Tight Constraint on Photon Mass from Pulsar Spindown

The simplest model for pulsar spindown is magnetic dipole radiation. For ${m}_{\gamma }=0$, the energy loss of the vacuum magnetic dipole radiation energy is given by $L={B}_{p}^{2}{R}^{6}{{\rm{\Omega }}}^{4}{\sin }^{2}\theta /6{c}^{3}$, where B_p is the polar magnetic field, and θ is the angle between the magnetic and rotational axes. The radiation energy originates from the rotational kinetic energy of the pulsar, causing the spindown of the pulsar.

We calculate the magnetic dipole radiation with ${m}_{\gamma }\ne 0$. Following Crandall & Wheeler (1984) (see Appendix B), we obtain the radiation power of the magnetic dipole field in vacuum, i.e.,

$$\begin{eqnarray}&&{L}_{m}=\displaystyle \frac{{m}^{2}{\rm{\Omega }}}{3}{\left(\displaystyle \frac{{{\rm{\Omega }}}^{2}}{{c}^{2}}-{\mu }^{2}\right)}^{1/2}\left(\displaystyle \frac{{{\rm{\Omega }}}^{2}}{{c}^{2}}+\displaystyle \frac{{\mu }^{2}}{2}\right)\end{eqnarray} \tag{ 3 }$$

for ${\rm{\Omega }}\gt \mu c$, and L_m = 0 for ${\rm{\Omega }}\leqslant \mu c$. Here m is the magnetic dipole moment. For $\mu =0$, the magnetic dipole radiation power reduces to the classical result, i.e., ${L}_{m,0}\,=(1/3){m}^{2}{{\rm{\Omega }}}^{4}/{c}^{3}$. We define

$$\begin{eqnarray}&&\eta \equiv \displaystyle \frac{{L}_{m}}{{L}_{m,0}}={\left(1-\displaystyle \frac{{\mu }^{2}{c}^{2}}{{{\rm{\Omega }}}^{2}}\right)}^{1/2}\left(1+\displaystyle \frac{{\mu }^{2}{c}^{2}}{2{{\rm{\Omega }}}^{2}}\right)\end{eqnarray} \tag{ 4 }$$

for ${\rm{\Omega }}\gt \mu c$, and $\eta =0$ for ${\rm{\Omega }}\leqslant \mu c$. η characterizes the correction of the nonzero photon mass effect. The $\eta \mbox{--}{m}_{\gamma }$ relation for the vacuum case is presented in Figure 1. In general, the observed period of the pulsar is very accurate with error $\ll 1\,{\rm{s}}$. Even for the WD pulsar in AR Scorpii, the relative uncertainty is of the order $\sim 1 \%$. The main uncertainty of the photon mass limit is from that of η. Observationally, quantifying η needs to independently measure L_m and ${L}_{m,0}$. Since pulsar spindown is naturally attributed to the magnetic dipole radiation in the vacuum case, one has ${L}_{m}\simeq \dot{E}=(2/5){{MR}}^{2}{\rm{\Omega }}\dot{{\rm{\Omega }}}$, which may be derived from pulsar mass M, radius R, and spin parameters Ω and $\dot{{\rm{\Omega }}}$. On the other hand, the magnetic dipole radiation without photon mass is given by ${L}_{m,0}\simeq {{\rm{\Omega }}}^{4}{R}^{6}{B}_{p}^{2}/6{c}^{3}$ (here we have adopted $\theta =\pi /2$; for the case with $\theta \ne \pi /2$, the constraint on the photon mass would be better), which requires an independent measurement of the polar cap magnetic field B_p at the surface. This field may be measured or constrained via other methods such as magnetohydrodynamic (MHD) pumping, Zeeman splitting,⁵ cyclotron lines, and properties of magnetar bursts. Finally, one has

$$\begin{eqnarray}&&\eta \simeq \displaystyle \frac{\dot{E}}{{L}_{m,0}}=\displaystyle \frac{12{c}^{3}M\dot{{\rm{\Omega }}}}{5{B}_{p}^{2}{R}^{4}{{\rm{\Omega }}}^{3}}.\end{eqnarray} \tag{ 5 }$$

Once η is derived from M, R, B_p, Ω, and $\dot{{\rm{\Omega }}}$ according to Equation (5), one can insert it into Equation (4) to calculate the photon mass.

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However, there is an immediate, most conservative upper limit on the photon mass that can be readily derived. According to Equations (3) and (4), as long as $\eta \gt 0$ is satisfied, which is true as long as a pulsar is observed to spin down, one should have $\mu \lt {\rm{\Omega }}/c$. A robust photon mass upper limit can be set to ${m}_{\gamma ,\mathrm{crit}}$, i.e.

$$\begin{eqnarray}&&{m}_{\gamma }\lt {m}_{\gamma ,\mathrm{crit}}\equiv h/{{Pc}}^{2},\end{eqnarray} \tag{ 6 }$$

which is shown as the a sharp cut-off of η in Figure 1. Equation (6) is essentially the result of the standard energy–momentum relation and does not depend on the detailed pulsar parameters other than the spin period P.

More generally, if η can be constrained from Equation (5), one can give a more stringent limit than that of Equation (6). In Table 1, we list the results of m_γ upper limits for the most conservative ($\eta \gt 0$) and three other assumed lower limit values, i.e., 0.1, 0.5, and 0.9. One can see that the improvement from the most conservative value even from $\eta \gtrsim 0.9$ is not significant. As a result, for the vacuum case one does not need to measure B_p to derive a robust constraint. As long as the pulsar is observed to spin down, i.e., $\eta \gt 0$, a conservative limit of ${m}_{\gamma }\lt {m}_{\gamma ,\mathrm{crit}}$ can be placed.

Table 1.  Upper Limits of the Photon Mass

Sources	Period / s	${m}_{\gamma }/{10}^{-49}\,{\rm{g}}$ in vacuum case				${m}_{\gamma }/{10}^{-49}\,{\rm{g}}$ in nonvacuum case
		η > 0	η > 0.1	η > 0.5	η > 0.9	η > 0.1	η > 0.5	η > 0.9
PSR J2144−3933	8.51	$\lt {m}_{\gamma ,\mathrm{crit}}=8.6$	<8.5	<8.1	<5.9	<20.3	<9.3	<3.1
1ES 1841−045	11.78	$\lt {m}_{\gamma ,\mathrm{crit}}=6.2$	<6.2	<5.8	<4.2	<14.7	<6.7	<2.2
WD in AR Scorpii	117	$\lt {m}_{\gamma ,\mathrm{crit}}=0.63$	<0.62	<0.59	<0.43	<1.48	<0.67	<0.22

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Since ${m}_{\gamma ,\mathrm{crit}}$ is inversely proportional to P, a pulsar with a longer period would give a more stringent constraint. Among neutron-star radio pulsars, that with the longest period is PSR J2144−3933 (Young et al. 1999) with $P=8.51\,{\rm{s}}$, $\dot{P}=0.475\times {10}^{-15}\,{\rm{s}}\,{{\rm{s}}}^{-1}$, and spin-down luminosity $\dot{E}\,=3.2\times {10}^{28}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,(I/{10}^{45}\,{\rm{g}}\,{\mathrm{cm}}^{2})$. Among magnetars, that with the longest period is 1ES 1841−045 (Dib et al. 2008) with $P\,=11.78\,{\rm{s}}$, $\dot{P}=3.93\times {10}^{-11}\,{\rm{s}}\,{{\rm{s}}}^{-1}$, and $\dot{E}=0.95\times {10}^{33}\,\mathrm{erg}\,{{\rm{s}}}^{-1}(I/{10}^{45}\,{\rm{g}}\,{\mathrm{cm}}^{2})$. Some magnetized WDs (Wickramasinghe & Ferrario 2000) with $B\sim {10}^{6}\mbox{--}{10}^{9}\,{\rm{G}}$ have periods around one hour (Ferrario et al. 1997). However, no spin-down measurements have been reported. The WD–M dwarf binary system, AR Scorpii, was recently reported to emit pulsed broadband emission (Marsh et al. 2016). The WD pulsar in AR Scorpii has measured spin-down parameters $P=117\,{\rm{s}}$, $\dot{P}=3.9\times {10}^{-13}\,{\rm{s}}\,{{\rm{s}}}^{-1}$, and $\dot{E}=1.5\times {10}^{33}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. The mean luminosity of AR Scorpii is $1.7\times {10}^{32}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, which includes thermal emission of $4.4\times {10}^{31}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ from the stellar components. Therefore, non-thermal emission from AR Scorpii could be comfortably powered by the magnetic dipole radiation of the WD (Geng et al. 2016; Marsh et al. 2016). Such a long-period WD pulsar can provide a stringent constraint on the photon mass.

In Figure 1, the blue, red, and green lines denote the periods of PSR J2144−3933, 1ES 1841−045 and the WD pulsar in AR Scorpii, respectively. The respective constrained upper limits of the photon mass are shown in Table 1: for PSR J2144−3933, one has ${m}_{\gamma }\lt 8.6\times {10}^{-49}\,{\rm{g}};$ for 1ES 1841−045, one has ${m}_{\gamma }\,\lt 6.2\times {10}^{-49}\,{\rm{g}};$ and for the WD pulsar in AR Scorpii, one has

$$\begin{eqnarray}&&{m}_{\gamma }\lt 6.3\times {10}^{-50}\,{\rm{g}}.\end{eqnarray} \tag{ 7 }$$

If the magnetic axis is parallel to the rotation axis, the magnetic dipole radiation is zero in vacuum. However, active pulsars are believed to be surrounded by a magnetosphere, from which a continuous outflow is launched from the open field-line regions as a pulsar wind (Goldreich & Julian 1969). The outflowing plasma exerts an electromagnetic torque on the pulsar, so that the pulsar spins down due to the existence of a pulsar wind (Harding et al. 1999; Xu & Qiao 2001; Contopoulos & Spitkovsky 2006; Tong et al. 2013).

To quantify the effect of nonzero photon mass on the wind spin-down rate, we first calculate the magnetic dipole field with ${m}_{\gamma }\ne 0$. We consider a statistic field solution (see Appendix A): $({{\rm{\nabla }}}^{2}-{\mu }^{2}){\boldsymbol{A}}=-4\pi {\boldsymbol{J}}/c$. The corresponding solution is (Jackson 1962)

$$\begin{eqnarray}{\boldsymbol{A}} & = & \displaystyle \frac{1}{c}\displaystyle \int {\boldsymbol{J}}({\boldsymbol{x}}{\boldsymbol{^{\prime} }})\displaystyle \frac{{e}^{-\mu | {\boldsymbol{x}}-{\boldsymbol{x}}{\boldsymbol{^{\prime} }}| }}{| {\boldsymbol{x}}-{\boldsymbol{x}}{\boldsymbol{^{\prime} }}| }{d}^{3}x^{\prime} \\ & = & -\,{\boldsymbol{m}}\times {\rm{\nabla }}\displaystyle \int \delta ({\boldsymbol{x}}{\boldsymbol{^{\prime} }})\displaystyle \frac{{e}^{-\mu | {\boldsymbol{x}}-{\boldsymbol{x}}{\boldsymbol{^{\prime} }}| }}{| {\boldsymbol{x}}-{\boldsymbol{x}}{\boldsymbol{^{\prime} }}| }{d}^{3}x^{\prime} .\end{eqnarray} \tag{ 8 }$$

Note that ${\boldsymbol{J}}=c({\rm{\nabla }}\times {\boldsymbol{ \mathcal M }})$, where ${\boldsymbol{ \mathcal M }}={\boldsymbol{m}}\delta ({\boldsymbol{x}})$ is the magnetization for the dipole approximation. According to ${\boldsymbol{B}}={\rm{\nabla }}\times {\boldsymbol{A}}$, one has (Jackson 1962)

$$\begin{eqnarray}{\boldsymbol{B}} & = & -\,{\rm{\nabla }}\times \left({\boldsymbol{m}}\times {\rm{\nabla }}\displaystyle \frac{{e}^{-\mu r}}{r}\right)\\ & = & [3{\boldsymbol{n}}({\boldsymbol{n}}\cdot {\boldsymbol{m}})-{\boldsymbol{m}}]\left(1+\mu r+\displaystyle \frac{{\mu }^{2}{r}^{2}}{3}\right)\displaystyle \frac{{e}^{-\mu r}}{{r}^{3}}\\ & & -\displaystyle \frac{2}{3}{\mu }^{2}{\boldsymbol{m}}\displaystyle \frac{{e}^{-\mu r}}{r}.\end{eqnarray} \tag{ 9 }$$

The magnetic field components in a spherical coordinate system are given by

$$\begin{eqnarray}{B}_{r} & = & 2m\displaystyle \frac{{e}^{-\mu r}}{{r}^{3}}\left(1+\mu r+\displaystyle \frac{{\mu }^{2}{r}^{2}}{3}\right)\cos \theta -\displaystyle \frac{2{\mu }^{2}m}{3}\displaystyle \frac{{e}^{-\mu r}}{r}\cos \theta ,\\ {B}_{\theta } & = & m\displaystyle \frac{{e}^{-\mu r}}{{r}^{3}}\left(1+\mu r+\displaystyle \frac{{\mu }^{2}{r}^{2}}{3}\right)\sin \theta +\displaystyle \frac{2{\mu }^{2}m}{3}\displaystyle \frac{{e}^{-\mu r}}{r}\sin \theta .\end{eqnarray} \tag{ 10 }$$

The field-line equation ${dr}/{B}_{r}={rd}\theta /{B}_{\theta }$ can be written as ${dr}/d\theta =2r(1+\mu r)\cot \theta /(1+\mu r+{\mu }^{2}{r}^{2})$. Interestingly, for $\mu \to \infty$, one has ${dr}/d\theta \simeq 2\cot \theta /\mu \to 0$ for $\theta \gg 0$, which means that the magnetosphere would approach a three-dimensional sphere, as shown in Figure 2. Integrating the field-line equation, one obtains $L{\sin }^{2}\theta ={{re}}^{\mu r}/(1+\mu r)$. Thus, the open field-line angle at the pulsar surface becomes $\sin {\theta }_{c}=\xi {(R/{R}_{\mathrm{LC}})}^{1/2}$ where

$$\begin{eqnarray}\xi & \equiv & {\left(\displaystyle \frac{1+\mu {R}_{\mathrm{LC}}}{1+\mu R}\right)}^{1/2}{e}^{\mu (R-{R}_{\mathrm{LC}})/2}\\ & \simeq & {\left(1+\displaystyle \frac{\mu c}{{\rm{\Omega }}}\right)}^{1/2}{e}^{-\mu c/2{\rm{\Omega }}}\end{eqnarray} \tag{ 11 }$$

denotes the correction factor with respect to the zero photon mass case, and ${R}_{\mathrm{LC}}\equiv c/{\rm{\Omega }}$ is the light cylinder radius. As shown in Figures 2 and 3, the larger the photon mass m_γ, the smaller the polar cap.

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Next, we calculate the plasma density in the magnetosphere. Since photon mass does not affect the Lorentz force density of the matter (Goldhaber & Nieto 1971), Ohm's law is still ${\boldsymbol{J}}=\sigma ({\boldsymbol{E}}+{\boldsymbol{\upsilon }}\times {\boldsymbol{B}}/c)$. For astrophysical force-free plasmas, due to $\sigma \to \infty$ and ${\boldsymbol{\upsilon }}={\boldsymbol{\Omega }}\times {\boldsymbol{r}}$, one has ${\boldsymbol{E}}=-{\boldsymbol{\Omega }}\times {\boldsymbol{r}}\times {\boldsymbol{B}}/c$ (Goldreich & Julian 1969; Ryutov 1997, 2007). Therefore, the charge density is given by $\rho =({\rm{\nabla }}\cdot {\boldsymbol{E}}+{\mu }^{2}\phi )/4\pi \,=-{\boldsymbol{\Omega }}\cdot {\boldsymbol{B}}/2\pi c+{\mu }^{2}\phi /4\pi$ (see Appendix A). On the other hand, according to the statistical field equation $\rho \,=(-{{\rm{\nabla }}}^{2}+{\mu }^{2})\phi /4\pi$, one has ${{\rm{\nabla }}}^{2}\phi =-4\pi {\rho }_{\mathrm{GJ}}$, where ${\rho }_{\mathrm{GJ}}\,\equiv -{\boldsymbol{\Omega }}\cdot {\boldsymbol{B}}/2\pi c$ is the classical Goldreich–Julian density (Goldreich & Julian 1969). The new plasma charge density ρ with ${m}_{\gamma }\ne 0$ is then given by

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}(\rho -{\rho }_{\mathrm{GJ}})+{\mu }^{2}{\rho }_{\mathrm{GJ}}\,=0.\end{eqnarray} \tag{ 12 }$$

For a neutron star white dwarf, if ${m}_{\gamma }\ll {\hslash }/{Rc}\simeq 3.5\times {10}^{-44}\,{\rm{g}}$ ($6.4\times {10}^{-47}\,{\rm{g}}$) (established with existing photon mass limits), $R\ll {R}_{\mathrm{LC}}$ is satisfied. The magnetic field strength and the charge density at the pole are ${B}_{p}\simeq 2m/{R}^{3}$ and $\rho \simeq {\rho }_{\mathrm{GJ}}$, respectively, which are consistent with the case of ${m}_{\gamma }=0$. The radius of the polar cap is ${r}_{p}=\xi R{({\rm{\Omega }}R/c)}^{1/2}$. The potential difference between the center and the edge of the polar cap is ${\rm{\Delta }}V\simeq {\rm{\Omega }}{r}_{p}^{2}{B}_{p}/2c\,={\xi }^{2}{{\rm{\Omega }}}^{2}{R}^{3}{B}_{p}/2{c}^{2}$. The net charged-particle flux from the polar cap is $\dot{N}\simeq \pi {r}_{p}^{2}{n}_{p}c={\xi }^{2}\chi {{\rm{\Omega }}}^{2}{R}^{3}{B}_{p}/2{ec}$, where ${n}_{p}=\chi {n}_{{}_{\mathrm{GJ}}}$ with $\chi \sim 1$ being the mean number density of the primary charged particles at the polar cap, which is essentially the Goldreich–Julian density. The spin-down power of the wind is therefore approximately

$$\begin{eqnarray}&&{L}_{w}\simeq 2e\dot{N}{\rm{\Delta }}V={\xi }^{4}\displaystyle \frac{\chi {{\rm{\Omega }}}^{4}{R}^{6}{B}_{p}^{2}}{2{c}^{3}}\equiv \eta {L}_{w,0},\end{eqnarray} \tag{ 13 }$$

where

$$\begin{eqnarray}&&\eta \equiv {L}_{w}/{L}_{w,0}={\xi }^{4}={\left(1+\displaystyle \frac{\mu c}{{\rm{\Omega }}}\right)}^{2}{e}^{-2\mu c/{\rm{\Omega }}},\end{eqnarray} \tag{ 14 }$$

and ${L}_{w,0}\simeq \chi {{\rm{\Omega }}}^{4}{R}^{6}{B}_{p}^{2}/2{c}^{3}$ is the spin-down power of the wind with ${m}_{\gamma }=0$. This result can be also derived from the method of Contopoulos et al. (1999) with $\chi =1/3$ (see Appendix C). Equation (14), even though not an explicit expression of m_γ, can be used to derive its upper limit when a lower limit of η is given (${m}_{\gamma }=0$ when $\eta =1$). The $\eta \mbox{--}{m}_{\gamma }$ relation for the nonvacuum wind spin-down case is presented in Figure 4. Similar to the vacuum case, the wind spin-down power rapidly falls off beyond a certain m_γ given a measured P. The difference from the vacuum case is that there is no absolute cutoff at ${m}_{\gamma ,\mathrm{crit}}=h/{{Pc}}^{2}$. In order to obtain $\eta \equiv {L}_{w}/{L}_{w,0}$ in Equation (14), where ${L}_{w}\simeq \dot{E}=(2/5){{MR}}^{2}{\rm{\Omega }}\dot{{\rm{\Omega }}}$ and ${L}_{w,0}\,\simeq \chi {{\rm{\Omega }}}^{4}{R}^{6}{B}_{p}^{2}/2{c}^{3}$, measurement of the magnetic field strength is necessary (unlike in the vacuum case).

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Once P, $\dot{P}$ and B_p are measured, one may constrain $\eta ={L}_{w}/{L}_{w,0}$ and derive the upper limit on the photon mass, as shown in Figure 5. We still take the data from PSR J2144−3933, 1ES 1841−045, and the WD pulsar in AR Scorpii as examples. We again assume $\eta \gt 0.1,0.5,0.9$. The upper limits on the photon mass using different sources are shown in Table 1. In particular, the WD pulsar in AR Scorpii (Marsh et al. 2016) has $P=117\,{\rm{s}}$, $\dot{P}=3.9\times {10}^{-13}\,{\rm{s}}\,{{\rm{s}}}^{-1}$ and $\dot{E}\,=1.5\times {10}^{33}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. An independent constraint on the WD surface magnetic field strength was set via the "MHD pumping" of the secondary star (M-dwarf) in the binary system (Buckley et al. 2017). The magnetic field of the WD would penetrate and dissipate in the M-dwarf atmosphere and give rise to additional optical emission. The dissipation power of magnetic energy at the M-dwarf can be estimated from⁶ ${P}_{\mathrm{MHD}}=({B}_{2}^{2}/8\pi )(4\pi {R}_{2}^{2}\delta )(2\pi /{P}_{{\rm{b}}})$, where ${R}_{2}\simeq 2.5\,\times {10}^{10}\,\mathrm{cm}$ is the radius of the M-dwarf star, ${P}_{{\rm{b}}}\simeq 118\,{\rm{s}}$ is the beat period, $\delta ={({\eta }_{\mathrm{tur}}{P}_{{\rm{b}}}/\pi )}^{1/2}\simeq 2\times {10}^{8}{\eta }_{\mathrm{tur},15}^{1/2}\,\mathrm{cm}$ is the dissipation depth of magnetic energy for the photospheric conditions of an M-type dwarf, and ${\eta }_{\mathrm{tur}}=({10}^{15}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}){\eta }_{\mathrm{tur},15}$ is the turbulent diffusivity (Buckley et al. 2017). If a fraction of the mean optical luminosity of AR Scorpii in excess of the combined stellar contributions is from MHD pumping, i.e., ${P}_{\mathrm{MHD}}\simeq \zeta {L}_{+}=1.3\times {10}^{32}\zeta \,\mathrm{erg}\,{{\rm{s}}}^{-1}$, where ζ is the fraction of the MHD pumping contribution, the magnetic field strength at the secondary star would be ${B}_{2}\simeq 204{\zeta }^{1/2}{\eta }_{\mathrm{tur},15}^{-1/4}\,{\rm{G}}$. For ${m}_{\gamma }\,\ll {\hslash }/{Rc}\simeq 6.4\times {10}^{-47}\,{\rm{g}}$ (established using existing photon mass limits), the magnetic dipole field of the WD would satisfy B ∝ r⁻³. The magnetic field strength at the pole of the WD would then be derived as ${B}_{p}\simeq {B}_{2}{(a/R)}^{3}\simeq 6.4\,\times {10}^{8}{\zeta }^{1/2}{\eta }_{\mathrm{tur},15}^{-1/4}{M}_{0.8}^{3.4}\,{\rm{G}}$, where $a\simeq 8\times {10}^{10}{M}_{0.8}^{1/3}\,\mathrm{cm}$ is the binary separation (Buckley et al. 2017; Marsh et al. 2016), and $M=(0.8{M}_{\odot }){M}_{0.8}$ is the WD mass. According to the Hamada–Salpeter relation, the radius–mass relation of WDs satisfies $R\simeq 5.5\times {10}^{8}{M}_{0.8}^{-0.8}\,\mathrm{cm}$. The dipole radiation luminosity with a zero-mass photon is ${L}_{w,0}\simeq {{\rm{\Omega }}}^{4}{R}^{6}{B}_{p}^{2}/6{c}^{3}\simeq 5.8\,\times {10}^{32}\zeta {\eta }_{\mathrm{tur},15}^{-1/2}{M}_{0.8}^{2}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. The dipole radiation luminosity with a nonzero-mass photon is ${L}_{w}\simeq \dot{E}=(2/5){{MR}}^{2}{\rm{\Omega }}\dot{{\rm{\Omega }}}\simeq 1.9\,\times {10}^{33}{M}_{0.8}^{-0.6}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. One therefore has $\eta \simeq 3.3{\zeta }^{-1}{\eta }_{\mathrm{tur},15}^{1/2}{M}_{0.8}^{-2.6}$. Here the turbulent diffusivity is taken as ${\eta }_{\mathrm{tur}}\,\simeq ({10}^{14}-{10}^{15})\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$ (e.g., Meintjes & Jurua 2006; Buckley et al. 2017) and the WD mass is taken as $M\simeq (0.8\mbox{--}1.3){M}_{\odot }$ (Marsh et al. 2016). Since $\zeta \lt 1$, one can derive $\eta \gt 0.3$, which is the most conservative constraint based on the uncertainties of all observed quantities. As a result, for the nonvacuum wind spin-down case we obtain

$$\begin{eqnarray}&&{m}_{\gamma }\lt 9.6\times {10}^{-50}\,{\rm{g}}.\end{eqnarray} \tag{ 15 }$$

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