**3. Expression for photon axion vertex in presence of uniform background magnetic field and material medium**

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,

$$i\Gamma\_{\nu}(k) = g\_{af} \, e \, Q\_f \Big| \frac{d^4 p}{(2\pi)^4} k^\mu \text{Tr} \left[ \gamma\_\mu \gamma\_5 iS(p)\gamma\_\nu iS(p') \right]. \tag{3.13}$$

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> μν*(*k*). Where Π*<sup>A</sup> μν*, is the axial polarization tensor, comes from the axial coupling of the axions to the leptons and it's:

$$i\Pi^{A}\_{\mu\nu}(k) = \text{g}\_{af} \, e \, \text{Q} \int \frac{d^4 p}{(2\pi)^4} \text{Tr} \left[ \gamma\_\mu \gamma\_5 iS(p)\gamma\_\nu iS(p') \right]. \tag{3.14}$$

In general the axial polarization tensor, Π*<sup>A</sup> μν* (some times called the VA response function), 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 one with magnetized medium effects, are given in the following expression,

$$i\Pi\_{\mu\nu}^{A}(k) = \mathcal{g}\_{af} \, e \, \mathcal{Q}\_f \int \frac{d^4 p}{(2\pi)^4} \text{Tr} \left[ \gamma\_\mu \gamma\_5 i S\_B^V(p) \gamma\_\nu i S\_B^V(p') + \gamma\_\mu \gamma\_5 S\_B^\eta(p) \gamma\_\nu i S\_B^V(p') \right]$$

$$+ \gamma\_\mu \gamma\_5 i S\_B^V(p) \gamma\_\nu S\_B^\eta(p') \Big]. \tag{3.15}$$

The pure magnetic field contribution to Π*<sup>A</sup> μν*(*k*) has been estimated in [(A. K. Ganguly , 2006; 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 part contribution to the same would be reported .

#### **3.1 Magnetized vacuum contribution**

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:

where R*μν*(*p*, *p*�

<sup>B</sup> in R*μν*, as R(*E*)

*μν* and R(*O*)

**4.0.1 Vertex function: even powers in B**

= 4*iη*−(*p*0)

The expression for the R*<sup>E</sup>*

R(*E*) *μν* ◦

out to be,

,*s*,*s*�

Introduction to Axion Photon Interaction

R*μν*(*p*, *p*�

in Particle Physics and Photon Dispersion in Magnetized Media

,*s*,*s*�

*k* ←

) contains the trace part. R*μν*(*p*, *p*�

external magnetic field with even and odd powers of B, can be presented as,

) = <sup>R</sup>(*E*)

*ν*

Fig. 1. One-loop diagram for the effective axion electromagnetic vertex .

details of this analysis can be found in [(A. K. Ganguly , 2006)].

*μν* (*p*, *p*�

*p*

*p* + *k* ≡ *p*�

We have denoted the pieces with even and odd powers in the external magnetic field strength

are also odd and even in powers of chemical potential, therefore, under charge conjugation they would transform as, B&*μ* ↔ (−*μ*)&(−B),i. e., both behave differently. More over their parity structures are also different. These properties come very useful while analyzing, the structure of axion photon coupling, using discrete symmetry arguments to justify the presence or absence of either of the two; that is the reason, why they should be treated separately. The

*εμναβ <sup>p</sup>αkβ*(<sup>1</sup> <sup>+</sup> tan(*eQf* <sup>B</sup>*s*)tan(*eQf*B*<sup>s</sup>*

The two point VA response function Π*A*(*k*), can be interpreted as a (one particle irreducible) two point vertex; with one point for the external axion line and the other one (Lorentz indexed) for the external photon line. But since the evaluations are done in presence of external magnetic field B they correspond to soft external photon line insertions. That is their four momenta *<sup>k</sup><sup>α</sup>* <sup>→</sup> 0 . If each soft external photon line contributes either +1 or -1 to the total spin ( angular momentum ) of the effective vertex, then, for an even order term in external field strength B the total spin of this piece would be a coherent sum of all the contributions from all the odd number of soft photon linesB. Now recall that in order arrive at the the

the same (with some sort of naivete) by *a*(*x*)*Fμν*(*x*) . Therefore, it is worth noting that, if

� )

× tan(*eQf* B*s*)tan(*eQf* B*s*

Because of the presence of *εμναβk<sup>β</sup>* and *εμναβ*<sup>⊥</sup> *<sup>k</sup>α*, it vanishes on contraction R(*E*)

expression for the effective interaction Lagrangian for *<sup>γ</sup>* <sup>−</sup> *<sup>a</sup>* from <sup>Π</sup>*<sup>A</sup>*

,*s*,*s*�

,*s*,*s*�

) + <sup>R</sup>(*O*)

*μν* (*p*, *p*�

*k* ←

*μν* . In addition to being just even and odd in powers of *eQf* B, they

*μν*, (that is the term with even powers of the magnetic field), comes

�

tan(*eQf* B*s*) − tan(*eQf* B*s*�

tan(*eQf* B*s*) + tan(*eQf* B*s*�)

)) + *εμναβ*<sup>⊥</sup> *<sup>k</sup>αkβ*<sup>⊥</sup>

)

*μν* with *kν*.

*μν*(*x*)–we need to multiply

. (4.22)

,*s*,*s* �

) is a polynomial in powers of the

) (4.21)

55

$$\begin{split} \Pi^{A}\_{\mu\nu}(k) = \frac{i g\_{af} \left(e \, \mathcal{Q}\_f\right)^2}{(4\pi)^2} \int\_0^\infty dt \int\_{-1}^{+1} dv \, e^{\Phi\_0} \left\{ \left(\frac{1-v^2}{2}k\_{\parallel}^2 - 2m\_e^2\right) \tilde{F}\_{\mu\nu} - (1-v^2)k\_{\mu\_{\parallel}}(\tilde{F}k)\_{\nu} \right. \\ \left. \left. + \mathcal{R}\left[k\_{\tilde{\nu}\_{\perp}}(k\tilde{F})\_{\mu} + k\_{\tilde{\mu}\_{\perp}}(k\tilde{F})\_{\nu}\right] \right\} \right\}, \end{split} \tag{3.16}$$

Where, *R* = 1−*v* sin Z*v* sin Z−cos Z cos Z*v* sin2 <sup>Z</sup> and *<sup>φ</sup>*<sup>0</sup> <sup>=</sup> *it* <sup>1</sup>−*v*<sup>2</sup> <sup>4</sup> *<sup>k</sup>*<sup>2</sup> || <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>−</sup> cos *<sup>v</sup>*Z−cos<sup>Z</sup> <sup>2</sup><sup>Z</sup> sin <sup>Z</sup> *<sup>k</sup>*<sup>2</sup> ⊥ . In the above expression, *<sup>F</sup>μν* <sup>=</sup> <sup>1</sup> <sup>2</sup> *�μνρσFρσ*, and *�*<sup>0123</sup> <sup>=</sup> 1 is the dual of the field-strength tensor, with Z = *eQf* B*t*. Therefore, following eqn. [3.13], the photon axion vertex in a purely magnetized vacuum, would be, <sup>Γ</sup>*ν*(*k*) = *<sup>k</sup>μ*Π*AB μν* (*k*) i.e.,

$$\begin{split} \Gamma^{\boldsymbol{\upsilon}}(k) &= \frac{i g\_{\boldsymbol{\sigma}\boldsymbol{f}} (\boldsymbol{e} \, \mathcal{Q}\_{f})^{2}}{(4\pi)^{2}} \int\_{0}^{\infty} dt \int\_{-1}^{+1} dv \, \epsilon^{\phi\_{0}} k^{\mu} \left\{ \left( \frac{1 - v^{2}}{2} k\_{\parallel}^{2} - 2 m\_{\varepsilon}^{2} \right) \tilde{\mathcal{F}}\_{\mu\nu} - (1 - v^{2}) k\_{\mu\_{\parallel}} (\tilde{\mathcal{F}} k)\_{\nu} \right. \\ &\left. + \mathcal{R} \left[ k\_{\nu\_{\perp}} (k \tilde{\mathcal{F}})\_{\mu} + k\_{\mu\_{\perp}} (k \tilde{\mathcal{F}})\_{\nu} \right] \right\} , \end{split} \tag{3.17}$$

This result is not gauge invariant. However following [(A. K. Ganguly , 2006; A. N. Ioannisian et al. , 1997)], one may integrate the first term under the integral, and arrive at the expression for, the Effective Lagrangian for loop induced axion photon coupling in a magnetized vacuum, to be given by,

$$
\mathcal{L}\_{a\gamma}^{B} = a A^{\nu} \Gamma\_{\nu}(k) \tag{3.18}
$$

In eqn.[3.18],we define the axion field by *<sup>a</sup>* and (*kF*)*<sup>ν</sup>* <sup>=</sup> *<sup>k</sup>μFμν* and (*Fk* )*<sup>ν</sup>* <sup>=</sup> *<sup>F</sup>νμkμ*. Finally the loop induced contribution to the axion photon effective Lagrangian is,

$$\mathcal{L}\_{a\gamma}^{\mathcal{B}} = -\frac{1}{32\pi^2} g\_{af} (e\mathcal{Q}\_f)^2 \left[ 4 + \frac{4}{3} \left( \frac{k\_{\parallel}^2}{m^2} \right) \right] aF\_{\mu\nu} \tilde{F}^{\mu\nu}. \tag{3.19}$$

Since we are interested in *<sup>ω</sup> <sup>&</sup>lt; <sup>m</sup>*, so the magnitude of the factor *<sup>k</sup>*|| *m* 2 *<<* 1, thus the order of magnitude estimate estimate of this contribution is of *O*(1). However some of the factors there are momentum dependent, so it may affect the dispersion relation for photon and axion.

#### **4. Contribution from the magnetized medium**

Having estimated the effective axion photon vertex in a purely magnetic environment, we would focus on the contribution from the magnetized medium. As before, one can evaluate the same by using the expression for a fermion propagator in external magnetic field and medium; the result is:

$$\Pi\_{\mu\nu}^{A\_{\beta}}(k) = (i g\_{af} e \mathbf{Q}\_f) \int \frac{d^4 p}{(2\pi)^4} \int\_{-\infty}^{\infty} ds \, e^{\Phi(p,s)} \int\_0^{\infty} ds' e^{\Phi(p',s')} \text{Tr} \left[$$

$$\left[\gamma\_\mu \gamma\_5 G(p,s) \gamma\_\nu G(p',s')\right] \eta\_F(p) + \left[\gamma\_\mu \gamma\_5 G(-p',s') \gamma\_\nu G(-p,s)\right] \eta\_F(-p)\right]$$

$$= (i g\_{af} e \mathbf{Q}\_f) \int \frac{d^4 p}{(2\pi)^4} \int\_{-\infty}^{\infty} ds \, e^{\Phi(p,s)} \int\_0^{\infty} ds' \, e^{\Phi(p',s')} \text{R}\_{\mu\nu}(p, p', s, s') \tag{4.20}$$

6 Will-be-set-by-IN-TECH

<sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> <sup>2</sup> *<sup>k</sup>*<sup>2</sup>

and *φ*<sup>0</sup> = *it*

Z = *eQf* B*t*. Therefore, following eqn. [3.13], the photon axion vertex in a purely magnetized

<sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> <sup>2</sup> *<sup>k</sup>*<sup>2</sup>

This result is not gauge invariant. However following [(A. K. Ganguly , 2006; A. N. Ioannisian et al. , 1997)], one may integrate the first term under the integral, and arrive at the expression for, the Effective Lagrangian for loop induced axion photon coupling in a magnetized vacuum,

> 4 + 4 3

of magnitude estimate estimate of this contribution is of *O*(1). However some of the factors there are momentum dependent, so it may affect the dispersion relation for photon and axion.

Having estimated the effective axion photon vertex in a purely magnetic environment, we would focus on the contribution from the magnetized medium. As before, one can evaluate the same by using the expression for a fermion propagator in external magnetic field and

*ds e*Φ(*p*,*s*)

<sup>−</sup><sup>∞</sup> *ds e*Φ(*p*,*s*)

*ηF*(*p*) +

 ∞ 0 *ds*� *e* Φ(*p*� ,*s*� ) Tr 

 ∞ 0

*γμγ*5*G*(−*p*�

*ds*� *e* Φ(*p*� ,*s*� )

,*s*�

R*μν*(*p*, *p*�

)*γνG*(−*p*,*s*)

,*s*,*s* �  *ηF*(−*p*) 

) (4.20)

 *k*<sup>2</sup> � *m*<sup>2</sup> �−2*m*<sup>2</sup> *e* 

<sup>2</sup> *�μνρσFρσ*, and *�*<sup>0123</sup> <sup>=</sup> 1 is the dual of the field-strength tensor, with

� <sup>−</sup> <sup>2</sup>*m*<sup>2</sup> *e* 

+*R* 

 <sup>1</sup>−*v*<sup>2</sup> <sup>4</sup> *<sup>k</sup>*<sup>2</sup> *<sup>F</sup>μν* <sup>−</sup>(<sup>1</sup> <sup>−</sup> *<sup>v</sup>*2)*kμ*� (*Fk*

*<sup>k</sup>ν*<sup>⊥</sup> (*kF*˜)*<sup>μ</sup>* <sup>+</sup> *<sup>k</sup>μ*<sup>⊥</sup> (*kF*)*<sup>ν</sup>*


*<sup>F</sup>μν* <sup>−</sup>(<sup>1</sup> <sup>−</sup> *<sup>v</sup>*2)*kμ*� (*Fk*

, (3.17)

*<sup>a</sup><sup>γ</sup>* = *aAν*Γ*ν*(*k*) (3.18)

 *<sup>k</sup>*|| *m* 2 )*<sup>ν</sup>*

⊥ . In the

)*<sup>ν</sup>*

)*<sup>ν</sup>* <sup>=</sup> *<sup>F</sup>νμkμ*. Finally the

*<<* 1, thus the order

*aFμνF*˜*μν*. (3.19)

, (3.16)

<sup>2</sup><sup>Z</sup> sin <sup>Z</sup> *<sup>k</sup>*<sup>2</sup>

*dv eφ*<sup>0</sup>

*dv eφ*<sup>0</sup> *k<sup>μ</sup>*

L*B*

<sup>32</sup>*π*<sup>2</sup> *ga f*(*eQf*)<sup>2</sup>

In eqn.[3.18],we define the axion field by *<sup>a</sup>* and (*kF*)*<sup>ν</sup>* <sup>=</sup> *<sup>k</sup>μFμν* and (*Fk*

loop induced contribution to the axion photon effective Lagrangian is,

Since we are interested in *ω < m*, so the magnitude of the factor

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

*γμγ*5*G*(*p*,*s*)*γνG*(*p*�

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

> ,*s* � )

∞

Π*<sup>A</sup>*

above expression, *<sup>F</sup>μν* <sup>=</sup> <sup>1</sup>

Where, *R* =

to be given by,

medium; the result is:

*μν* (*k*)=(*iga feQf*)

=(*iga feQf*)

<sup>Π</sup>*A<sup>β</sup>*

*μν*(*k*)= *iga f*(*e Qf*)<sup>2</sup> (4*π*)<sup>2</sup>

vacuum, would be, <sup>Γ</sup>*ν*(*k*) = *<sup>k</sup>μ*Π*AB μν* (*k*) i.e.,

(4*π*)<sup>2</sup>

LB

*<sup>a</sup><sup>γ</sup>* == <sup>−</sup> <sup>1</sup>

**4. Contribution from the magnetized medium**

+ *R* 

<sup>Γ</sup>*ν*(*k*) = *iga f*(*e Qf*)<sup>2</sup>

 ∞ 0 *dt* +1 −1

1−*v* sin Z*v* sin Z−cos Z cos Z*v* sin2 <sup>Z</sup>

> ∞ 0 *dt* +1 −1

*<sup>k</sup>ν*<sup>⊥</sup> (*kF*˜)*<sup>μ</sup>* <sup>+</sup> *<sup>k</sup>μ*<sup>⊥</sup> (*kF*)*<sup>ν</sup>*

where R*μν*(*p*, *p*� ,*s*,*s*� ) contains the trace part. R*μν*(*p*, *p*� ,*s*,*s*� ) is a polynomial in powers of the external magnetic field with even and odd powers of B, can be presented as,

$$\mathbf{R}\_{\mu\nu}(p\_\prime p\_\prime^\prime, \mathbf{s}\_\prime s^\prime) = \mathbf{R}\_{\mu\nu}^{(\ominus)}(p\_\prime p\_\prime^\prime, \mathbf{s}\_\prime s^\prime) + \mathbf{R}\_{\mu\nu}^{(\ominus)}(p\_\prime p\_\prime^\prime, \mathbf{s}\_\prime s^\prime) \tag{4.21}$$

Fig. 1. One-loop diagram for the effective axion electromagnetic vertex .

We have denoted the pieces with even and odd powers in the external magnetic field strength <sup>B</sup> in R*μν*, as R(*E*) *μν* and R(*O*) *μν* . In addition to being just even and odd in powers of *eQf* B, they are also odd and even in powers of chemical potential, therefore, under charge conjugation they would transform as, B&*μ* ↔ (−*μ*)&(−B),i. e., both behave differently. More over their parity structures are also different. These properties come very useful while analyzing, the structure of axion photon coupling, using discrete symmetry arguments to justify the presence or absence of either of the two; that is the reason, why they should be treated separately. The details of this analysis can be found in [(A. K. Ganguly , 2006)].

### **4.0.1 Vertex function: even powers in B**

The expression for the R*<sup>E</sup> μν*, (that is the term with even powers of the magnetic field), comes out to be,

$$\mathcal{R}^{(E)}\_{\mu\nu} \stackrel{\circ}{=} 4i\eta \\_ {(p\_0)} \left[ \varepsilon\_{\mu\nu\alpha\beta} p^\mu k^\delta (1 + \tan(eQ\_f \mathcal{B}s) \tan(eQ\_f \mathcal{B}s')) + \varepsilon\_{\mu\nu\alpha\beta} k^\mu k^{\beta\_\perp} \right.$$

$$\times \tan(eQ\_f \mathcal{B}s) \tan(eQ\_f \mathcal{B}s') \frac{\tan(eQ\_f \mathcal{B}s) - \tan(eQ\_f \mathcal{B}s')}{\tan(eQ\_f \mathcal{B}s) + \tan(eQ\_f \mathcal{B}s')} \right]. \tag{4.22}$$

Because of the presence of *εμναβk<sup>β</sup>* and *εμναβ*<sup>⊥</sup> *<sup>k</sup>α*, it vanishes on contraction R(*E*) *μν* with *kν*.

The two point VA response function Π*A*(*k*), can be interpreted as a (one particle irreducible) two point vertex; with one point for the external axion line and the other one (Lorentz indexed) for the external photon line. But since the evaluations are done in presence of external magnetic field B they correspond to soft external photon line insertions. That is their four momenta *<sup>k</sup><sup>α</sup>* <sup>→</sup> 0 . If each soft external photon line contributes either +1 or -1 to the total spin ( angular momentum ) of the effective vertex, then, for an even order term in external field strength B the total spin of this piece would be a coherent sum of all the contributions from all the odd number of soft photon linesB. Now recall that in order arrive at the the expression for the effective interaction Lagrangian for *<sup>γ</sup>* <sup>−</sup> *<sup>a</sup>* from <sup>Π</sup>*<sup>A</sup> μν*(*x*)–we need to multiply the same (with some sort of naivete) by *a*(*x*)*Fμν*(*x*) . Therefore, it is worth noting that, if

In the light of these estimates, it is possible to write down the axion photon mixing Lagrangian, for low frequency photons in an external magnetic field, in the following way:

*<sup>a</sup><sup>γ</sup>* + L<sup>B</sup>

*e*2 <sup>32</sup>*π*<sup>2</sup> *<sup>a</sup>*FF, ˜

� *k*� *ω* �2

<sup>2</sup> <sup>−</sup> <sup>32</sup> � *<sup>k</sup>*�

*ω*

�2 Λ∑ *f*

� 4 + 4 3 � *k*� *m*

<sup>32</sup>*π*<sup>2</sup> ·

Therefore, in the limit of |*k*⊥| → 0 and *<sup>ω</sup> << mf* , one can write the total axion photon effective

We would like to point out that, the in medium corrections doesn't alter the tensorial structure of the same. It remains intact. However the parameter *M* , doesn't remain so. Apart from numerical factors it also starts depending on the kinematic factors. It is worth noting that, all the terms generated by loop induced corrections do respect **CPT**. Additionally, as we have analyzed already the total spin angular momentum is also conserved. The tree level photon axion interaction term in the Lagrangian as found in the literature is of the following form,

*<sup>M</sup> <sup>a</sup>*F*μν*.*F*˜*ext*

The bounds on various axion parameters are obtained by using this Lagrangian. As we have seen the medium and other corrections can affect the magnitude of *M* . Since *M* is related to the symmetry breaking scale, a change in the estimates of *M* would have reflection on the symmetry breaking scale and other axion parameters. This is the primary motivation for our dwelling on this part of the problem before moving into aspects of axion electrodynamics, that

Now that we are equipped with the necessary details of axion interactions with other particles, we can write down the relevant part of the Lagrangian that describes the Axion photon interaction. The tree level Lagrangian that describes the axion photon dynamics is given by,

F*μν*F˜ *μν* +

1 2 �

*∂μa∂μ<sup>a</sup>* <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

*a a*2 �

, (5.32)

*ga f*(*Qf*)

1

*<sup>a</sup><sup>γ</sup>* = −*gaγγ*

*<sup>a</sup><sup>γ</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> 32*π*<sup>2</sup> *<sup>a</sup><sup>γ</sup>* <sup>+</sup> <sup>L</sup>B,*μ*,*<sup>β</sup> <sup>a</sup><sup>γ</sup>* . (4.28)

*ga f*(*eQf*)

*ga f*(*eQf*)

*ga f*(*Qf*) 2 ⎤ ⎦ *e*2

*μν* , (4.31)

<sup>2</sup>*a*FF. ˜

<sup>2</sup>*a*FF. (4.29) ˜

57

<sup>32</sup>*π*<sup>2</sup> *<sup>a</sup>*FF. (4.30) ˜

�2� ∑ *f*

(Λ)∑ *f*

<sup>L</sup>*Total <sup>a</sup><sup>γ</sup>* <sup>=</sup> <sup>L</sup>*vac*

in Particle Physics and Photon Dispersion in Magnetized Media

<sup>L</sup>*vac*

LB

LB,*μ*,*<sup>β</sup> <sup>γ</sup><sup>a</sup>* <sup>=</sup> <sup>32</sup>

Lagrangian using eqn. [4.29], in the following form.

� 4 + 4 3 �*k*� *m* �2 � ∑ *f*

Where each of the terms are given by,

Introduction to Axion Photon Interaction

<sup>L</sup>*Total <sup>a</sup><sup>γ</sup>* =−

⎡

affects photon polarization.

**5. Axion photon mixing**

<sup>L</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> 4

F*μν*F*μν* +

1 4*M*

⎣*gaγγ*+

we multiply <sup>Π</sup>A ( Even B) *μν* (*x*) with *a*(*x*)*Fμν*(*x*), the number of photon lines become odd and number of spin zero pseudoscalar is also odd. Since the effective Lagrangian can be related to the generating functional of the vertex for transition of photons to axion, then for this case it would mean, odd number of photons are going to produce a spin zero pseudoscalar. That is odd number of spin one photons would combine to produce a spin zero axion— *which is impossible*, hence such a term better not exist. Interestingly enough, *that is what* we get to see here.

#### **4.0.2 Vertex function: odd powers in B**

The nonzero contribution to the vertex function would be coming from R*<sup>O</sup> μν*. More precisely, from the following term,

$$ik^{\mu} \mathcal{R}\_{\mu\nu}^{(\mathcal{O})} = 8m^2 \eta\_+(p) \left[ k^{\mu} \varepsilon\_{\mu\nu 12} (\tan(eQ\_f \mathcal{B} \mathbf{s}) + \tan(eQ\_f \mathcal{B} \mathbf{s}')) \right],\tag{4.23}$$

Placing all the factors and integral signs, the vertex function Γ*ν*(*k*) can be written as,

$$\begin{split} \Gamma\_{\boldsymbol{\nu}}(k) = \left( g\_{af} e Q\_f \right) \left( 8m^2 k^{\mu} \varepsilon\_{\mu\nu 12} \right) \int \frac{d^4 p}{(2\pi)^4} \eta\_+(p) \int\_{-\infty}^{\infty} \int\_0^{\infty} \mathrm{d}s^{\prime} e^{\Phi(p,s) + \Phi(p^{\prime}, s^{\prime})} \\ \qquad \times \left[ \tan(e Q\_f \mathcal{B}s) + \tan(e Q\_f \mathcal{B}s^{\prime}) \right] \end{split} \tag{4.24}$$

Upon performing the gaussian integrals for the perpendicular momentum components, there after taking limit |*k*| → 0 and assuming photon energy *ω < mf* one arrives at,

$$
\Gamma\_{\nu}(k) = -16 (\mathcal{g}\_{af}(e\mathcal{Q}\_f)^2) \left(\frac{k^{\mu}\tilde{\mathbf{F}}\_{\mu\nu}}{16\pi^2}\right) \Lambda(k\_{\parallel}^2 k \cdot \boldsymbol{\mu}, \boldsymbol{\beta}, \mu). \tag{4.25}
$$

All the informations about the medium, are contained in Λ(*k*<sup>2</sup> � , *<sup>k</sup>*� · *<sup>u</sup>*, *<sup>β</sup>*, *<sup>μ</sup>*) and it is given by.

$$\Delta(k\_{\parallel}^2 \mathbf{k} \cdot \mathbf{u}, \boldsymbol{\beta}, \boldsymbol{\mu}) = \oint \mathbf{d}^2 \boldsymbol{p}\_{\parallel} \left[ n\_F(|\boldsymbol{p}\_0|, \boldsymbol{\mu}) + n\_F(|\boldsymbol{p}\_0|\_{\prime} - \boldsymbol{\mu}) \right] \left( \frac{m^2 \delta(p\_{\parallel}^2 - m^2)}{(k\_{\parallel}^2 + 2(\boldsymbol{p} \cdot \boldsymbol{k})\_{\parallel})} \right) \tag{4.26}$$

In the expression above the temperature of the medium ( *β* = 1/*T*), number density of the fermions (which in turn is related to *μ*), mass of the particles in the loop (m), energy and longitudinal momentum of the photon ( i.e. *<sup>k</sup>*||). The statistical factor has already been evaluated in [(A. K. Ganguly , 2006)], in various limits. So instead of providing the same we state the result obtained in the limits *m* � *μ*, and limitT → 0. The value of the same in this limit is

$$\mathrm{Lt}\_{T\to 0}\Lambda \simeq -\frac{1}{2} \frac{\left|\frac{\mu}{m}\right|}{\sqrt{\left(1 + \left[\frac{\mu}{m}\right]^2\right)}}\tag{4.27}$$

In the limit *<sup>μ</sup>* � *<sup>m</sup>*, the right hand side of Eqn. [4.27] <sup>∼</sup> <sup>1</sup> <sup>2</sup> and when *μ* ∼ *m*, it would turn out to be <sup>∼</sup> <sup>1</sup> 2 √2

In the light of these estimates, it is possible to write down the axion photon mixing Lagrangian, for low frequency photons in an external magnetic field, in the following way:

$$
\mathcal{L}\_{a\gamma}^{\text{Total}} = \mathcal{L}\_{a\gamma}^{\text{vac}} + \mathcal{L}\_{a\gamma}^{\text{B}} + \mathcal{L}\_{a\gamma}^{\text{B,\mu,\beta}}.\tag{4.28}
$$

Where each of the terms are given by,

8 Will-be-set-by-IN-TECH

number of spin zero pseudoscalar is also odd. Since the effective Lagrangian can be related to the generating functional of the vertex for transition of photons to axion, then for this case it would mean, odd number of photons are going to produce a spin zero pseudoscalar. That is odd number of spin one photons would combine to produce a spin zero axion— *which is impossible*, hence such a term better not exist. Interestingly enough, *that is what* we get to see

The nonzero contribution to the vertex function would be coming from R*<sup>O</sup>*

<sup>8</sup>*m*2*kμεμν*<sup>12</sup>

after taking limit |*k*| → 0 and assuming photon energy *ω < mf* one arrives at,

Lt*T*→0<sup>Λ</sup> � −<sup>1</sup>

Placing all the factors and integral signs, the vertex function Γ*ν*(*k*) can be written as,

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

Upon performing the gaussian integrals for the perpendicular momentum components, there

2) *k<sup>μ</sup>*F˜ *μν* 16*π*<sup>2</sup>

*nF*(|*p*0|, *μ*) + *nF*(|*p*0|, −*μ*)

In the expression above the temperature of the medium ( *β* = 1/*T*), number density of the fermions (which in turn is related to *μ*), mass of the particles in the loop (m), energy and longitudinal momentum of the photon ( i.e. *<sup>k</sup>*||). The statistical factor has already been evaluated in [(A. K. Ganguly , 2006)], in various limits. So instead of providing the same we state the result obtained in the limits *m* � *μ*, and limitT → 0. The value of the same in this

2

 *μ m* 

1 + *<sup>μ</sup> m* 2 

× 

(2*π*)<sup>4</sup> *<sup>η</sup>*+(*p*)

*μν* (*x*) with *a*(*x*)*Fμν*(*x*), the number of photon lines become odd and

*<sup>k</sup>μεμν*12(tan(*eQf* <sup>B</sup>*s*) + tan(*eQf* <sup>B</sup>*s*�

 Λ(*k*<sup>2</sup>

 ∞ −∞ *ds* ∞ 0 *ds*�

tan(*eQf* B*s*) + tan(*eQf* B*s*

*μν*. More precisely,

, (4.23)

(4.24)

(4.26)

(4.27)

)) 

*e*Φ(*p*,*s*)+Φ(*p*�

� )  ,*s*� )

� , *<sup>k</sup>* · *<sup>u</sup>*, *<sup>β</sup>*, *<sup>μ</sup>*). (4.25)

� , *<sup>k</sup>*� · *<sup>u</sup>*, *<sup>β</sup>*, *<sup>μ</sup>*) and it is given by.

� <sup>−</sup> *<sup>m</sup>*2)

� <sup>+</sup> <sup>2</sup>(*<sup>p</sup>* · *<sup>k</sup>*)� )

<sup>2</sup> and when *μ* ∼ *m*, it would turn out

*m*2*δ*(*p*<sup>2</sup>

(*k*<sup>2</sup>

we multiply <sup>Π</sup>A ( Even B)

from the following term,

Λ(*k*<sup>2</sup>

limit is

to be <sup>∼</sup> <sup>1</sup> 2 √2 � , *<sup>k</sup>* · *<sup>u</sup>*, *<sup>β</sup>*, *<sup>μ</sup>*)=

**4.0.2 Vertex function: odd powers in B**

*ikμ*R(*O*)

Γ*ν*(*k*)=(*ga feQf*)

*μν* = <sup>8</sup>*m*2*η*+(*p*)

Γ*ν*(*k*) == −16(*ga f*(*eQf*)

All the informations about the medium, are contained in Λ(*k*<sup>2</sup>

 <sup>d</sup><sup>2</sup> *<sup>p</sup>*� 

In the limit *<sup>μ</sup>* � *<sup>m</sup>*, the right hand side of Eqn. [4.27] <sup>∼</sup> <sup>1</sup>

here.

$$\begin{aligned} \mathcal{L}\_{a\gamma}^{vac} &= -g\_{a\gamma\gamma} \frac{e^2}{32\pi^2} a \text{F\"{F}},\\ \mathcal{L}\_{a\gamma}^{\mathcal{B}} &= \frac{-1}{32\pi^2} \left[ 4 + \frac{4}{3} \left( \frac{k\_{\parallel}}{m} \right)^2 \right] \sum\_{f} g\_{af} (eQ\_f)^2 a \text{F\"{F}}. \end{aligned}$$

$$\mathcal{L}\_{\gamma a}^{\mathcal{B},\mu,\beta} = \frac{32}{32\pi^2} \cdot \left( \frac{k\_{\parallel}}{\omega} \right)^2 (\Lambda) \sum\_{f} g\_{af} (eQ\_f)^2 a \text{F\"{F}}. \tag{4.29}$$

Therefore, in the limit of |*k*⊥| → 0 and *<sup>ω</sup> << mf* , one can write the total axion photon effective Lagrangian using eqn. [4.29], in the following form.

$$\mathcal{L}\_{a\gamma}^{Total} = - \left[ g\_{a\gamma\gamma} + \left( 4 + \frac{4}{3} \left( \frac{k\_{\parallel}}{m} \right)^2 \right) \sum\_{f} g\_{af} (Q\_f)^2 - 32 \left( \frac{k\_{\parallel}}{\omega} \right)^2 \Lambda \sum\_{f} g\_{af} (Q\_f)^2 \right] \frac{\varepsilon^2}{32\pi^2} a \text{F\"{}F.} \tag{4.30}$$

We would like to point out that, the in medium corrections doesn't alter the tensorial structure of the same. It remains intact. However the parameter *M* , doesn't remain so. Apart from numerical factors it also starts depending on the kinematic factors. It is worth noting that, all the terms generated by loop induced corrections do respect **CPT**. Additionally, as we have analyzed already the total spin angular momentum is also conserved. The tree level photon axion interaction term in the Lagrangian as found in the literature is of the following form,

$$\frac{1}{M} \mathbf{a} \mathbf{F}^{\mu \nu}. \tilde{F}^{ext}\_{\mu \nu}. \tag{4.31}$$

The bounds on various axion parameters are obtained by using this Lagrangian. As we have seen the medium and other corrections can affect the magnitude of *M* . Since *M* is related to the symmetry breaking scale, a change in the estimates of *M* would have reflection on the symmetry breaking scale and other axion parameters. This is the primary motivation for our dwelling on this part of the problem before moving into aspects of axion electrodynamics, that affects photon polarization.

#### **5. Axion photon mixing**

Now that we are equipped with the necessary details of axion interactions with other particles, we can write down the relevant part of the Lagrangian that describes the Axion photon interaction. The tree level Lagrangian that describes the axion photon dynamics is given by,

$$\mathcal{L} = -\frac{1}{4} \mathbf{F}^{\mu \nu} \mathbf{F}\_{\mu \nu} + \frac{1}{4M} \mathbf{F}^{\mu \nu} \tilde{\mathbf{F}}\_{\mu \nu} + \frac{1}{2} \left( \partial\_{\mu} a \partial^{\mu} a - m\_{a}^{2} a^{2} \right), \tag{5.32}$$

One can solve for the eigen values of the eqn. [5.35], from the determinantal equation,

In eqn. [5.36] *j* can take either of the two values + or −, and the roots are as follows:

 *<sup>m</sup>*<sup>2</sup> *a* 2

M*<sup>j</sup>* −*ig*B*ω ig*B*<sup>ω</sup> <sup>m</sup>*<sup>2</sup>

*<sup>a</sup>* + M*<sup>j</sup>*

The equations of motion for the photon field with polarization vector in the perpendicular

The remaining single physical degree freedom for the photon, with polarization along the external magnetic field, gets coupled with the axion; and the equation of motion turns out to

It is possible to diagonalize eqn.[6.39] by a similarity transformation. We would denote the

 ≡ *c* −*s s c*

−*s c M*<sup>11</sup> *<sup>M</sup>*<sup>12</sup>

*s c* =

*<sup>M</sup>*<sup>21</sup> *<sup>M</sup>*22 *<sup>c</sup>* <sup>−</sup>*<sup>s</sup>*

*M*<sup>+</sup> 0 0 *M*<sup>−</sup>

*s c*

 *<sup>A</sup>*� *a* 

*<sup>z</sup>* ) **I** + *M*2×<sup>2</sup>

cos *θ* −sin *θ* sin *θ* cos *θ*

*c s*

with the following forms for the elements of the matrix M2×2, given by: *M*<sup>11</sup> =

*<sup>M</sup>*<sup>21</sup> *<sup>M</sup>*22 *<sup>c</sup>* <sup>−</sup>*<sup>s</sup>*

 

2

+ (*g*B*ω*)

2 

<sup>=</sup> 0. (5.36)

= 0 . (6.38)

= 0. (6.39)

, (6.40)

*<sup>a</sup>*. The value of the parameter *θ* is

*M*<sup>+</sup> 0 0 *M*<sup>−</sup>

=

, (6.41)

, (6.42)

, (6.43)

. (5.37)

59

 

*a* 2 ±

(*ω*<sup>2</sup> + *∂*<sup>2</sup>

*O* = 

*MD* <sup>=</sup> *<sup>O</sup>TM*<sup>2</sup>×2*<sup>O</sup>* <sup>=</sup>

0, *<sup>M</sup>*<sup>12</sup> <sup>=</sup> *ig*B*ω*, *<sup>M</sup>*<sup>21</sup> <sup>=</sup> <sup>−</sup>*ig*B*<sup>ω</sup>* and lastly *<sup>M</sup>*<sup>22</sup> <sup>=</sup> <sup>−</sup>*m*<sup>2</sup>

*c s*

−*s c M*<sup>11</sup> *<sup>M</sup>*<sup>12</sup>

 *<sup>c</sup>*2*M*<sup>11</sup> <sup>+</sup> *<sup>s</sup>*2*M*<sup>22</sup> <sup>+</sup> <sup>2</sup>*csM*<sup>12</sup> *<sup>M</sup>*12(*c*<sup>2</sup> <sup>−</sup> *<sup>s</sup>*2) + *cs*(*M*<sup>22</sup> <sup>−</sup> *<sup>M</sup>*11) *<sup>M</sup>*12(*c*<sup>2</sup> <sup>−</sup> *<sup>s</sup>*2) + *cs*(*M*<sup>22</sup> <sup>−</sup> *<sup>M</sup>*11) *<sup>s</sup>*2*M*<sup>11</sup> <sup>+</sup> *<sup>c</sup>*2*M*<sup>22</sup> <sup>−</sup> <sup>2</sup>*csM*<sup>12</sup>

Now equating the components of the matrix equation [6.43], one arrives at:

*MD* =

in short. The diagonal matrix can further be written as,

(*ω*<sup>2</sup> + *∂*<sup>2</sup> *z* ) *A*⊥ 

*<sup>M</sup>*<sup>±</sup> <sup>=</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

in Particle Physics and Photon Dispersion in Magnetized Media

**6. Equation of motion**

be,

direction to the external magnetic filed is,

Introduction to Axion Photon Interaction

diagonalizing matrix by *O*, given by,

fixed from the equality,

leading to,

here *ma* , is the axion mass and other quantities have their usual meaning. This effective Lagrangian shows the effect of mixing of a spin zero pseudo-scalar with two photons. If one of the dynamical photon field in eqn. [5.32] is replaced by an external magnetic field, one would recover the Lagrangian given by eqn.[4.31]. This mixing part can give rise to various interesting observable effects; however in this section we would consider, the change in the state of polarization of a plane polarized light beam, propagating in an external magnetic field, due to axion photon mixing. In order to perform that analysis, we start with the equation of motion for the photons and the axions, in an external magnetic field B , that follows from the interaction part of the Lagrangian in eqn. [5.32], as we replace one of the dynamical photon field by external magnetic field field.

This system that we are going to study involve the dynamics of three field Degrees Of Freedom (DOF). As we all know, that the massless spin one gauge fields in vacuum have just two degrees of freedom; so we have those two DOF and the last one is for the spin zero pseudoscalar Boson. In this simple illustrative analysis, we would ignore the transverse component of the momentum *<sup>k</sup>*⊥. With this simplification in mind we have three equations of motion, one each for: *<sup>A</sup>*⊥(*z*), *<sup>A</sup>*||(*z*) and *<sup>a</sup>*(*z*)–i.e., the three dynamical fields. Where *<sup>A</sup>*⊥(*z*) , the photon/gauge field with polarization vector directed along the perpendicular direction to the magnetic field, *<sup>A</sup>*||(*z*) the remaining component of the photon/gauge field having polarization vector lying along the magnetic field and ��*a*(*z*)�� the pseudoscalar Axion field. These three equations can be written in a compcat form e g.,

$$
\left[ \left( \omega^2 + \partial\_z^2 \right) \mathbf{I} + \mathcal{M} \right] \begin{pmatrix} A\_\perp \\ A\_\parallel \\ a \end{pmatrix} = \mathbf{0}.\tag{5.33}
$$

where **I** is a 3 × 3 identity matrix and **M** is the short hand notation for the following matrix.

$$\mathcal{M} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & ig\mathcal{B}\omega \\ 0 & -ig\mathcal{B}\omega & -m\_a^2 \end{pmatrix} \\ \text{,} \\ \tag{5.34}$$

usually termed as axion photon mixing matrix or simply the mixing matrix. As can be seen from eqn.[5.33], the transverse gauge degree of freedom gets decoupled from the rest, and the other two i.e., the longitudinal gauge degrees of freedom and pseudoscalar degree of freedom are coupled with each other. It is because of this particular way of evolution of the transverse and the parallel components of the gauge field, even magnetized vacuum would show dichoric effect.

In the off diagonal element of the matrix [5.34] given by, <sup>±</sup>*ig*B*ω*, <sup>B</sup> <sup>=</sup> <sup>B</sup>*E*sin (*α*ˆ), is the transverse part of the external magnetic field <sup>B</sup>*<sup>E</sup>* and *<sup>α</sup>*<sup>ˆ</sup> is the angle between the wave vector *<sup>k</sup>* and the external magnetic field <sup>B</sup>*<sup>E</sup>* and lastly in a short hand notation, *<sup>g</sup>* <sup>=</sup> <sup>1</sup> <sup>M</sup> . The nondiagonal part of the 3**x**3 matrix, in eqn. [5.34] can be written as,

$$M\_{2 \times 2} = \begin{pmatrix} 0 & \text{ig}\mathcal{B}\omega \\ -\text{ig}\mathcal{B}\omega & -m\_a^2 \end{pmatrix} \cdot \tag{5.35}$$

One can solve for the eigen values of the eqn. [5.35], from the determinantal equation,

$$
\begin{vmatrix}
\mathbf{M}\_{\dot{j}} & -ig\mathcal{B}\omega \\
ig\mathcal{B}\omega & m\_a^2 + \mathbf{M}\_{\dot{j}}
\end{vmatrix} = 0.
\tag{5.36}
$$

In eqn. [5.36] *j* can take either of the two values + or −, and the roots are as follows:

$$M\_{\pm} = -\frac{m\_a^2}{2} \pm \sqrt{\left[\left(\frac{m\_a^2}{2}\right)^2 + \left(g\mathcal{B}\omega\right)^2\right]}.\tag{5.37}$$

#### **6. Equation of motion**

10 Will-be-set-by-IN-TECH

here *ma* , is the axion mass and other quantities have their usual meaning. This effective Lagrangian shows the effect of mixing of a spin zero pseudo-scalar with two photons. If one of the dynamical photon field in eqn. [5.32] is replaced by an external magnetic field, one would recover the Lagrangian given by eqn.[4.31]. This mixing part can give rise to various interesting observable effects; however in this section we would consider, the change in the state of polarization of a plane polarized light beam, propagating in an external magnetic field, due to axion photon mixing. In order to perform that analysis, we start with the equation of motion for the photons and the axions, in an external magnetic field B , that follows from the interaction part of the Lagrangian in eqn. [5.32], as we replace one of the dynamical photon

This system that we are going to study involve the dynamics of three field Degrees Of Freedom (DOF). As we all know, that the massless spin one gauge fields in vacuum have just two degrees of freedom; so we have those two DOF and the last one is for the spin zero pseudoscalar Boson. In this simple illustrative analysis, we would ignore the transverse component of the momentum *<sup>k</sup>*⊥. With this simplification in mind we have three equations of motion, one each for: *<sup>A</sup>*⊥(*z*), *<sup>A</sup>*||(*z*) and *<sup>a</sup>*(*z*)–i.e., the three dynamical fields. Where *<sup>A</sup>*⊥(*z*) , the photon/gauge field with polarization vector directed along the perpendicular direction to the magnetic field, *<sup>A</sup>*||(*z*) the remaining component of the photon/gauge field having polarization vector lying along the magnetic field and ��*a*(*z*)�� the pseudoscalar Axion field.

field by external magnetic field field.

show dichoric effect.

These three equations can be written in a compcat form e g.,

�

(*ω*<sup>2</sup> + *∂*<sup>2</sup>

M =

nondiagonal part of the 3**x**3 matrix, in eqn. [5.34] can be written as,

*M*2×<sup>2</sup> =

⎛ ⎝

*<sup>z</sup>* ) **I** + M

where **I** is a 3 × 3 identity matrix and **M** is the short hand notation for the following matrix.

00 0 0 0 *ig*B*ω* <sup>0</sup> <sup>−</sup>*ig*B*<sup>ω</sup>* <sup>−</sup>*m*<sup>2</sup>

usually termed as axion photon mixing matrix or simply the mixing matrix. As can be seen from eqn.[5.33], the transverse gauge degree of freedom gets decoupled from the rest, and the other two i.e., the longitudinal gauge degrees of freedom and pseudoscalar degree of freedom are coupled with each other. It is because of this particular way of evolution of the transverse and the parallel components of the gauge field, even magnetized vacuum would

In the off diagonal element of the matrix [5.34] given by, <sup>±</sup>*ig*B*ω*, <sup>B</sup> <sup>=</sup> <sup>B</sup>*E*sin (*α*ˆ), is the transverse part of the external magnetic field <sup>B</sup>*<sup>E</sup>* and *<sup>α</sup>*<sup>ˆ</sup> is the angle between the wave vector

> � <sup>0</sup> *ig*B*<sup>ω</sup>* <sup>−</sup>*ig*B*<sup>ω</sup>* <sup>−</sup>*m*<sup>2</sup>

*a* �

*<sup>k</sup>* and the external magnetic field <sup>B</sup>*<sup>E</sup>* and lastly in a short hand notation, *<sup>g</sup>* <sup>=</sup> <sup>1</sup>

� ⎛

⎜⎝

*A*⊥ *A*� *a*

*a*

⎞

⎞

⎟⎠ <sup>=</sup> 0. (5.33)

⎠ , (5.34)

. (5.35)

<sup>M</sup> . The

The equations of motion for the photon field with polarization vector in the perpendicular direction to the external magnetic filed is,

$$\left[\left(\omega^2 + \partial\_z^2\right)\right] \left(A\_\perp\right) = 0\,. \tag{6.38}$$

The remaining single physical degree freedom for the photon, with polarization along the external magnetic field, gets coupled with the axion; and the equation of motion turns out to be,

$$
\left[\left(\omega^2 + \partial\_z^2\right)\mathbf{I} + M\_{2\times 2}\right] \begin{pmatrix} A\_{\parallel} \\ a \end{pmatrix} = \mathbf{0}.\tag{6.39}
$$

It is possible to diagonalize eqn.[6.39] by a similarity transformation. We would denote the diagonalizing matrix by *O*, given by,

$$O = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \equiv \begin{pmatrix} c & -s \\ s & c \end{pmatrix} ,\tag{6.40}$$

in short. The diagonal matrix can further be written as,

$$M\_D = \mathbf{O}^T M\_{2 \times 2} \mathbf{O} = \begin{pmatrix} \mathbf{c} & \mathbf{s} \\ -\mathbf{s} & \mathbf{c} \end{pmatrix} \begin{pmatrix} M\_{11} & M\_{12} \\ M\_{21} & M\_{22} \end{pmatrix} \begin{pmatrix} \mathbf{c} - \mathbf{s} \\ \mathbf{s} & \mathbf{c} \end{pmatrix},\tag{6.41}$$

with the following forms for the elements of the matrix M2×2, given by: *M*<sup>11</sup> = 0, *<sup>M</sup>*<sup>12</sup> <sup>=</sup> *ig*B*ω*, *<sup>M</sup>*<sup>21</sup> <sup>=</sup> <sup>−</sup>*ig*B*<sup>ω</sup>* and lastly *<sup>M</sup>*<sup>22</sup> <sup>=</sup> <sup>−</sup>*m*<sup>2</sup> *<sup>a</sup>*. The value of the parameter *θ* is fixed from the equality,

$$M\_D = \begin{pmatrix} c & s \\ -s & c \end{pmatrix} \begin{pmatrix} M\_{11} & M\_{12} \\ M\_{21} & M\_{22} \end{pmatrix} \begin{pmatrix} c - s \\ s & c \end{pmatrix} = \begin{pmatrix} M\_+ & 0 \\ 0 & M\_- \end{pmatrix} . \tag{6.42}$$

leading to,

$$
\begin{pmatrix}
\varepsilon^2 M\_{11} + \varepsilon^2 M\_{22} + 2\varepsilon s M\_{12} & M\_{12}(\varepsilon^2 - s^2) + \varepsilon s (M\_{22} - M\_{11}) \\
 M\_{12}(\varepsilon^2 - s^2) + \varepsilon s (M\_{22} - M\_{11}) & \varepsilon^2 M\_{11} + \varepsilon^2 M\_{22} - 2\varepsilon s M\_{12}
\end{pmatrix} = \begin{pmatrix} M\_+ & 0 \\ 0 & M\_- \end{pmatrix},\tag{6.43}
$$

Now equating the components of the matrix equation [6.43], one arrives at:

Since,

form:

*< A*∗

*< A*∗

*< A*∗

it follows from there that,

Introduction to Axion Photon Interaction

*A*¯

 = *O* 

*<sup>A</sup>*||(*z*) *a*(*z*)

in Particle Physics and Photon Dispersion in Magnetized Media

cos2*θ* + *eik*�

*<sup>A</sup>*||(*z*)=

*<sup>A</sup>*⊥(*z*)=*<sup>e</sup>*

defined as an ensemble average of direct product of two vectors:

*<sup>A</sup>*||(*z*) *<sup>A</sup>*⊥(*z*)

Using eqn.[9.55] we arrive at the relation,

*e ik*+*z*

*<sup>A</sup>*||(*z*) =

*<sup>a</sup>*(*z*) = *e ik*+*<sup>z</sup>* <sup>−</sup> *<sup>e</sup> ik*� +*z* 

gauge fields take the follwing form,



*ρ*(*z*) = �

<sup>⊥</sup>(*z*)*A*⊥(*z*) *<sup>&</sup>gt;*=*<sup>&</sup>lt; <sup>A</sup>*<sup>∗</sup>

**10. Digression on stokes parameters**

 *<sup>A</sup>*||(*z*) *<sup>A</sup>*⊥(*z*)

*ρ*(*z*) → *ρ*�

 ⊗ 

perpendicular the || and ⊥ components, would convert:

(*z*) = �R(*α*)


> +*z* sin2*θ*

*e ik*+*z*

 = *O<sup>T</sup>*

> *eik*+*<sup>z</sup>* 0 0 *eik*� +*z O<sup>T</sup>*

cos*<sup>θ</sup>* sin*<sup>θ</sup> <sup>A</sup>*||(0) +

If we assume the axion field to be zero, to begin with, i.e., *a*(0) = 0, then the solution for the

cos2*θ* + *e*

Now we can compute various correlation functions with the photon field. The correlation functions of parallel and perpendicular components of the photon field take the following

cos4*<sup>θ</sup>* <sup>+</sup> sin4*<sup>θ</sup>* <sup>+</sup> 2 sin2*<sup>θ</sup>* cos2*<sup>θ</sup>* cos *k*<sup>+</sup> <sup>−</sup> *<sup>k</sup>*�

Various optical parameters like polarization, ellipticity and degree of polarization of a given light beam can be found from the the coherency matrix constructed from various correlation functions given above. The coherency matrix, for a system with two degree of freedom is

> ∗ � =

 *<sup>A</sup>*||(*z*) *<sup>A</sup>*⊥(*z*)

The important thing to note here is that, under any anticlock-wise rotation *α* about an axis

 ⊗ 

cos<sup>2</sup> *θei*(*k*⊥−*k*+)*<sup>z</sup>* + sin2*θ ei*(*k*⊥−*k*�

*<sup>A</sup>*||(0) +

*ik*� <sup>+</sup>*<sup>z</sup>*sin2*θ <sup>A</sup>*||(0)

*e ik*+*z*

> +)*z < A*∗

�*A*||(*z*)*A*<sup>∗</sup>

*<sup>A</sup>*||(*z*) *<sup>A</sup>*⊥(*z*)

�*A*<sup>∗</sup>

<sup>⊥</sup>(0)*A*⊥(0) *<sup>&</sup>gt;* (9.59)

*<sup>A</sup>*||(*z*/0) *a*(*z*/0)

*<sup>A</sup>*||(0) *a*(0)

*<sup>e</sup>ik*+*<sup>z</sup>* <sup>−</sup> *<sup>e</sup>ik*�

+*z* 

> +*z* cos2*θ*

sin2*θ* + *eik*�

*ik*⊥*zA*⊥(0). (9.58)

+ *z < A*∗




∗

⊥(*z*)�

(10.60)

⊥(*z*)�

<sup>R</sup>−1(*α*)� (10.61)

. (9.54)

. (9.55)

cos*θ* sin*θ a*(0) (9.56)

*a*(0) (9.57)

61


$$\tan(2\theta) = \frac{2M\_{12}}{M\_{11} - M\_{22}} = \frac{2\text{ig}\mathcal{B}\omega}{m\_a^2}.\tag{6.44}$$

Therefore upon using this similarity transformation, the coupled Axion photon differential equation can further be brought to the following form,

$$
\left[\left(\omega^2 + \partial\_z^2\right)\mathbf{I} + M\_D \right] \begin{pmatrix} \bar{A}\_{\parallel} \\ \bar{a} \end{pmatrix} = 0. \tag{6.45}
$$

#### **7. Dispersion relations**

Defining the wave vectors in terms of *ki*'s, as:

$$\begin{aligned} k\_{\perp} &= \omega \\ k\_{+} &= \sqrt{\omega^2 + \mathcal{M}\_{+}} \\ k\_{-} &= -\sqrt{\omega^2 + \mathcal{M}\_{+}} \end{aligned} \tag{7.46}$$

and

$$\begin{aligned} k\_+' &= \sqrt{\omega^2 + M\_-} \\ k\_-' &= -\sqrt{\omega^2 + M\_-} \end{aligned} \tag{7.47}$$

#### **8. Solutions**

The solutions for the gauge field and the axion field, given by [6.45] as well as the solution for eqn. for *<sup>A</sup>*<sup>⊥</sup> in *<sup>k</sup>* space can be written as,

$$A\_{||}(z) = A\_{||\_{+}}(0)e^{ik\_{+}z} + A\_{||\_{-}}(0)e^{-ik\_{-}z} \tag{8.48}$$

$$
\vec{a}(z) = \vec{a}\_{+}(0) \, e^{i\vec{k}\_{+}'z} + \vec{a}\_{-}(0) \, e^{-i\vec{k}\_{-}'z} \tag{8.49}
$$

$$A\_{\perp}(z) = A\_{\perp\_{+}}(0)e^{i\vec{k}\_{\perp}z} + A\_{\perp\_{-}}(0)e^{-i\vec{k}\_{\perp}z} \tag{8.50}$$

#### **9. Correlation functions**

The solutions for propagation along the +ve z axis, is given by,

$$
\bar{A}\_{||}(z) = \bar{A}\_{||\_{+}}(0)e^{ik\_{+}z} \tag{9.51}
$$

$$
\vec{a}(z) = \vec{a}\_+(0) \ e^{i k'\_+ z} \tag{9.52}
$$

that can further be written in the following form,

$$
\begin{pmatrix}
\bar{A}\_{||}(z) \\
\bar{a}(z)
\end{pmatrix} = \begin{pmatrix}
e^{ik\_+z} & 0 \\
0 & e^{ik\_+'z}
\end{pmatrix} \begin{pmatrix}
\bar{A}\_{||}(0) \\
\bar{a}(0)
\end{pmatrix}.
\tag{9.53}
$$

Since,

12 Will-be-set-by-IN-TECH

*M*<sup>11</sup> − *M*<sup>22</sup>

*<sup>z</sup>* ) **I** + *MD*

*<sup>k</sup>*<sup>⊥</sup> = *<sup>ω</sup> k*+ =

*k*<sup>−</sup> = −

*k*� <sup>+</sup> =

*k*� <sup>−</sup> <sup>=</sup> <sup>−</sup>

The solutions for the gauge field and the axion field, given by [6.45] as well as the solution for

*ik*�

*ik*+*<sup>z</sup>* + *A*¯

<sup>+</sup>*<sup>z</sup>* <sup>+</sup> *<sup>a</sup>*¯−(0) *<sup>e</sup>*


−*ik*�

*<sup>A</sup>*⊥(*z*) =*A*⊥<sup>+</sup> (0)*eik*⊥*<sup>z</sup>* <sup>+</sup> *<sup>A</sup>*⊥− (0)*e*−*ik*⊥*<sup>z</sup>* (8.50)


*ω*<sup>2</sup> + *M*<sup>+</sup>

*<sup>ω</sup>*<sup>2</sup> <sup>+</sup> *<sup>M</sup>*<sup>−</sup>

Therefore upon using this similarity transformation, the coupled Axion photon differential

<sup>=</sup> <sup>2</sup>*ig*B*<sup>ω</sup> m*2 *a*

 *A*¯ � *a*¯  . (6.44)

= 0. (6.45)

*ω*<sup>2</sup> + *M*<sup>+</sup> (7.46)

*<sup>ω</sup>*<sup>2</sup> + *<sup>M</sup>*<sup>−</sup> (7.47)


<sup>+</sup>*<sup>z</sup>* (9.52)

. (9.53)

<sup>−</sup>*ik*−*<sup>z</sup>* (8.48)

<sup>−</sup>*<sup>z</sup>* (8.49)

tan(2*θ*) = <sup>2</sup>*M*<sup>12</sup>

equation can further be brought to the following form,

Defining the wave vectors in terms of *ki*'s, as:

eqn. for *<sup>A</sup>*<sup>⊥</sup> in *<sup>k</sup>* space can be written as,

**9. Correlation functions**

*A*¯

The solutions for propagation along the +ve z axis, is given by,

that can further be written in the following form,

*A*¯ ||(*z*) *a*¯(*z*)


*a*¯(*z*) = *a*¯+(0) *e*

*A*¯

 = 


*a*¯(*z*) = *a*¯+(0) *eik*�

*eik*+*<sup>z</sup>* 0 0 *eik*� +*z*

*A*¯


**7. Dispersion relations**

and

**8. Solutions**

(*ω*<sup>2</sup> + *∂*<sup>2</sup>

$$
\begin{pmatrix}
\bar{A}\_{||}(z/0) \\
\bar{a}(z/0)
\end{pmatrix} = O^T \begin{pmatrix}
A\_{||}(z/0) \\
a(z/0)
\end{pmatrix} . \tag{9.54}
$$

it follows from there that,

$$
\begin{pmatrix} A\_{||}(z) \\ a(z) \end{pmatrix} = \mathcal{O} \begin{pmatrix} \epsilon^{ik\_+ z} & 0 \\ 0 & \epsilon^{ik\_+' z} \end{pmatrix} \mathcal{O}^T \begin{pmatrix} A\_{||}(0) \\ a(0) \end{pmatrix} . \tag{9.55}
$$

Using eqn.[9.55] we arrive at the relation,

$$A\_{||}(z) = \left[e^{i\vec{k}\_{+}z}\cos^{2}\theta + e^{i\vec{k}\_{+}^{\prime}z}\sin^{2}\theta\right]A\_{||}(0) + \left[e^{i\vec{k}\_{+}z} - e^{i\vec{k}\_{+}^{\prime}z}\right]\cos\theta\sin\theta\,a(0)\tag{9.56}$$

$$a(z) = \left[e^{ik\_+z} - e^{ik'\_+z}\right] \cos\theta \sin\theta \, A\_{||}(0) + \left[e^{ik\_+z} \sin^2\theta + e^{ik'\_+z} \cos^2\theta\right] a(0) \tag{9.57}$$

If we assume the axion field to be zero, to begin with, i.e., *a*(0) = 0, then the solution for the gauge fields take the follwing form,

$$A\_{||}(z) = \left[e^{ik\_+z}\cos^2\theta + e^{ik\_+'z}\sin^2\theta\right]A\_{||}(0)$$

$$A\_{\perp}(z) = e^{i\mathbf{k}\_{\perp}z}A\_{\perp}(0). \tag{9.58}$$

Now we can compute various correlation functions with the photon field. The correlation functions of parallel and perpendicular components of the photon field take the following form:

$$ = \left[\cos^{4}\theta + \sin^{4}\theta + 2\sin^{2}\theta\cos^{2}\theta\cos\left[(k\_{+}-k\_{+}^{\prime})z\right]\right]$$

$$ = \left[\cos^{2}\theta e^{i(k\_{\perp}-k\_{+})z} + \sin^{2}\theta e^{i(k\_{\perp}-k\_{+}^{\prime})z}\right]$$

$$ = \tag{9.59}$$

#### **10. Digression on stokes parameters**

Various optical parameters like polarization, ellipticity and degree of polarization of a given light beam can be found from the the coherency matrix constructed from various correlation functions given above. The coherency matrix, for a system with two degree of freedom is defined as an ensemble average of direct product of two vectors:

$$\rho(\mathbf{z}) = \langle \begin{pmatrix} A\_{||}(\mathbf{z}) \\ A\_{\perp}(\mathbf{z}) \end{pmatrix} \otimes \begin{pmatrix} A\_{||}(\mathbf{z}) \ A\_{\perp}(\mathbf{z}) \end{pmatrix}^\* \rangle = \begin{pmatrix} \langle A\_{||}(\mathbf{z}) A\_{||}^\*(\mathbf{z}) \rangle \ \langle A\_{||}(\mathbf{z}) A\_{\perp}^\*(\mathbf{z}) \rangle \\ \langle A\_{||}^\*(\mathbf{z}) A\_{\perp}(\mathbf{z}) \rangle \ \langle A\_{\perp}(\mathbf{z}) A\_{\perp}^\*(\mathbf{z}) \rangle \end{pmatrix} \tag{10.60}$$

The important thing to note here is that, under any anticlock-wise rotation *α* about an axis perpendicular the || and ⊥ components, would convert:

$$\rho(z)\rightarrow\rho'(z)=\langle\mathcal{R}(a)\begin{pmatrix}A\_{||}(z)\\A\_{\perp}(z)\end{pmatrix}\otimes\begin{pmatrix}A\_{||}(z)&A\_{\perp}(z)\end{pmatrix}^\*\mathcal{R}^{-1}(a)\rangle\tag{10.61}$$

The ellipticity angle, *χ*, following [10.67], can be shown to be equal to,

and the polarization angle can be shown to be equal to.

in Particle Physics and Photon Dispersion in Magnetized Media

Introduction to Axion Photon Interaction

gets additional increment by twice the rotation angle, i.e.,

the degree of polarization is usually expressed by,

where I*PT* is the total intensity of the light beam.

stokes parameters and they turn out to be

cos2*<sup>θ</sup>* cos [(*k*<sup>⊥</sup> <sup>−</sup> *<sup>k</sup>*+) *<sup>z</sup>*]

cos2*<sup>θ</sup>* sin [(*k*<sup>⊥</sup> <sup>−</sup> *<sup>k</sup>*+) *<sup>z</sup>*]

cos4*<sup>θ</sup>* <sup>+</sup> sin4*<sup>θ</sup>* <sup>+</sup> 2 sin2*<sup>θ</sup>* cos2*<sup>θ</sup>* cos *k*<sup>+</sup> <sup>−</sup> *<sup>k</sup>*�

cos4*<sup>θ</sup>* <sup>+</sup> sin4*<sup>θ</sup>* <sup>+</sup> 2 sin2*<sup>θ</sup>* cos2*<sup>θ</sup>* cos *k*<sup>+</sup> <sup>−</sup> *<sup>k</sup>*�

I= 

Q= 

U=2

V=2

tan2*<sup>χ</sup>* <sup>=</sup> <sup>V</sup>

From the relations given above, it is easy to see that, under the frame rotation,

*p* =

**11. Evaluation of ellipticity (***χ***) and polarization (***ψ***) angles**

tan2*<sup>ψ</sup>* <sup>=</sup> <sup>U</sup>

<sup>R</sup>(*α*) = cos 2*<sup>α</sup>* sin 2*<sup>α</sup>*

tan(2*χ*) → tan(2*χ*)

the Tangent of *χ*, i.e., tan*χ* remains invariant, however the tangent of the polarization angle

It is worth noting that the two angles are not quite independent of each other, in fact they are ralated to each other. Finally we end the discussion of use of stokes parameters by noting that,

> Q<sup>2</sup> + U2 + V<sup>2</sup> I*PT*

Now we would proceed further from the formula given in the previous sections, to evaluate the ellipticity and polarization angles for a beam of plane polarized light propagating in the z direction. Since we are interested in finding out the effect of axion photon mixing, we need the expressions for the Stokes parameters with the Axion photon mixing effect and with that we would evaluate the ellipticity angle *χ* and polaraization angle *ψ* at a distance *z* from the source. Using the expressions for the correlators (i.e., eqns. [9.59] ) , one can evaluate the

> + *z < A*∗

> + *z < A*∗

> > + *z < A*∗

+ *z < A*∗

<sup>+</sup> sin2*θ*cos *k*<sup>⊥</sup> <sup>−</sup> *<sup>k</sup>*�

<sup>+</sup> sin2*θ*sin *k*<sup>⊥</sup> <sup>−</sup> *<sup>k</sup>*�

−sin 2*α* cos 2*α*

tan(2*ψ*) → tan(2*α* + 2*ψ*). (10.71)





<sup>Q</sup><sup>2</sup> <sup>+</sup> U2 , (10.68)

<sup>Q</sup> (10.69)

(10.70)

63

(10.72)

<sup>⊥</sup>(0)*A*⊥(0) *<sup>&</sup>gt;*

<sup>⊥</sup>(0)*A*⊥(0) *<sup>&</sup>gt;*

where R(*α*) is the rotation matrix. Now from the relations between the components of coherency matrix and the stokes parameters:

$$\mathcal{I} =  + ,$$

$$\mathcal{Q} =  - ,$$

$$\mathcal{U} = 2\text{Re} < \mathcal{A}\_{||}^{\*}(z)A\_{\perp}(z)>,$$

$$\mathcal{V} = 2\text{Im} < \mathcal{A}\_{||}^{\*}(z)A\_{\perp}(z)>. \tag{10.62}$$

It is easy to establish that,

$$\rho(z) = \frac{1}{2} \begin{pmatrix} \mathcal{I}(z) + \mathcal{Q}(z) & \mathcal{U}(z) - i\mathcal{V}(z) \\ \mathcal{U}(z) + i\mathcal{V}(z) & \mathcal{I}(z) - \mathcal{Q}(z) \end{pmatrix} \tag{10.63}$$

Therefore, under an anticlock wise rotation by an angle *α*, about an axis perpendicular to the plane containing *<sup>A</sup>*�(*z*) and *<sup>A</sup>*⊥(*z*), the density matrix transforms as: *<sup>ρ</sup>*(*z*) <sup>→</sup> *<sup>ρ</sup>*� (*z*); the same in the rotated frame would be given by,

$$\rho'(z) = \frac{1}{2}\mathcal{R}(\mathfrak{a}) \begin{pmatrix} \mathcal{I}(z) + \mathcal{Q}(z) & \mathcal{U}(z) - i\mathcal{V}(z) \\ \mathcal{U}(z) + i\mathcal{V}(z) & \mathcal{I}(z) - \mathcal{Q}(z) \end{pmatrix} \mathcal{R}^{-1}(\mathfrak{a}) \,. \tag{10.64}$$

For a rotation by an angle *α*–in the anticlock direction– about an axis perpendicular to *A*� and *<sup>A</sup>*<sup>⊥</sup> plane, the rotation matrix R(*α*) is,

$$\mathcal{R}(\alpha) = \begin{pmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{pmatrix}. \tag{10.65}$$

From the relations above, its easy to convince oneself that, in the rotated frame of reference the two stokes parameters, Q and U get related to the same in the unrotated frame, by the following relation.

$$
\begin{pmatrix} \mathbf{Q}'(z) \\ \mathbf{U}'(z) \end{pmatrix} = \begin{pmatrix} \cos 2\alpha & \sin 2\alpha \\ -\sin 2\alpha & \cos 2\alpha \end{pmatrix} \begin{pmatrix} \mathbf{Q}(z) \\ \mathbf{U}(z) \end{pmatrix} \tag{10.66}
$$

The other two parameters, i.e., I and V remain unaltered. It is for this reason that some times I and V are termed invariants under rotation.

For a little digression, we would like to point out that, in a particular frame, the Stokes parameters are expressed in terms of two angular variables *χ* and *ψ* usually called the ellipticity parameter and polarization angle, defined as,

$$\begin{aligned} \mathbf{I} &= \mathbf{I}\_p \\ \mathbf{Q} &= \mathbf{I}\_p \cos 2\psi \cos 2\chi \\ \mathbf{U} &= \mathbf{I}\_p \sin 2\psi \cos 2\chi \\ \mathbf{V} &= \mathbf{I}\_p \sin 2\chi. \end{aligned} \tag{10.67}$$

14 Will-be-set-by-IN-TECH

where R(*α*) is the rotation matrix. Now from the relations between the components of

<sup>⊥</sup>(*z*)*A*⊥(*z*) *<sup>&</sup>gt;*,

<sup>⊥</sup>(*z*)*A*⊥(*z*) *<sup>&</sup>gt;*,


 Q(*z*) U(*z*) 

V = I*p*sin 2*χ*. (10.67)

(10.63)

(10.66)

(*z*); the same

<sup>R</sup>−1(*α*) . (10.64)

. (10.65)




I(*z*) + Q(*z*) U(*z*) − *i*V(*z*) U(*z*) + *i*V(*z*) I(*z*) − Q(*z*)

I(*z*) + Q(*z*) U(*z*) − *i*V(*z*) U(*z*) + *i*V(*z*) I(*z*) − Q(*z*)

> cos *α* sin *α* −sin *α* cos *α*

Therefore, under an anticlock wise rotation by an angle *α*, about an axis perpendicular to the

For a rotation by an angle *α*–in the anticlock direction– about an axis perpendicular to *A*� and

From the relations above, its easy to convince oneself that, in the rotated frame of reference the two stokes parameters, Q and U get related to the same in the unrotated frame, by the

> cos 2*α* sin 2*α* −sin 2*α* cos 2*α*

The other two parameters, i.e., I and V remain unaltered. It is for this reason that some times

For a little digression, we would like to point out that, in a particular frame, the Stokes parameters are expressed in terms of two angular variables *χ* and *ψ* usually called the

> Q = I*p*cos 2*ψ* cos 2*χ* U = I*p*sin 2*ψ* cos 2*χ*

I = I*<sup>p</sup>*

plane containing *<sup>A</sup>*�(*z*) and *<sup>A</sup>*⊥(*z*), the density matrix transforms as: *<sup>ρ</sup>*(*z*) <sup>→</sup> *<sup>ρ</sup>*�

R(*α*) =

 =

coherency matrix and the stokes parameters:

It is easy to establish that,

in the rotated frame would be given by,

*ρ*�

*<sup>A</sup>*<sup>⊥</sup> plane, the rotation matrix R(*α*) is,

following relation.

(*z*) = <sup>1</sup> 2 R(*α*) 

> Q� (*z*) U� (*z*)

ellipticity parameter and polarization angle, defined as,

I and V are termed invariants under rotation.

I=*< A*∗

Q=*< A*∗

U=2Re *<* A∗

V=2 Im *<* A∗

*<sup>ρ</sup>*(*z*) = <sup>1</sup> 2  The ellipticity angle, *χ*, following [10.67], can be shown to be equal to,

$$\tan 2\chi = \frac{\mathcal{V}}{\sqrt{\mathcal{Q}^2 + \mathcal{U}^2}} \; \; \; \tag{10.68}$$

and the polarization angle can be shown to be equal to.

$$
\tan 2\psi = \frac{\mathcal{U}}{\mathcal{Q}} \tag{10.69}
$$

From the relations given above, it is easy to see that, under the frame rotation,

$$\mathcal{R}(\alpha) = \begin{pmatrix} \cos 2\alpha & \sin 2\alpha \\ -\sin 2\alpha & \cos 2\alpha \end{pmatrix} \tag{10.70}$$

the Tangent of *χ*, i.e., tan*χ* remains invariant, however the tangent of the polarization angle gets additional increment by twice the rotation angle, i.e.,

$$
\tan(2\chi) \to \tan(2\chi)
$$

$$
\tan(2\psi) \to \tan(2\alpha + 2\psi).\tag{10.71}
$$

It is worth noting that the two angles are not quite independent of each other, in fact they are ralated to each other. Finally we end the discussion of use of stokes parameters by noting that, the degree of polarization is usually expressed by,

$$p = \frac{\sqrt{\mathbf{Q}^2 + \mathbf{U}^2 + \mathbf{V}^2}}{\mathbf{I}\_{P\_T}} \tag{10.72}$$

where I*PT* is the total intensity of the light beam.

#### **11. Evaluation of ellipticity (***χ***) and polarization (***ψ***) angles**

Now we would proceed further from the formula given in the previous sections, to evaluate the ellipticity and polarization angles for a beam of plane polarized light propagating in the z direction. Since we are interested in finding out the effect of axion photon mixing, we need the expressions for the Stokes parameters with the Axion photon mixing effect and with that we would evaluate the ellipticity angle *χ* and polaraization angle *ψ* at a distance *z* from the source. Using the expressions for the correlators (i.e., eqns. [9.59] ) , one can evaluate the stokes parameters and they turn out to be

$$\mathcal{I} = \left[\cos^4\theta + \sin^4\theta + 2\sin^2\theta\cos^2\theta\cos\left[(k\_+ - k\_+')\,\boldsymbol{z}\right]\right] < A\_{\parallel}^\*(0)A\_{\parallel}(0)> + < A\_{\parallel}^\*(0)A\_{\perp}(0)>$$

$$\mathcal{Q} = \left[\cos^4\theta + \sin^4\theta + 2\sin^2\theta\cos^2\theta\cos\left[(k\_+ - k\_+')\,\boldsymbol{z}\right]\right] < A\_{\parallel}^\*(0)A\_{\parallel}(0)> - < A\_{\perp}^\*(0)A\_{\perp}(0)>$$

$$\mathcal{U} = 2\left(\left[\cos^2\theta\cos\left[(k\_\perp - k\_+)\,\boldsymbol{z}\right]\right] + \sin^2\theta\cos\left[(k\_\perp - k\_+')\,\boldsymbol{z}\right]\right) < A\_{\parallel}^\*(0)A\_{\perp}(0)>$$

$$\mathcal{V} = 2\left(\left[\cos^2\theta\sin\left[(k\_\perp - k\_+)\,\boldsymbol{z}\right]\right] + \sin^2\theta\sin\left[(k\_\perp - k\_+')\,\boldsymbol{z}\right]\right) < A\_{\parallel}^\*(0)A\_{\perp}(0)>\tag{11.73}$$

*<sup>χ</sup>* <sup>=</sup> <sup>1</sup> 96*ω*

photons get reflected, the axions are lost, they don't get reflected from the mirror.

tan(2*ψ* +

 sin2 

at the expression for *ψ*. Once substituted the polarization angle turns out to be.

*ψ* = <sup>B</sup>*Ez* 2

<sup>Q</sup> <sup>=</sup> <sup>−</sup>2*θ*<sup>2</sup>

angle from the expression

Introduction to Axion Photon Interaction

in Particle Physics and Photon Dispersion in Magnetized Media

following relation,

order *θ*2. Once this is done, we arrive at:

 <sup>B</sup>*m*<sup>2</sup> *a* 

*M*

The expression of the ellipticity angle *χ* as given by eqn. [11.78], found to be consistent with the same in (R. Cameron et al. , 1993). It should however be noted that, for interferometer based experiments, if the path length between the mirrors is given by *l*, and there are n reflections that take place between the mirrors then *χ*(*nl*) = *nχ*(*l*), i.e. the coherent addition of ellipticity per-pass. The reason is the following: every time the beam falls on the mirror the

Having evaluated the ellipticity parameter, we would move on to calculate the polaraization

tan(2*ψ*) = <sup>U</sup>

However there is little subtlety involved in this estimation; recall that the beam is initially polarized at an angle 45*<sup>o</sup>* with the external magnetic field. So to find out the final polarization after it has traversed a length z, we need to rotate our coordinate system by the same angle and evaluate the cumulative change in the polarization angle. We have already noted in the previous section, the effect of such a rotation on the stokes parameters and hence on the polarization angle; so following the same procedure, we evaluate the angle Ψ from the

> *π* <sup>2</sup> ) = <sup>U</sup>

We have already noted (eq. [11.75]) that for the magnitudes of the parameters of interest, the stokes parameter U ∼ 1; and that makes the angle 2*ψ* inversely proportional to Q, where the proportionality constant turns out to be unity. Therefore we need to evaluate just Q, using the approximations as stated before. Recalling the fact that, the mixing angle *θ* is much less than one, we can expand all the *θ* dependent terms in the expression for Q, and retain terms up to

Now one can substitute the necessary relations given in eqns. [11.77] in eqn. [11.80] to arrive

We would like to point out that, the angle of polarization as given by [11.81] also happens to be consistent with the same given in reference [(R. Cameron et al. , 1993)] where the authors had evaluated the same using a different method. In the light of this, we conclude this section by noting that, all the polarization dependent observables related to optical activity can be obtained independently by various methods, for the parameter ranges of interest or

(*k*<sup>+</sup> − *k* � +)*z*

2

<sup>16</sup>*M*2*<sup>ω</sup>* . (11.81)

Q .

<sup>2</sup>

*z*3. (11.78)

65

<sup>Q</sup>. (11.79)

, (11.80)

Till this point, the expressions, we obtain are very general i. e., no approximations were made. However for predicting or explaining the experimental outcome one would have to choose some initial conditions and make some approximations to evaluate the physical quantities of interest. In that spirit, in this analysis we would take the initial beam of light to be plane polarized, with the plane of polarization making an angle *<sup>π</sup>* <sup>4</sup> with the external magnetic field. And their amplitude would be assumed to be unity; therefore under this approximation *<sup>A</sup>*||(0) = *<sup>A</sup>*⊥(0) = <sup>√</sup> 1 2 .

It is important to note that, for axion detection through polarization measurements or, astrophysical observations, the parameter *θ <<* 1. Also we can define another dimension full parameter, *δ* = *<sup>g</sup> m*<sup>2</sup> *a* . With the current experimental bounds for Axion mass and coupling constant *δ <<* 1. So we can safely take cos*θ* ∼ 1 and sin*θ* ∼ *θ*. Now going back to eqns., (7.46) and (7.47) one can see that the dispersion relations for the wave vectors are given by,

$$\begin{aligned} k\_{\perp} &\simeq \omega \\ k\_{+} &\simeq \omega + \frac{(g\mathcal{B}\omega)^{2}}{2m\_{a}^{2}\omega} \\ k\_{+}' &\simeq \omega - \frac{m\_{a}^{2}}{2\omega} - \frac{(g\mathcal{B}\omega)^{2}}{2m\_{a}^{2}\omega} \\ \theta &= \frac{g\mathcal{B}\omega}{m\_{a}^{2}} \end{aligned} \tag{11.74}$$

Since the ratio *<sup>g</sup> m*<sup>2</sup> *a* = *δ <<* 1, we can always neglect their higher order contributions in any expansion involving *δ*. Therefore making the same, *Q* can be shown to be close to zero and the Stokes parameter U turns out to be:

$$\mathbf{U} = \mathbf{1} + \mathcal{O}(\delta^n) \text{ when } n \ge 1 \text{ ...} \tag{11.75}$$

Before proceeding further, we note the following relations,

$$\begin{split}k\_{+} - k\_{\perp} &= \frac{m\_{a}^{2}\theta^{2}}{2\omega} \\ k\_{+}^{\prime} - k\_{\perp} &= -\frac{m\_{a}^{2}}{2\omega}, \\ k\_{+} - k\_{+}^{\prime} &\simeq \frac{m\_{a}^{2}}{2\omega}. \end{split} \tag{11.76}$$

they would be useful to find out the other Stokes parameter V. In terms of these, V comes out to be,

$$\mathbf{V} = \sin(-\frac{m\_a^2 \theta^2 z}{2\omega}) + \theta^2 \sin(m\_a^2 z / 2\omega) \tag{11.77}$$

If we retain terms of order *θ*<sup>2</sup> only, in eqn. [11.77], then, we find, V = <sup>1</sup> 48 *θ*2*m*<sup>6</sup> *az*3 *<sup>ω</sup>*<sup>3</sup> , where an overall sign has been ignored. Finally substituting the values of *θ* and other quantities, the ellipticity angle *χ* is turns out to be

16 Will-be-set-by-IN-TECH

Till this point, the expressions, we obtain are very general i. e., no approximations were made. However for predicting or explaining the experimental outcome one would have to choose some initial conditions and make some approximations to evaluate the physical quantities of interest. In that spirit, in this analysis we would take the initial beam of light to be plane

field. And their amplitude would be assumed to be unity; therefore under this approximation

It is important to note that, for axion detection through polarization measurements or, astrophysical observations, the parameter *θ <<* 1. Also we can define another dimension

constant *δ <<* 1. So we can safely take cos*θ* ∼ 1 and sin*θ* ∼ *θ*. Now going back to eqns., (7.46) and (7.47) one can see that the dispersion relations for the wave vectors are given by,

> 2*m*<sup>2</sup> *<sup>a</sup><sup>ω</sup>* ,

*a* <sup>2</sup>*<sup>ω</sup>* <sup>−</sup> (*g*B*ω*)<sup>2</sup> 2*m*<sup>2</sup> *aω*

expansion involving *δ*. Therefore making the same, *Q* can be shown to be close to zero and

*<sup>k</sup>*<sup>+</sup> <sup>−</sup> *<sup>k</sup>*<sup>⊥</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup>

<sup>+</sup> <sup>−</sup> *<sup>k</sup>*<sup>⊥</sup> <sup>=</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

they would be useful to find out the other Stokes parameter V. In terms of these, V comes out

overall sign has been ignored. Finally substituting the values of *θ* and other quantities, the

<sup>+</sup> � *<sup>m</sup>*<sup>2</sup> *a* <sup>2</sup>*<sup>ω</sup>* .

<sup>2</sup>*<sup>ω</sup>* ) + *<sup>θ</sup>*2sin(*m*<sup>2</sup>

*k*<sup>+</sup> − *k*�

*<sup>a</sup>θ*2*z*

*k*�

<sup>V</sup> <sup>=</sup> sin(<sup>−</sup> *<sup>m</sup>*<sup>2</sup>

If we retain terms of order *θ*<sup>2</sup> only, in eqn. [11.77], then, we find, V = <sup>1</sup>

*<sup>k</sup>*<sup>+</sup> � *<sup>ω</sup>* <sup>+</sup> (*g*B*ω*)<sup>2</sup>

<sup>+</sup> � *<sup>ω</sup>* <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

*<sup>θ</sup>*<sup>=</sup> *<sup>g</sup>*B*<sup>ω</sup> m*2 *a*

*<sup>k</sup>*⊥�*ω*,

*k*�

Before proceeding further, we note the following relations,

. With the current experimental bounds for Axion mass and coupling

= *δ <<* 1, we can always neglect their higher order contributions in any

*aθ*2 2*ω*

*a*

<sup>U</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>O</sup>*(*δn*) when *<sup>n</sup>* <sup>≥</sup> 1 .. (11.75)

<sup>2</sup>*<sup>ω</sup>* , (11.76)

*<sup>a</sup> z*/2*ω*) (11.77)

48 *θ*2*m*<sup>6</sup> *az*3

*<sup>ω</sup>*<sup>3</sup> , where an

<sup>4</sup> with the external magnetic

(11.74)

polarized, with the plane of polarization making an angle *<sup>π</sup>*

1 2 .

*m*<sup>2</sup> *a*

*<sup>A</sup>*||(0) = *<sup>A</sup>*⊥(0) = <sup>√</sup>

full parameter, *δ* = *<sup>g</sup>*

Since the ratio *<sup>g</sup>*

to be,

*m*<sup>2</sup> *a*

ellipticity angle *χ* is turns out to be

the Stokes parameter U turns out to be:

$$\chi = \frac{1}{96\omega} \left( \frac{\left( \mathcal{B}m\_a^2 \right)}{M} \right)^2 z^3. \tag{11.78}$$

The expression of the ellipticity angle *χ* as given by eqn. [11.78], found to be consistent with the same in (R. Cameron et al. , 1993). It should however be noted that, for interferometer based experiments, if the path length between the mirrors is given by *l*, and there are n reflections that take place between the mirrors then *χ*(*nl*) = *nχ*(*l*), i.e. the coherent addition of ellipticity per-pass. The reason is the following: every time the beam falls on the mirror the photons get reflected, the axions are lost, they don't get reflected from the mirror.

Having evaluated the ellipticity parameter, we would move on to calculate the polaraization angle from the expression

$$\tan(2\psi) = \frac{\mathcal{U}}{\mathcal{Q}}\,\,\,\,$$

However there is little subtlety involved in this estimation; recall that the beam is initially polarized at an angle 45*<sup>o</sup>* with the external magnetic field. So to find out the final polarization after it has traversed a length z, we need to rotate our coordinate system by the same angle and evaluate the cumulative change in the polarization angle. We have already noted in the previous section, the effect of such a rotation on the stokes parameters and hence on the polarization angle; so following the same procedure, we evaluate the angle Ψ from the following relation,

$$\tan(2\psi + \frac{\pi}{2}) = \frac{\mathcal{U}}{\mathcal{Q}}.\tag{11.79}$$

We have already noted (eq. [11.75]) that for the magnitudes of the parameters of interest, the stokes parameter U ∼ 1; and that makes the angle 2*ψ* inversely proportional to Q, where the proportionality constant turns out to be unity. Therefore we need to evaluate just Q, using the approximations as stated before. Recalling the fact that, the mixing angle *θ* is much less than one, we can expand all the *θ* dependent terms in the expression for Q, and retain terms up to order *θ*2. Once this is done, we arrive at:

$$\mathbf{Q} = -2\theta^2 \left( \sin^2 \left( \frac{(k\_+ - k\_+')z}{2} \right) \right),\tag{11.80}$$

Now one can substitute the necessary relations given in eqns. [11.77] in eqn. [11.80] to arrive at the expression for *ψ*. Once substituted the polarization angle turns out to be.

$$
\psi = \frac{\left(\mathcal{B}^E z\right)^2}{16M^2 \omega}.\tag{11.81}
$$

We would like to point out that, the angle of polarization as given by [11.81] also happens to be consistent with the same given in reference [(R. Cameron et al. , 1993)] where the authors had evaluated the same using a different method. In the light of this, we conclude this section by noting that, all the polarization dependent observables related to optical activity can be obtained independently by various methods, for the parameter ranges of interest or

For the problem in hand we have two vectors and one tensor at our disposal, frame velocity of the medium *uμ*, 4 momentum of the photon *k<sup>μ</sup>* and external magnetic field strength tensor <sup>F</sup>*μν*. To describe the dynamics of the 4 component gauge field, we need to expand them in an

> *<sup>ν</sup>* = ⎛

�−*IμI<sup>μ</sup>* <sup>=</sup> <sup>|</sup>

�−*u*˜*μu*˜*<sup>μ</sup>* <sup>=</sup> *<sup>K</sup>*

The negative sign under the square roots are taken to make the vectors real. The Gauge field

The form factor *<sup>A</sup>*||(*k*) is associated with the gauge degrees of freedom and would be set to zero. It is easy to see that, this construction satisfies the Lorentz Gauge condition *kμA<sup>μ</sup>* = 0 .

*�μ*⊥*ν*⊥30*b*(1)*<sup>ν</sup> <sup>I</sup>*

*�μ*⊥*ν*⊥30*b*(1)*μ<sup>I</sup>*

�

*Ma*

� *ib*(2) *<sup>μ</sup> u*˜*<sup>μ</sup>* �

As in the previous case, in this case too we would assume the wave propagation to be in the z direction. and a generic solution written as Φ*i*(*t*, *z*) for all the dynamical degrees of freedom

> � ⎛

⎜⎜⎝

*A*1(*k*) *A*2(*k*) *AL*(*k*) *a*(*k*)

⎞

*μ* �

*<sup>k</sup>*<sup>2</sup> <sup>−</sup> <sup>Π</sup>*<sup>L</sup>*

*ν* �

�

N*LAL*(*k*)

In eqn. [12.86] we have made use of the additional vector, *u*˜*<sup>ν</sup>* = *g*˜*νμu<sup>μ</sup>* (*u<sup>μ</sup>* = (1, 0, 0, 0)).

N1 <sup>=</sup> <sup>1</sup> � −*b* (1) *<sup>μ</sup> b*(1)*<sup>μ</sup>*

N2 <sup>=</sup> <sup>1</sup>

<sup>N</sup>*<sup>L</sup>* <sup>=</sup> <sup>1</sup>

The equations of motion for the axions and photon form factors are given by,

�

N2*A*2(*k*) +

�

<sup>⎝</sup>*b*(2)*<sup>ν</sup>* <sup>−</sup> (*u*˜*μ<sup>b</sup>*

<sup>=</sup> <sup>1</sup> *BzK*<sup>⊥</sup>

*K*| *<sup>ω</sup>K*⊥*Bz*

> | *K*|

(2) *<sup>μ</sup>* ) *<sup>u</sup>*˜<sup>2</sup> *<sup>u</sup>*˜ *ν* ⎞

(1) *<sup>α</sup>* <sup>+</sup> *<sup>A</sup>*2(*k*)N2 *<sup>I</sup><sup>α</sup>* <sup>+</sup> *AL*(*k*)N*Lu*˜*<sup>α</sup>* <sup>+</sup> *<sup>k</sup>α*N�*A*||(*k*). (12.88)

N1*A*1(*k*) = −

*A*2(*k*) = 0 ,

*AL*(*k*) =

⎤ ⎦ = � *<sup>k</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

� *iN*2*b* (2) *<sup>μ</sup> I<sup>μ</sup>* � *a*

*iNL* � *b* (2) *<sup>μ</sup> u*˜*<sup>μ</sup>* � *a*

Φ*i*(0, *z*). As we had done before, now we

⎟⎟⎠ <sup>=</sup> 0. (12.90)

*Ma*

*Ma*

�

,

,

*a*. (12.89)

<sup>⎠</sup> , and *<sup>k</sup>μ*. (12.86)

67

, (12.87)

orthonormal basis. One can construct the basis in terms of the following 4-vectors,:

*<sup>b</sup>*(1)*<sup>ν</sup>* <sup>=</sup> *<sup>k</sup>μ*F*μν*, *<sup>b</sup>*(2)*<sup>ν</sup>* <sup>=</sup> *<sup>k</sup>μ*F˜ *μν*, *<sup>I</sup>*

in Particle Physics and Photon Dispersion in Magnetized Media

Introduction to Axion Photon Interaction

or photon field now can be expanded in this new basis,

*A*2(*k*) − *i*Π*pN*1*N*<sup>2</sup>

(*k*<sup>2</sup> <sup>−</sup> <sup>Π</sup>*T*(*k*))*A*1(*k*) + *<sup>i</sup>*Π*pN*1*N*<sup>2</sup>

*Aα*(*k*) = *A*1(*k*)N1*b*

�

⎡ ⎣ � *ib*(2) *<sup>μ</sup> I<sup>μ</sup>* �

would be assumed to be of the form, Φ*i*(*t*, *z*) = *e*−*iω<sup>t</sup>*

may express Eqs. (12.89), in real space in the matrix form

�

(*ω*<sup>2</sup> + *∂*<sup>2</sup>

*<sup>z</sup>* )**I** − *M*

*Ma*

�

*<sup>k</sup>*<sup>2</sup> <sup>−</sup> <sup>Π</sup>*T*(*k*)

instrument sensitivity, the results obtained using stokes parameters turns out to be consistent with the alternative ones.

#### **12. Axion electrodynamics in a magnetized media**

In the earlier section we have detailed the procedure of getting axion photon modified equation of presence of tree level axion photon interaction Lagrangian. And this equation of motion would be valid in vacuum, but in nature most of the physical processes take place in the presence of a medium, ideal vacuum is hardly available. Therefore to study the axion photon system and their evolution one needs to take the effect of magnetized vacuum into account. This could be done by taking an effective Lagrangian, that incorporates the magnetized matter effects. This Lagrangian is provided in [(A. K. Ganguly P.K. Jain and S. Mandal , 2009)].

In momentum space this effective Lagrangian is given by:,

$$\mathcal{L} = \frac{1}{2} \left[ -A\_{\mu} k^{2} \tilde{\mathcal{g}}^{\mu \nu} A\_{\nu} + A\_{\mu} \tilde{\Pi}^{\mu \nu} A\_{\nu} + i \frac{\mathcal{F}^{\mu \nu} k\_{\mu} A\_{\nu} a}{M\_{a}} - a(k^{2} - m\_{a}^{2}) a \right]. \tag{12.82}$$

The notations in eqn. [12.82] are the following, *g*˜*μν* = � *<sup>g</sup>μν* <sup>−</sup> *<sup>k</sup>μk<sup>ν</sup> k*2 � , <sup>F</sup>˜ *μν* is the field strength tensor of the external field, <sup>1</sup> *Ma* � <sup>1</sup> *<sup>M</sup>* the axion photon coupling constant, <sup>Π</sup>˜ *μν* is polarization tensor including Faraday contribution and is given by,

$$
\tilde{\Pi}^{\mu\nu}(k) = \Pi\_T(k)\mathbb{R}^{\mu\nu} + \Pi\_L(k)\mathbb{Q}^{\mu\nu}(k) + \Pi\_\mathcal{P}(k)\mathbb{P}^{\mu\nu}.\tag{12.83}
$$

Usually in the thermal field theory notations, the cyclotron frequency is given by, *ω<sup>B</sup>* = *eB m* and plasma frequency (in terms of electron density *ne* and temperature *T*) in written as, *ω<sup>p</sup>* =

� 4*παne m* � <sup>1</sup> <sup>−</sup> <sup>5</sup>*<sup>T</sup>* 2*m* � . In terms of these expressions, the longitudinal form factor Π*<sup>L</sup>* , transverse form factor Π*<sup>T</sup>* and Faraday form factor Π*<sup>p</sup>* along with their projection operators *Qμν*, *Rμν* and *Pμν* are given by,

$$\begin{aligned} \Pi\_L(k) &= k^2 \omega\_p^2 \left(\frac{1}{\omega^2} + 3\frac{|\vec{k}|^2}{\omega^4} \frac{T}{m}\right), \quad \Pi^p(k) = \frac{\omega \omega\_B \omega\_p^2}{\omega^2 - \omega\_B^2} \text{ and } \Pi\_T = \omega\_p^2 \left(1 + \frac{|\vec{k}|^2}{\omega^2} \frac{T}{m}\right), \\\ \text{where} \qquad \begin{cases} \qquad Q\_{\mu\nu} = \frac{a\_\mu \vec{\mu}\_\nu}{\hbar^2} \\\qquad R\_{\mu\nu} = \vec{\mathfrak{g}}\_{\mu\nu} - Q\_{\mu\nu} \\\qquad P\_{\mu\nu} = i \varepsilon\_{\mu\_\perp \nu a \not\equiv} \frac{k^2}{|\vec{k}|} \mu^6. \end{cases} \end{aligned}$$

The equations of motion for Gauge pseudoscala fields that follows from the Lagrangian (12.82) are the following:

$$\left(-k^{2}\tilde{\mathbf{g}}\_{\rm av} + \tilde{\Pi}\_{\rm av}(k)\right)A^{\vee}(k) = -i\frac{k^{\mu}\mathcal{F}\_{\mu a}a}{2M\_{a}}\tag{12.84}$$

$$\left(k^2 - m^2\right)a = \mathrm{i}\frac{b\_{\mu}^{(2)}A^{\mu}(k)}{2M\_a}.\tag{12.85}$$

18 Will-be-set-by-IN-TECH

instrument sensitivity, the results obtained using stokes parameters turns out to be consistent

In the earlier section we have detailed the procedure of getting axion photon modified equation of presence of tree level axion photon interaction Lagrangian. And this equation of motion would be valid in vacuum, but in nature most of the physical processes take place in the presence of a medium, ideal vacuum is hardly available. Therefore to study the axion photon system and their evolution one needs to take the effect of magnetized vacuum into account. This could be done by taking an effective Lagrangian, that incorporates the magnetized matter effects. This Lagrangian is provided in [(A. K. Ganguly P.K. Jain and S.

Usually in the thermal field theory notations, the cyclotron frequency is given by, *ω<sup>B</sup>* = *eB*

and plasma frequency (in terms of electron density *ne* and temperature *T*) in written as, *ω<sup>p</sup>* =

form factor Π*<sup>T</sup>* and Faraday form factor Π*<sup>p</sup>* along with their projection operators *Qμν*, *Rμν*

The equations of motion for Gauge pseudoscala fields that follows from the Lagrangian (12.82)

�

*<sup>A</sup>ν*(*k*) = <sup>−</sup>*<sup>i</sup>*

� *a* = *i b* (2) *<sup>μ</sup> Aμ*(*k*) 2*Ma*

, Π*p*(*k*) =

�

*kα* |*K*| *uβ*.

<sup>−</sup>*k*2*g*˜*αν* <sup>+</sup> <sup>Π</sup>˜ *αν*(*k*)

� *<sup>k</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>F</sup>˜ *μνkμAν<sup>a</sup> Ma*

�

Π˜ *μν*(*k*) = Π*T*(*k*)*Rμν* + Π*L*(*k*)*Qμν*(*k*) + Π*p*(*k*)*Pμν* . (12.83)

. In terms of these expressions, the longitudinal form factor Π*<sup>L</sup>* , transverse

*ωωBω*<sup>2</sup> *p <sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *B*

*<sup>g</sup>μν* <sup>−</sup> *<sup>k</sup>μk<sup>ν</sup> k*2 �

*<sup>M</sup>* the axion photon coupling constant, <sup>Π</sup>˜ *μν* is polarization

and Π*<sup>T</sup>* = *ω*<sup>2</sup>

*<sup>k</sup>μ*F˜*μα<sup>a</sup>* 2*Ma*

*p* � 1 + | *k*| 2 *ω*<sup>2</sup> *T m*

<sup>−</sup> *<sup>a</sup>*(*k*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

*<sup>a</sup>*)*a* �

, <sup>F</sup>˜ *μν* is the field strength

. (12.82)

*m*

�

(12.84)

. (12.85)

with the alternative ones.

Mandal , 2009)].

� 4*παne m* � <sup>1</sup> <sup>−</sup> <sup>5</sup>*<sup>T</sup>* 2*m* �

<sup>L</sup> <sup>=</sup> <sup>1</sup> 2 �

tensor of the external field, <sup>1</sup>

and *Pμν* are given by,

where

are the following:

<sup>Π</sup>*L*(*k*)=−*k*2*ω*<sup>2</sup>

*p* � 1 *<sup>ω</sup>*<sup>2</sup> <sup>+</sup> <sup>3</sup> | *k*| 2 *ω*<sup>4</sup> *T m*

⎧ ⎪⎨

⎪⎩

**12. Axion electrodynamics in a magnetized media**

In momentum space this effective Lagrangian is given by:,

*μνA<sup>ν</sup>* + *Aμ*Π˜ *μνA<sup>ν</sup>* + *i*

<sup>−</sup>*Aμk*<sup>2</sup> *<sup>g</sup>*˜

The notations in eqn. [12.82] are the following, *g*˜*μν* =

tensor including Faraday contribution and is given by,

*<sup>Q</sup>μν* <sup>=</sup> *<sup>u</sup>*˜*μu*˜*<sup>ν</sup> u*˜2 *Rμν* = *g*˜*μν* − *Qμν* , *<sup>P</sup>μν* = *<sup>i</sup>�μ*⊥*ναβ*

�

*Ma* � <sup>1</sup>

For the problem in hand we have two vectors and one tensor at our disposal, frame velocity of the medium *uμ*, 4 momentum of the photon *k<sup>μ</sup>* and external magnetic field strength tensor <sup>F</sup>*μν*. To describe the dynamics of the 4 component gauge field, we need to expand them in an orthonormal basis. One can construct the basis in terms of the following 4-vectors,:

$$b^{(1)\nu} = k\_{\mu} \mathcal{F}^{\mu\nu}, \quad b^{(2)\nu} = k\_{\mu} \mathcal{F}^{\mu\nu}, \quad I^{\nu} = \left( b^{(2)\nu} - \frac{(\vec{u}^{\mu} b\_{\mu}^{(2)})}{\vec{u}^2} \vec{u}^{\nu} \right), \quad \text{and } k^{\mu}. \tag{12.86}$$

In eqn. [12.86] we have made use of the additional vector, *u*˜*<sup>ν</sup>* = *g*˜*νμu<sup>μ</sup>* (*u<sup>μ</sup>* = (1, 0, 0, 0)).

$$\mathbf{N}\_{1} = \frac{1}{\sqrt{-b\_{\mu}^{(1)}b^{(1)\mu}}} = \frac{1}{\overline{B\_{z}K\_{\perp}}}$$

$$\begin{split} \mathbf{N}\_{2} &= \frac{1}{\sqrt{-I\_{\mu}I^{\mu}}} = \frac{|\vec{K}|}{\omega K\_{\perp}B\_{z}}\\ \mathbf{N}\_{L} &= \frac{1}{\sqrt{-\vec{u}\_{\mu}\vec{u}^{\mu}}} = \frac{K}{|\vec{K}|} \end{split} \tag{12.87}$$

The negative sign under the square roots are taken to make the vectors real. The Gauge field or photon field now can be expanded in this new basis,

$$A\_{\mathfrak{a}}(k) = A\_1(k)\mathbf{N}\_1\mathbf{b}\_{\mathfrak{a}}^{(1)} + A\_2(k)\mathbf{N}\_2\mathbf{l}\_{\mathfrak{a}} + A\_L(k)\mathbf{N}\_L\tilde{\boldsymbol{\mu}}\_{\mathfrak{a}} + k\_{\mathfrak{a}}\mathbf{N}\_{\parallel}A\_{\parallel}(k). \tag{12.88}$$

The form factor *<sup>A</sup>*||(*k*) is associated with the gauge degrees of freedom and would be set to zero. It is easy to see that, this construction satisfies the Lorentz Gauge condition *kμA<sup>μ</sup>* = 0 . The equations of motion for the axions and photon form factors are given by,

$$\left(k^{2} - \Pi\_{T}(k)\right)A\_{2}(k) - i\Pi\_{p}N\_{1}N\_{2}\left[\varepsilon\_{\mu\_{\perp}\nu\_{\perp}\mathfrak{D}}b^{(1)\nu}I^{\mu}\right]\mathcal{N}\_{1}A\_{1}(k) = -\frac{\left(iN\_{2}b\_{\mu}^{(2)}I^{\mu}\right)a}{M\_{a}},$$

$$\left(k^{2} - \Pi\_{T}(k)\right)A\_{1}(k) + i\Pi\_{p}N\_{1}N\_{2}\left[\varepsilon\_{\mu\_{\perp}\nu\_{\perp}\mathfrak{D}}b^{(1)\mu}I^{\nu}\right]A\_{2}(k) = 0,$$

$$\left(k^{2} - \Pi\_{L}\right)A\_{L}(k) = \frac{iN\_{L}\left(b\_{\mu}^{(2)}\tilde{u}^{\mu}\right)a}{M\_{d}},$$

$$\left[\frac{\left(ib\_{\mu}^{(2)}I^{\mu}\right)}{M\_{d}}N\_{2}A\_{2}(k) + \frac{\left(ib\_{\mu}^{(2)}\tilde{u}^{\mu}\right)}{M\_{d}}N\_{L}A\_{L}(k)\right] = \left(k^{2} - m^{2}\right)a. \tag{12.89}$$

As in the previous case, in this case too we would assume the wave propagation to be in the z direction. and a generic solution written as Φ*i*(*t*, *z*) for all the dynamical degrees of freedom would be assumed to be of the form, Φ*i*(*t*, *z*) = *e*−*iω<sup>t</sup>* Φ*i*(0, *z*). As we had done before, now we may express Eqs. (12.89), in real space in the matrix form

$$
\mathbb{E}\left[ (\omega^2 + \partial\_z^2) \mathbf{I} - M \right] \begin{pmatrix} A\_1(k) \\ A\_2(k) \\ A\_L(k) \\ a(k) \end{pmatrix} = 0. \tag{12.90}
$$

and the polarization angle,*ψ* would be given by,:

in Particle Physics and Photon Dispersion in Magnetized Media

Introduction to Axion Photon Interaction

*ω<sup>s</sup>* =

to take up advanced level investigations in this direction.

**13. Acknowledgment**

matrix of the following type,

� � � � � � *ωB* � *ω*4 *<sup>p</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *pm*<sup>2</sup> *a* �

tan (*ψ* + *π*/2) = −cot(Δ*z*). (12.95)

� � � � � �

, (12.96)

69

*M*2cos*θ*

when z is the path length traversed by the beam, in the magnetized media. We would like to emphasize here that, even in the limit of weak external magnetic field, it may not be prudent to ignore the contribution of Faraday effect. If we define a new energy scale *ωs*, such that

(B*E*)2sin2*<sup>θ</sup>*

We conclude here by noting that in this write up, we have tried to provide a comprehensive study of axion photon mixing and the associated observables of a photon beam. We have employed the coherency matrix formulation for studying the polarization properties; Starting with tree level axion photon interaction Lagrangian, we have demonstrated explicitly, how to construct the Stokes parameters from there. From there we have shown how to calculate the ellipticity angle and polarization angle from the Stokes Parameters. The relevant findings or questions pertaining to the current or proposed experiments in this area involve inclusion of matter effects, consideration of very strong magnetic field, dynamics of very high energy photon in such a scenario. Except the last, we have discussed the issues relevant for the first two. We end here by hoping that this elementary write up would help those who would like

Many of the ideas I have presented here, took its shape during my collaborations with Prof. P. K. Jain and Dr. Subhayan Mandal. I am acknowledge them here in this note. I also would

Here we out line diagonalization of a 3 × 3 matrix given by eqn. (12.92), i.e., a symmetric

Generalizing it to a hermitian matrix of the kind we have is trivial, so we would concentrate on diagonalizing the type given by eqn. (14.97). As noted already, the Cayley-Hamilton characterictic equation for this matrix looks like, |**X3** − *λ***i**| = 0. for the i'th eigen value. Or

*a b* 0 *bcd* 0 *d g* ⎤

� � � � � �

⎦ . (14.97)

= 0 (14.98)

⎡ ⎣

like to thank my wife, Dr. Archana Puri for her patience and understanding.

**14. Appendix: Constructing the orthogonal matrix for diagonalization**

**X3** =

*a* − *λ<sup>i</sup> b* 0 *b c* − *λ<sup>i</sup> d* 0 *d g* − *λ<sup>i</sup>*

for that matter, for any of the three eigen values, one should have: � � � � � �

then for *ω<sup>S</sup>* � *ω*, to estimate *χ*, one should consider the Faraday effect simultaneously.

where **I** is a 4 × 4 identity matrix and the modified mixing matrix, because of magnetized medium, turns out to be,

$$M = \begin{pmatrix} \Pi\_T & -iN\_1 N\_2 \Pi\_p \varepsilon\_{\mu\_\perp \nu\_\perp 30} b^{(1) \mu} I^{\nu} & 0 & 0\\ iN\_1 N\_2 \Pi\_p \varepsilon\_{\mu\_\perp \nu\_\perp 30} b^{(1) \nu} I^{\mu} & + \Pi\_T & 0 & -i \frac{N\_2 b\_\mu^{(2)} I^{\mu}}{M\_4} \\ 0 & 0 & \Pi\_L & -i \frac{N\_2 b\_\mu^{(2)} I^{\mu}}{M\_4} \\ 0 & i \frac{N\_2 b\_\mu^{(2)} I^{\mu}}{M\_4} & i \frac{N\_2 b\_\mu^{(2)} \bar{u}^{\mu}}{M\_4} & m\_a^2 \end{pmatrix}. \tag{12.91}$$

Solving this problem exactly is a difficult task, however in the low density limit one can usually ignore the effect of longitudinal field and Π*L*. Again if we assume the *ω* � *ωp*, then we can simplify the faraday contribution further. Incorporating these effects, the mixing matrix in this case turns out to be a 3 × 3 matrix, given by:

$$M = \begin{pmatrix} \omega\_p^2 & \imath \omega\_B \omega\_p^2 \cos \theta' / \omega & 0\\ -\imath \omega\_B \omega\_p^2 \cos \theta' / \omega & \omega\_p^2 & -\imath \jmath \mathcal{B} \omega\\ 0 & \imath \jmath \mathcal{B} \omega & m\_a^2 \end{pmatrix} \tag{12.92}$$

The angle *θ*� is the angle between the magnetic field and the photon momentum*k*, The other symbols are the same as used in the previously. This matrix can be diagonalized and one can obtain the exact result. The method of exact diagonalization of this matrix is relegated to the appendix.

The matrix given by eqn. [12.92] has been diagonalized and its eigen values have been evaluated perturbatively [(A. K. Ganguly P.K. Jain and S. Mandal , 2009)], in the limit *<sup>g</sup>*B*<sup>ω</sup>* � *<sup>ω</sup>Bω*<sup>2</sup> *<sup>p</sup> cosθ*� *<sup>ω</sup>* � |*m*<sup>2</sup> *<sup>a</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>p</sup>*|. The construction of the density (or coherency ) matrix from there is a straight forward exercise as illustrated before. Therefore instead of repeating the same here we would provide the values of the stokes parameters, computed from various components of the density matrix (2). In this analysis we assume plane polarized light, with the following initial conditions *a*(0) = 0 and *A*1(0) = *A*2(0) = <sup>√</sup> 1 2 . That is the initial angle the beam makes with the direction of *I<sup>μ</sup>* is *π*/4. The resulting stoke parameters are,

$$\begin{aligned} \mathbf{Q} &= -\sin\left(\Delta z\right) \\ \mathbf{V} &= \frac{\left(g\mathcal{B}\right)^2 \omega^3 \sin\left(\frac{\Delta z}{2}\right) \cos\left(\frac{\Delta z}{2} - \frac{\pi}{4}\right)}{\sqrt{2}\omega\_B \,\omega\_p^2 \cos\theta' \left(m\_a^2 - \omega\_p^2\right)} \end{aligned} \qquad \qquad \mathbf{U} = \cos\left(\Delta z\right) \,\tag{12.93}$$

where in eqn. [12.93], the parameter Δ is given by, Δ = −2 *ωBω*<sup>2</sup> *p*cos*θ*� *<sup>ω</sup>*<sup>2</sup> . Since V is associated with circular/ elliptic polarization, we can see from eqn. [12.93] that, even if one starts with a plane polarized wave, to begin with, it can become circularly or elliptically polarized light because of axion photon interaction and faraday effect. The ellipticity of the propagating wave turns out to be,

$$\chi = \frac{1}{2} \tan^{-1} \left( \frac{(g\mathcal{B})^2 \omega^3 \sin\left(\frac{\Delta z}{2}\right) \cos\left(\frac{\Delta z}{2} - \frac{\pi}{4}\right)}{\sqrt{2} \omega\_B \,\omega\_p^2 \cos \theta' \left(m\_a^2 - \omega\_p^2\right)} \right). \tag{12.94}$$

(2) See for instance equation. [5.14], in [(A. K. Ganguly P.K. Jain and S. Mandal , 2009)]

and the polarization angle,*ψ* would be given by,:

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

where **I** is a 4 × 4 identity matrix and the modified mixing matrix, because of magnetized

*iN*1*N*2Π*p�μ*⊥*ν*⊥30*b*(1)*νI<sup>μ</sup>* <sup>+</sup>Π*<sup>T</sup>* <sup>0</sup> <sup>−</sup>*<sup>i</sup>*

0 *i*

*ω*2

*<sup>p</sup>* cos *θ*�

matrix in this case turns out to be a 3 × 3 matrix, given by:

<sup>−</sup>*iωBω*<sup>2</sup>

⎛ ⎝

*<sup>a</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup>

<sup>V</sup> <sup>=</sup> (*g*B)

*<sup>χ</sup>* <sup>=</sup> <sup>1</sup> 2 tan−<sup>1</sup>

the following initial conditions *a*(0) = 0 and *A*1(0) = *A*2(0) = <sup>√</sup>

� <sup>Δ</sup>*<sup>z</sup>* 2 � cos � <sup>Δ</sup>*<sup>z</sup>* <sup>2</sup> <sup>−</sup> *<sup>π</sup>* 4 �

*<sup>p</sup>* cos *θ*� (*m*<sup>2</sup>

<sup>2</sup> *ω*3sin

where in eqn. [12.93], the parameter Δ is given by, Δ = −2

⎛ ⎝ (*g*B)

(2) See for instance equation. [5.14], in [(A. K. Ganguly P.K. Jain and S. Mandal , 2009)]

<sup>√</sup>2*ω<sup>B</sup> <sup>ω</sup>*<sup>2</sup>

*M* =

<sup>Π</sup>*<sup>T</sup>* <sup>−</sup>*iN*1*N*2Π*p�μ*⊥*ν*⊥30*b*(1)*μI<sup>ν</sup>* 0 0

0 0 Π*<sup>L</sup>* −*i*

*Ma i*

*<sup>p</sup>* cos *θ*�

*NL b* (2) *<sup>μ</sup> <sup>u</sup>*˜*<sup>μ</sup> Ma <sup>m</sup>*<sup>2</sup>

/*ω* 0

*<sup>p</sup>* −*ig*B*ω*

*<sup>p</sup>*|. The construction of the density (or coherency ) matrix from

*ωBω*<sup>2</sup> *p*cos*θ*�

*<sup>a</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *p*) *a*

1 2

*<sup>p</sup>*) , U <sup>=</sup> *cos*(Δ*z*) , (12.93)

⎞

⎞

*N*<sup>2</sup> *b* (2) *<sup>μ</sup> <sup>I</sup><sup>μ</sup>*

Solving this problem exactly is a difficult task, however in the low density limit one can usually ignore the effect of longitudinal field and Π*L*. Again if we assume the *ω* � *ωp*, then we can simplify the faraday contribution further. Incorporating these effects, the mixing

*<sup>p</sup> iωBω*<sup>2</sup>

/*ω ω*<sup>2</sup>

The angle *θ*� is the angle between the magnetic field and the photon momentum*k*, The other symbols are the same as used in the previously. This matrix can be diagonalized and one can obtain the exact result. The method of exact diagonalization of this matrix is relegated to the

The matrix given by eqn. [12.92] has been diagonalized and its eigen values have been evaluated perturbatively [(A. K. Ganguly P.K. Jain and S. Mandal , 2009)], in the limit

there is a straight forward exercise as illustrated before. Therefore instead of repeating the same here we would provide the values of the stokes parameters, computed from various components of the density matrix (2). In this analysis we assume plane polarized light, with

*<sup>a</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup>

with circular/ elliptic polarization, we can see from eqn. [12.93] that, even if one starts with a plane polarized wave, to begin with, it can become circularly or elliptically polarized light because of axion photon interaction and faraday effect. The ellipticity of the propagating wave

> � <sup>Δ</sup>*<sup>z</sup>* 2 � cos � <sup>Δ</sup>*<sup>z</sup>* <sup>2</sup> <sup>−</sup> *<sup>π</sup>* 4 �

*<sup>p</sup>* cos *θ*� (*m*<sup>2</sup>

<sup>2</sup> *ω*3sin

<sup>√</sup>2*ω<sup>B</sup> <sup>ω</sup>*<sup>2</sup>

the beam makes with the direction of *I<sup>μ</sup>* is *π*/4. The resulting stoke parameters are,

Q = −sin (Δ*z*) , I = 1,

<sup>0</sup> *ig*B*<sup>ω</sup> <sup>m</sup>*<sup>2</sup>

*N*<sup>2</sup> *b* (2) *<sup>μ</sup> <sup>I</sup><sup>μ</sup> Ma*

⎞

⎟⎟⎟⎟⎟⎠

⎠ , (12.92)

. That is the initial angle

*<sup>ω</sup>*<sup>2</sup> . Since V is associated

⎠ . (12.94)

. (12.91)

*NL b* (2) *<sup>μ</sup> <sup>u</sup>*˜*<sup>μ</sup> Ma*

*a*

medium, turns out to be,

⎛

⎜⎜⎜⎜⎜⎝

*M* =

appendix.

*<sup>g</sup>*B*<sup>ω</sup>* � *<sup>ω</sup>Bω*<sup>2</sup>

turns out to be,

*<sup>p</sup> cosθ*� *<sup>ω</sup>* � |*m*<sup>2</sup>

$$
\tan\left(\psi + \pi/2\right) = -\cot\left(\Delta z\right). \tag{12.95}
$$

when z is the path length traversed by the beam, in the magnetized media. We would like to emphasize here that, even in the limit of weak external magnetic field, it may not be prudent to ignore the contribution of Faraday effect. If we define a new energy scale *ωs*, such that

$$
\omega\_s = \left| \frac{\omega\_B \left( \omega\_p^4 - \omega\_p^2 m\_a^2 \right) M^2 \cos \theta}{(\mathcal{B}^E)^2 \sin^2 \theta} \right| \,, \tag{12.96}
$$

then for *ω<sup>S</sup>* � *ω*, to estimate *χ*, one should consider the Faraday effect simultaneously.

We conclude here by noting that in this write up, we have tried to provide a comprehensive study of axion photon mixing and the associated observables of a photon beam. We have employed the coherency matrix formulation for studying the polarization properties; Starting with tree level axion photon interaction Lagrangian, we have demonstrated explicitly, how to construct the Stokes parameters from there. From there we have shown how to calculate the ellipticity angle and polarization angle from the Stokes Parameters. The relevant findings or questions pertaining to the current or proposed experiments in this area involve inclusion of matter effects, consideration of very strong magnetic field, dynamics of very high energy photon in such a scenario. Except the last, we have discussed the issues relevant for the first two. We end here by hoping that this elementary write up would help those who would like to take up advanced level investigations in this direction.

#### **13. Acknowledgment**

Many of the ideas I have presented here, took its shape during my collaborations with Prof. P. K. Jain and Dr. Subhayan Mandal. I am acknowledge them here in this note. I also would like to thank my wife, Dr. Archana Puri for her patience and understanding.

#### **14. Appendix: Constructing the orthogonal matrix for diagonalization**

Here we out line diagonalization of a 3 × 3 matrix given by eqn. (12.92), i.e., a symmetric matrix of the following type,

$$\mathbf{X\_3} = \begin{bmatrix} a \ b \ 0 \\ b \ c \ d \\ 0 \ d \ g \end{bmatrix}. \tag{14.97}$$

Generalizing it to a hermitian matrix of the kind we have is trivial, so we would concentrate on diagonalizing the type given by eqn. (14.97). As noted already, the Cayley-Hamilton characterictic equation for this matrix looks like, |**X3** − *λ***i**| = 0. for the i'th eigen value. Or for that matter, for any of the three eigen values, one should have:

$$
\begin{vmatrix}
a - \lambda\_i & b & 0 \\
b & c - \lambda\_i & d \\
0 & d & g - \lambda\_i
\end{vmatrix} = 0\tag{14.98}
$$

which is trivial to check. Next we start from,

in Particle Physics and Photon Dispersion in Magnetized Media

Introduction to Axion Photon Interaction

variable *λ*3. To do that we would make use of the following tricks,

square bracket in eqns. (14.110) to get a function of only *λ*3. i.e.

As one uses eqns. (14.111) in eqn. (14.109) one arrives at,

�

**15. Proof: V's actually diagonalize the mixing matrix**

⎡ ⎣

*u*<sup>1</sup> *u*<sup>2</sup> *u*<sup>3</sup> *v*<sup>1</sup> *v*<sup>2</sup> *v*<sup>3</sup> *w*<sup>1</sup> *w*<sup>2</sup> *w*<sup>3</sup>

*u*1*a* + *bv*<sup>1</sup> *u*1*b* + *v*1*c* + *w*1*d v*1*d* + *gw*<sup>1</sup> *u*2*a* + *bv*<sup>2</sup> *u*2*b* + *v*2*c* + *w*2*d v*2*d* + *gw*<sup>2</sup> *u*3*a* + *bv*<sup>3</sup> *u*3*b* + *v*3*c* + *w*3*d v*3*d* + *gw*<sup>3</sup>

⎤ ⎦ *<sup>T</sup>* ⎛ ⎝

*a b* 0 *bcd* 0 *d g* ⎞ ⎠ ⎡ ⎣

> ⎤ ⎦ ⎛ ⎝

so

Similarly one can show that,

**V1** · **V2** = (*c* − *λ*3)

similar fashion it can be shown that,

⎡ ⎣

some cancellations,

Lets start from:

[(*<sup>g</sup>* <sup>−</sup> *<sup>λ</sup>*1)(*<sup>g</sup>* <sup>−</sup> *<sup>λ</sup>*2)] <sup>=</sup> *<sup>g</sup>*<sup>2</sup> <sup>−</sup> *<sup>g</sup>*(*λ*<sup>1</sup> <sup>+</sup> *<sup>λ</sup>*2) + *<sup>λ</sup>*1*λ*2. (14.109)

*λ*1*λ*<sup>2</sup> = [*λ*1*λ*<sup>2</sup> + *λ*2*λ*<sup>3</sup> + *λ*3*λ*1] − *λ*3(*λ*<sup>2</sup> + *λ*1) (14.110)

*λ*<sup>1</sup> + *λ*<sup>2</sup> = *a* + *c* + *g* − *λ*<sup>3</sup> *<sup>λ</sup>*1*λ*<sup>2</sup> <sup>=</sup> *gc* <sup>+</sup> *ga* <sup>+</sup> *ac* <sup>−</sup> *<sup>d</sup>*<sup>2</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup> <sup>−</sup> *<sup>λ</sup>*3(*<sup>a</sup>* <sup>+</sup> *<sup>c</sup>* <sup>+</sup> *<sup>g</sup>* <sup>−</sup> *<sup>λ</sup>*3). (14.111)

*<sup>g</sup>*<sup>2</sup> <sup>−</sup> *<sup>g</sup>*(*λ*<sup>1</sup> <sup>+</sup> *<sup>λ</sup>*2)+(*λ*1.*λ*2)=(*λ*<sup>3</sup> <sup>−</sup> *<sup>a</sup>*)(*λ*<sup>3</sup> <sup>−</sup> *<sup>c</sup>*) <sup>−</sup> *<sup>b</sup>*<sup>2</sup> <sup>−</sup> *<sup>d</sup>*2. (14.112)

*<sup>b</sup>*2(*<sup>g</sup>* <sup>−</sup> *<sup>λ</sup>*1)(*<sup>g</sup>* <sup>−</sup> *<sup>λ</sup>*2) = *<sup>b</sup>*2[(*λ*<sup>3</sup> <sup>−</sup> *<sup>a</sup>*)(*λ*<sup>3</sup> <sup>−</sup> *<sup>c</sup>*) <sup>−</sup> *<sup>b</sup>*<sup>2</sup> <sup>−</sup> *<sup>d</sup>*2]. (14.113)

*<sup>d</sup>*2(*<sup>a</sup>* <sup>−</sup> *<sup>λ</sup>*1)(*<sup>a</sup>* <sup>−</sup> *<sup>λ</sup>*2) = *<sup>d</sup>*2[(*λ*<sup>3</sup> <sup>−</sup> *<sup>g</sup>*)(*λ*<sup>3</sup> <sup>−</sup> *<sup>c</sup>*) <sup>−</sup> *<sup>b</sup>*<sup>2</sup> <sup>−</sup> *<sup>d</sup>*2]. (14.114)

**V1V2** = **V2V3** = **V3V1** = 0. (14.116)

*u*<sup>1</sup> *u*<sup>2</sup> *u*<sup>3</sup> *v*<sup>1</sup> *v*<sup>2</sup> *v*<sup>3</sup> *w*<sup>1</sup> *w*<sup>2</sup> *w*<sup>3</sup>

> *u*<sup>1</sup> *u*<sup>2</sup> *u*<sup>3</sup> *v*<sup>1</sup> *v*<sup>2</sup> *v*<sup>3</sup> *w*<sup>1</sup> *w*<sup>2</sup> *w*<sup>3</sup>

⎤

⎞ ⎠ �

⎦ = (15.117)

= 0, (14.115)

71

(*<sup>a</sup>* <sup>−</sup> *<sup>λ</sup>*3)(*<sup>g</sup>* <sup>−</sup> *<sup>λ</sup>*3)(*<sup>c</sup>* <sup>−</sup> *<sup>λ</sup>*3) <sup>−</sup> *<sup>b</sup>*2(*<sup>g</sup>* <sup>−</sup> *<sup>λ</sup>*3) <sup>−</sup> *<sup>d</sup>*2(*<sup>a</sup>* <sup>−</sup> *<sup>λ</sup>*3)

Eqn. (14.109) is a function of *λ*<sup>1</sup> and *λ*2, and we need to convert it to a function of a single

Now one can use the relations (14.101, 14.102 and 14.102), to replace the expressions inside the

Finally as we substitute in eqn. (14.108), the results of eqns. (14.113) and (14.114), we get after

because the expression inside the square bracket of eqn. (14.115) after the first = sign, is zero, as can be seen by expanding the determinant, i.e., eqn. (14.98) after taking *λ<sup>i</sup>* to be *λ*3. In a

*λ*<sup>1</sup> + *λ*<sup>2</sup> = [*λ*<sup>1</sup> + *λ*<sup>2</sup> + *λ*3] − *λ*<sup>3</sup>

Which when written in algebraic form looks like,

$$\left(\lambda^3 - \lambda^2 \left(a + c + g\right) + \lambda \left(gc + ga + ac - d^2 - b^2\right) + \left(ad^2 + gb^2 - gac\right)\right) = 0 \tag{14.99}$$

Recalling that, the three roots of eqn. (14.99) satisfies the following relations

$$
\lambda\_1 + \lambda\_2 + \lambda\_3 = (\underline{a} + \mathfrak{c} + \mathfrak{g}) \tag{14.100}
$$

$$
\lambda\_1 \lambda\_2 + \lambda\_2 \lambda\_3 + \lambda\_3 \lambda\_1 = \left( gc + ga + ac - d^2 - b^2 \right) \tag{14.101}
$$

$$
\lambda\_1 \lambda\_2 \lambda\_3 = -\left(ad^2 + gb^2 - \text{grad}\right) \tag{14.102}
$$

We should have for any value of *i*(1, 2*or*3),

$$
\begin{bmatrix} a - \lambda\_i & b & 0 \\ b & c - \lambda\_i & d \\ 0 & d & g - \lambda\_i \end{bmatrix} \begin{pmatrix} u\_i \\ v\_i \\ w\_i \end{pmatrix} = 0,\tag{14.103}
$$

with corresponding eigen-vector

$$\mathbf{V\_{i}} = \begin{pmatrix} u\_{i} \\ v\_{i} \\ w\_{i} \end{pmatrix} \\ \text{ \tag{14.104}}$$

All that we need to prove is ,

$$\mathbf{V}\_{\mathbf{i}} \cdot \mathbf{V}\_{\mathbf{j}} = \delta\_{\mathbf{i}\mathbf{j}}.\tag{14.105}$$

when suitably normalized. Next, assuming the eigen vectors to be normalized, we would demonstrate the necessary identities they need to satisfy. The proof should follow by explicit use of the values of *λ<sup>i</sup>* 's in (14.105) (which is laborious ) or by some other less laborius method. Here we explore the last option. We write down the generic eqns. satisfied by the components of the eigen vectors

$$\begin{array}{l} (a - \lambda)u + bv = 0 \\ bu + (c - \lambda)v + dw = 0 \\ dv + (g - \lambda)w = 0. \end{array} \tag{14.106}$$

It's easy to find out the nontrivial solns of (14.106) (for any of the three eigenvalues) by inspection and they are:

$$\begin{array}{l} u = -b(g - \lambda) \\ v = (a - \lambda)(g - \lambda) \\ w = -d(a - \lambda). \end{array} \tag{14.107}$$

All that is to be shown is **V**<sup>1</sup> · **V**<sup>2</sup> = 0 and other similar relations. We would prove the previous relation, others can be done using similar method. To begin with note that,

$$\mathbf{V}\_1 \cdot \mathbf{V}\_2 = \left[ b^2 (\mathbf{g} - \lambda\_1)(\mathbf{g} - \lambda\_2) + d^2 (\mathbf{a} - \lambda\_1)(\mathbf{a} - \lambda\_2) \right]$$

$$+ (\mathbf{g} - \lambda\_1)(\mathbf{g} - \lambda\_2)(\mathbf{a} - \lambda\_1) \times (\mathbf{a} - \lambda\_2) \Big|\_{\mathbf{a}} \tag{14.108}$$

which is trivial to check. Next we start from,

$$[(\mathbf{g} - \lambda\_1)(\mathbf{g} - \lambda\_2)] = \mathbf{g}^2 - \mathbf{g}(\lambda\_1 + \lambda\_2) + \lambda\_1 \lambda\_2. \tag{14.109}$$

Eqn. (14.109) is a function of *λ*<sup>1</sup> and *λ*2, and we need to convert it to a function of a single variable *λ*3. To do that we would make use of the following tricks,

$$
\lambda\_1 + \lambda\_2 = [\lambda\_1 + \lambda\_2 + \lambda\_3] - \lambda\_3$$

$$
\lambda\_1 \lambda\_2 = [\lambda\_1 \lambda\_2 + \lambda\_2 \lambda\_3 + \lambda\_3 \lambda\_1] - \lambda\_3(\lambda\_2 + \lambda\_1) \tag{14.110}$$

Now one can use the relations (14.101, 14.102 and 14.102), to replace the expressions inside the square bracket in eqns. (14.110) to get a function of only *λ*3. i.e.

$$
\lambda\_1 + \lambda\_2 = a + c + g - \lambda\_3$$

$$
\lambda\_1 \lambda\_2 = gc + ga + ac - d^2 - b^2 - \lambda\_3 (a + c + g - \lambda\_3). \tag{14.111}$$

As one uses eqns. (14.111) in eqn. (14.109) one arrives at,

$$g^2 - g(\lambda\_1 + \lambda\_2) + (\lambda\_1 \lambda\_2) = (\lambda\_3 - a)(\lambda\_3 - c) - b^2 - d^2. \tag{14.112}$$

so

22 Will-be-set-by-IN-TECH

*gc* <sup>+</sup> *ga* <sup>+</sup> *ac* <sup>−</sup> *<sup>d</sup>*<sup>2</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup>

�

�

⎤ ⎦ ⎛ ⎝ *ui vi wi*

⎞

� + �

*λ*<sup>1</sup> + *λ*<sup>2</sup> + *λ*<sup>3</sup> = (*a* + *c* + *g*) (14.100)

*gc* <sup>+</sup> *ga* <sup>+</sup> *ac* <sup>−</sup> *<sup>d</sup>*<sup>2</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup>

*ad*<sup>2</sup> <sup>+</sup> *gb*<sup>2</sup> <sup>−</sup> *gac*

⎞

*ad*<sup>2</sup> <sup>+</sup> *gb*<sup>2</sup> <sup>−</sup> *gac*

�

⎠ = 0, (14.103)

⎠ , (14.104)

**Vi** · **Vj** = *δij*. (14.105)

*dv* + (*g* − *λ*)*w* = 0. (14.106)

*w* = −*d*(*a* − *λ*). (14.107)

�

�

�

= 0 (14.99)

(14.101)

(14.102)

(14.108)

Which when written in algebraic form looks like,

�

*λ*1*λ*<sup>2</sup> + *λ*2*λ*<sup>3</sup> + *λ*3*λ*<sup>1</sup> =

⎡ ⎣

Recalling that, the three roots of eqn. (14.99) satisfies the following relations

*λ*1*λ*2*λ*<sup>3</sup> = −

**Vi** =

⎛ ⎝ *ui vi wi*

when suitably normalized. Next, assuming the eigen vectors to be normalized, we would demonstrate the necessary identities they need to satisfy. The proof should follow by explicit use of the values of *λ<sup>i</sup>* 's in (14.105) (which is laborious ) or by some other less laborius method. Here we explore the last option. We write down the generic eqns. satisfied by the components

> (*a* − *λ*)*u* + *bv* = 0 *bu* + (*c* − *λ*)*v* + *dw* = 0

It's easy to find out the nontrivial solns of (14.106) (for any of the three eigenvalues) by

All that is to be shown is **V**<sup>1</sup> · **V**<sup>2</sup> = 0 and other similar relations. We would prove the previous

+ (*g* − *λ*1)(*g* − *λ*2)(*a* − *λ*1) × (*a* − *λ*2)

*<sup>b</sup>*2(*<sup>g</sup>* <sup>−</sup> *<sup>λ</sup>*1)(*<sup>g</sup>* <sup>−</sup> *<sup>λ</sup>*2) + *<sup>d</sup>*2(*<sup>a</sup>* <sup>−</sup> *<sup>λ</sup>*1)(*<sup>a</sup>* <sup>−</sup> *<sup>λ</sup>*2)

relation, others can be done using similar method. To begin with note that,

�

**V**<sup>1</sup> · **V**<sup>2</sup> =

*u* = −*b*(*g* − *λ*) *v* = (*a* − *λ*)(*g* − *λ*)

*a* − *λ<sup>i</sup> b* 0 *b c* − *λ<sup>i</sup> d* 0 *d g* − *λ<sup>i</sup>*

*<sup>λ</sup>*<sup>3</sup> <sup>−</sup> *<sup>λ</sup>*<sup>2</sup> (*<sup>a</sup>* <sup>+</sup> *<sup>c</sup>* <sup>+</sup> *<sup>g</sup>*) <sup>+</sup> *<sup>λ</sup>*

We should have for any value of *i*(1, 2*or*3),

with corresponding eigen-vector

All that we need to prove is ,

of the eigen vectors

inspection and they are:

$$b^2(\mathcal{g} - \lambda\_1)(\mathcal{g} - \lambda\_2) = b^2[(\lambda\_3 - a)(\lambda\_3 - c) - b^2 - d^2].\tag{14.113}$$

Similarly one can show that,

$$d^2(a - \lambda\_1)(a - \lambda\_2) = d^2[(\lambda\_3 - \mathfrak{g})(\lambda\_3 - \mathfrak{c}) - b^2 - d^2].\tag{14.114}$$

Finally as we substitute in eqn. (14.108), the results of eqns. (14.113) and (14.114), we get after some cancellations,

$$\mathbf{V\_1 \cdot V\_2} = (\boldsymbol{\varepsilon} - \lambda\_3) \left[ (\boldsymbol{a} - \lambda\_3)(\boldsymbol{g} - \lambda\_3)(\boldsymbol{\varepsilon} - \lambda\_3) - b^2(\boldsymbol{g} - \lambda\_3) - d^2(\boldsymbol{a} - \lambda\_3) \right] = 0,\tag{14.115}$$

because the expression inside the square bracket of eqn. (14.115) after the first = sign, is zero, as can be seen by expanding the determinant, i.e., eqn. (14.98) after taking *λ<sup>i</sup>* to be *λ*3. In a similar fashion it can be shown that,

$$\mathbf{V\_1V\_2} = \mathbf{V\_2V\_3} = \mathbf{V\_3V\_1} = 0.\tag{14.116}$$

#### **15. Proof: V's actually diagonalize the mixing matrix**

Lets start from:

$$
\begin{bmatrix} u\_1 \ u\_2 \ u\_3 \\ v\_1 \ v\_2 \ v\_3 \\ w\_1 \ w\_2 \ w\_3 \end{bmatrix}^T \begin{pmatrix} a & b & 0 \\ b & c & d \\ 0 & d & g \end{pmatrix} \begin{bmatrix} u\_1 \ u\_2 \ u\_3 \\ v\_1 \ v\_2 \ v\_3 \\ w\_1 \ w\_2 \ w\_3 \end{bmatrix} = \tag{15.117}
$$
  $\begin{bmatrix} u\_1a + bv\_1 \ u\_1b + v\_1c + w\_1d \ v\_1d + gw\_1 \\ u\_2a + bv\_2 \ u\_2b + v\_2c + w\_2d \ v\_2d + gw\_2 \\ u\_3a + bv\_3 \ u\_3b + v\_3c + w\_3d \ v\_3d + gw\_3 \end{bmatrix} \begin{pmatrix} u\_1 \ u\_2 \ u\_3 \\ v\_1 \ v\_2 \ v\_3 \\ w\_1 \ w\_2 \ w\_3 \end{pmatrix}$ 

Imposed by CP Conservation in the Presence of Instantons," *Physical Review D* 16,

73

[arXiv:hep-ph/9606475]. For a more extensive review of the strong *CP* problem, see

*Physics Reports* Vol 198, 1. G.G. Raffelt, in *Proceedings of Beyond the Desert*, edited by H.V. Klapder-Kleingrothaus and H. Paes (Institute of Physics, Bristol, 1998), p. 808. G.G. Raffelt (1998) , in *Proceedings of 1997 European School of High-Energy Physics*, edited by

N. Ellis and M. Neubert (CERN, Geneva, 1998), p. 235. Report No. hep-ph/9712538.

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D. V. Galtsov and N. S. Nikitina (1972), "Photoneutrino processes in a strong field," *Zh. Eksp.*

L. L. . DeRaad, K. A. Milton and N. D. Hari Dass (1976), "Photon Decay Into Neutrinos in a

A. N. Ioannisian and G. G. Raffelt (1997), "Cherenkov radiation by massless neutrinos in a magnetic field," *Physical Review D* 55, 7038. [arXiv:hep-ph/9612285]. C. Schubert (2000), "Vacuum polarization tensors in constant electromagnetic fields. Part 2,"

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(Kluwer Academic, Dordrecht, 1991); G.S. Bisnovatyi-Kogan and S.G. Moiseenko

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in *CP Violation*, edited by C. Jarlskog (World Scientific, Singapore, 1989).

M. S. Turner (1990) , "Windows on the Axion," *Physics Reports,* Vol 197, 67.

J. Preskill M. B. Wise and F. Wilczek (1983) *Physics Letters,*B 120, 127.

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Introduction to Axion Photon Interaction

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Vol 43, 103.

Vol 321, 1457.

*Teor. Fiz.* Vol. 62, 2008.

82, 664.

*Physical Review Letters,* Vol 40, 279.

in Particle Physics and Photon Dispersion in Magnetized Media

Now if we recall (14.106), we see that,

$$au\_1 + bv\_1 = \lambda\_1 u\_1$$

$$\begin{split} bu\_1 + cv\_1 + dw\_1 &= \lambda\_1 v\_1 \\ dv\_1 + gw\_1 &= \lambda\_1 w\_1. \end{split} \tag{15.118}$$

Similarly,

$$au\_2 + bv\_2 = \lambda\_2 u\_2$$

$$bu\_2 + cv\_2 + dw\_2 = \lambda\_2 v\_2$$

$$dv\_2 + gw\_2 = \lambda\_2 w\_2. \tag{15.119}$$

And

$$a u\_3 + b v\_3 = \lambda\_3 u\_3$$

$$b u\_3 + c v\_3 + d w\_3 = \lambda\_3 v\_3$$

$$d v\_3 + g w\_3 = \lambda\_3 w\_3. \tag{15.120}$$

So we can substitute eqns. (15.118) to (15.120) in eqns. (15.118), to get:

$$
\begin{aligned}
\begin{bmatrix}
u\_1a+bv\_1 \ u\_1b+v\_1c+w\_1d \ v\_1d+gw\_1 \\
u\_2a+bv\_2 \ u\_2b+v\_2c+w\_2d \ v\_2d+gw\_2 \\
u\_3a+bv\_3 \ u\_3b+v\_3c+w\_3d \ v\_3d+gw\_3
\end{bmatrix}
\begin{pmatrix}
u\_1 \ u\_2 \ u\_3 \\ v\_1 \ v\_2 \ v\_3 \\ w\_1 \ w\_2 \ w\_3
\end{pmatrix} = \begin{bmatrix}
u\_1\lambda\_1 \ v\_1\lambda\_1 \ w\_1\lambda\_1 \\
u\_2\lambda\_2 \ v\_2\lambda\_2 \ w\_2\lambda\_2 \\
u\_3\lambda\_3 \ v\_3\lambda\_3 \ w\_3\lambda\_3
\end{bmatrix}
\text{(15.121)} \\
\times \begin{pmatrix}
u\_1 \ u\_2 \ u\_3 \\ v\_1 \ v\_2 \ v\_3 \\ w\_1 \ w\_2 \ w\_3
\end{bmatrix} = \begin{bmatrix}
\lambda\_1 \ 0 & 0 \\ 0 \ \lambda\_2 \ 0 \\ 0 \ 0 \ \lambda\_3
\end{bmatrix}
\end{aligned}
$$

So we have checked that, the transformation matrix, constructed from the orthogonal vectors, diagonalize the mixing matrix.

#### **16. References**

Weinberg S. (1975). The U(1) problem. *Physical Review D* 11, 3583. (1975)


24 Will-be-set-by-IN-TECH

*bu*<sup>1</sup> + *cv*<sup>1</sup> + *dw*<sup>1</sup> = *λ*1*v*<sup>1</sup>

*bu*<sup>2</sup> + *cv*<sup>2</sup> + *dw*<sup>2</sup> = *λ*2*v*<sup>2</sup>

*bu*<sup>3</sup> + *cv*<sup>3</sup> + *dw*<sup>3</sup> = *λ*3*v*<sup>3</sup>

⎤ ⎦ ⎛ ⎝

So we have checked that, the transformation matrix, constructed from the orthogonal vectors,

Belavin A.A., Polyakov A.M., Shvarts, A. S.and Tyupkin, yu. S. (1975). Inatanton Solutions In

't Hooft, G. (1976). Symmetry Breaking Through Bell-Jackiw anomalies *Physical Review Letters,*

't Hooft, G. (1976). Computation of the quantum effects due to a four dimensional pseudoparticle. *Physical Review D* 14, 3432 (1978); (E) ibid. 18, 2199, (1978). R.J. Crewther (1978), "Effects of topological charge in gauge theories. *Acta Phys. Austriaca*

R. J. Crewther, P. Di. Vecchia, G. Veneziano, E. Witten (1980) ,"Chiral estimates of the electric

R. D. Peccei and H. R. Quinn (1977) , "CP Conservation in the Presence of Instantons," *Physical*

dipole moment of the neutron in quantum chromodynamics ", *Physics Letters B,* 88,

*Review Letters,* Vol. 38, 1440. R. D. Peccei and H. R. Quinn(1977) , "Constraints

*u*<sup>1</sup> *u*<sup>2</sup> *u*<sup>3</sup> *v*<sup>1</sup> *v*<sup>2</sup> *v*<sup>3</sup> *w*<sup>1</sup> *w*<sup>2</sup> *w*<sup>3</sup>

> × ⎛ ⎝

⎞ ⎠ =

*u*<sup>1</sup> *u*<sup>2</sup> *u*<sup>3</sup> *v*<sup>1</sup> *v*<sup>2</sup> *v*<sup>3</sup> *w*<sup>1</sup> *w*<sup>2</sup> *w*<sup>3</sup>

So we can substitute eqns. (15.118) to (15.120) in eqns. (15.118), to get:

Weinberg S. (1975). The U(1) problem. *Physical Review D* 11, 3583. (1975)

Nonabelian Gauge Theories. *Physics Letters B* 59, 85, (1975)

V. Baluni(1979) "CP violating effects in QCD", *Physical Review D* 19, 2227.

*u*1*a* + *bv*<sup>1</sup> *u*1*b* + *v*1*c* + *w*1*d v*1*d* + *gw*<sup>1</sup> *u*2*a* + *bv*<sup>2</sup> *u*2*b* + *v*2*c* + *w*2*d v*2*d* + *gw*<sup>2</sup> *u*3*a* + *bv*<sup>3</sup> *u*3*b* + *v*3*c* + *w*3*d v*3*d* + *gw*<sup>3</sup>

*au*<sup>1</sup> + *bv*<sup>1</sup> = *λ*1*u*<sup>1</sup>

*au*<sup>2</sup> + *bv*<sup>2</sup> = *λ*2*u*<sup>2</sup>

*au*<sup>3</sup> + *bv*<sup>3</sup> = *λ*3*u*<sup>3</sup>

*dv*<sup>1</sup> + *gw*<sup>1</sup> = *λ*1*w*1. (15.118)

*dv*<sup>2</sup> + *gw*<sup>2</sup> = *λ*2*w*2. (15.119)

*dv*<sup>3</sup> + *gw*<sup>3</sup> = *λ*3*w*3. (15.120)

⎡ ⎣

> ⎞ ⎠ =

*u*1*λ*<sup>1</sup> *v*1*λ*<sup>1</sup> *w*1*λ*<sup>1</sup> *u*2*λ*<sup>2</sup> *v*2*λ*<sup>2</sup> *w*2*λ*<sup>2</sup> *u*3*λ*<sup>3</sup> *v*3*λ*<sup>3</sup> *w*3*λ*<sup>3</sup>

> ⎡ ⎣

*λ*<sup>1</sup> 0 0 0 *λ*<sup>2</sup> 0 0 0 *λ*<sup>3</sup> ⎤

⎤ ⎦

⎦(15.121)

Now if we recall (14.106), we see that,

Similarly,

And

⎡ ⎣

**16. References**

diagonalize the mixing matrix.

Vol.37, 8, (1976).

*Suppl.* 19, 47.

123; (E) ibid. B91, 487 (1980).

Imposed by CP Conservation in the Presence of Instantons," *Physical Review D* 16, 1791.


**4** 

Kihyeon Cho

*Republic of Korea* 

**The e-Science Paradigm for Particle Physics<sup>1</sup>**

Research in the 21st century is increasingly driven by the analysis of large amounts of data within the e-Science paradigm. e-Science is the data centric analysis of science experiments unifying experiment, theory, and computing. According to Simon C. Lin and Eric Yen (Lin & Yen, 2009), e-Science or data-intensive science unifies theory, experiment, and simulations using exploration tools that link a network of scientists with their datasets. Results are

In this chapter, we use the concept of e-Science to combine experiment, theory and computing in particle physics in order to achieve a more efficient research process. Particle physics applications are generally regarded as a driver for developing this global e-Science

According to Tony Hey at Microsoft (Hey, 2006), thousands of years ago science focused on experiments to describe natural phenomena. In the last few hundreds of years, science became more theoretical. In the last few decades, science has become more computational, focusing on simulations. Today, science can be described as more data-intensive in nature, requiring a combination of experiment, theory, and computing. Attempts have been made to realize this e-Science concept. One e-Science application is the Worldwide Large Hadron Collider Computing Grid (WLCG), which realizes Ian Foster's definition of a grid (Foster et al., 2001). The grid is the combination of computing resources from multiple administrative domains to reach a common goal (Cho & Kim, 2009). As the global e-Science infrastructure is rapidly established, we must take advantage of worldwide e-Science progress. Highenergy physics has advanced the e-Science paradigm by successfully unifying experiments,

We apply the e-Science concept to particle physics and show an example of this paradigm. As shown in Fig. 1, we construct a unified research model of experiment-theory-computing

This is not a simple collection of experiments, computing, and theory, but a fusion of research in order to achieve a more efficient research process. We apply this concept to the

1 This chapter is based on the paper titled "Collider physics based on e-Science paradigm of experimentcomputing-theory" by K. Cho et al. in Computer Physics Communication Vol. 182, pp. 1756-1759 (2011).

**1. Introduction** 

infrastructure.

analyzed using a shared computing infrastructure.

theory, and computing (Cho et al., 2011).

in order to probe the Standard Model and search for new physics.

*Korea Institute of Science and Technology Information* 

