---
abstract: 'The PVLAS collaboration has recently reported the observation of a rotation of the polarization plane of light propagating through a transverse static magnetic field. Such an effect can arise from the production of a light, $m_A\sim$ meV, pseudoscalar coupled to two photons with coupling strength $g_{A\gamma}\sim 5\times 10^{-6}$ GeV$^{-1}$. Here, we review these experimental findings, discuss how astrophysical and helioscope bounds on this coupling can be evaded, and emphasize some experimental proposals to test the scenario.'
address: 'Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D–22607 Hamburg, Germany'
author:
- Andreas Ringwald
title: '[-1cmDESY 05-229]{} Axion interpretation of the PVLAS data?[^1]'
---

There are various proposals in the literature in favour of the existence of light pseudoscalar particles beyond the Standard Model which have, so far, remained undetected, due to their weak coupling to ordinary matter. Such light particles would arise if there was a global continuous symmetry in the theory that is spontaneously broken in the vacuum. A well known example is the axion [@Weinberg:1978ma], which arises from a natural solution to the strong $CP$ problem. It appears as a pseudo Nambu-Goldstone boson of a spontaneously broken Peccei-Quinn symmetry [@Peccei:1977hh], whose scale $f_A$ determines its mass, ${m_A} = [z^{1/2}/(1+z)]\,
         m_\pi f_\pi/ f_A= { 0.6\,  {\rm meV}} 
         \times
         ( 
         10^{10}\, {\rm GeV}/{ f_A}  
        )
$ in terms of the mass $m_\pi$ and decay constant $f_\pi$ of the pion and the current quark mass ratio $z=m_u/m_d$. Only invisible axion models [@Kim:1979if; @Zhitnitsky:1980tq], where $f_A\gg 247$ GeV, are viable experimentally [@Eidelman:2004wy].

Clearly, it is of great interest to set stringent constraints on the properties of such light pseudoscalars. The interactions of axions and similar light pseudoscalars with Standard Model particles are model dependent, i.e. not a function of $1/f_A$ only. The most stringent constraints to date come from their coupling to photons, $g_{A\gamma}$, which arises via the axial anomaly [@Bardeen:1977bd], $$\label{eq:ax_ph}
        {\mathcal L}_{\rm int} = 
-\frac{1}{4}\,{ g_{A\gamma}}\,A\ F_{\mu\nu} \tilde{F}^{\mu\nu} 
=
 { g_{A\gamma}}\,A\ {\mathbf E}\cdot {\mathbf B}\, ;
        \hspace{5ex}
        { g_{A\gamma}} = -\frac{\alpha}{2\pi { f_A}} 
\left( { \frac{E}{N}} - \frac{2}{3}\,\frac{4+z}{1+z}\right)
\,,$$ where $A$ is the pseudoscalar field, $F_{\mu\nu}$ ($\tilde{F}^{\mu\nu}$) the (dual) electromagnetic field strength tensor, $\alpha$ the fine-structure constant, and $E/N$ the ratio of electromagnetic over color anomalies. As illustrated in Fig. \[fig:ax\_ph\], two quite distinct invisible axion models, namely the KSVZ [@Kim:1979if] (or hadronic) and the DFSZ [@Zhitnitsky:1980tq] (or grand unified) one, lead to quite similar $g_{A\gamma}$. The strongest constraints currently involve cosmological and astrophysical considerations. Only the laser experiments in Fig. \[fig:ax\_ph\] aim also at the production of axions in the laboratory.

![Exclusion region in mass $m_A$ vs. axion-photon coupling $g_{A\gamma}$ for various current and future experiments. The laser experiments [@Cameron:mr; @Zavattini:2005tm; @Ringwald:2001cp; @Ringwald:2003ns] aim at axion production and detection in the laboratory. The galactic dark matter experiments [@Eidelman:2004wy] exploit microwave cavities to detect axions under the assumption that axions are the dominant constituents of our galactic halo, and the solar experiments search for axions from the sun [@Andriamonje:2004hi]. The constraint from horizontal branch (HB) stars [@Eidelman:2004wy; @Raffelt:1999tx] arises from a consideration of stellar energy losses through axion production. \[fig:ax\_ph\]](ax_ph_lim_taup05.eps){width="15.5cm"}

Let us discuss such laser experiments in some detail. The most straightforward ones exploit photon regeneration. They are based on the idea [@Sikivie:ip] to send a polarized laser beam, with average power $\langle P\rangle$ and frequency $\omega$, along a superconducting dipole magnet of length $\ell$, such that the laser polarization is parallel to the magnetic field. In the latter, the photons may convert into axions via a Primakoff process. If another identical dipole magnet is set up in line with the first magnet, with a sufficiently thick wall between them to absorb the incident laser photons, then photons may be regenerated from the pure axion beam in the second magnet and detected with an efficiency $\epsilon$. The expected counting rate of such an experiment is given by $$\label{eq:ax_counting_rate}
\frac{{\rm d}N_\gamma}{{\rm d}t} =
{\frac{\langle P\rangle }{\omega}}\ 
\frac{N_r+2}{2}\,
\frac{1}{16} \left( g_{A\gamma}\,{B}\,\ell\right)^4 
\sin^2
\left( \frac{m_A^2\,\ell }{4\,\omega }
\right) 
\left( \frac{m_A^2\,\ell }{4\,\omega }
\right)^2  
\approx  { \frac{\langle P\rangle }{\omega}}\ 
\frac{N_r+2}{2}\,
\frac{1}{16} \left( g_{A\gamma}\,{B}\,\ell\right)^4 
\eta         
\,,$$ if one makes use of the possibility of putting the first magnet into an optical cavity with a total number $N_r$ of reflections. For $m_A \ll \sqrt{2\,\pi\,\omega/\ell} = 4\times 10^{-4}\, {\rm eV}
    \sqrt{({ \omega}/1\, {\rm eV})  (10\, {\rm m}/\ell )  }$, the approximate sign in (\[eq:ax\_counting\_rate\]) applies and the expected counting rate for a photon regeneration experiment is independent of the axion mass. A pilot photon regeneration experiment was performed by the Brookhaven-Fermilab-Rutherford-Trieste (BFRT) collaboration [@Cameron:mr]. It employed an optical laser of wavelength $\lambda =2\pi/\omega = 514$ nm and power $\langle P\rangle = 3$ W for $t=220$ minutes in an optical cavity with $N_r=200$, and used two superconducting dipole magnets with $B = 3.7$ T and $\ell = 4.4$ m. No signal of photon regeneration was found, which leads, taking into account a detection efficiency of $\eta =0.055$, to a $2\,\sigma$ upper limit of $g_{A\gamma}<6.7\times 10^{-7}$ GeV$^{-1}$ for axion-like pseudoscalars with mass $m_A< 10^{-3}$ eV.

Another possibility to probe $g_{a\gamma}$ is to measure changes in the polarization state when photons have traversed a transverse magnetic field [@Maiani:1986md]. In particular, the real production of axions leads to a rotation of the polarization plane of an initially linearly polarized laser beam by an angle $$\begin{aligned}
\label{ax_rot}
\epsilon &=& N_r\,\frac{g_{A\gamma}^2\,B^2\,\omega^2}{m_A^4}\,
\sin^2\left( \frac{m_A^2\,\ell}{4\,\omega}\right)\,
\sin 2\,\theta
\approx \frac{N_r}{16} \left( g_{A\gamma}\,B\,\ell \right)^2\,\sin 2\,\theta 
\,,\end{aligned}$$ where $\theta$ is the angle between the light polarization direction and the magnetic field component normal to the light propagation vector. The BFRT collaboration has also performed a pilot polarization experiment along these lines, with the same laser and magnets described before. For $\ell = 8.8$ m, $B=2$ T, and $N_r=254$, an upper limit on the rotation angle $\epsilon< 3.5\times 10^{-10}$ rad was set, leading to a limit $g_{A\gamma}< 3.6\times 10^{-7}$ GeV$^{-1}$ at the 95% confidence level, provided $m_A<1$ meV [@Cameron:mr]. Similar limits have been set from the absence of ellipticity in the transmitted beam. The overall envelope of the constraints from the BFRT collaboration [@Cameron:mr] is shown in Fig. \[fig:ax\_ph\] and labelled by “Laser (BFRT)” (cf. Ref. [@Eidelman:2004wy]).

Recently, the PVLAS experiment [@Zavattini:2005tm], consisting of a Fabry-Pérot cavity of very high finesse ($N_r\approx 44\, 000$), immersed in a magnetic dipole with $\ell =1$ m and $B=5$ T, reported the observation of a rotation of the polarization plane of light propagating through a transverse static magnetic field [@Zavattini:2005tm]. If interpreted in terms of the production of a light neutral pseudoscalar, the PVLAS collaboration finds a region $1.7\times 10^{-6}\, {\rm GeV}^{-1}\,{\mbox{\,\raisebox{.3ex}
    {$<$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}\,}}\,g_{A\gamma}\,{\mbox{\,\raisebox{.3ex}
    {$<$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}\,}}\,1.0\times 10^{-5}\, {\rm GeV}^{-1}$ for $0.7\, \textrm{meV}\,{\mbox{\,\raisebox{.3ex}
    {$<$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}\,}}\, m_A\,{\mbox{\,\raisebox{.3ex}
    {$<$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}\,}}\, 2.0\, \textrm{meV}$, from a combination of the $g_{A\gamma}$ vs. $m_A$ curve corresponding to the PVLAS rotation signal (cf. Eq. (\[ax\_rot\])) with the BFRT limits on the same quantities.

Clearly, a pseudoscalar with these properties is hardly compatible with a genuine QCD axion. For the latter, a mass $m_A\sim 1$meV implies a symmetry breaking scale $f_A\sim 6\times 10^{9}$GeV. According to (\[eq:ax\_ph\]), one needs then an extremely large ratio $|E/N|\sim 3\times 10^7$ of electromagnetic and color anomalies in order to arrive at an axion-photon coupling in the range suggested by PVLAS. This is far away from the predictions of any model conceived so far [@Cheng:1995fd]. Moreover, such a pseudoscalar must have very peculiar properties in order to evade the strong constraints on $g_{A\gamma}$ from stellar energy loss considerations (“HB stars" in Fig. \[fig:ax\_ph\]) and from its non-observation in helioscopes such as the CERN Axion Solar Telescope (“Solar (CAST)" in Fig. \[fig:ax\_ph\]) [@Raffelt:2005mt]. Pseudoscalar production in stars may be hindered, for example, if the $A\gamma\gamma$ vertex is suppressed at keV energies due to low scale compositeness of $A$ [@Masso:2005ym] or if, in stellar interiors, $A$ acquires an effective mass larger than the typical photon energy, $\sim$ keV [@jaeckel:tbp].

In any case, an independent and decisive experimental test of the finding of PVLAS is urgently needed. One opportunity is offered by high luminosity $e^+e^-$ colliders, e.g. a possible super-$B$ factory at KEK, where one may search for events with a single photon plus missing transverse energy in the final state [@Masso:1997ru]. The best and most timely possibilities, however, are offered by dedicated photon regeneration experiments, either based on ordinary optical lasers (e.g. [@Duvillaret:2005sv]) or on (soft) X-rays from free-electron lasers (FEL) at DESY and SLAC [@Ringwald:2001cp]. In fact, as can be seen in Fig. \[fig:ax\_ph\], the region of parameter space implied by PVLAS could be probed in a matter of minutes if one sets up a photon regeneration experiment exploiting the already operating FEL at DESY’s TESLA Test Facility, which provides tunable radiation from the vacuum-ultraviolet (VUV) to soft X-rays, $\omega=10$–$200$ eV, with an average power $\langle P\rangle = 20$–$40$ W, together with two superconducting dipole magnets of the type used in DESY’s electron proton collider HERA ($B=5$ T, $\ell =10$ m) [@Ringwald:2001cp]. The tuning of the FEL for fixed photon flux would allow a precision determination of $m_A$. Such an experiment could also serve as a test facility for an ambitious large scale photon regeneration experiment with sensitivity exceeding CAST [@Ringwald:2003ns], based on the recycling of all the 400 dipole magnets of HERA after its decommissioning in mid of 2007.

[99]{}

Weinberg S 1978 [*Phys. Rev. Lett.*]{}  [**40**]{} 223; Wilczek F 1978 [*Phys. Rev. Lett.*]{}  [**40**]{} 279 Peccei R. and Quinn H 1977 [*Phys. Rev. Lett.*]{}  [**38**]{} 1440; 1977 [*Phys. Rev.*]{} D [**16**]{} 1791 Kim J 1979 [*Phys. Rev. Lett.*]{}  [**43**]{} 103; Shifman M, Vainshtein A and Zakharov V 1980 [*Nucl. Phys.*]{} B [**166**]{} 493 Zhitnitsky A 1980 [*Sov. J. Nucl. Phys.*]{}  [**31**]{} 260; Dine M and Fischler W 1983 [*Phys. Lett.*]{} B [**120**]{} 137 Eidelman S [*et al.*]{} \[Particle Data Group Collaboration\] 2004 [*Phys. Lett.*]{} B [**592**]{} 1 Bardeen W and Tye S 1978 [*Phys. Lett.*]{} B [**74**]{} 229; Kaplan D 1985 [*Nucl. Phys.*]{} B [**260**]{} 215; Srednicki M 1985 [*Nucl. Phys.*]{} B [**260**]{} 689 Cameron R [*et al.*]{} \[BFRT Collaboration\] 1993 [*Phys. Rev.*]{} D [**47**]{} 3707 Zavattini E [*et al.*]{} \[PVLAS Collaboration\] 2005 hep-ex/0507107 Ringwald A 2001 hep-ph/0112254; Rabadan R, Ringwald A and Sigurdson K 2005 hep-ph/0511103 Ringwald A 2003 [*Phys. Lett.*]{} B [**569**]{} 51 Zioutas K [*et al.*]{} \[CAST Collaboration\] 2005 [*Phys. Rev. Lett.*]{}  [**94**]{} 121301 Raffelt G 1999 [*Ann. Rev. Nucl. Part. Sci.*]{}  [**49**]{} 163 Sikivie P 1983 [*Phys. Rev. Lett.*]{}  [**51**]{} 1415; Van Bibber K [*et al.*]{} 1987 [*Phys. Rev. Lett.*]{}  [**59**]{} 759 Maiani L [*et al*]{} 1986 [*Phys. Lett.*]{} B [**175**]{} 359; Raffelt G and Stodolsky L 1988 [*Phys. Rev.*]{} D [**37**]{} 1237 Cheng S, Geng C and Ni W 1995 [*Phys. Rev.*]{} D [**52**]{} 3132

Raffelt G 2005 hep-ph/0504152 Masso E and Redondo J 2005 [*JCAP*]{} [**0509**]{} 015 Jaeckel J, Masso E, Redondo J, Ringwald A and Takahashi F 2005 in preparation

Masso E and Toldra R 1997 [*Phys. Rev.*]{} D [**55**]{} 7967; Kleban M and Rabadan R 2005 hep-ph/0510183 Duvillaret L [*et al.*]{} 2005 CERN-SPSC-2005-034

[^1]: Talk presented at the ninth International Conference on Topics in Astroparticle and Underground Physics, TAUP 2005, Zaragoza, Spain, September 10-14, 2005.
