was introduced in order to solve the strongCP problem, has since then also been postulated as acandidate for the dark matter. A great number of ideasand experiments for the search this of particle have beenproposed [4], [5].Related to that, in a series of recent publications [6]one of us showed that an axion-photon system displaysa continuous axion - photon duality symmetry when anexternal magnetic field is present and when the axionmass is neglected. This allows one to analyze the behav-ior of axions and photons in external magnetic fields interms of an axion-photon complex particle. For example,the deflection of light from magnestars has been recentlystudied using these techniques [7].Together with this, the cosmic string solutions whichcontain a magnetic flux in their core have been exten-sively studied [8]. In particular, we concentrate here oncosmic strings that are generated by breaking of a local U (1) symmetry in the abelian Higgs model. This admitsstring solutions in the form of the Nielsen and Olesenvortex lines [9].In this letter we show that the coupling of axion-photoncomplex particles to the magnetic flux of the cosmicstring renders the cosmic string unstable. This resultsin 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 therelevant light pseudoscalar coupling to the photon, L = − F µν F µν + ∂ µ φ∂ µ φ − m a φ −− g φ(cid:15) µναβ F µν F αβ , (1)and, following Ref. [10] (and references therein), spe-cialize to the case where we consider an electromagneticfield with propagation along the x and y directions anda strong magnetic field pointing in the z -direction to bepresent. The magnetic field may have an arbitrary spacedependence in x and y , but it is assumed to be timeindependent. ∗ Electronic address: [email protected] † Electronic address: [email protected]
For the small perturbations, we consider only smallquadratic terms in the Lagrangian for the axion and theelectromagnetic 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 backgroundfield and the axion and photon fields reduces to L I = − βφE z , (2)where β = gB ( x, y ). Choosing the temporal gaugefor the photon excitations and considering only the z -polarization for the electromagnetic waves (since onlythis polarization couples to the axion) we get the fol-lowing 2+1 dimensional effective Lagrangian L = 12 ∂ µ A∂ µ A + 12 ∂ µ φ∂ µ φ − m a φ + βφ∂ t A , (3)where A is the z -polarization of the photon, so that E z = − ∂ t A .Without assuming any particular x and y -dependencefor β , but still insisting that it will be static, we see thatin the m a = 0 case (the validity of this assumption will bediscussed 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 alsosince the interaction term, after dropping a total timederivative, can also be expressed in an O (2) symmetricway as follows: L I = 12 β ( φ∂ t A − A∂ t φ ) . (4)Defining now the axion-photon complex field, Ψ, asΨ = 1 √ φ + iA ) (5)and plugging this into the Lagrangian results in L = ∂ µ Ψ ∗ ∂ µ Ψ − i β (Ψ ∗ ∂ t Ψ − Ψ ∂ t Ψ ∗ ) , (6) a r X i v : . [ h e p - t h ] O c t where Ψ ∗ is the charge conjugation of Ψ. From this weobtain the equation of motion for Ψ ∂ µ ∂ µ Ψ + iβ∂ t Ψ = 0 . (7)Writing separately the time and space dependence ofaxion-photon field as Ψ = e − iωt ψ ( (cid:126)x ) and considering themagnetic field of an infinitely thin cosmic string lyingalong the z axis, reduces Eq. (7) to[ − (cid:126) ∇ + gB ωδ ( x ) δ ( y )] ψ ( (cid:126)x ) = ω ψ ( (cid:126)x ) , (8)where B is magnetic flux of the cosmic string. Trans-forming to Fourier space, (cid:126)k φ ( (cid:126)k ) + gB ωψ (0) = ω φ ( (cid:126)k ) , (9)yields the solution φ ( (cid:126)k ) = − gB ωψ (0) (cid:126)k − ω . (10)Following Thorn [12], who solved a similar problem ofa non relativistic Schr¨odinger equation with a two dimen-sional delta function, we integrate the latter over (cid:126)k (cid:90) d k (2 π ) φ ( (cid:126)k ) = ψ (0) = − (cid:90) gB ωψ (0) (cid:126)k − ω d k (2 π ) , (11)to obtain an eigenvalue condition on ω − gB ω π (cid:90) d k(cid:126)k − ω . (12)In order to stop this integral from diverging we intro-duce a cutoff at | (cid:126)k | = Λ. Hence,1 = − gB ω π ln (cid:18) − Λ ω (cid:19) ≈ − gB ω π ln (cid:18) − Λ ω (cid:19) , (13)where in the last step we assume Λ (cid:29) | ω | . By manip-ulating the latter to the form2 πB gω exp (cid:18) πB gω (cid:19) = 2 πiB g Λ , (14)we find that ω is described by Lambert’s W function[11] ω = 2 πB g W (cid:16) πiB g Λ (cid:17) , (15) where W ( z ) satisfies z = W ( z )e W ( 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 inaxion 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 B g/ π by η and evaluate themagnitudes of η and Λ.Recent results from the CAST collaboration, thatsearches for solar produced axions, has set an upper limiton the magnitude of the axion-photon coupling constantof g < . × − GeV − for an axion mass of m a (cid:46) .
02 eV [13]. Along with this, in Planck units the mag-netic flux of a cosmic string is given by B = 2 πn/ √ α ,where n is an integer, so called the string winding num-ber. Therefore, η = gn/ √ α (cid:46) n × − GeV − .To evaluate the order of the cutoff, Λ, we understandfrom dimensional analysis that it is inversely propor-tional to the only length scale of the system, which isthe cosmic string radius. Studying the structure of vor-tex lines, Nielsen and Olesen [9] showed that the stringwidth is inversely proportional to the symmetry breakingscale of the theory. Thus, for GUT scale strings, we takeΛ ∼ GeV.The real and imaginary parts of ω as functions of η aredepicted, for Λ = 10 GeV, in Fig. 1. (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) Η (cid:64) GeV (cid:45) (cid:68) (cid:45) (cid:180) (cid:45) (cid:180) (cid:180) (cid:180) Ω (cid:64) GeV (cid:68)
FIG. 1: The real and imaginary parts of ω , for Λ = 10 GeV.The real part is described by the solid line, while the dashedline represents the imaginary part of ω . One notes thatIm[ ω ] goes to − Λ already at the scale η ∼ / Λ, sinceIm[1 /W (1 . i )] ≈ −
1. On the other hand, it would requirea very large value of η in order to make the real part of ω negligible (which could be misleading from the graph) sinceonly for η ∼
10 GeV − , with Λ ∼ GeV, do we getRe[ ω ] ∼ − GeV. For the largest allowed value of η (with n = 1), Re[ ω ] ≈ . × GeV.
Taking the limit η → ∞ , the asymptotic values of thereal and imaginary parts of ω are given bylim η →∞ Re[ ω ] → , lim η →∞ Im[ ω ] → − Λ . (16)Hence, the imaginary part of ω is bounded from aboveby − Λ, rendering the instability extremely strong, fora cutoff of the GUT scale. In particular, this analysisshows that the instability strength is independent of theaxion-photon coupling constant magnitude, but is ratherdetermined by the symmetry breaking scale of the cosmicstring theory.The effective energy of the axion-photon particles isgiven by ω . From the properties of ω , we see that theparticles effective energy will be complex as well, witha very large and negative real part (in the order − × GeV ) and a negligible imaginary part (in the order10 − GeV ). Hence, as long as the axion-photon particlesare localized around the string they will be in a boundstate. However, the perturbation will exponentially growin 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 obtainedfor the more general case of a wave function ψ ( (cid:126)x ) with ageneral momentum as well. To address the general situa-tion, we write Eq. (13) in an invariant form with respectto boosts in the z direction. The electromagnetic fieldis unchanged by this transformation since the boost ispointing along the direction of the magnetic field. Thequantity that will transform under the boost is ω . How-ever, there is an invariant quantity in the form of ω − k z .Thus, Eq. (13) in an arbitrary frame becomes1 = − gB (cid:112) ω − k z π ln (cid:18) − Λ ω − k z (cid:19) (17)and we find ω to be ω = π ( B g ) W (cid:16) πiB g Λ (cid:17) + k z / . (18)Therefore, ω is complex for any k z and our previous results appear in all modes.Lastly, we turn to discuss the validity of the m a = 0 ap-proximation we made at the beginning of this letter andshow that our conclusions are valid for massive axions aswell. In order to verify the massless axion approxima-tion we compare the Compton wavelength of the axion, ∼ /m a , with the localization length of the axion-photonparticles wave function. The axion mass is known to berestricted to the region 3 × − eV > m a (cid:38) − eV[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 | / √− ω | , which is much smallerthan the compton wavelength of the axion and thereforemaking the axion mass irrelevant to our problem.In conclusion, we have shown that axionic and electro-magnetic excitations will be extremely unstable in thepresence of a magnetic flux carrying cosmic string. Theaxions and the photons will be trapped in a bound stateas long as they are localized in the vicinity of the string,but as the perturbation becomes significant extremelyrapidly, strong axion and gamma ray bursts will quicklyemanate from the string, taking their energy from it andthus bringing its existence to an early end. We also notethat our conclusions are true for any value of the axion-photon coupling constant and are determined solely bythe symmetry breaking scale of the cosmic string theory.The same conclusions are of course valid as well foraxion like particles, which have no lower bound on theirmass and are usually considered very light [15], thus mak-ing the m = 0 assumption we made even more valid. Acknowledgment
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