Pinch technique: Theory and applications

Abstract

We review the theoretical foundations and the most important physical applications of the Pinch Technique (PT). This general method allows the construction of off-shell Green's functions in non-Abelian gauge theories that are independent of the gauge-fixing parameter and satisfy ghost-free Ward identities. We first present the diagrammatic formulation of the technique in QCD, deriving at one loop the gauge independent gluon self-energy, quark-gluon vertex, and three-gluon vertex, together with their Abelian Ward identities. The generalization of the PT to theories with spontaneous symmetry breaking is carried out in detail, and the profound connection with the optical theorem and the dispersion relations are explained within the electroweak sector of the Standard Model. The equivalence between the PT and the Feynman gauge of the Background Field Method (BFM) is elaborated, and the crucial differences between the two methods are critically scrutinized. A variety of field theoretic techniques needed for the generalization of the PT to all orders are introduced, with particular emphasis on the Batalin-Vilkovisky quantization method and the general formalism of algebraic renormalization. The main conceptual and technical issues related to the extension of the technique beyond one loop are described, using the twoloop construction as a concrete example. Then the all-order generalization is thoroughly examined, making extensive use of the field theoretic machinery previously introduced; of central importance in this analysis is the demonstration that the PT-BFM correspondence persists to all orders in perturbation theory. The extension of the PT to the non-perturbative domain of the QCD Schwinger-Dyson equations is presented systematically, and the main advantages of the resulting self-consistent truncation scheme are discussed. A plethora of physical applications relying on the PT are finally reviewed, with special emphasis on the definition of gauge-independent off-shell form-factors, the construction of non-Abelian effective charges, the gauge-invariant treatment of resonant transition amplitudes and unstable particles, and finally the dynamical generation of an effective gluon mass.

Baryons as relativistic three-quark bound states

“…Dyson-Schwinger equations [184][185][186] are the quantum equations of motion of a quantum field theory; see [147,[187][188][189][190][191][192] for reviews of their applications to QCD. The starting point for the derivation of DSEs is the same as in lattice QCD, namely the generating functional or partition function (3.10).…”

A covariant representation of the Ball–Chiu vertex

“…Having the same simple Ward identity also for the gluon loop case is possible, but requires more sophisticated techniques. It can be achieved using either the background field method ('BFM') [13,14,15] with Feynman gauge for the quantum field, or the pinch technique [16,17]. Although very different, those two methods turn out to lead to precisely the same Green's functions [18,19].…”

Production of massive color-octet vector bosons at next-to-leading order

“…These cancellations do not occur in the calculation of the NLO corrections to coloron production, because of vertex corrections involving the 3-point non-Abelian colored-boson vertices. As we describe in Section IV, we use the "pinch technique" [27] to divide the problematic non-Abelian vertex corrections into two pieces -a "pinched" piece whose UV divergence contributes to the renormalization of the coloron wavefunction (and, ultimately, a renormalization of the coloron coupling) and an "unpinched" part whose UV divergence (when combined with an Abelian vertex correction) cancels against the UV divergences in quark wavefunction renormalization. As we show, once the UV divergences are properly accounted for, the IR divergences cancel in the usual way: the IR divergences arising from real quark or gluon emission cancel against the IR divergences in the virtual corrections, and the IR divergences arising from collinear quarks or gluons in the initial state are absorbed in the properly defined parton distribution functions (PDFs).…”