Off-shell renormalization in the presence of dimension 6 derivative operators. Part I. General theory

Abstract

We evaluate the one-loop β functions of all dimension 6 parity-preserving operators in the Abelian Higgs-Kibble model. No on-shell restrictions are imposed; and the (generalized) non-polynomial field redefinitions arising at one-loop order are fully taken into account. The operator mixing matrix is also computed, and its cancellation patterns explained as a consequence of the functional identities of the theory and power-counting conditions.

Off-shell renormalization of spontaneously broken effective gauge theories

“…The decoupling of the unphysical Lagrange multiplier X 1 is guaranteed by the invariance of the vertex functional under the following constraint BRST differential [25,31]…”

“…so that the most general solution is X 2 = 1 v φ † φ − v 2 2 + η, η being a scalar field of mass m. However it can be proven that in perturbation theory the correlators of the field η with any gauge-invariant operator are zero [25], and so one can safely set η = 0. sσ , sχ. We remark that one can avoid the introduction of the antifield for the gauge connection A µ as well as for the antighostω since their BRST transformation is linear in the quantum fields and thus it is not subject to an independent renormalization.…”

“…The vertex functional Γ obeys the ST identity (equivalently the BV bracket) for the model at hand [24,25]:…”

“…This happens because the dependence of the vertex functional Γ on the field X 2 can be strongly restricted by a set of functional identities that involve external sources with a better UV behaviour, thus reducing the number of independent UV divergences [22,23,24,25].…”

“…After completing the renormalization of the X-theory one eventually goes on-shell with the X 2field and its companion X 1 , the field that plays the role of the Lagrange multiplier enforcing the constraint (1.10). It can be proven that one gets back the original amplitudes in the φ -formalism [25]. The latter are however decomposed into different pieces corresponding to different sectors of the starting X-theory that exhibit in a more transparent way the relations between the UV divergences of the non-power-counting renormalizable theory.…”