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Asymptotic Pade-Approximant Methods and QCD Current Correlation Functions

Asymptotic Pade-approximant methods are utilized to estimate the leading-order unknown (i.e., not-yet-calculated) 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 polynomial) may be extracted from prior-order contributions by use of the renormalization-group (RG) equation appropriate for each correlator. We find surprisingly good agreement between asymptotic Pade-approximant predictions and RG-determinations of such coefficients for each correlation function considered, although such agreement is seen to diminish with increasing numbers of quark flavours. The RG-determined coefficients we obtain are then utilized in conjunction with asymptotic Pade-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 order $(α_s^4)$ contribution to $R(s) \equiv[σ(e^+ e^- \to hadrons) / σ(e^+ e^- \to μ^+ μ^-)]$ for three, four, and five flavours.