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 us 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 τ decay width. Using the optimal expansions based on conformal mapping and the contourimproved prescription of renormalization-group resummation, we obtain α s ðm 2 τ Þ ¼ 0.314 AE 0.006, which implies α s ðm 2 Z Þ ¼ 0.1179 AE 0.0008.

Resurgence in the O(4) sigma model

“…where the first few coefficients arep m coefficient. This method suggests that the precision ofp (2) m between m = 7 and m = 51 gradually decreases from 88 digits to 6 digits. Again, the coefficients (4.17) suggest that they come from perturbative expansion around some saddle point but we could not recognize any resurgence here.…”

“…Important phenomena such as confinement and dynamical mass generation are inherently non-perturbative, and cannot be accessed from the few known perturbative coefficients. Perturbation theory in QCD is expected to be asymptotic, with coefficients growing factorially, see e.g [1,2]. This factorial growth can be traced to the proliferation of Feynman diagrams [3,4], or to integrals in specific renormalon diagrams, when loop momenta lie in various IR and UV domains [5].…”

“…For the actual calculation of (3.4), instead of the Padé approximants, we used the conformal mapping method. This method is often applied even to real physical problems [2] to approximate physical quantities better from available perturbative data. In our case we can make a high precision comparison between the perturbative and the exact results getting a deeper insight into the mathematical structure of the deviation.…”

“…Let us study the series of transformations {x [0] n } = {x n } =⇒ {x [1] n } =⇒ {x [2] n } =⇒ • • • , (B.11)…”

Hyperasymptotic approximation to the plaquette and determination of the gluon condensate

“…We finally mention that the nonzero value of G 2 PV shows that the PV regularization of the perturbative sum, even if computed exactly, would differ from the Montecarlo simulation of the plaquette by a term of O(a 4 Λ 4 ). This may affect the conjecture that the resummation technique of the perturbative expansion proposed in [38] for the Adler function would not need such nonperturbative corrections. This should be further investigated.…”