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

摘要

We discuss Pad\'e-improvement of known four-loop order results based upon anasymptotic three-parameter error formula for Pad\'e-approximants. We derive anexplicit formula estimating the next-order coefficient $R_4$ from the previouscoefficients in a series $1+R_1 x + R_2x^2 + R_3x^3$. We show that such anestimate 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 togenerate a [2$|$2] Pad\'e-summation of the QCD $\beta$-function and anomalousmass dimension in order to demonstrate both the relative insensitivity of theevolution of $\alpha_s(\mu)$ and the running quark masses to higher ordercorrections, as well as a somewhat increased compatibility of the presentempirical range for $\alpha_s(m_\tau)$ with the range anticipated via evolutionfrom 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 ofthe all-orders $\beta$-function which incorporates known two-, three-, andfour-loop contributions necessarily correspond to ultraviolet fixed points,regardless of the unknown five-loop term. Pad\'e-improvement of higher-orderperturbative expressions is presented for the decay rates of the Higgs into twogluons and into a $b \bar{b}$ pair, and is used to show the relativeinsensitivity of these rates to higher order effects. However,Pad\'e-improvement of the purely-perturbative component of scalar/pseudoscalarcurrent correlation functions is indicative of large theoretical uncertaintiesin QCD sum rules for these channels, particularly if the continuum-thresholdparameter $s_0$ is near 1 GeV$^2$.