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Xiv:1705.07030v1 [hep-ph] 19 May 2017

Three Loop Massive Operator Matrix Elements and Asymptotic Wilson Coefficients with Two Different Masses

J. Ablingera , J. Blu ¨mleinb , A. De Freitasb , A. Hasselhuhna ,1 C. Schneidera , and F. Wißbrocka,b,c a

Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenbergerstraße 69, A–4040, Linz, Austria b

Deutsches Elektronen–Synchrotron, DESY, Platanenallee 6, D-15738 Zeuthen, Germany c

IHES, 35 Route de Chartres, F-91440 Bures-sur-Yvette, France. Abstract

Starting at 3-loop order, the massive Wilson coefficients for deep-inelastic scattering and the massive operator matrix elements describing the variable flavor number scheme receive contributions of Feynman diagrams carrying quark lines with two different masses. In the case of the charm and bottom quarks, the usual decoupling of one heavy mass at a time no longer holds, since the ratio of the respective masses, η = m2c /m2b ∼ 1/10, is not small enough. Therefore, the usual variable flavor number scheme (VFNS) has to be generalized. The renormalization procedure in the two–mass case is different from the single mass case derived in [1]. We present the moments N = 2, 4 and 6 for all contributing operator matrix elements, expanding in the ratio η. We calculate the analytic results for general values of the Mellin variable N in the flavor non-singlet case, as well as for transversity and the matrix (3) element Agq . We also calculate the two-mass scalar integrals of all topologies contributing to the gluonic operator matrix element Agg . As it turns out, the expansion in η is usually inapplicable for general values of N . We therefore derive the result for general values of the mass ratio. From the single pole terms we derive, now in a two-mass calculation, the corresponding contributions to the 3-loop anomalous dimensions. We introduce a new general class of iterated integrals and study their relations and present special values. The corresponding functions are implemented in computer-algebraic form.

1

Present address: Institut f¨ ur Theoretische Teilchenphysik Campus S¨ ud, Karlsruher Institut f¨ ur Technologie (KIT), D-76128 Karlsruhe, Germany.

1

Introduction

The heavy flavor corrections to deep-inelastic scattering for pure photon exchange are known to leading [2] and next-to-leading order (NLO) [3]2 . The present accuracy of the deep-inelastic world data requires next-to-next-to leading order (NNLO) QCD analyses in order to determine the strong coupling constant αs (MZ2 ) [5–7] to ∼ 1% accuracy at NNLO, to obtain highly accurate values for the charm and bottom quark masses mc and mb , and to make precise determinations of the parton distribution functions. All of this is in turn needed to describe precision measurements at the LHC [8] and at facilities planned for the future [9, 10]. In the region of large scales Q2  m2 , analytic expressions for the heavy flavor Wilson coefficients have been obtained at NLO [11, 12]. A factorization relation valid in this asymptotic region was given in Refs. [11,13]. For the structure function F2 (x, Q2 ), the asymptotic corrections are sufficient at scales Q2 /m2 > ∼ 10, cf. [11]. The massless corrections at NNLO to the deepinelastic structure functions are available [14–16], while for the corresponding massive corrections in the asymptotic limit, a series of moments has been calculated in the single heavy mass case [1] for all contributing terms in neutral current deep-inelastic scattering. The calculation of the general expressions for the Wilson coefficients is still underway. The asymptotic Wilson coefficients for the structure function FL (x, Q2 ) have been completed [17, 18]. Here the first genuine two-mass contributions emerge at fourth order in the coupling constant. In the case of the structure function F2 (x, Q2 ), all corrections to the color factors O(NF TF2 CA,F ) have been obtained in [19, 20], which provides the complete results for two out of five contributing Wilson coefficients, cf. also [18]. The flavor non-singlet corrections have been calculated in Ref. [21] and the flavor pure singlet terms in Ref. [22]. The massive operator matrix elements (OMEs) calculated in [18,19,21,22] are also needed to describe the variable flavor number scheme (VFNS) in the case of a single heavy quark transition [13], for which also the gluonic contributions Agq,Q and Agg,Q are required and have been calculated at 3-loop order in [23] and in [20, 24, 25], respectively.3 Technical aspects of these calculations have been described in [27–29]. Heavy quark corrections to charged current deep-inelastic processes have been dealt with in Refs. [30]. In the calculations mentioned above, besides internal massless fermion lines, only a single heavy mass is attached to massive fermion lines. However, starting at 3-loop order, there are also diagrams with two different masses attached to the massive lines. In the present paper, we consider corrections of this type. As before in the single heavy mass case [1], a series of finite moments for all massive OMEs and the Wils