Effective field theories for muonic hydrogen

Experimental measurements of muonic hydrogen bound states have recently started to take place and provide a powerful setting in which to study the properties of QCD. We profit from the power of effective field theories (EFTs) to provide a theoretical framework in which to study muonic hydrogen in a model independent fashion. In particular, we compute expressions for the Lamb shift and the hyperfine splitting. These expressions include the leading logarithmic O(mμα6) terms, as well as the leading O(mμα5mμ2ΛQCD2)${\cal O}\left( {{m_\mu }{\alpha ^5}{{m_\mu ^2} \over {\Lambda _{{\rm{QCD}}}^2}}} \right)$ hadronic effects. Most remarkably, our analyses include the determination of the spin-dependent and spin-independent structure functions of the forward virtualphoton Compton tensor of the proton to O(p3) in HBET and including the Delta particle. Using these results we obtain the leading hadronic contributions to the Wilson coeffcients of the lepton-proton four fermion operators in NRQED. The spin-independent coeffcient yields a pure prediction for the two-photon exchange contribution to the muonic hydrogen Lamb shift, which is the main source of uncertainty in our computation. The spindependent coeffcient yields the prediction of the hyperfine splitting. The use of EFTs crucially helps us organizing the computation, in such a way that we can clearly address the parametric accuracy of our result. Furthermore, we review in the context of NRQED all the contributions to the energy shift of O(mμα5, as well as those that scale like mrα6× logarithms.