The three loop slope of the Dirac form factor and the 1S Lamb shift in hydrogen

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 nS energy level is found to be 3.016/n^3 kHz. Using the QED calculations of the 1S Lamb shift, we extract a precise value of the proton charge radius r_p=0.883 \pm 0.014 fm.