Radiative decays of heavy-light quarkonia through $M1$, and $E1$ transitions in the framework of Bethe-Salpeter equation

In this work we study the radiative decays of heavy-light quarkonia through M1 and E1 transitions, that involve quark-triangle diagrams with two hadron vertices, and are difficult to evaluate in BSE-CIA. We have expressed the transition amplitude, $M_{fi}$ as a linear superposition of terms involving all possible combinations of $++$, and $--$ components of Salpeter wave functions of final and initial hadron, with coefficients being related to results of pole integrals over complex $\sigma$- plane. We evaluate the decay widths for $M1$ transitions ($^3S_1 \rightarrow ^1S_0 +\gamma$), and $E1$ transitions ($^3S_1 \rightarrow ^1P_0 +\gamma$ and $^1P_0 \rightarrow ^3S_1 +\gamma$). We have used algebraic forms of Salpeter wave functions obtained through analytic solutions of mass spectral equations for ground and excited states of $0^{++},1^{--}$, and $0^{-+}$ heavy-light quarkonia in approximate harmonic oscillator basis, to calculate their decay widths. The input parameters used by us were obtained by fitting to their mass spectra. We have compared our results with experimental data and other models, and found reasonable agreements.