The renormalization of the effective gauge theory with spontaneous symmetry breaking: the $SU(2)\times U(1)$ case

We formulate the electroweak chiral Lagrangian in its mass eigenstates, and study the its one-loop renormalization and provide its renormalization group equations to the same order, so as to complete it as the low energy effective theory of the standard model below a few TeV. In order to make our computation consistent, we have provided a modified power counting rule to estimate the contributions of higher loop and higher operators. As one of the application of its renormalization group equations, we analyze the solution to the effects of the Higgs scalar. We find that similar to the SU(2) case, that the triple anomalous couplings are sensitive to the quartic couplings (here $\al_5$). While the quadratic anomalous couplings are not sensitive, due to the large leading contributions and the accidental cancellation. The differences in the triple anomalous couplings between the direct method and renormalization group equation method are well within the detection power of the LHC and LC, if the Higgs scalar is not too heavy (say, 300 or 400 GeV). We also suggest a new mechanism to generate the negative $S$ parameter through the radiative corrections of the anomalous couplings. Comparison of the renormalization group equation method and direct methods is provided in the full theory, the standard model, to reveal the basic differences of them. The problem of the unitarity violation is also addressed for our assumption in the modified power counting rule.