Renormalization of constrained SU(2)L×U(1)Y×Ũ(1) models

A simple renormalization framework for constrained SU(2${)}_{\mathrm{L}}$×U(1${)}_{\mathrm{Y}}$×Ũ(1) theories is presented. If the relation ${{\tan }^2\mathit{\phi }=(m_W^2/{\cos }^2{\theta }_W\mathit{-}{m}_{Z1}}^2$)/(${\mathrm{m}}_{\mathrm{Z}2}^2$-${\mathrm{m}}_{\mathrm{W}}$ ${}_{}{}^2{}_{}{}^{}$ ${/\mathrm{c}\mathrm{o}\mathrm{s}}^2$ ${\mathrm{\theta }}_{\mathrm{W}}$) is regarded as exact (the m’s are the physical masses and φ is the mixing angle of the neutral vector bosons), it is found that the definition of ${\sin }^2$ ${\theta }_W$ must be carefully chosen to ensure consistency and avoid the emergence of radiative corrections of O(α/φ), i.e., nonanalytic terms in the neighborhood of φ=0. The formulation developed in this paper prevents the occurrence of such potentially catastrophic terms and leads to a definition of ${\sin }^2$ ${\theta }_W$ in terms of $G_{\mu }$, α, and $m_W^2$ which is very close to the corresponding SU(2${)}_{\mathrm{L}}$×U(1${)}_{\mathrm{Y}}$ expression. A strategy to incorporate approximately the O(α) terms in the neutral currents of these theories is outlined. The discussion identifies in a simple way the mathematical origin of the potential nonanalytic terms and emphasizes the role that the $m_t$ dependence of the radiative corrections may have in the future in determining the tenability of these theories.