Diagrams / Overlapping rainbow

Overlapping rainbow

Perturbative order$\alpha^{2}$
Topologyoverlapping
Symmetry factorCounts overcounting in the perturbation series. If a diagram has internal symmetries—ways to relabel internal lines and get the same diagram back—the symmetry factor divides that out. Most QED diagrams have S=1; the vacuum polarization bubble has S=1/2 because swapping the two internal fermion lines gives the same diagram.1.0
Divergence UV divergent IR divergent
Gauge invariant aloneNo

Contributes to

Electron propagator

Two virtual photons whose arcs cross: the first photon is emitted at $v_1$ and absorbed at $v_3$, while the second is emitted at $v_2$ and absorbed at $v_4$, with ordering $v_1, v_2, v_3, v_4$ along the fermion line. This overlapping topology cannot be decomposed into a product of one-loop subdiagrams — it is irreducibly two-loop.