# Solar Radio Emissions and Ultralight Dark Matter
## 3 Conversion in Solar Plasma
 \tag{13}\]  
The off-diagonal matrix \(M_{1}\) is suppressed by the coupling constant, either \(\epsilon\) or \(g_{\sigma\gamma\gamma}\), and thus can be treated as a perturbative term. If we start with a \(X\) particle, \((A,X)=(0,1)\) initially, then solving the equation of motion (Equation (12)) to the first order of \(M_{1}\), the probability that \(X\) eventually converts to a photon \(A\), namely, \((A,X)=(1,0)\), is  
\[P_{X\rightarrow\gamma}=\left|\int_{r_{0}}^{r}dr^{\prime}\Delta_{AX}(r^{\prime })\exp\left\{i\int_{r_{0}}^{r^{\prime}}dr^{\prime\prime}\frac{1}{2k_{r}} \left[\omega_{p}^{2}(r^{\prime\prime})-m_{X}^{2}\right]\right\}\right|^{2}. \tag{14}\]  
We can solve the integration using the saddle-point method  
\[\left|\int dr^{\prime}\ g(r^{\prime})\mathrm{e}^{-ih(r^{\prime})}\right|\approx g (r_{c})\sqrt{\frac{2\pi}{|h^{\prime\prime}(r_{c})|}},\ \ \ h(r^{\prime})\equiv\int_{r_{0}}^{r^{\prime}}dr^{ \prime\prime}\frac{1}{2k_{r}}\left[\omega_{p}^{2}(r^{\prime\prime})-m_{X}^{2}\right] \tag{15}\]  
where we have used the expression of norm to remove the redundant phase factor. \(r_{c}\) is the position when \(h(r)\) reaches its local maximum/minimum \(h^{\prime}(r_{c})=0\), that is, the position of resonant \(X\)-\(A\) conversion when \(\omega_{p}(r_{c})=m_{X}\). Only the second derivative plays an important role, the expression of which is  
\[h^{\prime\prime}(r_{c})=\frac{\omega_{p}}{k_{r}}\left.\frac{\partial\omega_{p}(r )}{\partial r}\right|_{r=r_{c}}. \tag{16}\]  
The length of the resonant layer \(\delta r_{c}\) can be estimated as the length that makes \(h(r)\approx h(r_{c})+1/2\cdot(r-r_{c})^{2}h^{\prime\prime}(r_{c})\) change by \(2\pi\), so we obtain  
\[\delta r_{c}\sim\sqrt{\frac{4\pi}{|h^{\prime\prime}(r_{c})|}}\sim\sqrt{\frac{ r_{c}\overline{v_{r}}}{\omega_{p}}}. \tag{17}\]  
The resonant layer is very thin, \(\delta r_{c}/r_{c}\sim 10^{-6}\), for the typical values \(r_{c}=R_{\odot}\), \(\omega_{p}=2\pi\cdot 100\) MHz, and \(v_{r}=10^{-3}c\). Applying the saddle-point approximation as shown is Equation (15), we solve Equation (14) and obtain  
\[P_{X\rightarrow\gamma}=\Delta_{AX}^{2}(r_{c})\cdot\frac{2\pi k_{r}}{\omega_{p }}\left|\frac{\partial\omega_{p}}{\partial r}\right|_{r=r_{c}}^{-1}. \tag{18}\]  
Then, for each case we have  
\[P_{a\rightarrow\gamma}=\pi\frac{g_{\pi\gamma\gamma}^{2}|B_{T}|}{m_{a}}\frac{ 1}{v_{r}(r_{c})}\left|\frac{\partial\ln\omega_{p}^{2}}{\partial r}\right|_{r =r_{c}}^{-1},\ \ \ \text{(axion case)} \tag{19}\]  
and  
\[P_{A^{\prime}\rightarrow\gamma}\simeq\frac{2}{3}\pi\epsilon^{2}m_{A^{\prime} }\frac{1}{v_{r}(r_{c})}\left|\frac{\partial\ln\omega_{p}^{2}}{\partial r} \right|_{r=r_{c}}^{-1},\ \ \ \text{(dark photon case)}. \tag{20}\]  
Note that we have an extra prefactor \(2/3\) in the dark photon case. This is because the converted photons polarize in three directions with equal probability but the longitudinal mode cannot propagate out of the plasma. In comparison, the photons converted from axions polarize only in one transverse mode parallel to \(B_{T}\) as mentioned above. Furthermore, the conversion probabilities for the two cases are related via the following replacement  
\[\sqrt{\frac{2}{3}}\epsilon m_{A^{\prime}}^{2}\Leftrightarrow g_{\pi\gamma \gamma}|\mathbf{B}_{T}|\omega,\ \ \text{($\omega\simeq m_{a}$, non-relativistic)}. \tag{21}\]  
This relation is useful when we know the conversion rate of one case and directly calculate that of the other case.