Axion-photon mixing in quantum field theory and vacuum energy

TL;DR

This paper investigates axion-photon mixing within quantum field theory, revealing vacuum energy effects that resemble dark energy and suggesting potential experimental detection of axion-like particles through vacuum polarization phenomena.

Contribution

It introduces a quantum field theory framework for axion-photon mixing, showing vacuum energy corrections and their cosmological implications, which were not previously analyzed in this context.

Findings

Vacuum energy from axion-photon mixing mimics cosmological constant behavior.

Corrections to oscillation formulas depend on magnetic field strength.

Possible detection of vacuum polarization effects in laboratory experiments.

Abstract

We analyze axion--photon mixing in the framework of quantum field theory. The condensate structure of the vacuum for mixed fields induces corrections to the oscillation formulae and leads to non-zero energy of the vacuum for the component of the photon mixed with the axion. This energy generates a new effect of the vacuum polarization and it has the state equation of the cosmological constant, $w = -1$. This result holds for any homogeneous and isotropic curved space-time, as well as for diagonal metrics. Numerical estimates of the corrections to the oscillation formulae are presented by considering the intensity of the magnetic field available in the laboratory. Moreover, we estimate the vacuum energy density induced by axion--photon mixing in the Minkowski space-time. A value compatible with that of the energy density of the universe can be obtained for axions with a mass of $(10^{-3}-10^{-2}) eV$ in the presence of the strong magnetic fields that characterize astrophysical objects such as pulsars or neutron stars. In addition, a value of the energy density less than that of the Casimir effect is obtained for magnetic fields used in experiments such as PVLAS. The vacuum polarization induced by this energy could be detected in next experiments and it might provide an indirect proof of the existence of the axion--photon mixing. The quantum field theory effects presented in this work may lead to new methods for studying axion-like particles.

Figures

Equations

$$L$$

$$\displaystyle\left(\omega-i\partial_{z}+\textit{M}\right)\left(\begin{array}[]{c}\gamma_{\|}\\ a\\ \end{array}\right)=0\,.$$

$$\displaystyle\textit{M}=-\frac{1}{2\omega}\,\left(\begin{array}[]{cc}\omega_{P}^{2}&-g\omega B_{T}\\ -g\omega B_{T}&m_{a}^{2}\\ \end{array}\right),$$

$$\displaystyle\left(\begin{array}[]{c}\gamma^{\prime}_{\|}(z)\\ a^{\prime}(z)\\ \end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\\ \end{array}\right)\left(\begin{array}[]{c}\gamma_{\|}(z)\\ a(z)\\ \end{array}\right)\,,$$

$${\gamma }_{\Vert }(z)={\gamma }_{\Vert }^{\prime }(z)\cos \vartheta +a^{\prime }(z)\sin \vartheta$$

$$a(z)=-{\gamma }_{\Vert }^{\prime }(z)\sin \vartheta +a^{\prime }(z)\cos \vartheta ,$$

$${\gamma }_{\Vert }(z)=G_{\vartheta }^{-1}(t){\gamma }_{\Vert }^{\prime }(z)G_{\vartheta }(t)$$

$$a(z)=G_{\vartheta }^{-1}(t)a^{\prime }(z)G_{\vartheta }(t).$$

$${\omega }_{k,{\gamma }_{\Vert }^{\prime }}={\omega }_k+{\Delta }_{+},$$

$${\omega }_{k,a^{\prime }}={\omega }_k+{\Delta }_{-},$$

$${\Delta }_{\pm }=-\frac{{\omega }_P^2+m_a^2}{4{\omega }_k}\pm \frac{1}{4{\omega }_k}\sqrt{({\omega }_P^2-m_a^2{)}^2+(2g{\omega }_k{B}_T{)}^2}.$$

$${\alpha }_{\mathbf{k},{\gamma }_{\Vert }}(t)$$

$${\alpha }_{\mathbf{k},a}(t)$$

$$|{\Sigma }_{\mathbf{k}}|\equiv \frac{1}{2}\left(\sqrt{\frac{{\omega }_{k,{\gamma }_{||}^{\prime }}}{{\omega }_{k,a^{\prime }}}}+\sqrt{\frac{{\omega }_{k,a^{\prime }}}{{\omega }_{k,{\gamma }_{||}^{\prime }}}}\right),$$

$$|{{\rm Y}}_{\mathbf{k}}|\equiv \frac{1}{2}\left(\sqrt{\frac{{\omega }_{k,{\gamma }_{||}^{\prime }}}{{\omega }_{k,a^{\prime }}}}-\sqrt{\frac{{\omega }_{k,a^{\prime }}}{{\omega }_{k,{\gamma }_{||}^{\prime }}}}\right),$$

$${}_{{\gamma }_{||},a}⟨0(\vartheta ,t)|{\alpha }_{\mathbf{k},{\gamma }_{||}^{\prime }}^{†}{\alpha }_{\mathbf{k},{\gamma }_{||}^{\prime }}|0(\vartheta ,t){⟩}_{{\gamma }_{||},a}={\sin }^2\vartheta |{{\rm Y}}_{\mathbf{k}}{|}^2,$$

$$|0(\vartheta ,t){⟩}_{\gamma ,a}=|0(\vartheta ,t){⟩}_{{\gamma }_{||},a}\otimes |0{⟩}_{{\gamma }_{\perp }^{\prime }},$$

$${\mathcal{P}}_{\gamma \to \sigma }^{\mathbf{k}}(t)$$

$${\mathcal{P}}_{\gamma \to \gamma }^{\mathbf{k}}(t)$$

$${\mathcal{P}}_{\gamma \to a}^{\mathbf{k}}(t)$$

$${\psi }_i^C(\mathbf{x},t)=\frac{1}{\sqrt{V}}\sum _{\mathbf{k}}\left[u_{\mathbf{k},i}(\mathbf{x},t){\alpha }_{\mathbf{k},i}+u_{\mathbf{k},i}^{\ast }(\mathbf{x},t){\alpha }_{\mathbf{k},i}^{†}\right],$$

$$T_{\mu \nu }(x)$$

$${\Xi }_{\mu \nu }^{\Vert }$$

$${\rho }_{\Vert }$$

$$p_{\Vert }$$

$${\Xi }_{\mu \nu }^{\perp }(x)$$

$${\rho }_{\Vert }=\frac{\Delta m^2{\sin }^2\theta }{2}\frac{1}{V}\sum _{\mathbf{k}}\left(|{\zeta }_{\mathbf{k},1}(t){|}^2-|{\zeta }_{\mathbf{k},2}(t){|}^2\right),$$

$$d{s}^2$$

$${\rho }_{\Vert }=\frac{\Delta m^2{\sin }^2\theta }{4{\pi }^2}{\int }_0^{K}dk{k}^2\left(1/2{\omega }_{k,{\gamma }_{\Vert }^{\prime }}-1/2{\omega }_{k,a^{\prime }}\right),$$

$${\rho }_{\Vert }$$

$$K<\frac{\sqrt{2m_{a^{\prime }}^2+g^2{B}_T^2}}{2}.$$

$${\gamma }_{\Vert }^{\prime }(x)=\int \frac{d^{3}k}{(2\pi {)}^{\frac{3}{2}}}\frac{1}{\sqrt{2{\omega }_{k,{\gamma }_{\Vert }^{\prime }}}}\left({\alpha }_{\mathbf{k},{\gamma }_{\Vert }^{\prime }}(t)+{\alpha }_{-\mathbf{k},{\gamma }_{\Vert }^{\prime }}^{†}(t)\right)e^{i\mathbf{k}\cdot \mathbf{x}}$$

$${\pi }_{\gamma }^{\prime }(x)=i\int \frac{d^{3}k}{(2\pi {)}^{\frac{3}{2}}}\sqrt{\frac{{\omega }_{k,{\gamma }_{\Vert }^{\prime }}}{2}}\left({\alpha }_{\mathbf{k},{\gamma }_{\Vert }^{\prime }}^{†}(t)-{\alpha }_{-\mathbf{k},{\gamma }_{\Vert }^{\prime }}(t)\right)e^{i\mathbf{k}\cdot \mathbf{x}}.$$

$$G_{\vartheta }(t)=exp\left[-i\vartheta \int d^{3}x\left({\pi }_{\gamma }^{\prime }(x)a^{\prime }(x)-{\gamma }_{\Vert }^{\prime }(x){\pi }_a^{\prime }(x)\right)\right],$$

Full text

The quantum field theory effects presented in this work may lead to new methods for studying axion-like particles.

I Introduction

The study of ultra-light particles such as axions, which have very small cross-sections for strong and weak interactions, is a promising field of research that could allow the connection between particle physics and cosmology. Axions were first proposed to solve the strong CP problem peccei ; peccei1 . They can be described as boson scalar fields and are expected to have masses in the range of $10^{-6}$ to $10^{-2}$ eV. Recently, interest has increased in ultra-light axions (ULAs) as candidates for dark matter. ULAs naturally arise in the compactifications of some string landscapes Witten1984 ; Hu2000 ; Svrcek2006 ; Arvanitaki2010 ; Marsh2016 ; Hui2017 and they are expected to have masses of $\sim 10^{-22}$ eV because they are initially massless and acquire it via a symmetry-breaking mechanism Preskill1983 ; Abbott1983 ; Dine1983 . ULAs and axions have been studied using N-body simulations Woo2009 ; Schive2014 ; Schive2014b ; Mocz2017 ; Veltmaat2018 , interferometry cap2015 , globular clusters Hui2016 ; Emami2018 , galactic pulsars Khmeinitsky:2013lxt ; idm2017 ; idm2018 , galactic dynamics idm2018b , and the EDGES 21-cm absorption signal bak ; lamb , as well as other probes.

A further property of axions and axion-like particles (ALPs) is axion–photon mixing and oscillation in the presence of strong magnetic fields raffelt . This phenomenon is negligible in the case of ULAs. However it could be detected for axions with masses of $10^{-3}$ and $10^{-2}$ eV. Many experimental studies have attempted to detect axions PVLAS1 ; CAST ; CAST1 ; CAST2 ; ADMX ; ADMX1 and the conversion of photons into axions has also been extensively studied, although without positive results 11 ; 12 ; 13 ; 16 ; 17 . Despite these efforts, the lack of their direct detection demands the consideration of possible experimental setups that could demonstrate the existence of axions and ULAs.

In addition to the need to detect the presence of axions, complete theoretical understandings of the behavior of axions and axion–photon mixing are essentials.

In the framework of quantum field theory (QFT), it has been shown that a non-perturbative vacuum structure is associated with the mixing of neutrinos and bosons such as kaons Blasone:1998hf -CapolupoPLB2004 . In particular, it was found that the conventional treatment of neutrino mixing and meson mixing, where the flavor states (i.e., $|\nu_{e}\rangle$, $|\nu_{\mu}\rangle$ and $\nu_{\tau}\rangle$ for neutrinos) are defined in the Fock space of the energy-eigenstates, is affected by the problem of total probability non-conservation, thereby demonstrating that the mixed states should be treated independently from the energy-eigenstates Blasone:2005ae . In fact, the Fock space of the mixed states is unitary inequivalent to the Fock space of the energy-eigenstates and corrections to the oscillation formulae have been presented. The quantum mechanical result is reproduced only in the relativistic limit of QFT Blasone:1998hf -CapolupoPLB2004 . The contributions to the energy of the universe induced by the condensate structure of the vacuum of mixed particles have also been analyzed Capolupo:2006etS ; Capolupo:2006et2 .

The general theoretical results obtained previously cannot be applied immediately to axion–photon mixing and this phenomenon requires specific adjustments. Indeed, except for axion–photon mixing, all known mixed systems are characterized by the mixing of particles with the same statistic. Moreover, vacuum polarization occurs only in the conversion of photons in axions. Therefore, QFT effects on polarization must be considered.

In this paper, we consider axion–photon mixing in the QFT framework. We present new oscillation formulae for the axion–photon system, demonstrate the presence of QFT vacuum polarization induced by the condensate structure of the vacuum for mixed particles, and analyze axion–photon mixing in curved space-time. We show that the QFT corrections to the amplitudes of the oscillation formulae, which are negligible for neutrinos and kaons, are detectable in principle in the case of axion–photon mixing. Moreover, we show that a non zero vacuum energy with the state equation of the cosmological constant, $w=-1$, characterizes axion–photon mixing. This energy marks only one component of the photon field and therefore it produces further vacuum polarization. These results are obtained for a homogeneous and isotropic universe, and for diagonal metrics. The values of the vacuum energy compatible with that estimated for dark energy can be obtained for axions with masses of $10^{-3}$ to $10^{-2}$ eV in the presence of strong magnetic fields. In addition, energy density values smaller than that of the Casimir effect might be obtained in the laboratory. This energy could be detected in next experiments by means of the analysis of the vacuum polarization.

These contributions are zero in the quantum mechanics mixing treatment. We also verify that all of the QFT effects presented in this study for axions are negligible in the ULA case, where their very low mass inhibits the oscillation with photons.

The paper is organized as follows. In Section II, we present the QFT treatment of axion–photon mixing and show that the non-perturbative vacuum structure of mixed fields leads to modifications of the standard oscillation formulae and to a new polarization effect of the vacuum. In Section III, we study axion–photon mixing in curved space. We show that in any homogeneous background and for diagonal metrics, the vacuum energy of the mixed fields assumes a state equation similar to that for the cosmological constant. Moreover, we analyze the energy of the vacuum condensate in flat space-time. We give our conclusions in Section IV.

II Axion–photon mixing

The ALP–photon system is described by the Lagrangian density:

[TABLE]

where the first two terms are the Lagrangian densities of the free photon and axion, respectively, the third term is the Heisenberg–Euler term due to loop correction in QED Heisenberg -Dobrich , and the last term is the interaction of two photons with the axion pseudoscalar field $a$ in the presence of a magnetic field PDG . This term is responsible for axion–photon mixing. Moreover, $\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$ is the dual electromagnetic tensor, $g_{a\gamma\gamma}\equiv g\equiv g_{\gamma}\alpha/\pi f_{a}$ is the axion–photon coupling with the dimension of the inverse of the energy, $g_{\gamma}\sim 1$, $\alpha=1/137$ and $f_{a}$ is the decay constant for ALPs. In our treatment, we neglect the Heisenberg–Euler term and the birefringence of fluids in a transverse magnetic field (Cotton–Mouton effect).

We consider a monochromatic laser beam propagating along the z-axes through a region permeated by a magnetic field. We select the y-axis along the projection of B perpendicular to the z-axes (this configuration is typical in laser beams used in experiments conducted to test the existence of axions PVLAS1 -ADMX1 ). In this case, the photon polarization state $\gamma_{x}=\gamma_{\bot}$ decouples and the propagation equations can be written as:

[TABLE]

The magnetic field B coincides with the purely transverse field ${B}_{T}$, and M is the mixing matrix:

[TABLE]

where $\omega_{P}=(4\pi\alpha N_{e}/m_{e})^{1/2}$ is the plasma frequency. Here $N_{e}$ is the electron density and $m_{a}$ is the axion mass. In the case of propagation in the vacuum, we have $\omega_{P}=0$. However, the transition rate can be enhanced by filling the conversion region with a buffer gas van because the presence of the gas induces an effective photon mass given by: $m_{\gamma}=\frac{\hbar}{c^{2}}\omega_{P}$.

The matrix M can be diagonalized by a rotation to primed fields:

[TABLE]

where $\theta=\frac{1}{2}\arctan\displaystyle{\left(\frac{2g\omega B_{T}}{m_{a}^{2}-\omega^{2}_{P}}\right)}$ is the mixing angle, $\gamma_{\|}$ and $a$ are the fields associated with the mixed particles, and $\gamma^{\prime}_{\|}$ and $a^{\prime}$ are the “free” fields with definite masses ($m_{\gamma}$ in the presence of plasma, or zero for the photon and $m_{a}$ for the axion).

In order to illustrate the axion–photon mixing in the field-theoretical framework, we invert the mixing relations and consider the following relations:

[TABLE]

where $\vartheta=-\theta$. Notice that any polarization component of a neutral vectorial field behaves as a neutral scalar field. Since only the $\gamma_{\|}$ component of the electromagnetic field mixes with the axion field, then we can treat $\gamma_{\|}$ as a neutral scalar field. Therefore, axion–photon mixing can be considered as the mixing of two neutral scalar fields, and thus the formalism presented in Ref.CapolupoPLB2004 for kaons and the $B_{0}-\bar{B}_{0}$ system can be applied to the present phenomenon, while considering its intrinsic properties.

We quantize the fields $\gamma^{\prime}_{\|}(x)$ and $a^{\prime}(x)$ in the usual manner, and we recast Eq. (15) by means of the generator $G_{\vartheta}(t)$ (see Appendix 1 for details) in the form:

[TABLE]

In the finite volume, $G_{\vartheta}(t)$ is a unitary operator, which allows us to define the vacuum state for mixed fields $\gamma_{||}$ and $a$ as: $|0(\vartheta,t)\rangle_{\gamma_{||},a}\equiv G^{-1}_{\vartheta}(t)\;|0\rangle_{\gamma_{||}^{\prime},a^{\prime}}\,.$ This vacuum in the infinite volume limit is orthogonal to $|0\rangle_{\gamma^{\prime}_{||},a^{\prime}}$, i.e.: $\lim\limits_{V\rightarrow\infty}\,{}_{\gamma_{||}^{\prime},a^{\prime}}\langle 0|0(\vartheta,t)\rangle_{\gamma_{||},a}=0\,.$

The annihilation/creation operators for $\gamma_{\|}$ and $a$ are defined as $\alpha_{{\bf k},\gamma_{\|}}(\vartheta,t)\equiv G^{-1}_{\vartheta}(t)\;\alpha_{{\bf k},\gamma_{\|}^{\prime}}(t)\;G_{\vartheta}(t)$ and $\alpha_{{\bf k},a}(\vartheta,t)\equiv G^{-1}_{\vartheta}(t)\;\alpha_{{\bf k},a^{\prime}}(t)\;G_{\vartheta}(t)$, such that $\alpha_{{\bf k},\gamma_{\|}}(\vartheta,t)|0(\vartheta,t)\rangle_{\gamma_{\|},a}=\alpha_{{\bf k},a}(\vartheta,t)|0(\vartheta,t)\rangle_{\gamma_{\|},a}=0$, where $\alpha_{{\bf k},\gamma_{\|}^{\prime}}(t)=\alpha_{{\bf k},\gamma_{\|}^{\prime}}\,e^{-i\omega_{k,\gamma_{\|}^{\prime}}t}$, $\alpha_{{\bf k},a^{\prime}}(t)=\alpha_{{\bf k},a^{\prime}}\,e^{i\omega_{k,a^{\prime}}t}$, are the annihilation/creation operators for the free fields $\gamma_{\|}^{\prime}$ and $a^{\prime}$, respectively. The energies $\omega_{k,\gamma_{\|}^{\prime}}$ and $\omega_{k,a^{\prime}}$ are given by:

[TABLE]

with $\omega_{k}$ photon energy and

[TABLE]

Explicitly, we have:

[TABLE]

where $|\Sigma_{{\bf k}}|$ and $|\Upsilon_{{\bf k}}|$ are the Bogoliubov coefficients:

[TABLE]

which satisfy the following relation: $|\Sigma_{{\bf k}}|^{2}-|\Upsilon_{{\bf k}}|^{2}=1\,.$

One of the main properties of the vacuum $|0(\vartheta,t)\rangle_{\gamma_{||},a}$ is its structure of condensed particles. For any $t$, the condensation density of $|0(\vartheta,t)\rangle_{\gamma_{||},a}$ is given by:

[TABLE]

and a similar result can be obtained by considering $N_{{\bf k},a^{\prime}}=\alpha_{{\bf k},a^{\prime}}^{{\dagger}}\alpha_{{\bf k},a^{\prime}}$.

We note that the condensation density also originates in neutrino and boson mixing cases Blasone:1998hf -CapolupoPLB2004 . However, in the present case, the axion only mixes with one of the photon components. This behavior together with photon–axion conversion induces polarization of the vacuum.

The proposed mechanism can further affect photon vacuum polarization because the condensate structure of the physical vacuum state defined in Eq. (24) only characterizes the parallel component of the photon. Indeed, the vacuum for axions and photons is given by the following product:

[TABLE]

where $|0(\vartheta,t)\rangle_{\gamma_{||},a}$ is the vacuum for the mixed component of the electromagnetic field $\gamma_{||}$ and $a$, and $|0\rangle_{\gamma^{\prime}_{\bot}}$ is the vacuum for the unmixed component ${\gamma^{\prime}_{\bot}}$. The vacuum $|0\rangle_{\gamma^{\prime}_{\bot}}$ at the difference of $|0(\vartheta,t)\rangle_{\gamma_{||},a}$ has a trivial structure and it does not exhibit a condensed structure. As we show in the following, the vacuum energy is equal to zero for ${\gamma^{\prime}_{\bot}}$. By contrast, $|0(\vartheta,t)\rangle_{\gamma_{||},a}$ lifts the zero point energy to a non-zero and positive value, which leads to the production of a foam of condensed particles in the direction parallel to the photon propagation. This condensate is absent in the direction perpendicular to photon propagation and it derives from the properties of QFT. Indeed, it does not appear in the quantum mechanics mixing treatment.

We now derive the QFT oscillation formulae for the axion–photon system. Assuming an initial state of the photon $|\alpha_{{\bf k},\gamma_{\|}}\rangle=\alpha^{{\dagger}}_{{\bf k},\gamma_{\|}}(0)|0(\vartheta,0)\rangle_{\gamma_{\|},a}\,$, the oscillation formulae are obtained by computing the expectation value of the momentum operator for mixed fields ${\cal P}_{\sigma}(t)=\int d^{3}x\,\pi_{\sigma}(x)\,\nabla\,\phi_{\sigma}(x)$ at $t\neq 0$ normalized to its initial value:

[TABLE]

Explicitly, we have:

[TABLE]

where $\Omega_{k}^{\pm}=\frac{\omega_{k,a^{\prime}}\pm\omega_{k,\gamma_{\|}^{\prime}}}{2}$. Eqs. (27) and (28) are the relative population densities of the photons and axions in the beam, respectively. These equations indicate the presence of a high-frequency oscillation term that is proportional to: $|\Upsilon_{\bf k}|^{2}$. This term is maximal at low momenta and tends to zero for large momenta. It depends on the axion mass, coupling constant, and magnetic field. Moreover, this term is negligible in the case of very low axion masses but in principle, it is detectable in the case of “heavy” axions, as shown in Figs (1) and (2). In these figures, we consider laser experiments with a photon energy $\omega\simeq 1.5eV$. We assume a coupling constant $g_{a\gamma}\sim 1.4\times 10^{-11}GeV^{-1}$ giannotti and an axion mass $m_{a}\in[1-10^{-2}]eV$. Notice that a class of axion models allows axion masses up to $0.2eV$ iaxo ; DiLuzio . Figures (1) and (2) show that the presence of plasma reduces the effect of the QFT corrections. In any case, the values obtained for $|\Upsilon_{\bf k}|^{2}$ are much higher than the corresponding field-theoretical corrections for the other mixed systems.

Indeed, the values do not exceed $10^{-26}$ for mixed bosons, such as $K^{0}-\bar{K}^{0}$, $D^{0}-\bar{D}^{0}$, $B^{0}-\bar{B}^{0}$, and $B^{0}_{s}-\bar{B}^{0}_{s}$, and $10^{-18}$ for neutrinos CapolupoPLB2004 . Therefore, in principle, the axion–photon system could also represent an interesting system for testing the differences between QFT and quantum mechanics.

The presence of the rapidly oscillating component in Eqs. (27) and (28) shows that axions and photons behave as two coupled harmonic oscillators. The origin of this term and the presence of Bogoliubov coefficients in the oscillation formulae are due to the condensate structure of the vacuum for mixed fields. This structure is a genuine field theory phenomenon.

It should be noted that other rapidly oscillating terms were observed in the quantum field calculation for axion helioscopes and they correspond to the emission of “wrong direction” photons Raffelt . Thus, their nature is completely different from that of the terms presented in this paper.

III Axion–photon mixing in curved space and vacuum energy

Now, we analyze the energy of $|0(\vartheta,t)\rangle_{\gamma_{||},a}$ and we show that its state equation has an adiabatic index, $w=-1$. In particular, we study axion–photon mixing in curved space-time. In general, cosmic magnetic fields are not uniform on the typical scales of astrophysical objects and $B$ does not have a fixed orientation, but for simplicity, we consider a homogeneous magnetic field which coincides with its transverse component, $B=B_{T}$.

We quantize the unmixed scalar fields $\psi^{C}_{i}=\gamma^{\prime C}$, $a^{\prime C}$ in curved space-time. The field operators $\psi_{i}$ can be expanded in terms of the corresponding annihilation and creation operators as:

[TABLE]

where $u_{{\bf k},i}({\bf x},t)=\zeta_{{\bf k},i}(t)e^{i{\bf k}\cdot{\bf x}}$, and the mode functions $\zeta_{{\bf k},i}(t)$ have analytical expressions only in particular cases. Eq. (29) can be obtained starting from the unmixed scalar fields in flat space-time $\psi_{i}$, by using the generator $J$ of the curvature: $\psi^{C}_{i}({\bf x},t)=J^{-1}({\bf x},t)\,\psi_{i}({\bf x},t)\,J({\bf x},t)$. The form of $J$ depends on the particular metric analyzed Birrell . The vacuum state for $\psi^{C}_{i}$ is defined as $|0({\bf x},t)\rangle^{C}_{\gamma_{\|}^{\prime},a^{\prime}}=J^{-1}({\bf x},t)|0\rangle_{\gamma_{\|}^{\prime},a^{\prime}}\,.$ Then, the curved mixed vacuum is given by: $|0(\vartheta,x)\rangle^{C}_{\gamma_{\|},a}=(G^{C}_{\vartheta}(x))^{-1}\;J^{-1}(x)|0\rangle_{\gamma_{\|}^{\prime},a^{\prime}}\,,$ where $G^{C}_{\vartheta}(x)$ is the mixing generator for fields in curved space-time. This generator has a structure similar to the generator in flat space-time (see Eq. (42) in Appendix 1), but it is expressed in terms of the field operators in curved space-time reported in Eq. (29).

For any curved space, the covariant derivative for scalar fields reduces to the ordinary derivative, so the energy momentum tensor density for unmixed scalar fields in curved space time $\psi^{C}_{i}$ is given by:

[TABLE]

The contributions to the pressure and energy density are obtained by computing the expectation value of $T_{\mu\nu}(x)$ on the curved mixed vacuum: $|0(\vartheta,x)\rangle^{C}_{\gamma_{\|},a},$ i.e.,

[TABLE]

The symbol, $:...:$, in Eq. (31) denotes the normal ordering with respect to the curved vacuum for unmixed fields, $|0({\bf x},t)\rangle^{C}_{\gamma_{\|}^{\prime},a^{\prime}}$. In the case of a homogeneous and isotropic universe, as well as for diagonal metrics, the off-diagonal components of $\Xi^{\|}_{\mu\nu}(x)$ are zero Capolupo:2006etS , so the condensate behaves as a perfect fluid, and thus we can define the energy density and pressure as: $\rho_{\|}=g^{00}\;\Xi^{\|}_{00}$, and $p_{\|}=-g^{jj}\;\Xi^{\|}_{jj}$ (no summation on the index $j$ is intended). Explicitly, we have (see Appendix 2 for the derivation):

[TABLE]

Then, the state equation is $w=-1$, which is similar to that for the cosmological constant.

We note that nontrivial values of $\rho_{\|}$ and $p_{\|}$ only characterize the component parallel to the magnetic field of the photon polarization because the mixing only involves $\gamma_{\|}$ and $a$. By contrast, the vacuum energy density and pressure of $\gamma_{\bot}$ are equal to zero:

[TABLE]

where $|0({\bf x},t)\rangle^{C}_{\gamma_{\bot}}=J^{-1}(x)|0\rangle_{\gamma_{\bot}}$ is the vacuum for the component perpendicular to the magnetic field of the photon polarization $\gamma_{\bot}^{C}$, and $|0\rangle_{\gamma_{\bot}}$ is the vacuum for $\gamma_{\bot}$ in the Minkowski metric. The presence of $\rho_{\|}\neq 0$ for $\gamma_{\|}$ indicates further polarization of the vacuum, which represents a genuine QFT effect. Indeed, $\rho_{\|}$ is zero in the quantum mechanics axion-photon mixing framework. If we denote $\Delta m^{2}=|m_{a^{\prime}}^{2}-m_{\gamma^{\prime}}^{2}|$, then we have:

[TABLE]

where $V$ is the volume.

It should be noted that around astrophysical objects such as pulsars or neutron stars where strong magnetic fields are produced (see below), the Schwarzschild metric should be considered:

[TABLE]

In this case, the Klein–Gordon equation is given in terms of the confluent Heun functions Bezerra and the explicit form of the energy density $\rho_{\|}$ is difficult. In order to derive an estimate of $\rho_{\|}$, we consider the Minkowski metric for simplicity, which means that we neglect the gravitational effects induced by the metric in Eq. (III), i.e., we assume that $r>>2M$. This assumption is reasonable for many astrophysical objects with strong magnetic fields. For example, magnetars have a Schwarzschild radius of about $6km$, and thus the gravitational effect can be neglected for $r>>6km$. In the Minkowski metric, the energy density $\rho_{\|}$ becomes:

[TABLE]

where $K$ is the cut-off on the momenta. By considering the expressions for $\omega_{k,\gamma_{\|}^{\prime}}$ and $\omega_{k,a^{\prime}}$ given in Eqs. (18) and (19), and by solving the integral given above analytically, we obtain:

[TABLE]

For real values of $x$, the function $\tanh^{-1}(x)$ is defined in the domain: $-1<x<1$, (indeed, for $x\in R$, we can write $\tanh^{-1}(x)=\frac{1}{2}\log\left(\frac{1+x}{1-x}\right)$), then Eq. (38) imposes a condition on the cut-off $K$ on the momenta given by: $-1<\frac{\sqrt{m_{a^{\prime}}^{4}+4g^{2}B_{T}^{2}K^{2}}}{m_{a^{\prime}}^{2}+g^{2}B_{T}^{2}}<1,$ which implies the following upper bound on $K$,

[TABLE]

We note that the magnetic fields of pulsars and neutron stars are around $10^{12}G$, those of white dwarfs are around $10^{6}G$, and in astrophysical objects, such as active galactic nuclei and quasars, the magnetic field strength varies in the range of values: $[10^{6}-10^{17}]G$. Other interesting systems include magnetars, which are neutron stars characterized by extremely powerful magnetic fields of $[10^{14}-10^{15}]$ G. In these systems, the intense magnetic field powers the emission of electromagnetic radiation, particularly X-rays and gamma rays.

In order to obtain an estimate of $\rho_{\|}$ around the astrophysical systems mentioned above, we consider an axion mass of $m_{a^{\prime}}\sim 2\times 10^{-2}eV$, $g\sim 10^{10}GeV^{-1}$, $B_{T}\sim(10^{15}-10^{16})G$ (which can be obtained for different astrophysical objects), a photon energy of $\omega\sim 100eV$, and cut-off $K\sim 10^{-2}eV$ (which satisfies Eq. (39)). Thus, we obtain $\rho\sim 10^{-47}GeV^{4}$, which is of the same order as the estimated value of the dark energy. This contribution to the vacuum energy is produced around astrophysical objects with high magnetic field. The highly magnetized volume around these objects is very small compared with intergalactic scales. Moreover, the energy density is not made much larger than that of dark energy, so this contribution is negligible around such astrophysical systems. In addition, these objects are generally bound inside galaxies where dark energy and vacuum energy are generally not observable. We also note that the plasma frequencies, photon energies $\omega$, and the mixing angle $\theta_{a}$ depend on the particular system considered. In general, small fluctuations from the parameter values lead to different values for $\rho$.

Let us now consider terrestrial experiments. Strongest continuous magnetic field yet produced in a laboratory is $45T$. Due to the presence of the term $\tanh^{-1}\left(\frac{m_{a^{\prime}}^{2}}{m_{a^{\prime}}^{2}+g^{2}B_{T}^{2}}\right)$ in Eq. (38), for $g\sim 10^{10}GeV^{-1}$, the energy density $\rho_{\|}$ has a finite value only for axion mass values less than $3\times 10^{-7}eV.$ For example, for $m_{a^{\prime}}\sim 2\times 10^{-7}eV$ and $\omega\sim 10eV$, we have $\rho\sim 10^{-66}GeV^{4}$. This energy density is less than the Casimir energy density. Indeed, for a length $L$ between parallel plates of about $(10^{-5}-10^{-4})m$, the energy density generated by the Casimir effect is $E=\frac{\pi^{2}}{1440L^{4}}=(10^{-44}-10^{-48})GeV^{4}$, respectively. However, the vacuum polarization induced by $\rho_{\|}$ could be detected in experiments similar to PVLAS PVLAS and might represent an indirect proof of the existence of the axion–photon mixing.

It should be noted that the new effects revealed in this work by analyzing axion–photon mixing in the QFT framework (i.e., corrections to the oscillation formulae and a non-trivial vacuum energy that produces further vacuum polarization) are not expected in the usual quantum mechanics treatment of axion–photon mixing.

IV Conclusions

We considered many aspects of the QFT formalism for axion–photon mixing. The unitary inequivalence between the space for the mixed field states and the state space where the unmixed field operators are defined, affects the oscillation formulae and leads to polarization of the vacuum. These effects, that are not expected in previous studies of axion–photon mixing in the framework of quantum mechanics, in principle, they can be detected in laser beam experiments. In order to connect our findings with large-scale structures such as cosmology or astrophysics frameworks, we considered axion–photon mixing in curved space-time. We showed that the vacuum energy induced by axion-photon mixing for a homogeneous and isotropic universe as well as for diagonal metrics has a state equation with the adiabatic index $w=-1$. We provided a numerical estimate of the energy density for the vacuum condensate for the Minkowski metric. A value compatible with that for dark energy can be obtained for axions with masses of $10^{-3}-10^{-2}eV$, in regions of the universe with strong magnetic fields. However, this energy does not contribute to the cosmological dark energy, and it is a purely local phenomenon that only appears in the magnetized volume around astrophysical objects where other fields would have a larger impact on the energy-momentum tensor. For example, the energy density of magnetic fields $B$ with values $(10^{15}-10^{16})G$ is $\rho_{B}\in(10^{-10}-10^{-8})GeV^{4}$.

On the other hand, in principle, the energy density of the vacuum condensate induced by axion–photon mixing and QFT vacuum polarization can be detected in laser beam experiments for ALPs with masses less than $10^{-7}eV$.

The theoretical results presented in this paper could open new interesting scenarios in the research into axions and axion-like particles.

Acknowledgments

AC, GL, and AS acknowledge partial financial support from MIUR and INFN. AC, IDM and GL acknowledge the COST Action CA1511 Cosmology and Astrophysics Network for Theoretical Advances and Training Actions (CANTATA) supported by COST (European Cooperation in Science and Technology).

Appendix A: Axion–photon mixing in the Minkowski metric - useful formulae

The Fourier expansions of the free fields $\gamma^{\prime}_{\|}$ and their conjugate momenta $\pi^{\prime}_{\gamma}$ in flat space-time are:

[TABLE]

Similar expansions hold for the free axion fields $a^{\prime}(x)$ and their conjugate momenta $\pi^{\prime}_{a}$. In Eqs. (Appendix A: Axion–photon mixing in the Minkowski metric - useful formulae) and (Appendix A: Axion–photon mixing in the Minkowski metric - useful formulae), we have $\alpha_{{\bf k},\gamma_{\|}^{\prime}}(t)=\alpha_{{\bf k},\gamma_{\|}^{\prime}}(0)e^{-i\omega_{k,\gamma_{\|}^{\prime}}t}$, and similarly, the time evolution of the annihilators for axions is: $\alpha_{{\bf k},a^{\prime}}(t)=\alpha_{{\bf k},a^{\prime}}(0)e^{-i\omega_{k,a^{\prime}}t}$.

The form of the generator $G_{\vartheta}(t)$ of the mixing transformation is:

[TABLE]

which can be written as:

[TABLE]

The orthogonality between $|0(\vartheta,t)\rangle_{\gamma_{||},a}$ and $|0\rangle_{\gamma_{||}^{\prime},a^{\prime}}$ in the infinite volume limit can be shown easily by considering the explicit form of the product $\,{}_{\gamma_{||}^{\prime},a^{\prime}}\langle 0|0(\vartheta,t)\rangle_{\gamma_{||},a}$. Indeed, we have $\lim\limits_{V\rightarrow\infty}\,{}_{\gamma_{||}^{\prime},a^{\prime}}\langle 0|0(\vartheta,t)\rangle_{\gamma_{||},a}=\lim\limits_{V\rightarrow\infty}\,e^{\frac{V}{(2\pi)^{3}}\int d^{3}k\,\ln\left(\frac{1}{1+|\Upsilon_{\bf k}|^{2}\sin^{2}\vartheta}\right)}\,=\,0~{},$ since the logarithm is negative for any values of ${\bf k}$, $\theta$ and $m_{\gamma},m_{a}$ (note that $0\leq\vartheta\leq\pi/4$).

The explicit form of the momentum operator for mixed fields is:

[TABLE]

where ${\sigma}=\gamma_{\|},a$.

Appendix B: Axion–photon mixing in curved space - derivation of $\rho_{\|}$ and $p_{\|}$

For a homogeneous and isotropic universe as well as for diagonal metrics, the energy density $\rho_{\|}$ and pressure $p_{\|}$ of the vacuum condensate induced by axion–photon mixing are given by:

[TABLE]

Let us denote with $\psi_{\sigma}(\vartheta,x)$ the mixed fields, where $\sigma=\gamma_{\|},a$. For any homogeneous and isotropic metric, by considering the mixing relations in Eq. (LABEL:mixingRel), at any cosmic time, we have:

[TABLE]

i.e., these operators are invariant under the action of the generator $G^{C}_{\vartheta}(t)$, which implies that the kinetic and gradient terms of the mixed vacuum are equal to zero:

[TABLE]

On the contrary, the mass terms in Eqs. (Appendix B: Axion–photon mixing in curved space - derivation of $\rho_{\|}$ and $p_{\|}$) and (Appendix B: Axion–photon mixing in curved space - derivation of $\rho_{\|}$ and $p_{\|}$) generate non-vanishing contributions because $m_{\gamma}\neq m_{a}$. From the relations given above, we obtain Eqs. (III) and (III) for $\rho_{\|}$ and $p_{\|}$.