The pinch technique at two-loops: The case of mass-less Yang-Mills theories

The generalization of the pinch technique beyond one loop is presented. It is shown that the crucial physical principles of gauge-invariance, unitarity, and gauge-fixing-parameter independence are instrumental for accomplishing this task, and is explained how the aforementioned requirements single out at two loops exactly the same algorithm which has been used to define the pinch technique at one loop, without any additional assumptions. The two-loop construction of the pinch technique gluon self-energy, and quark-gluon vertex are carried out in detail for the case of mass-less YangMills theories, such as perturbative QCD. We present two different but complementary derivations. First we carry out the construction by directly rearranging two-loop diagrams. The analysis reveals that, quite interestingly, the well-known one-loop correspondence between the pinch technique and the background field method in the Feynman gauge persists also at two-loops. Since we use dimensional regularization the entire construction does not depend on the value of the space-time dimension d. The renormalization (when d = 4) is discussed in detail, and is shown to respect the aforementioned correspondence. Second, we present an absorptive derivation, exploiting the unitarity of the S-matrix and the underlying BRS symmetry; at this stage we deal only with tree-level and one-loop physical amplitudes. The gauge-invariant sub-amplitudes defined by means of this absorptive construction correspond precisely to the imaginary parts of the n-point functions defined in the full two-loop derivation, thus furnishing a highly nontrivial self-consistency check for the entire method. Various future applications are briefly discussed. e-mail: [email protected];@uv.es