Axion mediated photon to dark photon mixing

The interaction between the dark/mirror sector and the ordinary sector is considered, where the two sectors interact with each other by sharing the same QCD axion field. This feature makes the mixing between ordinary and dark/mirror photons in ordinary and dark electromagnetic fields possible. Perturbative solutions of the equations of motion describing the evolution of fields in ordinary and dark external magnetic fields are found. User-friendly quantities such as transition probability rates and Stokes parameters are derived. Possible astrophysical and cosmological applications of this model are suggested.


damian.ejlli@lngs.infn.it
One main problem of quantum chromodynamics (QCD) is that it preserves the charge-parity (CP) symmetry that, instead is observed to be broken in weak interactions. In general, if we are not concerned with the violation of CP symmetry or time (T) symmetry, in any gauge theory we can introduce in the Lagrangian density a term L ∝ θ αβ ǫ µνρσ F α µν F β ρσ , where θ αβ is a constant matrix and F α µν is a gauge field tensor. In the case of QCD, the P, T and CP violating term in the Lagrangian density is L θ ∝θ G µν,aG a,µν whereθ is the effective angle of the theory and G a,µν is the gluon field tensor. The CP violating term induces electric dipole moments in baryons, where for example in the case of the neutron, theoretical estimates give d n (θ) ≃ 10 −16θ e cm [1] while experimentally is found d n < 2.9 × 10 −26 e cm [2]. Such a small experimental value for d n implies an effective angleθ 10 −10 .
The solution of the strong CP problem (smallθ value) is based on the Peccei-Quinn mechanism [3] and the existence of a new particle, the axion, has been proposed. In this mechanismθ becomes a dynamical field with an effective potential V (a) for the axion field a induced by non-perturbative QCD effects. The vacuum expectation value (VEV) of the axion field a = −θf is minimum for the effective potential and the CP violating term in the effective Lagrangian is dynamically cancelled. The axion scale, f , is a free parameter of the mechanism and is model dependent. Originally f was taken to coincide with the electroweak scale [4] but the non observation of axions in experiments would suggest that its scale could in principle be much larger than the electroweak scale. This fact has been implemented in the so called invisible axion models, namely the KSVZ axion model [5] and DFSZ axion model [6].
Another possibility that solves the strong CP problem is based on mirror symmetry, M-symmetry 1 see Ref. [8]. The general idea of this model is based on the assumption that exist a parallel sector of mirror or dark particles which has the same group and coupling constants analogous to the standard model sector. In this model the total standard model (SM) Lagrangian is invariant under M-symmetry. More precisely, the gauge group of the theory is G × G ′ where G is the ordinary group of the SM of particles G = SU (3) × SU (2) × U (1) with fermion fields Ψ i = q i , l i ,ū i ,d i ,ē i and Higgs doublet φ = (H 1 , H 2 ) T and G ′ = SU (3) ′ × SU (2) ′ × U (1) ′ is the mirror gauge group 2 with analogous particle content Ψ ′ i = q ′ i , l ′ i ,ū ′ i ,d ′ i ,ē ′ i and Higgs doublet φ ′ = (H ′ 1 , H ′ 2 ) T . Here, q i , i = 1, 2, 3 is the left handed quark doublet, l i is the left handed lepton doublet,ū i is the right handed quark singlet (u, c, t),d i is the right handed quark singlet (d, s, b) andē i is the right handed anti-lepton singlet. Here fermions are represented as Weyl spinors.
In the case when M-parity is an exact symmetry, the particle physics must be the same in both sectors. For example, for the Yukawa theory, we would have that ordinary and mirror sectors have E} are the Yukawa couplings (3 × 3 complex matrices) and are equal in both sectors. Since the Yukawa couplings are the same, this would imply that quark and lepton mass matrices have the same form, namely On the other hand, the total renormalizable Higgs potential in this model has the following form V tot = V + V ′ + V mix , where V is the standard model Higgs potential and V ′ is the mirror/dark sector Higgs potential with the same pattern as its standard model counterpart. The mixing potential comes out due to gauge symmetry of the theory and has a quartic interaction term of the form where the coupling constant κ is real due to M-symmetry. Its worth to stress that the mixing term V mix would bring the two sectors in equilibrium in the early universe due to the decay φ † φ → φ ′ † φ ′ unless κ < 10 −8 [9].
M-parity can be spontaneously broken with the introduction of a real scalar singlet η with odd parity, namely under the M-parity it changes the sign η → −η. In this case the interaction Lagrangian between the two sectors includes a term L I ∝ η(φ † φ − φ ′ † φ ′ ) that would introduce a renormalizable term ∆V to the total potential V . If η has a non-zero VEV, η = µ, it would induce differences in mass-squared terms of φ and φ ′ . This difference implies that VEVs v 1,2 are different from v ′ 1,2 and therefore different weak interaction scales where f a gets contribution from both ordinary and dark sectors, f a = f 2 + f ′2 , with f ′ being the axion decay constant in the dark sector and f being the axion decay constant in the ordinary sector. Consequently, the axion mass m a gets contribution from ordinary and dark sectors where m a is given by [8] where N is the colour anomaly of U (1) PQ current, K and K ′ are respectively the gluon condensates of ordinary and dark sectors which are respectively related to the ordinary and dark QCD scales Λ, Λ ′ through K ∼ Λ 3 , K ′ ∼ Λ ′3 and V, V ′ are respectively the quark condensates of ordinary and dark sectors with V ∼ Λ 3 and V ′ ∼ Λ ′3 . Here M and M ′ are respectively the mass matrices of light quarks of ordinary and dark sectors where M = diag(m u , m d ) and The characteristic of this model is that the axion field a couples to ordinary sector as DFSZ-like axion while it couples to the dark sector as the original axion or Weinberg-Wilczek (WW) axion [4] in case when f ′ ≫ f or Λ ′ ≫ Λ. In this case while the axion behaves as DFSZ-like axion with respect to the ordinary sector its mass given in (1) gets contribution from a small term coming from the ordinary sector and a much larger term coming from the dark sector. On the other hand the axion field couples to photons 3 with two different coupling constants g aγ and g ′ aγ which are given by where z = m u /m d , z ′ = m ′ u /m ′ d and α S is the fine structure constant.
The model which we briefly discussed above and which solves the strong CP problem in both sectors has several applications in astrophysics and cosmology. Before proceeding further, it is necessary to stress since now that apart from interacting with the same axion field a, the two sectors also interact gravitationally but this interaction is not important for the purposes of this paper and will not be considered in what follows. In particular, in this paper we are mostly interested in interaction of the axion field with ordinary and dark photon fields. Therefore, let us consider the following model where the effective interaction Lagrangian density is given by 4 where F µν is the electromagnetic field tensor of ordinary sector and F ′ µν is the electromagnetic field tensor of the dark sector. We may note the appearance of axion field a in the second and sixth terms in (3) which make possible the mixing of ordinary photons with dark photons mediated by a. The last term in (3) is the interaction Lagrangian of photons and dark photons respectively with ordinary and dark media. Such a term essentially corresponds to forward scattering of photons and dark photons in media which is encoded in the index of refraction. Generally, the Lagrangian density in such case involves a non local photon and dark photon polarization tensors in position space and is given by where A µ , A ′ µ are respectively the ordinary and dark photon fields and Π µν , Π ′µν are respectively photon polarization tensors of ordinary and dark photons in ordinary and dark media. The interaction Lagrangian L med gives rise to effective masses for ordinary and dark photons in ordinary and dark media. a Figure 1: Axion mediated photon to dark photon transition in ordinary and dark external magnetic fields. The external magnetic fields are denoted with cross symbols.
The equations of motions of (3) for the fields A ν , A ′ν and a in case when particles propagate in ordinary and dark media are respectively given by where we have included the effect of L med in Eqs. (4) as effective masses m γ and m ′ γ for the ordinary and dark photons. Next, we assume that media is magnetized, namely there is respectively an external magnetic field in ordinary and dark sectors where photons and dark photons propagate through. Moreover in this work we assume that external magnetic fields B e and B ′ e are perpendicular with respect to the 4 In this work we adopt the metric with signature ηµν = diag(1, −1, −1, −1).
direction of propagation of ordinary and dark photons. Let us consider ν = i and expand the fields A i , A ′i and a in Fourier modes as where e λ i is the i-th component of the polarization vector of a photon with helicity λ. Now we can use the expansion (5) in Eqs. (4) and consider ordinary and dark photons propagating along the observer's z axis, namely k = (0, 0, k) and B e = (B e , 0, 0), Here we are adopting a rather simplified assumption that both ordinary and dark external magnetic fields have the same direction. Moreover, we also assume that spatial variation of external magnetic fields are much larger than photon and dark photon wavelengths and consequently we may use WKB approximation in (5). In this case one can linearize Eqs. (4) and get the following system of linear differential equations where I is the unit matrix,

is a five component field and M is the mixing matrix which is given by
The photon states labelled with (+) are the states which are perpendicular to the external magnetic fields B e and B ′ e while the states labelled with (×) are those which are parallel to the external magnetic fields. The elements of the mixing matrix M are given by: where n + , n × are the total indexes of refraction of photons in ordinary medium for the polarization states (+), (×) and n ′ + , n ′ × are the total indexes of refraction of dark photons in dark medium for the states (+), (×).
Until now the only assumptions made in deriving Eq. (6) have been that photons and dark photons propagate in transverse external magnetic fields and used the WKB approximation for field expansion (5). The next step is to find the solution of Eq. (6) for given magnetized media. Indeed, the solution which one finds for Eq. (6) will depend if either the elements of M depend explicitly on time or not. For time independent M the solution is rather straightforward while for time dependent M a perturbative approach is usually enough. Therefore, in case M is time independent, we find the following exact solution for Eq. (6) × e −ikt A ′ × (k, 0) + e iλ 4 t sin 2 (φ) + cos 2 (φ) e iλ 5 t cos 2 (θ) + e iλ 2 t sin 2 (θ) e −ikt a(k, 0), 5 The entries of the mixing matrix M have been made real by global transformations of the fields in Ψ.
where we took the initial time to be zero in (8), namely t in = 0 and λ i (i = 1, 2, .., 5) are the eigenvalues of the mixing matrix M . The are given by , where φ and θ are two mixing angles and are respectively given by In the case when M is time dependent, in general it is not possible to find exact closed solutions but one might attempt to look for solutions by using perturbation theory. In this regard we can split the mixing matrix in the following way 6 , M a (t) is a diagonal matrix and M 1 (t) is a small perturbation matrix given by In the interaction picture, Eq. (6) becomes i∂ t Ψ int (t) = M int (t)Ψ int (t). By using standard iterative procedure, we find the following perturbative solution for Ψ int (t) to first and second order in the perturbation matrix M int (t) where Ψ int (t) + higher order terms. Performing several algebraic operations we get the following solutions for the fields up to the second order in perturbation theory in the Schrödinger picture where we have definedM ,M a (t) = dt (k(t) − M a (t)) with λ = (+, ×) and ∆M 1 (t) =M × (t) −M a (t) and ∆M 2 (t) =M ′ × (t) −M a (t). The solutions (8) and (12) which we found in cases of time independent and time dependent mixing matrix M , represent the main result in this work since we can easily extract from them other important quantities such as induced phase shifts of the fields and transition probability rates. In order to calculate these quantities, let us first concentrate in the case of (exact) solutions (8). Here we mostly concentrate in the additional effects to the mixing problem with the introduction of the dark photon. As in the case of photon-axion mixing, the introduction of the dark photon in the mixing problem give rise to birefringence and dichroism effects in external magnetic fields. In case of weak mixing with respect to θ, namely θ ≪ 1, we find the following additional phase shift 7 from the imaginary part of the first term in A × (k, t) in (8), of combined effects of the axion and dark photon on the photon polarization to second where we made a field redefinition Ψ(k, t) → e −i(M × −k) Ψ(k, t) and ∆M β = M × − β. In case when M ′ aγ → 0 we can easily recover the usual (additional) phase shift induced by the axion field. The additional phase shift induced by the dark photon is given by Aside the additional phase shift induced by dark photon and consequently the generation of birefringence effect, there is also a rotation of the polarization plane of light. This is due to the fact that photons convert into axions and dark photons in external magnetic fields and thus the magnitude of the mixed state A × (k, t) is reduced by a factor in the weak mixing case and for A ′ × (k, 0) = 0 = a(k, 0). From (14) we can easily recover the contribution of the axion field for M ′ aγ → 0. As a consequence of (14), the polarization plane of light for initially |A + (k, 0)| = |A × (k, 0)| is rotated by a quantity (to second order in θ) α a+γ ′ (t) = θ 2 sin 2 (∆M β t/2) + O(θ 4 ).
In the case when θ = π/4 or maximal mixing or resonant mixing with respect to the axion field, the first term in A × (k, t) in (8) has no imaginary part and therefore there is not induced additional birefringence effect. However, the first term in A × (k, t) has a real part which deviates from unity and it has not dark photon contribution but only axion contribution.
In the case of weak and maximal mixing with respect to φ, namely φ ≪ 1 and φ = π/4, the first term in A × (k, t) in (8) has in both cases real and imaginary parts. The general expression of the induced phase shift of combined effects of the axion and dark photon on the photon state A × (k, t), for arbitrary φ is given by where obviously λ 2 and λ 5 depend on the strength of φ and consequently also Φ a+γ ′ (t) depends on φ.
In case when φ ≪ 1 or φ = π/4, expression (16) is different from zero only when θ = π/4. In such cases birefringence effects would manifest. Similar considerations can be done for the rotation of the polarization plane in complete analogy to the case of weak and maximal mixing with respect to θ. The mixing of the dark photon with the axion and photon plays a crucial role on the transition probability rate between the states A × , A ′ × and a. As one might expect, such a mixing for example changes the transition probability between an initial photon state A × (k, 0) to an axion final state a(k, t) and vice-versa. The expression for such a transition is given by where λ ij ≡ λ i − λ j are the differences between the i-th and j-th eigenvalues of M and we used the expression of a(k, t) in (8) for A ′ × (k, 0) = 0 = a(k, 0). We may observe from (17) that the presence of the dark photon in the mixing problem, manifest through the terms cos 2 (φ) and λ 25 . In the limit when M ′ aγ → 0 we recover the usual photon-axion transition probability rate.
Other interesting effects which manifest in the mixing process are that now it is possible for photons to oscillate in dark photons and also the transition of dark photons into axions. In these cases the transition probability (or oscillation probability) rates are given by where we assumed that a(k, 0) = 0 in the calculation of P γ ′ γ and A × (k, 0) = 0 in the calculation of P aγ ′ . We may observe from (18) that in the limit when M aγ ′ → 0, P γ ′ γ → 0 and P aγ ′ → 0. Our discussion on the birefringence and dichroism effects and transition probability rates has been done in the case when the mixing matrix M is time independent. In the case when M is time dependent we can derive analogous expressions for phase shifts and transition probability rates. However, since we found expressions for the fields in (12) up to second order in the perturbation matrix M 1 , not all effects that we studied in the case of time independent M manifest to second order. For example already to second order we can calculate the transition probability rates from (12) and already to first order shows up the acquired phase shift of A × (t) from the axion field. However, neither the phase shift induced by the dark photon nor the contribution of the dark photon to the transition probability rate P γa (t) manifest up to second order in the perturbation theory. The latter quantity manifest starting from the third order of iteration while the former manifest starting at the fourth order. For example the transition probability rate of photon-axion up to the third order is given by where again we used A ′ × (0) = 0 = a(0). The first term in (20) is the contribution of only axion field to P γa , the second term gives the contribution of the axion and dark photon to transition probability while the last term is the contribution of the axion field at third order which is a negligible quantity with respect to the first and second term. The second term in (20) reflects the fact that contribution of the dark photon to P γa is in general negligible with respect to the axion contribution at first order. It is worth to recall that the perturbative approach used here is valid only when the magnitude of off-diagonal terms in M (t) are much smaller than the magnitude of diagonal terms. For more general cases of relative strength between the elements of M , our perturbative approach is not valid anymore.
The model of photon-axion-dark photon mixing which we discussed above may have several applications that we suggest in what follows. However, before applying it to a concrete example, it is important to recall that in order to have photon-dark photon mixing there must necessarily exist an external dark magnetic field in addition to the ordinary one. Obviously, laboratory experiments looking for axions and dark photons are ruled out since one can generate in the laboratory an ordinary magnetic field but not a dark magnetic field. This fundamental observations tells us that we must look for this effect elsewhere possibly in astrophysical or cosmological situations where both ordinary and dark magnetic fields coexist.
One possibility where to apply our model exist in cosmology and more precisely in the context of CMB physics. Indeed, since we are assuming the existence of a dark/mirror sector, based on Msymmetry one would expect that both sectors have the same cosmology and same evolution. In order to avoid any conflict with the Bing Bang Nucleosynthesis (BBN), the two sectors must have different initial conditions and different temperatures T = T ′ at the reheating epoch [9]. The BBN bound on the number of effective neutrino species puts very stringent limits on the temperature of the dark CMB which must be T ′ < 0.64 T [10] where T is the temperature of ordinary CMB.
Since ordinary and dark CMB's evolve with different temperatures and because they do not come in thermal equilibrium with each other, based on M-symmetry also the dark CMB experiences a decoupling epoch which happens to be slightly earlier than ordinary decoupling epoch. Therefore based on Msymmetry there must exist also a large scale dark magnetic field complementary to ordinary large scale magnetic field. Consequently we would have two CMB's, one ordinary and one dark where each of them interact with their respective large scale magnetic field. Based on this assertion, we may use our earlier formalism of photon-axion-dark photon mixing in order to study the effects which the dark CMB has on ordinary CMB.
One possible effect that the dark CMB has on ordinary CMB due to photo-axion-dark photon mixing, is that it generates for example temperature anisotropy. In order to see this, first we must recall that temperature anisotropy of ordinary CMB is essentially generated at decoupling time or afterwards due to several process. Here we consider the case that the CMB acquires temperature anisotropy starting at decoupling epoch and continues evolving until at the present epoch. Therefore, we must study the evolution of temperature anisotropy in this time interval and consequently all quantities of interest would evolve and time. So, based on this observation we must use our time dependent formalism developed earlier.
Let us consider two observation directions, one parallel to external magnetic fields and one perpendicular to external fields. In the direction perpendicular to external magnetic fields the intensity of the CMB would be I ⊥ γ (t) ∝ |A + (t)| 2 + |A × (t)| 2 where A + (t) and A × (t) are given by the perturbative solutions up to the second order in (12). In the direction parallel to external magnetic fields, there is not generation of axions or dark photons but only the mixing of the states A + and A × due to the Faraday effect. In this work we did not consider it because we studied only the case when external magnetic fields are perpendicular with respect to the photon/dark photon direction of propagation while the Faraday effect is a longitudinal effect. However, it is known that the Faraday effect does not change the intensity of light but only its polarization state. Consequently, the intensity of light parallel to external magnetic fields would be I || γ (t) ∝ |A + (0)| 2 + |A × (0)| 2 . The proportionality factor in I ⊥ γ and I || γ in general is a factor which takes into account the dilution of particle number density in an expanding universe and it cancels out in the final result. The intensity of light in the parallel direction thus is equal to the intensity of light in an unperturbed universe. Now considering the CMB in thermal state at ordinary decoupling time where its intensity is given by the black body formula, one can derive the following relation between the differential intensity and temperature changes δI γ (t 0 )/I γ (t 0 ) = [xe x /(e x − 1)]δT /T 0 where T 0 is the present day temperature of the CMB and x = 2πν 0 /T 0 with ν 0 being the CMB frequency at present epoch. In our case we have that for two directions of observations separated by 90 • in the sky. By using the perturbative expressions for A + (t) and A × (t) in (12) and expression (21) we get the following relation where we took a(0) = 0 and defined At this point we make the assumption that at ordinary decoupling time the dark CMB is roughly speaking in thermal state. Moreover by averaging over the polarization states at initial time t in = 0, which we choose to coincide with ordinary decoupling time we get, |A × (0) By collecting all terms and using (22), we get the following expression which relates the temperature anisotropy with photon and dark photon intensities We must stress that expression (23) has been derived by using expansion of fields up to the second order in perturbation theory where M 1 (t) is much smaller than M 0 (t) and by taking the temperature difference between the direction parallel and perpendicular with respect to external magnetic fields. The first term on the left hand side of (23) reflects the change in the photon intensity due to photon-axion mixing while the second term is the contribution of conversion of dark photons into photons. In general, the relative strength of the two terms would depend on many parameters and in most cases the axion contribution is the dominant term. Concluding, in this work we proposed and studied the mechanism of photo-axion-dark photon mixing in external ordinary and dark magnetic fields. As a consequence of this mixing, dark photons can interact with ordinary photons via the same axion field. Our axion behaves like a WW axion in the dark sector while it interacts with ordinary sector as DFSZ-like axion. Then we solved equations of motion for time depended and time independent mixing matrix where in the former case perturbative solutions have been found while in the latter case exact solutions have been found. By using standard calculation technique, we derived user friendly quantities such as phase shifts, angle of rotation of the polarization plane of light and transition probabilities. The derived results can be applied in cases when photons/dark photons propagate through a magnetized media which might be constant in time or it may evolve in time.
With the introduction of the dark photon in the mixing problem, the usual expressions of photon-axion transition probability rates, phase shifts etc., are modified. This could have significant impact in those situations where is present an external dark magnetic field and it is required to know the magnitude of these quantities in order to confront with observations.
Our results have been derived by neglecting the weak gravitational interaction between the two sectors and considered their interaction via the same axion. In our model photons and dark photons interact solely through the axion field. In principle, one could include in the interaction Lagrangian a kinetic mixing between photons and dark photons, namely L I ∝ ǫF µν F ′µν , which is not forbidden by M-symmetry. The inclusion of such term can be easily accommodated in our formalism but is beyond the main purpose of this paper and will be studied elsewhere.
In order for the photo-axion-dark photon mechanism to work there must coexist in the same place both ordinary and dark magnetic fields. The only possibility to apply it happens to be in astrophysical and cosmological situations. In this work we applied our mechanisms in the context of CMB physics and showed that it can generate CMB temperature anisotropy at ordinary post decoupling epoch. The same mechanism would also generate spectral distortion and polarization of the CMB which in this work we did not study. In an astrophysical context, our model could be used in order to calculate the generated flux of photons in ordinary and dark magnetic fields by dark stars and/or other dark objects which emit dark photons, where the generated flux might contribute to well know galactic or extragalactic backgrounds.