J. Ablinger, A. Behring, J. Blümlein, G. Falcioni, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, F. Wißbrock
We present recent results on newly calculated 2- and 3-loop contributions to the heavy quark parts of the structure functions in deep-inelastic scattering due to charm and bottom.
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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.
We calculate the massive two--loop pure singlet Wilson coefficients for heavy quark production in the unpolarized case analytically in the whole kinematic region and derive the threshold and asymptotic expansions. We also recalculate the corresponding massless two--loop Wilson coefficients. The complete expressions contain iterated integrals with elliptic letters. The contributing alphabets enlarge the Kummer-Poincaré letters by a series of square-root valued letters. A new class of iterated integrals, the Kummer-elliptic integrals, are introduced. For the structure functions $F_2$ and $F_L$ w…
We calculate the massless unpolarized Wilson coefficients for deeply inelastic scattering for the structure functions $F_2(x,Q^2), F_L(x,Q^2), x F_3(x,Q^2)$ in the $\overline{\sf MS}$ scheme and the polarized Wilson coefficients of the structure function $g_1(x,Q^2)$ in the Larin scheme up to three--loop order in QCD in a fully automated way based on the method of arbitrary high Mellin moments. We work in the Larin scheme in the case of contributing axial--vector couplings or polarized nucleons. For the unpolarized structure functions we compare to results given in the literature. The polarize…
A survey is given on the status of 3-loop heavy flavor corrections to deep-inelastic structure functions at large enough virtualities $Q^2$.
We report on recent progress in the calculation of the 3-loop massive Wilson coefficients in deep-inelastic scattering at general values of $N$ for neutral and charged current reactions in the asymptotic region $Q^2 \gg m^2$.
We calculate the polarized massive two--loop pure singlet Wilson coefficient contributing to the structure functions $g_1(x,Q^2)$ analytically in the whole kinematic region. The Wilson coefficient contains Kummer--elliptic integrals. We derive the representation in the asymptotic region $Q^2 \gg m^2$, retaining power corrections, and in the threshold region. The massless Wilson coefficient is recalculated. The corresponding twist--2 corrections to the structure function $g_2(x,Q^2)$ are obtained by the Wandzura--Wilczek relation. Numerical results are presented.
We present our most recent results on the calculation of the heavy flavor contributions to deep-inelastic scattering at 3-loop order in the large $Q^2$ limit, where the heavy flavor Wilson coefficients are known to factorize into light flavor Wilson coefficients and massive operator matrix elements. We describe the different techniques employed for the calculation and show the results in the case of the heavy flavor non-singlet and pure singlet contributions to the structure function $F_2(x,Q^2)$.
We report on our latest results in the calculation of the three-loop heavy flavor contributions to the Wilson coefficients in deep-inelastic scattering in the asymptotic region $Q^2 \gg m^2$. We discuss the different methods used to compute the required operator matrix elements and the corresponding Feynman integrals. These methods very recently allowed us to obtain a series of new operator matrix elements and Wilson coefficients like the flavor non-singlet and pure singlet Wilson coefficients.
We use the large N_f self consistency formalism to compute the $O(1/N_f)$ critical exponent corresponding to the renormalization of the flavour non-singlet twist two Wilson operators which arise in the operator product expansion of currents in deep inelastic processes. Expanding the $d$-dimensional expression in powers of $ε$ $=$ $(4-d)/2$ the coefficients of $ε$ agree with the known two loop structure of the corresponding renormalization group function and we deduce analytic expressions for all moments, $n$, at three and higher orders in perturbation theory in the $\overline{\mbox{MS}}$ schem…
We report on recent results obtained for the 3-loop heavy flavor Wilson coefficients in deep-inelastic scattering (DIS) at general values of the Mellin variable $N$ at larger scales of $Q^2$. These concern contributions to the gluonic ladder-topologies, the transition matrix elements in the variable flavor scheme of $O(n_f T_F^2)$ and $O(T_F^2)$, and first results on higher 3-loop topologies. The knowledge of the heavy flavor Wilson coefficients at 3-loop order is of importance to extract the parton distribution functions and $α_s(M_Z^2)$ in complete NNLO QCD analyses of the world precision da…
We calculate the $O(α_s^2)$ heavy flavor corrections to charged current deep-inelastic scattering at large scales $Q^2 \gg m^2$. The contributing Wilson coefficient are given as convolutions between massive operator matrix elements and massless Wilson coefficients. Foregoing results in the literature are extended and corrected. Numerical results are presented for the kinematic region of the HERA data.
We present recent analytic results for the 3-loop corrections to the massive operator matrix element $A_{Qg}^{(3)}$for further color factors. These results have been obtained using the method of arbitrarily large moments. We also give an overview on the results which were obtained solving all difference and differential equations for the corresponding master integrals that factorize at first order.
The logarithmic contributions to the massive twist-2 operator matrix elements for deep-inelastic scattering are calculated to $O(α_s^3)$for general values of the Mellin variable $N$.
In the color glass condensate framework the saturation scale measured in deep inelastic scattering of high energy hadrons and nuclei can be determined from the correlator of Wilson lines in the hadron wavefunction. These same Wilson lines give the initial condition of the classical field computation of the initial gluon multiplicity and energy density in a heavy ion collision. In this paper the Wilson line correlator in both adjoint and fundamental representations is computed using exactly the same numerical procedure that has been used to calculate gluon production in a heavy ion collision. I…
In the asymptotic limit $Q^2 \gg m^2$, the non-power corrections to the heavy flavour Wilson coefficients in deep--inelastic scattering are given in terms of massless Wilson coeffcients and massive operator matrix elements. We start extending the existing NLO calculation for these operator matrix elements by calculating the O($ε$) terms of the two--loop expressions and having first investigations into the three--loop diagrams needed to O($α_s^3$).
We calculate the $O(α_s^3)$ 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 $Q^2 \gg m^2$. 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 …
The complete three-loop unpolarized and polarized massive operator matrix elements and asymptotic Wilson coefficients
本文研究了三阶非极化和极化重子算符矩阵元及渐近威金系数，提供了精确的数值表示和适用于QCD拟合的拆分函数与质量修正。
We report on recent progress in the calculation of the 3-loop massive Wilson coefficients in deep-inelastic scattering at general values of $N$ for neutral and charged current reactions in the asymptotic region $Q^2 \gg m^2$. Four new out of eight massive operator matrix elements and Wilson coefficients have been obtained recently. We also discuss recent results on Feynman graphs containing two-massive fermion lines and present complete results for the bubble topologies for all processes.
We calculate the quarkonic $O(α_s^2)$ massive operator matrix elements $ΔA_{Qg}(N), ΔA_{Qq}^{\rm PS}(N)$ and $ΔA_{qq,Q}^{\rm NS}(N)$ for the twist--2 operators and the associated heavy flavor Wilson coefficients in polarized deeply inelastic scattering in the region $Q^2 \gg m^2$ to $O(\varepsilon)$ in the case of the inclusive heavy flavor contributions. The evaluation is performed in Mellin space, without applying the integration-by-parts method. The result is given in terms of harmonic sums. This leads to a significant compactification of the operator matrix elements and massive Wilson coef…
In this note we evaluate Wilson coefficients for "deep inelastic scattering" (DIS) in N=4 SYM theory at NLO in perturbation theory, using as a probe an R-symmetry conserved current. They exhibit uniform transcendentality and coincide with the piece of highest transcendentality in the corresponding QCD Wilson coefficients. We extract from the QCD result a NNLO prediction for the N=4 SYM Wilson coefficient, and comment on the features of its Regge limit asymptotics.
In the asymptotic limit $Q^2 \gg m^2$, the heavy flavor Wilson coefficients for deep--inelastic scattering factorize into the massless Wilson coefficients and the universal heavy flavor operator matrix elements resulting from light--cone expansion. In this way, one can calculate all but the power corrections in $(m^2/Q^2)^k, k > 0$. The heavy flavor operator matrix elements are known to ${\sf NLO}$. We present the last 2--loop result missing in the unpolarized case for the renormalization at 3--loops and first 3--loop results for terms proportional to the color factor $T_F^2$ in Mellin--space.…
We verify the operator product expansion (OPE) of deep inelastic scattering (DIS) on the lattice and present first results of a non-perturbative calculation of the Wilson coefficients.
The paper presents the QCD description of the small $x$ behaviour of parton distribution functions in the leading twist of Wilson operator product expansion. The smooth transition between the cases of the soft and hard Pomerons is obtained. The results are in qualitative agreement with deep inelastic experimental data.
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, $η= m_c^2/m_b^2 \sim 1/10$, 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…
We provide a fast and precise Mellin-space implementation of the $O(α_s)$ heavy flavor Wilson coefficients for charged current deep inelastic scattering processes. They are of importance for the extraction of the strange quark distribution in neutrino-nucleon scattering and the QCD analyses of the HERA charged current data. Errors in the literature are corrected. We also discuss a series of more general parton parameterizations in Mellin space.
We compute the logarithmic contributions to the polarized 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 polarized massive operator matrix elements needed in the variable flavor number scheme. The calculation is performed in the Larin scheme. For the massive operator matrix elements $A_{qq,Q}^{(3),\rm PS}$ and $A_{qg,Q}^{(3),\rm S}$ the complete results are presented. The expressions are given in Mellin-$N$ space and in momentum fraction $z$-spac…
The $O(α_s^2)$ massive operator matrix elements for unpolarized and polarized heavy flavor production at asymptotic values $Q^2 >> m^2$ are calculated in Mellin space without applying the integration-by-parts method. We confirm previous results given in Refs. \cite{BU1,BU2}, however, obtain much more compact representations.
The three-loop single-mass heavy flavor corrections to deep-inelastic scattering
本文报告了深度非弹性散射中重味Wilson系数和算符矩阵元三圈计算的进展，涵盖非极化与极化情形的单质量及大部分双质量贡献，并给出结构函数数值结果。
We calculate the light cone wave functions for a virtual photon to split into quark-antiquark states, including for the first time quark masses at one loop accuracy. These wave functions can be used to calculate cross sections for several precision probes of perturbative gluon saturation at the Electron-Ion Collider. Using these wave functions we derive, for the first time, the dipole picture DIS cross sections at one loop for longitudinal and transverse virtual photons including quark masses. The quark masses are renormalized in the pole mass scheme, satisfying constraints from the requiremen…
We calculate the O($\eps$)--term of the two--loop massive operator matrix elements for twist 2--operators, which contribute to the heavy flavour Wilson coefficients in unpolarized deep--inelastic scattering in the asymptotic limit $Q^2 \gg m^2.$ Our calculation was performed in Mellin space using Mellin--Barnes integrals and generalized hypergeometric functions. The O($\eps$)--term contributes in the renormalization at 3--loop order.
We apply the soft-collinear effective theory (SCET) to deep inelastic scattering near the endpoint region. The forward scattering amplitude, and the structure functions are shown to factorize as a convolution of the Wilson coefficients, the jet functions, the parton distribution functions. The behavior of the parton distribution functions near the endpoint region is considered. It turns out that it evolves with the Altarelli-Parisi kernel even in the endpoint region, and the parton distribution function can be factorized further into a collinear part and the soft Wilson line. The factorized fo…
The results of a recent next-to-leading order QCD analysis of the world data on polarized deep inelastic scattering are reported. New parameterizations are derived for the quark and gluon distributions, accounting for the massive Wilson coefficient for the charm quarks, and the value of $α_s(M_z^2)$ is determined with correlated errors. We obtain $α_s^{\rm NLO}(M_Z^2)= 0.1132 {array}{l} + 0.0056 -0.0095 {array}$. Limits on potential higher twist contributions to the structure function $g_1(x,Q^2)$ are derived. We also compare to the results obtained by other groups.
We present details on calculation of next-to-next-to-leading order QCD corrections to massive charged-current coefficient functions in deep-inelastic scattering. Especially we focus on the application to charm-quark production in neutrino scattering on fixed target that can be measured via the dimuon final state. We construct a fast interface to the calculation so for any parton distributions the cross sections can be evaluated within milliseconds by using the pre-generated interpolation grids. We discuss agreements of various theoretical predicitons with the NuTeV and CCFR dimuon data and the…
Lattice calculations could boost our understanding of Deep Inelastic Scattering by evaluating moments of the Nucleon Structure Functions. To this end we study the product of electromagnetic currents between quark states. The Operator Product Expansion (OPE) decomposes it into matrix elements of local operators (depending on the quark momenta) and Wilson coefficients (as functions of the larger photon momenta). For consistency with the matrix elements, we evaluate a set of Wilson coefficients non-perturbatively, based on propagators for numerous momentum sources, on a 24^3 x 48 lattice. The use…
We calculate moments of the $O(α_s^3)$ heavy flavor contributions to the Wilson coefficients of the structure function $F_2(x,Q^2)$ in the region $Q^2\gg m^2$. The massive Wilson coefficients are obtained as convolutions of massive operator matrix elements (OMEs) and the known light flavor Wilson coefficients. The calculation of moments of the massive OMEs involves a first independent recalculation of moments of the fermionic contributions to all 3--loop anomalous dimensions of the unpolarized twist--2 local composite operators stemming from the light--cone expansion \cite{url}.
We study the effect of scheme dependence upon the NLO QCD analysis of the world data on polarized DIS. The reliability of an analysis at NLO is demonstrated by the consistency of our polarized densities with the NLO transformation rules relating them to each other. We stress the importance of the chiral JET scheme in which all the hard effects are consistently absorbed in the Wilson coefficient functions.
The effects of the soft gluon emission in hard scattering processes at the phase boundary are resummed in the soft-collinear effective theory (SCET). In SCET, the soft gluon emission is decoupled from the energetic collinear part, and is obtained by the vacuum expectation value of the soft Wilson-line operator. The form of the soft Wilson lines is universal in deep inelastic scattering, in the Drell-Yan process, in the jet production from e+e- collisions, and in the gamma* gamma* -> pi0 process, but its analytic structure is slightly different in each process. The anomalous dimensions of the s…
We examine diffractive proton-proton scattering. Using a functional integral approach we derive the scattering amplitudes, which are governed by the expectation value of lightlike Wegner-Wilson loops. This expectation value is then evaluated using a cumulant expansion and the model of the stochastic vacuum. From the scattering amplitudes we calculate total and differential cross sections for high centre of mass energy and small momentum transfer and compare with experiments.
UV Massive Resonance from IR Double Copy Consistency
本研究提出一类自洽大质量共振双拷贝（CMRDC）模型，通过红外EFT威尔逊系数的双拷贝一致性重构紫外大质量留数，结合色对偶自举与Padé外推，为从红外涌现紫外结构提供新路径。
We report on results for the heavy flavor contributions to $F_2(x,Q^2)$ in the limit $Q^2\gg m^2$ at {\sf NNLO}. By calculating the massive $3$--loop operator matrix elements, we account for all but the power suppressed terms in $m^2/Q^2$. Recently, the calculation of fixed Mellin moments of all $3$--loop massive operator matrix elements has been finished. We present new all--$N$ results for the $O(n_f)$--terms, thereby confirming the corresponding parts of the $3$--loop anomalous dimensions. Additionally, we report on first genuine $3$--loop results of the ladder--type diagrams for general va…
The event rates for WIMP-nucleus and neutrino-nucleus scattering processes, expected to be detected in ton-scale rare-event detectors, are investigated. We focus on nuclear isotopes that correspond to the target nuclei of current and future experiments looking for WIMP- and neutrino-nucleus events. The nuclear structure calculations, performed in the context of the deformed shell model, are based on Hartree-Fock intrinsic states with angular momentum projection and band mixing for both the elastic and the inelastic channels. Our predictions in the high-recoil-energy tail show that detectable d…
A QCD analysis of the world data on polarized deep inelastic scattering is presented in next--to--leading order, including the heavy flavor Wilson coefficient in leading order in the fixed flavor number scheme. New parameterizations are derived for the quark and gluon distributions and the value of $α_s(M_z^2)$ is determined. The impact of the variation of both the renormalization and factorization scales on the distributions and the value of $α_s$ is studied. We obtain $α_s^{\rm NLO}(M_Z^2) = 0.1132~~\begin{array}{l} + 0.0056 \\ -0.0095 \end{array}$. The first moments of the polarized twist--…
We present O(g^4) calculations of both planar and non-planar Wilson loops for various actions in the presence of sea quarks. In particular, the plaquette, the static potential and the static self energy are calculated to this order for massive Wilson, Sheikholeslami-Wohlert and Kogut-Susskind fermions, including the mass and n_f dependence. The results can be used to obtain alpha_{MS} and m_b(m_b) from lattice simulations. We compare our perturbative calculations to simulation data of the static potential and report excellent qualitative agreement with boosted perturbation theory predictions f…
Diffractive deep inelastic scattering at NLO in the dipole picture
该研究在深度非弹性散射偶极图像下完成衍射结构函数的完整NLO计算，结合光锥微扰论与色玻璃凝聚理论，验证快度发散可因子化且其余发散抵消，完善了相关理论框架。
The lower moments of the unpolarized and polarized deep-inelastic structure functions of the nucleon are calculated on the lattice. The calculation is done with Wilson fermions and for three values of the hopping parameter $κ$, so that we can perform the extrapolation to the chiral limit. Particular emphasis is put on the renormalization of lattice operators. The renormalization constants, which lead us from lattice to continuum operators, are computed perturbatively to one loop order as well as non-perturbatively.
We report on recent results obtained for the massive operator matrix elements which contribute to the massive Wilson coefficients in deep-inelastic scattering for $Q^2 \gg m_i^2$ in case of sub-processes with two fermion lines and different mass assignment.
We consider the fermion spectrum in the strong coupling vortex phase of a lattice fermion-scalar model with a global $U(1)_L\times U(1)_R$, in 2D, in the context of a recently proposed two-cutoff lattice formulation. The fermion doublers are made massive by a strong Wilson-Yukawa coupling, but in contrast with the standard formulation of these models, in which the light fermion spectrum was found to be massive and vectorlike, we find massless undoubled fermions with chiral quantum numbers at finite lattice spacing. When the global symmetry is gauged, this model is expected to give rise to a ch…
The paper presents a theoretical approach to the description of the relativistic scattering of a massive (neutral) lepton on a nucleus, in which the latter retains its integrity. The measurable cross section of this process includes the elastic (or coherent) contribution, when the nucleus remains in its original quantum state and the inelastic (incoherent) contribution, when the nucleus goes into another (excited) quantum state. Transition from the elastic scattering regime to the inelastic scattering regime is regulated automatically by the dependence of the form factors on the momentum trans…
We report on progress in the calculation of 3-loop corrections to the deep-inelastic structure functions from massive quarks in the asymptotic region of large momentum transfer $Q^2$. Recently completed results allow us to obtain the $O(a_s^3)$ contributions to several heavy flavour Wilson coefficients which enter both polarised and unpolarised structure functions for lepton-nucleon scattering. In particular, we obtain the non-singlet contributions to the unpolarised structure functions $F_2(x,Q^2)$ and $x F_3(x,Q^2)$ and the polarised structure function $g_1(x,Q^2)$. From these results we als…
We calculate moments of the $O(α_s^3)$ heavy flavor contributions to the Wilson coefficients of the structure function $F_2(x,Q^2)$ in the region $Q^2\gg m^2$. The massive Wilson coefficients are obtained as convolutions of massive operator matrix elements (OMEs) and the known light flavor Wilson coefficients. The calculation of moments of the massive OMEs involves a first independent recalculation of moments of the fermionic contributions to all 3--loop anomalous dimensions of the unpolarized twist--2 local composite operators stemming from the light--cone expansion. The expressions for the m…
Renormalization-group methods in soft-collinear effective theory are used to perform the resummation of large perturbative logarithms for deep-inelastic scattering in the threshold region x->1. The factorization theorem for the structure function F_2(x,Q^2) for x->1 is rederived in the effective theory, whereby contributions from the hard scale Q^2 and the jet scale Q^2(1-x) are encoded in Wilson coefficients of effective-theory operators. Resummation is achieved by solving the evolution equations for these operators. Simple analytic results for the resummed expressions are obtained directly i…
Computing the soft anomalous dimension with massless particles using the method of regions
本文提出用区域法处理涉及类光威尔逊线的关联函数，以计算含无质量粒子的软反常维度，并以两圈两有质量一无质量情形为例验证了方法的有效性。
The small-$x$ deep inelastic scattering in the saturation region is governed by the non-linear evolution of Wilson-line operators. In the leading logarithmic approximation it is given by the BK equation for the evolution of color dipoles. In the next-to-leading order the BK equation gets contributions from quark and gluon loops as well as from the tree gluon diagrams with quadratic and cubic nonlinearities. We calculate the gluon contribution to small-x evolution of Wilson lines (the quark part was obtained earlier).
We investigate the power-suppressed corrections to fragmentation functions in flavour-singlet deep inelastic lepton scattering, to complement the previous results for the non-singlet contribution. Our method is a dispersive approach based on an analysis of Feynman graphs containing massive gluons. As in non-singlet deep inelastic scattering we find that the leading corrections are proportional to 1/Q^2.
We study two-loop anomalous dimension matrices in QCD and related gauge theories for products of Wilson lines coupled at a point. We verify by an analysis in Euclidean space that the contributions to these matrices from diagrams that link three massive Wilson lines do not vanish in general. We show, however, that for two-to-two processes the two-loop anomalous dimension matrix is diagonal in the same color-exchange basis as the one-loop matrix for arbitrary masses at absolute threshold and for scattering at ninety degrees in the center of mass. This result is important for applications of thre…
Single-scale quantities, like the QCD anomalous dimensions and Wilson coefficients, obey difference equations. Therefore their analytic form can be determined from a finite number of moments. We demonstrate this in an explicit calculation by establishing and solving large scale recursions by means of computer algebra for the anomalous dimensions and Wilson coefficients in unpolarized deeply inelastic scattering from their Mellin moments to 3-loop order.
We examine diffractive proton-proton scattering p p -> p X and photo- and electroproduction of ρ^0 mesons γ^(*) p -> ρ^0 X, where X denotes a proton or a final state, into which the proton can go by diffractive dissociation. Using a functional integral approach we derive the scattering amplitudes, which are governed by the expectation values of light-like Wegner-Wilson loops, which are then evaluated using the model of the stochastic vacuum. For the proton, we assume a quark-diquark structure. From the scattering amplitudes we calculate total and differential cross sections for high centre of …
In deep-inelastic processes the heavy flavor Wilson coefficients factorize for $Q^2 \gg m^2$ into the light flavor Wilson coefficients of the corresponding process and the massive operator matrix elements (OMEs). We calculate the $O(α_s^2)$ and $O(α_s^3)$ massive OME for the flavor non-singlet transversity distribution. At $O(α_s^2)$ the OME is obtained for general values of the Mellin variable $N$, while at $O(α_s^3)$ the moments $N = 1$ to 13 are computed. The terms $\propto T_F$ of the 3--loop transversity anomalous dimension are obtained and results in the literature are confirmed. We disc…
Recent 3-Loop Heavy Flavor Corrections to Deep-Inelastic Scattering
本文报告了深度非弹性散射中重味贡献的三圈QCD修正及变味数方案下算符矩阵元的计算进展，并提出了一种无需计算$N$空间表达式即可获得逆Mellin变换的新方法。
The massive 3-loop fermion-loop corrections $\propto C_A N_f T_F^2$ and $C_F N_f T_F^2$ to the massive operator matrix elements $A_{Qg}$, $A_{Qq}^{\rm{PS}}$, $A_{qq,Q}^{\rm{PS}}$, $A_{qq,Q}^{\rm{NS}}$ and $A_{qq,Q}^{\rm{NS,TR}} have been obtained for general values of $N$. Thereby the corresponding contributions to the asymptotic heavy flavor Wilson coefficients of the structure function $F_2(x,Q^2)$ and of transversity in the region $Q^2 \geq 10 \cdot m^2$ are known. Our method is based on direct integration, avoiding the integration-by-parts technique, which is advantageous due to the compac…
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