Pade-Improvement of QCD Running Coupling Constants, Running Masses, Higgs Decay Rates, and Scalar Channel Sum Rules

We discuss Pad\'e-improvement of known four-loop order results based upon an asymptotic three-parameter error formula for Pad\'e-approximants. We derive an explicit formula estimating the next-order coefficient $R_4$ from the previous coefficients in a series $1+R_1 x + R_2x^2 + R_3x^3$. We show that such an estimate is within 0.18% of the known five-loop order term in the O(1) $\beta$-function, and within 10% of the known five-loop term in the O(1) anomalous mass-dimension function $\gamma_m(g)$. We apply the same formula to generate a [2$|$2] Pad\'e-summation of the QCD $\beta$-function and anomalous mass dimension in order to demonstrate both the relative insensitivity of the evolution of $\alpha_s(\mu)$ and the running quark masses to higher order corrections, as well as a somewhat increased compatibility of the present empirical range for $\alpha_s(m_\tau)$ with the range anticipated via evolution from the present empirical range for $\alpha_s(M_z)$. For $3 \leq n_f \leq 6$ we demonstrate that positive zeros of any [2$|$2] Pad\'e-summation estimate of the all-orders $\beta$-function which incorporates known two-, three-, and four-loop contributions necessarily correspond to ultraviolet fixed points, regardless of the unknown five-loop term. Pad\'e-improvement of higher-order perturbative expressions is presented for the decay rates of the Higgs into two gluons and into a $b \bar{b}$ pair, and is used to show the relative insensitivity of these rates to higher order effects. However, Pad\'e-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$^2$.