High Precision Computation of the Lamb Shift of the Muonic Hydrogen Atom Within the Relativistic Framework

The measurement of the muonic hydrogen atom Lamb shift is one of the most fundamental ways to obtain the proton charge radius experimentally. The theoretical calculation of the muonic hydrogen atom Lamb shift is indispensable to extract the proton charge radius from the Lamb shift. In this work, we calculate the muonic hydrogen atom Lamb shift within the relativistic framework. The relativistic effect is described with the Breit-Pauli hamiltonian. The vacuum polarization effect is realized with the effective Uehling potential. With a high precision few-body method, Gaussian expansion method, we calculate the 2S_ 1/2^ and 2P_1/2^ levels of the muonic hydrogen atom. We find that the muonic hydrogen atom Lamb shift with considering the relativistic effect is about 0.09 meV larger than the one without considering the relativistic effect and the former one is closer to the experimental value.