**Introduction to Axion Photon Interaction in Particle Physics and Photon Dispersion in Magnetized Media**

Avijit K. Ganguly *Banaras Hindu University (MMV), Varanasi, India* 

#### **1. Introduction**

20 Will-be-set-by-IN-TECH

48 Particle Physics

[15] [LEP Higgs Working Group for Higgs boson searches and ALEPH Collaboration],

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Symmetries, global or local, always play an important role in the conceptual aspects of physics be in broken or unbroken phase. Spontaneous breaking of the continuous symmetries always generates various excitations with varying mass spectra. Axion is one of that type, generated via spontaneous breaking of a global Chiral U(1) symmetry named after its discoverers, Peccei and Queen. This symmetry is usually denoted by U(1)*PQ*. To give a brief introduction to this particle and its origin we have to turn our attention to the development of the standard model of particle physics and its associated symmetries. The standard model of particle physics describes the strong, weak and electromagnetic interactions among elementary particles. The symmetry group for this model is, SUc(3) × SU(2) × U(1). The strong interaction ( Quantum Chromo Dynamics (QCD)) part of the Lagrangian has SU(3) color symmetry and it is given by,

$$\mathcal{L} = -\frac{1}{2g^2} \text{Tr } \mathbf{F}^a\_{\mu\nu} \mathbf{F}^{\mu\nu}\_a + \overline{q}(i\mathcal{D} - m)q. \tag{1}$$

It was realized long ago that, in the limit of vanishingly small quark masses (chiral limit), Strong interaction lagrangian has a global U(2)<sup>V</sup> × U(2)<sup>A</sup> symmetry. This symmetry group would further break spontaneously to produce the hadron multiplets. The vector part of the symmetry breaks to iso-spin times baryon number symmetry given by U(2)<sup>V</sup> = SU(2)<sup>I</sup> × U(1)B. In nature baryon number is seen to be conserved and the mass spectra of nucleon and pion multiplets indicate that the isospin part is also conserved approximately.

So one is left with the axial vector symmetry. QCD being a nonabelian gauge it is believed that this theory is confining in the infrared region. The confining property of the theory is likely to generate condensates of antiquark quark pairs. Thus *u*- and *d* quark condensates would have non-zero vacuum expectation values, i.e.,

$$<0|\bar{u}(0)u(0)|0> = <0|\bar{d}(0)d(0)|0> \neq 0\,. \tag{2}$$

and they would break the *U*(2)*<sup>A</sup>* symmetry. Now if the axial symmetry is broken, we would expect nearly four degenerate and massless pseudoscalar mesons. Interestingly enough, out of the four we observe three light pseudoscalar Nambu Goldstone (NG) Bosons in nature, i.e., the pions. They are light, m*<sup>π</sup>* � 0, but the other one (with approximately same mass) is not

ruled out. However modified versions of the same with their associated axions are still of interest with the symmetry breaking scale lying between EW scale and 1012 GeV. Since the axion photon/matter coupling constant, is inversely proportional to the breaking scale of the PQ symmetry, *fa* and is much larger than the electroweak scale *fa* � *fw*, the resulting axion turns out to be very weakly interacting. And is also very light (*ma* <sup>∼</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>a</sup>* ) therefore it is often called "the invisible axion model" [(M.Dine et al. , 1981; J. E. Kim , 1979)]. For very good

There are various proposals to detect axions in laboratory. One of them is the solar axion experiment. The idea behind this is the following, if axions are produced at the core of the Sun, they should certainly cross earth on it's out ward journey from the Sun. From equation [5], it can be established that in an external magnetic field an axion can oscillate in to a photon and vice versa. Hence if one sets up an external magnetic field in a cavity, an axion would convert itself into a photon inside the cavity.This experiment has been set up in CERN, and is usually referred as CAST experiment[(K. Zioutas et al.,, 2005)]. The conversion

and the axion flux. Since inside the sun axions are dominantly produced by Primakoff and compton effects. One can compute the axion flux by calculating the axion production rate via primakoff & compton process using the available temp and density informations inside the sun. Therefore by observing the rate of axion photon conversion in a cavity on can estimate the axion parameters. The study of solar axion puts experimental bound on *M* to

The same can be estimated from astrophysical observations. In this situation, it possible to estimate the rate at which the axions would draw energy away form the steller atmosphere

<sup>−</sup> → *γ* + *a* , *e*<sup>−</sup> + *γ* → *e*

Axions being weakly interacting particles, would escape the steller atmosphere and the star would lose energy. Thus it would affect the age vs luminosity relation of the star. Comparison of the same with observations yields bounds on e.g., axion mass *ma* and *M*. A detailed survey of various astrophysical bounds on the parameters of axion models and constraints on them,

In the astrophysical and cosmological studies, mentioned above, medium and a magnetic field are always present. So it becomes important to seek the modification of the axion coupling to photon, in presence of a medium or magnetic field or both. Particularly in some astrophysical situations where the magnetic component, along with medium (usually referred as magnetized medium) dominates. Examples being, the Active Galactic Nuclei (AGN), Quasars, Supernova, the Coalescing Neutron Stars or Nascent Neutron Stars, Magnetars etc. . The magnetic field strength in these situations vary between, B ∼ 106 <sup>−</sup> 1017 G, where some

&

*γplasmon* → *γ* + *a* , *γ* + *γ* → *a*. (7)

*<sup>e</sup>*/*<sup>e</sup>* � 4.41 <sup>×</sup> 1013 <sup>G</sup> (8)

by calculating the axion flux (i.e. is axion luminosity) from the following reactions[7]

*<sup>M</sup>* ), axion mass

51

− + *a* (6)

rate inside the cavity, would depend on the value of the coupling constant ( <sup>1</sup>

be, *<sup>M</sup> <sup>&</sup>gt;* 1.7 <sup>×</sup> 1011GeV [(Moriyama et al. , 1985),(Moriyama et al. , 1998b)].

are significantly above the critical, Schwinger value[( J. Schwinger , 1951)]

<sup>B</sup>*<sup>e</sup>* <sup>=</sup> *<sup>m</sup>*<sup>2</sup>

*e*<sup>+</sup> + *e*

can be found in ref. [(G .G .Raffelt , 1996)].

introduction to this part one may refer to[(R. Peccei , 1996)].

in Particle Physics and Photon Dispersion in Magnetized Media

Introduction to Axion Photon Interaction

to be found. Eta meson though is a pseudoscalar meson, but it has mass much greater than the pion ( m*<sup>η</sup>* � m*π*). So the presence of another light pseudoscalar meson in the hadronic spectrum, seem to be missing. This is usually referred in the literature [(Steven Weinberg , 1975)] as the U(1)*<sup>A</sup>* problem.

#### **1.1 Strong CP problem and neutron dipole moment**

Soon after the identification of QCD as the correct theory of strong interaction physics, instanton solutions [(Belavin Polyakov Shvarts and Tyupkin , 1975)] for non-abelian gauge theory was discovered. Subsequently, through his pioneering work, 't Hooft [('t Hooft , 1976a),('t Hooft , 1976b)] established that a *θ* term must be added to the QCD Lagrangian. The expression of this piece is,

$$\mathcal{L}\_{\theta} = \theta \frac{\text{g}^2}{32\pi^2} F\_a^{\mu\nu} \tilde{F}\_{a\mu\nu}. \tag{3}$$

But in the presence of this term *the axial symmetry* is no more a realizable symmetry for QCD. This term violates Parity and Time reversal invariance, but conserves charge conjugation invariance, so it violates CP. Such a term if present in the lagrangian would predict neutron electric dipole moment. The observed neutron electric dipole moment [(R. J. Crewther, 1978)] is <sup>|</sup>*dn*<sup>|</sup> *<sup>&</sup>lt;* <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>26</sup> ecm and that requires the angle *<sup>θ</sup>* to be extremely small [*dn* � *<sup>e</sup>θmq*/*M*<sup>2</sup> *N* indicating [(V. Baluni , 1979; R. J. Crewther et. al. , 1980)] *θ <* 10−9]. This came to be known as the strong CP problem. In order to overcome this problem, Pecci and Queen subsequently Weinberg and Wilckzek [(R. Peccei and H. Quinn , 1977; S. Weinberg , 1978; F. Wilczek , 1978)] postulated the parameter *θ* to be a dynamical field with odd parity arising out of some chiral symmetry breaking taking place at some energy scale *fPQ*. With this identification the *θ* term of the QCD Lagrangian now changes to,

$$\mathcal{L}\_a = \frac{g^2}{32\pi^2} a F\_a^{\mu\nu} \mathcal{F}\_{a\mu\nu\nu} \tag{4}$$

where *a* is the axion field. They[(R. Peccei and H. Quinn , 1977; S. Weinberg , 1978; F. Wilczek , 1978)] also provided an estimate of the mass of this light pseudoscalar boson. Although these ultra light objects were envisioned originally to provide an elegant solution to the strong CP problem [(R. Peccei and H. Quinn , 1977),WW,wilczek] (see (R. Peccei , 1996)] for details) but it was realized later on that their presence may also solve some of the outstanding problems in cosmology, like the dark matter or dark energy problem (related to the closure of the universe). Further more their presence if established, may add a new paradigm to our understanding of the stellar evolution. A detailed discussion on the astrophysical and cosmological aspects of axion physics can be found in [(M.S. Turner , 1990; G. G. Raffelt , 1990; G. G. Raffelt , 1997; G .G .Raffelt , 1996; J. Preskill et al , 1983)]. In all the models of axions, the axion photon coupling is realized through the following term in the Lagrangian,

$$
\mathcal{L} = \frac{1}{M} \, a \, \mathbf{E} \cdot \mathcal{B}. \tag{5}
$$

Where *M* ∝ *fa* the axion coupling mass scale or the symmetry breaking scale and *a* stands for the axion field. The original version of Axion model, usually known as Peccei-Queen Weinberg-Wilczek model (PQWW), had a symmetry breaking scale that was close to weak scale, *fw*. Very soon after its inception, the original model, associated with the spontaneous breakdown of the global PQ symmetry at the Electro Weak scale (EW) *fw*, was experimentally 2 Will-be-set-by-IN-TECH

to be found. Eta meson though is a pseudoscalar meson, but it has mass much greater than the pion ( m*<sup>η</sup>* � m*π*). So the presence of another light pseudoscalar meson in the hadronic spectrum, seem to be missing. This is usually referred in the literature [(Steven Weinberg ,

Soon after the identification of QCD as the correct theory of strong interaction physics, instanton solutions [(Belavin Polyakov Shvarts and Tyupkin , 1975)] for non-abelian gauge theory was discovered. Subsequently, through his pioneering work, 't Hooft [('t Hooft , 1976a),('t Hooft , 1976b)] established that a *θ* term must be added to the QCD Lagrangian.

> <sup>32</sup>*π*<sup>2</sup> *<sup>F</sup>μν <sup>a</sup> F*˜

But in the presence of this term *the axial symmetry* is no more a realizable symmetry for QCD. This term violates Parity and Time reversal invariance, but conserves charge conjugation invariance, so it violates CP. Such a term if present in the lagrangian would predict neutron electric dipole moment. The observed neutron electric dipole moment [(R. J. Crewther, 1978)] is <sup>|</sup>*dn*<sup>|</sup> *<sup>&</sup>lt;* <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>26</sup> ecm and that requires the angle *<sup>θ</sup>* to be extremely small [*dn* � *<sup>e</sup>θmq*/*M*<sup>2</sup>

indicating [(V. Baluni , 1979; R. J. Crewther et. al. , 1980)] *θ <* 10−9]. This came to be known as the strong CP problem. In order to overcome this problem, Pecci and Queen subsequently Weinberg and Wilckzek [(R. Peccei and H. Quinn , 1977; S. Weinberg , 1978; F. Wilczek , 1978)] postulated the parameter *θ* to be a dynamical field with odd parity arising out of some chiral symmetry breaking taking place at some energy scale *fPQ*. With this identification the *θ* term

> <sup>32</sup>*π*<sup>2</sup> *aFμν <sup>a</sup> F*˜

where *a* is the axion field. They[(R. Peccei and H. Quinn , 1977; S. Weinberg , 1978; F. Wilczek , 1978)] also provided an estimate of the mass of this light pseudoscalar boson. Although these ultra light objects were envisioned originally to provide an elegant solution to the strong CP problem [(R. Peccei and H. Quinn , 1977),WW,wilczek] (see (R. Peccei , 1996)] for details) but it was realized later on that their presence may also solve some of the outstanding problems in cosmology, like the dark matter or dark energy problem (related to the closure of the universe). Further more their presence if established, may add a new paradigm to our understanding of the stellar evolution. A detailed discussion on the astrophysical and cosmological aspects of axion physics can be found in [(M.S. Turner , 1990; G. G. Raffelt , 1990; G. G. Raffelt , 1997; G .G .Raffelt , 1996; J. Preskill et al , 1983)]. In all the models of axions, the axion photon

*<sup>a</sup>μν*. (3)

*<sup>a</sup>μν*, (4)

*<sup>M</sup> <sup>a</sup>* **<sup>E</sup>** · B. (5)

*N*

<sup>L</sup>*<sup>θ</sup>* <sup>=</sup> *<sup>θ</sup> <sup>g</sup>*<sup>2</sup>

<sup>L</sup>*<sup>a</sup>* <sup>=</sup> *<sup>g</sup>*<sup>2</sup>

<sup>L</sup> <sup>=</sup> <sup>1</sup>

Where *M* ∝ *fa* the axion coupling mass scale or the symmetry breaking scale and *a* stands for the axion field. The original version of Axion model, usually known as Peccei-Queen Weinberg-Wilczek model (PQWW), had a symmetry breaking scale that was close to weak scale, *fw*. Very soon after its inception, the original model, associated with the spontaneous breakdown of the global PQ symmetry at the Electro Weak scale (EW) *fw*, was experimentally

coupling is realized through the following term in the Lagrangian,

1975)] as the U(1)*<sup>A</sup>* problem.

The expression of this piece is,

of the QCD Lagrangian now changes to,

**1.1 Strong CP problem and neutron dipole moment**

ruled out. However modified versions of the same with their associated axions are still of interest with the symmetry breaking scale lying between EW scale and 1012 GeV. Since the axion photon/matter coupling constant, is inversely proportional to the breaking scale of the PQ symmetry, *fa* and is much larger than the electroweak scale *fa* � *fw*, the resulting axion turns out to be very weakly interacting. And is also very light (*ma* <sup>∼</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>a</sup>* ) therefore it is often called "the invisible axion model" [(M.Dine et al. , 1981; J. E. Kim , 1979)]. For very good introduction to this part one may refer to[(R. Peccei , 1996)].

There are various proposals to detect axions in laboratory. One of them is the solar axion experiment. The idea behind this is the following, if axions are produced at the core of the Sun, they should certainly cross earth on it's out ward journey from the Sun. From equation [5], it can be established that in an external magnetic field an axion can oscillate in to a photon and vice versa. Hence if one sets up an external magnetic field in a cavity, an axion would convert itself into a photon inside the cavity.This experiment has been set up in CERN, and is usually referred as CAST experiment[(K. Zioutas et al.,, 2005)]. The conversion rate inside the cavity, would depend on the value of the coupling constant ( <sup>1</sup> *<sup>M</sup>* ), axion mass and the axion flux. Since inside the sun axions are dominantly produced by Primakoff and compton effects. One can compute the axion flux by calculating the axion production rate via primakoff & compton process using the available temp and density informations inside the sun. Therefore by observing the rate of axion photon conversion in a cavity on can estimate the axion parameters. The study of solar axion puts experimental bound on *M* to be, *<sup>M</sup> <sup>&</sup>gt;* 1.7 <sup>×</sup> 1011GeV [(Moriyama et al. , 1985),(Moriyama et al. , 1998b)].

The same can be estimated from astrophysical observations. In this situation, it possible to estimate the rate at which the axions would draw energy away form the steller atmosphere by calculating the axion flux (i.e. is axion luminosity) from the following reactions[7]

&

$$e^{+} + e^{-} \to \gamma + a \quad e^{-} + \gamma \to e^{-} + a \tag{6}$$

$$
\gamma\_{plasmon} \to \gamma + a \quad \gamma + \gamma \to a. \tag{7}
$$

Axions being weakly interacting particles, would escape the steller atmosphere and the star would lose energy. Thus it would affect the age vs luminosity relation of the star. Comparison of the same with observations yields bounds on e.g., axion mass *ma* and *M*. A detailed survey of various astrophysical bounds on the parameters of axion models and constraints on them, can be found in ref. [(G .G .Raffelt , 1996)].

In the astrophysical and cosmological studies, mentioned above, medium and a magnetic field are always present. So it becomes important to seek the modification of the axion coupling to photon, in presence of a medium or magnetic field or both. Particularly in some astrophysical situations where the magnetic component, along with medium (usually referred as magnetized medium) dominates. Examples being, the Active Galactic Nuclei (AGN), Quasars, Supernova, the Coalescing Neutron Stars or Nascent Neutron Stars, Magnetars etc. . The magnetic field strength in these situations vary between, B ∼ 106 <sup>−</sup> 1017 G, where some are significantly above the critical, Schwinger value[( J. Schwinger , 1951)]

$$\mathcal{B}\_{\varepsilon} = m\_{\varepsilon}^{2}/e \simeq 4.41 \times 10^{13} \text{ G} \tag{8}$$

the fermion species). We would like to add that, for the sake of brevity at places, we may use *g* instead of *gaγγ* at some places in the rest of this paper. Therefore the additional contribution to the axion photon effective lagrangian from the new vertex would add to the existing one

In order to estimate the loop induced *γ* − *a* coupling, one can start with the Lagrangian given by Eqn. [2.9]. Defining *p*� = *p* + *k* the effective vertex for the *γ* − *a* coupling turns out to be,

*γμγ*5*iS*(*p*)*γνiS*(*p*�

*γμγ*5*iS*(*p*)*γνiS*(*p*�

) 

) 

*μν* (some times called the VA response function),

*B*(*p*)*γνiS<sup>V</sup>*

*<sup>B</sup>* (*p*)*γνS<sup>η</sup>*

*<sup>B</sup>* (*p*� )

> *B*(*p*� )

. (3.15)

) +*γμγ*5*S<sup>η</sup>*

+*γμγ*5*iS<sup>V</sup>*

*μν*(*k*) has been estimated in [(A. K. Ganguly , 2006;

. (3.13)

. (3.14)

*μν*(*k*).

53

(2*π*)<sup>4</sup> *<sup>k</sup>μ*Tr

The effective vertex given by [3.13], is computed from the diagram given in [Fig.1]. In eqn. [3.13] *S*(*p*) is the in medium fermionic propagator in external magnetic field, computed to all orders in field strength. The structure of the same can be found in [(A. K. Ganguly , 2006)]. One can easily recognize that, eqn. [3.13], has the following structure, Γ*ν*(*k*) = *kμ*Π*<sup>A</sup>*

would have contributions from pure magnetic field background, as well as magnetic field plus medium, i.e., magnetized medium. The contribution from only magnetic field and the

D. V. Galtsov , 1972; L. L.DeRaad *et al*. , 1976; A. N. Ioannisian et al. , 1997; C. Schubert , 2000)]. The expression of the would be provided in the next section, after that the thermal

The VA response function in a magnetic field Π*<sup>A</sup>* has been evaluated in [(A. K. Ganguly , 2006; D. V. Galtsov , 1972; L. L.DeRaad *et al*. , 1976; A. N. Ioannisian et al. , 1997; C. Schubert , 2000)], with varying choice of metric; we have reevaluated it according to our metric convention *gμν* ≡ diag (+1, −1, −1, −1). The expression for the same according our convention is:

*<sup>B</sup>* (*p*)*γνiS<sup>V</sup>*

*<sup>B</sup>* (*p*�

*μν*, is the axial polarization tensor, comes from the axial coupling of the axions to the

**3. Expression for photon axion vertex in presence of uniform background**

*d*<sup>4</sup> *p*

 *d*<sup>4</sup> *p* (2*π*)<sup>4</sup> Tr

*γμγ*5*iS<sup>V</sup>*

one with magnetized medium effects, are given in the following expression,

i.e.,eqn. [2.11].

Where Π*<sup>A</sup>*

leptons and it's:

*i*Π*<sup>A</sup>*

**magnetic field and material medium**

Introduction to Axion Photon Interaction

*i*Π*<sup>A</sup>*

In general the axial polarization tensor, Π*<sup>A</sup>*

*μν*(*k*)= *ga f e Qf*

**3.1 Magnetized vacuum contribution**

The pure magnetic field contribution to Π*<sup>A</sup>*

part contribution to the same would be reported .

*i*Γ*ν*(*k*) = *ga f e Qf*

in Particle Physics and Photon Dispersion in Magnetized Media

*μν*(*k*)= *ga f e Qf*

 *d*<sup>4</sup> *p* (2*π*)<sup>4</sup> Tr

[(M. Ruderman , 1991; Duncan & Thompson , 1992)]. In view of this observation and the possibilities of applications of axion physics to these astrophysical as well as cosmological scenarios, it is pertinent to find out the effect of medium and magnetic field to axion photon coupling.

As we already have noted, the axion physics is sensitive to presence of medium and magnetic field. In most of the astrophysical or cosmological situations these two effects are dominant. In view of this it becomes reasonable to study how matter and magnetic field effect can affect the axion photon vertex. Modification to axion photon vertex in a magnetized media was studied in [(A. K. Ganguly , 2006)]. In this document we would present that work and discuss new correction to *a* − *γ* vertex in a magnetized media. In the next section that we would focus on axion photon mixing effect with tree level axion photon vertex and show how this effect can change the polarization angle and ellipticity of a propagating plane polarized light beam passing through a magnetic field. After that we would elaborate on how the same predictions would get modified if the same process takes place in a magnetized media. This particular study involves diagonalisation of a 3 × 3 matrix, so at the end we have added an appendix showing how to construct the diagonalizing matrix to diagonalize a 3 × 3 symmetric matrix.

#### **2. The loop induced vertex**

The axion-fermion ( lepton in this note ) interaction1 — with *g*� *a f* = *Xf mf* / *fa* the Yukawa coupling constant, *Xf* , the model-dependent factors for the PQ charges for different generations of quarks and leptons [(G .G .Raffelt , 1996)], and fermion mass *mf* – is given by, [(M.Dine et al. , 1981)],

$$\mathcal{L}\_{af} = \frac{\mathcal{S}\_{af}^{'}}{m\_f} \sum\_{f} (\bar{\Psi}\_f \gamma\_\mu \gamma\_5 \Psi\_f) \,\partial^\mu a\_\prime \tag{2.9}$$

The sum over f, in eqn. [2.9], stands for sum over all the fermions, from each family. Although, in some studies, instead of using [2.9], the following Lagrangian has been employed,

$$\mathcal{L}\_{af} = -2\mathrm{i}g\_{af}' \sum\_{f} (\bar{\Psi}\_f \gamma\_5 \Psi\_f) a\_\prime \tag{2.10}$$

but, Raffelt and Seckel [( G. Raffelt , 1988)] has pointed out the correctness of using [2.9]. We for our purpose we will make use of [2.9]. We would like to note that the usual axion photon mixing Lagrangian in an external magnetic field turns out to be,

$$\mathcal{L}\_{a\gamma} = -g\_{a\gamma\gamma} \frac{e^2}{32\pi^2} a \text{F\"F}^{\text{Ext}}.\tag{2.11}$$

In equation [2.11] the axion photon coupling constant is described by,

$$g\_{a\gamma\gamma} = \frac{1}{f\_a} \left[ A\_{PQ}^{em} - A\_{PQ}^c \frac{2(4+z)}{3(1+z)} \right] \,\text{.}\tag{2.12}$$

with *z* = *mu md* , where *mu* and *md* are the masses of the light quarks. Anomaly factors are given by the following relations, *Aem PQ* <sup>=</sup> Tr(*Q*<sup>2</sup> *<sup>f</sup>*)*Xf* and *<sup>δ</sup>abAem <sup>c</sup>* = Tr(*λaλbXf*) (and the trace is over

<sup>1</sup> Some of the issues related axion fermion coupling had been reviewed in [(A. K. Ganguly , 2006)], one can see the references there.

4 Will-be-set-by-IN-TECH

[(M. Ruderman , 1991; Duncan & Thompson , 1992)]. In view of this observation and the possibilities of applications of axion physics to these astrophysical as well as cosmological scenarios, it is pertinent to find out the effect of medium and magnetic field to axion photon

As we already have noted, the axion physics is sensitive to presence of medium and magnetic field. In most of the astrophysical or cosmological situations these two effects are dominant. In view of this it becomes reasonable to study how matter and magnetic field effect can affect the axion photon vertex. Modification to axion photon vertex in a magnetized media was studied in [(A. K. Ganguly , 2006)]. In this document we would present that work and discuss new correction to *a* − *γ* vertex in a magnetized media. In the next section that we would focus on axion photon mixing effect with tree level axion photon vertex and show how this effect can change the polarization angle and ellipticity of a propagating plane polarized light beam passing through a magnetic field. After that we would elaborate on how the same predictions would get modified if the same process takes place in a magnetized media. This particular study involves diagonalisation of a 3 × 3 matrix, so at the end we have added an appendix showing how to construct the diagonalizing matrix to diagonalize a 3 × 3 symmetric matrix.

Yukawa coupling constant, *Xf* , the model-dependent factors for the PQ charges for different generations of quarks and leptons [(G .G .Raffelt , 1996)], and fermion mass *mf* – is given by,

The sum over f, in eqn. [2.9], stands for sum over all the fermions, from each family. Although,

*a f* ∑ *f*

but, Raffelt and Seckel [( G. Raffelt , 1988)] has pointed out the correctness of using [2.9]. We for our purpose we will make use of [2.9]. We would like to note that the usual axion photon

L*a<sup>γ</sup>* = −*gaγγ*

*e*2

2(4 + *z*) 3(1 + *z*)

, where *mu* and *md* are the masses of the light quarks. Anomaly factors are given

*<sup>f</sup>*)*Xf* and *<sup>δ</sup>abAem*

<sup>1</sup> Some of the issues related axion fermion coupling had been reviewed in [(A. K. Ganguly , 2006)], one

in some studies, instead of using [2.9], the following Lagrangian has been employed,

L*a f* = −2*ig*�

mixing Lagrangian in an external magnetic field turns out to be,

In equation [2.11] the axion photon coupling constant is described by,

*PQ* <sup>=</sup> Tr(*Q*<sup>2</sup>

*fa Aem PQ* <sup>−</sup> *<sup>A</sup><sup>c</sup> PQ*

*gaγγ* <sup>=</sup> <sup>1</sup>

*a f* =

(Ψ¯ *<sup>f</sup> γμγ*5Ψ*f*) *∂<sup>μ</sup> a*, (2.9)

(Ψ¯ *<sup>f</sup> γ*5Ψ*f*)*a*, (2.10)

<sup>32</sup>*π*<sup>2</sup> *<sup>a</sup>*FF˜ Ext. (2.11)

*<sup>c</sup>* = Tr(*λaλbXf*) (and the trace is over

, . (2.12)

*Xf mf* / *fa*

 the

The axion-fermion ( lepton in this note ) interaction1 — with *g*�

<sup>L</sup>*a f* <sup>=</sup> *<sup>g</sup>*� *a f mf* ∑ *f*

coupling.

**2. The loop induced vertex**

[(M.Dine et al. , 1981)],

with *z* = *mu*

*md*

by the following relations, *Aem*

can see the references there.

the fermion species). We would like to add that, for the sake of brevity at places, we may use *g* instead of *gaγγ* at some places in the rest of this paper. Therefore the additional contribution to the axion photon effective lagrangian from the new vertex would add to the existing one i.e.,eqn. [2.11].
