Off-shell renormalization in the presence of dimension 6 derivative operators. Part III. Operator mixing and β functions

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 powercounting conditions.

Background field method and generalized field redefinitions in effective field theories

“…The kernels have been computed to the accuracy required to renormalize dim.6 operators in Ref. [25]. Let us denote by a bar the UV divergent part of a given amplitude or of the full vertex functional.…”

“…( 10) denotes the UV divergent part of the vertex functional at zero background. It has been evaluated in [25] for the relevant sector of operators up to dimension 6. Eq.…”

“…is in general deformed in a non-linear (and gauge-dependent) way, unlike in the powercounting renormalizable case where only multiplicative Z-factors arise both for background and quantum fields; ii) in the Landau gauge no deformation of the tree-level backgroundquantum splitting happens, to all orders in the loop expansion; iii) also the background fields renormalize non-linearly, similarly to what happens to quantum fields when no backgrounds are switched on [25][26][27]; iv) the redefinition of the background fields is background gauge invariant. This follows from non-trivial cancellations between the non gauge-invariant contributions to the background-quantum splitting and the non gauge-invariant terms in the generalized field redefinitions of the quantum fields; v) both the background and the quantum field redefinitions need to be taken into account in order to ensure gauge-invariance of the coupling constants renormalization.…”

Decoupling Limits in Effective Field Theories via Higher Dimensional Operators

“…The construction of the gauge-invariant dynamical counter-part of a scalar particle has been studied in [19][20][21][22].…”

Background Field Method and Generalized Field Redefinitions in Effective Field Theories

“…is in general deformed in a non-linear (and gauge-dependent) way, unlike in the power-counting renormalizable case where only multiplicative Z-factors arise both for background and quantum fields; ii) in the Landau gauge no deformation of the tree-level background-quantum splitting happens, to all orders in the loop expansion; iii) also the background fields renormalize nonlinearly, similarly to what happens to quantum fields when no backgrounds are switched on [25][26][27]; iv) the redefinition of the background fields is background gauge invariant. This follows from non-trivial cancellations between the non gauge-invariant contributions to the backgroundquantum splitting and the non gauge-invariant terms in the generalized field redefinitions of the quantum fields; v) both the background and the quantum field redefinitions need to be taken into account in order to ensure gauge-invariance of the coupling constants renormalization.…”