Gauge-invariant scalar and vector operators in the $SU\left(2\right)\times U\left(1\right)$ Higgs model: Ward identities and renormalization

TL;DR

The paper develops a BRST-invariant, gauge-invariant framework for the $SU(2)\times U(1)$ Higgs model by introducing gauge-invariant composite operators and localizing nonlocal expressions via a Stueckelberg-like field. Employing Algebraic Renormalization, it derives a complete set of Ward identities (including non-integrated identities linked to the residual $U_{EM}(1)$ symmetry) and determines the invariant counterterms, establishing a renormalizable action for correlation functions of the composites. A key result is the non-renormalization of the current-like operator $O_{\mu}$ (i.e., $\gamma_{O_μ}=0$) and the relation that the transverse part of $\langle O_{μ}O_{ν}\rangle$ is fixed by $\langle X_{μ}X_{ν}\rangle$, while the longitudinal part is a constant, implying no propagating mode. The work also proves a non-renormalization theorem for propagators in Landau gauge, shows how tadpoles and the effective potential are constrained by an integrated $H$-equation, and provides a detailed mapping between renormalization factors of composite operators and fundamental fields. These results offer a gauge-invariant handle on the electroweak sector with potential non-perturbative applications and lattice comparisons, paving the way for incorporating fermions and further SM extensions.

Abstract

In this work, we investigate the renormalization of the gauge-invariant composite operators proposed in \cite{Dudal:2023jsu} to describe the $SU(2)\times U(1)$ Higgs model from a gauge-invariant perspective. To establish the relationship between the counterterms, we use the Algebraic Renormalization approach \cite{Piguet:1995er}. Therefore, we also derive the set of Ward identities of the model after introducing these composite operators. Using these Ward identities, we demonstrate important exact relations between the correlation functions of elementary fields and composite fields.