Abstract
The quanta of scalar fields like the dilaton (\(\phi \)) of scale symmetry origin and those of pseudoscalar fields like the axion (\(\phi '\)) of Peccei–Quinn symmetry origin couple to di-photons through dimension-5 operators. In a magnetized medium (MM), they in principle can interact with the two transverse (\(A_{\parallel ,\perp }\)) and one longitudinal (\(A_{L}\)) degree of freedom of photons (\(\gamma \)) as long as the total spin is conserved. However, of \(\phi \) and \(\phi '\), only one interacts with \(A_{L}\). We found that the ambient external magnetic field B and media break the intrinsic Lorentz symmetry of the system affecting the dispersion relation of the propagating modes. The boost and the rotational symmetry along and around B are however the ones that are preserved. Invoking C, P, and T symmetries, we analyze the mixing dynamics of \(\phi \gamma \) and \(\phi '\gamma \) systems and the structural difference in their mixing pattern. It is noted that while the \(\phi \gamma \) mixing matrix is \(3\times 3\), the \(\phi ^{\prime }\gamma \) is governed by a \(4\times 4\) mixing matrix. Using the exact solutions of both systems in MM, we estimate the strength of the electromagnetic (EM) signals available due to these interactions, which are found to be different in strength. We conclude by commenting on (a) the possibility of detecting this difference in polarimetric observables of the EM signal, (b) the implications of these different mixing patterns with respect to the minimum detectable signal for astrophysical observations, and (c) the variation in the energy of the dispersed photons of different polarization with the variation in B.
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1 Introduction
Scalar or pseudoscalar bosons like the dilaton (\(\phi (x)\)) [1, 2] or axion (\(\phi '(x)\)) [3,4,5,6,7,8,9,10] are postulated to arise out of symmetry breaking through quantum effects in field theory. In addition, unified theories like string theory [11,12,13,14,15,16] predict the existence of similar particles that may appear through moduli compactification. Of the two bosons mentioned above, the dilaton is supposed to restore the scaling symmetry and the quantum chromodynamics (QCD) axion is supposed to cure the \(U_{A}\)(1) anomaly, whereas the string theory axion is supposed to break the shift symmetry of the theory. Both of these particles, though still elusive to experimental verification, nonetheless offer remarkable solutions to the outstanding problems of physics. This makes any effort towards their verification highly meaningful. As such, the detection of EM signals of their presence has become a coveted goal among scientists worldwide.
In terms of electromagnetic field strength tensors \(f_{\mu \nu }\) (defined in terms of gauge potentials \(A_{\mu }\) as \(f_{\mu \nu }(k)= k_{\mu }A_{\nu }-k_{\nu }A_{\mu }\)) and its dual \({\tilde{f}}^{\mu \nu }= \frac{1}{2}\epsilon ^{\lambda \sigma \mu \nu }f_{\lambda \sigma }\), the interaction of photons \((\gamma )\) with \(\phi \) or \(\phi '\) is governed by a dimension-five anomalous interaction Lagrangian of the form
for the pseudoscalar–photon interaction, where \(g_{\phi \gamma \gamma }\) and \(g_{\phi '\gamma \gamma }\) are the anomaly-induced coupling constants between \(\phi \) or \(\phi '\) with photons. We denote \(\phi \) and \(\phi '\) collectively with a subscript i as \(\phi _i\) (such that for scalars, \(\phi _{i}= \phi \), and for pseudoscalars, \(\phi _{i}= \phi '\)). As a result of this anomalous coupling, the lifetime of these particles, having mass \(m_{\phi _i}\), against decay into two photons, is as follows:
As an aside, we note that except for the axions of quantum chromodynamics (QCD) origin, the mass \(m_{\phi '}\) and the coupling constant \(g_{\phi ' \gamma \gamma }\) for axions of other theories are not related to each other. The same is true for dilatons. In this work, they (scalar and pseudoscalar) are collectively referred to as axion-like particles (ALP).
Ever since the cosmological production mechanisms of these particles were realized [17], numerical estimates of the parameters (\(m_{\phi _i}\) and \(g_{\phi _i \gamma \gamma }\)) began being constrained from various astronomical observational facts. One such fact involves the number density of photons \(n_{\gamma }\) at different epochs of cosmological evolution. For instance, cosmological Big Bang nucleosynthesis puts a very restrictive bound on the number density of photons \(n_{\gamma }\) during primordial deuterium synthesis [18,19,20]. Therefore, processes that lead to a change in \(n_{\gamma }\), such as \(\phi _{i} \rightarrow \gamma \gamma \), must be fine-tuned to fulfill this criterion. This in turn imposes a bound on parameters \(m_{\phi _i}\) and \(g_{\phi _i \gamma \gamma }\) so that there is no change in \(n_{\gamma }\) due to (pseudo)scalar decay into di-photons during cosmic deuterium synthesis.
The allowed regions of the parameters \(g_{\phi _i \gamma \gamma }\) and \(m_{\phi _i}\) obtained from various laboratory-based experiments, such as the one reported in [21], as well as astrophysical observations [22] lead us to the estimates of lifetime \(\tau _{i}\) of the (pseudo)scalars which can be compared with the age of the universe. These estimates do not disturb the magnitude of baryon-to-photon number density ratio \(\frac{n_{B}}{n_{\gamma }}\sim 10^{-11}\) at the current epoch.
Coming back to the main objective of this investigation, we recall that the identification of ALP like dark matter candidates would provide an elegant solution to dual problems: one in the realm of cosmology and the other in the area of physics beyond the Standard Model of Particle Physics.
However, this task of ALP identification becomes difficult because of the twofold objective associated with it, namely (i) identification of their type, i.e., ALP or not, and (ii) identification of their nature, i.e., scalar or pseudoscalar. There are two possible ways to search for them: (a) collider-based experiments realizable in the laboratory or (b) spectro-polarimetry-based experiments/events realizable in terrestrial laboratory or astrophysical situations. The collider-based search is difficult because the colliding beam must have high luminosity in order to obtain a high signal-to-noise ratio. However, creating a high-luminosity beam in itself is costly and technically challenging. Moreover, in some channels it becomes even more difficult if the final-state decay products involve scalar or pseudoscalar interactions in the initial or intermediate states where they have identical interaction parameters. For instance, the lifetime observations (estimates) in the di-photon or di-lepton decay channels for \(\phi \) and \(\phi '\) for nearly the same numerical strength and size of \(m_{\phi _i}\) and \(g_{{\phi _i}\gamma \gamma }\) would offer very limited scope to distinguish one from the other.
As the discussion of collider-based search is beyond the scope of this work, we will focus on the second method. It was introduced in the seminal work of [23] and has been followed by many others in the decades since then.
Depending on the (astrophysical) objects, i.e., their environments and physical situations, detecting and distinguishing signals due to \(\phi \) from \(\phi '\) using spectro-polarimetric techniques may be as difficult as with their collider-based counterparts. Nevertheless, it can be achieved with less effort if we appropriate the symmetry properties of the dark matter candidates (i.e., \(\phi \) and \(\phi '\)) that we are looking for and the background media under consideration, before analyzing the spectro-polarimetric signals from their collective interactions. However, to fully appreciate such an approach and understand its distinction (from that of the available ones), a general symmetry-based analysis of the system is necessary. In the next few sections of this work, we present a complementary route for discussing some of these issues present in the towering volume of G.G. Raffelt [24].
1.1 Motivation: symmetries and their consequences
We start by recollecting the obvious fact that any Lagrangian transforms like a scalar under Lorentz transformation. Hence, the sum of the spin angular momentum assignments of the fields (background and dynamical) constituting each of their terms must add up to zero. Therefore, the Lagrangian for scalar or pseudoscalar–photon interacting systems in an ambient background EM field (\(\bar{F}^{\mu \nu }\)) should also conform with this observation. That is, the (pseudo)scalar–photon interaction Lagrangian (IL) in an ambient \(\bar{F}_{\mu \nu }\) field should also respect this. Stated differently, the \(\phi \gamma \) IL and \(\phi '\gamma \) IL should respect this when subjected to the replacement \(f_{\mu \nu } \rightarrow \bar{F}_{\mu \nu } + f_{\mu \nu }\) in the Lagrangians given in Eq. (1.1), giving the ILs \(g_{\phi \gamma \gamma }\phi {\bar{F}}_{\mu \nu }f^{\mu \nu }\) and \(g_{\phi '\gamma \gamma }\phi ' \tilde{{\bar{F}}}_{\mu \nu }f^{\mu \nu }\).
The characteristics of the scalar–photon (\(\phi \gamma \)) or pseudoscalar–photon (\(\phi '\gamma \)) IL in any ambient background EM field depends on the nature of the background field itself, that is, whether it is an electric or magnetic type, and the discrete symmetries of the dynamical fields, i.e., those of the photons and the scalars/pseudoscalars involved.
Furthermore, not all the terms present in a Lagrangian endowed with gauge fields contribute to the equations of motion, unless gauge-fixed, due to the associated gauge ambiguities. Expressing the degrees of freedom of a gauge-fixed Lagrangian in terms of a set of orthonormal four-vectors representing the modes or degrees of freedom, one obtains a transformed Lagrangian, such that each term of the field equations obtained from this transformed Lagrangian has a set of definite discrete symmetry assignments. That is, each term in every equation of motion must transform identically either individually—under charge (C) conjugation, parity (P), and time(T) reversal transformations—or under the action of a set of their combinations. If the Lagrangian has some additional unitary internal symmetries associated with it, then each term in the field equations should also transform identically under these unitary symmetry transformations, thus preserving the covariance of the field equations.
This results in the following consequences: In a magnetized vacuum (\(\textbf{MV}\)) (i.e., in an ambient magnetic field B, such that only \(B=B_{z}= \bar{F}^{12}\ne 0\)), only one degree of freedom describing a particular polarization state of the photons out of the two possible states (polarized transversely, along and orthogonal to the ambient (external) magnetic field B) interacts with the scalars or pseudoscalars at the \(\phi _{i}\gamma \) interaction vertex; the other one does not. This interaction depends on the CP symmetry of the polarization state of the photon and that of the interacting fields \(\phi \) or \(\phi '\). The dynamics of the two states or the transverse degrees of freedom of the photons can be described in two ways: one in terms of gauge fields, and the other in terms of gauge-invariant form factors \(\Psi \) and \(\tilde{\Psi }\), defined as \(\Psi = \bar{F}^{\mu \nu }f_{\mu \nu }\) and \(\tilde{\Psi }=\tilde{\bar{F}}^{\mu \nu }f_{\mu \nu }\), in the notations of [25]. In this notation, the degree of freedom of the photons having a plane of polarization (POP) orthogonal to the ambient magnetic field B is denoted by \(\Psi \). Similarly, for the photons having POP along B, this will be described by \(\tilde{\Psi }\).
Under the operation of charge conjugation C and parity transformation P, the tensors \(\bar{F}^{\mu \nu }\) and \(f^{\mu \nu }\) both turn out to be odd and even (see Sect. 2 for more details)[30]. Hence, the combination \(\bar{F}^{\mu \nu }f_{\mu \nu }\) is CP even, and the other one, i.e. \(\tilde{\bar{F}}^{\mu \nu }f_{\mu \nu }\), is PT even and therefore CP odd [23, 31]. Since the scalar \(\phi (x)\) is always CP even, then at the level of the equation of motion (EOM) in a \(\textbf{MV}\), for the \(\phi \gamma \) interaction, the CP even \(\Psi \) couples only to CP even \(\phi (x)\), and the remaining CP odd form factor \(\tilde{\Psi }\) propagates freely. In a similar situation, when the interaction of \(\phi '(x)\) with \(\gamma \) is considered, with the former being a pseudoscalar, i.e., P odd but C even, CP odd will couple only to the CP odd form factor \(\tilde{\Psi }\) of the photon. The CP even \(\Psi \) would propagate freely. This is just the reverse of what happens for the \(\phi \gamma \) interaction in \(\textbf{MV}\). That is, in \(\textbf{MV}\), the role of \(\Psi \) in the \(\phi '\gamma \) interaction is similar to that of \(\tilde{\Psi } \) in the \(\phi \gamma \) interaction, and vice versa. Hence, one can conclude that, in a \(\textbf{MV}\), as the form of the interaction changes from \(\phi \gamma \rightarrow \phi '\gamma \), the roles of \(\Psi \) and \(\tilde{\Psi } \) are interchanged. Hence, the mixing dynamics of both systems \(\phi \gamma \) or \(\phi '\gamma \) are governed by a 2\(\times \)2 mixing matrix. This was originally pointed out by Raffelt and Stodolosky in their seminar paper [23].
The propagation modes of the \(\phi \gamma \) system in \(\textbf{MV}\) are obtained by diagonalizing the mixing matrix. The CP even propagating modes for this system in the massless limit are \(\left( \Psi \pm \Phi \right) \), where \(\Phi = \left( |k_{\perp }|B\phi \right) \) when \(k_{\perp } = K\sin \Theta \) and \(\Theta \) is the polar angle between propagation vector (\(\vec {k}\)) and the magnetic field (\(\vec {B}\)). The symbol K stands for the magnitude of the propagation three-vector (|k|). Additionally, \(k_{\perp }\) has an azimuthal symmetry \(k_{\perp } = \left( k^{2}_{x} + k_{y}^{2}\right) ^{\frac{1}{2}}\) (where the z-direction is chosen as that along the direction of the magnetic field). Their corresponding dispersion relations (for four-momentum \(k_{\mu }\) when \(k^{2} = k_{\mu }k^{\mu }\)) satisfy \(k^2 \pm g_{\phi \gamma \gamma }|k_{\perp }|B=0\), [25].
Remarkably, the propagating modes for the \(\phi '\gamma \) system in a similar situation are CP odd and are given by \(\left( \tilde{\Psi } \pm \Phi \right) \), but their dispersion relation remains the same as for the \(\phi \gamma \) system; that is, \(k^2 \pm g_{\phi '\gamma \gamma }|k_{\perp }|B=0\). It can be seen that the dispersion relations (DRs) for both cases are indeed invariant under boost and rotation about the direction of the magnetic field. This happens because the presence of the external field breaks the Lorentz symmetry of the system. It was argued in Sect. 2 that the action integral (\(S=\int L \textrm{d}^4x\)) remains invariant only under the action of the generators of boost and rotation along and around the third direction of the Lorentz group, so as to keep the action (S) invariant; all other generators for the remaining Lorentz symmetry transformations must vanish. The dispersion relation mentioned above is a manifestation of this. As a result, a \(\textbf{MV}\) is found to be optically active and dichroic in the presence of (pseudo)scalar–photon interactions in a magnetic field.Footnote 1
Therefore, for near-degenerate magnitudes of the coupling constants and masses of the candidate particles, properly distinguishing one from the other may become an arduous task for an unfavorably oriented magnetic field or even for some astrophysical situations. One possible way to avoid this impasse and refine the study of mixing physics is through incorporation of matter effects [35,36,37,38,39,40,41], by including a term like \(A^{\mu }(k)\Pi _{\mu \nu }(k,\mu , T)A^{\nu }(k)\) in the effective Lagrangian (\(L_{\textrm{eff}}\)) of the system, with \(\Pi _{\mu \nu }(k,\mu , T)\) being the in-medium photon self-energy tensor that depends on temperature (T), chemical potential (\(\mu \)), and four-momentum (\(k^{\mu }\)). We recall that chemical potential (\(\mu \)) brings the effect of C symmetry into the system.
The introduction of matter effects brings additional subtlety in the mixing dynamics by introducing additional spin-zero degrees of freedom, called longitudinal degrees of freedom (\(A_{L}\)), for photons [82]. Both scalars and pseudoscalars having spin zero can now mix with \(A_{L}\) because of spin conservation, thus obtaining a \(4\times 4\) mixing matrix and thereby spoiling the cure that we were looking for.
However, a PT symmetry analysis of the terms of the field equations for both systems expressed in terms of the photon form factors as introduced in [38] shows that because of PT symmetry, the pseudoscalar–photon mixing matrix is \(3\times 3\) and the scalar–photon mixing matrix is \(2\times 2\). This, however, leaves unanswered the question of whether all the elements of the \(3\times 3\) pseudoscalar mixing matrix are finite, or just some of them. The answer is no! Not all the elements are nonzero. This can be understood from the fact that all the nonzero elements that are present in the mixing matrix are due to the interaction of the one transverse degree of the photon with the pseudoscalar. This is because, since the form factors were expressed in an orthogonal basis, with a medium undergoing only EM interaction, the longitudinal one would not mix with the transverse one, and the transverse ones would not mix with each other. An interesting way to understand this in terms of the quantum fluctuations of the EM degrees of freedom can be found in reference [24]. However, we provide a complementary point of view in Sect. 2.
The refined study, described previously, accounts only for mixing of two degrees of freedom for scalar–photon and three degrees of freedom for pseudoscalar–photon systems, leaving two degrees of freedom for scalar–photon and one degree of freedom for pseudoscalar–photon system free. In other words, although the mixing patterns for both have changed, we have yet to achieve maximum mixing for either. This issue can be further cured by incorporation of the parity-violating part of the photon self-energy tensor in the system, following the steps mentioned previously, the necessary details of which are similarly relegated to the next section.
The investigations discussed in this article are presented as follows: In Sect. 2, we begin with the introduction of the tree-level Lagrangian for scalar–photon (\(\phi \gamma \)) and pseudoscalar–photon (\(\phi ^{\prime }\gamma \)) interactions in the MV environment. Subsequently, the symmetries (\(\textbf{C, P, T}\) and Lorentz symmetries) and their consequences as regards the mixing phenomena of these fields with photons are discussed. In Sect. 3, a brief discussion on the solutions of the EOM is presented in a MV. The expressions of the polarimetric observables using these solutions are then provided in Sect. 4. Section 5 is dedicated to explaining the structure of the interaction Lagrangian by introducing the photon self-energy tensor corresponding to matter and magnetized matter backgrounds in the context of the \(\phi _{i}\gamma \) interactions. In Sect. 6, we report on the EOM and the exact solutions obtained for the \(\phi \gamma \) interaction in a magnetized medium using diagonalization of the \(3\times 3\) mixing matrix. In the following subsections, the perturbative and exact solutions of the EOM for the \(\phi ^{\prime }\gamma \) system are reported. They are obtained using a unitary transformation by \(4\times 4\) unitary matrices evaluated for that purpose. Section 7 is dedicated to the mixing analysis of the available degrees of freedom for \(\phi _{i}\gamma \) interactions using discrete symmetry arguments, and in Sect. 8 we discuss the astrophysical applications of such interaction using a model of eclipsing binary stars. In Sect. 9, we provide the results of our numerical investigations on \(\phi _{i}\gamma \) interactions in a magnetized medium. Then, in Sect. 10, we provide the conclusions and an overall discussion of the ways of differentiating scalar–photon interactions from pseudoscalar–photon interactions in a magnetized medium. Finally, we provide Appendices A, B, and C for the details of the steps used to arrive at the results presented in this article. Appendix D is introduced to relate the EOMs and the form of the mixing matrix presented in the book by Prof. G. G. Raffelt [24] to similar entities used in this work. Lastly, the size of the contributions to the effective Lagrangian arising from the different form factors associated with the field-independent and field-dependent tensorial basis is explained in Sect. 5.2 and Appendix E.
2 Interaction dynamics
2.1 Tree-level Lagrangian
The non-minimally coupled \(\phi _{i}\gamma \gamma \) interacting Lagrangian (as stated earlier, \(\phi _{i}\) stands for either a scalar (\(\phi \)) or a pseudoscalar (\(\phi '\)) field) in an ambient magnetic field (B) can be expressed as a sum over three parts:
where \( L_{\textrm{free},\phi _{i}} \)in Eq. (2.1) stands for the free-field Lagrangian, given by
where \(L^{B}_{\textrm{int},\phi _{i}}\) stands for the interactive part due to the ambient magnetic field and is given by
and lastly, the Lagrangian for the \(\phi _{i}\gamma \gamma \) interactive part \( L_{\textrm{int},\phi _{i}} \)—the last term in the equation—is given by
2.2 Symmetries
2.2.1 Discrete symmetries
The EOMs for the (pseudo)scalar–photon system in \(\textbf{MV}\), following from the Lagrangian (2.1), can be cast in matrix form in terms of the gauge-invariant variables
as
Here, \(\Phi = |k_{\perp }| B\phi \), \(\Phi ^{\prime }= |k_{\perp }|B\phi ^{\prime }\) and \(B_{T} = Bsin\Theta \). The matrix Eq. (2.6) stands for the EOM for the photon–pseudoscalar system, and the one given by Eq. (2.7) stands for the EOM of the photon–scalar system. Under \(\textbf{CP}\) transformation, the bilinear field variables \(\Psi \) and \({\tilde{\Psi }} \) and the fields \(\Phi \) and \(\Phi ^{\prime }\) transform as follows:
From the EOMs (2.6) and (2.7), it is easy to see that \(\textbf{CP}\) even \(\Phi \) (or \(\phi \)) couples to \(\textbf{CP}\) even \({\bar{F}_{\mu \nu }f^{\mu \nu }}\), and CP odd \(\Phi ^{\prime }\) (or \(\phi ^{\prime }\)) couples to CP odd \( \tilde{\bar{F}}_{\mu \nu }f^{\mu \nu }\).
Alternatively, the dynamics of the discrete symmetries associated with the various degrees of freedom of the photons can be followed by analyzing the discrete symmetries of each term forming the interaction vertex, upon expressing the gauge field \(A_{\mu }(k)\) in an orthogonal four-vector basis \( b^{(1)}_{\alpha }\), \(I_{\alpha }\), \(\tilde{u}_{\alpha }\), \(k^{\mu }\) along with the associated electromagnetic form factors \(A_{\parallel }, A_{\perp }, A_{L}\), and \(A_{gf}\). The gauge field—upon expansion in this basis, following [38]—assumes the form
Here, \(N_i\)s are the normalization constants and \(A_i\)s are the form factors already introduced above. We next set \(A_{gf}=0\) to comply with the Lorentz gauge condition \(k_{\mu }. A^{\mu }(k)= 0\). The longitudinal component of the photon (\(A^{\mu }_{L}\)) remains finite in a medium. In our case, this is also finite following [26, 28]. The basis vectors introduced in Eq. (2.10) are defined in terms of the medium four-velocity vector \(u^{\mu }~ =~(1,0,0,0)\), photon four-momentum \(k^{\mu }\), and the nonzero external field tensor in the one–two direction \(\bar{F}^{\mu \nu }\) as
The normalization constants \(N_{1}\) \(N_{2}\) and \(N_{L}\) in Eq. (2.11) are given by
where \( k_{\perp } = (k^2_1 + k^2_2)^{\frac{1}{2}}\). By expressing \(k_1 = k_{\perp } \cos ({{\varvec{\Phi }}})\) and \(k_2 = k_{\perp } \sin (\mathbf{\Phi })\) (where \({{\varvec{\Phi }}}\) is the azimuthal angle), the azimuthal symmetry of \(k_{\perp }\) can be established.
Now, to find the directions associated with the form factors, we note that the four-vector \(B^{\mu }= \epsilon ^{\mu \nu \lambda \rho }u_{\nu }\bar{F}_{\lambda \rho }\) has a nonzero component in only one spatial direction, which we identify as the third one. Correspondingly, the four-vector that is orthogonal to \(B^{\mu }\), \(k^{\mu }\) and \(u^{\mu }\) simultaneously turns out to be \({\tilde{B}}^{\mu } = \frac{1}{2} \epsilon ^{\mu \tau \kappa \sigma }k_{\tau }u_{\kappa }B_{\sigma }\). With this form in hand, it is straightforward to verify that the four-vector \(b^{(1)}_{\mu }\) is orthogonal and \(I^{\mu }\) and \( \tilde{u}^{\mu }\) are parallel to \({\tilde{B}}^{\mu } \). In other words, the four vectors \(k^{\mu }\), \(I^{\mu }\), \(\tilde{u}^{\mu }\), and \({\tilde{B}}^{\mu }\) form a set of four orthogonal vectors in four dimensions.
Coming to the issue of dynamics, we can find how individual EM form factors of the photons (given by Eqs. (2.8) and (2.9)) couple to \(\phi \) or \(\phi ^{\prime }\) in an ambient external field by expressing the gauge potential \(A_{\mu }(k)\) in the form provided in (2.10) and substituting this in the expression for the field strength tensor \(f_{\mu \nu }\) appearing in both the (field strength) bilinears \(\tilde{\bar{F}}_{\mu \nu }f^{\mu \nu }\) and \({\bar{F}_{\mu \nu }f^{\mu \nu }}\).
Upon performing this we obtain
Using Eq. (2.13), in the pseudoscalar–photon interaction Lagrangian, \(L^{B}_{\textrm{int},\phi '} = g_{\phi '\gamma \gamma } \phi ' \tilde{\bar{F}}^{\mu \nu }f_{\mu \nu }\), we obtain
Equation (2.15) shows that, in principle, \(A_L \) and \(A_{\perp }\) can mix with \(\phi '\) in a background magnetic field. But since \(A_L\) comes into existence only in the presence of a medium, in magnetized vacuum, \(\phi '\) will mix only with \(A_{\perp }\), while \(A_{\parallel }\) will remain free. A similar exercise for the \(\phi \gamma \) system with the interaction Lagrangian
yields
implying the interaction of \( A_{\parallel }\) with \(\phi \) only in the ambient magnetic field. In other words, mixing is possible only between \(A_{\parallel }\) and \(\phi \), with \(A_{\perp }\) remaining free. Hence, the mixing matrix for both systems remains \(2\times 2\) in \(\textbf{MV}\). Furthermore, the mixing matrix for the \(\phi \gamma \) system, even in a medium, would remain \(2\times 2\) because of nonexistent \(A_{L}-\phi \) interaction at the Lagrangian level, as was verified in Eq. (2.17). However, since the form factor \(A_{L}\) for the longitudinal degree of freedom is physically realized only in a medium, the mixing matrix for the \(\phi '\gamma \) system in a medium is \(3\times 3\). This happens because \(\phi '\) couples directly to the form factors \(A_{\perp }\) and \(A_L\) at the level of the interaction Lagrangian (Eq. (2.15)), so at the level of the EOM, they all would drive each other through their coupling with \(\phi \). This will become clearer when we discuss the consequences of magnetized matter effects.
2.2.2 Lorentz symmetry
In the Lagrangian \(L_{\phi _{i}}\), except for \(L^{B}_{\textrm{int},\phi _{i}}\), all terms respect Lorentz and gauge symmetry and remain invariant under charge conjugation (C) along with parity (P) and time-inversion (T) transformations. Re-normalizability of the theory, however, is compromised by the presence of dim-5 operators \(L^{B}_{\textrm{int}, \phi _{i}} \), and \( L_{\textrm{int},\phi _{i}}\).
The response of this system to any symmetry transformation can be investigated by subjecting the dynamical fields constituting the action
to the intended symmetry transformations. For instance, under the action of an infinitesimal Lorentz transformation (\(\Lambda \)) around the identity ( I) given by
the dynamical field strength tensor \(f_{\mu \nu }\) would change in the following way:
and the background electromagnetic field strength tensor \({\mathcal {F }}^{\mu \nu } \) would be held constant. It can be further shown that under an infinitesimal Lorentz transformation given by (2.19) in a background EM field (such that \( \bar{F}^{21} =\tilde{\bar{F}}^{03} \ne 0\)), the change in the action (2.18) is as follows:
Having the action (given by Eq. (2.18)) invariant under transformation (2.19), i.e., \(\delta S =0\), is possible if two out of the six components of the antisymmetric parameters \(\omega _{\mu \nu }\) (only \(\omega _{03}\) and \(\omega _{12}\)) are nonzero [43], while the other four are identically zero, implying that the action given by Eq. (2.18) remains invariant only under the action of two generators of the Lorentz group: (i) the generators of boost (\(\mathbf{K_3}\)) and (ii) rotation (\(\mathbf{J_3}\)) along the third direction. These two generators of the Lorentz symmetry are unbroken, while the other four generators are broken. Therefore, the only space-time symmetries remaining preserved are the boost along the third (i.e., z)-direction and rotation in the one–two (xy) plane. The remainder of the symmetries are all broken. Hence, the EOMs and the dispersion relations following therefrom should respect these conditions. Thus, it is not surprising that the dispersion relations (obtained from the mixing matrices appearing in Eqs. (2.6) and (2.7)) corresponding to the photons having a plane of polarization (POP) parallel to the magnetic field, when expressed in terms of the four-vector k and its component orthogonal to B (that is, \(k_{\perp }\)), turn out to be
for massless ALPs or massive ALPs (given in Eq. (3.1)). That is, these relations remain invariant under boost and rotation around the z-axis, a consequence claimed on the basis of reduced Lorentz symmetry.
This results in \(\textbf{MV}\) being birefringent and dichroic. In the remainder of this work, though we may not explicitly discuss the fate of the space-time/Lorentz symmetry of the Lagrangian upon incorporation of other corrections due to magnetized or unmagnetized medium effect, it however can be shown to remain compromised due to the appearance of the velocity four-vector of the center of mass of the material medium \(u^{\mu }\) or the external field B or both, in the description of the effective Lagrangian.
3 Solutions of field equation
3.1 Solutions in magnetized vacuum
Solutions of the field equations require the use of dispersion relations. The dispersion relation for the \(\phi _{i}\gamma \) system having a plane of polarization at rest pointed along the magnetic field is given by
and \(k^{2} = 0\) for the photons polarized orthogonal to the magnetic field.
Since the gauge symmetry remains intact, in the following we will study the dynamics of these systems in terms of gauge-invariant variables noted earlier in [25]. To find the solutions of the variables appearing in either of the matrix equations (2.6) or (2.7) for the \(\phi \gamma \) or \(\phi '\gamma \) system in \(\textbf{MV}\), it is sufficient to solve for the first of the two; the solution for the second can be obtained from the first due to the symmetry \(\Psi \leftrightarrow \tilde{\Psi }\). Hence, we start with the solutions of (2.7). These solutions in [44] in terms of constants \(A_{0} \), \(A_{1} \), and \(A_{2} \) are given by
where \(\omega \), \(\omega _+\), \(\omega _{-}\) appearing in Eq. (3.2) follow from the dispersion relations and are given by
The absence of the (pseudo)scalar at the initial stage (according to the physics of curvature radiation) favors the boundary conditions \(\Phi (0,0)= 0 \), \(\Psi (0,0)=1\), and \(\tilde{\Psi }(0,0)=1\). These boundary conditions yield \(A_{0}=1\), \(A_{1}= \cos (\theta ) \), and \( A_{2} = -\sin (\theta )\). The angle \(\theta \) in Eq. (3.2) is given by \(\theta = \frac{1}{2}\tan ^{-1}(\frac{2g_{\phi \gamma \gamma } B_{T}\omega }{m_{\phi _{i}}^2}) \). With these conditions, the solutions for \(\Psi \) finally turn out to be
where \(a_x^2(t)= 1+ 2\sin ^2\theta \cos ^2\theta \left( \cos (\omega _+-\omega _-\right) t - 1)\).
4 Observables
4.1 Polarimetric observables
The Stokes variables as obtained from the coherency matrix can be expressed in terms of the solutions as
It may be noted that V appearing in Eq. (4.1) is a measure of circular polarization. Other polarimetric observables including the ellipticity angle, polarization angle, and degree of linear polarization follow from the expressions of I, U, Q, and V. The polarization angle (represented by \(\psi _{\phi _{i}}\)) is defined in terms of U and Q as
The ellipticity angle (denoted by \(\chi _{\phi _{i}}\)) is defined as
The polarization fraction \(\Pi ^{P}_{\phi _{i}}\) of the radiation in terms of parameters Q and I is given by
And lastly, the degree of linear polarization (represented as \(P_{L}\)) is given by
4.1.1 Polarimetric observables for magnetized vacuum
Using the solutions given in Eq. (3.2), in Eq. (4.1) one can obtain the expressions for the Stokes parameters for a scalar–photon system, as follows:
Similarly, the parameters for the pseudoscalar–photon system can be obtained by exchanging the solutions of \(\Psi \) with \(\tilde{\Psi }\) and vice versa in Eq. (3.2). As a result, the Stokes parameters for \(\phi \gamma \) (expressed with suffix s) can be related to those of the \(\phi '\gamma \) system in the following form:
As a result, Stokes Q, V, polarization angle \(\psi _{\phi _{i}}\), and ellipticity parameter \(\chi _{\phi _{i}}\) pick up a sign as one moves from the \(\phi \gamma \) to the \(\phi '\gamma \) system, provided other parameters remain the same. Therefore, distinguishing one from the other becomes difficult when \(\chi _{\phi _{i}}\) and \(\psi _{\phi _{i}}\) both tend to zero. Making the degree of linear polarization \({ P_{L\phi _{i}}}\) work to distinguish \(\phi \) from \(\phi '\) is even more difficult, since \(P_{L\phi _{i}}\) remains the same.
5 The effective Lagrangian (\(L_{\textrm{eff}}\))
In this section we start with a description of the effective Lagrangian (\(L_ {\textrm{eff}} \)), incorporating the effects of vacuum with B field (i.e., magnetized vacuum) and magnetized media in terms of the photon polarization tensor \(\Pi _{\mu \nu }(k, eB,T,\mu )\) following the notations of (2.3). In general, it can be a function of magnetic field eB and medium \((T,\mu )\) plus magnetic field eB. However, out of all these, the only pieces that are functions of even eB, even \(\mu \) and odd eB, odd \(\mu \) turn out to be nonvanishing [45,46,47]. In these terms, the corresponding effective Lagrangian for the \(\phi _{i}\gamma \) system is \(L_{\textrm{eff},\phi _{i}} = L_{\textrm{eff},\phi _{i}}^{\mathrm{free-med}} + L_{\textrm{eff},\phi _{i}}^{\textrm{int}}\), where
By virtue of the underlying discrete symmetry (C, P, and T) of quantum electrodynamics, \(\Pi ^{\mathrm{(even)}}_{\mu \nu }(k, \mu , T, eB)\) can be functions of \((\mu )^{(2n)}\) and \((eB)^{(2m)}\), with n and m being integers including zero. And in the case of \(\Pi ^{\mathrm{(odd)}}_{\mu \nu }(k, \mu , T, eB)\), the powers of \(\mu \) and eB would be simultaneously odd, so that their total power when added together would be even. In the absence of a medium, the contributions to the photon polarization tensor in a MV should be nonzero. Next we try to estimate the contribution to the \(L_{\textrm{eff},\phi _{i}}\) from \(\Pi _{\mu \nu }^{\textrm{vac}, eB \mathrm{even/odd}}(k, \mu , T, eB)\) in the following two sections. The purpose of this exercise is to determine the dominant contributing factor to the mixing dynamics. Keeping this in mind, we start by estimating the MV contributing to \(L_{\textrm{eff},\phi _{i}}\) in the next subsection.
5.1 Contribution to \(L_{\textrm{eff}}\) from magnetized vacuum
The contribution of magnetized vacuum to \(L_{\textrm{eff}}\) can be estimated from the size of the correction to the photon polarization tensor from magnetized vacuum. Introducing two projectors \(\tilde{{g}}^{\parallel }_{\mu \nu }= \left( g^{\parallel }_{\mu \nu } - \frac{k_{\mu _{\parallel }}k_{\nu _{\parallel }}}{k^{2}_{\parallel }} \right) \) and \(\tilde{{g}}^{\perp }_{\mu \nu } = \left( g^{\perp }_{\mu \nu } - \frac{k_{\mu _{\perp }}k_{\nu _{\perp }}}{k^{2}_{\perp }} \right) \) (where the suffixes \(\parallel \) and \(\perp \) correspond to directions along and orthogonal to the vector eB) in the limit of \(\frac{eB}{m_{e}^2}\ll 1\) and \(\left( \frac{eB}{m_{e}^2} \right) ^2 \frac{k^2_{\perp }}{m_{e}^2}\ll 1\) (where \(m_{e}\) is the mass of the electron or positron) [50, 51],
5.2 Matter and magnetized matter effect
The magnetic field-dependent parts of the polarization tensor \( \Pi ^{\mathrm{(even/odd)}}_{\mu \nu }(k, eB,\mu , T)\) can further be expressed in terms of the scalar form factors times the tensors constructed out of the vectors in hand for the system under consideration. The form factors and the basis tensors for this decomposition were carried out in [47] and are provided as follows:
In Eq. (5.4), \(\Pi ^{(j)}(k, \mu , T, eB)\) are the scalar form factors and \(\mathcal{Z}^{(j)}_{\mu \nu }\) are the tensorial basis to expand the polarization tensor, given by
One can find six independent components in the decomposition above, given by Eqs. (5.5–5.10). In the standard notations, the field-independent symmetric parts of the polarization tensor can be written in terms of two form factors and the same field-independent basis tensors, as follows:
where \(\Pi _T\) and \(\Pi _{L}\) are the transverse and the longitudinal form factors for the photon polarization tensor.
Now, comparing Eqs. (5.5) and (5.6) with (5.11), one can ensure that \(\mathcal{Z}_{\mu \nu }^{(1)} = R_{\mu \nu } \) and \(\mathcal{Z}_{\mu \nu }^{(2)} = Q_{\mu \nu } \). Therefore, \(\Pi ^{(1)}\) and \(\Pi ^{(2)}\) of Eq. (5.4) would correspond to \(\Pi _{T}\) and \(\Pi _{L}\), respectively. Henceforth, we will denote \(\Pi ^{(1)}\) and \(\Pi ^{(2)}\) as \(\Pi _{T}\) and \(\Pi _{L}\). Since these two form factors depend on even powers of eB, they can have contributions from field-dependent and field-independent terms. In an unmagnetized medium in the long wavelength limit (LWL), i.e., in the \(\lim _{k \rightarrow 0}\), they turn out to be \(\sim \omega ^{2}_{p}\), where \(\omega _{p}\) is the plasma frequency and is equal to \(\sqrt{4\pi \alpha (\frac{n_{e}}{m_e})}\). For low temperatures, \(\omega ^{2}_{p} \sim e^2\sqrt{m_{e}}T^{3/2}e^{-\frac{(m_{e}-\mu )}{T}}\)[72].
The field-dependent parts (even in eB) of \(\Pi _{T}\) and \(\Pi _{L}\) under the same approximation are
and
In the limit \(m^{2}_{e}>k^{2}_{0}>T^{2}>eB\), it is easy to convince oneself that the contribution to \(\Pi _{T, L}\) from the unmagnetized medium is the dominant one.
The form factors associated with the symmetric field-dependent parts of the tensor \(\mathcal{Z}_{\mu \nu }^{(3)}\) and \(\mathcal{Z}_{\mu \nu }^{(4)}\) happen to be \(\Pi ^{(3)}\) and \(\Pi ^{(4)}\), respectively. The details of their evaluations are provided in Appendix E of this article. The results of that analysis establish that their contribution is subdominant.
We recall that in the realm of this calculation, \(\mu \) is considered to be less than \(m_{e}\); hence it does not appear as a small expansion parameter. Therefore, by the arguments presented above, they would be subdominant. This justifies the retention and use of \(\Pi _{T,L} \sim \omega ^{2}_{p}\).
We would like to point out that the contributions to \(L_{\textrm{eff}}\) coming from the odd field-dependent tensorial basis \(\mathcal{Z}^{(5)}_{\mu \nu }\) and \(\mathcal{Z}^{(6)}_{\mu \nu }\) happen to be \(A^{\mu }\mathcal{Z}^{(5)}_{\mu \nu }\Pi ^{(5)}A^{\nu }\) and \(A^{\mu }\mathcal{Z}^{(6)}_{\mu \nu }\Pi ^{(6)}A^{\nu }\), where \(\Pi ^{(5),(6)}\) are the scalar form factors.
This contribution can be related to \(\Pi ^{\mu \nu \mathrm{(odd)}}(k, \mu , T, eB)\) evaluated in [45]. Out of the two contributions proportional to \(\mathcal{Z}_{\mu \nu }^{(5,6)}\), the leading one would come from the term proportional to \(\mathcal{Z}_{\mu \nu }^{(6)}\) only in the LWL, i.e., \(k\rightarrow 0\).
The contribution from the photon self-energy tensor \(\Pi ^{\mu \nu \mathrm{(odd)}}(k, \mu , T, eB)\) estimated to linear order in \(\frac{eB}{m^{2}_{e}}\) provides same contribution to \(L_{\textrm{eff}.}\). This is denoted by \(\Pi ^{p}_{\alpha \nu }(k)\) and was found to be of the form \( \Pi ^{p}_{\mu \nu }(k)= \Pi ^{p}(k)P_{\mu \nu }\), as evaluated in [45], where the projection operator in the last expression stands for \(P_{\mu \nu } = i\epsilon _{\mu \nu \delta _{\parallel }\beta }\frac{k^{\beta }}{\mid k \mid }u^{\tilde{\delta }_\parallel }\), and \(\Pi ^{p}(k)\) is the associated form factor. Non-subscript Greek indices in these expressions can have values lying between 0 and 3; however, those with subscript \(\parallel \) (e.g., \(\mu _{\parallel }\)) mean that they can assume only two values, either 0 or 3. The tilde on \(\delta _{\parallel }\), i.e. \(\tilde{\delta }_{\parallel }\), signifies that if \(\delta _{\parallel } = 0\), the \(\tilde{\delta }_{\parallel }\) should be 3, and vice versa.
To make the connection with the existing literature, we need to pay give greater consideration to the Lorentz indices and manipulate them.
All the interplay between \(\delta _{\parallel }\) and \(\tilde{\delta }_{\parallel }\) stated above can be accomplished if we define \(\tilde{u}^{\delta _{\parallel }} = \frac{1}{2}\epsilon ^{\delta _{\parallel }\sigma \rho \tau }u_{\sigma }\frac{\bar{F}_{\rho \tau }}{|F_{\rho \tau }|}\). With this manipulation, it can be verified that \(P^{\mu \nu }\) is identical to \(\mathcal{Z}_{\mu \nu }^{(6)}\) modulo, a factor of \(\frac{i}{|k|}\). The form factor \(\Pi ^{p}(k)\) to order eB(in terms of Larmor frequency \(\omega _{B} = \frac{eB}{m_{e}}\)) is
and is also called the Faraday term in the literature. We may do the same at places in this work. If we compare field-dependent contributions from \(\Pi ^{(1)}, \Pi ^{(2)}\) or \(\Pi _{T}\) and \(\Pi _{L}\) (according to our new identification), given by Eqs. (5.12) and (5.13) with Eq. (5.14), we see that the contribution from Eq. (5.14) remains dominant in the LWL and \(\left( \frac{T^{2}}{m^{2}_{e}}\right) \left( \frac{eB}{T^{2}} \right) < 1\) limit. Hence, the contributions from Eq. (5.3) become subdominant under the same approximations. This justifies our choice of terms for the effective Lagrangian. In most astrophysical situations, this is the limit that is realized.
In this work, if we ignore the contributions to the electron pole mass and width from the non-relativistic medium and weak magnetic field effects, the system can be considered to be conservative. For such a system, the mixing matrix is supposed to be Hermitian. Thus, those terms of the polarization tensor that are even in powers of eB should appear in the diagonal elements of the mixing matrix, and those that are odd in powers of eB should appear in the off-diagonal elements of the mixing matrix. Therefore, the odd eB pieces contribute to mixing between various degrees of freedom such as \(\perp \) to \(\phi \), \(\parallel \) to \(\phi \), \(\parallel \) to \(\perp \), and longitudinal to \(\phi \). The even eB pieces do not. To obtain an estimate of the mixing angles between various degrees of freedom, one needs to formulate the problem in terms of Euler angles. This analysis can be extended to evaluate those estimates; we however do not perform this here. There is one important thing worth noting here: due to the inclusion of the parity-violating piece, the oscillation probabilities involve superposition of more than one oscillating piece, making analytical estimation of the oscillation length very difficult. Thus, a comparison of the mean free path of the system and oscillation length for all values of the parameters analytically is no longer possible. This is our main motivation behind the choice of the form factors for our analysis.
6 Systems with dim-5 interactions
6.1 Scalar–photon system in magnetized medium
The EOMs for the coupled \(\phi \gamma \) interacting system, including the Faraday term, can be cast in a compact matrix form as
where I is the \(4\times 4\) identity matrix, and M is a \(4\times 4\) mixing matrix given by
]
It must be noted that \(A_{L}\) are not mixed up with other degrees of freedom in a magnetized medium. Therefore, we will not be considering the longitudinal degrees of freedom of photons in further calculations.
6.1.1 Unitary matrix that diagonalizes the \(\phi \gamma \) mixing matrix (exact result)
Solutions for the field equations (6.1) can be obtained by diagonalizing the mixing matrix (see Appendix A for details) provided in Eq. (6.2) by a unitary transformation using the following unitary matrix:
where \(\mathcal{{N}}^{(\textrm{i})}_{\textrm{vn}}\) (when \( i = 1, 2, 3\)) are the normalization constants which can be found in Appendix A. Performing the same and considering the initial condition that at the origin there are no scalars (i.e., \(\phi (\omega ,0)= 0\)), the final solutions for the electromagnetic form factors can be written (in terms of the elements of the orthonormal eigenvectors (\(\bar{u}_{i}, \bar{v}_{i}\)) of the mixing matrix) as
Similarly, the perpendicular \(A_{\perp }(\omega ,z) \) component turns out to be
The variables \( \Omega _\parallel \), \( \Omega _\perp \), and \(\Omega _\phi \) introduced in Eqs. (6.4) and (6.5) are functions of the roots of the \(3 \times 3\) matrix M. They are given by
The energy of the photon in Eq. (6.6) is given by \(\omega \). In the final solutions, i.e. in Eqs. (6.4) and (6.5), the initial conditions for the two form factors of the photons at the origin are denoted by \({ A_{\parallel }(\omega , 0)}\) and \({ A_{\perp }(\omega , 0)}\). Their magnitude can be estimated according to the process under consideration. The expressions for the Stokes parameter are as follows:
6.2 Pseudoscalar–photon system in magnetized medium
In this section we study \(\phi '\gamma \) mixing in a magnetized medium when the parity-violating part of the photon self-energy or polarization tensor \(\Pi ^{p}_{\mu \nu } (k, \mu , T, eB)\) is included in the effective Lagrangian, \(L_{\textrm{eff},\phi '}\).
We carry out the analysis to point out the distinct analytical features of the mixing matrix that make the \(\phi ^{\prime }\gamma \) mixing dynamics different from the \(\phi \gamma \) case. We then obtain the estimates of polarimetric observables arising out this mixing, for compact astrophysical objects.
Following these estimations, we next evaluate the same observables, for \(\phi \gamma \) system, for same physical situation in the same physical parameter range and point out the difference in their magnitude from that for the \(\phi '\gamma \) system, arising due to the difference in their respective mixing dynamics. Our final objective is to determine whether these differences are detectable for the current or proposed experiments.
6.3 Perturbative solutions for the \(\phi ^{\prime }\gamma \) system
The EOMs for the \(\phi '\gamma \) system obtained from the Lagrangian \(L_{\textrm{eff},\phi '}\), as before, can be expressed in compact matrix notations as
where I is an identity matrix and matrix \(\textbf{M}^{\prime }\) is the \(4\times 4\) mixing matrix. The same, in terms of its elements is given by
Note that the projection operator \(P_{\mu \nu }\) appearing in the \(\textbf{M}^{\prime }_\textbf{12}\) and \(\textbf{M}^{\prime }_\textbf{21}\) elements of the mixing matrix \(\textbf{M}^{\prime }\) is a complex one that makes the matrix \(\textbf{M}^{\prime }\) Hermitian, which is otherwise expected even on general grounds.
6.3.1 Unitary matrix that diagonalizes the \(\phi ^{\prime }\gamma \) mixing matrix in the low-energy limit
Equation (6.11) with \(\textbf{M}^{\prime }\) given by (6.12) can be diagonalized perturbatively (when the (perturbation) parameter \(\Pi _{L}\) is small) by a unitary transformation using a unitary matrix, given by
Variables introduced in (6.13) are defined as \(\xi _i = \frac{Lw_i}{d_i(\lambda _{3}^0-\lambda _{i}^0)}\) (i can take values 1,2,4).
Here, u\(_j\), v\(_j\), w\(_j\), and x\(_j\) are the components of the \(j^{th}\) column vectors of U, and
is the corresponding normalization constant (\(j=1, 2, 3, 4\)).
In order to obtain the solutions in terms of form factors, we apply the boundary conditions wherein, at the origin, there is no pseudoscalar field, i.e., \(\phi '(0)=0\). The necessary steps to arrive at the solution are provided in [42], i.e,
The component \(A_{\perp }(\omega ,z)\) is given by
where \( \Omega _\parallel = \left( \omega -\frac{\lambda ^{0}_1}{2\omega }\right) \), \( \Omega _\perp = \left( \omega -\frac{\lambda ^{0}_2}{2\omega }\right) \), \( \Omega _L = \left( \omega -\frac{\lambda ^{0}_3}{2\omega }\right) \), and \( \Omega _{\phi '} =\left( \omega -\frac{\lambda ^{0}_4}{2\omega }\right) \). The difference between the \(\phi \gamma \) and \(\phi ^{\prime }\gamma \) systems is the appearance of the contribution of the longitudinal form factor \(A_L\) of the photon. Using the solutions obtained in (6.14) and (6.15) for the \(\phi \gamma \) interaction, we provide the polarimetric variables for pseudoscalar–photon interaction taking magnetized matter effect into account, as follows:
The complete expressions for the coefficients of \(A_{\parallel }, A_{\perp }, A_{L},A_{\parallel }A_{\perp }, A_{\parallel }A_{L}\), and \(A_{\perp }A_{ L}\) are explicitly defined in [42].
6.4 Exact solutions for the \(\phi ^{\prime }\gamma \) system
The EOMs for \(\phi ^{\prime }\gamma \) interaction in a magnetized medium given by Eq. (6.11) can also be written as
The form of matrix \(M_{H}\) and the steps to reduce it to a real symmetric matrix are provided in Appendix C of this article.
To obtain the exact solutions of the EOMs for the \(\phi ^{\prime }\gamma \) interaction, we need to diagonalize the \(4\times 4\) mixing matrix \(M_{H}\). The details of the steps involved in performing this are provided in Appendix C. However, the form of the unitary matrix used in diagonalizing \(M_{H}\) is presented in the next subsection.
6.4.1 Unitary matrix that diagonalizes the \(\phi ^{\prime }\gamma \) mixing matrix (exact result)
The Hermitian mixing matrix \({M_{H}}\) can be diagonalized by a unitary transformation. A unitary matrix denoted as \(\textbf{U}_{H}\)(H stands for Hermitian)
would serve the purpose, with the following expressions for the respective elements of the unitary matrix \(\textbf{U}_{H}\):
where the suffix i in Eqs. (6.22)–(6.25)) can take values ranging from 1 to 4. The parameters \(h(\lambda _{i})= \prod _{j=1, i\ne j}^{4}(\lambda _{j}-\lambda _{i})\) appearing in the same equations are normalization constants written in terms of the eigenvalues (\(\lambda _{1}, \lambda _{2}, \lambda _{3}\), and \(\lambda _{4}\)) of the mixing matrix \(M_{H}\).
The solutions of Eq. (6.20) with appropriate boundary conditions given by choosing appropriate values of \(A_{\parallel }(\omega ,0)\), \(A_{\perp }(\omega ,0)\), \(\phi ^{\prime }(\omega ,0)\), and \(A_{L}(\omega ,0)\) can be used to obtain the solutions of the form factors. The solutions for the photons polarized parallel to \(\tilde{B}_{\mu }\) are given by
Here, the characteristic frequencies for the pseudoscalar photon system are denoted by \(\Omega _{\parallel }\), \(\Omega _{\perp }\), \(\Omega _{L}\) and \(\Omega _{\phi ^{\prime }}\) where \( \Omega _\parallel = \left( \omega -\frac{\lambda _1}{2\omega }\right) \), \( \Omega _\perp = \left( \omega -\frac{\lambda _2}{2\omega }\right) \), \( \Omega _L = \left( \omega -\frac{\lambda _3}{2\omega }\right) \), and \( \Omega _{\phi '} =\left( \omega -\frac{\lambda _4}{2\omega }\right) \).
Similarly, the solutions for the photons polarized perpendicular to \(B_{\mu }\) (transverse component) represented by the form factor for the gauge field \(A_{\perp }(\omega ,z)\) are given by
The solution for the pseudoscalar field in terms of \(A_{\parallel } (\omega , 0)\), \(A_{\perp } (\omega , 0)\), \(\phi ^{\prime } (\omega , 0)\). and \(A_{L} (\omega , 0)\) is given by,
Similarly, the solution for the longitudinal form factor for the photon at any point z from the origin is given by
The solutions show that even if the pseudoscalar field is zero at the origin, it can be generated due to interaction at any point z.
7 Mixing
7.1 Analysis of mixing
In this section, we deliberate on the origin of the couplings between various degrees of freedom, following the mixing dynamics, from the point of view of symmetry[24]. We begin our discussion on the consequences of including just the matter effects, and then on including the matter plus the magnetized matter effects, with respect to the mixing dynamics.
We begin with the demonstration of the observations made in Sect. 1.1, with the incorporation of matter effects by including \(A^{\mu }(k)\Pi _{\mu \nu }(k,\mu , T)A^{\nu }(k)\) in the effective Lagrangian \(L_{\textrm{eff}, \phi _i}\), the mixing matrix of the scalar–photon system is 2 \(\times \) 2, but the matrix of the pseudoscalar–photon system is 3 \(\times \) 3.
This feature can be easily verified by setting the parity-violating part \(\Pi ^{p}=0\) in Eqs. (6.2) and (6.12) for the \(\phi \gamma \) and \(\phi '\gamma \) systems. This is because Eq. (6.2) has both effects incorporated into it, so by setting \(\Pi ^{p}=0\), one retains only the matter effects.
Next we address the question of the coupling between different degrees of freedom and their discrete symmetry assignments. The emerging 2 \(\times \) 2 mixing matrix structure for the \(\phi \gamma \) system is due to mixing between \(A_{\parallel }\) and \(\phi \). This follows from the expression of the Lagrangian (\(L^{B}_{\textrm{int},\phi }\)) given by Eq. (2.17), denoting the \(\phi \gamma \) interaction in an external magnetic field. It follows from there that the only form factor that has nonzero interaction with the scalar \(\phi \) is \(A_{\parallel }\). No other form factor has any interaction with \(\phi \) at the level of \(L^{B}_{\textrm{int},\phi }\). Hence, the mixing matrix is 2 \(\times \) 2. Investigating this issue using active and passive Lorentz transformations may shed additional light on the form of coupling.
The other interesting issue in this mixing is that although one would have naively expected the longitudinal degree of freedom of the photon and the scalar \(\phi \) to mix with each other for having the same spin assignments \(s_z = 0 \),= that remains forbidden, because at the level of the Lagrangian \(L^{B}_{\textrm{eff},\phi _i}\), they have no interaction. Thus, the mixing matrix for the \(\phi \gamma \) system turns out to be \(2\times 2\).
Lastly, we examine another subtlety in these mixing dynamics, following from the discrete symmetry arguments. We begin with the observation that the discrete symmetries of the associated degrees of freedom of electromagnetic form factors appearing in the description of the gauge potential given by (2.10), including \(A_{\parallel }\), are odd under time (T) reversal and even under parity (P) inversion symmetry transformations (for details consult Table [1] in the supplemental material [42] of this article). And \(\phi \), being a scalar, remains even under PT transformation. So at the level of field equations, the coupling of the PT odd \(A_{\parallel }\) with the PT even scalar \(\phi \) is a bit bizarre.
However, a careful inspection of the field equations reveals that these two degrees of freedom with uneven PT symmetry appear in the field equations after being multiplied by specific PT-dependent factors. It so happens that after PT transformation, the products in the field equations have retained same PT symmetry. Stated differently, if we consider the IL of Eq. (2.17), i.e.,
the product \( b^{(1)\nu }b^{(1)}_{\nu }\) is even under time-reversal symmetry transformation (T), and so are the transformation properties of \(N_{1}\) and \(\phi \), but \(A_{\parallel }\) is odd under \(\textbf{T}\). Therefore, Eq. (7.1) would not have been T-symmetric unless the multiplicative factor i was absent. Therefore, here the factor i is the T-dependent multiplicative factor that was discussed in the preceding paragraph.
Next we consider the issue of the effect of a magnetized medium on \(\phi \gamma \) mixing dynamics. This effect is obtained through the incorporation of the parity-violating part \(\Pi ^{p}\) in the effective Lagrangian. Since this causes the two transverse degrees of freedom of the photon \(A_{\parallel }\) and \(A_{\perp }\) to couple directly with each other, and as \(\phi \) also couples directly with \(A_{\parallel }\), the appearance of \(\Pi ^{p}\) causes \(A_{\perp }\) to couple indirectly to \(\phi \). The longitudinal degrees of freedom of the photon, however, remain decoupled. Thus, the mixing matrix for the \(\phi \gamma \) system is \( 3 \times 3 \) on incorporation of the \(\textbf{P}\)-violating \(\Pi ^{p}\).
We now turn our attention to the pseudoscalar (axion) photon system. This system has nonzero interaction between \(A_{\perp }\) and \(A_{L}\) at the level of the tree-level interaction Lagrangian, as can be checked from Eq. (2.15). Therefore, the inclusion of matter effects makes direct mixing possible between \(\phi '\), \(A_L\), and \(A_{\perp }\), turning the mixing matrix into a \(3\times 3\) matrix.
On inclusion of magnetized matter effects along with unmagnetized matter effects, the mixing matrix for the \(\phi '\gamma \) becomes \( 4 \times 4\), for the following reason: Because of matter effects alone, \(\phi '\) couples directly to \(A_L\) and \(A_{\perp }\) at the level of \(L^{B}_{\textrm{int}, \phi '}\). As we already have pointed out, upon the inclusion of the parity-violating \(\Pi ^{p}\) part, \(A_{\perp }\) further couples directly with \(A_{\parallel }\), and hence there is an indirect mixing between \(i\phi '\) and the photon EM form factor for transverse polarization (\(A_{\parallel }\)), resulting in all the degrees of freedom for the pseudoscalar–photon system interacting with each other. The essence of this is captured through the elements of the \(4 \times 4 \) mixing matrix.
Coming to the question of the discrete symmetry assignments of the degrees of freedom and their coupling in the field equations for the the pseudoscalar–photon system, we note the existence of coupling between different degrees of freedom with uneven PT symmetry assignments. This can be understood as follows. Due to odd PT transformation assignment, \(A_L\) (see Table [1] in [42]) can couple only to the product \(i \times \phi '\) in any equation because, as noted before, even though \(\phi '\) is even under PT, i (i.e., \(\sqrt{-1}\)) being odd under T and even under P makes \(i \times \phi '\) odd under PT, thus favoring the coupling between the two at the level of field equations. This can be verified from the last relation of the equations presented at (S.4.3) in the supplementary section [42]. Similar conclusions follow for the other equations. The preceding analysis of mixing dynamics is true in general, but the magnitude of the signal will depend on the fraction of the total energy that each component has at the initial state. An analysis to this effect can be found in [27], and an interesting analysis corroborating this observation can be found in [29].
We conclude this discussion with the observation that even the difference in the mixing pattern between \(\phi \gamma \) and \(\phi ^{\prime }\gamma \) systems having ALPs as secondary and photons as primary components in a magnetized medium can have far-reaching consequences concerning the polarimetric signals. Hence, this can serve as an ideal astrophysical laboratory to distinguish one type of interaction from the other. A detailed discussion on the possible nonzero elements of the mixing matrix is, however, left for a future communication [52].
7.2 Searches so far
Based on the physics principles mentioned above, systematic studies to find ALP through laboratory- or astrophysics-based experiments have been going on for some time now. The ongoing and proposed laboratory-based searches for ALP can be found in [53,54,55,56,57,58,59,60,61,62]. Among those, the CERN Axion Solar Telescope (CAST) collaboration [53] is a prominent one. This collaboration is engaged in detecting ALP produced at the interior of the Sun via the Primakoff process [63, 64], based on energy loss arguments from the stellar interior. This collaboration had earlier reported the following bounds on ALP parameters: \(m_{\phi '} \le 0.02 \) eV and \(g_{\phi '\gamma \gamma } < 1.16 \times 10^{-10}\) GeV\(^{-1}\) [53]; and their last improvement to this result reads \(g_{\phi '\gamma \gamma } < 0.66 \times 10^{-10}\) GeV\(^{-1}\), in the same mass range [54].
In the astrophysical front, bounds on ALP parameters are also obtained by studying the cooling rate of stellar objects such as supergiants, red giants, helium core-burning stars, white dwarfs, and neutron stars [24, 65,66,67]. An up-to-date introduction to the relevant issues can be found in [68]. The interesting part is that the astrophysical bound on axion photon coupling based on measuring the ratio of the number of horizontal branch stars to that of red giant branch stars in the globular cluster also provides \(g_{\phi '\gamma \gamma } < 0.66 \times 10^{-10}\) GeV\(^{-1}\) at the \(95\%\) confidence [22], the same as that of [54]. Most of these bounds were obtained by taking only the matter effects into account in the axion–photon effective Lagrangian. That is when the mixing occurs between only three degrees of freedom(\(\phi ^{\prime }\), \(A_{\perp }\), and \(A_{L}\)), leaving \(A_{\parallel }\) free.
However, according to current observations, the data suggest additional cooling at various stages of their evolution, referred to as cooling anomalies. This has sparked renewed interest in a model-dependent axion-induced cooling [69]. However, a possible way to achieve this is through mixing of all the degrees of freedom of the photon with ALPs, which will open additional channels of energy transport via conversion of photons into axions.Footnote 2 This can also be achieved by the incorporation of the parity-violating part of the photon self-energy tensor \(\Pi ^{p}_{\mu \nu }\) that emerges from magnetized matter effects in the pseudoscalar photon effective Lagrangian \(L_{\textrm{eff}}\). This happens to change the mixing dynamics and hence the mixing matrix for the \(\phi '\gamma \) system. The mixing matrix for the \(\phi ' \gamma \) system turns out to be 4\(\times \)4, implying complete mixing between all three degrees of freedom of the (in-medium) photon with a single degree of freedom of \(\phi '\).
For the \(\phi \gamma \) system, the same procedure changes the mixing matrix from \(2\times 2\) to a \(3\times 3\) matrix [70]. That is, the mixing occurs only between the two transverse polarization states of the photon and \(\phi \). The longitudinal degree of freedom of the photons is decoupled and propagates freely. The spectro-polarimetric signatures of these features are the ones investigated in the following sections of this work.
8 Astrophysical application
In this section we focus on a possible astrophysical scenario that is likely to provide the distinct signatures that would distinguish scalar–photon mixing from pseudoscalar–photon mixing using magnetized media effects. The choice of the astrophysical system considered here for obtaining a reasonably strong signal is an eclipsing compact binary system. This choice is due to the strong surface magnetic field (\(10^{9}{-}10^{14}\) Gauss) associated with them. We expect that the \(\phi _{i}\gamma \) interaction in such a strong field would generate a sufficiently strong signal to be detected by the future space-borne X-ray observatories.
Since the polarization angle \((\psi _{\phi _{i}})\) and ellipticity angle \((\chi _{\phi _{i}})\) have emerged as reliable observables for polarimetric signatures, we would consider estimating their contribution for signals for the kind of situation discussed here.
Schematic diagram of the binary configuration, as considered in the text
The binary system considered here is assumed to be composed of a neutron star (primary) and its “companion,” which may be a white dwarf or neutron star, represented by compact star A and compact star B, respectively, as shown in Fig. 1. Both compact stars A and B orbiting around their center of mass are situated at the origin of the coordinate system. The primary is imagined to be an orthogonal rotor having its axial dipole field direction nearly orthogonal to the dipole axis of the companion B. We consider companion B to be an axisymmetric rotor. At the onset of the eclipsing phase with respect to the observer, an electromagnetic radiation beam from the primary grazes past the polar cap of the companion, to the observer. This happens during a finite fraction of their orbital cycle (when the primary is in position \(P^{\prime }\)). This scenario is depicted in Fig. 1. On the other hand when the primary approaches its orbital position \(P^{0}\) as shown in Fig. 1, it will beam its radiation directly to the observer. In this scenario, any difference in the observed values of \((\psi _{\phi _{i}})\) and \((\chi _{\phi _{i}})\) between the direct beaming phase and the grazing phase can be attributed to propagation in the magnetized environment of the companion. If scalars or pseudoscalars do exist in nature, their interaction in the magnetized environment of the companion would also contribute to the difference in the measured values of \(\psi _{\phi _{i}}\) and \( \chi _{\phi _{i}}\), the study of which happens to be the main goal of this investigation, as mentioned previously.
Since the EM beam, according to our physical picture, grazes past the polar cap of the companion, the magnetized path length that photons travel would be \(\sim \) twice the polar cap radius \((R_{PC})\) of the companion. Assuming the magnetic field to be dipolar, the polar cap opening angle for the last open field lines at the surface of the compact star can be estimated from dipole field equations. In terms of light-cylinder radius \(R_{LC}\), this is found to be
Therefore, the polar cap radius of the companion in terms of its radius R and period P is
In Eq. (8.2), we have used the relation \(R_{LC}= \frac{P}{2\pi }\).
Assuming period \(P \sim 100\) s and radius \(R=10\) km, for the companion, the polar cap radius is \(R_{pc}=6.5\) m. We further assume the primary and the companion to be sufficiently separated that they do not to pass through each other’s light cylinder during their orbital motion. The companion considered here is cold enough to have no charged particles in its magnetosphere. Hence, the availability of plasma around the polar cap should be due to the charged particles pulled out of the compact star surface due to the rotating dipole field. Therefore, the plasma frequency will be given by \(\omega ^{2}_{p} = 4\pi \alpha (\frac{n_{GJ}}{m_e})\). The Goldreich–Julian number density [71] given by \(n_{GJ} = \frac{\vec {\Omega }.\vec {B}}{2\pi }\) for an orthogonal rotor can in principle be very small, depending on the angle between \(\vec {\Omega }\) and \(\vec {B}\), so we consider the plasma frequency to be of the order of \(10^{-3}\) eV generated from secondary pair production and the surface magnetic field \(B_{S} \sim 10^{10}\) G. The choice of these fiducial parameters ensure that the particle density in the stellar magnetosphere will be negligible and the narrow EM beam will just pass through the plasma close to the surface of the companion. This scenario implies that the interaction with the magnetized media is confined only in the polar cap region for the propagating EM beam.
9 Results
In this section we discuss the signatures of \(\gamma \phi \) and \(\gamma \phi ^{\prime }\) interactions on the spectro-polarimetric observables I, Q, U, V, \(\chi _{\phi _{i}},\psi _{\phi _{i}}\), and \(P_{L\phi _{i}}\) on the EM radiation coming from the astrophysical sources. We also wish to establish whether the contributions to the polarimetric variables are the same or different for these two types of interactions when the interaction parameters remain the same. We further attempt to understand the origin of the difference in the size of their contributions based on the number of degrees of freedom participating in mixing.
[Low energy] Plots of Stokes parameters I, Q, U, and V (along the ordinates) against \(\omega /\omega _{B}\) (along the abscissas) using exact solutions of equations of motion for \(\phi \gamma \) interactions presented in the panels above. Stokes parameter I was transformed in the following way: It was initially shifted by a numerical amount of magnitude 1.999999999, followed by scaling by a factor of \(10^{+8}\). Stokes parameter Q was transformed as follows: It was scaled by a factor of \(10^{+2}\) without any numerical shifting. In the same spirit, Stokes parameter U was shifted by a numerical amount of magnitude 1.9990, followed by scaling by a factor of \(10^{+4}\). Similarly, Stokes parameter V was transformed as follows: It was scaled by the scaling factor \(10^{+8}\). The parameters chosen for this estimations are as follows: photon energy \(\omega \) in the range of \(1-100\) keV; mass of scalar and pseudoscalar particles of \(m_{\phi ,\phi ^{\prime }} = 1.0 \times 10^{-11}\) GeV; coupling constant of \(g_{\phi \gamma \gamma } = 1.0 \times 10^{-11}\) GeV; magnetic field \(\sim 10^{10}\) Gauss; plasma frequency = \(1.0 \times 10^{-12} \) GeV; and photon path length (z) equal to 12 m
[Low energy] Plots of Stokes parameters I, Q, U, and V (along the ordinates) against \(\omega /\omega _{B}\) (along the abscissas) for \(\phi ^{\prime }\gamma \) interactions using the perturbative solution of equations of motion. Stokes parameter I was transformed in the following way: It was numerically shifted by an amount of magnitude 1.999999999, followed scaling by a factor of \(10^{+7}\). Stokes parameter Q was scaled by a scaling factor of \(10^{+2}\). In the same spirit, Stokes parameter U was transformed as follows: It was numerically shifted an amount of magnitude 1.9990, followed by scaling by a factor of \(10^{+4}\). Similarly, Stokes parameter V was scaled by a factor of \(10^{+2}\). The plots are in the same parameter space as those used in Fig. 2
The current and proposed space-based polarimetric experiments to detect polarimetric signals from astrophysical sources include APEX [73], IXPE [74], e-ASTOGRAM [75], and ASTRO-2020 [76, 77], to name but a a few. They are expected to cover the soft X-ray band of the EM spectra. Furthermore since the energy distribution of the observed cosmic \(\gamma \)-ray spectra is seen to peak in the range of 1–10 keV [75], we have confined our analysis to a slightly broader energy domain of 1–100 keV to ensure coverage of the peak emission energy range.
9.1 Numerical estimates in the low-energy range (1–100 keV)
Assuming the physical scenario mentioned already, we have estimated the Stokes parameters for each type of interaction (i.e., \(g_{\phi \gamma \gamma }\phi F_{\mu \nu } F^{\mu \nu }\)/ \(g_{\phi ^{\prime }\gamma \gamma } \phi ^{\prime } F_{\mu \nu } F^{\mu \nu }\)), keeping \(\omega _{p}\), path length z, magnetic field B, \(g_{\phi _{i}\gamma \gamma }\), and \(m_{\phi _i}\) fixed. We have further estimated the difference in the polarization angle, \(\Delta \psi =(\psi _{\phi ^{\prime }} - \psi _{\phi })\) and the ellipticity angle \( \Delta \chi =(\chi _{\phi ^{\prime }} - \chi _{\phi } )\) acquired by a light beam due to pseudoscalar \(g_{\phi ^{\prime }\gamma \gamma } \phi ^{\prime } F_{\mu \nu } F^{\mu \nu }\) and scalar \(g_{\phi \gamma \gamma }\phi F_{\mu \nu } F^{\mu \nu }\) interactions for the same parameters as the EM beam passes through the magnetosphere of the companion star of a binary system.
The numerical magnitude of these parameters determined as follows: The coupling constant \(g_{\phi _i\gamma \gamma }\) was considered to be \(g_{\phi _i\gamma \gamma }=1.0\times 10^{-11}\) GeV\(^{-1}\) for both the scalar (\(\phi \)) and pseudoscalar (\(\phi ^{\prime }\)) interactions, and their mass at \(m_{\phi _{i}} = 1.0 \times 10^{-11}\) GeV. We considered the path length of the photon to be 12 m, keeping the magnetic field (B) \(\sim 1.0 \times 10^{10}\) Gauss and plasma frequency \( {~ \omega _p} = 1.0 \times 10^{-12} \) GeV. The energy band of investigation was kept in the soft X-ray region (1–100 keV). The evolution of the Stokes parameters with the energy \(\omega \) (in units of \(\omega _{B}\)) of the photon for both interactions is plotted in Figs. 2 and 3 using exact solutions for \(\phi \gamma \) (provided in Eqs. (6.4), (6.5)) and perturbative solutions for \(\phi ^{\prime }\gamma \) interactions (provided in Eqs. (6.14), (6.15)), respectively. The magnitudes of the two initial amplitudes of the orthogonal modes of the photon beam in the perturbative and exact numerical solutions are considered to be identical, which is not too drastic an assumption if the system is isotropic and homogeneous. In a more realistic situation, the same can be extracted from the direct beaming phase of the radiation. The results show an achromatic pattern in particular around 100 keV. The numerical results lead us to believe that the use of detectors operative in this energy band would provide optimal performance.
The plots were obtained by estimating the respective quantities numerically, maintaining certain identities (e.g., \( {I}^2= { Q}^2+{ U}^2+{V}^2\)) to accuracy of at least one part in \(10^{-7}\) to \(10^{-10}\).
[Low energy] In the left panel: Plot of ellipticity angle difference \(\Delta \chi \) (along the ordinates in radians) versus \(\omega /\omega _{B}\) (along the abscissas) between \(\phi ^{\prime }\gamma \) and \(\phi \gamma \) interactions using exact solutions of \(\phi \gamma \) interactions and perturbative solutions of \(\phi ^{\prime }\gamma \) interactions. In the right panel: Plot of linear polarization angle difference \(\Delta \psi \) versus \(\omega /\omega _{B}\) between \(\phi ^{\prime }\gamma \) and \(\phi \gamma \) interactions using exact solutions of \(\phi \gamma \) and perturbative solutions of \(\phi ^{\prime }\gamma \) interactions. The parameters for these plots are the same as those used in Fig. 2
[Medium energy] In the left panel: Plots of ellipticity angle difference \(\Delta \tilde{\chi }\) (along the ordinates in radians) versus \(\omega /\omega _{B}\) (along the abscissas), where the difference in \(\Delta \tilde{\chi }\) represents the difference in the ellipticity angle generated due to the \(\phi ^{\prime }\gamma \) and \(\phi \gamma \) interaction using the exact solutions of equations of motion for both \(\phi \gamma \) and \(\phi ^{\prime }\gamma \) interactions. In the right panel: Plot of linear polarization angle difference \(\Delta \tilde{\psi }\) versus \(\omega /\omega _{B}\) between \(\phi \gamma \) and \(\phi ^{\prime }\gamma \) interaction using the exact solutions of equations of motion for both interactions. The parameters for these plots are same as those used in Fig. 2
Using these Stokes parameters, the ellipticity angle and the linear polarization angles were estimated for both the \(\phi \gamma \) and \(\phi ^{\prime }\gamma \) systems. From there we evaluated the angle difference (\(\Delta \chi , \Delta \psi \)) due to \(\phi ^{\prime }\gamma \) and \(\phi \gamma \) interactions. The important features in the plots for \(\Delta \psi \) versus \(\omega /\omega _{B}\) and \(\Delta \chi \) versus \(\omega /\omega _{B}\) are (a) their spectral dependence and (b) the variations in the magnitude of the difference of the ellipticity angle \(\Delta \chi \) and polarization angle \(\Delta \psi \)—that is, the variation in angle difference (AD) with \(\omega /\omega _{B}\). It is worth noting that, while \(\Delta \psi \) may be affected by geometrical effects such as the rotation of the coordinate axis of the observer frame, \(\Delta \chi \) is free from that problem, because \(\chi _{\phi _{i}}\) remains invariant under rotation about the propagation direction. The variation in AD, particularly \(\Delta \chi \) with \(\omega /\omega _{B}\), may be used for generating compact star models and dark matter identificationFootnote 3 through multi-wavelength spectro-polarimetric studies, because of the distinct nature of the variational pattern in AD with energy, as is evident from Fig. 4.
9.2 Numerical estimates in the high-energy range (1–500 keV)
We further estimated AD (\(\Delta \tilde{\chi }\), \(\Delta \tilde{\psi }\)) using exact solutions of the EOMs for both scalar–photon (Eqs. (6.4) and (6.5)) and pseudoscalar–photon systems (given in equations from (6.26)–(6.29)) for a broad energy range of 1–500 keV.
As can be seen in Fig. 5, the magnitude of the ellipticity AD (\(\Delta \tilde{\chi } = \chi _{\phi ^{\prime } } - \chi _{\phi }\)) is reduced and is tending asymptotically towards zero in the higher-energy range. However the magnitude of \(\Delta \tilde{\chi }\) increases by decreasing the plasma frequency i.e., from \(10^{-12}\) GeV to \(10^{-16}\) GeV (Fig. 6). In contrast, there is no such significant change seen in the (\(\Delta \tilde{\psi }= \psi _{\phi ^{\prime } } - \psi _{\phi }\)) as compared with the AD obtained using the perturbative solution of \(\phi ^{\prime }\gamma \) interaction presented in Fig. 4. Thus, numerical features similar to these may be considered in pursuit of dark matter identification of ALP type. Furthermore, the plot suggests that the numerical feature in this parameter range would favor this pursuit for ALP such as dark matter.
[Medium energy] In the left panel: Plots of ellipticity angle difference \(\Delta \tilde{\chi }\) (along the ordinates in radians) versus \(\omega /\omega _{B}\) (along the abscissas). In the right panel: Plot of linear polarization angle difference \(\Delta \tilde{\psi }\) versus \(\omega /\omega _{B}\) between \(\phi \gamma \) and \(\phi ^{\prime }\gamma \) interaction. The plasma frequency \(\omega _{p}\) is chosen as \(\sim 10^{-16}\) GeV, while the other parameters for these plots are the same as those used in Fig. 2
9.3 Statistical analysis with data for numerical solutions
It may be noted in passing that an identity involving the ellipticity angle \(\chi _{\phi _{i}}\), polarization angle \(\psi _{\phi _{i}}\), and polarization fraction \(\Pi _{\phi _{i}}^P\) can be derived from their constitutive relations, as follows:
where the expressions for \(\chi _{\phi _{i}}\), \(\psi _{\phi _{i}}\), \(\Pi _{\phi _{i}}^P\), and \(P_{L\phi _i}\) can be found in Eqs. (4.2)–(4.5). One can in principle identify the interaction type by using the observational data in models of compact stars and verifying Eqs. (9.1) and (9.2). Although this identity in principle should be satisfied for each \(\omega \), the observational data seem to fall short of that expectation. One reason behind this may be the use of energy band spectra by the detectors, when this relation is supposed to be satisfied for the line spectrum. This brings us to explore the use of numerical data from EOMs in static form in order to draw conclusions.
However, for correct identification of the nature of the interaction, the magnitude of the variations in \(\Delta \psi _P\) and \(\Delta \chi \) should be greater than the half-power diameter (HPD) or the angular resolution of the detectors in a given energy band.
Stokes parameter I (along the ordinate) versus photon energy \(\omega \) (along the abscissa) for the parameters \(m_{\phi } = m_{\phi ^{\prime }} = 1.0\times 10^{-11}\) GeV\(^{-1}\), \(g_{\phi ,\phi ^{\prime }\gamma \gamma } = 1.0\times 10^{-11}\) GeV\(^{-1}\), B \(\sim \) 10\(^{12}\) Gauss, and \(\omega _{p} = 1.0 \times 10^{-10}\) GeV in energy range of 1–100 keV, path length \(z = 22.3\) m. The plot in the left panel is for the \(\phi \gamma \) interaction using exact solutions of the equations of motion. The ordinate of the left panel was transformed in the following way: It was shifted by a numerical factor of magnitude 1.9999990 followed by scaling by a factor of \(10^{+7}\). The abscissa (in GeV unit) was scaled by a factor of \(10^{+5}\). Similarly, the plot in the right panel is for the \(\phi ^{\prime }\gamma \) interaction using the exact solutions of equations of motion. The abscissa (in GeV) in the right panel is scaled by a factor of \(10^{+5}\)
Stokes parameter Q (along the ordinate) versus photon energy \(\omega \) (along the abscissa) for the same parameter space as that used in Fig. 7. The plot in the left panel is for the \(\phi \gamma \) interaction using exact solutions of equations of motion. Similarly, the plot in the right panel is for the \(\phi ^{\prime }\gamma \) interaction using exact solutions of equations of motion. The abscissas (in GeV unit) in both panels are scaled by a factor of \(10^{+5}\)
Stokes parameter U (along the ordinate) versus photon energy \(\omega \) (along the abscissa) for the same parameter space used in Fig. 7. The plot in the left panel is for the \(\phi \gamma \) interaction using exact solutions of equations of motion, and the plot in the right panel is for the \(\phi ^{\prime }\gamma \) interaction using exact solutions of equations of motion. The abscissas (in GeV) in both panels are scaled by a factor of \(10^{+5}\)
Stokes parameter V (along the ordinate) versus photon energy \(\omega \) (along the abscissa) for the same parameter space as that used in Fig. 7. The plot in the left panel is for the \(\phi \gamma \) interaction using exact solutions. Similarly, the plot in the right panel is for the \(\phi ^{\prime }\gamma \) interaction using exact solutions. The ordinate in the left panel was scaled by a factor of \(10^{+6}\), while the abscissas (in GeV) in both panels were scaled by a factor of \(10^{+5}\)
Plots of degree of linear polarization vs. counts. The parameters for these plots are \(m_{\phi } = m_{\phi ^{\prime }} = 1.0\times 10^{-11}\) GeV\(^{-1}\), \(g_{\phi ,\phi ^{\prime }\gamma \gamma } = 1.0\times 10^{-11}\) GeV \(^{-1}\), B \(\sim \) 10\(^{12}\) Gauss, and \(\omega _{p} = 1.0 \times 10^{-10}\) GeV in the energy range of 1–100 keV, path length \(z = 22.3\) m
If the angular resolution of the detectors in the soft X-ray region (1–100 keV) following [78, 79] falls at \(4^{\prime \prime }\), they could in principle be detected with the current available resolution of onboard detectors. Furthermore, the differences in polarization angles \(\Delta \psi =(\psi _{\phi ^{\prime }} - \psi _{\phi })\) and in ellipticity angles \(\Delta \chi =(\chi _{\phi ^{\prime }} - \chi _{\phi })\) due to two different interactions also seem to come in the observable range in some energy intervals, as may be observed from the plot in Fig. 4.
In the face of the discrepancy between the observed data and the identities presented in Eqs. (9.1) and (9.2), we have tried to estimate the confidence level on the numerical size of the observables obtained from the exact numerical solutions. This analysis is relegated to the next subsection.
9.4 Normal distributions from numerical solutions
We have plotted the Stokes parameters in Figs. 7, 8, 9, and 10 using the exact numerical solutions obtained by diagonalizing (6.2) and (6.12) exactly for the following values of the system parameters: \(m_{\phi } = m_{\phi ^{\prime }} = 1.0\times 10^{-11}\) GeV\(^{-1}\), \(g_{\phi ,\phi ^{\prime }\gamma \gamma } = 1.0\times 10^{-11}\) GeV\(^{-1}\), B \(\sim \) 10\(^{12}\) Gauss, \(\omega _{p} = 1.0 \times 10^{-10}\) GeV, in the energy range of 1–100 keV, path length \(z = 22.3\) m.
The oscillating behavior of the Stokes parameters as a function of energy is evident from the plots displayed in Figs. 7, 8, 9, and 10. As noted earlier, none of the current space-based observations confirm this, and we believe that the proposed observations also would not be able to detect this. The chief reason behind this being the space-based detectors is the sensitivity over an energy band, not a single line.
Thus, in the spirit of the analysis of the observational X-ray polarization data, we initially divided the window of the energy band under consideration, i.e. 1–100 keV, into 10, 000 intervals and evaluated the parameters I, Q, and U in each of these intervals. This was followed by the estimation of the averages of the Stokes parameters denoted by \({\bar{ I}}, {\bar{ Q}}\), and \({ \bar{U}}\). From there, the average degree of linear polarization \({P_{L\phi _{i}}}\) was estimated to derive the first set of samples. This procedure was followed in sequence by increasing N to \(N^\prime \) such that \(N^{\prime } = N + \Delta N\), where \(\Delta N = 100\). We continued this process until \(N^{\prime } =N+ n\Delta N\), where \(n = 400\). Thus, we generated a set of data points for \({P}_{L\phi _{i}}\).
These samples were further distributed on frequency distribution plots for both systems (\(\gamma \phi \), \(\gamma \phi ^\prime \)) by choosing the appropriate bin size along the x-axis. These frequency distributions were then plotted as shown in Fig. 11 for both the scalar–photon (left panel) and pseudoscalar–photon (right panel) systems.
Assuming the obtained frequency distributions of the samples as normal distributions, we have estimated the (mean) average value and the variance of the \(P_{L\phi _{i}}\) obtained in the energy interval of \(1-100\) keV. The mean \(P_{L\phi _{i}}\) for the \(\gamma \phi \) and \(\gamma \phi ^\prime \) systems is \(\mu _{\phi } = 0.414\) and \(\mu _{\phi ^{\prime }} = 0.435 \) and the corresponding variance is \(\sigma ^{2}_{\phi } = 0.275 \times 10^{-6} \) and \(\sigma ^{2}_{\phi ^{\prime }} = 0.258 \times 10^{-6}\), respectively.
Statistically, this means that for the \(\gamma \phi \) system, the range of \(P_{L\phi _{i}}\) lies between \(0.414 - 0.275 \times 10^{-6}\) and \(0.414 + 0.275 \times 10^{-6}\), and for the \(\gamma \phi ^\prime \) system it is \(0.435 - 0.258 \times 10^{-6}\) to \(0.435 + 0.258 \times 10^{-6}\) in the one sigma level. As can be seen from the statistical evaluations of the linear polarization \(P_{L\phi _{i}}\), with an increase in the numbers of degrees of freedom of the system (\(\phi _{i}\gamma \)), the polarization amplitude increases. This happens for both systems when they operate in the same parameter range. The confidence level analysis for other interaction parameters similar to this may be performed for designing detectors for space-borne experiments.
9.4.1 Minimum detectable polarization
Many of the astrophysical polarimeters use minimum detectable polarization (MDP) to assess detector efficiency. MDP is defined as the degree of polarization corresponding to the amplitude of modulation that has only a \(1\%\) probability of being detected by chance. This is typically expressed as
where \(\mu \) is the modulation factor of the detector, which can vary between 0 and 1, and N is the sample size. Instruments like APEX have MDP of \(\sim 1\%\) at 5.2 keV [73]. The MDP in terms of mean \(\mu _{\phi _{i}}\) and \(\sigma _{\phi _{i}}\) for the observed data of \(P_{L\phi _{i}}\), where the background is zero and the modulation factor is 1, can be expressed as
In the case of our investigation, we found that \(MDP_{\phi ^{\prime }} > MDP_{\phi }\) for the parameter range under consideration in this work. Thus, in the same parameter space, with an increase in the number of degrees of freedom of the system, the detectability of \(P_{L\phi _{i}}\) seems to be increased. This information should be considered when searching for ALPs in compact star models using observational data.
10 Conclusions
This investigation deals with an alternative to the method presented in [24] for analyzing \(\phi \gamma \) and \(\phi ^{\prime }\gamma \) interactions in various backgrounds, and highlights the salient features in each method.
Because the interactions take place in various backgrounds, the dispersion relation is affected. We have demonstrated that for magnetized vacuum, this change is consistent with modified Lorentz symmetry, implying that similar modifications to the dispersion relation in other backgrounds would also respect the modification to the Lorentz symmetry realized there.
In this paper we have compared the mixing dynamics of \(\frac{1}{M}\phi FF\) and \(\frac{1}{M} \phi ' F \tilde{F}\) with the inclusion of the photon self-energy correction (PSEC) evaluated in a weakly magnetized finite-density medium to linear order in the \(eB/m^{2}_{e}\). The inclusion of the PSEC modifies the mixing dynamics of the available degrees of freedom among themselves; as a result, the mixing matrix for \(\frac{1}{M}\phi FF\) becomes \(3 \times 3\) and that for \(\frac{1}{M} \phi ' F \tilde{F}\) becomes \(4 \times 4 \) from \(2 \times 2\), as realized in \(\textbf{MV}\). That is, for the \(\phi \gamma \) system, there is a mixing between only three degrees of freedom, but for the \(\phi '\gamma \) system there is a mixing among all four available degree of freedom: the pseudoscalar, the longitudinal, and the two transverse degrees of freedom of the photon. As a result, when PSEC is considered, the spectro-polarimetric signal undergo substantial modification. Here we have provided the exact solution for each case.
We also numerically estimated the difference in the polarimetric observables due to the change in the mixing pattern for both the \(\phi \gamma \) and \(\phi '\gamma \) systems and plotted them in figures. These include plots generated in the low-energy region, using the exact results of the \(\phi \gamma \) system and perturbative solutions of the \(\phi ^{\prime }\gamma \) system. In addition, we plotted observables against energy evaluated using the exact solutions of both \(\phi \gamma \) and \(\phi ^{\prime }\gamma \) systems for a moderately higher energy range. Our results indicate that the differences in the observables are within the detector resolution limit.
Our findings indicate that the magnetized environment of compact astrophysical systems may be a good place to look for the dark matter signatures coming from both scalars and pseudoscalars. Looking for dark matter signals from such systems would complement the searches going on elsewhere [40, 80] and hence should be pursued. Lastly, we hope that the results of this work can be used to study the magnetized plasma effects on ALP photon interaction in curved space-time[83].
Data Availability Statement
This manuscript has no associated data. [Author’s comment: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.]
Code Availability Statement
Code/software will be made available on reasonable request. [Author’s comment: The code/software generated during and/or analysed during the current study is available from the corresponding author on reasonable request.]
Notes
Any modification to \(\Pi _{T}\) and \(\Pi _{L}\) due to any higher-order corrections from magnetic field eB does not do that.
We have verified this numerically.
The rest of the variables are the elements of M defined in Eq. (13.6).
References
Y.M. Cho, J.H. Kim, Phys. Rev. D 79, 023504 (2009)
T. Damour, J.F. Donoghue, Phys. Rev. D 82, 084033 (2010)
R.D. Peccei, H.R. Quinn, Phys. Rev. Lett. 38, 1440 (1977)
R.D. Peccei, H.R. Quinn, Phys. Rev. D 16, 1791 (1977)
R.D. Peccei (1989) Report No. arXiv:hep-ph/9606475; for a more extensive review of the strong CP problem, see in CP Violation, ed. by C. Jarlskog (World Scientific, Singapore, 1989)
S. Weinberg, Phys. Rev. Lett. 40, 223 (1978)
F. Wilczek, Phys. Rev. Lett. 40, 279 (1978)
Y. Chikashige, R. Mohapatra, R.D. Peccei, Phys. Rev. Lett. 45, 1926 (1980)
Y. Chikashige, R. Mohapatra, R.D. Peccei, Phys. Lett. B 98, 265 (1981)
G. Gelmini, M. Roncadelli, Phys. Lett. B 99, 411 (1981)
P. Svrcek, E. Witten, JHEP 06, 061 (2006)
A. Arvanitaki, S. Dimopoulos, S. Dubovosky, N. Kaloper, J. March-Russel, Phys. Rev. D 81, 123530 (2010)
J.D. Conlon, J.D.E. Marsh, Phys. Rev. Lett. 111, 151301 (2013)
D.J.E. Marsh, Phys. Rep. 643, 1 (2016)
J. Maharana, A. Das, Phys. Lett. B 699, 264 (2011)
J.P. Conlon, F. Quevedo, JCAP 08, 019 (2007)
F.D. Steffen, Eur. Phys. J. C 59, 537 (2009), section (2.1) and the references their in
S. Sarkar, Rep. Prog. Phys. 59(12), 1493 (1996)
E. Kolb, M.S. Turner, Phys. Rev. D 34, 2197 (1986)
S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, New York, 1972)
CAST Collaboration, V. Anastassopoulos, S. Aune, et al., Nat. Phys. 13, 584–590 (2017)
A. Ayala, I. Domanguez, M. Giannotti, A. Mirizzi, O. Straniero, Phys. Rev. Lett. 113, 191302 (2014)
G. Raffelt, L. Stodolsky, Phys. Rev. D 37, 1237 (1988)
G.G. Raffelt, Stars as Laboratories for Fundamental Physics: The Astrophysics of Neutrinos, Axions, and Other Weakly Interacting Particles (Chicago University Press, Chicago, 1996)
A.K. Ganguly, M.K. Jaiswal, Phys. Rev. D 90, 026004 (2014)
A. Chaubey, M.K. Jaiswal, A.K. Ganguly, Phys. Rev. D 102, 123029 (2020)
A. Chaubey, M.K. Jaiswal, A.K. Ganguly, Phys. Rev. D 107, 023008 (2023)
S.V. Chavez, A.E. Shabad, Phys. Rev. D 86, 105040 (2012)
S.V. Chavez, A.E. Shabad, C. Muller, EPJC 81, 331 (2021)
S.L. Cherkas, K.K. Batrakov, D. Matsukevich, Phys. Rev. D 66, 065011 (2002)
S. Adler, Ann. Phys. 67, 599 (1971)
V. Bauch, A. Zuk, T. Kahniashvili, Phys. Rev. D 37, 1237 (1988)
A.K. Ganguly, R. Parthasarathy, Phys. Rev. D 68, 106005 (2003)
P. Majumdar, Mod. Phys. Lett. A 19, 1319 (2004)
N.V. Mikheev, G. Raffelt, L. Vasilevskaya, Phys. Rev. D 58, 055008 (1998)
R. Das, P. Jain, J.P. Ralston, R. Saha, Pramana 70, 439 (2008)
A.K. Ganguly, Ann. Phys. 321(6), 1457 (2006)
A.K. Ganguly, P. Jain, S. Mandal, Phys. Rev. D 79, 115014 (2009)
H. Tercas, J.D. Rodrigues, J.T. Mendonca, Phys. Rev. Lett. 120, 181803 (2018)
F. Miniati, G. Gregory, B. Revile, S. Sarkar, Phys. Rev. Lett. 121, 021301 (2018)
K. Choi, H. Kim, T. Sekiguchi, Phys. Rev. Lett. 121, 031102 (2018)
Please check with the supplementary document for this work
E.J. Ferrer, V. de la Incera, Phys. Rev. D 49, 6 (1994)
A.K. Ganguly, M.K. Jaiswal, J. Korean Phys. Soc. 72(1), 6 (2018)
A.K. Ganguly, S. Konar, P.B. Pal, Phys. Rev. D 60, 105014 (1999)
J.C. DOlivo, J.F. Nieves, S. Sah, Phys. Rev. D 67, 025018 (2003)
H. Perez Rojas, A.E. Shabad, Ann. Phys. (N.Y.) 121, 432–455 (1979)
H.P. Rojas, Z. Eksp, Teor. Fiz. 76, 3–16 (1979)
A. Chaubey, A.K. Ganguly, to be communicated
F. Karbstein, Phys. Rev. D 88, 085033 (2013)
L.F. Urrutia, Phys. Rev. D 17, 8 (1978)
A.K. Ganguly, to be communicated
K. Zioutas et al. [CAST Collaboration], Phys. Rev. Lett. 94, 121301 (2005)
K. Zioutas et al. [CAST Collaboration], Nat. Phys. 13, 584 (2017)
E. Armengaud et al., arXiv:1401.3233 [INSPIRE]
XENON100 Collaboration, E. Aprile et al., Phys. Rev. D. 90, 062009 (2014) [Erratum ibid. 95, 029904 (2017)] arXiv:1404.1455
DARWIN Collaboration, J. Aalbers et al., JCAP 11, 017 (2016)
MADMAX Working Group Collaboration, A. Caldwell et al., Phys. Rev. Lett. 118, 091801 (2017)
I.G. Irastorza, J. Redondo, arXiv:1801.08127
A.J. Millar, J. Redondo, F.D. Steffen, JCAP 10, 006 (2017)
E. Masaki, A. Aoki, J. Soda, Phys. Rev. D 96, 043519 (2017)
A. Kartavtsev, G. Raffelt, H. Vogel, JCAP 024(01), 1701 (2017)
H. Primakoff, Phys. Rev. 81, 899 (1951)
R. Chanda, J.F. Nieves, P.B. Pal, Phys. Rev. D 37, 2714 (1988)
A. Ringwald, arXiv:1506.04259
M. Giannotti, arXiv: 1508.07576
M. Giannotti, I. Irastorza, J. Redondo, A. Ringwald, JCAP 05, 057 (2016)
M. Giannotti, I.G. Irastorza, J. Redondo, A. Ringwaldd, K. Saikawad, JCAP 10, 010 (2017)
K. Saikawa, T.T. Yanagida, arXiv:1907.07662v1
A.K. Ganguly, M.K. Jaiswal, to be communicated
P. Goldreich, W.H. Julian, ApJ 157 (1969)
E. Braaten, D. Segel, Phys. Rev. D 48, 4 (1993)
E. Silver, H. Schnopper, Bragg crystal polarimeters, in X-Ray Polarimetry: A New Window in Astrophysics. ed. by R. Belazzini, E. Costa, G. Matt, G. Tagliaferri (Cambridge University Press, Cambridge, 2010)
M.C. Weisskopf et al., RINP 6, 1179 (2016)
V. Tatischeff et al., arXiv:1706.07031v2
B. Rani et al., arXiv:1903.04607v1
Z. Wadiasingh et al., arXiv:1903.05648
S.L. O Dell et al., arXiv:1010.4892 [astro-ph.IM], https://doi.org/10.1117/12.862315
H.S. Krawczynski, arXiv:1904.09313 (2019)
J.R. Espinosa, C. Grojean, G. Panico, A. Pomarol, O. Pujols, G. Servant, Phys. Rev. Lett. 115, 251803 (2015)
A. Hernandez-Galeana, Phys. Rev. D 093, 006 (2007)
C.M. Kim, S.P. Kim, arXiv:2210.12890
D.J. Schwarz, J. Gowswami, A. Basu, Phys. Rev. D 103, L081306 (2021)
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The original online version of this article was revised: the title was incorrectly given as ‘Final-REVISION “Differentiating dilatons from axions by their mixing with photons” but should have been ‘Differentiating dilatons from the axions by their mixing with photons’.
Appendices
Appendix A
1.1 Diagonalizing the 3 \(\times \) 3 mixing matrix
Our objective here is to obtain the analytical expression for the diagonalizing (unitary) matrix U, for the Hermitian matrix M given in Sect. 6 Eq. (6.2). We express the elements of (U) in terms of algebraic expressions in order to maintain the desired numerical accuracy (O(10\(^{-15}\))). To this end, we need to solve for the characteristic equation obtained from \(Det(\textbf{M}- \lambda {I}) = 0\) to find the eigenvalues (roots) of M, and then use these to find the corresponding eigenvectors. Finally, using the eigenvectors, we construct the unitary matrix U that diagonalizes M. The characteristic equation for this \(3 \times 3\) Hermitian matrix M, for obvious reasons, turns out to be a cubic equation, having real roots.
The cubic equation that follows from the characteristic equation can be written in terms of parameters b, c, and d as
where the parameters b, c, and d are functions of the elements of mixing matrix M, denoted by
Next we introduce the variables P and Q, where \(P= \left( \frac{3c-b^2}{9}\right) \) and \(2Q = \left( \frac{2b^3}{27} -\frac{bc}{3} +d \right) \); the roots are
The orthonormal eigenvectors \(\textbf{X}_j\) of M are to be found from the matrix relation \(\left[ \textbf{M}-\lambda _j\right] \left[ \mathbf{X_{j}}\right] = 0.\) In terms of its elements, the normalized column vector \(\left[ \mathbf{X_{j}}\right] \) can be denoted as
Following standard methods, one can evaluate these elements in terms of the roots \(\lambda _j\) and elements of the mixing matrix M. From this we obtain the following:
Here, \(\mathcal{{N}}^{(\textrm{j})}_{\textrm{vn}}\) is the normalization constant, and j can take values from 1 to 3. Using these eigenvectors, the unitary matrix U is
The obtained unitary matrix (6.3) diagonalizes the 3\(\times \)3 mixing matrix \(\textbf{M}\). We used the diagonalized mixing matrix \(\textbf{M}_{D}\) to produce the solution of the field equation.
Appendix B
1.1 Diagonalizing the \(4\times 4\) mixing matrix pertubatively
Diagonalizing the matrix \(\textbf{M}^{\prime }\) given in (6.12) with its full generality is a tedious task, which involves finding the roots of a fourth-order polynomial. Therefore, this was initially performed perturbatively in [38]. In the following we briefly discuss that procedure.
In order to find the approximate eigenvalues and eigenvectors, we can write \(\textbf{M}^{\prime }\) as a sum of two parts,
The first matrix on the rhs of Eq. (12.1) is \(\textbf{M}^{ \prime 0}\), and the second is \(\textbf{M}^{ \prime \delta } \), constructed with the \(M^{\prime }_{34}\) and \(M^{\prime }_{43}\). In the limit \(m_{\phi '} \rightarrow 0\), the eigenvalues of the \(\textbf{M}^{ \prime 0} \) matrix can be expressed in terms of the variables X and Y (defined as \(X = \frac{F}{(m^2_{\phi '} - \omega ^2_p)}\) and \(Y = \frac{G}{(m^2_{\phi '} - \omega ^2_p)}\)) as follows:
The eigenvectors of \(\textbf{M}^{ \prime 0}\), defined in terms of \({\tilde{\lambda }_1} = \sqrt{X^2 +\frac{Y^2}{4}}- \frac{Y^2}{2}\), \({\tilde{\lambda }_2} = -\sqrt{X^2 +\frac{Y^2}{4}}- \frac{Y^2}{2}\), and \({\tilde{\lambda }_3} = 1 + {Y^2}\), are
In Eq. (12.3) the variables \(d_1 = \sqrt{1 + \frac{\tilde{\lambda }^2_1}{X^2}}\), \(d_2 = \sqrt{1 + \frac{\tilde{\lambda }^2_2}{X^2}}\) and \(d_3 = d_4 =1.\)
As mentioned before, we estimated the approximate eigenvalues and eigenvectors of matrix \(\textbf{M}^{\prime }\) by treating \(\textbf{M}^{\prime \delta }\) as a perturbation over \(\textbf{M}^{\prime 0}\). The eigenvectors under this approximation are as follows:
In the above equations, \(v_1 = i \frac{\tilde{\lambda }_1}{X} \), \(v_2 = i \frac{\tilde{\lambda }_2}{X} \), \(v_4 = -iY \), and \(w_1 = Y\frac{\tilde{\lambda }_1}{X} \), \(w_2 = Y\frac{\tilde{\lambda }_2}{X} \), \(w_4 = 1\). We use these eigenvectors to construct the unitary matrix \(\textbf{U}\) to diagonalize the mixing matrix \(\textbf{M}^{\prime }\) by unitary transformation. The constructed unitary matrix is given as follows:
Variables introduced in (12.5) are defined as follows: \(\xi _i = \frac{Lw_i}{d_i(\lambda _{3}^0-\lambda _{i}^0)}\) (i can take the values 1, 2, 4).
Here, u\(_j\), v\(_j\),w\(_j\), and x\(_j\) are the components of the \(j^{th}\) column vectors of U, and
is the corresponding normalization constant (\(j=1, 2, 3, 4\)).
Appendix C
1.1 Diagonalizing the \(4\times \)4 mixing matrix exactly
1.1.1 Converting a real symmetric matrix (M) into a Hermitian matrix (\({ M_{H}}\)) using a global unitary transformation
In order to obtain the solutions of the EOMs of the \(\phi ^{\prime }\gamma \) interaction in terms of propagating degrees of freedom, we need to diagonalize the Hermitian mixing matrix (\({M}_{H}\)), given by
where \(F=\Pi _{p}N_{1}N_{2}\epsilon _{\mu \nu \delta \beta } \frac{k^{\beta }}{|k|}u^{\tilde{\delta _{\parallel }}}b^{(1)\mu }I^{\nu }\), \(G= -g_{\phi '\gamma \gamma }N_{2}b_{\mu }^{(2)}I^{\mu } \), and \(L= -g_{\phi '\gamma \gamma }N_{L}b_{\mu }^{(2)} {\tilde{u}^{\mu }}\), obtained from the EOMs provided in section (5) of the supplemental document of this article[42]. This can be performed with less effort if this matrix can be transformed into a real symmetric matrix (M) that can be diagonalized with a unitary matrix \(\textbf{U}\) whose form is known to us.
To do this, we begin with the compact form of \(M_{H}\) given by
such that the elements in (13.2) are as follows: \(a =\alpha = \omega ^{2}_{p} \), \(b = m^{2}_{\phi _{i}} \), \(c = \Pi _{L} = \omega _{p}^{2}\left( 1-\frac{\omega _{p}^{2}}{\omega ^{2}} \right) \), \(d = G = g_{\phi \gamma \gamma }\omega B \sin (\pi /4) \), \(e = F = \omega _{B}\omega ^{2}_{p}\cos (\pi /4)/\omega \), and \(\delta = L = g_{\phi \gamma \gamma }\omega _{p}B \sin (\pi /4) \). The real symmetric \(4 \times 4\) matrix M is defined in the following form:
which can be diagonalized with a known unitary matrix \(\textbf{U}\). Now, to transform matrix M into a Hermitian matrix \(M_{H}\), we would use a global transformation. To this end, we start with a unitary matrix \(\mathcal{{Q}}\) and its conjugate \(\mathcal{{Q}}^{\dagger }\) given by
such that \(\mathcal {Q}^{\dagger }\mathcal {Q}=\)1. The unitary matrix to diagonalize matrix M given by Eq. (13.3) is known and will be denoted by \(\textbf{U}\). The elements and structure of this matrix will be discussed later in Sect. 13.3.
Under a global rotation by \(\mathcal{Q}\) and \(\mathcal{Q}^{\dagger }\), the matrix M transforms itself as
which after simplification yields
where \({ M_{H}} \) is a Hermitian matrix. We need to diagonalize \(M_{H}\) to find the solutions of the propagating modes.
1.2 Characteristic equation
To diagonalize the matrix \({ M_{H}}\), we need to obtain its eigenvalues (\(\lambda _{i}\) for \(i = 1,2,3,4\)), which can be obtained by solving the characteristic equation, given as
After a tedious but straightforward simplification, we obtain the characteristic equation in the form of a quartic polynomial, given by
The eigenvalues of the matrix (13.1) are given by the roots of the characteristic equation of the matrix \({M}_{H}\). The characteristic Eq. (13.8) of \({M}_{H}\) in shorthand notation is given by
where the expressions for the coefficients \(a_{i}\) (for \(i = 0\) to 3) are given by
The roots of the given quartic Eq. (13.9) are given by
where the variables \(Z_{1}, Z_{2}, Z_{3}\), and \(Z_{4}\) are given in terms of the variables \(\mathcal{X}, N\), and K as
where N and K follow from
The variable \(\mathcal{X}\) is to be evaluated from the value of \(\mathcal{X}^{2}\) that happens to form the roots of the resolving cubic equation given by
Out of the three possible roots of the resolvent cubic equation written in terms of \(\mathcal{X}^{2}\), the one that yields real value for \(\mathcal{X}\) is to be considered. The variables N and K are to be evaluated using the variables \(\mathcal{X}\), \(\mathcal{X}^{2}\), \(\tilde{G}\), and H expressed in the following form. The variables \(\tilde{G}\) and H happen to be the coefficients of the resolvent cubic of power two and zero, respectively.
The roots of the resolvent cubic equation are obtained by introducing the variables \(\tilde{P}\) and \(\tilde{Q}\), where \(\tilde{P}= \left( \frac{3C-B^2}{9}\right) \) and \(2\tilde{Q} = \left( \frac{2B^3}{27} -\frac{BC}{3} +D \right) \). Therefore, the roots are obtained as follows:
where
and the variables H, \(\tilde{G}\), and I in terms of the coefficients of the characteristic Eq. (13.9) are given by
The criteria for selecting one particular root out of the three given in Eq. (11.5) are to ensure that the substitution of the particular root keeps the roots of the quartic equation real.
In fact, the roots obtained in this way happen to be the eigenvalues of the matrix \(M_{H}\) (given by (13.6)), and the real symmetric matrix M is established from the identities \(Det(\mathcal{Q}^{\dagger }M\mathcal{Q}) = Det M = Det M_{H}\).
1.3 Construction of the unitary matrix \(\textbf{U}_{H}\) to diagonalize \({M_{H}}\)
Our next objective is to construct the unitary matrix that diagonalizes the matrix \({ M_{H}}\). This matrix (denoted as \(\textbf{U}_{H}\)) can diagonalize the matrix \({ M_{H}}\) following the transformation \(\mathbf{{U}^{\dagger }_{H}}{M_{H}{} \textbf{U}_{H}=\mathbf{M_D}}\). Since the diagonal elements of M and \(M_{H}\) are the same, we should have Eq. (13.3),
Using Eq. (13.32), we can further find the unitary matrix \(\textbf{U}\) that diagonalizes the real symmetric matrix M, using Eq. (13.32) by the following identifications \(\mathcal{Q}\textbf{U}_{H}=\textbf{U}\) and \((\mathcal{Q}{} \textbf{U}_{H})^{\dagger }=\textbf{U}^{\dagger }\) or \(\textbf{U}_{H}= \mathcal{Q}^{\dagger }{} \textbf{U}\).
Since the unitary matrix \(\textbf{U}\) used to diagonalizeFootnote 4 the real symmetric matrix M as reported in [81] is of the form
with variables \(\Delta \) and h given by
and
(for \(i= 1, 2, 3, 4\)), the expression for \(\textbf{U}_{H}\)Footnote 5 following the argument that is \(\textbf{U}_{H}= \mathcal {Q}^{\dagger }{} \textbf{U}\) turns out to be
Appendix D
The pseudoscalar (axion)–photon mixing matrix presented in [24] happens to be a \(3\times 3\) matrix. This equation is obtained in a temporal gauge, with specific choice of basis for the gauge potential. We outline a procedure here (maintaining the principle of covariance) so that using this, one would be able to reproduce a \(4 \times 4\) mixing matrix from the equations for \(\phi ^{\prime }\gamma \) evolution, using the temporal gauge condition starting with Eq. (5.24) of [24].
To begin with, let us denote the vector potential in the form
The variables in this equation were already defined in Eq. (2.10). Then, on contracting Eq. (16.1) with \(u^{\mu }\), we obtain
which happen to be our choice of the gauge condition (i.e, axial gauge). The form factors associated with the components of the gauge field can be obtained by contracting \(\bar{A}^{\mu }\) with \(b^{(1)}_{\mu }\), \(I_{\mu }\), and \(\tilde{u}_{\mu }\), resulting in
The polarization tensor corresponding to the Faraday term is proportional to \(i\epsilon _{\mu \nu \alpha \beta } k^{\beta }u^{\beta }\). This form of the gauge fields used in Eq. (5.24) of [24] along with the polarization tensor part of the Faraday term discussed would produce a \(4\times 4\) mixing equation for the \(\phi ^{\prime }\gamma \) system.
Incidentally, another issue related to the CP property of longitudinal degrees of freedom mentioned in the seminal work of Raffelt and Stodolsky [23] can be verified using our formalism directly. To prove this, we start with \(\tilde{F}^{\mu \nu }f_{\mu \nu }\) given by
Under CP, the lhs acquires a negative sign, so under this transformation, the rhs should also acquire a negative sign. Since \(A_{\perp }\) is odd under CP, to maintain consistency, \(A_{L}\) should also pick up a negative sign. Therefore, one can see that \(A_{L}\) is odd under CP.
Appendix E
The expressions of the form factors of the polarization tensor can be evaluated in terms of Lorentz scalars (provided in [47]) represented by \(I_{i} = p, s, t\) for \(i = 1, 2, 3\) and are defined as
where the electron and positron thermal distribution functions in the presence of magnetic field B (at a temperature T, momentum \(p_{3}\), chemical potential \(\mu \), and relativistic energy \( \mathcal {E}_{l} (p_{3}) = (m_{e}^{2} + p^{2}_{3} + 2eBl)^{\frac{1}{2}}\)) at Landau levels l are given by
Here, the temperature is set by the variable \(\beta = \frac{1}{T}\). The expressions corresponding to \(i = 1\) and 3, i.e., for p and t, are
where the first term inside the bracket of the integrand is defined as \( \Delta ^{(1)}_{l,l^{\prime }} = \frac{k^{2}}{k_{0}^{2}-k^{2}_{3}}F^{(2)}_{l,l^{\prime }}{\left( \frac{k^{2}_{\perp }}{eB}\right) } \) and \( \Delta ^{(3)}_{l,l^{\prime }} = F^{(2)}_{l,l^{\prime }}{\left( \frac{k^{2}_{\perp }}{eB}\right) } \). In the the second term, \(\alpha _{l,l^{\prime }} = {(k_{0}^{2}-k^{2}_{3}) + 2eB(l^{\prime } - l)}\), \(\Theta ^{(1)}_{l,l^{\prime }} = \frac{k^{2}}{k_{0}^{2}-k^{2}_{3}}\left[ \lambda _{l,l^{\prime }}F^{(2)}_{l,l^{\prime }}{\left( \frac{k^{2}_{\perp }}{eB}\right) } - 4k^{2}_{\perp }N^{(1)}_{l,l^{\prime }}\left( \frac{k^{2}_{\perp }}{eB}\right) \right] \) and \(\Theta ^{(3)}_{l,l^{\prime }} = \left[ \lambda _{l,l^{\prime }}F^{(2)}_{l,l^{\prime }}{\left( \frac{k^{2}_{\perp }}{eB}\right) } + 4k^{2}_{\perp }N^{(1)}_{l,l^{\prime }}\left( \frac{k^{2}_{\perp }}{eB}\right) \right] \) when \(\lambda _{l,l^{\prime }} = (k_{0}^{2}-k^{2}_{3}) + 2eB(l+l^{\prime })\). In the LWL, i.e., when \(k_{\parallel , \perp } \rightarrow 0\), the term \( N^{(1)}_{l,l^{\prime }} \left( \frac{k^{2}_{\perp }}{eB}\right) \rightarrow 0\)[47, 48], \(\Delta ^{(1)}_{l,l^{\prime }} = \Delta ^{(3)}_{l,l^{\prime }} = F^{(2)}_{l,l^{\prime }} (0) = (\delta _{l,l^{\prime - 1}} + \delta _{l-1,l^{\prime }})\) [48] and \(\Theta ^{(1)}_{l,l^{\prime }} = \Theta ^{(3)}_{l,l^{\prime }} = \lambda _{l,l^{\prime }}F^{(2)}_{l,l^{\prime }} (0)\). In this kinematic limit, the expressions of the form factors p and t will be identical. We use this finding in our further calculations. Lastly, the term \(\Pi ^{0}\) corresponds to the form factor at \(T = 0\).
The transverse form factor (i.e., for \(i = 2\)) Eq. (15.1) can be written in the following form:
where the expressions of \( F_{l,l^{\prime }}^{(1)}\left( \frac{k^{2}_{\perp }}{eB}\right) \) and \(\Theta ^{(2)}_{l,l^{\prime }}\) can be found in [47, 48]. After performing some lengthy algebra to solve the integral of Eq. (15.6), we obtain the following form of s for the field-dependent contribution:
The expression for the scalar function that estimates the longitudinal polarization tensor \(\Pi ^{L}_{\mu \nu }\) is [47]
Using similar techniques as were used for the evaluation of Eq. (15.7) provided in detail in [49]) the field-dependent part of q, i.e., \(q_{(eB)^{2}}\), we obtained
One can find the values of the scalar coefficients (given by Eq. (15) in [47] as p, q, s, and t for the symmetric part of the polarization tensor) making up the photon two-point function after carrying out the loop momentum integration. These parameters or variables are related to the form factors \(\Pi ^{(j)}s\) (given in Sect. 5.2 of this article) in the following ways:
An external magnetic field for which only \(F^{12} \ne 0, k\tilde{F}^{2}k = k_{\parallel }^{2} B^{2}, (u.k)^{2} = k^{2}_{0}\), and \(kF^{2}k = k^{2}_{\perp }B^{2}\). Both s and q are found to be the series of even powers in eB and \(\mu \) with \((eB)^{0}\) as the leading contribution in the series. Therefore, to incorporate the magnetic field-dependent part, we retained the series up to quadratic order in eB as higher-order contributions. The remaining higher-order terms will tend to zero in the limit \(\frac{eB}{T^{2}}<< 1\) and \(\frac{k^2_{\perp }}{m_{e}^2}<<1\).
The transverse and longitudinal field-independent parts of the form factor are \(\Pi _{T}(k, T, (eB)^{0}) \sim \Pi _{L}(k, T, (eB)^{0})\sim \omega ^{2}_{p}\). However, the expression of the field eB-dependent part of \(\Pi _{T} (k, \mu , T, (eB)^{2})\) in the LWL is
In the same way, the field-dependent longitudinal part of \( \Pi ^{(eB)^{2}}_{L}\) is
The limit of validity of Eqs. (15.14) and (15.15) for the range of parameters that we are interested in is relegated to the discussion after Eq. (15.23).
The expression for \(\Pi ^{(3)}\) can be estimated by subtracting (the form factor) s of Eq. (15.12) from (the form factor) p given in Eq. (15.10). The expression of \(\Pi ^{(3)}\) after this exercise is
Thus, in the LWL, \(\Pi ^{(3)} = \frac{(p-s)}{k_{\parallel }^{2}B^{2}}\). Using this expression in Eq. (15.13), we further arrive at
or, using the equality between p and t under LWL, we obtain
Now, if we use the equation for \(\Pi ^{(3)}\) in Eq. (15.18) in the LWL, we arrive at
If we use the expression for s from Eq. (15.12) in Eq. (15.19), we obtain \(\Pi ^{(2)}\mathcal{Z}^{(2)\mu \nu }=\left( \frac{k_{3}}{k_{\parallel }}\right) ^{2}\hat{b}_{1}^{\mu }\hat{b}_{1}^{\nu }\Pi _{L}\), where \(\hat{b}_{1}^{\mu }, \hat{b}_{1}^{\nu }\) are unit vectors in the transverse directions (see Eq. (2.11) in the main body of this paper). This contribution becomes sub-leading when \(k_{0} \ll m_{e}\) and \(k_{\perp } \rightarrow 0\) (i.e., in LWL).
To find the contribution of \(\Pi ^{(3)}\), it should be noted that \(A^{\mu }\mathcal{Z}^{(3)}_{\mu \nu }A^{\nu } \rightarrow k^{2} A_{\mu }F^{\mu }_{\lambda } F^{\lambda \nu }A_{\nu }\) due to the choice of our gauge (i.e., \(k_{\mu }A^{\mu } = 0\)), and for nonzero spatial parts of the external field \(F_{\mu \nu }\), this reduces to
Now if we use the result (i.e., \(p = t\)) obtained by imposing the LWL on the scalar form factors and use the expression for \(\Pi _{L}\) in order to solve for \(\Pi ^{(3)}\), it can be shown after some tedious algebra that
and we obtain
Therefore, \(A_{\mu } \Pi ^{(3)\mu \nu }A_{\nu } = \frac{|\vec {k}^{2}|}{k^{2}_{\parallel }}A_{\mu } \delta ^{\mu _{\perp }}{^{\nu _{\perp }}}A_{\nu }\) \(\Pi _{L}\). In the LWL, this contribution becomes sub-leading.
For the range of parameters such that \(m_{e}^{2}> T^{2} > eB\), the field-dependent term in \(\Pi _{L}\) contributes as
So this term is subdominant in the limit under consideration and can be neglected. The form factor corresponding to Eq. (15.14) is also subdominant in the LWL. This justifies our action of neglecting the contributions to \(L_{\textrm{eff}}\) originating with the tensorial basis given by Eqs. (5.7)–(5.9). The correction to the \(L_{\textrm{eff}}\) following from the odd field-dependent (tensorial basis) part of the polarization tensor follows from \(\Pi ^{(6)/(5)}_{\mu \nu }; i.e., \Pi ^{(6)/(5)} \mathcal{Z}^{(6)/(5)}_{\mu \nu }\). In real-time formalism of finite-temperature field theory, this was estimated in [45]. In the weak field limit, that expression turns out to be
The convention of performing the summation over repeated appearances of the Lorentz indices \(\alpha _{\parallel }\) and \(\tilde{\alpha }_{\parallel }\) following Einstein’s convention is explained in [45]. The dominant contribution to this expression appears for the Lorentz indices \(\beta = 0\) and \(\alpha = 3\), and corresponds to the contribution following from \( \Pi ^{(6)}\mathcal{Z}^{(6)}_{\mu \nu }\). The other one that follows from \( \Pi ^{(5)}\mathcal{Z}^{(5)}_{\mu \nu }\) can be recovered or identified for the contribution for \(\beta = 3\) and \(\alpha _{\parallel } = 0\). However, as was already noted and also explained in [45], this contribution is subdominant in the limit \(|\vec {k}| \rightarrow 0\), so this term is not retained in our analysis.
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Chaubey, A., Jaiswal, M.K., Singh, D. et al. Differentiating dilatons from the axions by their mixing with photons. Eur. Phys. J. C 84, 627 (2024). https://doi.org/10.1140/epjc/s10052-024-12851-1
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Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s10052-024-12851-1