M1 and E1 Transitions of Heavy-Light Quarkonia in Bethe-Salpeter Equation

We calculate the radiative decay widths of heavy-light quarkonia for the processes, \(H\rightarrow H'\gamma \) in the framework of \(4\times 4\) Bethe-Salpeter equation (BSE) under Covariant instantaneous ansatz, which is a Lorentz-invariant generalization of Instantaneous Approximation. The 3D transition amplitudes thus obtained for various processes are relativistically covariant. We use this framework to calculate the M1 transitions [1–2] \(0^{-+}\rightarrow 1^{--}\gamma \) , \(1^{--}\rightarrow 0^{-+}\gamma \) , and E1 transitions [1–2] involving axial vector mesons such as, \(1^{+-}\rightarrow 0^{-+}\gamma \) , \(0^{-+}\rightarrow 1^{+-}\gamma \) , \(1^{++}\rightarrow 1^{--}\gamma \) and \(1^{--}\rightarrow 1^{++}\gamma \) , for which very little data is available as of now. We make use of the general structure of the transition amplitude, expressed as a linear superposition of terms involving all possible combinations of \(++\) , and \(--\) components of Salpeter wave functions of the final and the initial hadrons, expressible in a covariant form in terms of transition form factors. We evaluate the decay widths for both the above mentioned M1 and E1 transitions. We have compared our results with experimental data [3], and other models.