The Gluonic Operator Matrix Elements at O(α_s^2) for DIS Heavy Flavor Production
aa r X i v : . [ h e p - ph ] J a n DESY 08-187SFB/CPP–08–107IFIC/08–68December 2008
The Gluonic Operator Matrix Elementsat O ( α s ) for DIS Heavy Flavor Production Isabella Bierenbaum , Johannes Bl¨umlein and Sebastian Klein Deutsches Elektronen–Synchrotron, DESY,Platanenallee 6, D–15738 Zeuthen, Germany
Abstract
We calculate the O ( α s ) gluonic operator matrix elements for the twist–2 operators, whichcontribute to the heavy flavor Wilson coefficients in unpolarized deeply inelastic scatteringin the region Q ≫ m , up to the linear terms in the dimensional parameter ε , ( D = 4 + ε ).These quantities are required for the description of parton distribution functions in thevariable flavor number scheme (VFNS). The O ( α s ε ) terms contribute at the level of the O ( α s ) corrections through renormalization. We also comment on additional terms, whichhave to be considered in the fixed (FFNV) and variable flavor number scheme, adoptingthe MS scheme for the running coupling constant. Present address: Instituto de Fisica Corpuscular, CSIC-Universitat de Val`encia, Apartado de Correros 22085,E-46071 Valencia, Spain.
Introduction
Both unpolarized and polarized deep-inelastic structure functions receive contributions from lightpartons and heavy quarks. In the unpolarized case, the charm quark contribution may amountto 25-35% in the small x region, [1]. Since the scaling violations in case of the heavy quarkcontributions differ significantly from those of the light partons in a rather wide range startingfrom lower values of Q , a detailed description of the heavy quark contributions is required. Inthe FFNS the corresponding Wilson coefficients were calculated to next-to-leading order (NLO)in a semi-analytic approach in [2]. Consistent QCD analyzes to 3-loop order require thedescription of both the light and the heavy flavor contributions at this level to allow for anaccurate measurement of the QCD scale Λ
QCD in singlet analyzes [4] and the measurement ofthe parton distribution functions. The calculation of the 3-loop heavy flavor Wilson coefficientsin the whole Q region is currently not within reach. However, as noticed in [6], a very precisedescription of the heavy flavor Wilson coefficients contributing to the structure function F ( x, Q )is obtained for Q > ∼ m Q , disregarding the power corrections ∝ ( m Q /Q ) k , k ≥
1, which coversthe main region for deep–inelastic physics at HERA. In this case, the Wilson coefficients areeven obtained in analytic form. The heavy flavor Wilson coefficients factorize into universalmassive operator matrix elements (OMEs) A ij ( µ /m Q ) and the light flavor Wilson coefficients c j ( Q /µ ) [7] in this limit, H i ( Q /µ , m Q /µ , z ) = A ji ( m Q /µ , z ) ⊗ c j ( Q /µ , z ) , i = 2 , L . (1)Here, µ denotes the factorization scale and z the longitudinal momentum fraction of the partonin the hadron.In the strict sense, only massless particles can be interpreted as partons in hard scatteringprocesses since the lifetime of these quantum-fluctuations off the hadronic background τ life ∝ / ( k ⊥ + m Q ) has to be large against the interaction time τ int ∝ /Q in the infinite momentumframe, [8]. In the massive case, τ life is necessarily finite and there exists a larger scale Q belowwhich any partonic description fails. From this it follows, that the heavy quark effects aregenuinely described by the process dependent Wilson coefficients. Since parton-densities are process independent quantities, only those pieces out of the Wilson coefficients can be used todefine them for heavy quarks at all. Clearly this is impossible in the region close to threshold butrequires Q /m Q = r ≫
1, with r > ∼
10 in case of F ( x, Q ). For F L ( x, Q ) the corresponding ratioeven turns out to be r > ∼ µ ≫ m Q . Their use in observables is restricted to a region, in which the powercorrections can be safely neglected. This range may strongly depend on the observable consideredas the examples of F and F L show.For processes in the high p ⊥ region at the LHC, in which the above conditions are fulfilled, onemay use heavy flavor parton distributions by proceeding as follows. In the region Q > ∼ m Q theheavy flavor contributions to the F ( x, Q )–world data are very well described by the asymptoticrepresentation in the FFNS. For large scales one can then form a variable flavor representationincluding one heavy flavor distribution, [11]. This process can be iterated towards the nextheavier flavor, provided the universal representation holds and all power corrections can be safelyneglected. One has to take special care of the fact, that the matching scale in the coupling For a fast implementation of these corrections in Mellin space see [3]. For a determination of Λ
QCD effectively analyzing the scaling violations of the non-singlet world data to O ( a s ), ( a s = α s / (4 π )), cf. Ref. [5]. N f → N f +1 is to be performed, often differs rather significantlyfrom m Q , cf. [12],For the procedure outlined above, besides the quarkonic heavy flavor OMEs [6,13], the gluonicmatrix elements are required. These have been calculated to O ( a s ) in Ref. [11]. Here we verifythis calculation and extend it to the terms of O ( a s ε ), which enter the O ( a s ) matrix elementsthrough renormalization. The corresponding contributions for the quarkonic matrix elementswere calculated in [14]. The paper is organized as follows. In Section 2 we summarize the relations needed to describeheavy flavor parton densities out of parton densities of only light flavors in terms of massiveoperator matrix elements. Furthermore, we point out terms to be added to the FFNS descriptionin the MS scheme if compared to [2,6], which are of numerical relevance, cf. [16]. We also commenton the question of the effective number of flavors considering the renormalization of the process.In Section 3 the massive gluonic 2–loop operator matrix elements are presented and Section 4contains the conclusions.
In the asymptotic region Q ≫ m Q one may define heavy flavor parton densities. This is doneunder the further assumption that for the other heavy flavors the masses m Q i form a hierarchy m Q ≪ m Q ≪ etc . Allowing for one heavy quark of mass m Q and N f light quarks one obtainsthe following light and heavy-quark parton distribution functions in Mellin space, [11], f k ( N f + 1 , µ , N ) + f k ( N f + 1 , µ , N ) = A NS qq,Q N f , µ m Q , N ! · (cid:2) f k ( N f , µ , N ) + f k ( N f , µ , N ) (cid:3) + ˜ A PS qq,Q N f , µ m Q , N ! · Σ( N f , µ , N )+ ˜ A S qg,Q N f , µ m Q , N ! · G ( N f , µ , N ) , (2) f Q ( N f + 1 , µ , N ) + f Q ( N f + 1 , µ , N ) = A PS Qq N f , µ m Q , N ! · Σ( N f , µ , N )+ A S Qg N f , µ m Q , N ! · G ( N f , µ , N ) . (3)Here f k ( f ¯ k ) denote the light quark and anti–quark densities, f Q ( f ¯ Q ) the heavy quark densities,and G the gluon density. The flavor singlet, non-singlet and gluon densities for ( N f + 1) flavorsare given by Σ( N f + 1 , µ , N ) = " A NS qq,Q N f , µ m Q , N ! + N f ˜ A PS qq,Q N f , µ m Q , N ! + A PS Qq N f , µ m Q , N ! · Σ( N f , µ , N ) For the first few values of the Mellin moment N the pure-singlet and non-singlet quarkonic OMEs werecalculated to O ( a s ) in Refs. [15]. " N f ˜ A S qg,Q N f , µ m Q , N ! + A S Qg N f , µ m Q , N ! · G ( N f , µ , N ) (4)∆( N f + 1 , µ , N ) = f k ( N f + 1 , µ , N ) + f k ( N f + 1 , µ , N ) − N f + 1 Σ( N f + 1 , µ , N ) (5) G ( N f + 1 , µ , N ) = A S gq,Q N f , µ m Q , N ! · Σ( N f , µ , N )+ A S gg,Q N f , µ m Q , N ! · G ( N f , µ , N ) . (6)Here, A NS(PS , S) ij = h j | O NS(PS , S) i | j i = δ ij + ∞ X l =1 a ls A ( l ) , NS(PS , S) ij (7)are the operator matrix elements of the local twist–2 non-singlet (NS), pure singlet (PS) andsinglet (S) operators O NS(PS , S) j between on–shell partonic states | j i , j = q, g and A ij = N f ˜ A ij . (8)Note that in the pure-singlet case the term δ ij in (7) is absent. The normalization of thequarkonic and gluonic operators obtained in the light cone expansion can be chosen arbitrarily.It is, however, convenient to chose the relative factor such, that the non-perturbative nucleon-state expectation values, Σ( N f , µ , N ) and G ( N f , µ , N ), obeyΣ( N f , µ , N = 2) + G ( N f , µ , N = 2) = 1 (9)due to 4-momentum conservation. As a consequence, the OMEs fulfill the relations A NS qq,Q ( N f , N = 2) + N f ˜ A PS qq,Q ( N f , N = 2) + ˜ A PS Qq ( N f , N = 2) + A S gq,Q ( N f , N = 2) = 1 , (10) N f ˜ A S qg,Q ( N f , N = 2) + ˜ A S Qg ( N f , N = 2) + A S gg,Q ( N f , N = 2) = 1 . (11)The above scenario can be easily followed up to 2-loop order. Also here diagrams contributewhich carry two different heavy quark flavors. At this level, the heavy degree of freedom maybe absorbed into the coupling constant and thus being decoupled temporarily. Beginning with3-loop order the situation becomes more involved since there are graphs in which two differentheavy quark flavors occur in nested topologies, i.e. the corresponding diagrams depend on theratio ρ = m c /m b yielding power corrections in ρ . There is no strong hierarchy between thesetwo masses. The above picture, leading to heavy flavor parton distributions whenever Q ≫ m Q will not hold anymore, since one cannot decide immediately in case of the two-flavor graphs,whether they belong to the c – or the b –quark distribution. Hence, the partonic description canonly be maintained within a certain approximation by assuming ρ ≪ O ( a s ) was performed in a scheme, in which the heavy quark insertion in the external gluonlegs were absorbed into the strong coupling constant through the relationˆ a s = a s ( µ ) (cid:2) a s ( µ ) δa s ( µ , m Q ) (cid:3) (12) δa s ( µ , m Q ) = S ε ( β ( N f ) ε + 2 β ,Q ε (cid:18) m Q µ (cid:19) ε/ (cid:18) ζ ε + ζ ε (cid:19)) , (13) In the present case, these are the terms ∝ T F . N f light and one heavy flavor. Here ˆ a s denotes the bare coupling constant ˆ a s = ˆ g s / (16 π ), β ( N f ) = (11 / C A − (4 / T F N f , β ,Q = − (4 / T F , C A = 3 , T F = 1 / SU (3) c . The sphericalfactor S ε = exp[( ε/ γ E − ln(4 π )] is set to one in the MS scheme. The bare coupling constantis thus given byˆ a s = a s ( µ ) (cid:20) a s ( µ ) 2 β ( N f + 1) ε (cid:21) − a s ( µ ) β ,Q ln µ m Q ! + O ( ε ) . (14)If the above scheme is applied, cf. [6], the renormalized OMEs do not contain terms ∝ T F .However, to express a s in the MS scheme, only the first term in Eq. (14) has to be used, whilethe second remains as a prefactor of the 1–loop contributions. Hence, as has been lined outin [11] later, the latter term appears e.g. in A (2) Qg in front of A (1) Qg in the case Q ≫ m Q . This isencountered as well in the present paper for A (2) gg,Q . Additionally, one has to do the same for thecomplete heavy flavor Wilson coefficients, leading to the extra terms a s ( µ ) β ,Q ln µ m Q ! H (1) F i (cid:18) m Q µ , z (cid:19) (15)in the scattering cross section in the MS scheme. Here, H (1) F (cid:18) m Q Q , z (cid:19) = 8 T F (cid:26) β (cid:20) −
12 + 4 z (1 − z ) + m Q Q z (2 z − (cid:21) + (cid:20) −
12 + z − z + 2 m Q Q z (3 z − − m Q z (cid:21) ln (cid:18) − β β (cid:19)(cid:27) , (16) H (1) F L (cid:18) m Q Q , z (cid:19) = 16 T F (cid:20) z (1 − z ) β − m Q Q z ln (cid:12)(cid:12)(cid:12)(cid:12) β − β (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (17) β = s − m Q zQ (1 − z ) , (18)denote the leading order Wilson coefficients for massive quarks with the strong coupling constanttaken out. In the same manner the contributions ∝ T F in the non-1PI-contribution in A Qg ,Ref. [11], have to be removed, to avoid double counting if the asymptotic representation forthe heavy flavor Wilson coefficients is referred to. Since the light flavor Wilson coefficients arecalculated in the MS scheme, the same scheme has to be used for the massive OMEs. It shouldalso be thoroughly used for renormalization, as the case for light flavors, to derive consistentresults in QCD analyzes of deep-inelastic scattering data.In Refs. [2, 6, 11] another contribution, which belongs to the inclusive heavy flavor contribu-tions to the structure functions F ,L ( x, Q ), was not dealt with. To O ( a s ) these are heavy quarkloop insertions on the initial state gluon line for the 1st order light flavor Wilson coefficient c (1)2( L ) ,g ( x, Q ). The corresponding contribution is a s ( µ ) β ,Q ln µ m Q ! c (1) F i ,g ( z ) , (19) The running coupling constant including heavy flavors in the MOM–scheme was presented in [17] to O ( a s )recently, showing Applequist–Carrazone [18] decoupling of the heavy quark contributions. A (2) , NS qq,Q , resp. H (2) , NS2 ,L , and A (2) , S gg,Q need to beaccounted for. In the asymptotic case Q ≫ m Q they lead to +-functions, which regularize thesoft singularity, cf. [6, 11, 13, 14]. Here we always considered only one heavy quark contribution.The above expressions are derived for the FFNS. Charge– and mass–renormalization areperformed multiplicatively for the observables. As evident from Eq. (13), a s ( µ ) has to becalculated for N f + 1 flavors upon passing the N f + 1st flavor threshold. In the FFNS thestructure functions contain separate contributions of the strictly light and heavy flavors. Thecorresponding expressions for the Wilson coefficient contain anomalous dimensions which partlydepend on N f . In the case of the heavy flavor contributions to O ( a s ), [6, 11, 13, 14], (2–6), noclosed light fermion lines contribute, however. The evolution of the three light flavors proceedswith N f = 3. Due to this, there is no arbitrariness in the choice of N f as sometimes anticipatedin the literature. The description of heavy quark parton densities, Eqs. (2–6), requires the massive operator matrixelements given by the partonic on-shell expectation values h p | O K | p i , p = q, g , of the operators,cf. [20], O F, µ ,... ,µ n = i n − S [ ψγ µ D µ . . . D µ n ψ ] − trace terms , (20) O Vµ ,... ,µ n = 2 i n − S Sp[ F µ α D µ . . . D µ n − F αµ n ] − trace terms . (21)Here D µ = ∂ µ − ig s t a A aµ denotes the covariant derivative, t a are the generators of SU (3) c , ψ thequark fields, A aµ the gluon fields, F µν the gluonic field strength tensors, Sp the color trace, and S the operator which symmetrizes the Lorentz indices. The corresponding quarkonic operatormatrix elements were calculated in Refs. [6, 13] to O ( a s ) and O ( a s ε ) in [14], respectively.The renormalized gluonic operator matrix elements A gq,Q and A gg,Q to O ( a s ) are given by A gq,Q = a s " ˆ A (2) gq,Q + Z − , (2) gq ( N f + 1) − Z − , (2) gq ( N f )+ (cid:16) ˆ A (1) gg,Q + Z − , (1) gg ( N f + 1) − Z − , (1) gg ( N f ) (cid:17) Γ − , (1) gq + O ( a s ) , (22) A gg,Q = a s h ˆ A (1) gg,Q + Z − , (1) gg ( N f + 1) − Z − , (1) gg ( N f ) i a s " ˆ A (2) gg,Q + Z − , (2) gg ( N f + 1) − Z − , (2) gg ( N f ) + Z − , (1) gg ( N f + 1) ˆ A (1) gg,Q + Z − , (1) gq ( N f + 1) ˆ A (1) Qg + h ˆ A (1) gg,Q + Z − , (1) gg ( N f + 1) − Z − , (1) gg ( N f ) i Γ − , (1) gg ( N f )+ δa s ˆ A (1) gg,Q + O ( a s ) . (23)Here ˆ A ij are the operator matrix elements after mass–renormalization has been carried out. The Z –factors Z ij ( N f ) renormalize the ultraviolet singularities of the operators and Γ ij ( N f ) remove6he collinear singularities, cf. [6, 11, 13, 14]. The terms Z − gq ( g ) ( N f + 1) are equal to Z − gq ( N f + 1) = a s (cid:20) − ε γ (0) gq (cid:21) + a s (cid:20) ε (cid:18) − γ (1) gq − γ (0) gq δa s (cid:19) + 1 ε (cid:18) γ (0) gq β + 12 γ (0) gq γ (0) qq + 12 γ (0) gq γ (0) gg (cid:19)(cid:21) + O ( a s ) (24) Z − gg ( N f + 1) = 1 + a s (cid:20) − ε γ (0) gg (cid:21) + a s (cid:20) ε (cid:18) − γ (1) gg − γ (0) gg δa s (cid:19) + 1 ε (cid:18) γ (0) gg β + 12 γ (0) qg γ (0) gq + 12 γ (0) gg (cid:19)(cid:21) + O ( a s ) . (25) In Eqs. (24,25), γ ( l ) ij are the O ( a l +1 s ) anomalous dimensions and have to be taken - as well as β -at N f + 1 flavors. We adopt the notation ˆ γ ( l ) ij = γ ( l ) ij ( N f + 1) − γ ( l ) ij ( N f ) and define for later use f ( ε ) = (cid:18) m Q µ (cid:19) ε/ exp " ∞ X k =2 ζ k k (cid:16) ε (cid:17) k . (26)To the operator matrix element ˆ A (1) gg,Q necessarily only non-1PI diagrams contribute. The un-renormalized OME ˆ A (2) gq,Q is given by ˆ A (2) gq,Q = (cid:16) m Q µ (cid:17) ε " β ,Q ε γ (0) gq + ˆ γ (1) gq ε + a (2) gq,Q + a (2) gq,Q ε + O ( ε ) . (27)The constant and O ( ε ) contributions a (2) gq,Q and a (2) gq,Q read a (2) gq,Q = T F C F ( N + N + 2( N − N ( N + 1) (cid:16) ζ + S + S (cid:17) −
89 8 N + 13 N + 27 N + 16( N − N ( N + 1) S + 827 P ( N − N ( N + 1) ) , (28) a (2) gq,Q = T F C F ( N + N + 2( N − N ( N + 1) (cid:16) − S − S S − S + 4 ζ − ζ S (cid:17) + 29 8 N + 13 N + 27 N + 16( N − N ( N + 1) (cid:16) ζ + S + S (cid:17) − P S ( N − N ( N + 1) + 481 P ( N − N ( N + 1) ) , (29)with P = 43 N + 105 N + 224 N + 230 N + 86 (30) P = 248 N + 863 N + 1927 N + 2582 N + 1820 N + 496 . (31) In the following we drop the overall factor [1 + ( − N ] / S ~a ≡ S ~a ( N ) denote the (nested) harmonic sums, [21], S b,~a ( N ) = N X k =1 (sign( b )) k k | b | S ~a ( k ) . (32)The renormalized operator matrix element is given by A gq,Q = a s " β ,Q γ (0) gq (cid:16) m Q µ (cid:17) + ˆ γ (1) gq (cid:16) m Q µ (cid:17) + a (2) gq,Q − β ,Q γ (0) gq ζ + O ( a s ) . (33)Here the anomalous dimensions γ (0 , gq and γ (0 , gg are γ (0) gq = − C F N + N + 2( N − N ( N + 1) , (34) γ (0) gg = 8 C A (cid:20) S − N + N + 1( N − N ( N + 1)( N + 2) (cid:21) − β ( N f ) , (35)ˆ γ (1) gq = C F T F − N + N + 2( N − N ( N + 1) S + 329 8 N + 13 N + 27 N + 16( N − N ( N + 1) ! , (36)ˆ γ (1) gg = 8 C F T F (cid:20) −
43 1 N − N − N + 4 N − N + 1 − N + 1) + 4( N + 1) −
203 1 N + 2 (cid:21) + 163 C A T F (cid:20) N − −
193 1 N − N + 193 1 N + 1 − N + 1) −
233 1 N + 2 − S (cid:21) , (37)and ˆ γ (0) gg = (8 / T F . A closer look at Eqs. (33,40) reveals, that the terms ∝ ζ cancel. Thecoefficients of the un-renormalized OME ˆ A gg,Q are given byˆ A (1) gg,Q = − β ,Q ε f ( ε ) , (38)ˆ A (2) gg,Q = (cid:16) m Q µ (cid:17) ε " ε n γ (0) gq ˆ γ (0) qg + 2 β ,Q (cid:16) γ (0) gg + 2 β (cid:17)o + ˆ γ (1) gg ε + a (2) gg,Q + a (2) gg,Q ε + 4 β ,Q f ( ε ) ε + O ( ε ) . (39)The constant and O ( ε ) contributions a (2) gg,Q and a (2) gg,Q are a (2) gg,Q = T F C A ( − ζ S + 16( N + N + 1) ζ N − N ( N + 1)( N + 2) − N + 4727( N + 1) S + 2 P N − N ( N + 1) ( N + 2) ) + T F C F ( N + N + 2) ζ ( N − N ( N + 1) ( N + 2) − P ( N − N ( N + 1) ( N + 2) ) , (40)8 (2) gg,Q = T F C A ( − ζ S − ζ S + 16( N + N + 1)9( N − N ( N + 1)( N + 2) ζ + 2 N + 13( N + 1) S − S N + 1)+ 4 P ζ N − N ( N + 1) ( N + 2) − N + 256 N − N − N + 5481( N − N ( N + 1) S + P N − N ( N + 1) ( N + 2) ) + T F C F ( N + N + 2) ζ N − N ( N + 1) ( N + 2) + P ζ ( N − N ( N + 1) ( N + 2)+ P N − N ( N + 1) ( N + 2) ) , (41)where P = 15 N + 60 N + 572 N + 1470 N + 2135 N + 1794 N + 722 N − N − , (42) P = 15 N + 75 N + 112 N + 14 N − N + 107 N + 170 N + 36 N − N − N − , (43) P = 3 N + 9 N + 22 N + 29 N + 41 N + 28 N + 6 , (44) P = 3 N + 15 N + 3316 N + 12778 N + 22951 N + 23815 N + 14212 N + 3556 N − N + 288 N + 216 , (45) P = N + 4 N + 8 N + 6 N − N − N − N − N − , (46) P = 31 N + 186 N + 435 N + 438 N − N − N − N − N +88 N − N − N − . (47)The renormalized operator matrix element A gg,Q reads A gg,Q = a s T F ln (cid:16) m Q µ (cid:17) + a s " ( β ,Q (cid:16) γ (0) gg + 2 β (cid:17) + γ (0) gq ˆ γ (0) qg ) ln (cid:16) m Q µ (cid:17) + ˆ γ (1) gg (cid:16) m Q µ (cid:17) + a (2) gg,Q − ζ h β ,Q (cid:16) γ (0) gg + 2 β (cid:17) + γ (0) gq ˆ γ (0) qg i + O ( a s ) . (48)We agree with the results for a (2) gq,Q and a (2) gg,Q given in [11], which we presented in (28,40).The new terms a (2) gq,Q and a (2) gg,Q , (29, 41), contribute to all OMEs A (3) ij through renormalization.With respect to the mathematical structure, a (2) gq ( gg ) ,Q and a (2) gq ( gg ) ,Q , (28,40,29,41), belong to theclass being observed for two–loop corrections before, [22]. In the present case even only singleharmonic sums contribute. We checked our results for the moments N = 2 , . . . , MATAD , [23]. An additional check is provided by the sum rules in Eqs. (10,11), which are fulfilledby the renormalized OMEs presented here and in Refs. [6, 11, 13]. Moreover, we observe thatthese rules are obeyed on the unrenormalized level as well, even up to O ( ε ), [14].To describe the evolution of the parton distributions, Eqs. (2–6), the OMEs (33,48) have to9e supplemented by the corresponding 1PR terms A (2) Qg → A (2) Qg + a s β ,Q T F N + N + 2 N ( N + 1)( N + 2) ln µ m Q ! , (49) A (2) gg,Q → A (2) gg,Q + a s β ,Q ln µ m Q ! . (50)Eqs. (49,50) agree with the results presented in Ref. [11]. In applying these parton densities inother hard scattering processes this modification also affects part of the massless hard scatteringcross sections there, as outlined above. We calculated the massive gluonic operator matrix elements A gq,Q and A gg,Q , being required inthe description of heavy flavor parton densities at scales sufficiently above threshold, to O ( a s ε ).We confirm previous results given in [11] for the constant terms and obtained newly the O ( ε )terms which enter the 3-loop corrections to A ij via renormalization. We reminded of details ofthe charge renormalization and clarified that additional terms at O ( a s ) are to be included in thedata analysis in the FFNS and VFNS using the MS scheme. Acknowledgments.
We would like to thank S. Alekhin and E. Laenen for useful discussions.This work was supported in part by DFG Sonderforschungsbereich Transregio 9, Comput-ergest¨utzte Theoretische Teilchenphysik, Studienstiftung des Deutschen Volkes, the EuropeanCommission MRTN HEPTOOLS under Contract No. MRTN-CT-2006-035505, the Ministeriode Ciencia e Innovacion under Grant No. FPA2007-60323, CPAN (Grant No. CSD2007-00042),the Generalitat Valenciana under Grant No. PROMETEO/2008/069, and by the EuropeanCommission MRTN FLAVIAnet under Contract No. MRTN-CT-2006-035482.10 eferences [1] K. Lipka [H1 and ZEUS Collaborations], Nucl. Phys. Proc. Suppl. (2006) 128;S. Chekanov et al. [ZEUS Collaboration], arXiv:0812.3775 [hep-ex];J. Bl¨umlein and S. Riemersma, arXiv:hep-ph/9609394.[2] E. Laenen, S. Riemersma, J. Smith and W. L. van Neerven, Nucl. Phys. B (1993) 162;S. Riemersma, J. Smith and W. L. van Neerven, Phys. Lett. B (1995) 143 [arXiv:hep-ph/9411431].[3] S. I. Alekhin and J. Bl¨umlein, Phys. Lett. B (2004) 299 [arXiv:hep-ph/0404034].[4] S. Alekhin et al. , arXiv:hep-ph/0601012; arXiv:hep-ph/0601013; M. Dittmar et al. ,arXiv:hep-ph/0511119; H. Jung et al. , arXiv:0809.0549 [hep-ph].[5] J. Bl¨umlein, H. B¨ottcher and A. Guffanti, Nucl. Phys. B (2007) 182 [arXiv:hep-ph/0607200]; hep-ph/0407089.[6] M. Buza, Y. Matiounine, J. Smith, R. Migneron and W. L. van Neerven, Nucl. Phys. B (1996) 611 [arXiv:hep-ph/9601302].[7] E. B. Zijlstra and W. L. van Neerven, Nucl. Phys. B (1992) 525;J. A. M. Vermaseren, A. Vogt and S. Moch, Nucl. Phys. B (2005) 3 [arXiv:hep-ph/0504242].[8] S. D. Drell and T. M. Yan, Annals Phys. (1971) 578 [Annals Phys. (2000) 450].[9] J. Bl¨umlein, A. De Freitas, W. L. van Neerven and S. Klein, Nucl. Phys. B (2006) 272[arXiv:hep-ph/0608024].[10] M. Gluck, E. Reya and M. Stratmann, Nucl. Phys. B (1994) 37.[11] M. Buza, Y. Matiounine, J. Smith and W. L. van Neerven, Eur. Phys. J. C (1998) 301[arXiv:hep-ph/9612398].[12] J. Bl¨umlein and W. L. van Neerven, Phys. Lett. B (1999) 417 [arXiv:hep-ph/9811351].[13] I. Bierenbaum, J. Bl¨umlein and S. Klein, Nucl. Phys. B (2007) 40 [arXiv:hep-ph/0703285]; Phys. Lett. B (2007) 195 [arXiv:hep-ph/0702265].[14] I. Bierenbaum, J. Bl¨umlein, S. Klein and C. Schneider, Nucl. Phys. B (2008) 1[arXiv:0803.0273 [hep-ph]];I. Bierenbaum, J. Bl¨umlein and S. Klein, Acta Phys. Polon. B (2007) 3543[arXiv:0710.3348 [hep-ph]].[15] I. Bierenbaum, J. Bl¨umlein and S. Klein, Nucl. Phys. Proc. Suppl. (2008) 162[arXiv:0806.4613 [hep-ph]]; arXiv:0812.2427 [hep-ph].[16] S. Alekhin, J. Bl¨umlein, and S. Klein, in preparation.[17] K. G. Chetyrkin, B. A. Kniehl and M. Steinhauser, arXiv:0812.1337 [hep-ph] and referencestherein. 1118] T. Appelquist and J. Carazzone, Phys. Rev. D (1975) 2856.[19] A. Chuvakin, J. Smith and W. L. van Neerven, Phys. Rev. D (2000) 096004 [arXiv:hep-ph/9910250].[20] B. Geyer, D. Robaschik and E. Wieczorek, Fortsch. Phys. (1979) 75.[21] J. Bl¨umlein and S. Kurth, Phys. Rev. D (1999) 014018 [arXiv:hep-ph/9810241];J. A. M. Vermaseren, Int. J. Mod. Phys. A (1999) 2037 [arXiv:hep-ph/9806280].[22] J. Bl¨umlein, Nucl. Phys. Proc. Suppl. (2008) 232 [arXiv:0807.0700 [math-ph]] andreferences therein.[23] M. Steinhauser, Comput. Phys. Commun.134