# Solar Radio Emissions and Ultralight Dark Matter
## 2 Ultralight Dark Matter  
An axion is a pseudoscalar particle whose interaction with the electromagnetic part can be written as  
\[\mathcal{L}_{\rm axion}=-\frac{1}{4}F^{\mu\
u}F_{\mu\
u}+\frac{1}{2}\partial_{ \mu}a\partial^{\mu}a-\frac{1}{2}m_{a}^{2}a^{2}+\frac{1}{4}ag_{\sigma\gamma \gamma}F^{\mu\
u}\tilde{F}_{\mu\
u}. \tag{1}\]  
Figure 1: (a) Quiet-Sun electron density \(n_{e}\) and temperature \(T\) profiles in the solar atmosphere above the photosphere. Figure adapted from [67] based on the result of [68]. (b) \(n_{e}\) and \(T\) (thermal temperature of electrons) profiles in the solar-wind plasma with the distance extending up to about 1 AU. The \(n_{e}\) profile is based on the in situ measurements of the Parker solar probe [69] and then fitted by the relation \(n_{e}(r)\sim\left[3.3\times 10^{5}(r/R_{\odot})^{-2}+4.1\times 10^{6}(r/R_{ \odot})^{-4}+8.0\times 10^{7}(r/R_{\odot})^{-6}\right]\) cm\({}^{-3}\) given in [70]. The \(T\) profile is the relation \(T(r)\sim 418\) eV \(\times(r/R_{\odot})^{-0.74}\)[69] based on the in situ measurements.  
where \(a\) is the axion field and \(m_{a}\) is its mass. \(F^{\mu\
u}=\partial^{\mu}A^{\
u}-\partial^{\
u}A^{\mu}\) is the field strength of the standard model photon and \(\tilde{F}_{\mu\
u}=\varepsilon_{\mu\
u\alpha\beta}/2\cdot F^{\alpha\beta}\) is the dual field strength. \(g_{\alpha\gamma\gamma}\) is the coupling between the axion and the standard model photon. The interaction term in Equation (1) can be further simplified to \(-g_{\alpha\gamma\gamma}a\mathbf{E}\cdot\mathbf{B}\) in terms of the electric and magnetic fields.  
A dark photon is a \(U(1)\) gauge boson that, after electroweak symmetry breaking, interacts with the standard model photon via the following Lagrangian:  
\[\mathcal{L}_{dp}=-\frac{1}{4}F^{\mu\
u}F_{\mu\
u}-\frac{1}{4}F^{\mu\
u}F^{ \prime}_{\mu\
u}+\frac{1}{2}m^{2}_{A^{\prime}}A^{\prime 2}-\frac{\epsilon}{2}F^{ \prime\mu\
u}F_{\mu\
u}. \tag{2}\]  
where \(F^{\prime\mu\
u}=\partial^{\mu}A^{\prime\
u}-\partial^{\
u}A^{\prime\mu}\) is the field strength of dark photon and \(m_{A^{\prime}}\) is its mass. \(\epsilon\) is the kinetic mixing strength between the dark photon and the photon. To more conveniently study the conversion between a dark photon and a photon, we can diagonalize Equation (2) in the interaction basis with the following field redefinition:  
\[\begin{pmatrix}A\\ A^{\prime}\end{pmatrix}\rightarrow\begin{pmatrix}\frac{1}{\sqrt{1-\epsilon^{2 }}}&0\\ \frac{-\epsilon}{\sqrt{1-\epsilon^{2}}}&1\end{pmatrix}\begin{pmatrix}A\\ A^{\prime}\end{pmatrix}, \tag{3}\]  
and Equation (2) becomes  
\[\mathcal{L}_{dp}=-\frac{1}{4}F^{\mu\
u}F_{\mu\
u}-\frac{1}{4}F^{\prime\mu\
u}F^ {\prime}_{\mu\
u}+\frac{1}{2}m^{2}_{A^{\prime}}A^{\prime 2}-\epsilon m^{2}_{A^{ \prime}}A^{\prime}_{\mu}A^{\mu}. \tag{4}\]  
In solar plasma, photons will acquire an effective mass equal to the plasma frequency, \(m_{A}=\omega_{p}\). The plasma is charge neutral consisting of electrons and ions. Ignoring the contributions of ions due to their large masses, we only care about the plasma frequency of electrons, which in the non-relativistic case is  
\[\omega_{p}=\sqrt{\frac{4\pi\alpha n_{e}}{m_{e}}}. \tag{5}\]  
where \(\alpha\) is the fine structure constant, \(m_{e}\) is the electron mass, and \(n_{e}\) is the electron density. Therefore, for photons in plasma, we should add an effective mass term \(\mathcal{L}\supset 1/2\cdot\omega_{p}^{2}A^{2}\) in the Lagrangians (1) and (4).