The Pinch Technique Beyond One Loop: The Gauge-Independent Two-Loop Quark Self-Energy

It is shown how the pinch technique algorithm may be consistently extended beyond the one-loop level to obtain the gauge-independent two-loop fermion self-energy −iΣ̂(p) in QCD in the pinch technique approach. The starting point for the construction is the general diagrammatic representation of the two-loop quark self-energy in terms of renormalized one-loop twoand three-point function and tree level Bethe-Salpeter-type quark-gluon scattering kernel insertions in the one-loop quark self-energy. Using factors of longitudinal gluon four-momentum only from lowest order gauge field propagators and triple gauge vertices to trigger the relevant Ward identities, the function −iΣ̂(p) is explicitly constructed from the consideration of the two-loop QCD corrections to the Compton scattering of a photon off a quark. It is shown that the resulting pinch technique self-energy −iΣ̂(p) is gauge-independent at all momenta, does not shift the position of the propagator pole and is multiplicatively renormalizable by local counterterms. The demonstration of the gauge independence is based on an efficient diagrammatic method to deal with the several dozen two-loop diagrams involved. It is explicitly shown by this example that the general correspondence between the pinch technique n-point functions and those obtained in the background field method in the Feynman quantum gauge ξQ = 1 does not persist beyond one loop. 1 Present address: Rutherford Appleton Laboratory, Chilton, Didcot, Oxon., OX11 0QX, U.K. Email: [email protected]