Classical and quantum aspects of the extended antifield formalism

Starting from a solution to the classical Batalin-Vilkovisky master equation, an extended solution to an extended master equation is constructed by coupling all the observables, the anomaly candidates and the generators of global symmetries. The construction of the formalism and its applications in the context of the renormalization of generic and potentially anomalous gauge theories are reviewed. The main aim of the standard antifield formalism is the construction, for generic gauge theories, of the proper solution of the master equation. The master equation is formulated in terms of the antisymplectic structure for the fields and antifields, and its solution is the generator of the BRST differential. The coupling constants play a passive role in the usual discussions. The most important feature of the extended antifield formalism is to promote the coupling constants to active participants of the construction, by allowing the BRST differential to act on them, and by introducing, in an intermediate stage “anticoupling constants” which are their partners in an extended antibracket. The understanding that the physically relevant coupling constants are related to independent local BRST cohomology classes in ghost number 0 naturally leads to consider the constant ghosts coupled to the generators of generalized global symmetries (local BRST cohomology classes in negative ghost numbers) as generalized coupling constants and to include the couplings to anomaly and anomaly for anomaly candidates (local BRST cohomology classes in positive ghost numbers). The heart of the extended antifield formalism is the construction of an extended master equation for an extended action to which all these cohomology classes have been coupled and a characterization of the cohomology of the associated BRST differential, which takes into account in a systematic way higher order cohomological restrictions through the Lie-Massey brackets. This construction is shown to be a particular case of a general structure that is available as soon as one has a differential graded Lie algebra, i.e., a graded vector space with an even or odd Lie bracket and a differential that is a graded derivation of the bracket. Applications of these ideas in the context of renormalized quantum field theory are then discussed. More precisely: • The existence and relevance of higher order cohomological restrictions on anomalies and counterterms is demonstrated. As an illustration, the case of Yang-Mills theories with abelian factors is discussed. • It is shown that the use of the extended formalism guarantees stability independently of power counting restrictions (“renormalizability in the modern sense”) for generic gauge theories. • The anomalous Zinn-Justin equation for the renormalized effective action can be written to all orders as a functional differential equation. This allows to prove the existence of a quantum BRST differential and to extend the whole framework of algebraic renormalization to anomalous gauge theories. In particular, the existence of well defined quantum BRST cohomologies is proved. • The dependence of the quantum theory on the parameters of the gauge fixing is considered and the general structure of the Callan-Symanzik and the renormalization group equations is discussed. • A refined anomaly consistency condition for local BRST cohomology classes is derived. As an application, a new approach to the Adler-Bardeen theorem on the non renormalization of the non abelian gauge anomaly is proposed.