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We have quoted, to be specific, the τ decay result of Ref., which contains a careful treatment of the small, but (for a precision determination) numerically important, D>4 operator product expansion contributions, the inconsistent treatment of which created unquantified theoretical systematic errors in a number of the earlier τ-decay-based analyses. The results of the fifth order fixed order perturbation theory analysis, 0.1185-0.0009+0.0014, and of the more model-dependent extension, 0.1180±0.0008, both reported in Ref., are compatible within errors with that of Ref., but are based on assumed, rather than fitted, values for the relevant D=6 and 8 operator product expansion contributions. The ranges corresponding to the two results of Ref., however, are fully contained within the range listed in the table.
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The details on the treatment of αs G2/π reported here are not contained in Ref. We thank
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The details on the treatment of αs G2/π reported here are not contained in Ref. We thank G.P. Legage of the HPQCD Collaboration for providing us with this information.
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To be explicit, a self-consistent combination of 2-loop matching and 3-loop running was employed, with the nf=3→4 and nf=4→5 flavor thresholds taken to be at mc(mc)=1.25GeV and mb(mb)=4.25GeV, respectively. The use of 2-loop matching and 3-loop running rather than 3-loop matching and 4-loop running is inessential; the resulting value at the scale MZ is lowered by only 0.0002 if one uses the latter approach, a change which is small on the scale of the final quoted error.
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To be explicit, a self-consistent combination of 2-loop matching and 3-loop running was employed, with the nf=3→4 and nf=4→5 flavor thresholds taken to be at mc(mc)=1.25GeV and mb(mb)=4.25GeV, respectively. The use of 2-loop matching and 3-loop running rather than 3-loop matching and 4-loop running is inessential; the resulting value at the scale MZ is lowered by only 0.0002 if one uses the latter approach, a change which is small on the scale of the final quoted error.
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The definition of αV is, in fact, not clear from what is written in Ref. We thank the members of the HPQCD Collaboration for clarifying the procedure used in defining it.
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