Mellin moments of the O(αs3) heavy flavor contributions to unpolarized deep-inelastic scattering at Q2

Abstract

We calculate the O(α 3 s ) heavy flavor contributions to the Wilson coefficients of the structure function F 2 (x, Q 2 ) and the massive operator matrix elements (OMEs) for the twist-2 operators of unpolarized deeply inelastic scattering in the region Q 2 ≫ m 2 . The massive Wilson coefficients are obtained as convolutions of massive OMEs and the known light flavor Wilson coefficients. We also compute the massive OMEs which are needed to evaluate heavy flavor parton distributions in the variable flavor number scheme (VFNS) to 3-loop order. All contributions to the Wilson coefficients and operator matrix elements but the genuine constant terms at O(α 3 s ) of the OMEs are derived in terms of quantities, which are known for general values in the Mellin variable N . For the operator matrix elements A /µ 2 , m 2 i /µ 2 ) this approximation is only valid for Q 2 /m 2 > ∼ 800, [29]. The 3-loop corrections were calculated in Ref. [30].3 massive operator matrix elements A jk contributing to the heavy flavor Wilson coefficients for the structure function F 2 (x, Q 2 ) in the region Q 2 /m 2 > ∼ 10 to 3-loop order for fixed moments of the Mellin variable N. In case of the flavor non-singlet (NS) contributions, we also present the odd moments of the −-projection. We further calculate the operator matrix elements, which are required to define heavy quark densities in the VFNS [31]. Due to renormalization, higher order contributions in ε to corrections of lower order in a s , cf. [29,[31][32][33][34][35], and other renormalization terms, such as the anomalous dimensions and the expansion coefficients of the QCD β-function and mass anomalous dimensions, contribute. For these reasons, the present calculation yields also the moments of the complete 2-loop anomalous dimensions and the terms ∝ T F of the 3-loop anomalous dimensions γ ij (N). In the pure singlet (PS) case, γ +,PS qq (N), and for γ qg (N), these are the complete anomalous dimensions given in [18,19], to which we agree. Since the present calculation is completely independent by method, formalism, and codes, it provides a check on the previous results. Except for the constant part of the unrenormalized heavy flavor operator matrix elements, we obtain the heavy quark Wilson coefficients in the asymptotic region for all values of the Mellin variable N. The analytic continuation of these expressions to complex values of N can be performed with the help of the representations in [36] and those given for the anomalous dimensions and massless Wilson coefficients in [18,19,26].The paper is organized as follows. In Section 2, a brief outline of the basic formalism is given. The renormalization of the different massive operator matrix elements is described in Section 3. In Section 4, we present details on the unrenormalized and renormalized operator matrix elements. Technical details of the calculation and the main results are discussed in Section 5. Depending on the CPU time and storage size required, the moments up to N = 10, 12, and 14 of the different operator matrix eleme...

On higher-order flavour-singlet splitting and coefficient functions at large x

“…See Ref. [10] for the status of the third-order computation of the heavy-quark contributions to DIS.…”

The contributions to the gluonic massive operator matrix elements

“…Here F m=0 2 (x, Q 2 ) denotes the well-known massless contribution and the massive contribution in the presence of a single massive quark reads [5] …”

“…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]. In the MS scheme with the heavy quark mass m on-shell they are given by :…”

“…[5]. The unreormalized OMEÂ (3) gg,Q also receives contributions from the vacuum polarization insertions on the external lineŝ…”

“…At 3-loop order a series of moments has been calculated for all massive OMEs for N = 2... 10(14) contributing in the fixed and variable flavor scheme, [5]. The 3-loop heavy flavor corrections to F L (x, Q 2 ) in the asymptotic case were calculated in [16].…”