Canonical Transformations and Renormalization Group Invariance in the presence of Non-trivial Backgrounds

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We show that for a SU(N) Yang-Mills theory the classical background-quantum splitting is non-trivially deformed at the quantum level by a canonical transformation with respect to the Batalin-Vilkovisky bracket associated with the Slavnov-Taylor identity of the theory. This canonical transformation acts on all the fields (including the ghosts) and antifields; it uniquely fixes the dependence on the background field of all the one-particle irreducible Green's functions of the theory at hand. The approach is valid both at the perturbative and non-perturbative level, being based solely on symmetry requirements. As a practical application, we derive the renormalization group equation in the presence of a generic background and apply it in the case of a SU(2) instanton. Finally, we explicitly calculate the one-loop deformation of the background-quantum splitting in lowest order in the instanton background.