Massless Parton Asymptotics within Variable Flavour Number Schemes

In this note we formulate and investigate theoretical uncertainties for high Q deep inelastic heavy quark (charm, etc.) production rates which arise within collinear resummation techniques from variations of the a priori unknown charm input scale Q0 of O(αs) variable flavour number schemes. We show that Q0 variations constitute a source of considerable theoretical uncertainty of present O(αs) calculations within such schemes and we suggest to consider a scale optimization from higher order corrections. We also outline how the stability of the fixed order and collinearly resummed perturbation series for heavy quark production can be comparatively investigated by variation of Q0. The present discussion on the appropriate scheme for the perturbative treatment of the deep inelastic production of heavy quarks of mass m ≫ ΛQCD can be partly traced back to the question what is the effective expansion parameter for high Q predictions. While fixed order perturbation theory (FOPT) proceeds strictly stepwise in powers of (αs/2π) at all scales, variable flavor number schemes (VFNSs) are based upon the expectation that terms ∼ (αs/2π × lnQ /m) from collinear regions in the phase space have to be resummed [1] to all perturbative orders n for high Q when (αs/2π× lnQ /m) → O(1). Such terms are undebatedly present in the high Q limit of the perturbative partonic coefficient functions but their impact is less clear [2] on observable hadronic quantities like the charm component of the deep inelastic structure function F c 2 where the partonic coefficient functions have to be convoluted with modern, i.e. steep, parton distribution functions (−) q (x, μF ) and dominantly g(x, μ 2 F ), μf ∼ m. The question which ordering of the perturbation series optimizes its convergence can therefore not be answered a priori but only from explicit quantitative, i.e. numerical investigations [2, 3]; prominent tools for testing perturbative stability being K factor considerations [2] or scale variations [3]. At present both criteria indicate a well behaved fixed order perturbation series for relevant subasymptotic but large scales Q ≫ m [2–5]. As regards scale uncertainties, mainly variations of the mass factorization scale μF have been considered so far despite the fact that collinear resummation techniques introduce an additional arbitrary scale in the process set by the input scale Q0 for the heavy quark: Recently proposed variable flavour number schemes [6–10] for global PDF analyses are constructed upon the boundary condition q (nf +1) H (x,Q0) ∣