| The Analytic Continuation by Duality (ACD) method is described. Theory of the electroweak radiative corrections is explained. A short review of the technicolor and related theories is given. Application of the ACD technique to estimation of the electroweak Peskin-Takeuchi S-parameter is studied. The reliability of the ACD technique is checked, first on several toy models and then on the QCD-like technicolor theory. The ACD-derived S-estimates are compared with the exact values coming from dispersive relations. The following toy models were used: the one-loop perturbative contribution of a pair of heavy fermions, vector-meson dominance model, two Breit-Wigner resonances, Shifman, and cotangent models. (The latter two models include an infinite number of {dollar}delta{dollar}-function resonances.) A more realistic model with an infinite number of Breit-Wigner (finite-width) resonances is suggested. Dependence of the S-estimates on the upper cut-off R of the ACD integral is investigated. Contrary to some claims in the literature, this dependence proved to be strong. Since the dependence of the estimates on the routine used to fit {dollar}1/s{dollar} with polynomials has not been explored earlier, a detailed analysis of the fit routines is included. Analytic expressions for coefficients of the fitting polynomials are derived for best {dollar}Lsb1, Lsb2,{dollar} and {dollar}Lsb{lcub}infty{rcub}{dollar} approximations. The fit routine dependence is mild, but it is not negligible in most cases. A brief overview of other applications of the ACD technique is included. Earlier criticisms of the ACD technique as applied to QCD are discussed. Estimation of the total error of the ACD-derived results is identified as a serious shortcoming of the ACD technique. In conclusions, we find that the ACD method is not always reliable. This is especially true for the simplified ACD formula which neglects the momentum dependence of the coefficients of the asymptotic expansion of the spectral function. Because of the questionable reliability of the ACD estimates for known spectra the ACD estimates for unknown spectra, such as for walking technicolor, cannot be trusted. Therefore, the ACD method can only be recommended as a secondary means to check consistency of the results derived by other methods. |
