31 9 v 2 1 5 Ja n 20 01 Estimate of the Three - Loop MS Contribution

The three-loop contribution to the M S single-Higgs-doublet standard-model cross-section σ(W + L W − L → ZLZL) at s = (5MH) 2 is estimated via least-squares matching of the asymptotic Padé-approximant prediction of the next order term, a procedure that has been previously applied to QCD corrections to correlation functions and decay amplitudes. In contrast to these prior applications, the expansion parameter for the W + L W − L → ZLZL process is the non-asymptotically-free quartic scalar-field coupling of the standard model, suggesting that the least-squares matching be performed over the " infrared " µ 2 ≤ s region of the scale parameter. All three coefficients of logarithms within the three-loop term obtained by such matching are found to be within 6.6% relative error of their true values, as determined via renormalization-group methods. Surprisingly , almost identical results are obtained by performing the least squares matching over the µ 2 ≥ s region. Within standard-model single-Higgs-doublet electroweak physics, the cross-section for the scattering of two longitudinal W's into two longitudinal Z's at very high energy takes the form σ[s, L(µ), g(µ)] = 8π 3 9s H[s, L(µ), g(µ)], (1) where the scale-sensitive portion of the cross-section [1], H[s, L(µ), g(µ)] = g 2 (µ) {1 + [−4L(µ) − 10.0]g(µ) + 12L 2 (µ) + 68.667L(µ) + 93.553 + 2 3 ln s M 2 H g 2 (µ) + c 3 L 3 (µ) + c 2 L 2 (µ) + c 1 L(µ) + c 0 g 3 (µ) + ... , (2) depends on the renormalization scale µ explicitly through the logarithm L(µ) ≡ ln(µ 2 /s) (3) and implicitly through the M S quartic scalar-field coupling g(µ) = 6 λ MS (µ)/16π 2 .