Radiative corrections to the muonium hyperfine structure. II. Theα(Zα)2correction

This is the second of a series of papers on the radiative corrections of order ${\alpha }^2\left(Z\alpha \right)$, $\alpha (Z\alpha {)}^2$, 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 $+16.904\mathrm{}\hspace{0.5em}2\hspace{0.5em}\mathrm{}\left(11\right)\mathrm{}\alpha (Z\alpha {)}^2{E}_F/\pi$ 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 2 orders of magnitude.