Neutrino Emissivities as a Probe of the Internal Magnetic Fields of White Dwarfs

The hottest and brightest WDs correspond to the younger ones. During this stage, the dwarf cools mostly through neutrino emission. The dominant emission is from plasmon decay (Kantor & Gusakov 2007), with other processes like bremsstrahlung coming far second (Winget et al. 2004). Plasmon decay in a nonmagnetized medium has been extensively studied in the literature (Adams et al. 1963; Zaidi 1965), with the most comprehensive treatment given by the seminal paper of Braaten & Segel (1993), which we use as the basis of our calculations. Explicitly, for the energy regimes relevant to stellar cooling, plasmon processes proceed via Fermi interactions,

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{int}}=\displaystyle \frac{{G}_{{\rm{F}}}}{\sqrt{2}}\,\bar{e}{\gamma }^{\mu }({C}_{{\rm{V}}}-{C}_{{\rm{A}}}{\gamma }_{5})e\,{\bar{\nu }}_{a}{\gamma }_{\mu }(1-{\gamma }_{5}){\nu }_{a},\end{eqnarray} \tag{ 1 }$$

where G_F = 1.166 × 10⁻⁵ GeV⁻² is the Fermi constant and e and ν_a are spinors representing electrons and neutrinos of flavor a = e, μ, τ, respectively. The effective vector (V) and axial-vector (A) coupling constants include both neutral current and charged current. Altogether one finds

$$\begin{eqnarray}&&{C}_{{\rm{V}}}=\tfrac{1}{2}(4\sin {{\rm{\Theta }}}_{{\rm{W}}}+1)\quad \mathrm{and}\quad {C}_{{\rm{A}}}=+\tfrac{1}{2}\quad \mathrm{for}\,{\nu }_{e},\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}&&{C}_{{\rm{V}}}=\tfrac{1}{2}(4\sin {{\rm{\Theta }}}_{{\rm{W}}}-1)\quad \mathrm{and}\quad {C}_{{\rm{A}}}=-\tfrac{1}{2}\quad \mathrm{for}\,{\nu }_{\mu },{\nu }_{\tau }\end{eqnarray} \tag{ 2b }$$

in terms of the weak mixing angle Θ_W. One can obtain the squared matrix element for the transition $\gamma \to \bar{\nu }\nu$ and compute the decay rate

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{\gamma \to \nu \bar{\nu }}(\omega ) & = & \displaystyle \int \displaystyle \frac{{d}^{3}{{\boldsymbol{k}}}_{\nu }}{{\left(2\pi \right)}^{3}2{\omega }_{\nu }}\,\displaystyle \frac{{d}^{3}{{\boldsymbol{k}}}_{\bar{\nu }}}{{\left(2\pi \right)}^{3}2{\omega }_{\bar{\nu }}}\displaystyle \frac{| {{ \mathcal M }}_{\gamma \to \nu \bar{\nu }}{| }^{2}}{2\omega }\\ & & \times {\left(2\pi \right)}^{4}\,{\delta }^{4}(k-{k}_{\nu }-{k}_{\bar{\nu }}),\end{array}\end{eqnarray} \tag{ 3 }$$

where ${k}_{\nu ,\bar{\nu }}=({\omega }_{\nu ,\bar{\nu }},{{\boldsymbol{k}}}_{\nu ,\bar{\nu }})$ and (ω, k) are the wavevectors of outgoing neutrinos and incoming photons, and ${ \mathcal M }$ the relevant scattering amplitude. The energy loss (per volume per unit time) for each polarization can be computed as

$$\begin{eqnarray}&&{Q}_{\gamma \to \nu \bar{\nu }}=\int \displaystyle \frac{{d}^{3}{\boldsymbol{k}}}{{\left(2\pi \right)}^{3}}\omega {{\rm{\Gamma }}}_{\gamma \to \nu \bar{\nu }}(\omega ){f}_{B}(\omega ),\end{eqnarray} \tag{ 4 }$$

where f_B (ω) = 1/(e^ω/T − 1) is the Bose–Einstein distribution evaluated at the plasmon energy ω = ω_t , ω_l given in Appendix A. Integrating the energy loss over the star volume, one obtains the luminosity. We can identify three contributions to the plasmon emissivity ${Q}_{\gamma \to \nu \bar{\nu }}$, which depend on the polarization of the photon and whether the process is due to the axial vector or the vector coupling. These are, calling the fine structure constant α = e²/4π,

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{T} & = & 2\left(\sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{0}^{+\infty }\\ & & \times {dk}\,{k}^{2}{Z}_{t}(k){\left({\omega }_{t}^{2}-{k}^{2}\right)}^{3}{f}_{B}({\omega }_{t}),\end{array}\end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{A} & = & 2\left(\sum _{{\nu }_{\alpha }}{C}_{A}^{2}\right)\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{0}^{+\infty }\\ & & \times {dk}\,{k}^{2}{{\rm{\Pi }}}_{A}{\left({\omega }_{t},k\right)}^{2}{Z}_{t}(k)\left({\omega }_{t}^{2}-{k}^{2}\right){f}_{B}({\omega }_{t}),\end{array}\end{eqnarray} \tag{ 5b }$$

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{L} & = & \left(\sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{0}^{{k}_{\max }}\\ & & \times {dk}\,{k}^{2}{Z}_{l}(k){\omega }_{l}^{2}{\left({\omega }_{l}^{2}-{k}^{2}\right)}^{2}{f}_{B}({\omega }_{l}),\end{array}\end{eqnarray} \tag{ 5c }$$

which we refer to respectively as transverse, axial, and longitudinal. Full expressions for Z_ℓ , ω_ℓ , ω_t , and Π_A can be found in Appendix A. We display the plasmon emissivities ⁶ , Equations (5a)–(5c) in Figure 1. Note in particular the subdominance of the axial contribution. In the remainder of this paper, we make frequent use of nonrelativistic approximations characterized by p ≪ m_e and T ≪ m_e , where p is a typical electron momentum and m_e is the electron mass. For consistency, we examine in Appendix A.2 to what extent the relativistic results for plasmon cooling. Equations (5a)–(5c) are well approximated by their nonrelativistic limits for the parametric regimes relevant for WDs, thereby providing a cross check that relativistic corrections can be reasonably neglected in the remainder of the paper.

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At colder temperatures, WDs cool predominantly through photon surface emission. This can be estimated by Mestel's cooling law (Kaplan 1950; Mestel 1952; Shapiro & Teukolsky 1983). As the interior of the star is degenerate, electrons have a long mean free path. Therefore, there is a roughly homogeneous temperature all over the core (see, for example, Figure 1 of Bischoff-Kim et al. 2008 and Figure 4 of Córsico et al. 2014). On the other hand, the external layers are nondegenerate and are in radiative equilibrium. Thus, one can show that for typical values of a WD, the emission from the nondegenerate external layer is approximately determined by the temperature T in the transition region between the degenerate core and the nondegenerate external layers, approximately equal to the central temperature (Shapiro & Teukolsky 1983),

$$\begin{eqnarray}&&{L}_{\gamma }\simeq (2\times {10}^{6}\,\mathrm{erg}\,{{\rm{s}}}^{-1})\displaystyle \frac{M}{{M}_{\odot }}{T}^{3.5},\end{eqnarray} \tag{ 6 }$$

where M is the WD mass. As noted by Mestel, this can be understood as the WD radiating away its residual ion thermal energy. ⁷ This is a surface emission, and therefore depends on the global properties of the star. For comparison with other rates given per unit volume, it is useful to define the following volume-averaged emissivity, obtained by dividing the left and right terms of Equation (6) by the volume,

$$\begin{eqnarray}&&{Q}_{\gamma }=\vartheta \displaystyle \frac{{L}_{\odot }}{{M}_{\odot }}\rho {T}_{7}^{3.5},\end{eqnarray} \tag{ 7 }$$

where L_⊙ = 3.83 × 10³³ erg s⁻¹, ϑ = 1.7 × 10⁻³, and T₇ = T/10⁷ K (Raffelt 1986). ⁸

After this stage, the core crystallizes, and the WD becomes fainter and fainter. The crystallization phase has been recently observed (Tremblay et al. 2019) by the Gaia satellite (Prusti et al. 2016), which could improve the accuracy of methods used to determine the age of stellar populations.

The internal magnetic field B in WDs is poorly constrained by observations and could be much stronger than the one at the surface. The existence of a B field on the surface of magnetic white dwarfs (MWDs) is observed via spectroscopic measurements (Liebert et al. 2003; Landstreet et al. 2015; Landstreet & Bagnulo 2019; Ferrario et al. 2020). The presence of a large magnetic field, up to hundreds of megagauss, on the surface of MWDs had already been observed in the 1970s by detection of broadband circular and linear polarization (Kemp et al. 1970; Angel & Landstreet 1971). Another way of detecting such huge fields is by relying on the Zeeman effect, which can cause the splitting of Balmer lines (Angel & Landstreet 1970).

In this way, MWDs have been discovered with B fields of 10³–10⁹ G, in an ever-growing number, from the circa 70 in the early 2000s (Wickramasinghe & Ferrario 2000) to 600 and counting (Ferrario et al. 2015), thanks to SDSS (Kepler et al. 2013). The data suggest that 10%–20% of WDs show a surface magnetic field (see, e.g., Sion et al. 2014; Ferrario et al. 2020). However, we cannot constrain the B field in the core of WDs by spectropolarimetric observations, thus making viable the possibility of a very large magnetic field in the core, possibly even of WDs, which do not appear magnetic at the surface. The B field in the core of a WD could be as large as 10¹³ G (Angel 1978; Shapiro & Teukolsky 1983). Interestingly, MWDs in the past were considered to be WDs that got rid of the hydrogen envelope, and it was suspected that all WDs had large B fields (Imoto & Kanai 1971; Chanmugam & Gabriel 1972). However, this hypothesis has been abandoned since magnetic stars have been observed with H-dominated atmospheres. Nevertheless, it might still be the case that even WDs showing no magnetic field on their surface might host a large magnetic field in their core.

The problem of measuring the internal magnetic field arises for stars going through different stages of their lives. For example, the Sun may host a B field as large as 10⁷ G, despite having a much smaller field on the surface (Couvidat et al. 2003). In the case of the Sun, bounds can come from neutrinos (as they track the temperature, and the latter would be affected by a different energy density due to the presence of a B field), and from the oblateness of the Sun itself (Couvidat et al. 2003; Friedland & Gruzinov 2004). Helioseismology can only put bounds on the magnetic field down to the tachocline (Antia et al. 2000; Baldner et al. 2009; Barnabé et al. 2017). Concerning the core of RGs, a progenitor of WDs, the presence of strong magnetic fields (in excess of 10⁵ G or more) is inferred using asteroseismology. In Fuller et al. (2015), it is shown how certain oscillations (the so-called g-dominated mixed modes) observed at the surface of RGs are damped by the presence of a strong internal magnetic field above a critical value. This critical value turns out to be 10⁵ G. In a sample of 3000 RGs (Stello et al. 2016), about 20% show the presence of suppressed dipole modes. Assuming that this reveals the presence of magnetism, 20% of RGs might possess strong internal magnetic fields. Notice that these fields are likely confined to the stellar core, and do not extend to the stellar envelope/stellar surface.

Interestingly, asteroseismological observations of RGs suggest that a large fraction of lower mass WDs might have strong internal magnetic fields (Cantiello et al. 2016). The key observation is that their magnetic fields are likely buried below the surface and not detectable via Zeeman spectropolarimetry. A few caveats are nevertheless in order. Even if strong B fields are generated by convective cores during the main sequence, it is not clear if these fields are preserved all the way to the WD formation. There are many processes that could potentially destroy the B field as the star goes, e.g., through the He flash, or other later convective phases. Moreover, a problematic issue is the stability of the magnetic field itself. Both purely poloidal and toroidal fields are dynamically unstable, even though a combination of the two could be stable (Braithwaite & Nordlund 2006; Duez et al. 2010). This field can hide under the surface, or emerge at the surface, as long as it has both a poloidal and toroidal component. As the problem of magnetic field stability is well beyond the scope of this paper, we will neglect these issues.

A final comment should be about variable stars. There is the possibility that pulsating WDs might host a large B field underneath its surface, hidden even to asteroseismology. An order of magnitude estimate is enough to show that an extremely large B field would affect the amplitude of oscillations (Cox 1980). Moreover, if B fields affect oscillations, even a small variation over time of the B field would be enormously amplified by the oscillations. The possibility of using asteroseismology to detect the B field in a pulsating WD has been questioned (Heyl 2000; Loi & Papaloizou 2018). However, we stress that very large magnetic fields should be potentially bound by asteroseismology in pulsating WDs (see also Jones et al. 1989).

With these points in mind, the question left is to develop some understanding of the internal magnetic field structure of WDs. In Appendix B, we use virial arguments to examine an upper bound (see Equation (B8)) on the maximum strength of B that can be supported within some core regions while maintaining stellar stability. The strength of these arguments lies in their simplicity—as they are built on very basic physical principles, and thus in principle are very robust. However, by construction virial arguments can only constrain the integrated magnetic field within some regions. Applying them to the star as a whole yields a rather robust upper limit on the average value of B, which could, however, easily be avoided by much stronger fields that are localized in small regions inside the WD. Applying virial arguments to the subregions of the WD relies on a detailed understanding of its structure, leading to a loss of robustness due to modeling uncertainties. In the remainder of this section, we present an alternative way to constrain the local B in subregions of the WD, based on its impact on the neutrino emissivity.

Neglecting processes that depend on the ion content (such as crystallization), a nonmagnetized QED plasma is completely described by the electron number density (or alternatively the chemical potential) and the temperature. The presence of a macroscopic magnetic field, as we will review, can affect the plasma in several ways. First, it can generate QED nonlinear effects. Moreover, it can affect the electron wave functions, and modify the propagation of the electromagnetic field excitations. Some of these effects will modify the energy loss from plasmon decay, as shown in the 1970s (Canuto et al. 1970a, 1970b; Galtsov & Nikitina 1972; DeRaad et al. 1976; Skobelev 1976). We will follow a more recent treatment that corrects for several flaws in previous works (Kennett & Melrose 1998).

QED nonlinear effects are of two kinds. On the one hand, a critical field ${B}_{c}={m}_{e}^{2}/e\,=\,4.41\,\times \,{10}^{13}\,{\rm{G}}$ is so large that the macroscopic field could create electron-positron pairs. We will always consider fields smaller than this value. The second class of effects is related to the vacuum birefringence effect (Tsai & Erber 1975) as a large macroscopic field affects the refractive index, and can affect plasmon decay to neutrinos. ⁹ The magnetic field introduces an additional energy scale, the synchrotron frequency

$$\begin{eqnarray}&&{\omega }_{B}=\displaystyle \frac{{eB}}{{m}_{e}}={m}_{e}\displaystyle \frac{B}{{B}_{c}}\simeq 11.5\,{B}_{12}\,\mathrm{keV}.\end{eqnarray} \tag{ 8 }$$

As a rule of thumb, one can compare how the magnetic field and the plasma modify the refractive index, respectively. The equations for vacuum birefringence are given in Tsai & Erber (1975), and reduce for B ≲ B_c to ${n}_{\mathrm{vacuum}}^{2}-1\simeq \kappa {\alpha }^{2}{B}^{2}/{m}_{e}^{4}$, where κ is a constant that depends on the polarization (κ_∥ = 14/45 and κ_⊥ = 8/45 for the electric field parallel and perpendicular to the external magnetic field, respectively; Raffelt 1996). As the refractive index in an unmagnetized plasma is ${n}_{\mathrm{plasma}}^{2}-1=-{\omega }_{p}^{2}/{\omega }^{2}$, one finds that vacuum birefringence is negligible for (Meszaros & Ventura 1979; Pavlov et al. 1980)

$$\begin{eqnarray}&&\displaystyle \frac{\alpha }{4\pi }\displaystyle \frac{{\omega }_{B}^{2}}{{m}_{e}^{2}}\displaystyle \frac{{\omega }^{2}}{{\omega }_{p}^{2}}=\displaystyle \frac{\alpha }{4\pi }\displaystyle \frac{{B}^{2}}{{B}_{c}^{2}}\displaystyle \frac{{\omega }^{2}}{{\omega }_{p}^{2}}\lesssim 1,\end{eqnarray} \tag{ 9 }$$

which is safely satisfied for ω ≃ T, B ≲ B_c and the range of temperatures and densities considered here, see also Equation (A17).

The most important effect that B has in the range considered here is to force electrons on Landau levels with energies (notice that ω_B /m_e = B/B_c )

$$\begin{eqnarray}&&{E}_{\nu }=\sqrt{{p}_{\parallel }^{2}+{m}_{e}^{2}\left(1+2\nu \displaystyle \frac{B}{{B}_{c}}\right)},\end{eqnarray} \tag{ 10 }$$

with p_∥ the component of the momentum parallel to the magnetic field and $\nu =l+\tfrac{1}{2}+\sigma$, l ≥ 0 is an integer labeling the orbital angular momentum, $\sigma =\pm \tfrac{1}{2}$, g₀ = 1, and all other g_ν = 2. The electron density is then computed as (see, e.g., Lai & Shapiro 1991)

$$\begin{eqnarray}&&{n}_{e}=\displaystyle \frac{{m}_{e}^{2}}{{\left(2\pi \right)}^{2}}\displaystyle \frac{B}{{B}_{c}}\sum _{\nu =0}^{\infty }{g}_{\nu }{\int }_{-\infty }^{\infty }{{dp}}_{\parallel }{f}_{-}^{\nu }({E}_{\nu }),\end{eqnarray} \tag{ 11 }$$

where ${f}_{\pm }^{\nu }={\left[\exp \left(\tfrac{{E}_{\nu }\pm \mu }{T}\right)+1\right]}^{-1}$ for electrons (−) and positrons (+), respectively, and we neglected positrons. In Kennett & Melrose (1998) the extreme case where only the lowest level is occupied was considered. We shall in the following investigate the impact of B on the plasmon emission rate under this extreme assumption. By doing so we overestimate the effect of B for realistic values in WDs, as one can see by estimating the magnitude of B that would be needed to confine all electrons to the lowest Landau level. To show this we make the conservative assumption T = 0, which yields a maximal (Fermi) momentum p_∥ = p_F∥(ν) that is defined by the condition

$$\begin{eqnarray}&&{p}_{F\parallel }^{2}(\nu )+{m}_{e}^{2}\left(1+2\nu \displaystyle \frac{B}{{B}_{c}}\right)={\mu }^{2},\end{eqnarray} \tag{ 12 }$$

with μ the electron chemical potential. The highest occupied Landau level ${\nu }_{\max }$ corresponds to the maximal ν for which Equation (12) has a real solution in p_F∥. Hence, if we require a value for ${\nu }_{\max }$, we can demand ${p}_{F\parallel }({\nu }_{\max }+1)=0$ to identify the largest chemical potential μ for which only Landau levels up to ${\nu }_{\max }$ are filled, and express

$$\begin{eqnarray}&&{n}_{e}=\displaystyle \frac{{m}_{e}^{3}}{{\left(2\pi \right)}^{2}}{\left(\displaystyle \frac{2B}{{B}_{c}}\right)}^{3/2}\sum _{\nu =0}^{{\nu }_{\max }}{g}_{\nu }\sqrt{1+{\nu }_{\max }-\nu }.\end{eqnarray} \tag{ 13 }$$

Demanding ${\nu }_{\max }=0$ leads to ${n}_{e}={m}_{e}^{3}/{\left(2\pi \right)}^{2}\ {\left(2B/{B}_{c}\right)}^{3/2}$ or $B=\tfrac{1}{2}{B}_{c}{\left(4{\pi }^{2}{n}_{e}/{m}_{e}^{3}\right)}^{2/3}$, i.e., B/B_c ≃ 0.6 for Y_e ρ ≃ 10⁶ g cm⁻³, where Y_e is the number of electrons per baryon. Therefore, one needs incredibly large magnetic fields to keep the electrons at the lowest Landau level. We find that the value of the B field for this to happen is much larger than the one cited in Kennett & Melrose (1998), where it was assumed that the kinetic energy of the electron was due to the temperature, rather than the Fermi energy. However, for the strongest magnetic fields considered here, one still expects a sizeable fraction of the electrons in the lowest Landau levels.

As the general problem in which both the thermal motion of electrons (due to temperature or Fermi energy) and the magnetic fields are accounted for is very complicated (Melrose 1986; Kennett & Melrose 1998), we will limit ourselves to an electron cold plasma distribution, assuming that most electrons are in the lowest Landau energy level. In this way, we will be able to estimate the maximal variation in the plasmon decay neutrino flux due to the presence of the magnetic field. In particular, the effect will be large on the axial-vector coupling emission: since this is associated with coherent spin oscillations (see also Appendix A), the magnetic field breaks parity and aligns the spins, so that constructive interference is possible in the magnetized plasma for ν = 0.

Even with these simplifications, as the magnetic field breaks the isotropy of the medium, the problem of finding the dispersion relations of the electromagnetic field excitations becomes rapidly complicated (Swanson 2012). Therefore, we can estimate the effect the B field has on neutrino emission from plasmon decay by considering the decay of plasmons propagating parallel or orthogonal to the magnetic field. The explicit evaluation of Equation (3) is enormously simplified by using a long-wavelength approximation for both the vector and axial-vector polarization functions, and using the Stix form for the dielectric tensor (Stix 1962; Melrose 1986; Swanson 2012) one can obtain simple expressions of the refractive index for different wavevector directions and polarizations.

Taking these facts in combination, the different plasmon emission processes in a magnetized medium can be categorized according to (i) whether they proceed via the axial-vector or vector coupling (ii) the value of θ (i.e., the direction of propagation with respect to the magnetic field) and (iii) the polarization of the mode in question, which can be oriented relative to the direction of B. Expressions for general θ are quite complicated so in order to obtain some quantitative understanding of the size of effects, we examine various special cases below taken from Kennett & Melrose (1998).

Circularly polarized parallel propagation (θ = 0)—For θ = 0, the axial-vector and longitudinal contribution is not modified. The remaining contribution therefore proceeds via the vector coupling resulting in an effect to a modification of Q_T . This consists of two circularly polarized modes:

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\pm }(\theta =0) & = & \displaystyle \frac{1}{4\pi }\left(\sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{{\omega }_{\min }}^{\infty }\\ & & \times d\omega \ {\omega }^{8}{n}_{\pm }{\left(1-{n}_{\pm }^{2}\right)}^{3}{f}_{B}(\omega ),\end{array}\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}&&{n}_{\pm }^{2}=1-\displaystyle \frac{{\left({\omega }_{p}/\omega \right)}^{2}}{1\pm {\omega }_{B}/\omega },\end{eqnarray} \tag{ 15 }$$

and ${\omega }_{\min }$ corresponds to the refractive index n = k/ω being equal to 0, since below this cutoff the modes do not propagate. ¹⁰ This expression coincides with the one found in Canuto et al. (1970a), and is independent of the axial-vector coupling. Note that summing Q_±, taking the limit of zero magnetic field, and integrating over the plasmon direction, we find again Equation (A18), as expected.

Perpendicular propagation (θ = π/2), parallel polarization—A mode propagating perpendicular to B but polarized parallel to B is known as the ordinary mode (so-called because it has the same dispersion relation as though the anisotropy were not there). Both the axial-vector and vector contributions are modified by the B field for this case, giving rise to

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{o}\left(\theta =\displaystyle \frac{\pi }{2}\right) & = & \displaystyle \frac{1}{4\pi }\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{{\omega }_{\min }}^{\infty }d\omega \ {\omega }^{8}{n}_{o}{\left(1-{n}_{o}^{2}\right)}^{2}\\ & \times & \left[\left(\displaystyle \sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)(1-{n}_{o}^{2})+\left(\displaystyle \sum _{{\nu }_{\alpha }}{C}_{A}^{2}\right){n}_{o}^{2}\right]{f}_{B}(\omega ),\end{array}\end{eqnarray} \tag{ 16 }$$

where the refractive index is independent of B,

$$\begin{eqnarray}&&{n}_{o}^{2}=1-{\left(\displaystyle \frac{{\omega }_{p}}{\omega }\right)}^{2}.\end{eqnarray} \tag{ 17 }$$

Perpendicular propagation (θ = π/2), perpendicular polarization—This case is referred to as the extraordinary mode (the plasma dispersion relation receives magnetic field corrections), for which the vector (but not the axial-vector) contribution is modified:

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{x}\left(\theta =\displaystyle \frac{\pi }{2}\right) & = & \displaystyle \frac{1}{4\pi }\left(\displaystyle \sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{{\omega }_{\min }}^{\infty }d\omega \,{\omega }^{8}{n}_{x}(1-{n}_{x}^{2})\\ & & \times \left[{\left(S-1\right)}^{2}+{D}^{2}-\displaystyle \frac{4{D}^{2}S(S-1)}{{D}^{2}+{S}^{2}}\right]{f}_{B}(\omega ),\end{array}\end{eqnarray} \tag{ 18 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}{n}_{x}^{2}=\displaystyle \frac{{S}^{2}-{D}^{2}}{S},\,\,S=1-\displaystyle \frac{{\left({\omega }_{p}/\omega \right)}^{2}}{1-{\left({\omega }_{B}/\omega \right)}^{2}},\\ \,\,D=\displaystyle \frac{{\omega }_{B}{\omega }_{p}^{2}}{\omega ({\omega }^{2}-{\omega }_{B}^{2})}.\end{array}\end{eqnarray} \tag{ 19 }$$

We summarize the size of these contributions in Figure 2, which compares the plasmon processes in a magnetized medium to the plasmon processes in the unmagnetized medium. Neutrinos from plasmon decay might potentially constrain the internal B field of WDs. The colder part of the WDLF shows some excessive cooling (Giannotti et al. 2016), while the hot part does not show it (Hansen et al. 2015). Notice however that the WDLF uncertainty is large in the hottest range. From Figure 2 it can be seen that the plasmon decay is mostly modified at rather large temperatures. We will show in the next section that the emission from synchrotron radiation dominates plasmon decay for values of the magnetic field whose effect on plasmon decay is large. If in the future this part of the WDLF is measured precisely, a dedicated and computationally more challenging estimate of the B field effect on plasmon decay would be demanded, featuring, e.g., a warm plasma (Kennett & Melrose 1998).

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Similar to neutrino production from bremsstrahlung, which is the main neutrino production channel during the surface cooling phase, synchrotron emission is possible thanks to an external field. The B field forces the electrons to rotate around the magnetic field lines. The electron momentum is not conserved, allowing for the process $e\mathop{\to }\limits^{B}e\,\bar{\nu }\,\nu$. Neutrino pair synchrotron emission has been the subject of many studies, spanning several decades (see, e.g., Landstreet 1967; Iakovlev & Tschaepe 1981; Yakovlev et al. 2001). Using the notation introduced in Section 3.2, the emissivity reads as (Kaminker et al. 1992)

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\mathrm{syn}} & = & \left(\displaystyle \sum _{{\nu }_{\alpha }}{C}_{A}^{2}\right)\displaystyle \frac{{G}_{F}^{2}{\omega }_{B}^{6}}{30{\pi }^{3}}\displaystyle \frac{m{\omega }_{B}}{{\left(2\pi \right)}^{2}}\displaystyle \sum _{\nu =1}^{\infty }{\displaystyle \int }_{-\infty }^{+\infty }\\ & & \times {{dp}}_{\parallel }\left[{f}_{-}^{\nu }\left(1-{f}_{-}^{\nu -1}\right)+{f}_{+}^{\nu }\left(1-{f}_{+}^{\nu -1}\right)\right],\end{array}\end{eqnarray} \tag{ 20 }$$

where as above

$$\begin{eqnarray}\begin{array}{rcl}{f}_{\pm }^{\nu } & = & {\left[\exp \left(\displaystyle \frac{{E}_{\nu }\pm \mu }{T}\right)+1\right]}^{-1},\\ {E}_{\nu } & = & \sqrt{{p}_{\parallel }^{2}+{m}_{e}^{2}+2\nu {\omega }_{B}{m}_{e}}.\end{array}\end{eqnarray} \tag{ 21 }$$

Notice that all neutrino flavors are equally produced by the synchrotron process. It is useful to obtain two formulae that describe synchrotron emission in a regime relevant to WDs. We assume T ≪ m_e , so that positrons can again be neglected (${f}_{+}^{\nu }\ll {f}_{-}^{\nu }$). The degenerate limit $T\ll {p}_{F}^{2}/(2{m}_{e})={E}_{F}^{\mathrm{NR}}$ is valid, as ${p}_{F}\simeq {\left(3{\pi }^{2}{n}_{e}\right)}^{1/3}$ for ω_B ≪ E_F ≃ μ, and it is always large in WD cores (see Appendix A). The emissivity is

$$\begin{eqnarray}&&{Q}_{\mathrm{syn}}({\omega }_{B}\ll T\ll {E}_{F}^{\mathrm{NR}})=\left(\sum _{{\nu }_{\alpha }}{C}_{A}^{2}\right)\displaystyle \frac{{G}_{F}^{2}{\omega }_{B}^{6}}{60{\pi }^{3}}\displaystyle \frac{3}{2}\displaystyle \frac{T}{{E}_{F}^{\mathrm{NR}}}{n}_{e},\end{eqnarray} \tag{ 22a }$$

$$\begin{eqnarray}&&{Q}_{\mathrm{syn}}(T\ll {\omega }_{B}\ll {E}_{F}^{\mathrm{NR}})=\left(\sum _{{\nu }_{\alpha }}{C}_{A}^{2}\right)\displaystyle \frac{{G}_{F}^{2}{\omega }_{B}^{6}}{60{\pi }^{3}}\displaystyle \frac{3}{2}\displaystyle \frac{{\omega }_{B}}{{E}_{F}^{\mathrm{NR}}}{e}^{-{\omega }_{B}/T}{n}_{e}.\end{eqnarray} \tag{ 22b }$$

These limits are referred to as non-quantized degenerate and weakly quantized degenerate, respectively. In Figures 3 and 4, we display the full numerical results of Equation (20) together with the analytic estimates in Equations (22a) and (22b). The analytical approximation is precise up to large temperatures, where the energy loss should be interpolated between Equations (22a) and (22b). Given the strong dependence on the magnetic field, the analytical approximation of Equation (22b) is good enough to describe the region of transition from non-quantized to quantized, considering our accuracy goals. Synchrotron emission is compared with all other processes in Figure 5, from which it can be seen that synchrotron emission can enhance the energy loss from WDs.

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Magnetic fields as large as those discussed in the previous sections can have additional effects on the evolution of a WD. One of them is the heating of the star due to ohmic decay of the magnetic field (Cumming 2002; Ferrario et al. 2015). The resulting conversion of the magnetic field energy ¹¹ U_em = B²/2 into thermal energy is the dominant process in the evolution of a field smaller than 10¹² G (Heyl & Kulkarni 1998). We do not attempt to quantify the potential effect of ohmic heating on the surface temperature at later stages (Valyavin et al. 2014), but instead only estimate it at the level of energy gain and loss per unit volume in the core in order to compare it to other processes discussed in this work. As an example, we consider a hypothetical magnetic field in the well-known WD G117-B15A, for which we take the parameters (Kepler et al. 2000; Bischoff-Kim et al. 2008)

$$\begin{eqnarray}&&{Y}_{e}\rho \simeq {10}^{6}\ {\rm{g}}\,{\mathrm{cm}}^{-}3,R=9.6\times {10}^{8}\ \mathrm{cm},T=1.2\times {10}^{7}\ {\rm{K}}.\end{eqnarray} \tag{ 24 }$$

Assuming an exponential decay $B={B}_{0}\exp (-t/{t}_{\mathrm{ohm}})$ with t_ohm ≃ 3 × 10¹¹ yr (Cumming 2002; Ferrario et al. 2015) and an age much smaller than t_ohm, the energy density deposited by an initial magnetic field B₀ ≃ 3 × 10¹¹ G per unit of time is $-{Q}_{\mathrm{ohm}}={{dU}}_{\mathrm{em}}/{dt}\simeq {B}_{0}^{2}/{t}_{\mathrm{ohm}}\simeq 750\,\mathrm{erg}\,{\mathrm{cm}}^{-}3\,{{\rm{s}}}^{-1}$. Integrated over the volume, this corresponds to ≃7 × 10⁻⁴ L_⊙. Comparing this to the energy loss due to neutrino cooling obtained from Equation (22b) with Equation (24), one finds ∣Q_ohm/Q_syn∣ ≃ 7.6, and hence, the additional heating by ohmic decay of the B would clearly dominate over the enhancement of the cooling rate caused by the B field. Moreover, the temperature would be larger due to the heating produced in previous times as well.

The electron and positron distributions are

$$\begin{eqnarray}&&{f}_{-}(E)=\displaystyle \frac{1}{{e}^{(E-\mu )/T}+1},\quad {f}_{+}(E)=\displaystyle \frac{1}{{e}^{(E+\mu )/T}+1},\end{eqnarray} \tag{ A1 }$$

where $E=\sqrt{{p}^{2}+{m}_{e}^{2}}$, so that the net charge density

$$\begin{eqnarray}&&{n}_{c}=\displaystyle \frac{1}{{\pi }^{2}}{\int }_{0}^{\infty }{dp}\,{p}^{2}\left({f}_{-}(E)-{f}_{+}(E)\right)\end{eqnarray} \tag{ A2 }$$

depends only on the chemical potential μ and the temperature T, which are therefore the two parameters upon which the QED plasma depends on when ion features are negligible. The dispersion relations for plasmons depend on the typical electron velocity

$$\begin{eqnarray}&&{v}_{* }=\displaystyle \frac{{\omega }_{1}}{{\omega }_{p}},\quad v=\displaystyle \frac{p}{E},\end{eqnarray} \tag{ A3 }$$

where the plasma frequency is given by

$$\begin{eqnarray}&&{\omega }_{p}^{2}=\displaystyle \frac{4\alpha }{\pi }{\int }_{0}^{\infty }{dp}\displaystyle \frac{{p}^{2}}{E}\left(1-\displaystyle \frac{1}{3}{v}^{2}\right)\left({f}_{-}(E)+{f}_{+}(E)\right),\end{eqnarray} \tag{ A4 }$$

and

$$\begin{eqnarray}&&{\omega }_{1}^{2}=\displaystyle \frac{4\alpha }{\pi }{\int }_{0}^{\infty }{dp}\displaystyle \frac{{p}^{2}}{E}\left(\displaystyle \frac{5}{3}{v}^{2}-{v}^{4}\right)\left({f}_{-}(E)+{f}_{+}(E)\right).\end{eqnarray} \tag{ A5 }$$

The dispersion relations for transverse and longitudinal plasmons with energy ω and momentum k are, respectively,

$$\begin{eqnarray}\begin{array}{rcl}{\omega }_{t}^{2} & = & {k}^{2}+{\omega }_{p}^{2}\displaystyle \frac{3{\omega }_{t}^{2}}{2{v}_{* }^{2}{k}^{2}}\left(1-\displaystyle \frac{{\omega }_{t}^{2}-{v}_{* }^{2}{k}^{2}}{{\omega }_{t}^{2}}\displaystyle \frac{{\omega }_{t}}{2{v}_{* }k}\mathrm{log}\displaystyle \frac{{\omega }_{t}+{v}_{* }k}{{\omega }_{t}-{v}_{* }k}\right),\\ 0 & \leqslant & k\lt \infty \end{array}\end{eqnarray} \tag{ A6 }$$

and

$$\begin{eqnarray}&&{\omega }_{l}^{2}={\omega }_{p}^{2}\displaystyle \frac{3{\omega }_{l}^{2}}{{v}_{* }^{2}{k}^{2}}\left(\displaystyle \frac{{\omega }_{l}}{2{v}_{* }k}\mathrm{log}\displaystyle \frac{{\omega }_{l}+{v}_{* }k}{{\omega }_{l}-{v}_{* }k}-1\right),\quad 0\leqslant k\lt {k}_{\max },\end{eqnarray} \tag{ A7 }$$

where ${k}_{\max }$ is the momentum at which the longitudinal plasmon dispersion relation crosses the light cone,

$$\begin{eqnarray}&&{k}_{\max }={\left[\displaystyle \frac{3}{{v}_{* }^{2}}\left(\displaystyle \frac{1}{2{v}_{* }}\mathrm{log}\displaystyle \frac{1+{v}_{* }}{1-{v}_{* }}-1\right)\right]}^{1/2}{\omega }_{p}.\end{eqnarray} \tag{ A8 }$$

The total energy loss due to plasmon decay is given by

$$\begin{eqnarray}&&{Q}_{\gamma \to \nu \bar{\nu }}={Q}_{T}+{Q}_{A}+{Q}_{L},\end{eqnarray} \tag{ A9 }$$

where the contribution from transverse plasmons through vector and axial-vector couplings, and from longitudinal plasmons through vector coupling, are, respectively,

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{T} & = & 2\left(\displaystyle \sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{0}^{+\infty }\\ & & \times {dk}\,{k}^{2}{Z}_{t}(k){\left({\omega }_{t}^{2}-{k}^{2}\right)}^{3}{f}_{B}({\omega }_{t}),\end{array}\end{eqnarray} \tag{ A10a }$$

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{A} & = & 2\left(\displaystyle \sum _{{\nu }_{\alpha }}{C}_{A}^{2}\right)\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{0}^{+\infty }\\ & & \times {dk}\,{k}^{2}{Z}_{t}(k)\left({\omega }_{t}^{2}-{k}^{2}\right){{\rm{\Pi }}}_{A}{\left({\omega }_{t},k\right)}^{2}{f}_{B}({\omega }_{t}),\end{array}\end{eqnarray} \tag{ A10b }$$

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{L} & = & \left(\displaystyle \sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{0}^{{k}_{\max }}\\ & & \times {dk}\,{k}^{2}{Z}_{l}(k){\omega }_{l}^{2}{\left({\omega }_{l}^{2}-{k}^{2}\right)}^{2}{f}_{B}({\omega }_{l}),\end{array}\end{eqnarray} \tag{ A10c }$$

where the plasmon distribution is given by

$$\begin{eqnarray}&&{f}_{B}(\omega )=\displaystyle \frac{1}{{e}^{\omega /T}-1}.\end{eqnarray} \tag{ A11 }$$

To compute them, one needs the functions renormalizing the coupling (Braaten & Segel 1993; Raffelt 1996),

$$\begin{eqnarray}&&{Z}_{t}(k)=\displaystyle \frac{2{\omega }_{t}^{2}\left({\omega }_{t}^{2}-{v}_{* }^{2}{k}^{2}\right)}{3{\omega }_{p}^{2}{\omega }_{t}^{2}+\left({\omega }_{t}^{2}+{k}^{2}\right)\left({\omega }_{t}^{2}-{v}_{* }^{2}{k}^{2}\right)-2{\omega }_{t}^{2}\left({\omega }_{t}^{2}-{k}^{2}\right)}\end{eqnarray} \tag{ A12 }$$

for the transverse plasmon, and

$$\begin{eqnarray}&&{Z}_{l}(k)=\displaystyle \frac{2\left({\omega }_{l}^{2}-{v}_{* }^{2}{k}^{2}\right)}{3{\omega }_{p}^{2}-\left({\omega }_{l}^{2}-{v}_{* }^{2}{k}^{2}\right)}\end{eqnarray} \tag{ A13 }$$

for the longitudinal one. The axial-vector polarization function reads as

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Pi }}}_{A} & = & \displaystyle \frac{2\alpha }{\pi }\displaystyle \frac{{\omega }^{2}-{k}^{2}}{k}{\displaystyle \int }_{0}^{\infty }{dp}\displaystyle \frac{{p}^{2}}{{E}^{2}}\left(\displaystyle \frac{\omega }{2{vk}}\mathrm{log}\displaystyle \frac{\omega +{vk}}{\omega -{vk}}\right.\\ & & -\left.\displaystyle \frac{{\omega }^{2}-{k}^{2}}{{\omega }^{2}-{v}^{2}{k}^{2}}\right)\left({f}_{-}(E)-{f}_{+}(E)\right),\end{array}\end{eqnarray} \tag{ A14 }$$

where

$$\begin{eqnarray}&&{\omega }_{A}=\displaystyle \frac{2\alpha }{\pi }{\int }_{0}^{\infty }{dp}\displaystyle \frac{{p}^{2}}{{E}^{2}}\left(1-\displaystyle \frac{2}{3}{v}^{2}\right)\left({f}_{-}(E)-{f}_{+}(E)\right).\end{eqnarray} \tag{ A15 }$$

In the nonrelativistic limit, the dispersion relations for transverse and longitudinal plasmons are, respectively, ¹³

$$\begin{eqnarray}&&{\omega }_{t}^{2}={\omega }_{p}^{2}+{{\boldsymbol{k}}}^{2}\quad \mathrm{and}\quad {\omega }_{l}^{2}={\omega }_{p}^{2}\,.\end{eqnarray} \tag{ A16 }$$

The plasma frequency in the limit T ≪ m_e is given in terms of the electron density n_e by

$$\begin{eqnarray}&&{\omega }_{p}^{2}=\displaystyle \frac{4\pi \alpha \,{n}_{e}}{{m}_{e}}{\left[1+\displaystyle \frac{1}{{m}_{e}^{2}}{\left(3{\pi }^{2}{n}_{e}\right)}^{2/3}\right]}^{-1/2}\,\simeq {\left(20\,\mathrm{keV}{\rho }_{6}^{1/2}\right)}^{2},\end{eqnarray} \tag{ A17 }$$

where ρ₆ = ρ/10⁶ g cm⁻³, and we assumed Y_e = 0.5. We shall also make use of the approximations (valid for T ≪ m_e ) Z_t ≃ 1, ${{\rm{\Pi }}}_{A}\simeq {\omega }_{p}^{2}/2{m}_{e}k({\omega }^{2}-{k}^{2})/{\omega }^{2}.$

With this in mind we obtain the following approximations:

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{T}^{\mathrm{NR}} & \simeq & 2\left(\sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)\displaystyle \frac{{G}_{F}^{2}{\omega }_{p}^{6}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{0}^{+\infty }{dk}\,{k}^{2}{f}_{B}({\omega }_{t})\\ & \simeq & \left(\sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)\displaystyle \frac{{G}_{F}^{2}{\omega }_{p}^{6}}{24{\pi }^{4}\alpha }\zeta (3){T}^{3},\end{array}\end{eqnarray} \tag{ A18 }$$

where in the second steps we approximate ω_t ≃ k, and

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{L}^{\mathrm{NR}} & \simeq & \left(\sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{0}^{{\omega }_{p}}{{dkk}}^{2}{\omega }_{p}^{2}{\left({\omega }_{p}^{2}-{k}^{2}\right)}^{2}{f}_{B}({\omega }_{p})\\ & \simeq & \left(\sum _{{\nu }_{\alpha }}{C}_{V}^{2}\right)\displaystyle \frac{{G}_{F}^{2}}{96{\pi }^{4}\alpha }\,\displaystyle \frac{T}{{\omega }_{p}}\,\displaystyle \frac{8{\omega }_{p}^{9}}{105},\end{array}\end{eqnarray} \tag{ A19 }$$

where in the second step we assume ω_p ≪ T. Notice however that for our conditions ω_p ≃ T, so one should use the first equality as a good approximation. The axial-vector contribution is

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{A}^{\mathrm{NR}} & = & 2\left(\sum _{{\nu }_{\alpha }}{C}_{A}^{2}\right)\displaystyle \frac{{G}_{F}^{2}{\omega }_{p}^{2}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{0}^{+\infty }{dk}\,{k}^{2}{{\rm{\Pi }}}_{A}{\left({\omega }_{t},k\right)}^{2}{f}_{B}({\omega }_{t})\\ & = & 2\left(\sum _{{\nu }_{\alpha }}{C}_{A}^{2}\right)\displaystyle \frac{{G}_{F}^{2}{\omega }_{p}^{6}}{96{\pi }^{4}\alpha }{\displaystyle \int }_{0}^{+\infty }{dk}\,\displaystyle \frac{{k}^{4}}{{\omega }_{t}^{4}}\displaystyle \frac{{\omega }_{p}^{4}}{4{m}_{e}^{2}}{f}_{B}({\omega }_{t}).\end{array}\end{eqnarray} \tag{ A20 }$$

We show the approximations from Equations (A18)–(A20) in Figure 1 from which we conclude that relativistic effects can be reasonably neglected for the regimes relevant for WDs.

That the agreement with nonrelativistic results being good is unsurprising, as can be seen by making the following estimates. If the WD is to be nonrelativistic and degenerate then ${p}_{F}\simeq {\left(3{\pi }^{2}{n}_{e}\right)}^{1/3}$, which gives

$$\begin{eqnarray}&&{p}_{F}\simeq 515\,\mathrm{keV}{\left({Y}_{e}{\rho }_{{}_{6}}\right)}^{1/3},\end{eqnarray} \tag{ A21 }$$

with ${\rho }_{{}_{6}}$ expressed in units 10⁶ g cm⁻³ and Y_e is the number of electrons per baryon. The latter are related by n_e = Y_e n_c = Y_e ρ/m_u , where ρ is the mass density and m_u is the atomic mass unit. This corresponds (taking Y_e ρ = 10⁶ g cm⁻³) to a nonrelativistic Fermi energy ${E}_{F}^{\mathrm{NR}}={p}_{F}^{2}/(2{m}_{e})\simeq 200\,\mathrm{keV}$ (and it is actually even smaller, as the average density of the profile is about 15% of the central density (Shapiro & Teukolsky 1983). The highest temperatures of relevance here are T ≃ 10⁸ K = 8.621 keV. Hence, we have that ${E}_{F}^{\mathrm{NR}}\ll {m}_{e}$ and T ≪ m_e .

These approximations also allow us to understand the relative subdominance of the axial contribution. If we neglect logarithmic corrections, we can see that ${Q}_{T}/{Q}_{A}={ \mathcal O }({m}_{e}^{2}{T}^{2}/{\omega }_{p}^{4})$, therefore, finding that in a nonrelativistic and nonmagnetized plasma the emission through the axial-vector coupling is highly suppressed. As discussed in Raffelt (1996; see also Vitagliano et al. 2017), one can heuristically think of the plasmon decay as dipole radiation from electrons coherently oscillating in space, thus contributing to vector emission, while the axial-vector emission corresponds to electron spin flips. As spins do not oscillate coherently in an unpolarized medium, the emission rate is suppressed. This also implies that in the nonmagnetized medium plasmon decay produces mostly ${\nu }_{e}{\bar{\nu }}_{e}$ pairs (Vitagliano et al. 2017).