Study of the renormalization of BRST invariant local composite operators in the $U(1)$ Higgs model

The renormalization properties of two local BRST invariant composite operators, $(O,V_mu)$, corresponding respectively to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are scrutinized in the $U(1)$ Higgs model by means of the algebraic renormalization setup. Their renormalization $Z$s factors are explicitly evaluated at one-loop order in the $overline{text{MS}}$ scheme by taking into due account the mixing with other gauge invariant operators. In particular, it turns out that the operator $V_mu$ mixes with the gauge invariant quantity $partial_ u F_{mu u}$, which has the same quantum numbers, giving rise to a $2 times 2$ mixing matrix. Moreover, two additional powerful Ward identities exist which enable us to determine the whole set of $Z$s factors entering the $2 times 2$ mixing matrix as well as the $Z$ factor of the operator $O$ in a purely algebraic way. An explicit check of these Ward identities is provided. The final setup obtained allows for computing perturbatively the full renormalized result for any $n$-point correlation function of the scalar and vector composite operators.