Scheme Variations of the QCD Coupling and Hadronic τ Decays.

The quantum chromodynamics (QCD) coupling α_{s} is not a physical observable of the theory, since it depends on conventions related to the renormalization procedure. We introduce a definition of the QCD coupling, denoted by α[over ^]_{s}, whose running is explicitly renormalization scheme invariant. The scheme dependence of the new coupling α[over ^]_{s} is parametrized by a single parameter C, related to transformations of the QCD scale Λ. It is demonstrated that appropriate choices of C can lead to substantial improvements in the perturbative prediction of physical observables. As phenomenological applications, we study e^{+}e^{-} scattering and decays of the τ lepton into hadrons, both being governed by the QCD Adler function.

Testing the Bethe ansatz with large N renormalons.

The ground-state energy of integrable asymptotically free theories can be conjecturally computed using the Bethe ansatz once the theory has been coupled to an external potential through a conserved charge. This leads to a precise prediction for the perturbative expansion of the energy. We provide a non-trivial test of this prediction in the non-linear sigma model and its supersymmetric extension, by calculating analytically the associated Feynman diagrams at next-to-leading order in the 1/N expansion, and at all loops. By investigating the large order behavior of the diagrams, we locate the position of the renormalons of the theory and we obtain an analytic expression for the large N trans-series associated to each. As a spin-off of our calculation, we provide a direct derivation of the beta function of these theories, at next-to-leading order in the 1/N expansion.