Pinch technique self - energies and vertices to all orders in perturbation theory

Daniele Binosi and Joannis Papavassiliou Departamento de F́ısica Teórica and IFIC, Centro Mixto, Universidad de Valencia-CSIC, E-46100, Burjassot, Valencia, Spain Abstract The all-order construction of the pinch technique gluon self-energy and quark-gluon vertex is presented in detail within the class of linear covariant gauges. The main ingredients in our analysis are the identification of a special Green’s function, which serves as a common kernel to all self-energy and vertex diagrams, and the judicious use of the Slavnov-Taylor identity it satisfies. In particular, it is shown that the ghost-Green’s functions appearing in this identity capture precisely the result of the pinching action at arbitrary order. By virtue of this observation the construction of the quarkgluon vertex becomes particularly compact. It turns out that the aforementioned ghost-Green’s functions play a crucial role, their net effect being the non-trivial modification of the ghost diagrams of the quark-gluon vertex in such a way as to reproduce dynamically the characteristic ghost sector of the background field method. The gluon self-energy is also constructed following two different procedures. First, an indirect derivation is given, by resorting to the strong induction method and the assumption of the uniqueness of the S-matrix. Second, an explicit construction based on the intrinsic pinch technique is provided, using the Slavnov-Taylor identity satisfied by the all-order three-gluon vertex nested inside the self-energy diagrams. The process-independence of the gluon self-energy is also demonstrated, by using gluons instead of quark as external test particles, and identifying the corresponding kernel function, together with its Slavnov-Taylor identity. Finally, the general methodology for carrying out the renormalization of the resulting Green’s functions is outlined, and various open questions are briefly discussed.