A new analytic calculation solves the renormalization constants Z_m and Z₁ = Z₂ of quantum electrodynamics in Feynman gauge, within the on-shell scheme and using dimensional regularization. The results are obtained in analytic form at two loops, closing a technical problem that until now was handled only numerically or through partial approximations.

In addition to the main values, the work provides analytic expressions for the expansion coefficients in the dimensional regularization parameter ε, from order 1/ε² through order ε⁵. This expansion range is far wider than what single-loop calculations usually require, and responds to the demands of future higher-order corrections in QED.

The value of these high-order coefficients is twofold. On the one hand, they will be needed for more precise multi-loop calculations in quantum electrodynamics, where the ε expansions must be carried several terms beyond the simple pole. On the other, they provide a useful check for multi-loop calculations in quantum chromodynamics (QCD), whose renormalization structure shares techniques and divergences with those of QED.

The availability of closed-form analytic results also facilitates the detection of errors in future automatic implementations of Feynman integrals, a field in which independent checks are increasingly necessary as the number of loops considered at colliders grows.