Large-order Behaviour due to Ultraviolet Renormalons in QCD

Ultraviolet renormalons, contrary to their infrared counterparts, lead to a universal contribution to the large-order behaviour of perturbative expansions in QCD. In this letter, we determine nature of the leading ultraviolet renormalon singularity for the inclusive hadroproduction cross section in ee annihilation, for hadronic τ decays and the moments of deep-inelastic scattering structure functions. We comment on the relevance of ultraviolet renormalons to estimates in low orders of perturbation theory. 1. Perturbative expansions of physical quantities in QCD are divergent, and as such they are related to a measurement only through the additional assumption that the series is asymptotic. Of the two known mechanisms that cause divergence of the series, instantons [1, 2] are unimportant in large orders and the asymptotic behaviour of the series expansion is determined by renormalons [3, 4]. According to whether the divergence arises from regions of small or large momentum in internal integrations, renormalons are classified as infrared (IR) or ultraviolet (UV). The growth of perturbative coefficients due to IR renormalons depends on whether one considers, for example, deep-inelastic structure functions, hadronic event shape observables or the hadronic total cross section in ee annihilation and is closely related to power corrections to these observables. For this reason they have recently been studied intensely [5]. UV renormalons are often regarded to be more complicated and have mostly been left aside, perhaps because they are Borel summable. However, the structure of UV renormalons is universal in the sense that it depends only on the theory under consideration, in our case QCD. Moreover, for some quantities of interest, such as the hadronic total cross section in ee annihilation, the sign-alternating UV renormalon behaviour determines the actual large-order behaviour of the series expansion. In this letter we determine the universal ultraviolet renormalon asymptotics in QCD and apply it to a number of observables of phenomenological interest. The universality of UV renormalons was recognised by Parisi [6], who noted that UV renormalons could be compensated by adding higher-dimension operators to the Lagrangian just as logarithmic UV divergences can be compensated by dimension-four counterterms. To state Parisi’s hypothesis precisely, we consider a quantity R(αs), expanded as R(αs) = A (