Radiative Corrections to the Muonium Hyperfine Structure. II. The $\alpha (Z\alpha)^2 $ Correction

This is the second of a series of papers on the radiative corrections of order $\alpha^2 (Z\alpha)$, $\alpha (Z\alpha )^2$, and various logarithmic terms of order $\alpha^4$, to the hyperfine structure of the muonium ground state. This paper deals with the $\alpha (Z\alpha)^2$ correction. Based on the NRQED bound state theory, we isolated the term of order $\alpha(Z\alpha)^2$ exactly. Our result $+16.904 2 (11) \alpha(Z\alpha)^2 E_F / \pi$ for the non-logarithmic part is consistent with the $\alpha (Z\alpha )^2$ part of Sapirstein's calculation and the recent result of Pachucki, and reduces the numerical uncertainty in the $\alpha (Z\alpha)^2$ term by two orders of magnitude.