Relativistic recoil effects in a muonic atom within a Grotch-type approach: General approach

Recently we calculated relativistic recoil corrections to the energy levels of the low lying states in muonic hydrogen induced by electron vacuum polarization effects. The results were obtained by Breit-type and Grotch-type calculations. The former were described in our previous papers in detail, and here we present the latter. The Grotch equation was originally developed for pure Coulomb systems and allowed to express the relativistic recoil correction in order $(Z\alpha)^4m^2/M$ in terms of the relativistic non-recoil contribution $(Z\alpha)^4m$. Certain attempts to adjust the method to electronic vacuum polarization took place in the past, however, the consideration was incomplete and the results were incorrect. Here we present a Groth-type approach to the problem and in a series of papers consider relativistic recoil effects in order $\alpha(Z\alpha)^4m^2/M$ and $\alpha^2(Z\alpha)^4m^2/M$. That is the first paper of the series and it presents a general approach, while two other papers present results of calculations of the $\alpha(Z\alpha)^4m^2/M$ and $\alpha^2(Z\alpha)^4m^2/M$ contributions in detail. In contrast to our previous calculation, we address now a variety of states in muonic atoms with a certain range of the nuclear charge $Z$.