Gauge Invariant Description of the $SU(2)$ Higgs model: Ward identities and Renormalization

Abstract

The renormalization properties of two local gauge invariant composite operators (O, R a µ ) corresponding, respectively, to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are analyzed to all orders in perturbation theory by means of the algebraic renormalization in the SU (2) Higgs model, with a single scalar in the fundamental representation, when quantized in the Landau gauge in Euclidean spacetime. The present analysis generalizes earlier results presented in the case of the U (1) Higgs model. A powerful global Ward identity, related to an exact custodial symmetry, is derived for the first time, with deep consequences at the quantum level. In particular, the gauge invariant vector operators R a µ turn out to be the conserved Noether currents of the above-mentioned custodial symmetry. As such, these composite operators do not renormalize, as expressed by the fact that the renormalization Z-factors of the corresponding external sources, needed to define the operators R a µ at the quantum level, do not receive any quantum corrections. Another consistency feature of our analysis is that the longitudinal component of the two-point correlation function R a µ (p)R b ν (−p) exhibits only a tree level non-vanishing contribution which, moreover, is momentum independent, being thus not associated to any physical propagating mode. Finally, we point out that the renowned non-renormalization theorem for the ghost-antighost-vector boson vertex in Landau gauge remains true to all orders, also in presence of the Higgs field.

Vector boson scattering from the lattice

“…But the off-shell properties and interactions of the bound states can in principle deviate at higher orders of the vev [7,9,13,16,17]. Correspondingly, such deviations have been observed for the form factor of the physical vector bosons in lattice simulations [18] and analytically for off-shell Higgs [9,16] and vector boson properties [17]. This can have potentially observable consequences.…”

“…Recent studies showed that higher orders in v can be accounted for analytically [9,14,16,17]. Testing the quality of this, as well as ensuring that no relevant surplus effects beyond this double-expansion arise, requires nonperturbative methods.…”