N3LO-Renormalon-Inspired Resummations and Fully Analytic
Infra-red Freezing in Perturbative QCD

Abstract

We make use of the recent calculation of d3 by Baikov, Chetrykin and Kuhn of N3LO QCD vacuum polarization to analyze the inclusive tau-decay ratio Rτ . We perform an all-orders resummation of the QCD Adler D function for the vector correlator, in which the part of perturbative coefficients containing the leading power of b, the first QCD beta-function equation coefficient, is resummed to all-orders. We match the resummation to the exactly
known next-to-leading order (NLO), next-NLO (N2LO) and next-N2LO (N3LO) results, we employ the Complete Renormalization Group Improvement (CORGI) approach in which all RG-predictable ultra-violet logarithms are resummed to all-orders, removing all dependence on the renormalization scale. Hence the NLO, N2LO and N3LO CORGI result can be obtained and compared with the “leading b” all-orders CORGI result. Using an appropriate weight function, we can numerically integrate these results for the Adler D function in the complex energy plane to obtain so-called "contour-improved" results for the ratio Re+e− and its tau-decay analogue Rτ . A table showing the differences of αs(M2τ ) and αs(M2 Z) extracted from NLO, N2LO and N3LO CORGI as well as all-orders CORGI results were made, together with αS(M2τ) and αS(M2Z) extracted directly from Fixed-Order-Perturbation Theory at NLO, N2LO and N3LO. We also compared the ALEPH data for
Rτ(s) with the all-orders CORGI result fitted at s = m2τ.

We then go on to study the analyticity in energy of the leading one-chain term in a skeleton expansion for QCD observables. We show that by adding suitable non-perturbative terms in the energy regions Q2 > Λ2 and Q2 < Λ2 (where Q2 = Λ2 is the Landau pole of the one loop
coupling), one can obtain an expression for the observables which is a holomorphic
function of Q2, for which all derivatives are finite and continuous at Q2 = Λ2. This function is uniquely constrained by the requirement of asymptotic freedom, and the finiteness as Q2→0, up to addition of a non-perturbative holomorphic function. This full analyticity replaces the piecewise analyticity and continuity exhibited by the leading one-chain term itself. Using The Analytic Perturbation Theory (APT) Euclidean functions introduced by Shirkov and collaborators, we finally matched the equations K(L)PT +K(L)NP and U(L)PT + U(L)NP with a resummation of coefficients extracted from their Borel Transform multiplied by the APT Euclidean functions in the one loop case. For D(L)PT + D(L)NP , it is shown that it freezes to 2/b. Considering the GDH Sum Rule, we construct an analytic function which fits well with data from Jefferson Laboratory (JLab) for 0 < Q < 2GeV.