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 $\alpha_s(m_Z^2)=0.1179 \pm 0.0008$.
Highlights
Perturbation theory is known to lead to divergent series for many quantities in quantum mechanics and quantum field theory (QFT)
To give the divergent series a precise meaning, Dyson proposed to interpret it as asymptotic to the exact function, which changed the entire philosophy of perturbation theory
According to the results mentioned above, this expansion has the best asymptotic convergence rate compared to other expansions, based on conformal mappings which map a part of the holomorphy domain onto the unit disk
Summary
Perturbation theory is known to lead to divergent series for many quantities in quantum mechanics and quantum field theory (QFT) This surprising fact was first noticed in 1952 by Freeman Dyson [1], who argued that the perturbation expansions in QED cannot be convergent since the expanded functions are singular at the expansion point. In the so-called hyperasymptotic theory, the power corrections can be interpreted as a first piece of a transseries, i.e., a sequence of truncated series, each of them exponentially small in the expansion parameter of the previous one, which allow to recover the expanded function from its asymptotic divergent expansion (see [12,13,14] and references therein).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have