On the Lamb shift in neutral muonic helium

The neutral muonic helium is an exotic atomic system consisting of an electron, muon and a nucleus. We consider it as a hydrogen-like atom with a compound nucleus that is also hydrogen-like system. There are a number of corrections to the Bohr energy levels, which all can be treated as contributions of generic hydrogen-like theory. While the form of those contributions is the same for all hydrogen-like atoms, their relative numerical importance differs from an atom to an atom. Here, the leading contribution to the electronic Lamb shift in the neutral muonic helium is found in a close analytic form together with the most important corrections. We believe that the Lamb shift in the neutral muonic hydrogen is measurable, at least through a measurement of the electronic 1$s-$2$s $ transition. We present a theoretical prediction for the 1$s-$2$s $ transitions with the uncertainty of 2 ppm (4 GHz), as well as for the 2$s-$2$p $ Lamb shift with the uncertainty of 0.6GHz.