Model independent determination of the muonic hydrogen Lamb shift and proton radius

We obtain a model independent expression for the muonic hydrogen Lamb shift. This expression includes the leading logarithmic ${\cal O}(m_{\mu}\alpha^6)$ terms, as well as the leading ${\cal O}(m_{\mu}\alpha^5 \frac{m_{\mu}^2}{m_{\rho}^2})$ hadronic effects. The latter are controlled by the chiral theory, which allows for their model independent determination. In this paper we give the missing piece for their complete expression including the pion and Delta particles. Out of this analysis and the experimental measurement of the muonic hydrogen Lamb shift we determine the electromagnetic proton radius: $r_p=0.8412(15)$ fm. This number is at 6.8$\sigma$ variance with respect to the CODATA value. The accuracy of our result is limited by uncomputed terms of ${\cal O}(m_{\mu}\alpha^5\frac{m_{\mu}^3}{m_{\rho}^3},m_{\mu}\alpha^6)$. This parametric control of the uncertainties allows us to obtain a model independent determination of the error, which is dominated by hadronic effects.