Abstract
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 $\partial_\nu F_{\mu\nu}$, 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.
Highlights
The renormalization properties of two local composite operators, ðO; VμÞ, which are invariant under the infinitesimal Becchi-Rouet-Stora-Tyutin (BRST) transformations, corresponding respectively to the gauge invariant respectively to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are scrutinized in the Uð1Þ Higgs model by means of the algebraic renormalization setup
In two previous works [1,2], the elementary excitations of the Uð1Þ Higgs model, namely, the Higgs particle and the vector massive gauge boson, have been investigated within a fully gauge invariant setup, relying on the introduction of two local operators ðO; VμÞ, which are invariant under the infinitesimal Becchi-Rouet-Stora-Tyutin (BRST) transformations, [3,4,5]1: OðxÞ
II we briefly review some basic features of the Uð1Þ Higgs model quantized in the Landau gauge
Summary
In two previous works [1,2], the elementary excitations of the Uð1Þ Higgs model, namely, the Higgs particle and the vector massive gauge boson, have been investigated within a fully gauge invariant setup, relying on the introduction of two local operators ðO; VμÞ, which are invariant under the infinitesimal Becchi-Rouet-Stora-Tyutin (BRST) transformations, [3,4,5]1: OðxÞ 1⁄4 ðh þ 2vh þ ρ2Þ; VμðxÞ1 2 ð−ρ∂μh þ h∂μρ þ v∂μρ þ eAμðv þ h2 þ 2vh þ ρ2ÞÞ; ð1Þ. In two previous works [1,2], the elementary excitations of the Uð1Þ Higgs model, namely, the Higgs particle and the vector massive gauge boson, have been investigated within a fully gauge invariant setup, relying on the introduction of two local operators ðO; VμÞ, which are invariant under the infinitesimal Becchi-Rouet-Stora-Tyutin (BRST) transformations, [3,4,5]1: OðxÞ 1⁄4 ðh þ 2vh þ ρ2Þ; VμðxÞ. Let us start with the scalar operator OðxÞ, Eq (1) It has dimension two, ghost number zero and is even under charge conjugation, Eq (21). We look at the most general scalar nonintegrated quantity with dimension two and even under charge conjugation, ΔðxÞ, such that sΔðxÞ 1⁄4 0 and Δ ≠ sΔfor some Δwith ghost number −1. It turns out that the most general expression for Δ is given by
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