2024 Kinetic theory formulation of the P - and CP -odd terms in the photon self-energy in a medium

In an optically active medium, such as a plasma that contains a neutrino background, the left-handed and right-handed polarization photon modes acquire diﬀerent dispersion relations. We study the propagation of photons in such a medium, which is otherwise isotropic, within the framework of the covariant collissionless Boltzmann equation incorporating a term that parametrizes the optical activity. Using the linear response approximation, we obtain the formulas for the components of the photon polarization tensor, expressed in terms of integrals over the momentum distribution function of the background particles. The main result here is the formula for the P - and CP -breaking component of the photon polarization tensor in terms of the parameter involved in the new term we consider in the Boltzmann equation to describe the eﬀects of optical activity. We discuss the results for some particular cases, such as long-wavelength and nonrelativistic limits, for illustrative purposes. We also discuss the generalizations of the P - and CP -breaking term we included in the Boltzmann equation. In particular we consider the application to a plasma with a neutrino background and establish contact with calculations of the photon self-energy in those systems in the framework of thermal ﬁeld theory.


Introduction and outline
It was shown sometime ago, in the context of thermal field theory (TFT), that the general expression for the photon self-energy in an isotropic medium, consistent with gauge and Lorentz invariance in four dimensions, may contain a term which signals P -and CP -symmetry breaking, either in the Lagrangian, or in the background, or both [1,2].The effect of this term is that the dispersion relation of the photon transverse modes, which would otherwise be degenerate, are split according to the polarization of the propagating mode.A consequence of this is the rotation of the linear polarization of an electromagnetic wave traveling in such media, or birefringence.These effects arise, for example, when photons propagate in a medium that contains a neutrino background [3,4,5].
On the other hand it has also been shown that the effect mentioned can be described in the context of the classical Maxwell equations in terms of an additional electromagnetic constant ζ, besides the standard dielectric (ǫ) and magnetic (µ) constants [6].A covariant version of this formulation has also been given [7].Following Ref. [6] we refer to ζ as the activity constant, and to this kind of medium as an optically active medium.
Our objective in the present work is to revisit the study the propagation of photons in such media, in the framework of the covariant, collisionless, Boltzmann equation, but incorporating a term that gives rise to the optical activity effects mentioned above.This approach, which lies somewhere in the middle between the two approaches mentioned, that is, TFT on one hand, and a purely phenomenological description on the other, could be more suitable than those two approaches in certain situations.
Recently, the optical activity effects on the cosmic microwave photons as they travel through the medium of the cosmic neutrino background have been considered in the context of the cosmic birefringence [8].Independently of whether or not the photon-neutrino interactions are responsible for the observed effects in this context, or whether another source of the optical activity may be required as suggested in Ref. [8], our work may be useful for further development in this area, and other astrophysical contexts in which optical activity plays a role.
While in the present work we restrict ourselves to the collisionless Boltzman equation, modified to incorporate the optical activity effects, the method we use can be extended to include the effect of collisions in an optically active plasma, for example by adapting the techniques used for this purpose in the framework of the Boltzmann equation for an ordinary plasma (see, e.g., Ref. [9] and references therein).For example, in the case of photons propagating in a plasma medium with a neutrino background, it may be important to include also the effects of collisional interactions.While such interactions typically lead to damping effects, under the appropriate circumstances can also lead to growth effects.An example of a growth effect was provided in Ref. [4], with regard to the evolution of a magnetic field perturbation in such a medium.This particular phenomena has been considered by Semikoz and Sokoloff [10] as a mechanism for the generation of large-scale magnetic fields in the early Universe as a consequence of the neutrino-plasma interactions.In this sense the present work provides the mechanism for considering the effects of collisions in an optically active medium on a firm ground.
In short, our proposal is to consider the relativistic collisionless Boltzmann equation for a given particle specie "a", of charge q a , p modified by adding a term proportional to the dual electromagnetic tensor Fµν , as we state precisely below, including the explanation of the various symbols that enter in this equation.The strategy is to use the linear response approach to obtain the expression for the induced current, and thereby the photon polarization tensor, in terms of the particle distribution functions, and study various aspects of the results that could be useful for applications, such as the corresponding dispersion relations and the interpretation in terms of the activity constant.In the present work we restrict ourselves to implement this program for an isotropic system, that is, the momentum distribution functions of all the particle species are isotropic in the rest frame of the system.The generalization of the approach to other cases, for example a two-stream component plasma is straightforward from a conceptual point of view, although of course the details will be different and they can be important in specific physical contexts.The outline of the rest of the paper is as follows.In Section 2 we present the covariant Boltzmann equation, that includes the P -and CP -breaking term to describe the effects of an optically active medium.For consistency, we discuss some particular features and consequences of the equation, including current conservation and the role of the discrete symmetries of the new term.In Section 3, we consider the solution of the equation, using the standard linearization method.The expression for the induced current is determined, in terms of integrals of the momentum distribution functions of the particles.There we establish contact with the photon polarization tensor, or equivalently the photon self-energy, in the TFT language, specifically as used in Ref. [1], and the formulas for the components of the polarization tensor are obtained.The results for the longitudinal and transverse components of the photon self-energy are the familiar ones.The formula for the P -and CP -breaking term is the new result here.In Section 4 we discuss some details of the results obtained, and consider specifically some particular cases (e.g., the long-wavelength and the nonrelativistic limit) that are useful in many applications and can serve as benchmark references for more general situations.We also point out possible generalizations of the P -and CP -breaking term we included in the Boltzmann equation, in particular how it applies in the context of a plasma with a neutrino gas as a background, and establish contact with calculations of the photon self-energy in such backgrounds in the framework of TFT [3].Possible avenues for extensions and exploration of the present work are mentioned in Section 5.

P -and CP -breaking kinetic equation
Without further preamble, the equation we consider is, We will discuss some generalizations of this equation in Section 4.2.In this expression, f a (x, p) is the number density of the particles specie "a", of charge q a , in the plasma, expressed as a function of the four-vectors x µ and p µ , and normalized as specified below [e.g., see Eq. (2.10)].Further, F µν is the dual electromagnetic field tensor, and we are using the shorthand symbols We use the conventions such that g µν has diagonal elements (1, −1, −1, −1) and ǫ 0123 = +1.
The parameter γ a is a phenomenological parameter, which in this approach is unknown.However Eq. (2.1), or its generalizations, as we will discuss, parametrize effectively the results of the calculations of optical activity in some systems, such as those considered in the references already cited (e.g., Refs, [3,4,5]).In this sense this approach serves as a bridge between those calculations on one hand, and a pure phenomenological description in terms of the electrodynamics equations on the other (e.g., Refs.[1,6,7]).
Before entering in the practical calculations, there are various aspects of these equations that are worth discussing.We consider them below.

Current conservation
The electromagnetic current density is where We have introduced the velocity four-vector of the medium, u µ , which has components in the medium's own rest frame.Notice that with since θ(p • u) = 0 for the negative solution p 0 = −E ap .Therefore Eq. (2.5) reduces to which is the conventional expression for the current density four-vector of each specie.In particular, its zeroth component is the particle number density of each specie, (2.10) We now consider the divergence of the current density.From Eq. (2.5), and using Eq.(2.1) where we have defined and In the second equality in Eq. (2.14) we have integrated by parts, dropping the surface term since f a vanishes for infinite momentum.Current conservation is a consequence of the fact that, while A µν a is antisymmetric, J aµν is, as we show below, symmetric, J aµν = J aνµ .
(2.16) Therefore, A µν a J aµν = 0 , (2.17) which together with Eq. (2.12) imply that The proof of Eq. ( 2.16) is by straightforward algebra.Taking derivative that appears in the integrand of Eq. (2.14) we obtain, and for the second term in Eq. (2.19) (2.20) In the last formula, the product of the two delta functions give zero because the two conditions, and cannot be satisfied simultaneously.Therefore we get which explicitly verifies Eq. (2.16).

Modification of the Boltzmann equation
If the distribution function is treated as a function only of p rather than p and p 0 separately, in other words we explicitly set it is well known that the covariant equation given in Eq. (1.1) is equivalent to (see, e.g., Ref. [11]) with (2.26) Equation (2.25) is the standard form of the Boltzmann equation for charged particles in an electromagnetic field.
In the case that f a satisfies Eq. (2.1) the corresponding equation, analogous to Eq. (2.25), is where F a is given above, and (2.28) The result given in Eq. (2.27) follows easily from the fact that the elements of Fµν are obtained from F µν by making the replacement which implies that G a is obtained from F a by making the same replacement.Thus, for example, in the presence of only a magnetic field B, the equation is (2.30) A quick observation that follows from Eq. (2.27) is that some discrete space-time symmetries are broken in the system when the Fµν term is present in the Boltzmann equation.This is obvious, for example for parity (P ), from the fact that E and B have opposite phase under a P transformation, and therefore the same holds for F and G.
3 Linearization of the kinetic equation and π µν

Linearization and the induced current
The dispersion relations for the propagating photons are obtained by linearizing the kinetic equation.We put where f a0 is the equilibrium distribution, and f a1 is linear in F µν .Substituting Eq. (3.1) in Eq. (2.1), and retaining only terms that are linear in F µν , gives The next step is to consider the momentum space equation corresponding to Eq. (3.2).Denoting the wave vector by k µ , the momentum space equation is obtained from Eq. (3.2) by making the replacements with the understanding that the functions on the right-hand side are the Fourier transforms of those on the left.In particular, in correspondence with Eq. (2.2).Furthermore, remembering that we are considering an isotropic system, f a0 is a function only of in which case where With these substitutions, the momentum space equation corresponding to Eq. (3.2) is, The induced current is obtained from Eqs. (2.4) and (2.9) using Eqs.(3.1) and (3.9).The terms containing the equilibrium distributions f a0 do not contribute.This is most easily seen by going to the medium's own rest frame.Due to the isotropy condition, the vector current density j a is zero, while the total j 0 is zero assuming that the charge density is zero in equilibrium.Thus, where The next step is to write Eq. (3.10) in terms of the vector potential rather than the field.We consider separately the two terms in Eq. (3.10).

f µν term
For the term with f µν , using we have which gives where I aµν is given in Eq. (3.11) and To arrive at Eq. (3.14) we have used again the fact that we are considering an isotropic system so f a0 is a function of p • u, and therefore For the sake of completeness, we mention that the contribution to the current from the term given in Eq. (3.14) is transverse by itself, as can be easily verified explicitly by multiplying by k µ and using Eq.(3.16).

f µν term
We now derive the relation analogous to Eq. (3.14) for the f µν term.From Eqs. (3.4) and (3.12), relabeling some of the Lorentz indices, it follows that aµλ , (3.17) where I aµν is defined in Eq. (3.11).Since I aµν is a symmetric in µ, ν, and depends only on k µ and u µ , it is of the form Only the C term contributes in Eq. (3.17) and therefore C a can in turn be written in terms of the tensor R λρ defined in Eq. (A.6) as aλρ . (3.20) This expression for π µν can be decomposed in terms of π T,L,P as given in Eq. (A.2).π P can be written by inspection, while π T,L can be obtained by projecting the term in square brackets in Eq. (3.21), which is symmetric and transverse, with R µν and Q µν , using Eqs.(A.10) and (A.9).Thus, where C a has been defined in Eq. (3.20), while We consider the evaluation of the integrals for C a and D a in the medium's rest frame, and thus we set We decompose k µ in the form k µ = (ω, κ) , (3.26) and in the integrands we will write p µ = (E ap , p) , (3.27) with the understanding that we are working in the rest frame of the medium.In particular f a0 is a function only of E ap , and (3.28) Thus, for example, the numerator of the integrand for D a in Eq. (3.24) while the denominator can be written as Substituting these in the expression for D a in Eq. (3.24), and similarly, Although we do not indicate it explicitly, it is understood that in these formulas for C a and D a , as well as in the formulas for π L,T,P given below, the singularity of the integrand at ω = κ • v a is to be handled by making the replacement as usual.From Eq. (3.22) we then obtain The corresponding expressions for ǫ ℓ,t obtained from Eq. (A.23) reproduce the standard classic results, e.g., and similarly for ǫ t .On the other hand, the formula for π P , and the corresponding formula for ǫ p obtained from Eq. (A.23), are new.

Dispersion relations in the long-wavelength limit
The longitudinal dispersion relation is not affected by the γ a terms, as already indicated.Therefore, we focus on the transverse ones (i.e., polarizations perpendicular to κ) which involve both π T and π P .Moreover, we consider specifically the long-wavelength limit, which is a particularly useful and representative of more general situations.
Using the fact that we are considering the case that the distribution functions f a0 are isotropic, by straightforward manipulation of the integrand, where For reference, recall that the plasma frequency (ω pl,a ) of each specie is given by For example, in the nonrelativistic limit, where n a0 is the equilibrium particle number density of the specie [see Eq. (2.10)].From Eq. (3.22) we then obtain in the long-wavelength limit and the well-known result where and As reviewed in Appendix A, the transverse dispersion relations are determined as the solutions of Substituting in this equation the results for π P and π T given in Eqs.(4.6) and (4.7), respectively, in the long-wavelength limit the equation becomes In the limit γ P → 0 we obtain the standard transverse solutions, for either polarization.In the more general situation, assuming the solutions are The dispersion relations are such that one polarization mode is damped (absorption by the medium) while the other one grows (emission by the medium).Which is one or the other depends on the sign of γ P , which in turn depends on the relative signs and values of the q a γ a terms in Eq. (4.9).

Generalization
Equation (2.1) is probably the simplest equation of the kind we are discussing, but there are some possible generalizations.Here we mention some of them.We will write them in the generic form The requirement is that Γ µν must contain, in some form, the dual tensor F αβ , and that it is antisymmetric in µ, ν so that the proof of current conservation given in Section 2.1 applies in this case as well, with the identification of A µν = −q a F µν + Γ µν in place of Eq. (2.13).
One possibility is where γ a is a constant parameter.The steps to arrive at Eq. (3.9) for fa1 apply also in this case, with the identification γ a = −i(k • u)γ (1)  a . (4.17) Thus for example, from Eqs. (4.6) and (4.9), we can see that in the long wavelength limit this gives a contribution to π P of the form κq a γ (1)  a Ω 2 a . (4.18) The main qualitative difference, relative to the case in which γ a is independent of ω, is that in the present case the corresponding contribution to π P is real, which in turn produces a real term in the dispersion relations of opposite sign for the (±) polarizations.
More generally, we can consider the equation The momentum space equation in the linear approximation for fa1 is again Eq. (3.9), but with γ a being the Fourier transform of Γ a (x − x ′ ), that is, γ a defined by writing The fact that Γ a (x − x ′ ) is real implies that

Neutrino background
An example case in which γ a has the form given in Eq. (4.17) is afforded by an electron plasma with a neutrino background.The calculation of π P in that case gives [3] π where n νe and n νe stand for the number densities of the electron neutrinos and antineutrinos, and ω pl,e is the electron plasma frequency.For simplicity let us consider the nonrelativistic limit, so that only the electrons (no positrons) are present, in which case (i.e., Eqs.(4.4) and (4.5)) where q e is the electron charge and n e0 is the equilibrium electron number density.On the other hand, for this case that we are considering, in the framework of the kinetic equation The main lesson here is that the kinetic approach allows us to parametrize the effects produced by the π P term in the photon polarization tensor in terms of the parameter γ a .In this framework, γ a is a phenomenological parameter that must be determined by other means, e.g., thermal field theory in the case of an electron plasma with neutrino background, as we have seen.Nevertheless, the kinetic approach allows us to study further the consequences of the presence of the π P term, such as the effects of external fields [10], streaming neutrino background [5] or collisional plasmas [9], among others.

Conclusions and Outlook
In this work we have proposed a method to study the propagation of photons in an optically active isotropic medium, based on the covariant collisionless Boltzmann equation.As shown in Section 2, the covariant Boltzmann equation can be modified by adding a term that gives rise to the optical activity effects, in a way that is consistent with the general requirements of current conservation and symmetry considerations.In Section 3, using the linear response method, we obtained an expression for the induced current, expressed in terms of integrals over the momentum distribution function of the background particles.There we established contact with the photon polarization tensor, or equivalently the photon self-energy, in the TFT language, specifically as used in Ref. [1], and the formulas for the components of the polarization tensor were obtained.The results for the longitudinal and transverse components of the photon self-energy, π L,T respectively, are the familiar ones.The new result here is the formula for the P -and CP -breaking component π P due to the new term we considered in the Boltzmann equation to describe the effects of optical activity.In Section 4 we discussed some details of the results obtained, and considered specifically some particular cases (e.g., the long-wavelength and the nonrelativistic limit) that are useful in practical applications and representative of more general situations.To emphasize the usefulness of the method, we pointed out how the P -and CP -breaking term we included in the Boltzmann equation can be generalized, in particular how it applies to a plasma with a neutrino gas as a background, and established contact with calculations of the photon self-energy in such contexts in the framework of TFT [3].The strength and advantages of the method here presented comes from its semiclassical standpoint, which in many circumstances is more suitable than the thermal field theory approach for incorporating other potentially important effects such as collisions, external fields, stream backgrounds and multicomponent plasmas.

A Notation and conventions
We use the notation and conventions used in Ref. [1], which we briefly review here for convenience.The momentum of the propagating photon is denoted by k µ , and u µ is the velocity four-vector of the medium, already introduced in Eq. (2.5).

A.1 Photon polarization tensor
In the context of TFT, the photon self-energy, π µν gives rise to a contribution to the effective Lagrangian of the photon and the corresponding field equation that leads to identify j µ = −π µν A ν , (A.1) .23) Using Eqs.(3.11) and (3.15), the integral formulas for C a and D a can be written in the form

π P = iκq e γ e Ω 2 3πm 2 e
the framework of the kinetic equation, the effects of the neutrino background can be parametrized in terms of a γ e parameter for the electron of the form γ e = −iωγ(1)  (n νe − n νe ) .(4.27) .21) Being a scalar, γ a is a function of the scalar variables ω and κ defined in Eq. (A.3), a fact that we indicate by writing it as γ a (ω, κ) when needed.In particular, Eq. (4.21) actually implies the condition