O(αs3) estimate for the longitudinal cross-section in e+e- annihilation to hadrons

Using Renormalization Group (RG) and Pade approximant methods, we estimate the O(αs3) contribution to the longitudinal cross-section (σL) in the e+e- annihilation to hadrons. The O(αs3) contribution is expressed as a quadratic polynomial in L≡ln(μ2/Q2). We find that the Pade approximant predictions for the coefficients L and L2 are within 3.5% of the true values extracted via RG methods. Incorporation of the full O(αs3) contribution (including the estimated RG-inaccessible coefficient) to σL yields good agreement of the measured cross-section with the current Particle Data Guide value of αs(MZ)=0.1185±0.020, indicative of the importance of higher order perturbative contribution(s). Incorporation of this contribution yields reduced renormalization scale dependence and near agreement of Principal of Minimal Sensitivity (PMS) and Fastest Apparent Convergent (FAC) values of the cross-section