The Logarithmic Contributions to the O(\alpha_s^3) Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering

We calculate the logarithmic contributions to the massive Wilson coefficients for deep-inelastic scattering in the asymptotic region $Q^2 \gg m^2$ to 3-loop order in the fixed-flavor number scheme and present the corresponding expressions for the massive operator matrix elements needed in the variable flavor number scheme. Explicit expressions are given both in Mellin-$N$ space and $z$-space.


Introduction
The heavy flavor corrections to deep-inelastic structure functions amount to sizeable contributions, in particular in the region of small values of the Bjorken variable x. Starting from lower values of the virtuality, over a rather wide kinematic range, their scaling violations are very different from those of the massless contributions. Currently the heavy flavor corrections are known in semi-analytic form to 2-loop (NLO) order [1]. The present accuracy of the deep-inelastic data reaches the order of 1% [2]. It therefore requires the next-to-next-to-leading order (NNLO) corrections for precision determinations of both the strong coupling constant α s (M 2 Z ) [3] and the parton distribution functions (PDFs) [4,5], as well as the detailed understanding of the heavy flavor production cross sections in lepton-nucleon scattering [6]. The precise knowledge of these quantities is also of central importance for the interpretation of the physics results at the Large Hadron Collider, LHC, [7].
In the kinematic region at HERA, where the twist-2 contributions to the deep-inelastic scattering (DIS) dominate cf. [8] 2 , i.e. Q 2 /m 2 > ∼ 10, with m = m c the charm quark mass, it has been proven in Ref. [10] that the heavy flavor Wilson coefficients factorize into massive operator matrix elements (OMEs) and the massless Wilson coefficients. The massless Wilson coefficients for the structure function F 2 (x, Q 2 ) are known to 3-loop order [11]. In the region Q 2 m 2 , where Q 2 = −q 2 , with q the space-like 4-momentum transfer and m the heavy quark mass, the power corrections O((m 2 /Q 2 ) k ), k ≥ 1 to the heavy quark structure functions become very small.
In Ref. [12] a series of fixed Mellin moments N up to N = 10, ..., 14, depending on the respective transition, has been calculated for all the OMEs at 3-loop order 3 . Also the moments of the transition coefficients needed in the variable flavor scheme (VFNS) have been calculated. Here, the massive OMEs for given total spin N were mapped onto massive tadpoles which have been computed using MATAD [14].
In the present paper, we calculate the logarithmic contributions to the unpolarized massive Wilson coefficients in the asymptotic region Q 2 m 2 to 3-loop order and the massive OMEs needed in the VFNS. These include the logarithmic terms log(Q 2 /m 2 ). In the following, we set the factorization and renormalization scales equal µ F = µ R ≡ µ and exhibit the log(m 2 /µ 2 ) dependence on the Wilson coefficients, besides their dependence on the virtuality Q 2 . The logarithmic contributions are determined by the lower order massive OMEs [15][16][17][18][19], the massand coupling constant renormalization constants, and the anomalous dimensions [20,21], as has been worked out in Ref. [12]. For the structure function F L (x, Q 2 ) the asymptotic heavy flavor Wilson coefficients at O(α 3 s ) were calculated in [22]. They are also presented here, for inclusive hadronic final states. In this case the corrections, however, become effective only at much higher scales of Q 2 [10] compared to the case of F 2 (x, Q 2 ). We first choose the fixed flavor number scheme to express the heavy flavor contributions to the structure functions F 2 (x, Q 2 ) and F L (x, Q 2 ). This scheme has to be considered as the genuine scheme in quantum field theoretic calculations since the initial states, the twist-2 massless partons can, at least to a good approximation, be considered as LSZ-states. The representations in the VFNS can be obtained using the respective transition coefficients within the appropriate regions, where one single heavy quark flavor becomes effectively massless. Here, appropriate matching scales have to be applied, which vary in dependence on the observable considered, cf. [23].
Two of the OMEs, A PS qq,Q (N ) and A qg,Q (N ), have already been calculated completely including the constant contribution in Ref. [24]. They and the corresponding massive Wilson coefficients contribute first at 2-and 3-loop order, respectively. For these quantities we also derive numerical results. The quantities being presented in the present paper derive from OMEs which were computed in terms of generalized hypergeometric functions [25] and sums thereof, prior to the expansion in the dimensional variable ε = D − 4, cf. [26][27][28]. Finally, they are represented in terms of nested sums over products of hypergeometric terms and harmonic sums, which can be calculated using modern summation techniques [29][30][31][32][33]. They are based on a refined difference field of [34] and generalize the summation paradigms presented in [35] to multi-summation. The results of this computation can be expressed in terms of nested harmonic sums [36,37]. The corresponding representations in z-space are obtained in terms of harmonic polylogarithms [38]. Here, the variable z denotes the partonic momentum fraction. The results in Mellin N -space can be continued to complex values of N as has been described in Refs. [26,39].
It is the aim of the present paper to provide a detailed documentation of formulae both in N -and z-space for all logarithmic contributions to the heavy flavor Wilson coefficients of the structure functions F 2 (x, Q 2 ) and F L (x, Q 2 ) and the massive OMEs needed in the variable flavor number scheme up to O(α 3 s ). Here, we refer to a minimal representation, i.e. we use all the algebraic relations between the harmonic sums and the harmonic polylogarithms, respectively, leading to a minimal number of basic functions. Based on the known Mellin moments [12] we also perform numerical comparisons between the different contributions to the Wilson coefficients and massive OMEs at O(α 3 s ) referring to the parton distributions [5]. The paper is organized as follows. In Section 2, we summarize the basic formalism. The Wilson coefficients L PS q,2 and L S g,2 are discussed in Section 3. As they are known in complete form we also present numerical results. In Section 4, the logarithmic contributions to the Wilson coefficients H PS q,2 and H S g,2 4 are derived. The corresponding Wilson coefficients for the longitudinal structure function F L (x, Q 2 ) in the asymptotic region are presented in Section 5. In Section 6, we compare the different loop contributions to the massive Wilson coefficients and OMEs for a series of Mellin moments in dependence on the virtuality Q 2 . Section 7 contains the conclusions. In Appendix A the massive OMEs needed in the VFNS are given in Mellin N -space. The asymptotic heavy flavor Wilson coefficients contributing to the structure function F 2 (x, Q 2 ) are presented in z-space in Appendix B, retaining all contributions except for the 3-loop constant part of the unrenormalized OMEs a (3) ij being not yet known. Likewise, in Appendix C and D, the asymptotic heavy flavor Wilson coefficients for the structure function F L (x, Q 2 ) and the massive OMEs are given in z-space.

The heavy flavor Wilson coefficients in the asymptotic region
We consider the heavy flavor contributions to the inclusive unpolarized structure functions F 2 (x, Q 2 ) and F L (x, Q 2 ) in deep-inelastic scattering, cf. [41,42], in case of single electro-weak gauge-boson exchange at large virtualities Q 2 . At higher orders in the strong coupling constant these corrections receive both contributions from massive and massless partons in the hadronic final state, which is summed over completely. In the latter case, the heavy flavor corrections are also due to virtual contributions. We consider the situation in which the contributions to the twist-2 operators dominate in the Bjorken limit. Here, no transverse momentum effects of the initial state contribute. In the present paper, we consider only heavy flavor contributions due to N F massless and one massive flavor of mass m. 5 The Wilson coefficients are calculable perturbatively and are denoted by C S,PS,NS i, (2,L) x, N F + 1, Here, x denotes the Bjorken variable, the index i refers to the respective initial state on-shell parton i = q, g being a quark or gluon, and S, PS, NS label the flavor singlet, pure-singlet and non-singlet contributions, respectively. In the twist-2 approximation the Bjorken variable x and the parton momentum fraction z are identical. Representations in momentum fraction space are therefore also called z-space representation in what follows. The massless flavor contributions in (1) may be identified and separated in the Wilson coefficients into a purely light part C i, (2,L) , and a heavy part by : (2,L) x, N F + 1, Q 2 µ 2 , m 2 µ 2 = C S,PS,NS i, (2,L) x, N F , Q 2 µ 2 +H S,PS i, (2,L) x, N F + 1, (2,L) x, N F + 1, The heavy flavor Wilson coefficients are defined by L i,j and H i,j , depending on whether the exchanged electro-weak gauge boson couples to a light (L) or heavy (H) quark line. From this it follows that the light flavor Wilson coefficients C i,j depend on N F light flavors only, whereas H i,j and L i,j may contain light flavors in addition to the heavy quark, indicated by the argument N F + 1. The perturbative series of the heavy flavor Wilson coefficients read H S g, (2,L) x, N F + 1, g, (2,L) x, N F + 1, H PS q, (2,L) x, N F + 1, q, (2,L) x, N F + 1, L S g, (2,L) x, N F + 1, g, (2,L) x, N F + 1, L S q, (2,L) x, N F + 1, (2,L) x, N F + 1, Here, we defined a s = α s /(4π). At leading order, only the term H g, (2,L) contributes via the photon-gluon fusion process, [44][45][46][47][48][49], At O(a 2 s ), the terms H PS q, (2,L) , L S q, (2,L) and L S g,(2,L) contribute as well. They result from the processes γ * + q(q) → q(q) + X , where X may contain heavy flavor contributions. L S q, (2,L) can be split into the flavor non-singlet and pure-singlet contributions L S q,(2,L) = L NS q,(2,L) + L PS q, (2,L) , (10) and at O(a 2 s ) only the non-singlet term contributes. The pure-singlet term emerges at 3-loop order.
The heavy quark contribution to the structure functions F (2,L) (x, Q 2 ) for one heavy quark of mass m and N F light flavors is then given by, cf. [15], in case of pure photon exchange 6 (2,L) x, N F + 1, (2,L) x, N F + 1, (2,L) x, N F + 1, (2,L) x, N F + 1, (2,L) x, N F + 1, The meaning of the argument (N F + 1) in Eqs. (11) in the massive Wilson coefficients shall be interpreted as N F massless and one massive flavor. N F denotes the number of massless flavors. The symbol ⊗ denotes the Mellin convolution, 7 The charges of the light quarks are denoted by e k and that of the heavy quark by e Q . The scale µ 2 is the factorization scale, and f k , f k , Σ and G are the quark, anti-quark, flavor singlet and gluon distribution functions, with An important part of the kinematic region in case of heavy flavor production in DIS is located at larger values of Q 2 , cf. e.g. [54,55]. As has been shown in Ref. [10], the heavy flavor Wilson coefficients H i,j , L i,j factorize in the limit Q 2 m 2 into massive operator matrix elements A ki and the massless Wilson coefficients C i,j , if one heavy quark flavor and N F light flavors are considered. The massive OMEs are process independent quantities and contain all the mass dependence except for the power corrections ∝ (m 2 /Q 2 ) k , k ≥ 1. The process dependence is implied by the massless Wilson coefficients. This allows the analytic calculation of the NLO 6 For the heavy flavor corrections in case of W ± -boson exchange up to O(α 2 s ) see [50][51][52][53]. 7 Note that the heavy flavor threshold in the limit Q 2 m 2 is again x and not x(1 + 4m 2 /Q 2 ), which is the case retaining also power corrections. heavy flavor Wilson coefficients, [10,17]. Comparing these asymptotic expressions with the exact LO and NLO results obtained in Refs. [44][45][46][47]49] and [1], respectively, one finds that this approximation becomes valid in case of F QQ 2 for Q 2 /m 2 > ∼ 10. These scales are sufficiently low and match with the region analyzed in deeply inelastic scattering for precision measurements. In case of F QQ L , this approximation is only valid for Q 2 /m 2 > ∼ 800, [10]. For the latter case, the 3-loop corrections were calculated in Ref. [22]. This difference is due to the emergence of terms ∝ (m 2 /Q 2 ) ln(m 2 /Q 2 ), which only vanish slowly in the limit Q 2 /m 2 → ∞.
In order to derive the factorization formula, one considers the inclusive Wilson coefficients C S,PS,NS i,j , which have been defined in Eq. (1). After applying the light cone expansion (LCE) [56] to the partonic tensor, or the forward Compton amplitude, corresponding to the respective Wilson coefficients, one arrives at the factorization relation, C S,PS,NS,asymp j, (2,L) N, N F + 1, (2,L) N, N F + 1, Here, µ refers to the factorization scale between the heavy and light contributions in C j,i and 'asymp' denotes the limit Q 2 m 2 . The C i,j are the light Wilson coefficients, cf. [11], taken at N F + 1 flavors. This can be inferred from the fact that in the LCE the Wilson coefficients describe the singularities for very large values of Q 2 , which can not depend on the presence of a quark mass. The mass dependence is given by the OMEs A ij , between partonic states. Eq. (14) accounts for all mass effects but corrections which are power suppressed, (m 2 /Q 2 ) k , k ≥ 1. This factorization is only valid if the heavy quark coefficient functions are defined in such a way that all radiative corrections containing heavy quark loops are included. Otherwise, (14) would not show the correct asymptotic Q 2 -behavior, [15,19]. An equivalent way of describing Eq. (14) is obtained by considering the calculation of the massless Wilson coefficients. Here, the initial state collinear singularities are given by evaluating the massless OMEs between off-shell partons, leading to transition functions Γ ij . The Γ ij are given in terms of the anomalous dimensions of the twist-2 operators and transfer the initial state singularities to the bare parton-densities due to mass factorization, cf. e.g. [10,15]. In the case at hand, something similar happens: The initial state collinear singularities are transferred to the parton densities except for those which are regulated by the quark mass and described by the OMEs. Instead of absorbing these terms into the parton densities as well, they are used to reconstruct the asymptotic behavior of the heavy flavor Wilson coefficients. Here, are the operator matrix elements of the local twist-2 operators between on-shell partonic states |j , j = q, g.
Expanding the above relations up to O(a 3 s ), we obtain, using Eqs. (16,17), the heavy flavor Wilson coefficients in the asymptotic limit, cf. [12] : L S g,(2,L) (N F + 1) = a 2 s A (1) gg,Q (N F + 1)N FC (1) g, (2,L) qg,Q (N F + 1) δ 2 + A (1) gg,Q (N F + 1) N FC (2) g,(2,L) (N F + 1) +A (2) gg,Q (N F + 1) N FC (1) g,(2,L) (N F + 1) + A (1) Qg (N F + 1) N FC g, (2,L) Qg (N F + 1) δ 2 +C (1) g,(2,L) (N F + 1) + a 2 s A (2) Qg (N F + 1) δ 2 + A (1) Qg (N F + 1) C (1),NS q,(2,L) (N F + 1) gg,Q (N F + 1)C (1) g,(2,L) (N F + 1) +C (2) g, (2,L) Qg (N F + 1) δ 2 + A (2) Qg (N F + 1) C (1),NS q,(2,L) (N F + 1) + A (2) gg,Q (N F + 1)C (1) g,(2,L) (N F + 1) + A (1) Qg (N F + 1) C (2),NS q,(2,L) (N F + 1) +C (2),PS q,(2,L) (N F + 1) + A (1) gg,Q (N F + 1)C (2) g,(2,L) (N F + 1) +C (3) g, (2,L) with δ 2 = 1 for F 2 and δ 2 = 0 for F L . Again, the argument (N F + 1) in the massive OMEs signals that these functions depend on N F massless and one massive flavor, while the setting of N F in the massless Wilson coefficients is a functional one. The above equations include radiative corrections due to heavy quark loops to the Wilson coefficients. Therefore, in order to compare e.g. with the calculation in Refs. [1], these terms still have to be subtracted. Since the light flavor Wilson coefficients were calculated in the MS-scheme, the same scheme has to be used for the massive OMEs. It should also be thoroughly used for renormalization to derive consistent results in QCD analyses of deep-inelastic scattering data and to be able to compare to other analyses of hard scattering data directly. This requests special attendance w.r.t. the choice of the scheme in which a s is defined, cf. [12]. The renormalized massive OMEs depend on the ratio m 2 /µ 2 , while the scale ratio in the massless Wilson coefficients is µ 2 /Q 2 . The latter are pure functions of the momentum fraction z, or the Mellin variable N , if one sets µ 2 = Q 2 . The mass dependence on the heavy flavor Wilson coefficients in the asymptotic region derives from the unrenormalized massive OMEŝ applying mass, coupling constant, and operator-renormalization, as well as mass factorization, cf. Ref. [12]. The renormalized massive OMEs obey then the general structure The subsequent calculations will be performed in the MS scheme for the coupling constant and the on-shell scheme for the heavy quark mass m. The transition to the scheme in which m is renormalized in the MS-scheme is described in Ref. [12]. The strong coupling constant is obtained as the perturbative solution of the equation to 3-loop order, where β k are the expansion coefficients of the QCD β-function and µ 2 denotes the renormalization scale. For simplicity we identify the factorization (µ F ) and renormalization (µ R ) scales from now on. In the subsequent sections we present explicit expressions of the asymptotic heavy flavor Wilson coefficients in Mellin-N space. They depend on the logarithms where µ ≡ µ F = µ R . Besides the Wilson coefficients (23-27) the massive OMEs are important themselves to establish the matching conditions in the variable flavor number scheme in describing the process of a single massive quark becoming massless 8 at large enough scales µ 2 , [12,15]. Here, the PDFs for N F + 1 massless quarks are related to the former N F massless quarks process independently. The corresponding relations to 3-loop order read, cf. also [15] 9 : Here, the N F -dependence of the OMEs is understood as functional and µ 2 denotes the matching scale, which for the heavy-to-light transitions is normally much larger than mass scale m 2 , [23]. We will present the corresponding OMEs in Appendix A. The results of the calculations being presented in the subsequent sections have been obtained making mutual use of the packages HarmonicSums.m [58] and Sigma.m [29].
3 The Wilson Coefficients L PS q,2 and L S g,2 The OMEs for these Wilson coefficients have been calculated in [24]. They contribute for the first time at 3-and 2-loop order, respectively, and stem from processes in which the virtual electro-weak gauge boson couples to a massless quark. As a shorthand notation we also define the functionγ 0 qg = −4 denoting the kinetic part of the leading order anomalous dimensions separating off the corresponding color factor. In Mellin-N space the Wilson coefficient L PS q,2 reads : with the polynomials For the massless 3-loop Wilson coefficients C k i,j we refer to Ref. [11]. Here and in the following, their expression will be kept symbolically. The corresponding z-space expressions are given in Appendix B.
Likewise the Wilson coefficient L S g,2 is given by : where In all the representations of the massive Wilson coefficients and OMEs in N -space we apply algebraic reduction [59]. The 2-loop term in (47) is purely multiplicative and induced by renormalization only, while the 3-loop contributions require the calculation of massive OMEs. The above Wilson coefficients depend on the harmonic sums apart of those defining the massless 3-loop Wilson coefficients [11] 10 . The harmonic sums are defined recursively by, cf. [36,37], In the above ζ l = ∞ k=1 1/k l , l ∈ N, l ≥ 2 denote the Riemann ζ-values, which are convergent harmonic sums in the limit N → ∞. In the constant part of the other Wilson coefficients it is expected that more complicated multiple zeta values emerge, which have been dealt with in [61].
In Eq. (47) denominator terms ∝ 1/(N −2) occur. They cancel in the complete expression and the rightmost singularity is located at N = 1 as expected for this Wilson coefficient. Let us now consider both the small-and large-x dominant terms for both Wilson coefficients. Those of the massless parts were given in [11] before. Both Wilson coefficients contain terms ∝ 1/(N − 1). For simplicity we consider the choice of scale Q 2 = µ 2 here. The expansion of the heavy flavor contribution, subtracting the massless 3-loop Wilson coefficients, denoted by L i , around N = 1(x → 0) and in the limit N → ∞(x → 1), setting Q 2 = µ 2 , are given by The corresponding limits for the contributions of the massless Wilson coefficients behave like 10 For the algebraically reduced representations see [60].   While the expression for L PS q,2 is the same in the MS-and on-mass-shell scheme to O(a 3 s ), L S g,2 , in its 3-loop contribution, changes by the term setting Q 2 = µ 2 . Here, we have identified the logarithms L M in both schemes symbolically. In applications, either the on-shell or the MS mass has to be used here. The corresponding expression in z-space reads with H a (z) harmonic polylogarithms, see Eq. (592).  Figure 3: The O(a 3 s ) contribution by L PS q,2 to the structure function F 2 (x, Q 2 ).
x  Table 1: Values of the structure function F 2 (x, Q 2 ) in the low x region using the PDFparameterization [5].
In Figure 1 we illustrate the contribution of L PS q,2 to the structure function F 2 (x, Q 2 ) using the PDFs of Ref. [5], cf. Eq. (11). Likewise Figures 2 and 3 show the corresponding contributions by L S g,2 at O(a 2 s ) and O(a 3 s ), respectively. Note that the O(a 2 s )-terms, cf. also Ref. [19] are smaller than those at O(a 3 s ), which is caused by terms ∝ 1/z in the 3-loop contribution to L S g,2 , which are absent at 2-loop order.
These contributions emerging on the 2-and 3-loop level are minor compared to the values of the structure function F 2 (x, Q 2 ), for which typical values are given in Table 1. 11 A global comparison of all heavy flavor contributions up to 3-loop order can presently only be performed using the known number of Mellin moments, cf. [12], given in Section 6. of the unrenormalized operator matrix element in the on-shell scheme can be expressed by harmonic sums and rational functions in N only. As before we reduce to a basis eliminating the algebraic relations [59]. It is given by : Note that the kinematic region at small x probed at HERA is limited to values with the polynomials The Wilson coefficient H S g,2 , except for the constant contribution a Qg , has a similar structure. It is given by : with the polynomials (234) The corresponding expressions in z-space are given in Appendix B. Note that our result for H S g,2 differs from the one given in Eq. (B.7) in z-space in Ref. [9] by the term in N -space. This result of [9] is based on the calculation carried out in Ref. [12], including the renormalization formulae derived there. We have checked, however, that our result Eq. (152) is in full agreement with Eq. (27) and the moments having been calculated by part of the present authors in Ref. [12]. The corresponding expression in z-space is presented in Appendix B.

The Asymptotic Wilson Coefficients for the Longitudinal Structure Function
The Wilson coefficients have been calculated in Ref. [22] for exclusive heavy flavor production, retaining three contributions only. In total also here five Wilson coefficients contribute and the expressions are slightly modified in the inclusive case of the complete structure function F L (x, Q 2 ), cf. [12]. In Mellin-N space they read : with with with and with The expressions in z-space are presented in Appendix C.
As has been outlined for the 2-loop results in Ref. [10] already, the scales at which the asymptotic expressions are dominating are estimated to be Q 2 /m 2 > ∼ 800. They are far outside the kinematic region in which the structure function F L (x, Q 2 ) can presently be measured in deepinelastic scattering. The corresponding expressions are therefore of merely theoretical character and cannot be used in current phenomenological analyses.

Comparison of Mellin Moments for the Wilson Coefficients and OMEs
In order to compare the relative impact of the different Wilson coefficients on the structure function F 2 (x, Q 2 ) we will consider the Mellin moments for N = 2 to 10 in the following, folded with the moments of the respective parton distribution functions in the flavor singlet case, i.e. the gluon G(x, Q 2 ) and quark-singlet density Σ(x, Q 2 ) for N F = 3 and characteristic values of Q 2 . Since only a series of Mellin moments has been calculated at large momentum transfer Q 2 in Ref. [12], a detailed numerical comparison is only possible in this way at the moment. The numerical results for the moments of the contributing parton densities are given in Table 2.
Note that for N ≥ 2 the moments for the singlet-distribution are mostly larger than those of the gluon. We apply these parton densities to study the relative contributions of the different Wilson coefficients, normalizing to H S g,2 within the respective order in a s using the following ratios: In the numerical examples we set e Q = e c = 2/3.   Table 3: Relative impact of the moments N = 2, ..., 10 of the individual massive Wilson coefficients, weighted by moments of the corresponding parton distributions [5], at O(a 2 s ) and O(a 3 s ) normalized to the contribution to H S g,2 for Q 2 = 20, 100 and 1000 GeV 2 .
Before we discuss quantitative results, a remark on the contributions by the color factor d abc d abc to the massless Wilson coefficients Refs. [62][63][64] and [11,57] used in the present analysis, is in order. For SU (N ) one obtains It emerges weighted by 1/N c and 1/N A for external quark and gluon lines, respectively, with N c = N and N A = N 2 − 1. In Refs. [62][63][64] this group-theoretic expression has been used, while in [11,57] a factor of 16 has been taken out and was absorbed into the Lorentz structure of the corresponding contribution to the Wilson coefficient. We agree with the N F -dependence as given in Refs. [62][63][64]. Furthermore, we note a typographical error in Eq. (4.13) of [11]. Here, the corresponding term reads correctly 12  Table 4: Relative impact of the moments N = 2, ..., 10 of the individual massive OMEs, weighted by moments of the corresponding parton distributions [5], at the different orders in a s normalized to the contribution to A Qg for Q 2 = 20 and 100 GeV 2 .
Let us now consider the relative impact of the individual massive Wilson coefficients. The ratios at O(a 2 s ) and O(a 3 s ) for different values of Q 2 and the moments N = 2 to 10 are given in Table 3. One first notes that at low values of Q 2 the moments of L S g,2 change sign, which is also the case for H PS q,2 in the whole region up to Q 2 = 1000 GeV 2 . At O(a 2 s ) L S g,2 is small for low moments and grows 24% for N = 10 compared to H S g,2 at Q 2 = 100 GeV 2 , with lower values at higher Q 2 . A comparable tendency is observed at O(a 2 s ). The fraction |R(H PS q,2 , H S g,2 )| moves between 25% and 170% comparing the moments N = 2 to 10 at Q 2 = 20 GeV 2 and upper values of ∼ 70% at Q 2 = 1000GeV 2 .
In the case of the comparison of the massive OMEs we normalize to A Qg with PDFs according to their appearance in the singlet and gluon transitions from N F → N F + 1 massless flavors in the variable flavor number scheme, cf. Eqs. (33)(34)(35): These ratios describe the relative impact, within the corresponding order in a s , of the massive OMEs in the variable flavor number scheme for the flavor singlet contributions.  Table 5: The same as Table 4 for Q 2 = 1000GeV 2 .
The numerical values for different scales of Q 2 are given in Tables 4 and 5. |A gg,Q /A Qg | rises from about 1 to higher values from N = 2 to 10, irrespectively of the values of Q 2 and the order in a s . The smallest contributions are |A qg,Q | and A PS qq,Q contributing the ratios R by ∼ 0.5% to 5% and form ∼ 5 to ∼ 10%, respectively, for N = 2 and 4, i.e. in the region dominated by lower values of the Bjorken variable x. The OMEs |A PS Q,q | and A gq,Q have contributions of 16-62% and 14-150%, respectively, for N = 2 and 4. Also the flavor non-singlet Wilson coefficient |A NS qq,Q | contributes in the flavor singlet transitions and is weighted by the distribution Σ here. Its relative impact rises with Q 2 and amounts from ∼ 4% to 370% for the R-ratio considering the lower moments N = 2 and N = 4 only.
Right after having obtained a series of moments for the massive OMEs at 3-loops in [12], it became clear that the logarithmic contributions are of comparable order to the constant term. Moreover, there is a strong functional dependence w.r.t. N , as displayed in Tables 2-5. To obtain a definite answer, the calculation of the constant parts of the unrenormalized OMEs a (3) ij as a function of N ∈ C is necessary. In particular predictions for the range of small values x 10 −4 appear to be rather difficult otherwise.

Conclusions
We have derived the contributions of the massive Wilson coefficients to the structure functions F 2 (x, Q 2 ) and F L (x, Q 2 ) in deep-inelastic scattering and the corresponding massive OMEs to 3-loop order in the asymptotic region Q 2 m 2 both in Mellin-N and z-space except for the constant parts a ij of the unrenormalized OMEs, which are not known for all quantities yet. Here, we retained both the scale-dependence due to the virtuality Q 2 and the factorization and renormalization scales µ 2 , which were set equal. This allows for scale variation studies in applications. Two of the Wilson coefficients, L PS q,2 and L S g,2 , are known in complete form, and the corresponding results for L NS q,2 will be given in [40]. In the variable flavor number scheme being applied to describe the process through which an initially massive quark transmutes into a massless one at high momentum scales, moreover, the matching coefficients A ij are needed. Here, A PS qq,Q and A qg,Q are known in complete form to 3-loop order and the results for A gq and A NS qq,Q are given in [65] and [40], respectively. We have given numerical results for the Wilson coefficients L PS q,2 and L S g,2 . Using the available Mellin moments we have performed a numerical comparison of the different Wilson coefficients and operator matrix elements inside the respective order in the coupling constant for the moments N = 2 to 10 and in the Q 2 range between 20 and 1000 GeV 2 . While some of the quantities studied are of minor importance, several others of the Wilson coefficients and OMEs are of similar size, which is varying in the kinematic range of experimental interest for present and future precision measurements. Even in case of the charm-quark contributions the logarithmic terms are not dominant over the constant contributions in wide kinematic ranges, as. e.g. at HERA.
The expression which were derived in the present paper are available in form of computerreadable files on request via e-mail to Johannes.Bluemlein@desy.de.

A The massive operator matrix elements in N -space
In this appendix we present the massive OMEs in Mellin-space to be used in the matching coefficients in the variable flavor number scheme Eqs. (32)(33)(34)(35). The corresponding representations in z-space are given in Appendix D. Thus far the OMEs A PS qq,Q and A qg,Q are known completely. The other OMEs are presented except for the 3-loop constant part a ij in the unrenormalized OMEs. The OMEs A NS qq,Q and A S gq,Q are presented elsewhere [40,65].

C The Longitudinal Wilson Coefficients in z-space
The Wilson coefficients for the longitudinal structure function F L (x, Q 2 ) in the asymptotic region in z-space are presented in the following. L PS q,L and L S g,L read : The pure-singlet Wilson coefficient H PS q,L reads :