The Complete $O(\alpha_s^2)$ Non-Singlet Heavy Flavor Corrections to the Structure Functions $g_{1,2}^{ep}(x,Q^2)$, $F_{1,2,L}^{ep}(x,Q^2)$, $F_{1,2,3}^{\nu(\bar{\nu})}(x,Q^2)$ and the Associated Sum Rules

We calculate analytically the flavor non-singlet $O(\alpha_s^2)$ massive Wilson coefficients for the inclusive neutral current non-singlet structure functions $F_{1,2,L}^{ep}(x,Q^2)$ and $g_{1,2}^{ep}(x,Q^2)$ and charged current non-singlet structure functions $F_{1,2,3}^{\nu(\bar{\nu})p}(x,Q^2)$, at general virtualities $Q^2$ in the deep-inelastic region. Numerical results are presented. We illustrate the transition from low to large virtualities for these observables, which may be contrasted to basic assumptions made in the so-called variable flavor number scheme. We also derive the corresponding results for the Adler sum rule, the unpolarized and polarized Bjorken sum rules and the Gross-Llewellyn Smith sum rule. There are no logarithmic corrections at large scales $Q^2$ and the effects of the power corrections due to the heavy quark mass are of the size of the known $O(\alpha_s^4)$ corrections in the case of the sum rules. The complete charm and bottom corrections are compared to the approach using asymptotic representations in the region $Q^2 \gg m_{c,b}^2$. We also study the target mass corrections to the above sum rules.


Introduction
Deep-inelastic scattering provides one of the most direct methods to measure the strong coupling constant from precision data on the scaling violations of the nucleon structure functions [1,2]. The present accuracy of these data also allows to measure the mass of the charm, cf. [3], and bottom quarks due to the heavy flavor contributions. The Wilson coefficients are known to 2-loop order in semi-analytic form [4][5][6] in the tagged-flavor case 2 , i.e. for the subset in which the hadronic final state contains at least one heavy quark, having been produced in the hard scattering process. The corresponding reduced cross section does not correspond to the notion of structure functions, since those are purely inclusive quantities and terms containing massless final states contribute as well. The heavy flavor contribution to inclusive deep-inelastic structure functions are described by five Wilson coefficients in the case of pure photon exchange [8][9][10]. In the asymptotic case Q 2 m 2 , where Q 2 = −q 2 denotes the virtuality of the exchanged gauge boson and m the mass of the heavy quark, analytic expressions for the Wilson coefficients have been calculated. A series of Mellin moments has been computed to 3-loop order in [10]. All logarithmic 3-loop corrections [11] as well as all N F terms are known [12,13]. Four out of five Wilson coefficients contributing to the unpolarized deep inelastic structure functions have been calculated to 3-loop order for general values of Mellin N [12,14,15] in the asymptotic region Q 2 m 2 . In the flavor non-singlet case also the asymptotic 3-loop contributions to the combinations of the polarized structure functions g NS 1 (2) [16] and the unpolarized charged current structure function xFν p 3 + xF νp 3 have been computed [17]. In the present paper, we calculate the complete 2-loop non-singlet heavy flavor corrections to the deep inelastic charged current structure functions F νp 1,2,3 and the neutral current structure functions F ep 1,2 and g ep 1 and a series of sum rules in the deep inelastic region, Q 2 > ∼ m 2 c . In the asymptotic case Q 2 m 2 the corresponding Wilson coefficients have been calculated in [11,[16][17][18] to O(α 2 s ) and in [14,16,17] to O(α 3 s ). Here the massless Wilson coefficients [19,20] to O(α 3 s ) enter. In the tagged flavor case the corresponding corrections to O(α 2 s ) have been calculated in [8,21] and in the asymptotic charged current case in [22] 3 .
The associated sum rules are the Adler sum rule [23], the unpolarized Bjorken sum rule [24], the polarized Bjorken sum rule [25], and the Gross-Llewellyn Smith sum rule [26]. A central observation in the inclusive case is that there are no logarithmic corrections for the associated sum rules at large Q 2 , which are present in the tagged flavor case [27,28], however. The complete massive O(α 2 s ) corrections to the structure functions improves the accuracy towards lower values of Q 2 . In the case of the sum rules, the corresponding contributions are found to be of the order of the known massless 4-loop corrections. We will also consider the target mass corrections to the sum rules, since they are relevant in the region of low Q 2 .
The paper is organized as follows. In Section 2 we present a general outline on the massive Wilson coefficients for the structure functions which will be considered. The O(α 2 s ) corrections to the polarized non-singlet neutral current structure functions g ep,NS 1 and g ep,NS 2 are derived in detail in Section 3 as an example. In Section 4 we discuss the corrections to the neutral current structure functions F ep,NS 1 (2) , and in Section 5 those to the non-singlet charged current structure functions F ν(ν)p, NS 1,2,3 . Detailed numerical results are presented for all the seven nonsinglet structure functions for experimental use. The heavy flavor O(α 2 s ) corrections and target mass corrections to the associated sum rules are computed in Section 6, comparing massless and massive effects and numerical results are presented for the target mass corrections. Section 7 contains the conclusions. The Appendices contain technical parts of the calculation.

The Wilson Coefficients
We consider the heavy flavor corrections to deep-inelastic structure functions, which are inclusive observables, i.e. the hadronic final state in the corresponding differential scattering cross sections is summed over completely. Under this condition the Kinoshita-Lee-Nauenberg theorem [29,30] is valid and no infrared singularities, which have to be eventually cured by arbitrary cuts, are present [10]. As we consider deep-inelastic scattering, both the scales Q 2 and W 2 = Q 2 (1 − x)/x + M 2 have to be large enough, to probe the interior of the nucleon. Here M denotes the nucleon mass, x = Q 2 /(Sy) is the Bjorken variable, with S = (p + l) 2 , and y = p.q/p.l the inelasticity, and p and l the incoming nucleon and lepton 4-momenta. One usually demands W 2 , Q 2 > ∼ 4 GeV 2 . To fully avoid the region of higher twist terms, a cut W 2 > ∼ 12.5 GeV 2 [31] is necessary.
The structure functions are then given by where F massless i (x, Q 2 ) is the fully massless part of the structure function and F massive i (x, Q 2 ) contains contributions due to a heavy quark mass m c or m b . Both quantities are inclusive. F massive i (x, Q 2 ) does not correspond to the so-called tagged flavor case, demanding a heavy quark in the hadronic final state. 4 In the asymptotic case Q 2 m 2 , the Wilson coefficients contributing to (2.1) were calculated for the non-singlet neutral current structure functions g NS 1,2 and F NS 1,2 and the non-singlet charged current structure function F NS 3 [16][17][18] to O(α 3 s ) (NNLO). The unpolarized and polarized neutral current non-singlet structure functions in the case of pure photon exchange are given by ⊗ ∆f k (x, µ 2 , N F ) + ∆fk(x, µ 2 , N F ) , (2.3) with i = 1, 2. Here N F is the number of active flavors, e k the electric charge of the massless quarks, and r 1 = 1 2 , r 2 = x; C NS q and ∆C NS q denote the corresponding massless Wilson coefficients and L NS q and ∆L NS q the massive ones, f k(k) and ∆f k(k) are the unpolarized (polarized) quark and anti-quark distribution functions, and µ 2 is the factorization scale. Here we follow again the convention used in [9], (2.26). The notion 'N F + 1' in (∆)L NS q means that the Wilson coefficient is calculated for N F massless and one massive flavor. 5 For the unpolarized charged current structure functions F 1,2,3 (x, Q 2 ) a second Wilson coefficient H W + −W − ,NS i,q contributes, which in the case of charm describes the flavor excitation in addition to or without heavy flavor pair production and possible virtual heavy quark corrections. This transition contributes already at tree level. Here θ c denotes the Cabibbo-angle [33]. The complete corrections to O(α s ) have been calculated in [34,35] 6 . At O(α 2 s ) the asymptotic heavy flavor corrections have been calculated in [18] and for xF νp 3 + xFν p 3 to O(α 3 s ) in Ref. [17]. Beyond the terms of O(α s ) we will use the results in the asymptotic case for the numerical illustrations given below. Note that the latter contributions are Cabibbo suppressed (2.5) Given the present experimental accuracy, this approximation is justified, leaving the full calculation for the future. For the transition (2.4) the momentum fraction of the massless quarks at tree-level changes, as well known, to because the corresponding Wilson coefficient is a δ-distribution. This is different at higher orders, where the Wilson coefficients are given by extended distributions, [34,35]. In the asymptotic region Q 2 m 2 the following representations hold for the Wilson coefficients with a s (µ 2 ) = α s (µ 2 )/(4π), A NS qq,Q the massive non-singlet operator matrix element (OME) [10,14] and C W + −W − ,NS F 3 ,q (N F ) the massless Wilson coefficient up to 3-loop order. Here we use the conventionf (2.10) In the following sections we calculate the Wilson coefficients (∆)L NS i,q to O(a 2 s ) in complete form in the deep-inelastic region. In Section 3 we present the main details of the calculation, which allows us to focus on the results in the other cases.

The polarized non-singlet structure functions
The polarized flavor non-singlet neutral current structure functions g NS 1,2 receive massless and massive QCD corrections, where the latter contribute starting at O(a 2 s ). In the following we will give a detailed discussion of the heavy flavor contributions to g NS 1 as an example. Main aspects of the calculation are given in Appendices A and B.
3.1 The structure function g NS 1 To O(a 2 s ) the non-singlet contribution for g 1 (x, Q 2 ) reads ∆d(x, µ 2 ) + ∆s(x, µ 2 ) . (3.1) Here ∆u v and ∆d v denote the polarized valence quark densities, ∆ū, ∆d and ∆d are the polarized sea quark distributions 7 , ⊗ denotes the Mellin convolution, and the massless Wilson coefficient is given by cf. e.g. [38]. Here P 0 qq is the leading order splitting function with the +-prescription being defined by 7 For a review on polarized deep-inelastic scattering, see [37].
The NLO non-singlet splitting functions P (1),NS,± qq (x) were calculated in [39] 8 , the quarkonic oneloop Wilson coefficient c (1) g 1 ,q for the structure function g 1 [40] is given by and c (2) g 1 ,q (z) has been calculated in Ref. [41]. The color factors are C A = N c , C F = (N 2 c − 1)/(2N c ), T F = 1/2 for SU (N c ) and N c = 3 in Quantum Chromodynamics. Here and in the following we set the factorization and renormalization scales both to µ.
The O(a 2 s ) Wilson coefficient ∆L NS g 1 ,q receives contributions from the Feynman diagrams shown in Figures 1 and 2. The diagrams of the Compton process, Figure 1, describe the real production of a heavy quark pair in the kinematic range z ≤ Q 2 /(Q 2 + 4m 2 ) of the parton momentum fraction, and contain no singularities, enabling their calculation in d = 4 dimensions. We obtain ∆L NS,(2),C In Appendix A the principal steps of the calculation of (3.9) are outlined. For this contribution to L NS g 1 ,q we agree with the result given in [21]. The inclusive scattering cross section, however, receives also contributions from the virtual corrections shown in Figure 2.  In the Bremsstrahlung corrections (a,b) to these diagrams, the heavy flavor correction is given by the one-loop polarization function Π QQ (k 2 = 0). The polarization insertion Π QQ (k 2 ) also appears in the virtual correction (c). For technical reasons we decompose Π QQ (k 2 ) = Π QQ (k 2 = 0) + [Π QQ (k 2 ) − Π QQ (k 2 = 0)] and combine the first term with the contributions due to (a,b). This yields the term ∆L NS,(2),massless with β 0,Q = −4T F /3. The term (3.11) corresponds to a heavy flavor contribution in the case of massless final states. There are also self-energy insertions contributing to (c), which, however, vanish for the term (3.11) since the corresponding graphs at 1-loop vanish and Π QQ (k 2 = 0) contributes multiplicatively. For the insertion Π QQ (k 2 = 0) this is not the case, cf. Appendix B.
The second term [Π QQ (k 2 ) − Π QQ (k 2 = 0)] is now used in the interference term calculating the form factor. The subtraction term allows to perform the calculation in d = 4 dimensions,  The massive Wilson coefficient is given by In the following we will use the values at NNLO in the on-shell scheme [3,42] for all numerical illustrations, both at O(a 2 s ) and O(a 3 s ), since we consider the present results as a part of our more general NNLO project, cf. [44], and would like to compare with numerical results given at O(a 3 s ) in [14][15][16][17]. The transformation to the MS scheme for the heavy quark masses is well-known [45]. In fitting the heavy quark masses from data one would use the corresponding formula. For the illustration given in the following, their equivalent value in the on-shell scheme has been used for brevity. In the numerical results given below, we choose for the factorization and renormalization scales µ 2 = Q 2 . In the calculation we used the codes HPLOG, CHAPLIN, HPL and HarmonicSums [46][47][48][49] at different steps. x (g NS 1 ) charm x complete, 1000 GeV 2 asymptotic, 1000 GeV 2 complete, 100 GeV 2 asymptotic, 100 GeV 2 massive 10 GeV 2 asymptotic 10 GeV 2  x (g NS 1 ) bottom x complete, 1000 GeV 2 asymptotic, 1000 GeV 2 complete, 100 GeV 2 asymptotic, 100 GeV 2 massive 10 GeV 2 asymptotic 10 GeV 2 In Figure 3 we illustrate the massless and massive contributions to the non-singlet structure function g NS 1 to O(a 2 s ) as a function of x and Q 2 using the parton distribution functions [43]. Due to the QCD evolution the peak of the function moves towards smaller values of x, keeping its valence-like profile. The contributions due to charm and bottom are illustrated in Figures 4 and 5. Here we also compare the asymptotic expressions with the complete results, which show differences for Q 2 ∼ 10 GeV 2 and become very close for Q 2 = 100 and 1000 GeV 2 for charm and at higher scales also for bottom.
The ratio of the heavy quark contributions to the complete structure function are illustrated in Figure 6. In the range of smaller values of x the fraction amounts to < +1.2%, while at larger values of x the corrections become negative amounting to −3. The asymptotic 3-loop corrections [16] at Q 2 = 1000 GeV 2 are even larger and contribute to O(2%) at lower values of x and amount to O(−6%) at large x.
Here and in the following we often will make the observation that the asymptotic expressions tend to agree better in the region of small x even at lower values of Q 2 , where this is not expected a priori. The reason for this is that the relevant effective scale, inside the corresponding integrals, is the hadronic mass squared W 2 , rather than Q 2 itself. 3.2 The structure function g NS 2 At leading twist, the structure function g 2 (x, Q 2 ) is obtained through the Wandzura-Wilczek relation Here g 2 (x, Q 2 ) denotes the non-singlet distribution, calculated using g 1 (x, Q 2 ) ≡ g NS 1 (x, Q 2 ), Eq. (3.1). The Wandzura-Wilczek relation has been derived for massless quarks in [50], see also [51,52], but possesses a much wider validity as has been shown in later years. It also holds for scattering off massive quarks [53] and for the target mass corrections [53,54], as well as for non-forward [55][56][57] and diffractive scattering [58,59] and heavy flavor production in photongluon fusion [60]. At leading twist the structure functions g 1 and g 2 are connected by an operator relation, cf. [55]. Representations in the covariant parton model were given in Refs. [51,[60][61][62].  x complete, 1000 GeV 2 asymptotic, 1000 GeV 2 complete, 100 GeV 2 asymptotic, 100 GeV 2 massive 10 GeV 2 asymptotic 10 GeV 2 In Figure 7 we illustrate the flavor non-singlet contribution at twist 2 to the structure function xg 2 (x, Q 2 ) for pure photon exchange up to O(α 2 s ). It takes values in the range +0.01 to −0.03, with only mild scaling violations varying Q 2 from 10 GeV 2 to 1000 GeV 2 . In Figures 8 and 9 we illustrate the heavy flavor corrections due to charm and bottom, respectively. The effect is of O(1%) in the case of charm. We also compare the exact results with those using the asymptotic x complete, 1000 GeV 2 asymptotic, 1000 GeV 2 complete, 100 GeV 2 asymptotic, 100 GeV 2 massive 10 GeV 2 asymptotic 10 GeV 2  x asymptotic O(α 3 s ), 1000 GeV 2 complete, 1000 GeV 2 asymptotic, 1000 GeV 2 complete, 100 GeV 2 asymptotic, 100 GeV 2 complete, 10 GeV 2 asymptotic, 10 GeV 2 representation, in which the power corrections are disregarded, cf. [16]. The effect is clearly visible at lower scales, and fully disappears at Q 2 ∼ 100 GeV 2 in the case of charm. In Figure 10 we illustrate the combined heavy flavor effect and also show the asymptotic 3-loop corrections, which turn out to be larger than the exact corrections.
The structure of the Wandzura-Wilczek relation implies that the associated sum rule for the first moment yields zero. However, this is not a prediction which derives from the light-cone expansion [52], since the corresponding moment does not contribute to it as a term. Rather the Wandzura-Wilczek relation, as an analytic continuation, is compatible with the result, which is also called (flavor non-singlet) Burkhardt-Cottingham sum rule [63]. It results from the fact that the imaginary part of g 2 ( q 2 , q 0 ) obeys a superconvergence relation. Unlike a series of other sum rules, it cannot be expressed as an expectation value of (axial)vector operators [64]. 4 The unpolarized non-singlet structure functions F NS 1,2 In the case of pure photon exchange, the unpolarized neutral current scattering cross section is parameterized by two deep-inelastic structure functions F 1,2 (x, Q 2 ) which obey in the absence of target mass corrections [65]. Here F L (x, Q 2 ) denotes the longitudinal structure function. In the following we will refer to the structure functions F 2 and F L . The calculation proceeds in a similar way to that outlined in Section 3.

L
In the case of the structure function F L , the Compton contribution is given by and we confirm the result given in [8]. The virtual contribution vanishes and the contribution corresponding to massless final states reads with [66] c (1) for large values of ξ leads toĈ (2) F L ,q , the corresponding massless 2-loop Wilson coefficient [67,68], as predicted by renormalization in Ref. [10,11] with no logarithmic term ∼ ln(ξ) left, unlike the case where we take just the term L NS,(2),C q,L into account, cf. [8,69], The non-singlet structure function for F L reads where u v and d v are the unpolarized valence quark densities andū andd the sea quark densities,   Figure 11. Dashed lines: asymptotic representation in Q 2 for the heavy flavor corrections; full lines: complete heavy flavor contributions. and the massless Wilson coefficient is given by (4.10) In Figure 11 we show the O(a 2 s ) corrections to the non-singlet structure function F NS L , including the complete charm and bottom quark corrections. During evolution this structure function grows towards small values of x. The absolute charm and bottom quark contributions are illustrated in Figures 12,13. In the present case, the corrections in the asymptotic limit are sufficiently close to the complete corrections only for Q 2 > ∼ 1000 GeV 2 in the case of charm. It is well known that for F L the asymptotic representation holds at very high scales only, which also applies to the non-singlet case. For the charm quark corrections the asymptotic representation holds at Q 2 ∼ 1000 GeV 2 . Below there are significant differences. The situation is correspondingly worse for the bottom quark corrections shown in Figure 13. In general the asymptotic corrections give larger negative corrections than found in the complete calculation. The relative heavy flavor corrections for F NS L are shown in Figure 14. They behave nearly constant in the small x region, amounting to −0.3 to −4% in the region Q 2 = 10 to 1000 GeV 2 , with larger asymptotic corrections.

F NS 2
For the structure function F 2 , we obtain the following Compton contribution This expression agrees with a result given in [8]. The virtual correction is the same as in the case of the structure function g NS 1 , Eq. (3.12), and the contribution with massless final states is given by: with [71] c (1) Up to O(a 2 s ) the non-singlet structure function F 2 (x, Q 2 ) reads 14) and the massless Wilson coefficient is given by with [38] C (1)  The bottom quark contributions shown in Figure 17 are about one order of magnitude smaller than those for charm quarks, still with clear differences between the exact and asymptotic result at Q 2 ∼ 100 GeV 2 . Figure 18 illustrates the relative contribution of the heavy flavor corrections up to O(a 2 s ). The corrections are rather flat in the small x region and amount to −0.1 to −0.6 % for x < 0.1 growing towards −2.5 % at large x from Q 2 = 10 GeV 2 to 1000 GeV 2 .  The corresponding flavor non-singlet combinations are given by Here, C NS F i ,q , L NS F i ,q , and H NS F i ,q denote the massless (C) and massive Wilson coefficients (L, H) for the coupling of the weak bosons to only massless quarks (C, L) and for charm excitation (H). We assume that the sea quark distributions obey The contributions due to the Wilson coefficients H NS F i ,q , i = 1, 2, 3 are Cabbibo suppressed. The combinations (5.2-5.4) are related to the unpolarized Bjorken sum rule [24], the Adler sum rule [23], and the Gross-Llewellyn Smith sum rule [26], respectively, by their first moments. First we consider these combinations themselves and turn to the sum rules later. Up to O(α s ) the single heavy quark excitations have been calculated in Refs. [34,35] correcting results in [36]. At two-loop order, only the asymptotic results for Q 2 m 2 are available [18] 9 , to which we refer in the following. We will limit our considerations to the case of the charm contributions.
In Figure 19 we illustrate the non-singlet structure function xF W + −W −    Figure 19.     Figure 19.   Here we also compare the asymptotic result against the complete ones. The charm contribution is found in the range of 0 to ∼ −12% at Q 2 = 4 GeV 2 to 0 to ∼ −8%, at Q 2 = 100 GeV 2 peaking around x ∼ 0.03.  In Figure 27 the charged current structure function xF W + +W − 3 including the charm quark corrections are shown. In this case also the asymptotic 3-loop corrections have been calculated [17]. As shown in Figures 28-30

The Sum Rules
In the following we discuss the corrections to the Adler sum rule [23], which have to vanish, and calculate the corrections to the polarized Bjorken sum rule [25], the unpolarized Bjorken sum rule [24], and the Gross-Llewellyn Smith sum rule [26], which are obtained as the first moments of the massive Wilson coefficients calculated in the previous sections. The combination of the parton distributions is partly different, as here differences between structure functions in the neutral current case are considered. But this affects only the normalization factor of the sum rules, which are known constants. As has been outlined in Refs. [16,17] up to 3-loop order, in the asymptotic region Q 2 m 2 the sum rules only modify the massless approximation by replacing the number of massless flavors from N F → N F + 1. Given the factorization of the massive Wilson coefficients [8,9], this holds for all orders in the coupling constant, since the first moment of the massive non-singlet OMEs vanish order by order in the coupling constant due to fermion number conservation. The 4-loop corrections to these sum rules have been calculated in Refs. [72][73][74]. Earlier Padé estimates were given in [75].
We emphasize that in the present paper the inclusive Wilson coefficients are calculated for deep-inelastic scattering, but not those in the flavor tagged case. The relations obtained do not smoothly transform into the photo-production limit Q 2 ≈ 0, both for the Wilson coefficients and the parton distribution functions, setting µ 2 = Q 2 . They are valid only up to a lower scale Q 2 0 , which usually should be at least of O(m 2 c ) or larger, also to stay outside the region of higher twist corrections. In the case of the sum rules discussed below, in the limit Q 2 /m 2 → 0 logarithmic contributions survive, while this is the not the case in the limit of large virtualities m 2 /Q 2 → 0. The photo-production region for the corresponding structure functions needs a separate treatment.
In the following we will discuss the complete massive corrections to the four sum rules in the deep-inelastic region. The power corrections of a single heavy quark c or b will be shown to basically interpolate between N F and N F + 1 massless flavors in the limit m 2 /Q 2 → 0, while at lower scales Q 2 , partly negative virtual corrections are possible. The sum rules are observables and we represent them choosing the factorization scale µ 2 = Q 2 . The scale matching can be performed analytically in Mellin space up to the respective order in a s in which the quantity is calculated, cf. [31].
To get closer to the unitary representation for the CKM matrix elements, we calculate the functions H F i ,q , (2.8, 2.9), allowing for massive charm quarks to be pair produced also for this Cabbibo suppressed term, but referring to massless s → c charged current transitions for the real and virtual corrections.
Finally, we also consider the target mass corrections to the deep-inelastic sum rules, as they are of relevance in the region of lower values of Q 2 .

The Adler sum rule
The Adler sum rule [23] states for three massless flavors. Here θ c denotes the Cabibbo angle [33]. The integral (6.1) neither receives QCD nor quark-or target mass corrections, cf. also [64,76]. The Compton contribution yields also vanishes, cf. (4.13, 3.6). For the charged current flavor excitation slow rescaling at tree level yields since the support of F 2 (z, Q 2 ) is z ∈ [0, 1]. For the first order massive QCD corrections given in [34,35] the first moment (6.1) vanishes. The corresponding O(a 2 s ) corrections have only been studied in the asymptotic case [18] and vanish. For massless quarks, the Adler sum rule has been checked at O(α 3 s ) in [77]. It seems that in the case of massless 4-loop corrections, the validity of the sum rule has not yet been checked perturbatively [78]. The target mass corrections are studied in Section 6.5.
In contrast, the QCD-, quark mass-and target mass corrections to the first moments of the structure functions g 1 , F 1 and F 3 do not vanish.

The polarized Bjorken sum rule
The polarized Bjorken sum rule [25] refers to the first moment of the flavor non-singlet combination  [74] in the massless case are given by choosing the renormalization scale µ 2 = Q 2 , cf. [41] for SU (3) c . Here N F denotes the number of active light flavors and the labels NS and SI refer to the genuine 'non-singlet' and 'singlet' contributions, respectively. The expression for general color factors was given in Ref. [74,82]. 10 The massless corrections for N F = 3 and N F = 4 are For the asymptotic massive corrections (2.7-2.8) only the first moments of the massless Wilson coefficientsĈ (2,3),NS g 1 ,q (N F ) contribute, since the first moments of the massive non-singlet OMEs vanish due to fermion number conservation, a property holding even at higher order. Therefore, any new heavy quark changes Eq. (6.7) by a shift in N F → N F + 1 only, for the asymptotic corrections.
We turn now to the heavy quark corrections, which are given by At low scales the corrections are negative and the interpolation to the asymptotic value 2 for N F → N F + 2 in ξ c proceeds very slowly. In Table 1 we illustrate the mass effects for the 2-loop terms. The massless prediction is only reached for considerably large values of Q 2 , namely for ξ c ∼ 24 in the case of N F = 4 and ξ c 500 for N F = 5.
0.9180 0.9205 -0.0025 -0.0008 100 0.9321 0.9335 -0.0014 -0.0011 10000 0.9587 0.9590 -0.0003 -0.0008 Table 2: In Table 2 we compare the values of the polarized Bjorken sum rule for different values of Q 2 to illustrate the effect of the heavy flavor contribution. The massive contribution turns out to be comparable in size to the massless 4-loop contribution. Due to surviving logarithms in ξ in the large ξ region in the tagged flavor case, different results are obtained [27,28]. However, the corresponding quantity does not describe the heavy flavor contributions to the structure functions, which are inclusive quantities.

The unpolarized Bjorken sum rule
The unpolarized Bjorken sum rule [24] is given by (ξ) approaches the asymptotic value of −2/3 given in (6.14). Its behaviour as a function of ξ c is shown in Figure 32. The massive 2-loop corrections are given by In the region ξ 1 one obtains (6.20) In Figure 33, C massive,(2) uBJ (ξ) is shown as a function of ξ. To O(â 2 s ) the unpolarized Bjorken sum rule reads In Table 3 we compare the values of the unpolarized Bjorken sum rule for different values of Q 2 to illustrate the effect of the heavy flavor contribution.

The Gross-Llewellyn Smith sum rule
The Gross-Llewellyn Smith sum rule [26] refers to the first moment of the flavor non-singlet combination  [73,74] in the massless case are given by choosing the renormalization scale µ 2 = Q 2 for SU (3) c . The expression for general color factors was given in Ref. [73,74]. Note that the QCD corrections to the Gross-Llewellyn Smith sum rule and to the polarized Bjorken sum rule [25] are identical up to O(â 2 s ). The excitation of charm basically interpolates between In Table 4   As in the case of the other sum rules, the charm corrections at O(â 2 s ) turn out to be of the same size as the massless O(â 4 s ) corrections.

The Target Mass Corrections to The Sum Rules
For the target mass corrections it has been shown [65] that the correction factor to the massless structure function F 2 (N, Q 2 ) in Mellin space is given by with M the nucleon mass, (∆)a (2) N +2j the (non-perturbative) moments of the massless PDFs and C 2 the moments of the Wilson coefficient contributing to F 2 . Here we consider the flavor-non singlet contribution (Fν p 2 − F νp 2 )/x which is relevant for the Adler-sum rule. Note that the first moment of C 2 , except for the tree-level contribution, vanishes, as has been proven to 3-loop order for the massless and massive Wilson coefficients (in the asymptotic region) by explicit calculations [14,71,77,88] and above for the massive contributions to the complete corrections at 2-loop order. One obtains In contrast, the first moments of the structure functions F 1 and F 3 do not vanish at higher orders in QCD both in the massless and massive cases [1,89]. Moreover, both in the unpolarized [53,64,89] and in the polarized cases, the target mass corrections are different for different structure functions, which are usually associated to other ones by current conservation, as in the case of F 4 (x, Q 2 ) and F 5 (x, Q 2 ). In the case of the unpolarized Bjorken sum rule, the target mass correction factor is given by [1,89] Here C (k) 1 denotes the kth moment of the Wilson coefficient contributing to the structure function Fν p 1 − F νp 1 . The target mass corrections to the polarized Bjorken sum rule are given by [53,54] (6.36) where ∆C 1 is the polarized flavor-non singlet Wilson coefficient corresponding to the structure function g ep 1 − g en 1 . For the target mass corrections to the Gross-Llewellyn Smith sum rule one obtains [1,89] and C k 3 are the moments of the Wilson coefficient contributing to the flavor non-singlet combination Fν p 3 + F νp 3 . In Figure 35 we illustrate the effect of the target mass corrections to the unpolarized and polarized Bjorken sum rule as well as the Gross-Llewellyn Smith sum rule, as a function of Q 2 /M 2 , accounting only for the operator matrix elements (∆)a (1,3) k . We refer to the unpolarized PDFs [70] at NNLO and polarized PDFs [43] at NLO, and α s at NNLO, to allow for a comparison of the different contributions up to NNLO.  Figure 35: The target mass corrections, normalized to the massless case, to the unpolarized (F 1 ) and polarized Bjorken sum rule (g 1 ) and the Gross-Llewellyn Smith sum tule (F 3 ).
The corrections diminish towards large virtualities Q 2 . At Q 2 ∼ 1 GeV 2 they amount +2.3%, 0.60 % and 0.33% for the unpolarized Bjorken sum rule, the Gross-Llewellyn Smith sum rule and the polarized Bjorken sum rule, respectively.

Conclusions
We have calculated the complete heavy flavor corrections to the flavor non-singlet deep-inelastic structure functions F 1,2 and g 1,2 in the neutral current case, and to F W + −W − 1,2 and F W + +W − 3 for charged current reactions. Here we considered the deep-inelastic region, which at least requests scales Q 2 > ∼ m 2 c or larger and W 2 > 4 GeV 2 . For the charged current non-singlet combinations of structure functions also the Cabibbo suppressed Wilson coefficient H i,q contributes, which we have considered in the asymptotic region starting at O(a 2 s ) as an approximation. Since the deep-inelastic structure functions are inclusive observables, the formerly considered tagged flavor case [8,21] is not sufficient. We have accomplished the calculation for the inclusive case, to which at O(a 2 s ) also virtual corrections and real corrections with massless final states, containing massive virtual corrections, contribute. We present detailed numerical results for the different unpolarized and polarized structure functions for the charm and bottom contribution in the neutral current case and the charm contributions in the charged current case, which are the most important. We compared in all cases to the formerly calculated asymptotic corrections in the region Q 2 m 2 , showing that except for the structure function F NS 2 (x, Q 2 ) this approximation holds only at higher scales, while for F nc,NS 2 (x, Q 2 ) a very good agreement for Q 2 25GeV 2 is obtained in the case of charm. In those cases in which the asymptotic 3-loops corrections are available, we have partly compared to these corrections as well. The O(a 2 s ) non-singlet heavy flavor effects are of the order of several per cent of the whole non-singlet structure function and are of relevance in precision measurements reaching this accuracy. The corrections will become even more important in the case of planned high-luminosity measurements at facilities like the EIC [90], future neutrino factories [91] or the LHeC [92].
We also investigated the heavy flavor corrections for deep-inelastic scattering sum rules, such as the Adler, polarized Bjorken, unpolarized Bjorken and Gross-Llewellyn Smith sum rule. While the corrections vanish in case of the Adler sum rule, finite corrections are obtained to the other three sum rules. They turn out to be of the same size as the massless O(a 4 s ) corrections which have been calculated recently and complete the picture from the side of the heavy quarks. Here it is important to refer to the inclusive rather than to the tagged heavy flavor case, since in the latter, logarithmic terms in the region of larger Q 2 would remain, after having already performed the renormalization completely (e.g. in the MS scheme) [10]. In the inclusive case, on the other hand, the transition from N F → N F + 1 proceeds smoothly. We also quantified the effect of target mass corrections to the deep inelastic sum rules. In general it turns out that for the sum rules the transition N F → N F + 1 proceeds slowly in ξ = Q 2 /m 2 . Therefore assuming scale matching at Q 2 = m 2 is, at least here, not appropriate.
This term can finally be integrated analytically to yield (3.9).

B The virtual corrections
The interaction of an on-shell fermion and the electromagnetic current is parameterized in terms of the Dirac and Pauli form factors F 1 (q 2 ), F 2 (q 2 ), respectively. In the space-like case one has where σ µν = i 2 γ µ , γ ν . The correction can be obtained by the subtracted dispersion relation for the Dirac form factor.
We will first perform the calculation in the time-like case and obtain then the space-like result by analytic continuation. One has which can be calculated from the diagrams in Figures 36, 37 and applying the Ward identity of Subsection B.1, with s = (p 1 + p 2 ) 2 .  Note that the integral (B.2) in the case of the unsubtracted dispersion relation diverges. The calculation proceeds in a similar way as in Refs. [93,94]. We will initially work in d dimensions and use quarks of mass m 0 for the external particles and m for the heavy quark in the loop. Later on we take the limit d = 4 and m 0 = 0. The Dirac form factor is projected to The contribution of the diagram in Figure 36 to the vertex Γ µ is given by where Π(k 2 ) denotes the vacuum polarization Its imaginary part is found putting the propagators on shell, cf. [93,94]. Note that when taking the cut a minus sign has to be introduced [95,96] 2Im The longitudinal parts of the photon polarization can be shown to vanish. One finally obtains Im F 1 (s) = α 2 with (p 2 − p 1 ) 2 = q 2 . In the limit of zero momentum transfer, q 2 → 0, the vertex function becomes Λ µ (0) = F 1 (0)γ µ . By comparing (B.23) and (B.22) one gets i.e.  In conclusion, the scattering amplitude of a massless quark and an off shell photon with momentum q is given by the sum of the virtual correction to the proper qq-gauge boson vertex, depicted in Figure 36, which we computed via dispersion relations up to an offset F 1 (0), and the self energies contributions above, which proves that the results of the subtracted dispersive approach Eqs. (B.15, B.2) give the complete renormalized form factor.