The last unknown contribution to hydrogen energy levels at order $m{\alpha }^7$, due to the slope of the Dirac form factor at three loops, is evaluated in a closed analytical form. The resulting shift of the hydrogen $\mathrm{nS}$ energy level is found to be ${3.016/n}^3\mathrm{kHz}$. Using the QED calculations of the $1S$ Lamb shift, we extract a precise value of the proton charge radius $r_p=0.883\mathrm{}\pm 0.014\mathrm{fm}$.

• Received 10 November 1999

DOI:https://doi.org/10.1103/PhysRevLett.84.1673