Abstract
A method of evaluation of spacelike QCD observables ${\cal D}(Q^2)$ is developed, motivated by the renormalon structure of these quantities. A related auxiliary quantity ${\widetilde {\cal D}}(Q^2)$ is introduced, which is renomalization scale independent only at the one-loop level, and a large-$\beta_0$-type renormalon motivated ansatz is made for the Borel transform of this quantity. This leads to a `dressed' Borel transform of the considered observable ${\cal D}(Q^2)$. From there, a Neubert-type distribution is obtained for the observable. The described method is then applied to the massless Adler function and the related decay ratio of the $\tau$ lepton semihadronic decays. Comparisons are then made with an evaluation method at higher truncated orders, developed in our earlier works, which is a renormalization scale invariant extension of the diagonal Pad\'e approximants.
Highlights
The perturbative QCD running coupling aðQ2Þ ≡ αsðQ2Þ=π, in the usual theoretical known renormalization schemes, has the peculiar property of having a significantly different regime of holomorphic behavior in the Q2-complex plane than the spacelike QCD observables DðQ2Þ such as current correlators, nucleon structure functions, and their sum rules
The perturbative QCD (pQCD) coupling aðQ2Þ has in general singularities along the negative semiaxis and, in addition, singularities outside the negative semiaxis, usually on the positive semiaxis 0 ≤ Q2 ≤ Λ2Lan
(18), namely at the one-loop level approximation. These considerations suggest that the Borel transform B1⁄2D ðuÞ of the auxiliary quantity DðQ2Þ has a one-loop type renormalon structure
Summary
The perturbative QCD (pQCD) running coupling aðQ2Þ ≡ αsðQ2Þ=π (where Q2 ≡ −q2), in the usual theoretical known renormalization schemes, has the peculiar property of having a significantly different regime of holomorphic (analytic) behavior in the Q2-complex plane than the spacelike QCD observables DðQ2Þ such as current correlators, nucleon structure functions, and their sum rules. For low spacelike scales Q2, jQ2j ≲ 1 GeV2, the argument μ2 1⁄4 κQ2 in the coupling aðκQ2Þ is either close to or within the regime of the Landau singularities, making the evaluation of aðκQ2Þ and of DðaðκQ2Þ; Q2Þev either unreliable or impossible This problem was addressed systematically first in Refs. In the usual perturbative QCD (pQCD) the evaluation has ambiguity due to the concurrence of the Landau singularities of the pQCD coupling aðQ2Þ 1⁄4 αsðQ2Þ=π and the IR renormalon This approach is applied to the evaluation of the massless Adler function DðQ2Þ in Sec. III B and the related (timelike) τ lepton semihadronic decay ratio rτ in Sec. III C, in two versions of QCD with IR-safe coupling (2δ [23,24] and 3δ AQCD [35]), and in pQCD in the corresponding schemes.
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