29 - Renormalons

Summary

Introduction

The renormalon problem is related to the well-known fact [372,375] (for more complete reviews, see for example [162,154]) that the QCD series is unfortunately divergent (no finite radius of convergence) like n!, which is the number of diagrams of nth order. Indeed, a given observable can be expressed as a power series of the coupling g as:

where the series are divergent:

and where the nth order grows like n!, such that it is not practicable to have a quantitative meaning of Eq. (29.1). For the approximation to be meaningful, the approximation should asymptotically approach the exact result in the complex g-plane, such that:

where the truncation error at order N should be bounded to the order g^N+1. If f_n behaves like in Eq. (29.2), K_N usually behaves as a^N N!N^b. The truncation error behaves similarly as the terms of the series. It first decreases until:

beyond which the approximation to F does not improve through the inclusion of higherorder terms. For N₀ ≫ 1, the approximation is good up to terms of the order:

Provided f_n ∼ K_n, the best approximation is reached when the series is truncated at its minimal term and the truncation error is given by the minimal term of the series.