In the absence of a tree-level scalar-field mass, renormalization-group methods permit the explicit summation of leading-logarithm contributions to all orders of the perturbative series within the effective potential for SU(2)xU(1) electroweak symmetry. This improvement of the effective potential function is seen to reduce residual dependence on the renormalization mass scale. The all-orders summation of leading-logarithm terms involving the dominant three couplings contributing to radiative corrections is suggestive of a potential characterized by a plausible Higgs boson mass of 216 GeV. However, the tree potential's local minimum at phi=0 is restored if QCD is sufficiently strong.

For any perturbative series that is known to k-subleading orders of perturbation theory, we utilize the process-appropriate renormalization-group ͑RG͒ equation in order to obtain all-orders summation of series terms proportional to ␣ n log nϪk ( 2 ) with kϭ͕0,1,2,3͖, corresponding to the summation to all orders of the leading and subsequent-three-subleading logarithmic contributions to the full perturbative series. These methods are applied to the perturbative series for semileptonic b decays in both MS and pole-mass schemes, and they result in RG-summed series for the decay rates which exhibit greatly reduced sensitivity to the renormalization scale . Such summation via RG methods of all logarithms accessible from known series terms is also applied to perturbative QCD series for vector-and scalar-current correlation functions, the perturbative static potential function, the ͑single-doublet standard-model͒ Higgs decay amplitude into two gluons, as well as the Higgs-mediated high-energy cross section for W ϩ W Ϫ →ZZ scattering. The resulting RG-summed expressions are also found to be much less sensitive to the renormalization scale than the original series for these processes.and the successive-order series coefficients within S͓x,L͔, as defined by Eq. ͑1.1͒, are ͓1͔T 0,0 ϭ1, T 1,0 ϭ4.25360, T 1,1 ϭ5, T 2,0 ϭ26.7848, T 2,1 ϭ36.9902, T 2,2 ϭ17.2917. ͑1.5͒The five active-flavor pole-mass expression for the same rate is obtained by replacing m b () with the renormalizationscale independent pole mass m b pole in Eqs. ͑1.4͒ and ͑1.2͒, as well as a concomitant alteration of the following series coefficients ͓1͔: PHYSICAL REVIEW D 66, 014010 ͑2002͒

Asymptotic Padé-approximant methods are utilized to estimate the leading-order unknown ͑i.e., not-yet-calculated͒ contributions to the perturbative expansions of two-current QCD correlation functions obtained from scalar-channel fermion and gluon currents, as well as from vector-channel fermion currents. Such contributions to the imaginary part of each correlator are polynomials of logarithms whose coefficients ͑other than the constant term within the polynomial͒ may be extracted from prior-order contributions by use of the renormalization-group ͑RG͒ equation appropriate for each correlator. We find surprisingly good agreement between asymptotic Padé-approximant predictions and RG determinations of such coefficients for each correlation function considered, although such agreement is seen to diminish with increasing n f . The RGdetermined coefficients we obtain are then utilized in conjunction with asymptotic Padé-approximant methods to predict the RG-inaccessible constant terms of the leading-order unknown contributions for all three correlators. The vector channel predictions lead to estimates for the O(␣ s 4 ) contribution to R(s)ϵ͓(e ϩ e Ϫ →hadrons)/(e ϩ e Ϫ → ϩ Ϫ )͔ for three, four, and five flavors. ͓S0556-2821͑99͒02510-2͔

We discuss Padé-improvement of known four-loop order results based upon an asymptotic threeparameter error formula for Padé-approximants. We derive an explicit formula estimating the nextorder coefficient R 4 from the previous coefficients in a series 1 + R 1 x + R 2 x 2 + R 3 x 3 . We show that such an estimate is within 0.18% of the known five-loop order term in the O(1) β-function, and within 10% of the known five-loop term in the O(1) anomalous mass-dimension function γ m (g). We apply the same formula to generate a [2|2] Padé-summation of the QCD β-function and anomalous mass dimension in order to demonstrate both the relative insensitivity of the evolution of α s (µ) and the running quark masses to higher order corrections, as well as a somewhat increased compatibility of the present empirical range for α s (m τ ) with the range anticipated via evolution from the present empirical range for α s (M z ). For 3 ≤ n f ≤ 6 we demonstrate that positive zeros of any [2|2] Padé-summation estimate of the all-orders β-function which incorporates known two-, three-, and four-loop contributions necessarily correspond to ultraviolet fixed points, regardless of the unknown five-loop term. Padé-improvement of higher-order perturbative expressions is presented for the decay rates of the Higgs into two gluons and into a bb pair, and is used to show the relative insensitivity of these rates to higher order effects. However, Padé-improvement of the purely-perturbative component of scalar/pseudoscalar current correlation functions is indicative of large theoretical uncertainties in QCD sum rules for these channels, particularly if the continuum-threshold parameter s 0 is near 1 GeV