#  BHU/HEP/2020 hep-ph/2002276

Magnetized matter effects on dilaton photon mixing

Ankur Chaubey,a Manoj K. Jaiswala and Avijit K. Gangulya ,
$a)$ Institute of Science, Department of Physics, Banaras Hindu University,
Varanasi- 221005, INDIA.
e-mail addresses<EMAIL_ADDRESS>corresponding author

###### Abstract

Dilatons ($\phi(x)$) are a class of bosonic scalar particles associated with
scaling symmetry and its compensation (under the violations of the same). They
are capable of interacting gravitationally with other massive bodies. As they
have coupling to two photons ($\gamma$), so they are (also) capable of
decaying to the two photons. However, the decay time is long and that makes
them good candidate for dark matter (DM). Furthermore due to two photon
coupling, they can produce optical signatures in a magnetic field. In vacuum
or plain matter they couple to one of the transversely polarized state of the
photon. But in a magnetized matter, they couple to both the transversely
polarized state of photon (due to emergence of a parity violating part of
photon self energy contribution- from a magnetized matter). Being spin zero
scalar, they could mix with spin zero longitudinal part of photons but they
don’t. A part of this work is directed towards understanding this issue of
mixing of scalar with various polarizations states of photon in a medium (
magnetized or unmagnetized ) due to the constraints from different discrete
symmetries e.g., Charge conjugation (${\bf C}$), Parity (${\bf P}$) and Time
reversal (${\bf T}$) associated with the interaction. Based on these symmetry
aided arguments, the structure of the mixing matrix is found to be $3\times
3$, like in the case of neutrino flavour mixing matrix.Thus there exists non-
zero finite probabilities of oscillation between different polarization states
of photon to dilaton. Our analytical and numerical analysis show no existence
of periodic oscillation length either in temporal or spatial direction for
most general values of the parameters in the theory. Possible astrophysical
consequences of these results, those can be detected through observations are
discussed.

##  1 Introduction

The issue of unification of four fundamental forces of nature by the
introduction of an additional scalar (fields)$(\phi(x))$ is probably one issue
that may hold key to the solution of dark energy/matter puzzle. These
postulated fields are found to make their appearance in two kinds of theories,
one is in quantum theories of unification and other is in higher dimensional
theories of cosmology/gravity.
In theories beyond the standard model (of particle physics), these additional
scalar fields often appear in theories of unification ([1] –[15]), like in
5-dimensional Kaluza Klien theory, in super-string theory and also in theories
of extended super gravity [1] etc. In higher dimensional theories, like the
String theory, the scalar fields – termed moduli (fields) ([2]–[12]), are
necessary to produce a four-dimensional effective theory from the original
higher dimensional theory, by compactifing the extra dimensions.
On the other hand, scalar fields ( both interacting and massive ) had also
been postulated independently, in models of cosmology to shed light on the
aspects of dark matter, they are called chameleon whose presence remains
imprinted in the cosmological observations of cosmic microwave background
radiations ([16] – [28]). Also, the compelling observational aspects of dark
matter physics has motivated particle-astrophysics community to construct
particle physics models with similar fields ([29] –[34]) and verify their
suitability in explaining the experimental data. A detailed description of the
possible candidates of dark matter can be found in [34]. And possible other
applications can be found in ([7]–[35]).
It is not so often that the solution of an unresolved issue in one area of
physics holds _the_ key to another unresolved issue in other area of the same.
The issues of unification of four forces of nature and the missing mass and
energy (dark matter and dark energy) problem of cosmology, might turn out to
represent such an event of rarity.
The notable feature that this field $\phi(x)$ exhibits in these theories lie
in the structure of their interaction with photons. This is given by, an
interaction Lagrangian of the form, $\frac{1}{M}\phi F^{\mu\nu}F_{\mu\nu}$,
where $M$, is the energy scale of the physics, involved. In place we may
write, $g_{\phi\gamma\gamma}=\frac{1}{M}$, where $g_{\phi\gamma\gamma}$ is the
dimension full coupling constant. This interaction term has most of the
desirable properties of an ideal Lagrangian except one. Although it is local,
remains invariant under Lorentz, gauge and CPT symmetry transformations,
however, is non-renormalizable, compromised for having interaction term of
mass dimension five. Incidentally, similar interaction terms are also possible
for loop induced Higgs photon interaction, given by,
$g^{\prime}HF^{\mu\nu}F_{\mu\nu}$, where $g^{\prime}$ is the effective
coupling constant, obtained after integrating out the heavy degrees of freedom
(dof), $H$ is the Higgs field, and the rest of the pieces have their usual
meaning.
Many of these fields can have coupling (very weak though, since
$\frac{m}{M}\ll 1$ ) to standard model particles, there by, may be detected
indirectly, through collider or astrophysical observations. Some of their
possible signatures are (i) existence of fifth force distinguishable from
gravity (ii) violation of Lorentz invariance[36] and (iii) spectro-
polarimetric signals those discussed in the phenomenal papers like, [37],
[38],[39] and [40] among many others of similar quality. A comprehensive list
of other possible signatures, can be found in ([41] – [49]).
A notable and important issue related to this kind of interaction, that is,
capable of distinguishing similar looking scalar fields from each other (owing
their origin to various symmetries ) – is related to the magnitude of their
mass ($m$). The same is not fixed by any symmetry argument. However the lower
limit to the same ($m$) is fixed by the torsion-balance fifth force
experiment, that is estimated to be greater than $10^{-2}eV$ [50]–[54]. And
the upper limit, on the same (i.e. $m$) can go upto few TeV [[55]] – depending
on the type of model (moduli) one is interested in. In cosmology for example,
a limit to the mass $m$ and energy scale $M$ is set from the estimate of their
life time to two photon decay, given by: $\tau_{\phi}=\frac{16\pi
M^{2}}{m^{3}}$ [56, 57, 58, 59], so that the produced photons do not interfere
with the big bang nucleosynthesis (bbn) constraints [60]. Although lately
there are proposals of new kind of dark matter termed as fuzzy dark matter[61]
aviailable in the literature, for which torsion balance bounds are not
applicable. The bounds on their masses are obtained from sachs-wolfe effect
[62, 63].
The laboratory based experimental search for these particles were initially
suggested in [39, 40] and some of the variants of the same were in [64, 65,
66, 67, 68]. Many of these lab based experiments have provided some bounds on
axion coupling constant with other fundamental particles like electron, photon
etc. These two experiments [67, 68] happen to be one of those. However the
launch of the satellite missions- EUVE [69], ROSAT [70], BeppoSAX [71], XMM-
Newton [72], Chandra [71, 73, 74] and Suzaku [75] sensitive to soft X-ray
emissions around $5-10$ KeV from the galaxy clusters– have opened up another
possibility of their astrophysical confirmation. Interestingly enough, ever
since their launch, evidence of 0.5-1.0 MeV lines [76], 3.5 KeV lines [77] and
511 keV lines ([78]–[80]) have been reported in the literature. Some of these
signals are believed to be due to Dark matter.
Though many of the laboratory and astrophysical EM signals are complementry to
each other however, for few the astrophysical ones are better than the
laboratory ones. The presence of strong coherent magnetic field over a large
length scale, an ambient plasma and abundance of highly energetic photons make
it convenient to look for EM signals from astrophysical sources to test many
types of interactions.
For the same reason, the EM signatures from astrophysical sources for
$\frac{1}{M}\phi F^{\mu\nu}F_{\mu\nu}$ interaction, is arguably better than
the ones from the laboratory. Although a large volume of literature [10],[37],
[38], [81], [82], [83] are already available on many aspects of the relevant
issues.

One of the notable aspect of these studies had been the incorporation of
magnetized vacuum effects by considering the Euler-Heisenberg Lagrangian. This
incorporation lifts the degeneracy between the two transversely polarized
photons and makes only one of the polarized states of photon mix with ALP like
particle leaving the other one free. This turns the vacuum dichoric and
birefringent that inprinciple can be detected if ALPs exist in nature.

In this note however, in this work we would like to point out another aspect,
that has rarely been considered important in such investigations, that is
background dependence of the dynamics of scalar or pseudoscalar photon
interaction.
Photons propagating in a magnetized vacuum with $\phi F^{\mu\nu}F_{\mu\nu}$
interaction have two transverse polarization states. One of them
$(|\gamma_{\perp}>)$ is orthogonal to $\rm{B}$ and the other one
$(|\gamma_{\parallel}>)$ lies on the ${k}$-$\rm{B}$ plane. Following the
reasoning of [38] (performed originally for axion photon system), the two
polarization states of photon, for this case too, would transform differently
under parity ${\bf P}$ and charge conjugation ${\bf C}$ symmetry
transformations. Since under ${\bf CP}$ transformation, the scalar and the
${\bf CP}$ even polarization state of photon would remain even, only these two
would couple during propagation; the ${\bf CP}$ odd polarized state of the
photon would propagate freely.
When medium induced corrections are considered, the contribution from the in-
medium polarization tensor $\Pi_{\mu\nu}(k)$ need to be taken into account.
The same, in absence of any parity violating interaction or ambient external
magnetic field $\rm{B}$, would be ${\bf CP}$ symmetric. Hence even in an
unmagnetized medium, the propagating modes of the scalar ($\phi$) photon
($\gamma$), system with dim-5 scalar photon interaction, would remain same as
in magnetized vacuum 111 However, that consideration of the ambient plasma
effects brings changes in the size of the contribution to the oscillation
probability $P_{{\gamma_{||}\to\phi}}$.
This picture, however changes, with the introduction of a new parity violating
interaction term to the effective Lagrangian ($L_{eff}$), that originates from
the magnetized medium induced corrections to the photon self energy tensor
(PSET), $\Pi_{\mu\nu}(k,T,\mu,e\rm{B})$ (sometimes called the Faraday term) in
the system. This tensor (PSET) in a magnetized medium, has parts thats even in
$e\rm{B}$ and a part that is odd in the same. It is the part that is odd in
$e\rm{B}$ is also odd in $\mu$ (chemical potential) and is parity violating.
This part was originally evaluated in [84].
An effective Lagrangian of the form
$A^{\mu}(-k)\Pi_{\mu\nu}(k,T,\mu,e\rm{B})A^{\nu}(k)$ constructed with the same
would change the mixing dynamics. The leading order magnetic field effects in
a magnetized plasma for $m^{2}>e\rm{B}$, can be obtained by retaining the
O($e\rm{B}$) piece from $A^{\mu}(-k)\Pi_{\mu\nu}(k,T,\mu,e\rm{B})A^{\nu}(k)$,
in the interaction Lagrangian $(L_{eff})$. We perform the same in this study.
This formalism was developed earlier in [38], [81], [85] to study the axion-
photon interaction dynamics. We have extended this formalism to the case of
scalar photon interaction dynamics by incorporating the correction mentioned
above.This causes further mixing between the two transverse polarization
states of the photon; therefore, the scalar and the two transversely polarized
states of the photon mix with each other. As a result $|\gamma_{\perp}>$ part
of a photon beam, unlike vacuum would evolve with propagation due to the
presence of PSET, and longitudinal dof would propagates freely. The effect of
the same on polarimetric signatures of scalar photon mixing case has been
discussed in [86].Though a similar approach, for polarimetric studies had been
considered in [85] for axion photon system also, but the conversion
probabilities of parallely or perpendicularly polarized photons to pseudo-
scalar axions remained unexplored.
We complement the same in this work by calculating these conversion
probabilities for dilaton-photon system.With the introduction of PSET, the
probability of conversion of perpendicularly polarized photon to dilaton and
vice versa turns out to be finite.Till so far this aspect remained unreported
in the literature. We explore the same here and its consequences.
We demonstrate here in this note that the inclusion of PSET, causes the
$\gamma-\phi$ mixing matrix to be $3\times 3$ instead of $2\times 2$, usually
encountered for a similar process taking place in magnetized vacuum, or
unmagnetized plasma. 222The situation is also different from axion photon
system when PSET is considered, where there is mixing between all four degrees
of freedom (three degrees of freedom of a photon in medium and single degree
of freedom of axion). Thus one has to deal with a $4\times 4$ mixing matrix to
study the evolution of the axion-photon system. Now, as a result of scalar
photon mixing, the two transverse degrees of freedom of photon, the
$\parallel$ and the $\perp$, can now oscillate into and out of scalar
(dilaton) mode in addition to the oscillations among themselves i.e.,
$\parallel\leftrightarrow\perp$. Under favourable circumstances signals of the
same may be within the future detector sensitivity.
The organisation of the document is as follows: in the next section that is in
section two, we have elaborated on the form of the action and propagators in
flat and curved space time. We have provided the logic behind sticking to the
description in flat space time because of pathological problems encountered in
curved spacetime results. Followed by that the tensorial structure of the
polarization tensor for photons in an unmagnetized medium is discussed. This
is followed by the description of the fermion propagator in coordinate space.
The parity violating part of the photon polarization tensor is discussed and
its tensorial structure is discussed next.
Section three contains a discussion of the effective Lagrangian. This is
followed by a brief discussion on the discrete symmetry transformation
properties of the gauge potentials and the terms of the equations of motion of
the $\gamma\phi$ system obtained from the effective Lagrangian under
consideration. In section four, we establish the unitary transformation matrix
which diagonalises the mixing matrix. In section five, using the same unitary
matrix obtained in section four, we diagonalize the equations of motion and
identify its similarity with Klein Gordon equation in the diagonal form. We
take this equation and following the procedure of [38], we estimates the
conversion probabilities of various modes into each other. In section six we
discuss the physics of appropriate astrophysical environments where the mixing
of the photons with scalar can take place. We identify some possible
signatures of this mixing from the EM signals coming out of these
astrophysical environments. The implications of this modified system is
discussed in section even. In section eight we provide possible implications
of our work for some of the DM signatures existing in the literatures followed
by an appendix where some technical details are elaborated.

##  2 The polarization tensor

Considering the form of coordinate space dilaton-photon interaction term
presented in [56] the effective action for scalar photon interaction including
medium corrections [[84]-[99]], in configuration space can be expressed as,

$\displaystyle S\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\int
d^{4}x\sqrt{-g}~{}\\!\\!\left(-\frac{1}{4}F_{\mu\nu}(x)F^{\mu\nu}(x)-\frac{g_{\phi\gamma\gamma}}{4}\phi(x)\,F_{\mu\nu}(x)F^{\mu\nu}(x)-\frac{1}{2}\int
d^{4}x^{\prime}\,A_{\mu}(x)\Pi^{\mu\nu}(x,x^{\prime})A_{\nu}(x^{\prime})\right.$
(2.1)
$\displaystyle+\left.\frac{1}{2}\partial_{\mu}\phi(x)\partial^{\mu}\phi(x)-\frac{1}{2}m_{\phi}^{2}\,\phi^{2}(x)\right).$

In eqn. (2.1) $g$ is the determinant of the space-time metric that assumes the
value -1 in Minkowski space. The same ( factor $\sqrt{-g}$ ) is multiplied by
$d^{4}x$ to maintain Lorentz invariance of the measure. In the same (i.e. eqn.
(2.1)) $\Pi^{\mu\nu}(x,x^{\prime})$ stands for the in-medium polarization
tensor in configuration space. The polarization tensor can be expressed,
following [90] as:,

$\displaystyle\Pi^{\mu\nu}(x,x^{\prime})=e^{2}tr\int\sqrt{-g}~{}d^{4}y^{\prime}\left(\gamma^{\mu}S_{F}(x,y^{\prime})\gamma^{\nu}S_{F}(y^{\prime},x)\right)\delta^{4}(x^{\prime}-y^{\prime}),$
(2.2)

the trace is defined over Dirac matrices. The in medium propagator in
coordinate space is defined, in terms of vacuum propagator:
$S^{v}_{F}(x^{\prime},x^{\prime\prime})=<x^{\prime}|\frac{1}{{\hbox
to0.0pt{/\hss}\partial}-m+i\epsilon}|x^{\prime\prime}>$ as,

$\displaystyle
S_{F}(x^{\prime},x^{\prime\prime})=S^{v}_{F}(x^{\prime},x^{\prime\prime})-f(p.u)\Big{[}S^{v}_{F}(x^{\prime},x^{\prime\prime})-S^{v*}_{F}(x^{\prime},x^{\prime\prime})\Big{]}$
(2.3)

where $f(p.u)$ happens to be the Fermi-distribution function in a frame moving
with four velocity $u^{\mu}$.
The curved spacetime Fermion propagator $S_{Fc}$ can also be represented as
[101],

$\displaystyle(i{\hbox
to0.0pt{/\hss}{D}}-m)S_{Fc}(x,x^{\prime})=\frac{i}{{\sqrt{-g}}}\delta(x,x^{\prime}).$
(2.4)

The reason behind using this alternative representation, is the inability of
expressing the asymptotic vacuum states uniquely curved space time. Here $D$
stands for the covariant derivative in curved space-time. It can further be
written in terms of the scalar green function

$\displaystyle\left(D^{2}+i\sigma^{\mu\nu}F_{\mu\nu}-\frac{R}{4}+m^{2}\right)G(x,x^{\prime})=\frac{-i}{\sqrt{-g}}\delta(x,x^{\prime})$
(2.5)

Finally, the Fermion propagator in curved space time in magnetic field is
follows from eqn. (2.5)

$\displaystyle S_{Fc}(x,x^{\prime})=\big{(}i{\hbox
to0.0pt{/\hss}{D}}+m\big{)}G(x,x^{\prime})$ (2.6)

where $R$ happens to be the Ricci scalar. One can express the above
expressions in momentum space, by taking the Fourier transform using two point
transforms [102],[103]:

$\displaystyle
f(x)=\int\frac{dk_{{\mu}^{\prime}}}{(2\pi)^{d}}g^{(-1/2)}(x^{\prime})exp(-ik_{\mu^{\prime}}\sigma^{\mu^{\prime}}(x,x^{\prime})){\tilde{f}}(k;x^{\prime}).$
(2.7)

A strong gravitational background is known to introduce some pathological
problem, in external electromagnetic field. For example the photon velocity
estimated for Euler-Heisenberg Lagrangian system, predicts superluminal
velocity of photon [104]. Therefore the search for particles like dilaton like
particles should be restricted to space where spacetime curvature is
negligible or flat. Usually the spacetime curvature at a distance $r$ from a
body of mass $M$, is given by the Kretschmann scalar [105],

$\displaystyle R=\frac{48~{}M^{2}}{r^{6}}.$ (2.8)

For a body with mass close to solar mass or less, the curvature is negligible
for most of the regions close to its surface. Moreover our region of interest
happens to be close to the light cylinder of the star, hence we have not
considered the curvature of the spacetime for our estimates.

Coming back to flat space time picture, $\Pi_{\mu\nu}(k,\mu,T,eB)$ has
contribution coming from (a) magnetized vacuum (b) unmagnetized medium and (c)
magnetized medium. Contribution from (c) can further be divided in two pieces
those, having algebraic structure that can be written as a polynomial, first
$(c_{1})$ that is even $eB$ even $\mu$ and second $(c_{2})$ that is a
polynomial odd in $eB$ and odd in $\mu$. The even $eB$ odd $\mu$ and odd $eB$
even $\mu$ parts make vanishing contribution to $\Pi_{\mu\nu}$. In parameter
region where $eB<<m^{2}$ and momentum $k<<m$, contributions from $(a)$ and
$(c_{1})$ are suppressed compared to $(c_{2})$[108, 107].
Though the contribution from (a), and $(c_{1})$ have their special
significance in the mixing dynamics for charge symmetric medium ($\mu=0$) but
inclusion of $(c_{2})$ makes a paradigm shift in structure of mixing. So to
initiate a discussion on effective Lagrngian we start with a discussion on the
structure of polarization tensor in an unmagnetized media in momentum space in
sub-section[2.1].

###  2.1  Polarization Tensor: Structure

The linear response to EM excitations of a medium at finite density,
temperature and an external field can studied by evaluating the in-medium
photon polarization tensor $\Pi_{\mu\nu}(k)$ by the techniques of quantum
statistical field theory [87, 88, 89, 90]. This tensor is supposed to have few
essential properties [87] e.g., it possesses a symmetry i.e.,

$\displaystyle\Pi_{\mu\nu}(k)=\Pi_{\nu\mu}(-k),$ (2.9)

called Bose symmetry. It should obey the Ward identity,

$\displaystyle k^{\mu}\Pi_{\mu\nu}(k)=0,$ (2.10)

ensuring gauge invariance (also called charge conservation law). The
requirement of unitarity demands that the polarization tensor should have the
property $\Pi_{\mu\nu}(k)=\Pi^{*}_{\nu\mu}(k)$. This when combined with Bose
symmetry(Eq.(2.9) yields:

$\displaystyle\Pi_{\mu\nu}(k)=\Pi^{*}_{\mu\nu}(-k).$ (2.11)

Therefore, in four dimensions, the tensor $\Pi_{\mu\nu}(k)$ now needs to be
constructed from the available 4-vectors and tensors with the system; namely
the medium centre of mass velocity 4-vector $u^{\mu}$, the photon 4-momentum
vector $k^{\mu}=({\omega,{\vec{k}}})$, the metric tensor $g^{\mu\nu}$, Levi-
Civita tensor $\epsilon_{\mu\nu\rho\lambda}$ and the Lorentz scalar form
factors such that, eqns. ((2.9)–(2.10)), are satisfied.
The tensorial structure of the in-medium photon self energy tensor in absence
of any external field is given by:

$\displaystyle\Pi_{\mu\nu}(k)=\Pi_{T}R_{\mu\nu}+\Pi_{L}Q_{\mu\nu}$ (2.12)
$\displaystyle\,\mbox{~{}~{}~{}~{}where,~{}~{}~{}~{}}\left\\{\begin{array}[]{c}R_{\mu\nu}=\tilde{g}_{\mu\nu}-Q_{\mu\nu},\\\
\,\,\,\,\,\,\,\,\,\,\,\,\tilde{g}_{\mu\nu}=\left(g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right),\\\
\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!Q_{\mu\nu}=\frac{{\tilde{u}_{\mu}}{\tilde{u_{\nu}}}}{{{\tilde{u}^{2}}}},\\\
\\!\\!\\!\\!\\!\\!\\!\tilde{u}_{\mu}=\tilde{g}_{\mu\nu}u^{\nu}.\\\
\end{array}\right.$ (2.16)

The dispersive part of $\Pi_{\mu\nu}(k)$, satisfies (2.11), in a charge $({\bf
C})$ symmetric ( $\mu_{\bar{f}}=\mu_{f}$ ) or asymmetric(
$\mu_{\bar{f}}\neq\mu_{f}$ ) medium automatically. This in turn dictates the
functional form of the form-factors $\Pi_{L}$ and $\Pi_{T}$ on the scalars
made out of $k^{2}$, (k.u) etc. The scalar form factor $\Pi_{L}(k)$,
corresponding to the longitudinal degree of freedom, is given by[91]:

$\displaystyle\Pi_{L}(k)=-\frac{k^{2}}{|{\vec{k}}|^{2}}\Pi_{\mu\nu}(k)u^{\mu}u^{\nu},$
(2.17)
$\displaystyle\mbox{~{}~{}~{}~{}where~{}~{}~{}~{}}u^{\mu}u^{\nu}\Pi_{\mu\nu}(k)=\omega^{2}_{p}\left(\frac{|{\vec{k}}|^{2}}{\omega^{2}}+3\frac{|{\vec{k}}|^{4}}{\omega^{4}}\frac{T}{m}\right).$

Similarly the transverse form factor $\Pi_{T}$ is given by the expressions:

$\displaystyle\Pi_{T}(k)$ $\displaystyle=$ $\displaystyle
R^{\mu\nu}\Pi_{\mu\nu}(k)\mbox{~{}~{}~{}~{}and~{}~{}~{}~{}}{\rm{R}}^{\mu\nu}\Pi_{\mu\nu}(k)=\omega^{2}_{p}\left(1+\frac{|{\vec{k}}|^{2}}{\omega^{2}}\frac{T}{m}\right).$
(2.18)

In the expressions above $\omega_{p}$ denotes the plasma frequency. In the
classical limit, to leading order in $\frac{T}{m}$, it is given by:

$\displaystyle\omega_{p}=\sqrt{\frac{4\pi\alpha
n_{e}}{m}\left(1-\frac{5T}{2m}\right)},$ (2.19)

where $n_{e}$, is the number density of electrons. A detailed structure of
photon polarization tensor $\Pi_{\mu\nu}(k)$ in vacuum, in medium and in
magnetized medium has been discussed in [92]. For more insight into it, one
can consult to this reference.
Recalling the transformation properties of the background EM field
$\bar{F}^{\mu\nu}$, and the 4-vectors $u^{\mu}$ and $k^{\mu}$ under ${\bf C}$
conjugation and ${\bf P}$ reversal operations [84], it’s easy to figure out
that under parity transformation the two orthogonal polarization eigen states
of a photon, described in the notation of [93] by (
$\left(\tilde{\bar{F}}_{\mu\nu}{f}^{\mu\nu}(k)\right)\triangleq|\perp>$ ) and
($\left(\bar{F}_{\mu\nu}{f}^{\mu\nu}(k)\right)\triangleq|\parallel>$), are at
odds with each other.
Now in the light of Eqn. (2.16) $\Pi_{\mu\nu}(k)$ is ${\bf C\rm{~{}and~{}}P}$
even, when the respective transformations are considered individually or
together. i.e.,

$\displaystyle{\bf P}^{-1}\Pi_{\mu\nu}(k){\bf P}=\Pi_{\mu\nu}(k),$ (2.20)
$\displaystyle{\bf C}^{-1}\Pi_{\mu\nu}(k){\bf C}=\Pi_{\mu\nu}(k),$ (2.21)
$\displaystyle\hskip 56.9055pt\left({\bf
CP}\right)^{-1}\Pi_{\mu\nu}(k)\left({\bf CP}\right)=\Pi_{\mu\nu}(k).$ (2.22)

Hence, recalling the issue of coupling of degrees of freedom in a unmagnetized
material medium that was initiated in the introduction, it is easy to realise
that in an unmagnetized medium the parity violating state ( i.e. $|\perp>$ )
would propagate freely but not the parity preserving state ($|\parallel>$).
This one would couple to $\phi(k)$, because of Eqn. (2.20)333 Since the ${\bf
CP}$ invariant background medium can not compensate for the parity odd
property of $|\perp>$, so that it can couple to $\phi(k)$.. Thereby, in a
material medium, the dynamics of the system remains as the same as it was in a
magnetized vacuum. The kinematics, however changes. In a plain material
medium, the magnitude of the oscillation probability undergoes modification
vis-a-vis the same in a magnetized vacuum.
In the presence of an external EM field, the photon polarization tensor in a
magnetized media can be expressed in terms of the rank two basis tensors
constructed out of the field strength tensor ${\bar{F}}^{\mu\nu}$, the Levi-
Civita tensor $\epsilon_{\mu\nu\lambda\sigma}$ along-with the other 4-vectors
and tensors mentioned before and the form factors those are Lorentz scalars
constructed using these 4-vectors and tensors. We deliberate on this in the
next sub section.

###  2.2 Photon polarization tensor in a magnetized medium: all orders in
(e$B$) .

Photon polarization tensor in a magnetized media in configuration space would
follow from, eqn. [2.2] where one needs to use the corresponding expressions
for in-medium Fermion propagators in an external magnetic field [2.3]. The
expression for the same is provided below. The Fermionic propagator in
magnetized vacuum is [95] ,

$\displaystyle iS^{v}_{F}(x^{\prime},x^{\prime\prime})=\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!-\frac{i\Phi(x^{\prime},x^{\prime\prime})}{(4\pi)^{2}}\int^{\infty}_{0}ds\frac{eB}{sin(eBs)}exp\left[-is\left(-\frac{e\sigma_{\mu\nu}F^{\mu\nu}}{2}+m^{2}-i\epsilon\right)\right]$
(2.23) $\displaystyle\times
exp\left[-\frac{i}{4}\left((x^{\prime}-x^{\prime\prime})^{\alpha}\left(eF~{}coth(eFs)\right)_{\alpha\beta}(x^{\prime}-x^{\prime\prime})^{\beta}\right)\right]$
$\displaystyle\times\left[\frac{\gamma^{\lambda}}{2}\left(eF~{}coth(eFs)+eF\right)_{\lambda\rho}(x^{\prime}-x^{\prime\prime})^{\rho}+m{\bf
I}\right]$

In eqn. [2.23] above the phase factor,$\Phi(x^{\prime},x^{\prime\prime})$ is
given by,

$\displaystyle\Phi(x^{\prime},x^{\prime\prime})=exp\left[ie\int^{x^{\prime}}_{x^{\prime\prime}}dx^{\mu}\left(A_{\mu}+\frac{1}{2}F_{\mu\nu}\left(x^{\prime}-x^{\prime\prime}\right)^{\nu}\right)\right],$
(2.24)

The symbol $F$ in eqn [2.24] stands for the field strength tensor $F^{\mu\nu}$
(suitably contracted when they appear in combination with functions of the
same or Dirac gamma matrices ), lastly quantity ${\bf I}$ in eqn. [2.23]
stands for 4$\times$4 unit matrix. Finally the in-medium propagator in
magnetized medium can be obtained by using the propagator [2.23] for
$S^{v}_{F}(x^{\prime},x^{\prime\prime})$ in eqn. [2.3].

###  2.3 Contribution to Photon self energy from magnetized medium: all odd
orders in (e$\rm{B}$)

In a magnetized medium, magnetic field $(e\rm{B})$ dependent extra
contributions appear in the expression for the photon polarization tensor.
They are of two types one of them is even the other one is odd in powers of
the field strength ${e\rm{B}}$. The one odd in ${e\rm{B}}$ turns out to be
also odd in $\mu$ (the chemical potential), so that this term remains even
under the operation of charge conjugation C. This apart, this term satisfies
the conditions given by Eqns. (2.9), (2.10) and Eqn. (2.11). This odd
${e\rm{B}}$ odd $\mu$, contribution to the polarization tensor violates
parity. The exact expression of the same is given by,

$\displaystyle\Pi^{p}_{\mu\nu}(k,\mu,T,e{\rm B})$ $\displaystyle=$
$\displaystyle
4ie^{2}\varepsilon_{\mu\nu\alpha_{\parallel}\beta}k^{\beta}\int\frac{d^{4}p}{(2\pi)^{4}}\eta_{-}(p)\int_{-\infty}^{\infty}ds\;e^{\Phi(p,s)}\int_{0}^{\infty}ds^{\prime}\;e^{\Phi(p^{\prime},s^{\prime})}$
(2.25) $\displaystyle\times$
$\displaystyle\Bigg{[}p^{\widetilde{\alpha}_{\parallel}}\tan
e{\rm{B}}s+p^{\prime\widetilde{\alpha}_{\parallel}}\tan
e{\rm{B}}s^{\prime}-{\tan e{\rm{B}}s\;\tan e{\rm{B}}s^{\prime}\over\tan
e{\rm{B}}(s+s^{\prime})}\;(p+p^{\prime})^{\widetilde{\alpha}_{\parallel}}\Bigg{]}\,.$

Here $e$ is the coupling constant for the U(1) gauge theory,
$\eta_{-}(p)=\eta_{F}(p)-\eta_{F}(-p)$, (with $\eta_{F}(p)$ is the statistical
factor involving Fermions and their antiparticles [84]) and $\rm{B}$, as
mentioned before is the background magnetic field. The functions ${\Phi(p,s)}$
and ${\Phi(p^{\prime},s)}$ are the contributions from Schwinger Propagator
[96], having loop momentum p and external momentum k, Fermion mass m and
parametric integrating variable s and s’. The symbol $p^{\prime}$ on r.h.s
stand for $(p+k)$ the loop momentum 4-vector
$p^{\widetilde{\alpha}_{\parallel}}$, appearing in (2.25) happens to be
components of momentum $p$ which take only the values 0 and 3 ( called the
$\parallel$ components); but with a difference. When ${\alpha_{\parallel}}$
and ${\widetilde{\alpha}_{\parallel}}$ appear together in any term and are
summed up, then, for ${\alpha_{\parallel}}$ equal to zero,
${\widetilde{\alpha}_{\parallel}}$ would take the value three and for
${\alpha_{\parallel}}$ equal to three, ${\widetilde{\alpha}_{\parallel}}$
would take the value zero. In the same equation, the symbol
$\varepsilon_{\mu\nu\alpha_{\parallel}\beta}$ is the completely antisymmetric
Levi Civita tensor, that takes the values 1 and -1, for even and odd
permutation of the indices and vanishes when any two two indices are the same.

$\displaystyle\Phi(p,s)$ $\displaystyle\equiv$ $\displaystyle
is\left(p_{\parallel}^{2}-{\tan(e{\rm{B}}s)\over
e{\rm{B}}s}\,p_{\perp}^{2}-m^{2}\right)-\epsilon|s|\,,$ (2.26)
$\displaystyle\Phi(p^{\prime},s^{\prime})$ $\displaystyle\equiv$
$\displaystyle
is^{\prime}\left({p^{\prime}}_{\parallel}^{2}-{\tan(e{\rm{B}}s^{\prime})\over
e{\rm{B}}s^{\prime}}\,{p^{\prime}}_{\perp}^{2}-m^{2}\right)-\epsilon|s^{\prime}|\,.$
(2.27)

This expression is exact to all odd orders in ${e\rm{B}}$, however performing
the integrals and arriving at compact form, from this expression is very
difficult. 444One can perform the integration with the following substitution:
express the four vector $P^{\mu}$, as
$p^{\mu}=\alpha_{(1)}u^{\mu}+\alpha_{(2)}n_{2}\cdot(k.F)^{\mu}+\alpha_{(3)}n_{3}\cdot{{B}}^{\mu}+\alpha_{(4)}n_{3}\cdot
q^{\mu}$, when $q^{\mu}=\epsilon^{\mu\nu\lambda\rho}F_{\lambda\rho}u_{\nu}$;
and $n_{i}$’s are normalization constants, so as to make the basis vectors
orthonormal.

####  2.3.1  In medium contribution to photon self energy to O (e$\rm{B}$)

The integral in the equation (2.25) describing parity violating PSET is little
difficult to evaluate exactly using analytical techniques. But perturbative
evaluation, to leading order in $e\rm{B}$ is possible [84]. The perturbative
expression can be expressed in terms of a scalar form factor
$\Pi^{p}(k,\mu,T,e{\rm B})$ and a projection operator ${P}_{\mu\nu}~{}$, so
$\Pi^{p}_{\mu\nu}(k,\mu,T,e{\rm B})$ to order ($e\rm{B}$) can be expressed in
the following form ([84], [85]):

$\displaystyle\Pi^{p}_{\mu\nu}(k)=\Pi^{p}(k)\left(i\epsilon_{\mu\nu\alpha_{\parallel}\beta}\frac{k^{\beta}}{|K|}u^{\tilde{\alpha}}\right)=\Pi^{p}(k)P_{\mu\nu},$
(2.28)

where,

$\displaystyle{P}_{\mu\nu}=i\epsilon_{\mu\nu\alpha\beta_{\parallel}}\frac{k^{\alpha}}{|K|}u^{\tilde{\beta}_{\parallel}}.$
(2.29)

The tensor ${P}_{\mu\nu}$ given by (2.29) is hermitian but it is odd under
parity transformation. The superscripts with ${\parallel}$, means, they can
take only values between $0$ and $3$. Furthermore in our notation,

$\displaystyle|K|=\left(\sum^{3}_{i=1}k^{2}_{i}\right)^{1/2}.$ (2.30)

The limit $k\to 0$ in Eqn. (2.28) should be taken in such a way that,

$\displaystyle\lim_{|k|\to 0}\left(\frac{k^{i}}{|k|}\right)\to 1.$ (2.31)

The scalar form factor $\Pi^{p}(k)$ appearing in Eq. (2.28) is given by

$\displaystyle\Pi^{p}(k)=\frac{\omega\omega_{B}\omega^{2}_{p}}{\omega^{2}-\omega^{2}_{B}},\mbox{~{}~{}where~{}~{}~{}}\omega_{B}=\frac{e\rm
B}{m}$ (2.32)

is the gyration frequency.

##  3 Effective Lagrangian with magnetized medium effects

To summarise the observations of the last section, we note that, the effects
of a medium to the propagation of excitations of interest can be considered by
evaluating the polarization tensor $\Pi_{\mu\nu}(k)$, following the methods of
finite temperature quantum field theory [85]. This takes into account the
correction arising out of temperature and density effects due to the
interactions amongst the particles fields that constitute the media.
The action in momentum space, as the quantum corrections due to ambient medium
and an external magnetic field to ${\cal O}(e\rm{B})$ are taken into account
can be obtained upon taking the Fourier transform of eqn. (2.1). The same turs
out to be,

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}k\left[\frac{1}{2}A^{\nu}(-k)\left(-k^{2}\tilde{g}_{\mu\nu}+\Pi_{\mu\nu}(k,\mu,T)+\Pi^{p}_{\mu\nu}(k,\mu,T,e\rm{B})\right)A^{\mu}(k)\right.$
(3.1)
$\displaystyle\left.+ig_{\phi\gamma\gamma}{\phi(-k)}\bar{F}_{\mu\nu}k^{\mu}A^{\nu}(k)+\frac{1}{2}\phi(-k)[k^{2}-m^{2}]\phi(k)\right].$

Here $\Pi_{\mu\nu}(k,\mu,T)$ is in medium polarization tensor and
$\Pi^{p}_{\mu\nu}(k,\mu,T,e\rm{B})$ is the correction due to magnetized medium
effects PEST as explained before. For the sake of compactness we would be
denoting $\Pi_{\mu\nu}(k,\mu,T)$ as $\Pi_{\mu\nu}(k)$ and
$\Pi^{p}_{\mu\nu}(k,\mu,T,e\rm{B})$ as $\Pi^{p}_{\mu\nu}(k)$ in subsequent
sections. We can find the equations of motion in momentum space by standard
variational principle. The equations of motion for photons is,

$\left[-k^{2}\tilde{g}_{\alpha\nu}+\Pi_{\alpha\nu}(k)+\Pi^{p}_{\alpha\nu}(k)\right]A^{\nu}(k)=-ig_{\phi\gamma\gamma}\bar{F}_{\mu\alpha}k^{\mu}{\phi(k)}.$
(3.2)

It can be simplified further in Lorentz gauge to,

$k^{2}A_{\alpha}(k)-\Pi_{\alpha\nu}(k)A^{\nu}(k)+\Pi^{p}_{\alpha\nu}(k)A^{\nu}(k)=ig_{\phi\gamma\gamma}\bar{F}_{\mu\alpha}k^{\mu}{\phi(k)},$
(3.3)

and other equation of motion for $\phi(k)$ is given by

$(k^{2}-m^{2})\phi(k)=ig_{\phi\gamma\gamma}\bar{F}_{\mu\alpha}k^{\mu}A^{\alpha}(k).$
(3.4)

###  3.1 Expanding the Gauge potential $A_{\mu}(k)$ in orthogonal basis
vectors

In order to capture the dynamics of the available degrees of freedom in a
medium, in this subsection, we need to expand the 4-vector potential
$A_{\mu}(k)$ in terms of the available 4-vectors at our disposal. The
available 4-vectors as was noted in section (2.1) are $k^{\mu}$, the
4-momentum of the particles and $u^{\mu}\equiv(1,{\bf 0})$ the center of mass
4-velocity of the medium and $\epsilon^{\mu\nu\lambda\rho}$. Using these two
and the constant external field strength tensor $\bar{F}^{\mu\nu}$ (where the
only nonzero component being $\bar{F}^{12}\neq 0$), two other vectors
$b^{(1)\mu}$ and $b^{(2)\mu}$ can be constructed. They are given by,

$b^{(1)\nu}=k_{\mu}\bar{F}^{\mu\nu}$ (3.5)

and

$b^{(2)\nu}=k_{\mu}\tilde{{\bar{F}}}^{\mu\nu},$ (3.6)

where,
$\tilde{{\bar{F}}}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}{\bar{F}}_{\alpha\beta}$.
The four-vector defining the longitudinal degree of freedom is usually given
by

$\tilde{u}^{\nu}=\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)u_{\mu}.$
(3.7)

Furthermore, the vector that is orthogonal to the vectors given in Eqn.(3.5)
and Eqn.(3.7) is:

$I^{\nu}=\left(b^{(2)\nu}-\frac{(\tilde{u}^{\mu}b^{(2)}_{\mu})}{\tilde{u}^{2}}\tilde{u}^{\nu}\right);$
(3.8)

It should be noted, that since the vectors given by, eqns.(3.5), (3.6) and
(3.7) are space like, so they have to be normalised without compromising the
hermitian character of the gauge fields. The explicit form of the
normalisation constants are given by,

$\displaystyle N_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-b^{(1)}_{\mu}b^{(1)^{\mu}}}}=\frac{1}{K_{\perp}\rm{B}},$
(3.9) $\displaystyle N_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-I_{\mu}I^{\mu}}}=\frac{K}{\omega
K_{\perp}\rm{B}}\mbox{~{}~{}~{}and lastly},$ (3.10) $\displaystyle N_{L}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-\tilde{u}_{\mu}\tilde{u}^{\mu}}}=\frac{\sqrt{k^{\mu}k_{\mu}}}{|\vec{k}|}.$
(3.11)

With these definitions, the gauge potential $A^{\mu}(k)$ can be expressed in
terms of these basis vectors and associated form factors as [93, 106]:

$A^{\nu}(k)=N_{1}A_{\parallel}(k)b^{(1)\nu}+N_{2}A_{\perp}(k)I^{\nu}+N_{L}A_{L}(k)\tilde{u}^{\nu}+A_{gf}(k)\frac{k^{\nu}}{k^{2}}.$
(3.12)

In order to get rid of the redundant degree of freedom of the gauge field, we
choose, $A_{gf}(k)=0$. The form factors $A_{\parallel}(k),~{}A_{\perp}(k)$ and
$A_{L}(k)$, in Eqn. (3.12) are Lorentz scalars made out of linear or nonlinear
combinations of the tensor and the 4-vectors discussed above, i.e.,
$\omega=k.u$, $|\vec{k}|=\sqrt{\omega^{2}-k^{2}}$,
$k_{\alpha}{\tilde{\bar{F}}^{\alpha\beta}}u_{\beta}$,
$k_{\alpha}{\bar{F}}^{\alpha\beta}{\bar{F}_{\beta\sigma}}k^{\sigma}$ etc.
It is important to note that, the magnetic field
${\cal{B}}_{\mu}=\frac{1}{2}\epsilon_{\mu\nu\lambda\rho}u^{\nu}{\bar{F}}^{\lambda\rho}$
is actually orthogonal to $b^{(1)}_{\nu}$, that is to say
${\cal{B}}^{\nu}\cdot b^{(1)}_{\nu}=0$, implying the direction of the
polarization vector $b^{(1)}_{\nu}$ is orthogonal to the external magnetic
field. Similar consideration would show that, the direction of polarization
vector $I_{\nu}$ is along the external magnetic field. However, to maintain
consistency with our previous work [93], we have denoted the associated form
factors for the corresponding polarization directions, as $A_{\parallel}(k)$,
$A_{\perp}(k)$.
In order to gain an insight of the equations of motions, it is instructive to
know the transformation properties of the vectors, tensors and the form
factors used for Eqn.(3.12), under C, P and T transformations. We briefly
discuss them next.

###  3.2  Properties of the basis vectors under C, P and T transformations:

The non-zero components of the four vectors under deliberation are,
$b^{(1)}_{\mu}=\left(0,b^{(1)}_{1},b^{(1)}_{2},0\right)$,
$b^{(2)}_{\mu}=\left(b^{(2)}_{0},0,0,b^{(2)}_{3}\right)$ and
$I_{\mu}=\left(I_{0},I_{i}\right)$ when $i=1,2,3$. In order to find out the
transformation properties of the same, we need to first identify the C, P and
T transformations of the 4-vectors $k^{\mu}$, $u^{\mu}$ and the tensor
$\bar{F}^{\mu\nu}$. For the sake of brevity we do not show the momentum
dependence of the form factors here.
The transformation properties of time ($k^{0}$) and space components ($k^{i}$,
for $i=$1,2,3 ) of wave propagation vector $k^{\mu}=(k^{0},k^{i})$ under C, P
and T transformations are given by:

$\displaystyle{\bf{C}}k^{0}{\bf{C}}^{-1}$
$\displaystyle=k^{0},~{}~{}{\bf{C}}k^{i}{\bf{C}}^{-1}$ $\displaystyle=+k^{i},$
(3.13) $\displaystyle{\bf{P}}k^{0}{\bf{P}}^{-1}$
$\displaystyle=k^{0},~{}~{}{\bf{P}}k^{i}{\bf{P}}^{-1}$ $\displaystyle=-k^{i},$
(3.14) $\displaystyle{\bf{T}}k^{0}{\bf{T}}^{-1}$
$\displaystyle=k^{0},~{}~{}{\bf{T}}k^{i}{\bf{T}}^{-1}$ $\displaystyle=-k^{i}.$
(3.15)

The centre of mass four velocity of the medium, defined as,
$u^{\mu}=\frac{dx^{\mu}}{d\tau}$ (when $d\tau$ is the differential proper time
interval), have the following transformation properties under time reversal,
parity and charge conjugation transformation:

$\displaystyle{\bf{C}}u^{0}{\bf{C}}^{-1}$
$\displaystyle=-u^{0},~{}~{}{\bf{C}}u^{i}{\bf{C}}^{-1}$
$\displaystyle=-u^{i},$ (3.16) $\displaystyle{\bf{P}}u^{0}{\bf{P}}^{-1}$
$\displaystyle=+u^{0},~{}~{}{\bf{P}}u^{i}{\bf{P}}^{-1}$
$\displaystyle=-u^{i},$ (3.17) $\displaystyle{\bf{T}}u^{0}{\bf{T}}^{-1}$
$\displaystyle=-u^{0},~{}~{}{\bf{T}}u^{i}{\bf{T}}^{-1}$
$\displaystyle=+u^{i}.$ (3.18)

The first property (3.16) follows from the observation, that the statistical
part of the thermal propagator in real time thermal quantum field theory
should remain invariant under the operation of charge conjugation, as
explained in [84]. The same ($u^{\mu}$) in the rest frame of the medium, is
given by $u^{\mu}=\left(1,0,0,0\right)$; for the remaining part of this paper
we shall assume this to be true.
The background field strength tensor $\bar{F}^{ij}$, transforms in the
following way under $\bf{CPT}$ separately as:

$\displaystyle{\bf{C}}{\bar{F}^{0i}}{\bf{C}}^{-1}$
$\displaystyle=-{\bar{F}^{0i}},~{}~{}{\bf{C}}{\bar{F}^{ij}}{\bf{C}}^{-1}$
$\displaystyle=-{\bar{F}^{ij}},$ (3.19)
$\displaystyle{\bf{P}}{\bar{F}^{0i}}{\bf{P}}^{-1}$
$\displaystyle=-{\bar{F}^{0i}},~{}~{}{\bf{P}}{\bar{F}^{ij}}{\bf{P}}^{-1}$
$\displaystyle=+{\bar{F}^{ij}},$ (3.20)
$\displaystyle{\bf{T}}{\bar{F}^{0i}}{\bf{T}}^{-1}$
$\displaystyle=-{\bar{F}^{0i}},~{}~{}{\bf{T}}{\bar{F}^{ij}}{\bf{T}}^{-1}$
$\displaystyle=-{\bar{F}^{ij}}.$ (3.21)

With these information we can look into the $\bf{C,P,\mbox{ and }T}$
transformation properties of the basis vectors. To begin with, we start with
vector $b^{(1)}_{\mu}$. It has only two nonzero components, and their
transformation properties are,

$\displaystyle{\bf{C}}b^{(1)0}{\bf{C}}^{-1}$
$\displaystyle=-b^{(1)0},~{}~{}{\bf{C}}b^{(1)i}{\bf{C}}^{-1}$
$\displaystyle=-b^{(1)i},$ (3.22) $\displaystyle{\bf{P}}b^{(1)0}{\bf{P}}^{-1}$
$\displaystyle=+b^{(1)0},~{}~{}{\bf{P}}b^{(1)i}{\bf{P}}^{-1}$
$\displaystyle=-b^{(1)i},$ (3.23) $\displaystyle{\bf{T}}b^{(1)0}{\bf{T}}^{-1}$
$\displaystyle=+b^{(1)0},~{}~{}{\bf{T}}b^{(1)i}{\bf{T}}^{-1}$
$\displaystyle=+b^{(1)i}.$ (3.24)

Similarly, one can write the transformation properties of $b^{(2)}_{\mu}$,
that has one time like and one space-like nonzero component. The time like
component of $b^{(2)}_{\mu}$ under $\bf{C,P,\mbox{ and }T}$ operations
transform as,

$\displaystyle{\bf{C}}b^{(2)0}{\bf{C}}^{-1}$
$\displaystyle=-b^{(2)0},~{}~{}{\bf{C}}b^{(2)i}{\bf{C}}^{-1}$
$\displaystyle=-b^{(2)i}.$ (3.25) $\displaystyle{\bf{P}}b^{(2)0}{\bf{P}}^{-1}$
$\displaystyle=+b^{(2)0},~{}~{}{\bf{P}}b^{(2)i}{\bf{P}}^{-1}$
$\displaystyle=-b^{(2)i},$ (3.26) $\displaystyle{\bf{T}}b^{(2)0}{\bf{T}}^{-1}$
$\displaystyle=-b^{(2)0},~{}~{}{\bf{T}}b^{(2)i}{\bf{T}}^{-1}$
$\displaystyle=-b^{(2)i}.$ (3.27)

Now using above equations, it is easy to establish that,

$\displaystyle{\bf{P}}\left(\tilde{u}\cdot b^{(2)}\right){\bf{P}}^{-1}$
$\displaystyle=$ $\displaystyle\left(\tilde{u}\cdot b^{(2)}\right).$ (3.28)

Recalling, the four vector $I^{\nu}$, to be given by,

$I^{\nu}=\left(b^{(2)\nu}-\frac{(\tilde{u}^{\mu}b^{(2)}_{\mu})}{\tilde{u}^{2}}\tilde{u}^{\nu}\right);$
(3.29)

the time and space like components of the same are found to be given by,

$I^{0}=\left(b^{(2){0}}-\frac{(\tilde{u}^{\mu}b^{(2)}_{\mu})}{\tilde{u}^{2}}\tilde{u}^{0}\right)\mbox{~{}and~{}}I^{i}=\left(b^{(2){i}}-\frac{(\tilde{u}^{\mu}b^{(2)}_{\mu})}{\tilde{u}^{2}}\tilde{u}^{i}\right),\mbox{~{}~{}(for
i$=$1,2,3)}.$ (3.30)

Using the ${\bf C}$, ${\bf P}$ and ${\bf T}$ transformation properties of the
individual components of $I^{\mu}$, $I^{0}$ and $I^{i}$ would transform under
the same as,

$\displaystyle{\bf{C}}I^{0}{\bf{C}}^{-1}$
$\displaystyle=-I^{0},~{}~{}{\bf{C}}I^{i}{\bf{C}}^{-1}$
$\displaystyle=-I^{i}.$ (3.31) $\displaystyle{\bf{P}}I^{0}{\bf{P}}^{-1}$
$\displaystyle=+I^{0},~{}~{}{\bf{P}}I^{i}{\bf{P}}^{-1}$
$\displaystyle=-I^{i},$ (3.32) $\displaystyle{\bf{T}}I^{0}{\bf{T}}^{-1}$
$\displaystyle=-I^{0},~{}~{}{\bf{T}}I^{i}{\bf{T}}^{-1}$
$\displaystyle=+I^{i}.$ (3.33)

Since the time like components of the gauge field under time reversal
transformation, remains the same, i.e.,

$\displaystyle{\bf{T}}A^{0}{\bf{T}}^{-1}=A^{0},$ (3.34)

therefore when the time reversal transformation is imposed on $A^{0}$,
following the definition of $A^{0}$ one should get,

$\displaystyle{\bf T}^{-1}A^{0}{\bf T}^{-1}={\bf
T}(N_{1}I^{0}A_{\parallel}+N_{L}\tilde{u}^{0}A_{L}){\bf
T}^{-1}=(N_{1}I^{0}A_{\parallel}+N_{L}\tilde{u}^{0}A_{L})$ (3.35)

that is the r.h.s remain invariant.
Now under time reversal transformation $I^{0}$ and $\tilde{u}^{0}$, picks up a
-ve sign. So to maintain overall neutrality, $A_{\parallel}$ and $A_{L}$ must
change sign under ${\bf T}$ transformation. Hence,

$\displaystyle\left({\bf{T}}A_{\parallel}{\bf{T}}^{-1}\right)=-A_{\parallel}\mbox{~{}and~{}}\left({\bf{T}}A_{L}{\bf{T}}^{-1}\right)=-A_{L}.$
(3.36)

We recall that the space like component of $A^{\mu}$ can be expressed as,

$\displaystyle
A_{i}=N_{1}b^{(1)}_{i}A_{\parallel}+N_{2}I_{i}A_{\perp}+N_{L}\tilde{u}_{i}A_{L}.$
(3.37)

The same, that is $A^{i}$, under T transformation pickup a $-ve$ sign i.e.,

$\displaystyle({\bf T}A_{i}{\bf T}^{-1})=-A_{i}.$ (3.38)

It then follows that, the right hand side of Eqn. (3.37) should respect
transformation Eqn. (3.38). Since $b^{(1)}_{i}$, $I_{i}$ and $\tilde{u}_{i}$
remains invariant under time reversal symmetry transformation therefore
$A_{\parallel}$, $A_{\perp}$ and $A_{L}$ should change sign under time
reversal transformation symmetry. The transformation rules of the form factors
can now be summarised as,

$\displaystyle{\bf C}A_{\parallel}{\bf C}^{-1}=+A_{\parallel},\,\,{\bf
C}A_{\perp}{\bf C}^{-1}=+A_{\perp},\,\,{\bf C}A_{L}{\bf C}^{-1}=+A_{L},$
(3.39) $\displaystyle{\bf P}A_{\parallel}{\bf P}^{-1}=+A_{\parallel},\,\,{\bf
P}A_{\perp}{\bf P}^{-1}=+A_{\perp},\,\,{\bf P}A_{L}{\bf P}^{-1}=+A_{L},$
(3.40) $\displaystyle{\bf T}A_{\parallel}{\bf T}^{-1}=-A_{\parallel},\,\,{\bf
T}A_{\perp}{\bf T}^{-1}=-A_{\perp},\,\,{\bf T}A_{L}{\bf T}^{-1}=-A_{L}.$
(3.41)

So the C, P and T transformations properties of the vectors, tensors and the
form factors can further be put in tabular form ( see Table [1]).

| $k_{\mu}$ | $u_{\mu}$ | $\tilde{u}_{\mu}$ | $b^{(1)}_{\mu}$ | $b^{(2)}_{\mu}$ | $I_{\mu}$ | $A_{\parallel}$ | $A_{\perp}$ | $A_{L}$ | $i$ | $\epsilon_{\mu\nu\rho\sigma}$ | $\bar{F}_{\mu\nu}$
---|---|---|---|---|---|---|---|---|---|---|---|---
C | ${\bf+}k_{\mu}$ | ${\bf-}u_{\mu}$ | ${\bf-}\tilde{u}_{\mu}$ | $-b^{(1)}_{\mu}$ | $-b^{(2)}_{\mu}$ | $-I_{\mu}$ | $+A_{\parallel}$ | $+A_{\perp}$ | $+A_{L}$ | $+i$ | $+\epsilon_{\mu\nu\rho\sigma}$ | ${\bf-}\bar{F}_{\mu\nu}$
P | ${\bf+}k^{\mu}$ | ${\bf+}u^{\mu}$ | ${\bf+}~{}\tilde{u}^{\mu}$ | $+b^{(1)\mu}$ | $+b^{(2)\mu}$ | $+I^{\mu}$ | $+A_{\parallel}$ | $+A_{\perp}$ | $+A_{L}$ | $+i$ | $-\epsilon_{\mu\nu\rho\sigma}$ | ${\bf+}\bar{F}^{\mu\nu}$
T | ${\bf+}k^{\mu}$ | ${\bf-}u^{\mu}$ | ${\bf-}\tilde{u}^{\mu}$ | $-b^{(1)\mu}$ | $-b^{(2)\mu}$ | $-I^{\mu}$ | $-A_{\parallel}$ | $-A_{\perp}$ | $-A_{L}$ | $-i$ | $-\epsilon_{\mu\nu\rho\sigma}$ | ${\bf-}\bar{F}^{\mu\nu}$
Table 1:  Transformation properties for the vectors, tensors and the EM form-
factors used to construct all the vectors, tensors and form factors used in
this work to expand $A^{\nu}(k)$, under ${\bf C}$, ${\bf P}$ and ${\bf T}$.

It can be shown that, each term appearing in the equations of motion of this
article when subjected to these transformations, transforms identically. We
will demonstrate the same later.

###  3.3 Dynamics Of The Degrees Of Freedom

In this section, we discuss the dynamics of the independent degrees of freedom
of the system. Recalling the fact that in a medium photon acquire an extra
degree of freedom in addition to the two transverse degrees of freedom,
therefore for the photon scalar interacting system, there should be four
degrees of freedom. Moreover, out of the three degrees of freedom of the
photon, the longitudinal degree of freedom has spin zero, and the other two
would be having spin one and minus one respectively. Therefore though naively
one may expect that, the longitudinal mode of photon would couple with the
scalar degree of freedom, because they both have same spin assignments,
however we would demonstrate in this subsection, by analysing the equations of
motion, that, this naive expectation doesn’t hold good.
Now we begin with Eqn.(3.3), and substitute the expression for the gauge
potential from Eqn.(3.12) in the same. The resulting equation (considering the
Faraday contribution to be $\Pi^{p}_{\mu\nu}(k)=-\Pi_{p}(k)P_{\mu\nu}$, when
the projection operator is defined as
$P_{\mu\nu}=i\epsilon_{\mu\nu\beta\delta_{\parallel}}\frac{k^{\beta}}{\mid
k\mid}u^{\tilde{\delta}_{\parallel}}$)555Where $\tilde{\delta}_{\parallel}$
can takes the value 0 or 3. When ${\delta}_{\parallel}=0$ then
$\tilde{\delta}_{\parallel}=3$ and vice versa. turns out to be,

$\displaystyle(k^{2}-\Pi_{T}(k))[A_{\parallel}(k)N_{1}b^{(1)}_{\alpha}+A_{\perp}(k)N_{2}I_{\alpha}+A_{L}(k)N_{L}\tilde{u}_{\alpha}]+\Pi_{T}(k)A_{L}(k)N_{L}\tilde{u}_{\alpha}$
(3.42) $\displaystyle-$
$\displaystyle\Pi_{L}(k)A_{L}(k)N_{L}\tilde{u}_{\alpha}-i\Pi^{p}(k)\epsilon_{\alpha\nu\beta\delta}\frac{k^{\beta}}{\mid{k}\mid}u^{\tilde{\delta}_{\parallel}}[A_{\parallel}(k)N_{1}b^{(1)\nu}+A_{\perp}(k)N_{2}I^{\nu}$
$\displaystyle+$ $\displaystyle
A_{L}(k)N_{L}\tilde{u}^{\nu}]=ig_{\phi\gamma\gamma}b^{(1)}_{\alpha}{\phi(k)},$

Next as we multiply equation (3.42) by the normalised basis vectors, we get
the equations of motion for different components of the form factors. For
example, if we multiply equation (3.42) by $b^{(1)\alpha}$, we find the
following equation,

$\displaystyle(k^{2}-\Pi_{T}(k))A_{\parallel}(k)N_{1}b^{(1)}_{\alpha}b^{(1)\alpha}-i\Pi^{p}(k)N_{2}\left[\epsilon_{\alpha\nu\beta\delta_{\parallel}}\frac{k^{\beta}}{\mid{k}\mid}u^{\tilde{\delta}_{\parallel}}b^{(1)\alpha}I^{\nu}\right]A_{\perp}(k)$
(3.43) $\displaystyle=$ $\displaystyle
ig_{\phi\gamma\gamma}b^{(1)}_{\alpha}b^{(1)\alpha}{\phi(k)}.$

Using the expression of the normalization constants, the same becomes

$\displaystyle(k^{2}-\Pi_{T}(k))A_{\parallel}(k)+i\Pi^{p}(k)N_{1}N_{2}\left[\epsilon_{\alpha\nu\beta\delta_{\parallel}}\frac{k^{\beta}}{\mid{k}\mid}u^{\tilde{\delta}_{\parallel}}b^{(1)\alpha}I^{\nu}\right]A_{\perp}(k)=\frac{ig_{\phi\gamma\gamma}\phi(k)}{N_{1}}.$
(3.44)

Similarly, multiplying Eqn. (3.42) by $I^{\nu}$ and $\tilde{u}^{\alpha}$
respectively, we get the following two equations:

$\displaystyle(k^{2}-\Pi_{T}(k))A_{\perp}(k)-i\Pi^{p}(k)N_{1}N_{2}\left[\epsilon_{\alpha\nu\beta\delta_{\parallel}}\frac{k^{\beta}}{\mid{k}\mid}u^{\tilde{\delta}_{\parallel}}b^{(1)\alpha}I^{\nu}\right]A_{\parallel}(k)=0,$
(3.45) $(k^{2}-\Pi_{L}(k))A_{L}(k)=0,$ (3.46)

those describe the dynamics of the three degrees of freedom of the photon.
Lastly, the equation of motion for the scalar field turns out to be,

$(k^{2}-m^{2})\phi(k)=-\frac{ig_{\phi\gamma\gamma}A_{\parallel}(k)}{N_{1}}.$
(3.47)

These four equations (3.44), (3.45), (3.46) and (3.47) describe the dynamics
of $\gamma\phi$ interaction in a magnetized medium. The correctness of the
above equations of motion can be established by performing PT transformation
on these equations.
To demonstrate it, let us choose the equation (3.45) and operate $({\bf PT})$
from left and $({\bf PT})^{-1}$ from right sides of the equation. Following
the transformation rules of table [1], under ${\bf PT}$, the first term of the
equation will pick up a negative sign due to presence of $A_{\perp}(k)$ which
is odd under the same. In the second term, the factors
$i,{u}^{\tilde{\delta}_{\parallel}},b^{(1)\alpha},I^{\nu}$ and
$A_{\parallel}(k)$ are ${\bf PT}$ odd and rest are ${\bf PT}$ even as shown
below:

$\displaystyle\begin{matrix}&({\bf PT})A_{\parallel}({\bf
PT})^{-1}=-A_{\parallel},&({\bf PT})A_{\perp}({\bf PT})^{-1}=-A_{\perp},\\\
&({\bf PT})i({\bf PT})^{-1}=-i,&({\bf PT})k^{\beta}({\bf
PT})^{-1}=+k^{\beta},\\\ &({\bf PT})u^{\tilde{\delta}_{\parallel}}({\bf
PT})^{-1}=-u^{\tilde{\delta}_{\parallel}},&({\bf PT})b^{(1)\mu}({\bf
PT})^{-1}=-b^{(1)\mu},\\\ &({\bf PT})I^{\nu}({\bf PT})^{-1}=-I^{\nu},&({\bf
PT})\epsilon_{\mu\nu\delta\beta}({\bf
PT})^{-1}=+\epsilon_{\mu\nu\delta\beta}.\end{matrix}$ (3.48)

The transformation properties of form factor $\Pi^{p}(k)$ under ${\bf C,P}$
and ${\bf T}$ can be figured out from the expression,

$\displaystyle\Pi^{p}(k)=\frac{{(k.u)}{(e\rm{B_{\parallel}}/m_{e})}}{\omega^{2}-(e\rm{B_{\parallel}}/m_{e})^{2}}\left(\frac{n_{e}}{m_{e}}\right).$
(3.49)

As can be seen from the equation (3.49) that, $\Pi^{p}(k)$ is invariant under
the ${\bf PT}$ transformation, i.e.,

$\displaystyle({\bf PT})\Pi^{p}(k)({\bf PT})^{-1}=\Pi^{p}(k).$ (3.50)

so collectively, the second term of equation (3.45) will also pick up a
negative sign. Therefore this equation remains invariant under ${\bf PT}$
transformation. Same can be established for other equations too using similar
logic.
The PT symmetric part of PSET, when included in the effective Lagrangian, can
in principle compensate for the $\bf P$ violation of $|A_{\perp}>$, when both
appear as a product, as they do in the equations of motion. Thus resulting
product of the two becomes ${\bf P}$ even, making mixing between $\phi(k)$ and
$|\perp>$ possible.
Now looking at the problem of mixing, we note that the initial mixing between
$\phi(k)$ and $|\parallel>$, due to $\phi FF$ coupling remains intact, but
introduction of the PSET term causes further mixing between the two orthogonal
transverse states, i.e., $|\parallel>$ and $|\perp>$; and lastly the
combination of the product of PSET term and $|\perp>$, as noted in the last
paragraph, causes mixing between, $|\perp>$ and $\phi(k)$. Thus the system
reduces to a system of three mutually coupled degrees of freedom, that evolves
following their respective equations of motion.
It may not be quite out of place, to mention here that, instead of
$\phi(k)F^{\mu\nu}F^{\mu\nu}$, if $a(k){\tilde{F}}^{\mu\nu}F^{\mu\nu}$
interaction is considered in a magnetized vacuum or an unmagnetized medium,
the role of scalar $\phi(k)$ gets interchanged with that of pseudoscalar [97];
and consequently, the role of $|\parallel>$ would get interchanged with
$|\perp>$ so that the symmetry remains intact. So the prediction from one can
be obtained from the prediction of the other.
This simple interrelation between the two, as noted already, however gets
modified as one includes the effect of PSET to, axion photon or scalar photon
systems.

Introducing,
$F=\Pi^{p}(k)N_{1}N_{2}[\epsilon_{\alpha\nu\beta\delta}\frac{k^{\beta}}{\mid{k}\mid}u^{\tilde{\delta}_{\parallel}}b^{(1)\alpha}I^{\nu}]$
and $G=\frac{g_{\phi\gamma\gamma}A_{\parallel}(k)}{N_{1}}$ for the sake of
brevity, the coupled set of equations (of motion) can be presented in matrix
form, as

$\displaystyle\left[\begin{matrix}(k^{2}-\Pi_{T}(k))&iF&0&-iG\cr-
iF&(k^{2}-\Pi_{T}(k))&0&0\cr 0&0&(k^{2}-\Pi_{L}(k))&0\cr
iG&0&0&(k^{2}-m_{\phi}^{2})\end{matrix}\right]\left[\begin{matrix}{A_{\parallel}(k)}\cr{A_{\perp}(k)}\cr{A_{L}(k)}\cr\phi(k)\end{matrix}\right]=0.$
(3.51)

The longitudinal degree of freedom of photon ($A_{L}(k)$) as can be seen, has
decoupled from the rest of the degrees of freedom.

###  3.4 Background dependent mixing pattern

In this section we try to understand the background’s influence on the mixing
dynamics of the transverse and longitudinal degrees of freedom of photon with
scalar or pseudoscalar as the interaction term in the Lagrangian changes from
$g_{\phi\gamma\gamma}\phi FF$ for scalar-photon system to
$g_{a\gamma\gamma}a\tilde{F}F$ for pseudoscalar-photon system.
As was noted in the introduction, that dynamics of the degrees of freedom of
photon in terms of the form factors $A_{\parallel}(k)$ and $A_{\perp}(k)$ in a
magnetized vacuum with $g_{\phi\gamma\gamma}\phi FF$ interaction, can be
anticipated from the evolution of these two form factors in pseudoscalar
photon system ($a\gamma\gamma$), with the identification of the role of
$A_{\parallel}(k)$ with $A_{\perp}(k)$ and $A_{\perp}$ with $A_{\parallel}$
when the interaction Lagrangian is $g_{a\gamma\gamma}a\tilde{F}F$.
So one can conclude that these system seem to have a symmetry that remains
invariant under the exchange of the scalar field with pseudoscalar and
parallely polarised state $A_{\parallel}$ with the perpendicularly polarised
state $A_{\perp}$ of the photon. Even in material medium, this symmetry
behaviour remains the same.
In presence of a magnetized medium, however, contributions from the photon
self energy tensor (PSET) $\Pi_{\mu\nu}(k,\mu,T,e\rm{B})$ that is odd in
powers of ${e\rm{B}}$ lifts this apparent degeneracy. The discrete symmetries
of nature, like charge conjugation (C) time reversal (T) and parity (P)
transformations play a major role in removing this apparent degeneracy. We
would like to come back to this issue in future publications.
Since the longitudinal degree of freedom $A_{L}(k)$, (with dimension-5 scalar-
photon interaction ) in a magnetized medium, is decoupled from the rest of the
degrees of freedom, therefore we won’t be considering it any further.
Therefore, the resulting matrix Eqn. (3.51) can be cast in the following form:

$\displaystyle\left[k^{2}{\bf I}-\left(\begin{matrix}\Pi_{T}(k)&iF&-iG\\\
-iF&\Pi_{T}(k)&0\\\
iG&0&m^{2}_{\phi}\end{matrix}\right)\right]\left[\begin{matrix}{A_{\parallel}(k)}\cr{A_{\perp}(k)}\cr\phi(k)\end{matrix}\right]=0.$
(3.52)

Here I is $3\times 3$ identity matrix. For the sake of compactness, we would
further like to denote the matrix, inside bracket, on the left hand side in
Eqn.(3.52) as,

$\left(\begin{array}[]{ccc}\Pi_{T}(k)&iF&-iG\\\ -iF&\Pi_{T}(k)&0\\\
iG&0&m^{2}_{\phi}\end{array}\right)={\bf M},$ (3.53)

and would like to discuss about the elements of the matrix ${\bf M}$ for the
physical situation under consideration in next few lines.
In the long wavelength limit, one can take $\Pi_{T}=\omega^{2}_{p}$, where
$\omega_{p}$ is the plasma frequency. With this identification, the other two
parameters $F$ and $G$, are given by
$F=\frac{\omega^{2}_{p}e\rm{B}\cos\bar{\theta}}{\omega m_{e}}$ and
$G=-g_{\phi\gamma\gamma}\rm{B}\sin\bar{\theta}\omega$. The angle
$\bar{\theta}$ here corresponds to the angle between the photon propagation
vector $\vec{k}$ and the magnetic field $\rm{B}$. Lastly the parameter $m_{e}$
is the electron mass. Now identifying $\rm{B}\cos\bar{\theta}=B_{\parallel}$
and $\rm{B}\sin\bar{\theta}=B_{\perp}$, the equation of motion can further be
written as,

$\left[\begin{array}[]{c}k^{2}{\bf
I}-\left(\begin{array}[]{ccc}\omega^{2}_{p}&i\frac{\omega^{2}_{p}eB_{\parallel}}{(\omega
m_{e})}&\,\,-ig_{\phi\gamma\gamma}B_{\perp}\omega\\\
-i\frac{\omega^{2}_{p}eB_{\parallel}}{(\omega m_{e})}&\omega^{2}_{p}&0\\\
ig_{\phi\gamma\gamma}B_{\perp}\omega&0&m^{2}_{\phi}\end{array}\right)\end{array}\right]\left[\begin{array}[]{c}A_{\parallel}(k)\\\
A_{\perp}(k)\\\ \phi(k)\end{array}\right]=0.$ (3.54)

Although, in this work, we will assume angle $\bar{\theta}$, to be $\pi/4$,
however depending on spatial geometry of the emission region this angle can
vary between zero to $2\pi$.
When $\bar{\theta}=n\pi$, for any integer $n$, the dilatons decouples from the
system and the resulting equation describe electrodynamics of magnetized
medium. This can be used to study the propagation of EM wave in magnetosphere
of pulsars, as was done in [98]. In the other limit, when
$\bar{\theta}=\pi/2$, the situation reduces to that of $\gamma-\phi$
interaction in an unmagnetized medium.
To obtain the solutions of Eqn. (3.54), first we need to have the knowledge of
the eigen values of the matrix on the l.h.s of Eqn.(3.54) inside square
bracket. Once that is done, one can find the corresponding eigen vectors and
from there obtain the unitary matrix, that can reduce the matrix ${\bf M}$, to
diagonal form.
The eigen values of matrix M can be obtained from the following cubic
equation, that follows from the characteristic equation,

$\displaystyle c_{1}{\bf E}_{i}^{3}+c_{2}{\bf E}_{i}^{2}+c_{3}{\bf
E}_{i}+c_{4}=0.$ (3.55)

The variables $c_{2},c_{3}$ and $c_{4}$ appearing in (3.55) are further
related to the elements of the mixing matrix ${\bf M}$, through the following
relations:

$c_{1}=1,{\mbox{~{}~{}~{}~{}~{}}}c_{2}=-(2\omega^{2}_{p}+m^{2}_{\phi}),$
(3.56) $\displaystyle c_{3}$ $\displaystyle=$
$\displaystyle\omega^{4}_{p}+2\omega^{2}_{p}m^{2}_{\phi}-\left(\frac{e\rm{B}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}\right)^{2}-(g_{\phi\gamma\gamma}\rm{B}_{\perp}\omega)^{2},$
(3.57) $\displaystyle c_{4}$ $\displaystyle=$
$\displaystyle-\omega^{4}_{p}m^{2}_{\phi}+\left(\frac{e\rm{B}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}\right)^{2}m^{2}_{\phi}+\omega^{2}_{p}(g_{\phi\gamma\gamma}\rm{B}_{\perp}\omega)^{2}.$
(3.58)

More details about the roots and their properties are provided in the
appendix. Here we just state the result. The roots are,

$\displaystyle{\bf E}_{1}$ $\displaystyle=$ $\displaystyle
2{\cal{R}}\cos(\alpha-\pi/3)-\frac{{c_{2}}}{3},$ (3.59) $\displaystyle{\bf
E}_{2}$ $\displaystyle=$ $\displaystyle
2{\bf\cal{R}}\cos(\alpha+\pi/3)-\frac{{c_{2}}}{3},$ (3.60) $\displaystyle{\bf
E}_{3}$ $\displaystyle=$
$\displaystyle-2{\bf\cal{R}}\cos(\alpha)-\frac{{c_{2}}}{3}.$ (3.61)

Where the variables $\alpha=\frac{1}{3}\cos^{-1}\left(\frac{{\cal
Q}}{{\bf\cal{R}}^{3}}\right)$ and
${\bf\cal{R}}=\sqrt{\left(-{\cal{P}}\right)}{\bf sgn}\left({\cal{Q}}\right)$,
when ${\cal{P}}=\frac{\left(3c_{3}-{c_{2}}^{2}\right)}{9}$ and
${\cal{Q}}=\left(\frac{{c_{2}}^{3}}{27}-\frac{{c_{2}}{c_{3}}}{6}+\frac{c_{4}}{2}\right)$.
This completes our knowledge of the roots.

##  4 The Unitary Diagonalizing matrix U

Before we go on to discuss oscillation between $\gamma$ and $\phi$, we need to
evaluate the phase space evolution of the individual fields. To achieve that,
we need to have the solutions for each one of them. In order to obtain the
same, the mixing matrix ${\bf M}$ (3.53) has to be transformed to its diagonal
form ${\bf M}_{D}$, by unitary matrices U i.e., ${\bf U}^{{\dagger}}{\bf
M}{\bf U}={\bf M}_{D}$, having eigenvalues, ${\bf E}_{1},{\bf
E}_{2}\mbox{~{}and ~{}}{\bf E}_{3}$ as the diagonal elements. The construction
of the unitary matrix ${\bf U}$ had been outlined in the Appendix. Therefore
we provide the final result here. Introducing the quantities
$u_{i}=(\omega^{2}_{p}-{\bf E}_{i})(m^{2}_{\phi}-{\bf E}_{i})$,
$v_{i}=i\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}(m^{2}_{\phi}-{\bf
E}_{i})$ and
$w_{i}=ig_{\phi\gamma\gamma}{\rm{B}}_{\perp}\omega(\omega^{2}_{p}-{\bf
E}_{i})$ related to the the eigenvectors of matrix ${\rm{\bf M}}$, the unitary
matrix $U$ can be written as,

$\displaystyle{\bf
U}=\left(\begin{matrix}&\frac{u_{1}}{\sqrt{u^{2}_{1}+v^{2}_{1}+w^{2}_{1}}}&\frac{u_{2}}{\sqrt{u^{2}_{2}+v^{2}_{2}+w^{2}_{2}}}&\frac{u_{3}}{\sqrt{u^{2}_{3}+v^{2}_{3}+w^{2}_{3}}}\\\
&\frac{v_{1}}{\sqrt{u^{2}_{1}+v^{2}_{1}+w^{2}_{1}}}&\frac{v_{2}}{\sqrt{u^{2}_{2}+v^{2}_{2}+w^{2}_{2}}}&\frac{v_{3}}{\sqrt{u^{2}_{3}+v^{2}_{3}+w^{2}_{3}}}\\\
&\frac{w_{1}}{\sqrt{u^{2}_{1}+v^{2}_{1}+w^{2}_{1}}}&\frac{w_{2}}{\sqrt{u^{2}_{2}+v^{2}_{2}+w^{2}_{2}}}&\frac{w_{3}}{\sqrt{u^{2}_{3}+v^{2}_{3}+w^{2}_{3}}}\end{matrix}\right).$
(4.1)

Introducing,
$\rm{\cal{N}}^{(i)}_{vn}=\frac{1}{\sqrt{u^{2}_{i}+v^{2}_{i}+w^{2}_{i}}}$ as
the normalization constant for the $i$’th eigen vector, we can further express
Eqn. (4.1) in the following form:

$\\\ {\bf U}=\left(\begin{array}[]{ccc}(\omega^{2}_{p}-{\bf
E}_{1})(m^{2}_{\phi}-{\bf
E}_{1})\rm{{\cal{N}}}^{(1)}_{vn}&(\omega^{2}_{p}-{\bf
E}_{2})(m^{2}_{\phi}-{\bf
E}_{2})\rm{{\cal{N}}}^{(2)}_{vn}&(\omega^{2}_{p}-{\bf
E}_{3})(m^{2}_{\phi}-{\bf E}_{3})\rm{{\cal{N}}}^{(3)}_{vn}\\\
i\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}(m^{2}_{\phi}-{\bf
E}_{1})\rm{{\cal{N}}}^{(1)}_{vn}&i\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}(m^{2}_{\phi}-{\bf
E}_{2})\rm{{\cal{N}}}^{(2)}_{vn}&i\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}(m^{2}_{\phi}-{\bf
E}_{3})\rm{{\cal{N}}}^{(3)}_{vn}\\\
ig_{\phi\gamma\gamma}{\rm{B}}_{\perp}\omega(\omega^{2}_{p}-{\bf
E}_{1})\rm{\cal{N}}^{(1)}_{vn}&ig_{\phi\gamma\gamma}{\rm{B}}_{\perp}\omega(\omega^{2}_{p}-{\bf
E}_{2})\rm{\cal{N}}^{(2)}_{vn}&ig_{\phi\gamma\gamma}{\rm{B}}_{\perp}\omega(\omega^{2}_{p}-{\bf
E}_{3})\rm{\cal{N}}^{(3)}_{vn}\end{array}\right).$ (4.2)

The explicit expression for the $\rm{\cal{N}}^{(i)}_{vn}$ is given by,

$\\\ \rm{\cal{N}}^{(i)}_{vn}=\frac{1}{\sqrt{(\omega^{2}_{p}-{\bf
E}_{i})^{2}(m_{\phi}^{2}-{\bf
E}_{i})^{2}+\left(\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}\right)^{2}(m_{\phi}^{2}-{\bf
E}_{i})^{2}+\left(g_{\phi\gamma\gamma}{\rm{B}}_{\perp}\omega\right)^{2}(\omega^{2}_{p}-{\bf
E}_{i})^{2}}},$ (4.3)

where ${\bf E}_{i}$ stands for the corresponding eigen value. The hermitian
conjugate matrix ${\bf U}^{\dagger}$ would follow from (4.2). The unitarity
relations, ${\bf U}^{\dagger}{\bf U}={\bf 1}$ has been verified numerically as
well as analytically, and ${\bf U}{\bf U}^{\dagger}={\bf 1}$, numerically.
Analytical verification of ${\bf U}{\bf U}^{\dagger}={\bf 1}$ is cumbersome,
so we have taken recourse to numerical verification, and checked that they are
satisfied.
There are few relations those are satisfied by the elements of ${\bf U}$, and
use of them makes it convenient to express the probability amplitudes in
compact notation. In order to derive them we first, rewrite Eqn. (4.2) with
respective identifications of the elements, as follows,

$\displaystyle{\bf
U}=\left(\begin{matrix}{\hat{u}_{1}}&{\hat{u}_{2}}&{\hat{u}_{3}}\cr{\hat{v}_{1}}&{\hat{v}_{2}}&{\hat{v}_{3}}\cr{\hat{w}_{1}}&{\hat{w}_{2}}&{\hat{w}_{3}}\cr\end{matrix}\right).$
(4.4)

The condition ${\bf U}^{\dagger}{\bf U}={\bf{1}}$ now implies, for i, j=1,2,3,

$\displaystyle{\hat{u}_{i}}{\hat{u}_{j}}^{*}+{\hat{v}_{i}}{\hat{v}_{j}}^{*}+{\hat{w}_{i}}{\hat{w}_{j}}^{*}=\left\\{\begin{matrix}1,~{}i=j\cr
0,~{}i\neq j.\end{matrix}\right.$ (4.5)

A similar exercise for ${\bf U}{\bf U}^{\dagger}={\bf{1}}$, further
establishes the following relations amongst the elements of ${\bf U}$. The off
diagonal terms of ${\bf U}{\bf U}^{\dagger}$ will give,

$\displaystyle\sum^{3}_{i=1}\hat{u}_{i}\hat{v}_{i}^{*}=\sum^{3}_{i=1}{\hat{u}_{i}}^{*}\hat{v}_{i}=\sum^{3}_{i=1}\mid\hat{u}_{i}\mid\mid\hat{v}_{i}\mid=0,$
(4.6)
$\displaystyle\sum^{3}_{i=1}\hat{v}_{i}\hat{w}_{i}^{*}=\sum^{3}_{i=1}{\hat{v}_{i}}^{*}\hat{w}_{i}=\sum^{3}_{i=1}\mid\hat{v}_{i}\mid\mid\hat{w}_{i}\mid=0,$
(4.7)
$\displaystyle\sum^{3}_{i=1}\hat{w}_{i}\hat{u}_{i}^{*}=\sum^{3}_{i=1}{\hat{w}_{i}}^{*}\hat{u}_{i}=\sum^{3}_{i=1}\mid\hat{w}_{i}\mid\mid\hat{u}_{i}\mid=0.$
(4.8)

and the diagonal entries of the same will yield,

$\displaystyle\sum^{3}_{i=1}\mid\hat{u}_{i}\mid\mid\hat{u}_{i}\mid=\sum^{3}_{i=1}\mid\hat{v}_{i}\mid\mid\hat{v}_{i}\mid=\sum^{3}_{i=1}\mid\hat{w}_{i}\mid\mid\hat{w}_{i}\mid=1.$
(4.9)

The variables $\hat{u}^{*}_{i},\hat{v}^{*}_{i}$, and $\hat{w}^{*}_{i}$
appearing in above expressions, represent the conjugates of the corresponding
elements of ${\bf U}$.

##  5 Conversion Probability

The conversion probability, of a photon of a particular polarization state to
the same of a different polarization state or scalars can be estimated from
the evolution (quantum evolution) equation of the corresponding polarized
states of the photon and the scalars. One can perform the same by promoting
the momentum variables to corresponding operators and components of the vector
potential $A^{\nu}(k)$ and $\phi(k)$ to the corresponding quantum states,
following [38, 122]. In order to follow the evolution of individual quantum
states, one needs to decouple them from each other by the following way. One
can multiply equation (3.54) by ${\bf U}^{\dagger}$ from left 666 Since ${\bf
U}^{-1}$ = ${\bf U}^{\dagger}$. , to reduce it to the following form,

$\left[\begin{array}[]{c}k^{2}\mathbf{I}-{\bf U}^{{\dagger}}{\bf M}{\bf
U}\end{array}\right]\left[\begin{array}[]{c}|A^{\prime}_{\parallel}(k)>\\\
|A^{\prime}_{\perp}(k)>\\\ |\phi^{\prime}(k)>\end{array}\right]=0,$ (5.1)

when ${\bf U}^{{\dagger}}{\bf MU}$ is a diagonal matrix. In the diagonal
representation, the propagating states are the diagonal states and they allow
principle of superposition. The matrix ${\bf U}$ is given in Eq. (4.2) and
${\bf U}^{{\dagger}}$ is the hermitian conjugate of the same. Here we have
denoted 777The set of unprimed and primed column vectors, at places,, may be
defined collectively as: $\left[{\bf
A(k)}\right]=\left(A_{\parallel}(k),A_{\perp}(k),\phi(k)\right)^{{\bf T}}$ and
$\left[{\bf
A(k)}^{\prime}\right]=\left(A^{\prime}_{\parallel}(k),A^{\prime}_{\perp}(k),\phi(k)\right)^{\bf
T}$, here the superscript T stands for transpose.

$\displaystyle\left[\begin{array}[]{c}|A^{\prime}_{\parallel}(k)>\\\
|A^{\prime}_{\perp}(k)>\\\ |\phi^{\prime}(k)>\end{array}\right]={\bf
U}^{\dagger}\left[\begin{array}[]{c}|A_{\parallel}(k)>\\\ |A_{\perp}(k)>\\\
|\phi(k)>\end{array}\right].$ (5.8)

The primed states corresponds to the propagating states and unprimed ones are
the physical states; They are related to each other by the unitary
transformation by ${\bf U}$ introduced earlier. For a beam of photon,
propagating in the z direction, following the principles stated already, one
can promote the momentum $k_{3}$ to the corresponding operator in $z$ space
and write (using natural units $\hbar=c=1$) $k^{2}\approx
2\omega(\omega-i\partial_{z})$. With this manipulations, the resulting
equations get transformed from Klien-Gordon to the form used by [38].
Recalling that ${\bf U}^{{\dagger}}{\bf M}{\bf U}={\bf M}_{D}$, where ${\bf
M}_{D}$ is the diagonal matrix, equation (5.8) can further be cast in the
form,

$\left[\begin{array}[]{c}(\omega-i\partial_{z})\mathbf{I}-\left[\begin{array}[]{ccc}\frac{{\bf
E}_{1}}{2\omega}&0&0\\\ 0&\frac{{\bf E}_{2}}{2\omega}&0\\\ 0&0&\frac{{\bf
E}_{3}}{2\omega}\end{array}\right]\end{array}\right]\left[\begin{array}[]{c}|{A^{\prime}}_{\parallel}(z)>\\\
|{A^{\prime}}_{\perp}(z)>\\\ |{\phi^{\prime}}(z)>\end{array}\right]=0.$ (5.9)

The matrix evolution Eqn. (5.9) is now easy to solve. Introducing the
variables, $\Omega_{\parallel}=\left(\omega-\frac{{\bf
E}_{1}}{2\omega}\right)$, $\Omega_{\perp}=\left(\omega-\frac{{\bf
E}_{2}}{2\omega}\right)$ and $\Omega_{\phi}=\left(\omega-\frac{{\bf
E}_{3}}{2\omega}\right)$, we can now directly write down the solutions for the
states vector $\left[|{\bf A}(z)>\right]$ ( where ${\bf A}(z)\equiv{\bf
A}(\omega,k_{\perp},z)$) in the following form,

$\left[\begin{matrix}{|{A}_{\parallel}(z)>}\cr{|{A}_{\perp}(z)>}\cr{|\phi(z)>}\end{matrix}\right]={\bf
U}\left[\begin{array}[]{ccc}e^{-i\Omega_{\parallel}z}&0&0\\\
0&e^{-i\Omega_{\perp}z}&0\\\ 0&0&e^{-i\Omega_{\phi}z}\end{array}\right]{\bf
U}^{\dagger}\left[\begin{matrix}{|{A}_{\parallel}(0)>}\cr{|{A}_{\perp}(0)>}\cr|{\phi(0)}>\end{matrix}\right].$
(5.10)

The elements of column vector $\left[|{\bf A}(0)>\right]$ in (5.10) and
$\left[|{\bf A_{L}}(0)>\right]$, are normalized such that,
$<A_{\parallel}(0)|A_{\parallel}(0)>=<A_{\perp}(0)|A_{\perp}(0)>=<\phi(0)|\phi(0)>=<A_{L}(0)|A_{L}(0)>=1$.
With the help of Eqn. (5.10) one can write down the solutions. And they are as
follows:

$\displaystyle|A_{\parallel}(\omega,z)>$ $\displaystyle=$
$\displaystyle\left(e^{-i\Omega_{\parallel}z}{\hat{u}}_{1}{\hat{u}}^{*}_{1}+e^{-i\Omega_{\perp}z}{\hat{u}}_{2}{\hat{u}}^{*}_{2}+e^{-i\Omega_{\phi}z}{\hat{u}}_{3}{\hat{u}}^{*}_{3}\right)|A_{\parallel}(\omega,0)>$
(5.11) $\displaystyle+$
$\displaystyle\left(e^{-i\Omega_{\parallel}z}{\hat{u}}_{1}{\hat{v}}^{*}_{1}+e^{-i\Omega_{\perp}z}{\hat{u}}_{2}{\hat{v}}^{*}_{2}+e^{-i\Omega_{\phi}z}{\hat{u}}_{3}{\hat{v}}^{*}_{3}\right)|A_{\perp}(\omega,0)>$
$\displaystyle+$
$\displaystyle\left(e^{-i\Omega_{\parallel}z}{\hat{u}}_{1}{\hat{w}}^{*}_{1}+e^{-i\Omega_{\perp}z}{\hat{u}}_{2}{\hat{w}}^{*}_{2}+e^{-i\Omega_{\phi}z}{\hat{u}}_{3}{\hat{w}}^{*}_{3}\right)|\phi(\omega,0)>,$

the perpendicular $|A_{\perp}(\omega,z)>$ component is given by,

$\displaystyle|A_{\perp}(\omega,z)>$ $\displaystyle=$
$\displaystyle\left(e^{-i\Omega_{\parallel}z}{\hat{v}}_{1}{\hat{u}}^{*}_{1}+e^{-i\Omega_{\perp}z}{\hat{v}}_{2}{\hat{u}}^{*}_{2}+e^{-i\Omega_{\phi}z}{\hat{v}}_{3}{\hat{u}}^{*}_{3}\right)|A_{\parallel}(\omega,0)>$
(5.12) $\displaystyle+$
$\displaystyle\left(e^{-i\Omega_{\parallel}z}{\hat{v}}_{1}{\hat{v}}^{*}_{1}+e^{-i\Omega_{\perp}z}{\hat{v}}_{2}{\hat{v}}^{*}_{2}+e^{-i\Omega_{\phi}z}{\hat{v}}_{3}{\hat{v}}^{*}_{3}\right)|A_{\perp}(\omega,0)>$
$\displaystyle+$
$\displaystyle\left(e^{-i\Omega_{\parallel}z}{\hat{v}}_{1}{\hat{w}}^{*}_{1}+e^{-i\Omega_{\perp}z}{\hat{v}}_{2}{\hat{w}}^{*}_{2}+e^{-i\Omega_{\phi}z}{\hat{v}}_{3}{\hat{w}}^{*}_{3}\right)|\phi(\omega,0)>,$

and lastly the evolution of the state $|\phi(\omega,z)>$ is given by

$\displaystyle|\phi(\omega,z)>$ $\displaystyle=$
$\displaystyle\left(e^{-i\Omega_{\parallel}z}{\hat{w}}_{1}{\hat{u}}^{*}_{1}+e^{-i\Omega_{\perp}z}{\hat{w}}_{2}{\hat{u}}^{*}_{2}+e^{-i\Omega_{\phi}z}{\hat{w}}_{3}{\hat{u}}^{*}_{3}\right)|A_{\parallel}(\omega,0)>$
(5.13) $\displaystyle+$
$\displaystyle\left(e^{-i\Omega_{\parallel}z}{\hat{w}}_{1}{\hat{v}}^{*}_{1}+e^{-i\Omega_{\perp}z}{\hat{w}}_{2}{\hat{v}}^{*}_{2}+e^{-i\Omega_{\phi}z}{\hat{w}}_{3}{\hat{v}}^{*}_{3}\right)|A_{\perp}(\omega,0)>$
$\displaystyle+$
$\displaystyle\left(e^{-i\Omega_{\parallel}z}{\hat{w}}_{1}{\hat{w}}^{*}_{1}+e^{-i\Omega_{\perp}z}{\hat{w}}_{2}{\hat{w}}^{*}_{2}+e^{-i\Omega_{\phi}z}{\hat{w}}_{3}{\hat{w}}^{*}_{3}\right)|\phi(\omega,0)>.$

The ways to arrive at these results can be found in [86]. We would like to end
this subsection with the following observation, that the states defined by
$|A_{\parallel}(\omega,0)>$, $|A_{\perp}(\omega,0)>$ and $|\phi(\omega,0)>$
are pure states. The corresponding states denoted by
$|A_{\parallel}(\omega,z)>$, $|A_{\perp}(\omega,z)>$ and $|\phi(\omega,z)>$
are the mixed states those evolve from the pure ones through propagation in
phase space through mixing. Even if any one of them is absent at the
beginning, it can be generated later through mixing much like the neutrinos.

###  5.1 Oscillation probability $P_{\gamma_{\parallel}\to\phi}$

The amplitude for the transition of a photon of energy $\omega$ in state
$|A_{\parallel}(\omega,0)>$ to $|\phi(\omega,z)>$ after traversing a distance
z, is given by $<A_{\parallel}(\omega,0)|\phi(\omega,z)>$. The probability of
the same, $P_{\gamma_{\parallel}\rightarrow\phi}(\omega,z)$ can be estimated
from the evolution equations obtained above by using the formula
$P_{\gamma_{\parallel}\rightarrow\phi}(\omega,z)=|<A_{\parallel}(\omega,0)|\phi(\omega,z)>|^{2}$.
The same same turns out to be,

$\displaystyle
P_{\gamma_{\parallel}\rightarrow\phi}=\left|\left(e^{-i\Omega_{\parallel}z}{\hat{u}}_{1}{\hat{w}}^{*}_{1}+e^{-i\Omega_{\perp}z}{\hat{u}}_{2}{\hat{w}}^{*}_{2}+e^{-i\Omega_{\phi}z}{\hat{u}}_{3}{\hat{w}}^{*}_{3}\right)\right|^{2}.$
(5.14)

After performing some lengthy algebra, one can observe that, the resulting
expression for Eqn. (5.14) contains a sum of three quadratic pieces, i.e.,
$\left[\left(|{\hat{u}}_{1}||{\hat{w}}_{1}|\right)^{2}+\left(|{\hat{u}}_{2}||{\hat{w}}_{2}|\right)^{2}+\left(|{\hat{u}}_{3}||{\hat{w}}_{3}|\right)^{2}\right]$
plus three other pieces involving the distance parameter $z$. Upon converting,
these quadratic pieces into a square of their sum and rearranging the
resultant expression, Eqn.(5.14) reduces to the following form:

$\displaystyle P_{\gamma_{\parallel}\rightarrow\phi}$ $\displaystyle=$
$\displaystyle\left[{\hat{u}}_{1}||{\hat{w}}_{1}|+|{\hat{u}}_{2}||{\hat{w}}_{2}|+|{\hat{u}}_{3}||{\hat{w}}_{3}|\right]^{2}$
(5.15) $\displaystyle-$ $\displaystyle
2|{\hat{u}}_{1}||{\hat{w}}_{1}||{\hat{u}}_{2}||{\hat{w}}_{2}|\left[1-\cos\left(\left(\Omega_{\perp}-\Omega_{\parallel}\right)z\right)\right]$
$\displaystyle-$ $\displaystyle
2|{\hat{u}}_{1}||{\hat{w}}_{1}||{\hat{u}}_{3}||{\hat{w}}_{3}|\left[1-\cos\left(\left(\Omega_{\parallel}-\Omega_{\phi}\right)z\right)\right]$
$\displaystyle-$ $\displaystyle
2|{\hat{u}}_{3}||{\hat{w}}_{3}||{\hat{u}}_{2}||{\hat{w}}_{2}|\left[1-\cos\left(\left(\Omega_{\phi}-\Omega_{\perp}\right)z\right)\right].$

At this stage, it is convenient to consider defining,
$\mathbb{A}=|{\hat{u}}_{1}||{\hat{w}}_{1}|$,
$\mathbb{B}=|{\hat{u}}_{2}||{\hat{w}}_{2}|$ and
$\mathbb{C}=|{\hat{u}}_{3}||{\hat{w}}_{3}|$. The
constraint,$|{\hat{u}}_{1}||{\hat{w}}_{1}|+|{\hat{u}}_{2}||{\hat{w}}_{2}|+|{\hat{u}}_{3}||{\hat{w}}_{3}|=0$,
that follows from equation (4.8), can now be recasted as,

$\displaystyle\mathbb{A}+\mathbb{B}+\mathbb{C}=0.$ (5.16)

Now we can make use of Eqn. (5.16) in (5.15), to verify that the perfect
square term vanishes owing to the constraint equation (5.16); and the
remaining $z$ dependent pieces, can be manipulated further to provide,

$\displaystyle P_{\gamma_{\parallel}\rightarrow\phi}$ $\displaystyle=$
$\displaystyle
4\mathbb{A}\left(\mathbb{A}+\mathbb{C}\right)\sin^{2}\left(\frac{\left(\Omega_{\perp}-\Omega_{\parallel}\right)z}{2}\right)$
(5.17) $\displaystyle+$ $\displaystyle
4\mathbb{B}\left(\mathbb{B}+\mathbb{A}\right)\sin^{2}\left(\frac{\left(\Omega_{\phi}-\Omega_{\perp}\right)z}{2}\right)$
$\displaystyle+$ $\displaystyle
4\mathbb{C}\left(\mathbb{C}+\mathbb{B}\right)\sin^{2}\left(\frac{\left(\Omega_{\parallel}-\Omega_{\phi}\right)z}{2}\right).$

For the sake of completeness, we provide the expressions for the new
variables, $\mathbb{A}$, $\mathbb{B}$ and $\mathbb{C}$, introduced earlier, in
terms of the parameters of the matrix ${\bf M}$. And they are:

$\displaystyle\mathbb{A}=\rm{\cal{N}}^{(1)}_{vn}\rm{\cal{N}}^{(1)}_{vn}\left(g_{\phi\gamma\gamma}{B}_{\perp}\omega\right)(\omega^{2}_{p}-{\bf
E}_{1})(\omega^{2}_{p}-{\bf E}_{1})(m^{2}_{\phi}-{\bf E}_{1}),$ (5.18)
$\displaystyle\mathbb{B}=\rm{\cal{N}}^{(2)}_{vn}\rm{\cal{N}}^{(2)}_{vn}\left(g_{\phi\gamma\gamma}{B}_{\perp}\omega\right)(\omega^{2}_{p}-{\bf
E}_{2})(\omega^{2}_{p}-{\bf E}_{2})(m^{2}_{\phi}-{\bf E}_{2}),$ (5.19)
$\displaystyle\mathbb{C}=\rm{\cal{N}}^{(3)}_{vn}\rm{\cal{N}}^{(3)}_{vn}\left(g_{\phi\gamma\gamma}{B}_{\perp}\omega\right)(\omega^{2}_{p}-{\bf
E}_{3})(\omega^{2}_{p}-{\bf E}_{3})(m^{2}_{\phi}-{\bf E}_{3}).$ (5.20)

The fact that equations (5.18–5.20 ) follow Eqn. (5.16), to a good accuracy,
has been verified numerically.

###  5.2 Oscillation probability $P_{\gamma_{\perp}\to\phi}$

The oscillation probability for the $A_{\perp}$ component of a photon to
scalar $\phi$ can be derived similarly. Therefore, instead of going through
the same set of arguments, we would provide the final result. Before we go to
the final expression, like before, we introduce the new set of variables,
$\mathbb{L}$, $\mathbb{M}$ and $\mathbb{N}$; defined as,
$\mathbb{L}=|{\hat{v}}_{1}||{\hat{w}}_{1}|$,
$\mathbb{M}=|{\hat{v}}_{2}||{\hat{w}}_{2}|$ and
$\mathbb{N}=|{\hat{v}}_{3}||{\hat{w}}_{3}|$. Their actual form, in terms of
the elements of the mixing matrix will be provided shortly. The expression for
$P_{\gamma_{\perp}\rightarrow\phi}$ is:

$\displaystyle P_{\gamma_{\perp}\rightarrow\phi}$ $\displaystyle=$
$\displaystyle
4\mathbb{L}\left(\mathbb{L}+\mathbb{N}\right)\sin^{2}\left(\frac{\left(\Omega_{\perp}-\Omega_{\parallel}\right)z}{2}\right)$
(5.21) $\displaystyle+$ $\displaystyle
4\mathbb{M}\left(\mathbb{M}+\mathbb{L}\right)\sin^{2}\left(\frac{\left(\Omega_{\phi}-\Omega_{\perp}\right)z}{2}\right)$
(5.22) $\displaystyle+$ $\displaystyle
4\mathbb{N}\left(\mathbb{N}+\mathbb{M}\right)\sin^{2}\left(\frac{\left(\Omega_{\parallel}-\Omega_{\phi}\right)z}{2}\right).$
(5.23)

The values of the parameters $\mathbb{L}$, $\mathbb{M}$ and $\mathbb{N}$ are
in terms of the parameters of the theory are given by,

$\displaystyle\mathbb{L}=\rm{\cal{N}}^{(1)}_{vn}\rm{\cal{N}}^{(1)}_{vn}\left(g_{\phi\gamma\gamma}{\rm{B}}_{\perp}\omega\right)\left(\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}\right)(\omega^{2}_{p}-{\bf
E}_{1})(m^{2}_{\phi}-{\bf E}_{1}),$ (5.24)
$\displaystyle\mathbb{M}=\rm{\cal{N}}^{(2)}_{vn}\rm{\cal{N}}^{(2)}_{vn}\left(g_{\phi\gamma\gamma}{\rm{B}}_{\perp}\omega\right)\left(\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}\right)(\omega^{2}_{p}-{\bf
E}_{2})(m^{2}_{\phi}-{\bf E}_{2}),$ (5.25)
$\displaystyle\mathbb{M}=\rm{\cal{N}}^{(3)}_{vn}\rm{\cal{N}}^{(3)}_{vn}\left(g_{\phi\gamma\gamma}{\rm{B}}_{\perp}\omega\right)\left(\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}\right)(\omega^{2}_{p}-{\bf
E}_{3})(m^{2}_{\phi}-{\bf E}_{3}).$ (5.26)

As before, following equation (4.7), equations (5.24–5.26) too have to satisfy
the constraint $\mathbb{L}+\mathbb{M}+\mathbb{N}=0$. We have tested the same
numerically to a good accuracy during the corresponding probability
evaluation.

###  5.3 Oscillation probability
$P_{\gamma_{\parallel}\rightarrow\gamma_{\perp}}$

Lastly, we provide the conversion probability for parallel component of photon
to perpendicular component of photon here
i.e.,$P_{\gamma_{\parallel}\rightarrow\gamma_{\perp}}$ . Introducing, as
before, the quantities, to be defined later below,
$\mathbb{P}=|{\hat{u}}_{1}||{\hat{v}}_{1}|$,
$\mathbb{Q}=|{\hat{u}}_{2}||{\hat{v}}_{2}|$ and
$\mathbb{R}=|{\hat{u}}_{3}||{\hat{v}}_{3}|$. The corresponding probablity for
conversion, $P_{\gamma_{\parallel}\rightarrow\gamma_{\perp}}$ turns out to be:

$\displaystyle P_{\gamma_{\parallel}\rightarrow\gamma_{\perp}}$
$\displaystyle=$ $\displaystyle
4\mathbb{P}\left(\mathbb{P}+\mathbb{R}\right)\sin^{2}\left(\frac{\left(\Omega_{\perp}-\Omega_{\parallel}\right)z}{2}\right)$
(5.27) $\displaystyle+$ $\displaystyle
4\mathbb{Q}\left(\mathbb{Q}+\mathbb{P}\right)\sin^{2}\left(\frac{\left(\Omega_{\phi}-\Omega_{\perp}\right)z}{2}\right)$
(5.28) $\displaystyle+$ $\displaystyle
4\mathbb{R}\left(\mathbb{R}+\mathbb{Q}\right)\sin^{2}\left(\frac{\left(\Omega_{\parallel}-\Omega_{\phi}\right)z}{2}\right),$
(5.29)

where,

$\displaystyle\mathbb{P}=\rm{\cal{N}}^{(1)}_{vn}\rm{\cal{N}}^{(1)}_{vn}\left(\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}\right)(\omega^{2}_{p}-{\bf
E}_{1})(m^{2}_{\phi}-{\bf E}_{1})^{2},$ (5.30)
$\displaystyle\mathbb{Q}=\rm{\cal{N}}^{(2)}_{vn}\rm{\cal{N}}^{(2)}_{vn}\left(\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}\right)(\omega^{2}_{p}-{\bf
E}_{2})(m^{2}_{\phi}-{\bf E}_{2})^{2},$ (5.31)
$\displaystyle\mathbb{R}=\rm{\cal{N}}^{(3)}_{vn}\rm{\cal{N}}^{(3)}_{vn}\left(\frac{e{\rm{B}}_{\parallel}}{m_{e}}\frac{\omega^{2}_{p}}{\omega}\right)(\omega^{2}_{p}-{\bf
E}_{3})(m^{2}_{\phi}-{\bf E}_{3})^{2}.$ (5.32)

Like before we have verified that, the condition,
$\mathbb{P}+\mathbb{Q}+\mathbb{R}=0$ is maintained to a good accuracy all
along during the course of the computation.
There is one important observation that follows from the probabilities derived
above; that is: The probability $P_{\gamma_{\parallel}\to\gamma_{\perp}}$ is
the only probability out of the three discussed above, that survives in the
limit $g_{\phi\gamma\gamma}\to 0$ and $m_{\phi}\to 0$. This can be related to
”rotation measure” (RM), that’s usually encountered in studies of optical
activity relating the angle of rotation of the plane of polarization of a beam
of plane polarized light, after travelling some distance L. The RM in this
case, can be defined by $\pi/\bar{\lambda}$, when $\bar{\lambda}$ is the
minimum distance that a plane polarized light beam needs to travel, to have
$P_{\gamma_{\parallel}\to\gamma_{\perp}}(\omega,\bar{\lambda})=1$. The non-
linear dependence of $\bar{\lambda}$ on the parameters of the theory indicates
that it is difficult to separate the contributions to
$P_{\gamma_{\parallel}\rightarrow\gamma_{\perp}}$ into parts originating from
(a) the magnetized plasma and (b) the one originating from magnetic field
induced - dilatonic interactions. Thus it should be considered as a total of
the two contributions mentioned above.
Lastly, the other important aspect of this analysis, is that the other
probabilities i.e., $P_{\gamma_{\parallel}\to{\phi}}$ and
$P_{{\phi}\to\gamma_{\parallel}}$, $P_{\gamma_{\perp}\to{\phi}}$, and
$P_{{\phi}\to\gamma_{\perp}}$ along with
$P_{\gamma_{\parallel}\to\gamma_{\perp}}$ and
$P_{\gamma_{\perp}\to\gamma_{\parallel}}$, turn out to be same as they should
be even otherwise.

Figure 1:  Plot of conversion probability of a parallely polarized photon into
a scalar dilaton in magnetic field $\rm{B}=10^{12}$ Gauss (in the left panel)
and the same in magnetic field $\rm{B}=10^{11}$ Gauss (in the right panel).
The solid line is for conversion probability in presence of magnetized media
and the dashed curve is the same in absence of self energy correction
$\Pi^{p}_{\mu\nu}$ from magnetized medium effects. The abscissa (energy of
photon ($\omega$) in GeV) is in units of $10^{-5}$ and the ordinates is
plotted in the units of $10^{-4}$. Here, the parameters used are: mass of
dilaton (scalar) particle $\phi$: $(m_{\phi})=1.0\times 10^{-12}$ GeV,
coupling constant $(g_{\phi\gamma\gamma})=1.0\times 10^{-11}$ GeV, photon path
length (z) = 1.2 Km, and plasma frequency $(\omega_{p})=1.96\times 10^{-2}$
eV.

Figure 2:  Plot of conversion probability of perpendicularly polarized photon
into scalar in magnetic field $\rm{B}=10^{12}$ Gauss (in the left panel) and
the same in magnetic field $\rm{B}=10^{11}$ Gauss (in the right panel). The
abscissa (energy of photon ($\omega$) in units of $10^{-5}$ GeV) and the
ordinate are plotted in units of $10^{-5}$. Here, the parameters used are:
mass of dilaton (scalar) particle $\phi$ $(m_{\phi})=1.0\times 10^{-12}$ GeV,
coupling constant $(g_{\phi\gamma\gamma})=1.0\times 10^{-11}$ GeV, photon path
length (z) = 1.2 Km and plasma frequency $(\omega_{p})=1.96\times 10^{-2}$ eV.

###  5.4 Conversion Probability $P_{\gamma_{\parallel}\to\phi}$ in an un-
magnetized medium.

In this subsection we provide the photon scalar conversion rate without
magnetized medium effects to compare its size with same due to magnetized
medium induced effects. The mixing matrix for the same is similar to that of
chamelion-photon mixing in an unmagnetized media, that is 2 $\times$ 2\. The
probability for the photon scalar transition obtained in [28] turns out to be
same as that of [38]. This probability of transition without any approximation
is given by:

$\displaystyle
P_{\gamma_{\parallel}\rightarrow\phi}=\frac{4{\rm{B_{\perp}}}^{2}{\omega}^{2}}{{(Mm_{ef}^{2})}^{2}+4{{\rm{B_{\perp}}}}^{2}{\omega}^{2}}\sin^{2}\left(\frac{\sqrt{{{(Mm_{ef}^{2})}^{2}+4{\rm{B_{\perp}}}^{2}{\omega}^{2}}}}{4\omega
M}z\right).$ (5.33)

When in Eqn. (5.33)
$m^{2}_{ef}=m^{2}_{\phi}-\omega^{2}_{p}-\frac{{\rm{B_{\perp}}}^{2}}{M^{2}}$
and $m_{\phi}$ is mass of scalar field, as stated before. In this case only
one degree of freedom of photon (i.e., $A_{\parallel}$) mix with the scalar,
and the other degrees of freedom ($A_{\perp}$, $A_{L}$) propagate freely. It
has two special features; One is, in the limit when the angle between $\rm{B}$
and $k$ $\to$ zero or $\pi$, the probability in (5.33) vanishes, indicating a
decoupling of fields. The second one is, in non-interacting limit; i.e.,
$g_{\phi\gamma\gamma}\to 0$ or $M\to\infty$, $P_{\gamma_{\parallel}\to\phi}\to
0$ for $\omega_{p}\neq m_{\phi}$. But in the same limit if
$\omega_{p}=m_{\phi}$, then the oscillation length diverges. It is straight
forward to verify the same. In the magnetized plasma case, however the
$P_{\gamma_{\parallel}\to\perp}$ and $P_{\gamma_{\perp}\to\parallel}$ remains
finite even if the angle between $\rm{B}$ and $k$ $\to$ zero or $\pi$. These
checks are useful to prove the consistency of the results.

##  6 Astrophysical Applications

Presence of very strong magnetic field and an ambient plasma in the
environment of astrophysical compact objects like white dwarves or neutron
stars, gamma ray bursters (GRB) etc., provides an opportunity to look for
astrophysical signatures of the kind of particles, we have been studying in
this article. In this work, we have focussed on the EM signals from the
compact objects that may bear possible signatures of dimension five, $\phi
F^{\mu\nu}F_{\mu\nu}$ interaction.
In this context we have tried to estimate the ratio of parallel component of
the electric flux with the perpendicular component . The estimate of this
ratio in dipole magnetic field of aligned rotor model is known. We have found
the same ratio in presence of the scalar field. So the difference between the
two when compared with observations, then one might be able to draw some
conclusions about the existence of the field $\phi$.
Following this point of view, we have taken the plasma frequency $\omega_{p}$
to be of the order $10^{-2}$ eV. We have further considered the photon path
length to be $z=1.2$Km [121]. For these numbers of $\omega_{p}$ and $z$, we
have estimated various oscillation probabilities in KeV energy range
(20-100)KeV as shown in Fig: [1] and Fig: [2]. The details that led to the
choice of these parameters have been provided in supplementary document of
this article.

###  6.1 Electric field

In this subsection we provide the expression for the electric field in a frame
of reference where the basis vectors are orthogonal to the propagation
direction $k_{\mu}$. We begin by noting that in the momentum space $k_{\mu}$
the electric field in four-component notation can be written as:

$\displaystyle{E}_{\mu}=\omega\tilde{A}_{\mu}-k_{\mu}(\tilde{A}.u).$ (6.1)

It should be noted that in Eqn. (6.1), the vector potential $\tilde{A}_{\mu}$
refer to the gauge fields for dynamical photon (in absence of scalar-photon
mixing). Expressing the vector potential using Eqn.(3.12) in the basis where
$A.k=0$, the electric field $E_{\mu}$ is,

$\displaystyle{E}_{\mu}=\omega\left[\tilde{A}_{\parallel}(k)\hat{b}^{(1)}_{\mu}+\tilde{A}_{\perp}(k)\hat{I}_{\mu}+\tilde{A}_{L}(k)\hat{\tilde{u}}_{\mu}\right]-k_{\mu}(\tilde{A}.u).$
(6.2)

The components of the electric fields vector can be written in terms of the
form factors of the vector potentials by contracting it with the corresponding
polarization vectors as,

$\displaystyle
E_{\parallel}=\hat{b}^{(1)}_{\mu}E^{\mu}=\omega\tilde{A}_{\parallel}(k),$
(6.3) $\displaystyle
E_{\perp}=\hat{I}_{\mu}E^{\mu}=\omega\tilde{A}_{\perp}(k),$ (6.4)
$\displaystyle E_{L}=\hat{\tilde{u}}_{\mu}E^{\mu}=\tilde{A}_{L}(k),$ (6.5)
$\displaystyle
k^{\mu}\hat{b}^{(1)}_{\mu}=k^{\mu}\hat{I}_{\mu}=k^{\mu}\hat{\tilde{u}}_{\mu}=0.$
(6.6)

The magnetic field also can be written in terms of these three components.
Therefore the total energy stored in the form of EM fields
$E_{tot}=\frac{1}{\mu_{0}}({E^{2}}+{B^{2}}$).

###  6.2 Intensities of Polarization modes

For highly Lorentz boosted electrons, the opening angle for the curvature
photons follow the relation, $\Gamma\simeq\frac{1}{\theta_{e}}$, when
$\theta_{e}$, is the opening angle. Therefore in this approximation, the
amplitudes of the two orthogonal modes of curvature photon in terms of
modified Bessel functions $K_{1/3}(\xi)$ and $K_{2/3}(\xi)$ would be given by
[147],

$\displaystyle|\tilde{A}_{\parallel}|$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{6}\Gamma}{\omega_{c}}K_{1/3}\left(\xi\right),$ (6.7)
$\displaystyle|\tilde{A}_{\perp}|$ $\displaystyle\simeq$
$\displaystyle\frac{2\sqrt{3}\Gamma}{\omega_{c}}K_{2/3}\left(\xi\right).$
(6.8)

The argument $(\xi)$ of the modified Bessel functions is defined as ,

$\displaystyle\xi=\frac{\omega
R_{c}}{3c}\left(\frac{1}{\Gamma^{2}}+\theta^{2}_{e}\right)^{3/2}.$ (6.9)

Using Eqn. (D.5) and the approximation considered above (i.e.,
$\Gamma\simeq\frac{1}{\theta_{e}}$ ) the same in terms of $\omega_{c}$ (in the
units of $\hbar=c=1$) turns out be $\xi=\frac{\omega}{0.7\omega_{c}}$. The
modified Bessel functions when $\omega<<\omega_{c}$, i.e., for small arguments
are given by,

$\displaystyle K_{2/3}\left(\frac{\omega}{\omega_{c}}\right)\simeq
2^{-1/3}\Gamma_{{}_{E}}\left(\frac{2}{3}\right)\xi^{-2/3}$ $\displaystyle
K_{1/3}\left(\frac{\omega}{\omega_{c}}\right)\simeq
2^{-2/3}\Gamma_{{}_{E}}\left(\frac{1}{3}\right)\xi^{-1/3},$ (6.10)

(here $\Gamma_{{}_{E}}$ is the Euler gamma function) one can express the
square of the ratio of the amplitude of the two polarization components as,

$\displaystyle\left|\frac{\tilde{A}_{\parallel}}{\tilde{A}_{\perp}}\right|^{2}\simeq
2^{-5/3}\left[\frac{(\Gamma_{{}_{E}}(1/3))}{(\Gamma_{{}_{E}}(2/3))}\right]^{2}\xi^{2/3}$
(6.11)

Since the emitted energy peaks at $\omega_{c}$, one considers $\omega\simeq
0.7\omega_{c}$, the ratio of the two amplitudes for this energy comes out as,
$2^{-5/3}\left[\frac{(\Gamma_{{}_{E}}(1/3))}{(\Gamma_{{}_{E}}(2/3))}\right]^{2}$.
It should be noted that this ratio is independent of the path traversed by the
radiation. Ideally, for the kind of situation under consideration, this is
what one would expect for the ratio of the intensities of the two polarization
states. However in presence of $\phi F^{\mu\nu}F_{\mu\nu}$ interaction, the
same will be modified. The square of the modified amplitudes ratio turns out
to be,

$\displaystyle\left|\frac{A_{\parallel}}{A_{\perp}}\right|^{2}\simeq
2^{-5/3}\left[\frac{(\Gamma_{{}_{E}}(1/3))}{(\Gamma_{{}_{E}}(2/3))}\right]^{2}\xi^{2/3}\times\left[\frac{1-P_{\gamma_{\parallel}\to\phi}(\omega,z)}{1-P_{\gamma_{\perp}\to\phi}(\omega,z)}\right].$
(6.12)

The last term inside the square bracket in Eqn. (6.12), is the modification
factor, where we have taken into account the modifications to the intensities
in the $\parallel$ ($\perp$) direction due to oscillation of the $\parallel$
($\perp$) mode into $\phi$ after travelling a distance $z$. It is to be noted
that the denominator of the modification factor in Eqn.(6.12), would remain
unity, unless the effect of the photon self energy correction in a magnetized
media to order $e{\rm{B}}$, is considered. In this work we have retained this
piece consistently in our formalism to find it’s contribution to the ratio of
the two polarization states. And in absence of scalar-photon interaction, the
ratio would be the same for pure curvature radiation. In presence of this
interaction ( dimension-five-scalar-photon ), the same would be given by
Eqn.(6.12). The contribution of this deviation in intensities can be observed
when we take the difference of the two i.e.,

$\displaystyle\Delta
F=\left|\frac{\tilde{A}_{\parallel}}{\tilde{A}_{\perp}}\right|^{2}-\left|\frac{A_{\parallel}}{A_{\perp}}\right|^{2}$
(6.13)

The same has been estimated numerically in the right panel of Fig.[3].

Figure 3:  In the left panel : Intensity of parallely and perpendicularly
polarized photons with scalar photon interaction in magnetized media vs energy
$(\omega)$. The dot curve represents $|A_{\parallel}|^{2}$ and the dashed
curve represents $|A_{\perp}|^{2}$ . In the right panel : Difference between
ratios of intensity of parallely and perpendicularly polarized photons(in
abscence and in presence of scalar photon interaction) $\Delta F$. Parameters
choosen are; $\Gamma$ (Lorentz boost factor) $=1.0\times 10^{6}$,
$\omega_{c}\sim 5\times 10^{-5}$ GeV, $\rm{B}\sim 10^{12}$ Gauss. The abscissa
(energy of photon ($\omega$) in GeV) is in units of $10^{-5}$ .

In absence of PEST in magnetized media ; the result would be same as that of
mixing with $2\times 2$ mixing matrix.

##  7 Result and discussion

In this work we have explored the behaviour of the magnetized matter on the
probabilities $P_{\gamma_{\parallel,\perp}\to\phi}$ for a electrodynamic
system and estimated their magnitudes numerically. To perform that we need to
compare the effects of magnetized matter effects with un-magnetized matter
effects in different magnetic field environments. We have numerically
estimated various probabilities for the astrophysical parameters given by:
plasma frequency of the stellar environment ($\omega_{p}$) = $1.96\times
10^{-2}$ eV, scalar photon coupling constant ($g_{\phi\gamma\gamma}$) =
$1.0\times 10^{-11}$ GeV-1, and the photon path length considered here is :
$z\sim 1.2$ Km. The results are plotted in Figs.[1] and [2]. There are three
major outcomes of our analysis. Firstly, $P_{\gamma_{\parallel}\to\phi}$ for
magnetized media, turns out to be different from
$P_{\gamma_{\parallel}\to\phi}$ for unmagnetized media, they admit frequency
dependent modification over each others contributions when the polarized
photon has propagated over the same distance. Qualitatively,
$P_{\gamma_{\parallel}\to\phi}$ conversion probability was found to be the
only non-zero probability when the correction of $\Pi^{p}_{\mu\nu}(k)$ is
absent. Secondly, the numerical strength of the probabilities of conversion
$P_{\gamma_{\parallel}\to\phi}$ (or $P_{\gamma_{\perp}\to\phi}$ ), increases
with increase in strength of the magnetic field. Since an analytical estimate
of the same is difficult, we have estimated the same numerically and have
shown both of them in a same panel (Fig.[1] and Fig.[2]). One interesting
feature of $P_{\gamma_{\parallel}\to\phi}$ is, it shows excess enhancement
over $P_{\gamma_{\parallel}\to\phi}$ estimated in an unmagnetized medium at
various energies. The third interesting feature is: the intensity of the other
orthogonally polarized mode of the photon ( denoted as $A_{\perp}$ ), no
longer remains the same when the effect of magnetized-self-energy correction
to photon is considered for studying (photon scalar) oscillation. In absence
of magnetized-self-energy correction, the intensity of $A_{\perp}$ mode
remains constant, provided no other absorption or enhancement mechanism is in
operation. That is to say, milli-charge pair production processes in a
magnetic field, ($\gamma+\gamma\to e_{m}^{+}+e_{m}^{-}$) that modifies the
$A_{\perp}$ spectra is considered absent.
The x-ray sources for compact stars can be classified into three categories;
one is due to bombardment of pair produced flux on the polar cap region of the
compact star, the second one is due to conduction heat coming out of the
surface of star having surface temperature $T\leq 10^{6}K$. And the last one
is due to the curvature radiation from the star.
The beams of EM radiations from first two sources follow the blackbody
radiation pattern and are unpolarized. However due to the presence of the
magnetic field, the isotropic beams of radiation get resolved into two
components, one of them becomes polarized parallel and the other one becomes
polarized perpendicular to the magnetic field. Total intensity of each of the
components becomes equal to the other.
On the other hand the amplitudes of the two polarized components of the
curvature radiation that we have discussed in this paper, denoted by
$\tilde{A}_{\parallel}$ and $\tilde{A}_{\perp}$, are different. As a result,
the spectra for their respective polarization, would look different from the
usual blackbody spectrum. The ratios of $\tilde{A}_{\parallel}$ and
$\tilde{A}_{\perp}$ would be given by Eqns. (6.7, 6.8). That yields
$\frac{\tilde{A}_{\parallel}}{\tilde{A}_{\perp}}\sim{(\frac{\omega}{0.7\omega_{c}})}^{1/3}$
for $\omega_{c}>>\omega$.
Once the scalar photon interaction is taken into account, and the same ratios
are estimated this pattern changes. If we take the difference between square
of these ratios without and with scalar photon modification i.e.,
$\left|\frac{\tilde{A}_{\parallel}}{\tilde{A}_{\perp}}\right|^{2}-\left|\frac{A_{\parallel}}{A_{\perp}}\right|^{2}$,
the resulting plot looks entirely different. The intensities of both the
polarization states (i.e., $A_{\parallel}$ and $A_{\perp}$) under the
circumstances mentioned above plotted separately, can be found in the left
panel of Fig.[3]. And the differences in their ratios of intensities (without
and with scalar photon interaction) can be found in the right panel of
Fig.[3]. Such an oscillating curve may bear the possible smoking gun signature
of dilaton-photon interaction from a compact star.

##  8 Outlook

Existence of DM was postulated to explain some astrophysical observations,
such as (i) Galaxy rotation curve (ii) observation of x rays from bullet
cluster (iii) weak micro lensing effect (iv) structure formation e.t.c.. The
candidate particles proposed, to explain the same, also predicts few
additional signatures steaming from ALP photon interaction [38] like,
supernova dimming, modification to distance duality relation, ALPs from
supernova, ALP photon conversion in magnetized domain etc., [122, 123, 124,
125, 126, 127, 128, 129, 130, 131] that happens to be some of them. So there
has been a surge for the search of ALP signals from the magnetized
environments of the compact stars [132] –[144] for their identifications.
Therefore to complement this surge in the interest we have explored ALP signal
from magnetized media in this work. In course of the investigation we find
that the formalism employed in this analysis, interpolates between two extreme
ends: one, the effective quantum statistical field theory of electrodynamics
(QSFED) in a magnetized environment, when the angle between $\rm{B}$ and $k$
is zero or $\pi$; and the other being the interacting theory of ALP with QSFED
at finite density when the angle between $\rm{B}$ and $k$ is $\pi/2$ that
follows from Eqn. (3.54). These are the checks that can be employed to
establish the consistency of our formalism for magnetized media.
It has been found that, in addition to $P_{\gamma_{\parallel}\to\phi}$ there
are two more additional probabilities of conversion i.e.,
$P_{\gamma_{\perp}\to\phi}$ and $P_{\gamma_{\parallel}\to\gamma_{\perp}}$
possible in a magnetized media according to the number of possible in-medium
polarization states of photon. Our analysis showed the existence of normal
modes corresponding to the three orthogonal directions of propagation of
photon.
Furthermore, we have noted in the body of the text that
$P_{\gamma_{\parallel}\to\phi}$ in a magnetized medium can be much larger than
the contribution of the same in an un magnetized medium in some energy range
for suitable values of other parameters. Since the produced ALPs would leave
the production region (of the star) fast because of its weak interaction with
the medium, this will lead to anomalous cooling of the star. The indication of
presence of ALP in the stellar environment comes from the estimate of $\Delta
F$ (i.e.,
$\left|\frac{\tilde{A}_{\parallel}}{\tilde{A}_{\perp}}\right|^{2}-\left|\frac{A_{\parallel}}{A_{\perp}}\right|^{2}$)
that becomes zero in absence of ALP and it becomes nonzero when ALP
interaction is present. These are the novel signatures that a magnetized
compact stars environment can offer for the verification of ALP.
To conclude, we have shown in this work that, effect of magnetized medium
brings non-trivial modifications to the ALP induced signals from compact
objects. Though the physics of these objects are complex, however it may be
possible to get a tale-tale signature of ALP like objects from the EM signals
of the compact stars. Some of these studies would be considered in separate
publications.

##  9  Acknowledgements

The authors would like to thank Prof. P. B. Pal for going through an earlier
version of this article.

##  10 Appendix

##  Appendix A Technical Details of constructing the unitary matrix

In order to describe the dynamics of scalar-photon interaction in magnetized
media in terms of different dof of mixing, the Hermitian mixing matrix
$M_{H}$, obtained from the equations of motion of scalar photon interaction,
given by,

$\displaystyle M_{H}=\left[\begin{matrix}\Pi_{T}(k)&iF&iG\cr-
iF&\Pi_{T}(k)&0\cr-iG&0&m_{\phi}^{2}\end{matrix}\right]$ (A.1)

needs to be diagonalize. Therefore, to do the same, we write $M_{H}$ as,

$\displaystyle M_{H}=\left[\begin{matrix}a&ib&ic\cr-ib&d&0\cr-
ic&0&e\end{matrix}\right]$ (A.2)

when $a=\Pi_{T}(k)$, $b=F$, $c=G$ and $e=m^{2}_{\phi}$. It can be shown that,
an unitary matrix $Q$ and its hermitian conjugate $Q^{\dagger}$ defined as,

$\displaystyle Q=\left[\begin{matrix}1&0&0\cr 0&i&0\cr
0&0&i\end{matrix}\right],Q^{\dagger}=\left[\begin{matrix}1&0&0\cr 0&-i&0\cr
0&0&-i\end{matrix}\right]$ (A.3)

can transform $M_{H}$ to real symmetric matrix $M_{RS}$ by,
$QM_{H}Q^{\dagger}=M_{RS}$, where $M_{RS}$ is, given by,

$\displaystyle M_{RS}=\left[\begin{matrix}a&b&c\cr b&d&0\cr
c&0&e\end{matrix}\right]$ (A.4)

Now, if for a real symmetric matrix $M_{RS}$, there exist a Unitary matrix
${\bf\tilde{U}}$ such that
${\bf\tilde{U}}^{\dagger}M_{RS}{\bf\tilde{U}}=M_{D}$, where $M_{D}$ is a
diagonalised matrix having the eigen values of matrix $M_{RS}$ as diagonal
elements. Now, using this fact, we can subsequently find out the unitary
matrix ${\bf{U}}$, that would diagonalize $M_{H}$. The argument goes as
follows. Since,

$\displaystyle{\bf\tilde{U}}^{\dagger}M_{RS}{\bf\tilde{U}}$ $\displaystyle=$
$\displaystyle M_{D}$ (A.5)

then, using $QM_{H}Q^{\dagger}=M_{RS}$ in equation (A.5), we get

$\displaystyle{\bf\tilde{U}}^{\dagger}QM_{H}Q^{\dagger}{\bf\tilde{U}}$
$\displaystyle=$ $\displaystyle M_{D}$ (A.6)
$\displaystyle{\bf{U}}^{\dagger}M_{H}{\bf{U}}$ $\displaystyle=$ $\displaystyle
M_{D}$ (A.7)

where we have defined $Q^{\dagger}{\bf\tilde{U}}={\bf{U}}$ and
$(Q^{\dagger}{\bf\tilde{U}})^{\dagger}={\bf{U}}^{\dagger}$. The eigen values
of the unitarily related matrices $M_{H}$ and $M_{RS}$ remain same. To obtain
$M_{D}$, we need unitary matrix ${\bf{U}}$ that follows from the unitary
matrix $\bf{\tilde{U}}$, constructed from the eigen vectors of $M_{RS}$.

##  Appendix B  Eigen vectors of $M_{RS}$

###  B.1 The characteristics equation

In this section we denote the eigen values of $M_{RS}$ as ${\bf
E}_{i},i=1,2,3$, then the characterstic equation for the same can be written
as,

$\displaystyle|M_{RS}-{\bf E}_{i}{\bf I}|=0.$ (B.1)

Where ${\bf I}$ is a 3$\times$3 identity matrix. Hence,

$\displaystyle\left|\begin{matrix}a-{\bf E}_{i}&b&c\cr b&d-{\bf E}_{i}&0\cr
c&0&e-{\bf E}_{i}\end{matrix}\right|=0$ (B.2)

The characteristics equation that follows from the determinant is,

$\displaystyle(a-{\bf E}_{i})(d-{\bf E}_{i})(e-{\bf E}_{i})-b^{2}(e-{\bf
E}_{i})-c^{2}(d-{\bf E}_{i})=0,$ (B.3)

that yields on simplification,

$\displaystyle{\bf E}_{i}^{3}c_{3}+{\bf E}_{i}^{2}c_{2}+{\bf
E}_{i}c_{1}+c_{0}=0.$ (B.4)

Where, the coefficients $c_{i}^{\prime}s$ are defined as :

$\displaystyle c_{3}$ $\displaystyle=$ $\displaystyle 1$ (B.5) $\displaystyle
c_{2}$ $\displaystyle=$ $\displaystyle-(a+d+e)$ (B.6) $\displaystyle c_{1}$
$\displaystyle=$ $\displaystyle(ae+de+ad-b^{2}-c^{2})$ (B.7) $\displaystyle
c_{0}$ $\displaystyle=$ $\displaystyle(b^{2}e+c^{2}d-aed)$ (B.8)

The nature of the roots of the cubic equation (B.4) must satisfy the following
properties,

$\displaystyle{\bf E}_{1}+{\bf E}_{2}+{\bf E}_{3}$ $\displaystyle=$
$\displaystyle-c_{2}$ $\displaystyle{\bf E}_{1}{\bf E}_{2}+{\bf E}_{2}{\bf
E}_{3}+{\bf E}_{3}{\bf E}_{1}$ $\displaystyle=$ $\displaystyle c_{1}$
$\displaystyle{\bf E}_{1}{\bf E}_{2}{\bf E}_{3}$ $\displaystyle=$
$\displaystyle-c_{0}.$ (B.9)

The roots of Eqn. (B.4) depends on the discriminant, ${\cal D}={\cal
Q}^{2}+{\cal P}^{3}$, where, variables ${\cal P}$ and ${\cal Q}$ are related
to the elements of Eqn. (B.4) through the relations,

$\displaystyle{\cal
P}=\frac{\left(3c_{1}-{c_{2}}^{2}\right)}{9}\mbox{~{}~{}~{}and~{}~{}~{}}{\cal
Q}=\left(\frac{{c_{2}}^{3}}{27}-\frac{{c_{2}}{c_{1}}}{6}+\frac{c_{0}}{2}\right).$
(B.10)

Furthermore, for hermitian matrices, roots are real, we should have ${\cal
Q}^{2}+{\cal P}^{3}\leq 0$. Finally, following [116, 117], the roots turns out
to be:

$\begin{array}[]{cc}{\bf E}_{1}=&{\cal R}\cos\alpha+\sqrt{3}{\cal
R}\sin\alpha-{c_{2}}/3,\\\ {\bf E}_{2}=&{\cal R}\cos\alpha-\sqrt{3}{\cal
R}\sin\alpha-{c_{2}}/3,\\\ {\bf E}_{3}=&\hskip-42.67912pt-2{\cal
R}\cos\alpha-{c_{2}}/3,\\\ \end{array}\mbox{~{}~{}With~{}}\\\
\left\\{\begin{array}[]{c}\alpha=\frac{1}{3}\cos^{-1}\left(\frac{{\cal{Q}}}{{\cal
R}^{3}}\right)\\\ \,\,\,\,\,{\cal R}=\sqrt{\left(-{\cal P}\ \right)}{\bf
sgn}\left({\cal{Q}}\right)\end{array}\right.$ (B.11)

For having real roots, one should have ${\cal P}<0$, and
$\mid\frac{\cal{Q}}{{\cal{R}}^{3}}\mid\leq 1$. Although for analytic
evaluation of the roots, this condition may not pose any problem, however
during numerical evaluation of the same, maintaining this ratio may become
difficult if the magnitudes of elements of the matrix ${\bf M}$ are close to
the precision available for the machine in use.
One can scale the matrix ${\bf M}$ by a suitable numerical factor to avoid
this difficulty.

###  B.2 Eigen function and Unitary matrix

The normalized eigen vector ${\bf\tilde{V}}_{i}$ of the real symmetric matrix
$M_{RS}$, can be represented as,

$\displaystyle{\bf\tilde{V}}_{i}=N_{i}\left[\begin{matrix}\tilde{u}_{i}\cr\tilde{v}_{i}\cr\tilde{w}_{i}\end{matrix}\right],$
(B.12)

where $N_{i}$ is the normalization constant. Therefore, using eigen value
equation, i.e., $(M_{RS}-{\bf E}_{i}{\bf I}){\bf\tilde{V}}_{i}=0$. Hence,

$\displaystyle\left[\begin{matrix}a&b&c\cr b&d&0\cr
c&0&e\end{matrix}\right]\left[\begin{matrix}\tilde{u}_{i}\cr\tilde{v}_{i}\cr\tilde{w}_{i}\end{matrix}\right]={\bf
E}_{i}\left[\begin{matrix}\tilde{u}_{i}\cr\tilde{v}_{i}\cr\tilde{w}_{i}\end{matrix}\right].$
(B.13)

We get,

$\displaystyle(a-{\bf E}_{i})\tilde{u}_{i}+b\tilde{v}_{i}+c\tilde{w}_{i}$
$\displaystyle=$ $\displaystyle 0$ (B.14) $\displaystyle
b\tilde{u}_{i}+(d-{\bf E}_{i})\tilde{v}_{i}$ $\displaystyle=$ $\displaystyle
0$ (B.15) $\displaystyle c\tilde{u}_{i}+(e-{\bf E}_{i})\tilde{w}_{i}$
$\displaystyle=$ $\displaystyle 0.$ (B.16)

Using the method discussed in [118], we can obtain analytical expressions of
the elements of eigen vector ${\bf\tilde{V}}_{i}$. They are,

$\displaystyle{\bf\tilde{V}}_{i}=N_{i}\left[\begin{matrix}\tilde{u}_{i}\cr\tilde{v}_{i}\cr\tilde{w}_{i}\end{matrix}\right]=N_{i}\left[\begin{matrix}(d-{\bf
E}_{i})(e-{\bf E}_{i})\cr-b(e-{\bf E}_{i})\cr-c(d-{\bf
E}_{i})\end{matrix}\right].$ (B.17)

Here, $N_{i}=\frac{1}{\sqrt{((d-{\bf E}_{i})(e-{\bf E}_{i}))^{2}+(b(e-{\bf
E}_{i}))^{2}+(c(d-{\bf E}_{i}))^{2}}}$, is a normalisation constant. We can
now construct the unitary matrix $\bf{\tilde{U}}$, using (B.17), as:

$\displaystyle{\bf\tilde{U}}=\left[\begin{matrix}N_{1}\tilde{u}_{1}&N_{2}\tilde{u}_{2}&N_{3}\tilde{u}_{3}\cr
N_{1}\tilde{v}_{1}&N_{2}\tilde{v}_{2}&N_{3}\tilde{v}_{3}\cr
N_{1}\tilde{w}_{1}&N_{2}\tilde{w}_{2}&N_{3}\tilde{w}_{3}\end{matrix}\right]$
(B.18)

###  B.3 Orthonormality check of ${\bf\tilde{V}}$

One can easily check the vectors are normalised i.e.,
${\bf\tilde{V}_{i}.\tilde{V}_{j}}=1$ when $i,j=1,2,3$ and i$=$j. Furthermore,
the verification of the orthogonal properties of vectors, provided below:

$\displaystyle{\bf\tilde{V}_{1}.\tilde{V}_{2}}$ $\displaystyle=$
$\displaystyle
N_{1}N_{2}(\tilde{u}_{1}\tilde{u}_{2}+\tilde{v}_{1}\tilde{v}_{2}+\tilde{w}_{1}\tilde{w}_{2})$
(B.19) $\displaystyle=$ $\displaystyle N_{1}N_{2}((d-{\bf E}_{1})(e-{\bf
E}_{1})(d-{\bf E}_{2})(e-{\bf E}_{2})+b^{2}(e-{\bf E}_{1})(e-{\bf E}_{2})$
$\displaystyle+$ $\displaystyle c^{2}(d-{\bf E}_{1})(d-{\bf E}_{2}))$

Now,

$\displaystyle(e-{\bf E}_{1})(e-{\bf E}_{2})=e^{2}-e({\bf E}_{1}+{\bf
E}_{2})+{\bf E}_{1}{\bf E}_{2}.$ (B.20)

We will convert the above equation having two functions ${\bf E}_{1}$ and
${\bf E}_{2}$ in to single function ${\bf E}_{3}$ by using the properties of
roots of cubic equation. In order to do that, we will try the following
substitutions,

$\displaystyle{\bf E}_{1}+{\bf E}_{2}$ $\displaystyle=$ $\displaystyle[{\bf
E}_{1}+{\bf E}_{2}+{\bf E}_{3}]-{\bf E}_{3}$ (B.21) $\displaystyle{\bf
E}_{1}{\bf E}_{2}$ $\displaystyle=$ $\displaystyle[{\bf E}_{1}{\bf E}_{2}+{\bf
E}_{2}{\bf E}_{3}+{\bf E}_{3}{\bf E}_{1}]-{\bf E}_{3}({\bf E}_{2}+{\bf
E}_{1}).$ (B.22)

Following the identities of equations (B.9) and make substitutions in Eqn.
(B.19), we obtain,

$\displaystyle{\bf\tilde{V}_{1}.\tilde{V}_{2}}=N_{1}N_{2}(a-{\bf
E}_{3})[(a-{\bf E}_{3})(d-{\bf E}_{3})(e-{\bf E}_{3})$
$\displaystyle-b^{2}(e-{\bf E}_{3})-c^{2}(d-{\bf E}_{3})].$ (B.23)

The terms inside the square brackets on the r.h.s of Eqn.(B.23) happens to be
zero due to equation (B.3) that follows from the characteristics equation.
Hence,

$\displaystyle{\bf\tilde{V}_{1}.\tilde{V}_{2}}=0.$ (B.24)

Following the same procedure one can verify that
${\bf\tilde{V}_{1}.\tilde{V_{2}}}={\bf\tilde{V_{2}}.\tilde{V_{3}}}={\bf\tilde{V_{3}}.\tilde{V_{1}}}=0$.

###  B.4 Analytical check of
${\bf\tilde{U}^{\dagger}}M_{RS}{\bf\tilde{U}}=M_{D}$

We have obtained the unitary matrix ${\bf\tilde{U}}$ , which suppose to
diagonalise the matrix $M_{RS}$. To check this claim analytically, we first
start unitary transformation of $M_{RS}$ by ${\bf\tilde{U}}$, given as
follows:

$\displaystyle{\bf\tilde{U}^{\dagger}}M_{RS}{\bf\tilde{U}}=M_{D}$ (B.25)

that implies,

$\displaystyle\left[\begin{matrix}N_{1}\tilde{u}_{1}&N_{1}\tilde{v}_{1}&N_{1}\tilde{w}_{1}\cr
N_{2}\tilde{u}_{2}&N_{2}\tilde{v}_{2}&N_{2}\tilde{w}_{2}\cr
N_{3}\tilde{u}_{3}&N_{3}\tilde{v}_{3}&N_{3}\tilde{w}_{3}\end{matrix}\right]\left[\begin{matrix}a&b&c\cr
b&d&0\cr
c&0&e\end{matrix}\right]\left[\begin{matrix}N_{1}\tilde{u}_{1}&N_{2}\tilde{u}_{2}&N_{3}\tilde{u}_{3}\cr
N_{1}\tilde{v}_{1}&N_{2}\tilde{v}_{2}&N_{3}\tilde{v}_{3}\cr
N_{1}\tilde{w}_{1}&N_{2}\tilde{w}_{2}&N_{3}\tilde{w}_{3}\end{matrix}\right]=M_{D}$
(B.26)

$\displaystyle\hskip-28.45274pt\left[\begin{matrix}N_{1}(\tilde{u}_{1}a+\tilde{v}_{1}b+\tilde{w}_{1}c)&N_{1}(\tilde{u}_{1}b+\tilde{v}_{1}d)&N_{1}(\tilde{u}_{1}c+\tilde{w}_{1}e)\cr
N_{2}(\tilde{u}_{2}a+\tilde{v}_{2}b+\tilde{w}_{2}c)&N_{2}(\tilde{u}_{2}b+\tilde{v}_{2}d)&N_{2}(\tilde{u}_{2}c+\tilde{w}_{2}e)\cr
N_{3}(\tilde{u}_{3}a+\tilde{v}_{3}b+\tilde{w}_{3}c)&N_{3}(\tilde{u}_{3}b+\tilde{v}_{3}d)&N_{3}(\tilde{u}_{3}c+\tilde{w}_{3}e)\end{matrix}\right]$
(B.27)
$\displaystyle\times\left[\begin{matrix}N_{1}\tilde{u}_{1}&N_{2}\tilde{u}_{2}&N_{3}\tilde{u}_{3}\cr
N_{1}\tilde{v}_{1}&N_{2}\tilde{v}_{2}&N_{3}\tilde{v}_{3}\cr
N_{1}\tilde{w}_{1}&N_{2}\tilde{w}_{2}&N_{3}\tilde{w}_{3}\end{matrix}\right]=M_{D}$
(B.28)

Using Eqns. (B.14), (B.15) and (B.16) in (B.28), we get,

$\displaystyle\hskip-28.45274pt\left[\begin{matrix}N_{1}\tilde{u}_{1}{\bf
E}_{1}&N_{1}\tilde{v}_{1}{\bf E}_{1}&N_{1}\tilde{w}_{1}{\bf E}_{1}\cr
N_{2}\tilde{u}_{2}{\bf E}_{2}&N_{2}\tilde{v}_{2}{\bf
E}_{2}&N_{2}\tilde{w}_{2}{\bf E}_{2}\cr N_{3}\tilde{u}_{3}{\bf
E}_{3}&N_{3}\tilde{v}_{3}{\bf E}_{3}&N_{3}\tilde{w}_{3}{\bf
E}_{3}\end{matrix}\right]\left[\begin{matrix}N_{1}\tilde{u}_{1}&N_{2}\tilde{u}_{2}&N_{3}\tilde{u}_{3}\cr
N_{1}\tilde{v}_{1}&N_{2}\tilde{v}_{2}&N_{3}\tilde{v}_{3}\cr
N_{1}\tilde{w}_{1}&N_{2}\tilde{w}_{2}&N_{3}\tilde{w}_{3}\end{matrix}\right]$
$\displaystyle=\left[\begin{matrix}{\bf\tilde{V}_{1}.\tilde{V}_{1}}{\bf
E}_{1}&{\bf\tilde{V}_{1}.\tilde{V}_{2}}{\bf
E}_{1}&{\bf\tilde{V}_{1}.\tilde{V}_{3}}{\bf
E}_{1}\cr{\bf\tilde{V}_{2}.\tilde{V}_{1}}{\bf
E}_{2}&{\bf\tilde{V}_{2}.\tilde{V}_{2}}{\bf
E}_{2}&{\bf\tilde{V}_{2}.\tilde{V}_{3}}{\bf
E}_{2}\cr{\bf\tilde{V}_{3}.\tilde{V}_{1}}{\bf
E}_{3}&{\bf\tilde{V}_{3}.\tilde{V}_{2}}{\bf
E}_{3}&{\bf\tilde{V}_{3}.\tilde{V}_{3}}{\bf E}_{3}\end{matrix}\right]$
$\displaystyle=M_{D}$ (B.29)

Now, if we use the orthonormality of the eigen vectors, we get,

$\displaystyle M_{D}=\left[\begin{matrix}{\bf E}_{1}&0&0\cr 0&{\bf E}_{2}&0\cr
0&0&{\bf E}_{3}\end{matrix}\right]$ (B.30)

##  Appendix C Data availability

The data used for plotting Fig. [1] and Fig [2] in this article we be shared
on request to the corresponding author.

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