Perturbative Expansions in QCD Improved by Conformal Mappings of the Borel Plane

Abstract

Perturbation expansions appear to be divergent series in many physically interesting situations, including in quantum field theories like quantum electrodynamics (QED) and quantum chromodynamics (QCD), where the perturbative coefficients exhibit a factorial growth at large orders. While this feature has little impact on physical predictions in QED, it can have nontrivial consequences in applications of perturbative QCD at moderate energies. In particular, it affects the theoretical error in the extraction of the strong coupling ${\alpha }_s$ from hadronic $\tau$ decays, despite progress of perturbative calculations available at present to four loops. We discuss a new type of perturbative expansion for QCD correlators, which uses instead of the standard powers of the coupling a new set of expansion functions. These functions are defined by means of an optimal conformal mapping of the Borel complex plane, which implements the known features of the high-order divergence in terms of the lowest Borel-plane singularities. The properties of the expansion functions resemble those of the expanded correlators, by exhibiting in particular the singular behaviour of the correlators at ${\alpha }_s=0$. We prove the good convergence properties of the new expansions on mathematical models that simulate the physical polarization function for light quarks and its derivative (the Adler function), in various prescriptions of renormalization-group summation.

arXiv commentreview based on previous works of authors; to appear in Perturbation Theory: Advances in Research and Applications, published by Nova Science Publishers, Inc

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Cited by in corpus (7)

• Physical Resurgent Extrapolation
• Uniformization and Constructive Analytic Continuation of Taylor Series
• On the Higher Loop Euler-Heisenberg Trans-Series Structure
• Conformal mapping of the Borel plane: going beyond perturbative QCD
• Renormalons in integrated spectral function moments and ${\alpha }_s$ extractions
• Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: ${\varphi }^3$ QFT in 6 Dimensions
• Renormalons beyond the Borel plane