CMB V modes from photon-photon forward scattering revisited

Recent literature has shown that photon-photon forward scattering mediated by Euler-Heisenberg interactions may generate some amount of circular polarization ($V$ modes) in the Cosmic Microwave Background (CMB) photons. However, there is an apparent contradiction among the different references about the predicted level of the amplitude of this circular polarization. In this work, we will resolve this discrepancy by showing that with a quantum Boltzmann equation formalism we obtain the same amount of circular polarization as using a geometrical approach that is based on the index-of-refraction of the cosmological medium. We will show that the expected amplitude of $V$ modes is expected to be $\approx$ 8 orders of magnitude smaller than the amplitude of $E$-polarization modes that we actually observe in the CMB, thus confirming that it is going to be challenging to observe such a signature. Throughout the paper, we also develop a general method to study the generation of $V$ modes from a photon-photon and photon-spin-1 massive particle forward scatterings without relying on a specific interaction, which thus represent possible new signatures of physics beyond the Standard Model.

Recent literature has shown that photon-photon forward scattering mediated by Euler-Heisenberg interactions may generate some amount of circular polarization (V modes) in the Cosmic Microwave Background (CMB) photons. However, there is an apparent contradiction among the different references about the predicted level of the amplitude of this circular polarization. In this work, we will resolve this discrepancy by showing that with a quantum Boltzmann equation formalism we obtain the same amount of circular polarization as using a geometrical approach that is based on the index-of-refraction of the cosmological medium. We will show that the expected amplitude of V modes is expected to be ≈ 8 orders of magnitude smaller than the amplitude of E-polarization modes that we actually observe in the CMB, thus confirming that it is going to be challenging to observe such a signature. Throughout the paper, we also develop a general method to study the generation of V modes from a photon-photon and photon-spin-1 massive particle forward scatterings without relying on a specific interaction, which thus represent possible new signatures of physics beyond the Standard Model.
The effects of the photon-photon forward scattering on the CMB polarization have been previously studied in [20,28,31,34]. The fundamental result is the production of CMB V modes for Faraday conversion of CMB linear polarization. However, there is an apparent contradiction between the different papers. For instance [28], working within the QBE formalism, predicts an amount of V -mode signal that is much larger than what predicted by [31,34], who work with a geometrical formalism focusing on the birefringence in the index-of-refraction of the cosmological medium. In this paper, we will show that the two formalisms are fully consistent and give the same prediction for the amount of V modes produced, correcting the formulae and estimates of [28].
The paper is organized as follows. In Sec. II, we will provide an introduction on how to use the quantum Boltzmann equation formalism to study the effects of the photon-photon forward scattering on CMB polarization. In Sec. III, we will study the CMB polarization mixing induced by the photon-photon forward scattering mediated by a generic interaction. In Sec. IV, we will derive the expected power-spectrum statistics of CMB V modes provided by the photon-photon forward scattering through Euler-Heisenberg interactions. Moreover, we will provide the expected amount of V modes for a generic interaction, as a function of free parameters. In Sec. V, we will investigate the CMB polarization mixing induced by a hypothetical photon-spin-1 massive particle forward scattering. Finally, Sec. VI contains our main conclusions.

Description of the formalism
We start our analysis by introducing the formalism adopted for the rest of this paper. We parametrize the intensity and the polarization of CMB radiation through a density matrix ρ ij , defined in terms of four Stokes parameters, in the following form [1] where the parameter I defines the intensity of unpolarized CMB radiation, Q and U define the CMB linear polarization and V refers to CMB circular polarization. The equations of motion for the Stokes parameters can be found through the so-called quantum Boltzmann equation, which is given in the literature as where k 0 is the energy of CMB photons, H I (t) is the (effective) interaction Hamiltonian (describing in our case, e.g., the photon-photon interactions), and D ij (k) = a † i (k)a j (k) is the photon number operator (a † i (k) and a j (k) being the creation and annihilation operators, see later more details). Within this formalism, the expectation value of a generic operator A is defined as [1] where ρ denotes the following density operator In Eq. (2) the first term on the right-hand side is the so-called forward-scattering term and the second term is the so-called damping term. In this work, we will focus on the forward scattering term, which is able to generate couplings between different polarization states. 1 In fact, Eq. (2) is derived adopting a perturbative approach so that increasing powers of the interaction Hamiltonian H I (t) reduce the strength of the corresponding term. For this reason, in any fundamental interaction in the perturbative regime in which the forward scattering term is nonzero, a priori it is expected to give the relevant physical effects on the CMB polarizations. Of course, this is not the case of the standard QED interaction between photons and electrons, where such a forward scattering term vanishes (see e.g. [1]), and all the relevant effects arise from the damping term only.
In particular, we are interested in the effects of the forward scattering of CMB photons with other (massless) spin-1 particles. Given the S (4) as the S-matrix element describing this process, the (effective) interaction Hamiltonian can be defined through [44] S (4) where H I can generally be written as H I = dp 1 dp 2 dp 3 dp 4 (2π where dp ≡ and M(p 1 , r; p 2 , s; p 3 , r ′ ; p 4 , s ′ ) is the Lorentz invariant amplitude of this interaction, as a function of photon momenta and photon polarization indices r, r ′ , s, s ′ = 1, 2, where 1 and 2 here stand for the two independent transverse polarizations. Moreover, a i (p) and a † i (p) denote respectively the annihilation and creation operators for photons obeying the following canonical commutation relation Inserting Eq. (6) in the forward scattering term of Eq. (2) we get [H I (0), D ij (k)] = dp 1 dp 2 dp 3 dp 4 (2π Using Eqs. (3) and (8), we obtain the following expectation value of the product of photon creation and annihilation operators Thus, using (10) we can perform the expectation value in (9) by employing Wick's theorem, and, after integrating out three of the momenta with the Dirac delta's, we find the following final form of our Boltzmann equation 2 where p and k indicate the momenta of the background (b) and of the line-of-sight observed (γ) photons respectively. In the next section, we will employ the latter equation to evaluate the effects of photon-photon forward scattering mediated by Euler-Heisenberg interactions on the CMB polarization field. However, before doing this, in the following subsections we will introduce a general parametrization of the photon-photon scattering amplitude. 2 In this equation, ρ γ ij refers to the density matrix of the observed CMB photons, while ρ b ij denotes the density matrix of the "background" target CMB photons. General photon-photon scattering amplitude In this subsection, by using symmetry considerations, we introduce a general amplitude describing the scattering of two massless spin-1 particles which does not rely on any specific photon-photon fundamental interaction. In fact, assuming to work in the context of quantum field theory (QFT) that is unitary and where all the interactions are local, we can employ the following general parametrization for the photon-photon Lorentz invariant scattering amplitude [44][45][46] where ǫ i µ ≡ ǫ µ (p i ) are the polarization vectors of incoming and outgoing photons, and M µνλσ (1234) ≡ M µνλσ (p 1 , p 2 , p 3 , p 4 ), with p 1 /p 3 and p 2 /p 4 denoting the four momenta of incoming/outgoing photons in the γ(p 1 ) + γ(p 2 ) → γ(p 3 ) + γ(p 4 ) scattering process. The four-rank tensor M µνλσ (1234) must respect the crossing and gauge symmetries. The gauge symmetry implies the following identities while, due to the crossing symmetry, M µνλσ (p 1 , p 2 , p 3 , p 4 ) is given by summing over all the 4! possible permutations of external photons with momenta p 1 , p 2 , p 3 and p 4 and simultaneously their corresponding vertex indices. We have depicted one of these terms in Fig. 1. We can expand M µνλσ (p 1 , p 2 , p 3 , p 4 ) in terms of a set of four-rank independent tensors T (i) µνλσ as [45] M µνλσ (1234) = where coefficients G i (s, t, u) are invariant scalar amplitudes that may depend on invariant kinematics, as the Mandelstam variables Moreover, the tensors T where the tensor basis is defined in the following equations Thus, using Eq. (12), we can express the photon-photon scattering amplitude as a function of the metric tensor g µν , the photon four-momenta and polarization vectors, and generic coefficients without specifying a given fundamental interaction.

QED case: Euler-Heisenberg amplitude
In the quantum electrodynamics (QED) context, photon-photon interactions are described by the so-called Euler-Heisenberg Lagrangian which is a low-energy effective Lagrangian describing multiple photon interactions. This reads [44,47,48] where and where α = e 2 /(4π) denotes the so-called fine-structure constant, m e is the electron mass and F µν = ∂ µ A ν − ∂ ν A µ is the well-known photon field strength. This Lagrangian can be also expressed in terms of the electric and magnetic fields as where This effective Lagrangian is derived by the photon-photon scattering process mediated by the one-loop box Feynman diagrams containing electrons in the internal lines (see, e.g., Fig. 2) in the low-energy limit where the external photons are soft with energies much lower than the electron mass m e . It is possible to show that the Feynman amplitude derived by the Euler-Heisenberg Lagrangian can be expressed in terms of the basis tensors f which is included in the general form (14).

III. POLARIZATION MIXING FROM PHOTON-PHOTON FORWARD SCATTERING
Now, the effect of the photon-photon forward scattering on the dynamics of CMB Stokes parameters is obtained by inserting Eq. (12) into the Boltzmann equation (11). Because we are finally interested in the polarization and FIG. 2: Three independent one-loop Feynman diagrams describing the photon-photon scattering for clockwise electron loop direction. In all the diagrams p1 and p2 denote the incoming photons momenta, while p3 and p4 refer to the outgoing momenta, [49].
intensity of CMB radiation, we first give the expression of the Stokes parameters in terms of the CMB density matrix. In this respect, the unperturbed CMB photon density matrix is written as [1] while the CMB radiation field perturbations are defined as where k c = ak is the comoving wavenumber of CMB photons, with a(η) denoting the scale-factor as a function of conformal time dη = dt/a(t), and I 0 (k) = (e k/T − 1) −1 is the Bose-Einstein distribution function describing the homogeneous (unperturbed) distribution of CMB photons. As above, the upper index γ refers to the observed photons.
In the same way, the background beam is described by and where this time the upper index"b" refers to the background photons (we use slightly different notations w.r.t. to Eq. (26) to easily keep track of the background target beam). Thus, using these definitions for the photon density matrices, we insert the general scattering amplitude (12) 4 into the Boltzmann equation (11) and sum over all the vector polarization indices. After some straightforward calculations, we find 3 These are analogous to the brightness perturbations defined in [1] apart for a factor 4, i.e. our ∆ γ I is a factor (1/4) the ∆ I quantities defined in [1] (see, e.g., Eq. (6.51) of [1]), so that, e.g., here ∆ γ I represents the temperature fluctuations (it would correspond to the quantity Θ defined, e.g., in [7] or in Eq. (5.3) of [50]. For a discussion of the various temperature variables that can be used, see, e.g., [51,52]). 4 It is understood that we are using the FRW metric ds 2 = −dt 2 + a 2 dx 2 in evaluating (12). that is expected, since there is no energy and momentum transfer in the forward scattering of photons and where the g i coefficients and the scalar functions s i are given in App. A. From the physical point of view, the set of coupled Eqs. (30), (31) and (32) just derived gives rise both to the transformation of Q modes into U modes and vice-versa (Faraday rotation), and the conversion of linear polarization to circular polarization and vice-versa (Faraday conversion). In this paper, we are interested only in the Faraday conversion effect, thus we decouple Q and U modes by assuming g 4 = 0 (or G 1 = G 2 ) and we remain with

Euler-Heisenberg case
In this subsection, we derive in our quantum Boltzmann equation formalism the linear-circular polarization mixing induced by Euler-Heisenberg interactions. Thus, substituting the Euler-Heisenberg Feynman amplitude (24) into (11) we get the following set of equations where the explicit expressions for the f i coefficients are given in App. B. Notice that these equations can be directly derived by (33), (34) and (35), once we identify Moreover, notice that by matching amplitudes (14) and (24), we get telling us that photon-photon scattering, as predicted by QED, leads only to Faraday conversion (moreover, in the low energy limit one also finds G 3 = 0, see [45]). Now, in order to perform the integral over p, we write the momenta and photon polarization vectors in the following general formk In this generic reference frame, we get Now, without losing generality, we fix the frame where the line-of-sight is aligned with the z-axis, i.e.k c z, as in the end we will work with quantities that are invariant under rotations. Thus, we get Hence, Eqs. (37), (38) and (39) become Notice that in Eqs. (30), (31) and (32) the source terms proportional to s 1 and s 2 are linear in the perturbations. However, as we have just shown such contributions at the end vanish, leaving therefore only the remaining source terms that are second-order in the cosmological fluctuations. Now, we can start to compare our results with previous calculations of this effect, i.e. [20,31,34]. For instance, in [20] the time evolution of the CMB Stokes parameter V obeys the following equation where θ and φ are the polar angles between the observed and background photons. Comparing Eqs. (47) and (48), we find that our results are fully consistent with [20], apart from different normalization conventions in the definition of CMB Stokes parameters.
We can show the consistency of our results with also [31,34]. In these works it is shown that the circular polarization of the radiation field is generally produced by Faraday conversion that occurs when a linearly polarized radiation propagates through a medium in which the axes perpendicular to the momentum of the incoming radiation have a different refraction index. In order to compare our results with [31,34], we need to expand the Q and U modes in Eq. (28) in terms of spin-weighted spherical harmonics s Y ℓm as (see, e.g., [3,6]) where a E ℓm and a B ℓm denote the coefficients in the harmonic sphere expansion of the so-called E and B modes that give an alternative (rotationally invariant) description of the CMB linear polarization. Thus, we have and Then, we also employ the fact that which follows from Eq. (49), together with the identities that follow from the reality condition on the E and B modes. Thus, after employing the angular decomposition of Q and U modes, we perform the momenta integration in Eqs. (45), (46) and (47), and we get 5 where a rad = π 2 /15 is the radiation energy density constant and T CMB is the CMB temperature. The last Eq. (57) is consistent with [31,34], after ignoring a B 2,−2 term with respect to a E 2,−2 , that is equivalent to assume pure E-mode polarization for CMB photons at the recombination epoch, which in turn is equivalent to neglect tensor perturbations from inflation. Notice that our effect turns out to be proportional to CMB quadrupolar anisotropies: this is what we would have expected as, in order to get the Faraday conversion of a linear polarized light-beam, we need to introduce some kind of birefringence in the propagation medium.
Notice also that we can obtain analogous equations for describing the Faraday conversion in the general photonphoton forward scattering case (14). In fact, after using the spin-weighted spherical harmonic expansion and performing the momenta integrals in Eqs. (30), (31) and (32) with the same prescriptions as before, we find Interestingly, the only difference between the general and the Euler-Heisenberg cases is in the overall coefficient G 1 + G 2 + 2G 4 , which in the Euler-Heisenberg case is fixed as (40), while in the general case is undetermined.

IV. POWER-SPECTRUM OF CIRCULAR POLARIZATION
In this section, we derive the expression of the expected CMB circular polarization angular power-spectrum induced by photon-photon forward scattering. We will assume Euler-Heisenberg interactions, but the final result will be generalized to any photon-photon interaction through (40). To this purpose, we first define the following quantities [3] which encode CMB linear polarization and in Fourier space can be expressed in terms of rotationally invariant quantities, namely E and B modes. Using Eqs. (55) and (56) and assuming a B 2,−2 ≪ a E 2,−2 , we get and These equations can be written in Fourier space as where K denotes the Fourier conjugate of x. Moreover, we also give the Fourier space expression of the CMB V -mode polarization induced by Euler-Heisenberg interactions. From Eq. (57) we get Now, we need to implement in the equations of motion of the CMB polarization fields also the standard radiation transport terms, as given in the literature (see e.g. [1, 3,6]). These take into consideration the contributions of the photon-electron Thomson scattering and projection effects. Thus, the Boltzmann equations (64), (65) and (66) get modified into where a * denotes convolution in Fourier space, τ ′ (η) is the so-called "differential optical depth" of Thomson scattering, defined as n e being the electron density, x e the ionization fraction and σ T = (8π/3) α 2 /m 2 e is the Thomson cross section; is the cosine of the angle between the observed CMB photon and the Fourier mode K, where ∆ In , ∆ Qn and ∆ V n represent the n-th order terms in the Legendre polynomial expansion of the corresponding quantities, T 0 CMB denotes the CMB temperature today, n γ (n e ) are the photon (electron) number densities, and z is the redshift.
In analogy with the standard CMB radiation transport solutions, the differential equations (67), (68) and (69) admit the following integral solutions where η 0 denotes the conformal time today with the condition e −τ (η0) = 1. We have also assumed to a first approximation e −τ (0) ≈ 0. Now, following e.g. Ref. [3], in order to obtain the expected value of the V -mode polarization today in then direction to the sky, we need to integrate over all the possible Fourier momenta as where ζ(K) is a random function used to describe the initial amplitude of primordial scalar perturbations from inflation. 6 After computing (78), we can define its harmonic sphere coefficients as Then, the V -mode angular power-spectrum reads Therefore, inserting Eq. (78) into (79) we get and thus the V -mode power-spectrum reads where P ζ (K) defined as denotes the scalar primordial power-spectrum from inflation. Eq. (82) can be simplified assuming that the circular polarization source terms are negligible in comparison with linear polarization ones. 7 Therefore, (82) reads where g(η) = τ ′ e −τ is the so-called visibility function and Now, due to the product of the two visibility functions in (84) and (85), the latter takes the relevant contributions for η ′ ≃ η. Thus, (84) becomes where x = K(η − η 0 ). Now, using the following integral (see, e.g., [53]) we can perform the angular integration in (86), obtaining where in the last step we have used the differential equation satisfied by the spherical Bessel functions together with the Bessel recurrence relation Finally, we rewrite the quantity Im a E * 2,−2 (K) * Π(K, η) = Im Here, a E 2,−2 (K − P) can be expressed in terms of the same quantity in the frame where K − P is aligned with the z-axis, as [34] where D ℓ m,m ′ (α, β, γ) is the well-known Wigner rotation matrix [54], and we have employed the fact that, since we have scalar perturbations, only the m = 0 term of a E 2,m (K − P z) gives a contribution. Thus, (91) reads where we have used the fact that the quantity a E 2,0 (K ′ z) depends only on the wave-number K ′ due to the invariance of E modes under rotations on the polarization plane. Thus, (88) finally gives Now, in order to give an order of magnitude estimate on the amount of the circular polarization produced by this effect, we employ in Eq. (94) the expected level of CMB linear polarization. To this purpose, we define CMB E and B modes as [3] ∆ γ E = − where ð andð are the so-called spin raising and lowering operators [3]. Using these definitions, together with Eqs. (75) and (76), we get [3] ∆ γ where again x = K(η − η 0 ) and Q(x) = (1 + ∂ 2 x ) 2 . Here, we have neglected the back-reaction terms due to the coupling with circular polarization. Now, following again [3], the expected E-mode angular power-spectrum is given by where and Thus, we get where we have used again (87), (89) and (90). Now, by the matching between Eqs. (94) and (102), we get the following approximate relation between the circular and linear CMB polarization fields x e (z) and taking the following value of the constant parameters n γ n e = 2 × 10 9 , m e = 5 × 10 5 eV , and the following redshift average where z rec indicates redshift at the recombination epoch. Therefore, we obtain Also, we know that the relative amplitude of the CMB E-mode polarization quadrupole left imprinted by scalar perturbations is of the order [55] Thus, Eq. (103) reads This result suggests that the amount of V modes produced by this effect is much smaller than the level of linear polarization that we actually observe in the CMB, in a way that our procedure of neglecting the back-reaction of V -mode source terms on linear polarization is a consistent and very good approximation. Moreover, the r.m.s. value of V modes is given approximately by Notice that employing Eq. (40), we can express (110) in terms of the G i general coefficients as that gives the order of magnitude of the expected amplitude of V modes from the photon-photon forward scattering mediated by a generic interaction. Now, confronting our result (110) with the one found in [28], it turns out that our result is about 4 orders of magnitude smaller. This discrepancy may be explained by the following considerations: first, in [28] the coupling between the observed CMB linear and circular polarization is realized through the CMB background intensity field. But, as also emphasized in [34], this is not possible, because in such a case the f 0 2 parameter of Eq. (49) in [28] would identically vanish, once we perform the underlying angular integrals. As we have explicitly shown in our work, Faraday conversion is possible only if the coupling is realized through the linear polarization field of the background photons. This leads to a 6 orders of magnitude discrepancy. Moreover, we notice that in [28] the matching between the linear and circular polarization fields is made by taking the average value of the parameter η EH (z) in their Eq. (49), which is related to our A L (η) apart for constant coefficients. These constant coefficients are such that η av EH is 2 orders of magnitude smaller thanĀ L (η rec ).

V. POLARIZATION MIXING FROM PHOTON-MASSIVE SPIN-1 PARTICLES FORWARD SCATTERING
In this section we investigate a new viable way to get circular polarization out of the forward scattering between CMB photons and massive spin-1 particles. An example of such cosmological candidates are the so-called hidden photons. These massive bosons are present in extensions of the Standard Model of particle physics containing a general new hidden U(1) gauge group [56][57][58][59]. The dominant interaction between the conventional photons and the hidden photons is realized through the gauge kinetic mixing between them. In the literature, different methods have been proposed to constrain the coupling and mass of hidden photons using astrophysical and cosmological observations [60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75]. Here, we will consider only a generic photon-spin-1 massive particle scattering, leaving the extension of our final set of equations for specific cases to future work.
First of all, it is well known that a massive spin-1 field satisfies the so-called Proca's equation [76], with a mass term that explicitly breaks gauge invariance. The polarization vector of such a field involves 3 independent components. Moreover, the polarization field is characterized by 8 parameters which describe all the possible independent polarization states (see App. C for a brief review). In particular, the polarization matrix of a massive photon can be written as [77][78][79] where ρ ij is a 3 × 3 matrix, λ i are the generators of the SU(3) group, tr(ρ) = I is the intensity field describing unpolarized massive photons, and T i are 8 generalized Stokes parameters describing the polarization state of a given system. However, since there is little knowledge about the physical polarization states of massive spin-1 particles, here we will focus only on their intensity, assuming an unpolarized background. As in the previous examples, we begin by writing down the scattering amplitude. The general scattering amplitude of two generic spin-1 particles has been found by [46] in the following form Since the expected amplitude of CMB circular polarization is expected to be almost 8 orders of magnitude smaller than the amplitude of CMB linear polarization, we have no hope to observe soon such a signature with CMB experiments.
Moreover, throughout this paper we have provided very general expressions extending the computations described above and the corresponding expected amount of V modes in the case of a generic photon-photon interaction, not relying on any specific fundamental interaction (111). Finally, we have investigated the possibility to get some amount of V modes from the forward scattering between CMB photons and spin-1 massive particles. Our final set of equations (120), (121) and (122) confirms that only polarized spin-1 massive particles can couple linear to circular polarization in the CMB. These latter results provide the basis to further investigate new cosmological signatures of physics beyond the standard model of particle physics. We leave further study in this direction for future research.

Appendix A COEFFICIENTS OF STOKES PARAMETERS IN THE CASE OF GENERAL PHOTON-PHOTON INTERACTION
Here, we provide the explicit expression of the coefficients in Eqs. (30), (31) and (32).
Now, by matching (139) with (140) we get the following definition of the parameters T i in terms of the photon polarization vector For massless photons with no longitudinal polarization states we have necessarily ǫ z = 0. As a consequence, we get T 1 = T 2 = T 6 = T 7 = 0, and T 3 is related to T 8 . Thus, only 3 independent degrees of freedom remain, which correspond to the usual Q, U and V Stokes parameters. However, if we include in the picture the possibility to have longitudinal polarization states, we need to describe the polarization state of a spin-1 particle with the T i parameters that play the role of generalized Stokes parameters.