Charm electroproduction viewed in the variable-flavour number scheme versus fixed-order perturbation theory

Abstract

Starting from fixed-order perturbation theory (FOPT) we derive expressions for the heavy-flavour components of the deep-inelastic structure functions (F i,H (x, Q 2 , m 2 H ), i = 2, L; H = c, b, t) in the variable-flavour number scheme (VFNS). These expressions are valid in all orders of perturbation theory. This derivation establishes a relation between the parton densities 1 supported by the Foundation for Fundamental Research on Matter (FOM) 2 on leave from ITP, SUNY at Stony Brook, New York 11794-3840, USA 1 parametrized at n f and n f + 1 light flavours. The consequences for the existing parametrizations of the parton densities are discussed. Further we show that in charm electroproduction the exact and asymptotic expressions for the heavy-quark coefficient functions yield identical results for F 2,c (x, Q 2 , m 2 c ) when Q 2 ≥ 20 (GeV/c) 2 . We also study the differences between the FOPT and the VFNS descriptions for F 2,c (x, Q 2 , m 2 c ). It turns out that the charm structure function in the VFNS is larger than the one obtained in FOPT over the whole Q 2 -range. Furthermore inspection of the perturbation series reveals that the higher order corrections in the VFNS are smaller than those present in FOPT for Q 2 ≥ 10 (GeV/c) 2 . Therefore the VFNS gives a better prediction for the charm structure function at large Q 2 -values than FOPT.2 c [10]. Although it was shown in NLO [11] that these logarithms lead to rather stable charm structure functions F i,c (x, Q 2 , m 2 c ) with respect to variations in the factorization and renormalization scales µ, their size still warrants some special treatment (for a discussion of the scale dependence of the charm con-4

CT10 next-to-next-to-leading order global analysis of QCD

“…3 The alternative TR 0 scheme [38,39] is used in the MSTW and HERAPDF fits, while the FONLL scheme [40,41] has been adopted by the NNPDF Collaboration. The BMSN scheme [42][43][44] and the fixed-flavor number (FFN) schemes are used by the ABM and GJR groups, respectively.…”

The contributions to the gluonic massive operator matrix elements

“…As has been shown in [26] this is not the case in the kinematic range at HERA. A smooth transition from the threshold region to asymptotic scales has been proposed in terms of the BMSN-scheme [8], (12) which is found to be in excellent agreement with the HERA data [27]. There is a series of other proposals to match between the threshold and asymtotic region [28][29][30], partly with a faster transition to the massless case.…”

“…The corresponding massive OMEs A gq,Q , A gg,Q were calculated to O(α 2 s ) in [8] and including also terms linear in ε in [9] correcting the previous result. The renormalized expressions A gq,Q and A gg,Q to O(a 3 s ) were derived in [5].…”

“…The renormalization group implies the following representation for the set of (n f + 1) (massless) parton densities expressed in terms of n f parton densities [8] :…”

“…The massless Wilson coefficients are known to 3-loop order [6]. At NLO the massive OMEs were calculated in [4,[7][8][9][10][11][12] in the unpolarized and polarized case, including the O(α 2 s ε) contributions, and in [13] for transversity. The heavy flavor corrections for charged current reactions are available at one loop and in the asymptotic case at two-loops [14,15].…”