Gauge Invariance of Green Functions: Background-Field Method versus Pinch Technique

Application of the background-field method to QCD and the electroweak Standard Model yields gauge-invariant effective actions giving rise to simple Ward identities. Within this method, we calculate the quantities that have been treated in the literature using the pinch technique. Putting the quantum gauge parameter equal to one, we recover the pinch-technique results as a special case of the background-field method. The oneparticle-irreducible Green functions of the background-field method fulfil for arbitrary gauge parameters the desirable theoretical properties that have been noticed within the pinch technique. Therefore the background-field formalism provides a general framework for the direct calculation of well-behaved Green functions. Within this formalism, the pinch technique appears as one of arbitrarily many equivalent possibilities. BI-TP. 94/17 May 1994 Supported by the Bundesministerium für Forschung und Technologie, Bonn, Germany. All known successful theories describing the interactions of elementary particles are gauge theories. However, in order to evaluate quantized gauge theories within perturbation theory, one has to break gauge invariance in intermediate steps by choosing a definite gauge. As a consequence, although the physical observables, i.e. the S-matrix elements, are gauge-independent, the Green functions, the building blocks of the S-matrix elements, are gauge-dependent in the conventional formalism. Before we proceed, we remind the reader of the notion of gauge invariance and gauge independence: gauge invariance means invariance under gauge transformations. The gauge invariance of the classical Lagrangian gives rise to Ward identities between the Green functions of the quantized theory. Gauge independence becomes relevant when quantization is done by fixing a gauge. It means independence of the method of gauge fixing. The gauge dependence of Green functions poses no problem as long as one calculates physical observables in a fixed order of perturbation theory. However, as soon as one does not take into account all contributions in a given order one will in general arrive at gaugedependent results. This happens usually if one tries to resum higher-order corrections via Dyson summation of self-energies or if one is only interested in particular contributions like definite formfactors, e.g. magnetic moments for off-shell particles, without taking into account the full set of diagrams. This has been frequently done in the literature. Motivated by these facts, various attempts have been made to define gauge-independent building blocks. In order to construct gauge-independent running couplings, several proposals for gauge-independent self-energies have been made [ 1, 2]. These were essentially obtained by considering four-fermion processes and shifting parts of the box and vertex diagrams to the self-energies to cancel the gauge-parameter dependence of the latter within the class of Rξ gauges. As one can shift arbitrary gauge-independent contributions between the different building blocks the resulting quantities are not unique. This freedom has been used to require certain desirable properties from the self-energies, like a decent asymptotic behaviour and the vanishing of the photon–Z-boson mixing at zero momentum transfer. It nevertheless resulted in different definitions of gauge-independent building blocks. All these ad-hoc treatments only refer to four-fermion processes and do not give a general prescription which is applicable to other vertex functions. Such a prescription is given by the so-called pinch technique (PT) [ 3, 4, 5, 6]. The PT is an algorithm for the construction of (within Rξ gauges) gauge-independent vertex functions by reorganizing parts of the Feynman diagrams contributing to a manifestly gauge-independent quantity, leaving only a trivial gauge dependence in the tree propagators. The results obtained via the PT directly fulfil the desirable properties that had to be explicitly enforced in the ad-hoc treatments mentioned above. But even more important, it turns out that the vertex functions constructed according to the PT fulfil the simple Ward identities related to the classical Lagrangian. However, the PT leaves many questions unanswered. So far, it has only been realized for specific vertex functions at the one-loop level. Its application to other vertex functions is not always clear and its generalization to higher orders is non-trivial and non-unique. Although the PT vertex functions are claimed to be process-independent this has to the best of our knowledge not been proven but only shown for specific examples. It is very unsatisfactory that no explanation exists for the fact that the PT rules yield