Conformal mapping of the Borel plane: going beyond perturbative QCD

Abstract

The power corrections in the Operator Product Expansion (OPE) of QCD correlators can be viewed mathematically as an illustration of the transseries concept, which allows to recover a function from its asymptotic divergent expansion. Alternatively, starting from the divergent behavior of the perturbative QCD encoded in the singularities in the Borel plane, a modified expansion can be defined by means of the conformal mapping of this plane. A comparison of the two approaches concerning their ability to recover nonperturbative properties of the true correlator was not explored up to now. In the present paper, we make a first attempt to investigate this problem. We use for illustration the Adler function and observables expressed as integrals of this function along contours in the complex energy plane. We show that the expansions based on the conformal mapping of the Borel plane go beyond finite-order perturbation theory, containing an infinite number of terms when reexpanded in powers of the coupling. Moreover, the expansion functions exhibit nonperturbative features of the true function, while the expansions have a tamed behavior at large orders and are expected even to be convergent. Using these properties, we argue that there are no mathematical reasons for supplementing the expansions based on the conformal mapping of the Borel plane by additional arbitrary power corrections. Therefore, we make the conjecture that they provide an alternative to the standard OPE in approximating the QCD correlator. This conjecture allows to slightly improve the accuracy of the strong coupling extracted from the hadronic $\tau$ decay width. Using the optimal expansions based on conformal mapping and the contour-improved prescription of renormalization-group resummation, we obtain ${\alpha }_s(m_{\tau }^2)=0.314\pm 0.006$, which implies ${\alpha }_s(m_Z^2)=0.1179\pm 0.0008$.

typos corrected; matches the published version

References in corpus (9)

• Hadronic Z- and tau-Decays in Order alpha_s^4
• alpha_s and the tau hadronic width: fixed-order, contour-improved and higher-order perturbation theory
• The Determination of alpha_s from Tau Decays Revisited
• ${\alpha }_s$ from $\tau$ decays: contour-improved versus fixed-order summation in a new QCD perturbation expansion
• Perturbative expansion of the QCD Adler function improved by renormalization-group summation and analytic continuation in the Borel plane
• Superasymptotic and hyperasymptotic approximation to the operator product expansion
• Higher-order perturbative coefficients in QCD from series acceleration by conformal mappings
• The case for duality violations in the analysis of hadronic $\tau$ decays
• Perturbative Expansions in QCD Improved by Conformal Mappings of the Borel Plane

Cited by in corpus (4)

• Precision physics with inclusive QCD processes
• Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $\tau$ decay data, and determination of pQCD coupling
• Resurgent representation of the Adler function in the large-${\beta }_0$ approximation of QCD
• Theoretical description of the plaquette with exponential accuracy