The Electron Self-Energy at 2 and 3 Loops

In these contributions, we report on the computation of the bare electron self-energy up to three loops in terms of iterated integrals over kernels of elliptic type. To systematically compute the master integrals, we use differential equations that are brought into canonical form such that the $\epsilon$-expansion can be easily calculated in terms of iterated integrals. Up to three loops, only sixteen different kernels are necessary to describe all master integrals, whereas at two loops, only seven are necessary. For a numerical evaluation of the iterated integrals, we use local series expansions, which can be analytically continued to any region of the relevant parameter space. Moreover, we show that all elliptic contributions can be traced back to the sunset elliptic curve, already appearing at two loops.