Gauge invariant operators in the SU(2) Higgs model: Ward identities and renormalization

The renormalization properties of two local gauge invariant composite operators (O, R-mu(a)) 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 space-time. 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-mu(a) 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-mu(a) at the quantum level, do not receive any quantum corrections. Using this Ward identity, one can also prove that the longitudinal component of the two-point correlation function ((R-mu(a)(p) R-v(b)(- p)) exhibits only a tree level nonvanishing contribution which, moreover, is momentum independent, thus it is not associated to any physical propagating mode. Finally, we point out that the renowned nonrenormalization theorem for the ghost-antighost-vector boson vertex in Landau gauge remains true to all orders, also in the presence of the Higgs field.

Automated summary

The renormalization properties of two local gauge invariant composite operators in the SU(2) Higgs model are analyzed to all orders in perturbation theory. A powerful global Ward identity, related to an exact custodial symmetry, is derived for the first time and has deep consequences at the quantum level. The conserved Noether currents of the custodial symmetry are found to be the gauge invariant vector operators. Using this Ward identity, it is proven that the longitudinal component of the two-point correlation function of these operators exhibits only a tree-level nonvanishing contribution, independent of momentum. The renowned nonrenormalization theorem for the ghost-antighost-vector boson vertex in Landau gauge also holds true in the presence of the Higgs field.