Gauge independent transverse and longitudinal self-energies and vertices   via the pinch technique
  In the context of the Standard Model we show how to apply the pinch technique to four-fermion amplitudes with non-conserved external charged currents,in order to construct to one-loop order gauge independent self-energies and vertices. We discuss the technical difficulties arising due to the presence of longitudinal contributions from the W and Goldstone boson ($\phi$) propagators, and derive gauge independent $WW$, $\phi W$, and $\phi\phi$ effective self-energies, and vertices. The quantities so constructed satisfy a set of Ward identities, whose validity enforces the gauge invariance of the physical amplitude considered; their derivation does not require knowledge of the explicit closed form of the gauge independent self-energies and vertices. Use of these Ward identities enables the decomposition of the amplitude in manifestly gauge independent transverse and longitudinal structures with distinct kinematic properties. Explicit one-loop calculations are carried out, and several applications of the results are briefly discussed.} 
Joannis Papavassiliou