Abstract

The full off-shell one loop renormalization for all divergent amplitudes up to dimension 6 in the Abelian Higgs-Kibble model, supplemented with a maximally power counting violating higher-dimensional gauge-invariant derivative interaction sim g ~ phi ^dagger phi (D^mu phi )^dagger D_mu phi , is presented. This allows one to perform the complete renormalization of radiatively generated dimension 6 operators in the model at hand. We describe in details the technical tools required in order to disentangle the contribution to ultraviolet divergences parameterized by (generalized) non-polynomial field redefinitions. We also discuss how to extract the dependence of the beta -function coefficients on the non-renormalizable coupling g in one loop approximation, as well as the cohomological techniques (contractible pairs) required to efficiently separate the mixing of contributions associated to different higher-dimensional operators in a spontaneously broken effective field theory.

Highlights

  • We discuss how to extract the dependence of the βfunction coefficients on the non-renormalizable coupling g in one loop approximation, as well as the cohomological techniques required to efficiently separate the mixing of contributions associated to different higherdimensional operators in a spontaneously broken effective field theory

  • In this paper we continue the study of the off-shell renormalization of the Abelian Higgs-Kibble model supplemented by the maximally power counting violating dimension 6 operator φ†φ(Dμφ)† Dμφ

  • We will show here how to evaluate the one-loop divergent coefficients associated to all dimension 6 operators which are radiatively generated, and without limiting ourselves to: (i) on-shell quantities that are customarily assumed in the Standard Model Effective Field Theories (SMEFT) literature, where the cancellations between one-loop anomalous dimensions of dimension 6 operators were originally discovered [1,2,3]; and (ii) the linearized approximation in the higher dimensional couplings

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Summary

Introduction

In this paper we continue the study of the off-shell renormalization of the Abelian Higgs-Kibble model supplemented by the maximally power counting violating dimension 6 operator φ†φ(Dμφ)† Dμφ. To systematically compute the (one-loop) UV coefficients in spontaneously broken effective field theories possessing (dimension 6) derivative operators, it is convenient to first renormalize an associated auxiliary model, the so-called X theory, which is obtained by describing the scalar physical degree of freedom in terms of the gauge-invariant field coordinate vX2. We hasten to emphasize that since we are working off-shell the effects of generalized field redefinitions, that are present already at one-loop order, and are not even polynomial for the model at hand [17], need to be correctly accounted for This is automatically done through the cohomologically trivial invariants of the X -theory. The paper ends with two appendices: Appendix A contains the list of all the independent invariants needed for renormalizing the theory, while the relevant X -theory divergent one-loop amplitudes up to dimension 6 are given in Appendix B

Notations and setup
Mapping on the external sources
Cohomologically trivial invariants
Tadpoles
The gauge invariant sector
The pure external sources sector
The θ1 and θ2 coefficients
The θ3 and θ5 coefficients
The θ4 and θ6 coefficients
The θ9 and θ10 coefficients
The λ2 and λ3 coefficients
The λ6 and λ7 coefficients
The λ8 and λ9 coefficients
The λ10 coefficient
Conclusions
Mixed field-external sources invariants
On-shell reduction
Comparison with the Warsaw basis
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