
We use the pinch technique formalism to construct the gaugeindependent offshell twoloop fermion selfenergy, both for Abelian (QED) and nonAbelian (QCD) gauge theories. The new key observation is that all contributions originating from the longitudinal parts of gauge boson propagators, by virtue of the elementary treelevel Ward identities they trigger, give rise to effective vertices, which do not exist in the original Lagrangian; all such vertices cancel diagrammatically inside physical quantities, such as current correlation functions or Smatrix elements. We present two different, but complementary derivations: First, we explicitly track down the aforementioned cancellations inside twoloop diagrams, resorting to nothing more than basic algebraic manipulations. Second, we present an absorptive derivation, exploiting the unitarity of the Smatrix, and the Ward identities imposed on treelevel and oneloop physical amplitudes by gauge invariance, in the case of QED, or by the underlying BecchiRouetStora symmetry, in the case of QCD. The propagatorlike subamplitude defined by means of this latter construction corresponds precisely to the imaginary parts of the effective selfenergy obtained in the former case; the real part may be obtained from a (twice subtracted) dispersion relation. As in the oneloop case, the final twoloop fermion selfenergy constructed using either method coincides with the conventional fermion selfenergy computed in the Feynman gauge.
