Two and Three Loop Heavy Flavor Corrections in DIS

Transcription

1 Two and Three Loop Heavy Flavor Corrections in DIS 1 Johannes Blümlein, DESY in collaboration with I. Bierenbaum and S. Klein based on: Introduction Renormalization of the OME s to 3 Loops Oǫ terms at Loops Towards Fixed Moments of A 3 Qg Conclusions - I. Bierenbaum, J. Blümlein, S. K., and C. Schneider arxiv: [math-ph]; arxiv: [hep-ph]. - I. Bierenbaum, J. Blümlein, and S. K., Phys. Lett. B ; Nucl. Phys. B ; arxiv: [hep-ph]; Acta Phys. Polon. B ; - J. Blümlein, A. De Freitas, W.L. van Neerven, and S. K., Nucl. Phys. B Johannes Blümlein Loopfest-08 U. Buffalo, NY, May 008

2 Deep Inelastic Scattering DIS: 1. Introduction k q k L µν Q := q, x := Q pq dσ dq dx W µνl µν Bjorken x P W µν LO: Q Q Hadronic tensor for heavy quark production via single photon exchange: W Q Q 1 µν q, P, s = d 4 ξ expiqξ P, s [Jµ em ξ, Jem ν 0] P, s 4π Q Q { unpol. { pol. = 1 x g µν q µq ν q M Pq ε µναβq α F Q Q L x, Q + x Q [ s β g Q Q 1 x, Q + P µ P ν + q µp ν + q ν P µ x s β sq Pq pβ g Q Q x, Q ] Q 4x g µν F Q Q x, Q.

3 3 Light quarks: In Bjorken limit, {Q, ν}, x fixed, at twist τ = level: F i x, Q = C j i x, Q }{{} µ f j x, µ, }{{} j }{{} structure functions parton densities, Wilson coefficients, non-perturbative perturbative Different flavor contributions to F x, Q : light flavor heavy flavor LO: x[f q + f q ] q=u,d,s a s [ C 1,H g NLO: a s [ C 1 q x[f q + f q ] + C 1 g ] xg a s ] xg [ C,H q,ns x[f q + f q ] + C,H q,ps xσ + C,H g ] xg N LO:.. 3-loop analysis: Oa 3 s heavy flavor Wilson coefficients needed.

4 4 Need for the calculation: Heavy flavor charm contributions to DIS structure functions are rather large [0 40 % at lower values of x]. Increase in accuracy of the perturbative description of DIS structure functions. QCD analysis and determination of Λ QCD, resp. α s MZ, from DIS data: δα s /α s < 1 %. Precise determination of the gluon and sea quark distributions. Derivation of variable flavor number scheme for heavy quark production to Oa 3 s. Goal: Calculation of the heavy flavor Wilson coefficients to higher orders for Q 5 GeV [sufficient in many applications].

5 5 F i x = , i = 1 x = , i = 0 x = , i = 19 x = , i = 18 x = , i = 17 x = , i = 16 x = , i = 15 x = , i = 14 x = 0.000, i = 13 x = 0.003, i = 1 H1 e + p ZEUS e + p BCDMS NMC F cc _ 4 i x= , i= x= , i=1 x= , i=0 x= , i=19 x= , i=18 x= , i=17 x=0.0000, i=16 x= , i=15 MRST004 MRST NNLO MRST004FF3 CTEQ6.5 CTEQ5F x = , i = 11 x = , i = 10 x = 0.013, i = 9 x = 0.00, i = 8 x = 0.03, i = 7 x = 0.050, i = x=0.0005, i=14 x=0.0006, i=13 x=0.0008, i=1 x=0.0010, i=11 x=0.0015, i=10 10 x = 0.080, i = 5 x = 0.13, i = x=0.00, i=9 x=0.003, i=8 1 x = 0.18, i = 3 x = 0.5, i = 10 3 x= , i=7 x=0.005, i= H1 PDF 000 extrapolation x = 0.40, i = 1 x = 0.65, i = High statistics in both cases Q / GeV H1 Collaboration ZEUS D H1 D H1 Displaced Track x=0.006, i= x=0.01, i=4 x=0.013, i=3 x=0.030, i= x=0.03, i=1 Q /GeV [Thompson, 007] F c c x, Q 0 40 % F x, Q for small values of x, but different scaling violations

6 6 Previous calculations: Unpolarized DIS : LO : [Witten, 1976; Babcock, Sivers, 1978; Shifman, Vainshtein, Zakharov 1978; Leveille, Weiler, 1979] NLO : [Laenen, Riemersma, Smith, van Neerven, 1993, 1995] asymptotic: [Buza, Matiounine, Smith, Migneron, van Neerven, 1996; Bierenbaum, Blümlein, Klein, 007] Observation: F c c x, Q is very well described by F c c x, Q Q m for Q > 10 m c. Polarized DIS : LO : [Watson, 198; Glück, Reya, Vogelsang, 1991; Vogelsang, 1991] NLO : asymptotic: [Buza, Matiounine, Smith, van Neerven, 1997; Bierenbaum, Blümlein, Klein, 008, to appear] Mellin space expressions: [Alekhin, Blümlein, 003]. Variable flavor number scheme at Oa s: [Buza, Matiounine, Smith, van Neerven, 1998] In the following, we report on results for unpolarized and polarized Heavy Flavor Wilson coefficients beyond NLO.

7 . The Method 7 massless RGE and light cone expansion in Bjorken limit {Q, ν}, x fixed: [ ] lim Jξ, J0 c N i,τξ, µ ξ µ1...ξ µn O µ 1...µ m ξ i,τ 0, µ. 0 i,n,τ Operators: flavor non-singlet 3, pure singlet and gluon; consider leading twist. RGE for collinear singularities = mass factorization of the structure functions into Wilson coefficients and parton densities: F i x, Q = C j i x, Q µ f j x, µ }{{} j }{{} non-perturbative perturbative Light-flavor Wilson coefficients: process dependent Q C,L;i fl = δ i,q + a l sc fl,l,l,i, µ l=1 i = q, g = Known up to Oa 3 s [Moch, Vermaseren, Vogt, 005.]

8 8 Heavy quark contributions given by heavy quark Wilson coefficient, H S,NS Q,L,i µ, m Q In the limit Q m Q [Q 10 m Q for F ]: massive RGE, derivative m / m acts on Wilson coefficients only: all terms but power corrections calculable through partonic operator matrix elements, i A l j, which are process independent objects! H S,NS Q,L,i µ, m µ = A S,NS m k,i µ }{{ } µ massive OMEs l=1 C S,NS Q,L,k µ. }{{} light parton Wilson coefficients holds for polarized and unpolarized case. OMEs obey expansion A S,NS m k,i = i O S,NS m k i = δ k,i + a l s AS,NS,l k,i, i = q, g [Buza, Matiounine, Migneron, Smith, van Neerven, 1996; Buza, Matiounine, Smith, van Neerven, 1997.] µ.

9 3. Renormalization 9 Unrenormalized massive operator matrix elements: need for: Â ij = δ ij + k=0 â k s Âk ij Mass renormalization Charge renormalization Renormalization of ultraviolet singularities Factorization of collinear singularities use MS scheme and decoupling formalism [Ovrut, Schnitzer 1981; Bernreuther, Wetzel 198]. Since the light-cone expansion is used, external legs obtain self-energy insertions due to heavy quarks.

10 10 Mass renormalization: on mass shell scheme for quarks [Tarrach 1981; Nachtmann, Wetzel 1981; Gray, Broadhurst, Graefe, Schilcher 1990] m ε/ [ 6 δm 1 = C F S ε m µ ε 4 + [ δm = C F Sε m ε m 1 µ ˆm = m + â s δm 1 + â s δm + Oâ 3 s ζ ε 18C F + C A 8T F N l + N h + 1 ε ε ] 45 C F + 91 C A 14T F N l + N h ] ] 199 +C F 8 51 ζ + 48 lnζ 1ζ 3 + C A ζ 4 lnζ + 6ζ 3 + T F [N l + 10ζ + N h 14ζ Charge renormalization: MS-scheme â s ε = Zgε, µ a s µ = a s µ [ 1 + δa s,1 a s µ + δa s, a sµ ] + Oa 4 s [ β 0 4β δa s,1 = S ε, δa s, = Sε 0 ε ε + β ] 1 ε

11 11 Operator renormalization: ultraviolet divergences, renormalized by Z factors. Generic formula in terms of anomalous dimensions γ ij,k. Has to be adapted e.g. for the various cases three-loop non-singlet, pure-singlet, etc. i, j, m, n {q, g}: { [ ] γ ij,0 1 1 Z ij N,a s,ε = δ i,j + a s S ε + a ε ssε ε γ im,0γ mj,0 + β 0 γ ij,0 + 1 } ε γ ij,1 { [ 1 1 +a 3 s S3 ε ε 3 6 γ in,0γ nm,0 γ mj,0 + β 0 γ im,0 γ mj,0 + 4 ] 3 β 0 γ ij,0 + 1 [ 1 ε 6 γ im,1γ mj,0 + γ im,0 γ mj,1 + ] 3 β 0γ ij,1 + β 1 γ ij,0 + γ ij, 3ε } Z PS qq = Z qq Z NS. The anomalous dimensions γ ij,k N are related to the splitting functions by γ ij,k N = 1 0 dzz N 1 P k ij z.

12 1 Mass factorization: Collinear singularities are factored into Γ NS, Γ ij,s and Γ qq,ps. For massless quarks: Γ NS = Z 1 NS, Γ ij,s = Z 1 ij,s, Γ qq,ps = Z 1 qq,ps Γ NS N, a s, ε = Γ ij,s N, a s, ε = Γ qq,ps N, a s, ε = 1 a s S ε γ NS,0 ε γ ij,0 δ ij a s S ε ε a s S ε + a s S ε + a ss ε [ 1 ε [ 1 ε [ 1 ε γ qg,0γ gq,0 + 1 ε γ qq,ps,1 1 γ NS,0 β 0γ NS,0 1 ] ε γ NS,1 1 γ ik,0γ kj,0 β 0 γ ij,0 1 ] ε γ ij,1 ]. Here: in each diagram at least one quark line is massive Γ matrices apply to parts of the diagrams with massless lines only at most loop sub-graphs mass factorization is different in various sub classes of contributing Feynman diagrams singularities contained in Γ NS, Γ ij,s, and Γ qq,ps are absorbed into the bare parton densities, which become scale dependent

13 13 The renormalized operator matrix elements are obtained removing the ultraviolet singularities and collinear singularities of the operator matrix elements, For example: A ij = Z 1 1 ik ÂklΓlj = δ ij + a s A 1 ij + a s A ij + a 3 s A3 ij. A Qg = Zqq 1 ÂPS Qq Γ 1 qg + Zqq 1 = A Qg =  ÂQgΓ 1 Qg + Z 1,1 qq  1 Qg + Z 1,1 qg gg + Zqg 1 Âgq,QΓ 1 mixing with ÂPS Qq and Âgg,Q at Oa 3 s. Âgq,Q starts contributing from Oa s at Oa 4 s to ÂQg qg + Zqg 1 Âgg,QΓ 1 gg.  1 Â1 gg,q + Z 1, qg + Qg + Z 1,1 qg Γ 1,1 gg.

14 14 For example: Renormalized gluonic massive operator matrix elements up to Oa s : Unrenormalized  1 m ε/ { Qg = S ε 1 } 0 ˆP  Qg to Oa s : µ qg + a 1 Qg ε + ε ā1 Qg + ε ā 1 Qg m Renormalized A Qg to Oa s:  Qg = S ε A 1 Qg = 1 A Qg = 1 8 a 1 Qg : Oε of the OME A1 Qg + 1 ε ε S ε µ { 1 P 0 qg ln ε [ 1 ε t H=Q { 1 0 ˆP qg P qq 0 P gg β 0 ˆP qg } 1 ˆP qg β 0 a 1 Qg a1 Qg m µ β 0,H m H µ { [ ]} m P qg 0 P qq 0 P gg 0 + β 0 ln +a 1 Qg P 0 qq P 0 gg ε/ ε ζ + ε3 4 ζ 3 P 1 qg ln µ 1 [ ] P qq 0 P gg 0 + β 0 + a Qg ; a 1 Qg 0 in the MS scheme. Enters A Qg For the renormalization of A 3 Qg : a Qg, a gg,q, a,ps Qq are needed. } ] + a Qg + ε ā Qg m µ through renormalization.

15 4. Calculation Techniques 15 Calculation in Mellin-space for space like q, Q = q : 0 x 1 use of generalized hypergeometric functions for general analytic results use of Mellin-Barnes integrals for numerical checks MB, [Czakon, 006] and some analytic results Summation of lots of new infinite one-parameter sums into harmonic sums. E.g.: S 1 is 1 i + j + N N = 4S,1,1 S 3,1 + S 1 3S,1 + 4S 3 S 4 ii + jj + N 3 i,j=1 S + S 1S + S S 1ζ 3 + ζ S 1 + S use of integral techniques and the Mathematica package SIGMA [C. Schneider, 007], [I. Bierenbaum, J. Blümlein, S. K., C. Schneider, arxiv: [math-ph]; arxiv: [hep-ph]] Partial checks for fixed values of N using SUMMER, [Vermaseren, Int. J. Mod. Phys. A ]. Algebraic and structural simplification of the harmonic sums [J. Blümlein, 003, 007]..

16 5. Results 16 Unpolarized case, Singlet, Oε a Qg = T F C F 3 + N + N + NN + 1N + N + N + 3N + 3N + P 1 N N + 1 ζ 3 + N + N 3 N N + S + N4 5N 3 3N 18N 4 N N + 1 S 1 N + 16S,1,1 8S 3,1 8S,1 S 1 + 3S 4 4 S 3 S 1 1 S S S 1 1 S ζ S ζ S 1 8 ζ 3 S N 3N 8 N S,1 + 3N + N + 1N + 3 N S N N N N 8 3N + N + 3 N N + 1 S 3 + N + N N + S S S 1 N ζ + N5 + N 4 8N 3 5N 3N N 3 N ζ N 5 N 4 11N 3 19N 44N 1 P N N S 1 + N + N 5 N N + N + N + + T F C A 16S,1,1 4S,1,1 8S 3,1 8S, 4S 3,1 β + 9S 4 16S,1 S 1 NN + 1N S 1 S 3 + 4β S 1 8β S + 1 S 8β S 1 + 5S 1 S + 1 S N N 4 4S N + 1 N +,1 + β 4β S N3 1N 7N NN + 1 N + S S N 5 10N 4 11N N + 4N + 16 N 1N N + 1 N + ζ 3 S 1 ζ 3 S ζ S 1 ζ 4β ζ 17 ζ 5 N 3 + 8N + 11N + NN + 1 N + S N 4 + N 3 + 7N + N + 0 N N + 3 β N N 4 + 9N 3 + 3N + 7N + 6 N + N 1 N 1N N + 1 N + S 3 8 N + 1 N + ζ S 1 P 5 NN N + 3 S 1 + P 6 NN N + 4 S 1 P 7 N 1N 5 N N + 5 P 3 N 1N 3 N N + 3 S P 4 N 1N 3 N N + ζ.

17 17 Unpolarized case, Pure Singlet and Non singlet, Oε a PS, Qq = C F T F 5N3 + 7N + 4N + 4N + 5N + S N 1N 3 N N + + ζ 4 3 N + N + 3S N 1N N ζ 3 N + P 9 + N 1N 5 N N + 4. qq,q = C 4 FT F 3 S S ζ 8 9 S 1ζ S S 1ζ + 3N + 3N + 9NN + 1 a NS, S + 3N4 + 6N N + 0N 1 18N N + 1 ζ S 1 + ζ 3 P 8 648N 4 N

18 18 Polarized case, Singlet Oε a N 1 Qg = T F C F 16S,1,1 8S 3,1 8S,1 S 1 + 3S 4 4 S 3 S 1 1 S 1 S1 4 8 S 1 ζ 3 S S 1 + S ζ S 1 ζ NN S,1 N N + 3N N N + 1N + N 13N + 3N + N N + 1 ζ 3 + N4 4N 3 3N + 0N + 1 N N + 1 S 1 + N + + T F C A N NN + 1 S 3 S S + 5S S S S 1 + S N4 + 48N N + 98N N N N P 1 N 3 N S + N3 6N N 36 N + N N + 1N + P 3 N 5 N N + N 1 S N S 1 ζ N + 1 S 1 + P N 3 N ζ 16S,1,1 4S,1,1 8S 3,1 8S, 4S 3,1 + 3 β 16S,1 S 1 4β S 1 + 8β S + 8β S 1 + 9S 4 S ζ β ζ S ζ S1 10 S 1 ζ ζ + N 1 16S NN + 1,1 4β + 16β S N 3 + 7N + 8N 6 3N 13 N N + 1 S 3 + N + NN + 1 S S 1 N + 3 N + 4N + 5 NN + 1 N + S ζ S 1 N + 1 N 19N N N + 1 ζ 3 N NN β P 4 N 3 N S N4 + N 3 5N 1N + N + NN N + P 6 P 7 + NN S 1 N + N 5 N N + S 1 P 5 N 3 N ζ [I. Bierenbaum, J. Blümlein and S. K., 008, to appear]

19 19 Polarized case, Pure Singlet and Non singlet, Oε a PS, 4N 1N + Qq = 3S 3N N ζ 3 + N + N3 + N + 1 S N 3 N ζ a NS, qq,q = a NS, qq,q. + + NN5 7N 4 + 6N 3 + 7N + 4N + 1 N N 5, Use of t Hooft Veltman scheme for γ 5. A finite renormalization has to be applied to maintain the Ward identities. The first moment of a Qg vanishes. = sum rule of H gx.

20 0 Â gg,q { = 1 T F C A ε 3 3 S 64N + N N 1NN + 1N + + +ε 8 3 ζ S ζ 3S N + N + 1ζ 3N 1NN + 1N + 9 ζ S 1 + Â gg,q unpolarized + 1 ε 4 56N N + 1 S S N + N + 1 9N 1NN + 1N + ζ 3 + N + 1 3N + 1 S 16P 1 9N 1N N + 1 N + P 3 7N 1N 3 N N + S 1 3N P 1 ζ + 56N 3 47N 175N N 1N N + 1 N + 38N P 5 81N 1NN + 1 S N 1N 4 N N + { 1 16N + N + +T F C F ε N 1N N P N + ε N 1N 3 N N + + +ε 4N + N + ζ N 1N N + 1 N + P 4 N 1N 4 N N + 4N + N + ζ 3 3N 1N N + 1 N + + P ζ N 1N 3 N N + + P 6 4N 1N 5 N N + Result obtained in terms of Γ and Ψ functions to all orders in ε to Oε 0 agreement with van Neerven et al. Oε new: term needed too to derive variable flavor number scheme to Oα 3 s. } }

21 1 A remark on the mathematical structure of the O1 and Oε terms: van Neerven et al. to O1: unpolarized: 48 basic functions; polarized: 4 basic functions. O1: {S 1, S, S 3, S, S 3 }, S,1 = basic objects. Oε: {S 1, S, S 3, S 4, S, S 3, S 4 }, S,1, S,1, S 3,1, S,1,1, S,1,1 = 6 basic objects These objects are in common to all single scale higher order processes. Str. Functions, DIS HQ, Fragm. Functions, DY, Hadr. Higgs-Prod., s+v contr. to Bhabha scatt.,... harmonic sums with index { 1} cancel holds even for each diagram [J. Blümlein, 004; J. Blümlein, V. Ravindran, 005,006; J. Blümlein, S. Klein, arxiv: [hep-ph]; J. Blümlein and S. Moch in preparation.] Expectation for 3 loops: weight 5 6 harmonic sums

22 6. Fixed moments of A 3 ij,q Contributing OMEs: Singlet A Qg A gg,q A gq,q Pure Singlet A PS Qq } mixing Non Singlet A NS,+ qq,q A NS, qq,q A NS,v qq,q All loop Oε terms in the unpolarized case are known: a Qg, a,ps Qq, a gg,q, a gq,q, a,ns qq,q. Unpolarized anomalous dimensions are known up to Oa 3 s [Moch, Vermaseren, Vogt, 004.] = All terms needed for the renormalization of unpolarized 3 Loop heavy OMEs are present. = Calculation will provide first independent checks on γ qg 3, γ qq 3,PS color projections of γ 3,NS±,v qq, γ 3 gg and γ 3 gq. Calculation proceeds in the same way in the polarized case. Known so far : a Qg, a,ps Qq, a,ns qq,q = a,ns qq,q. and on respective

23 3 3 loop OMEs are generated with QGRAF [Nogueira 1993] New operator insertions emerge: We consider first the terms with quark loops at least one heavy: # of diagrams for A 3 Qg 489 diagrams with two quark loops 1478 diagrams with one quark loop = number of diagrams can be reduced by using symmetry arguments.

24 4 First step: Calculation of fixed moments of A 3 Qg N, N =, 4, 6,... three loop self-energy type diagrams with an operator insertion Extension: additional scale compared to massive propagators: Mellin variable N Genuine tensor integrals due to µ 1... µ n p O µ1...µ n p = µ 1... µ n p S Ψγ µ1 D µ...d µn Ψ p = AN p N D µ = µ igt a A a µ, = 0. Construction of a projector to obtain the desired moment in N [undo -contraction] Color factors are calculated with [Ritbergen, Schellekens, Vermaseren 1998] Translation to suitable input for MATAD [Steinhauser, 001] Tests performed: Various loop calculations for N =, 4, 6,... were repeated agreement with our previous calculation; checked against result of A gg,q Status: automated chain to evaluate the OMEs Terms OT F d abcd abc vanish currently investigated: terms T F C F, T F C A terms

25 First Results 5 Non-singlet terms : OT F C F Â 3,NS m 3ε/ qq,qq = 8β 0,Q γ0,ns qq,q µ 3ε 3 A 3,NS± qq,q N : heavy quark loops 4β 1,NS 0,Qγqq,Q 3ε + γ,ns qq,qq 1β 0,Qa,NS qq,q 3ε + a 3,NS qq,qq. A 3,NS qq,qq = 1 6 ln3 m µ β0,q γ0,ns qq,q + 1 ln m µ β 0,Q γ 1,NS qq,q + 1 m ln µ γ,ns qq,qq +a 3,NS qq,qq + 4β 0,Qa,NS qq,q. [ ] γ 0,NS qq,q = 4C F S 1 3N + 3N + NN + 1 { γ 1,NS 8 qq,q = 4C F T F 3 S 40 9 S 1 + 3N4 + 6N N } + 0N 1 9N N + 1 γ,ns 18 qq,qq = C FTF 9 S S 18 7 S N N N N N + 16N 4 7N 3 N

26 6 Â 3,NS qq,qq = 1 ε ε ε ζ ζ ζ, Â 3,NS qq,qq 4 = 1 ε ε ε ζ ζ, Â 3,NS qq,qq 6 = 1 ε ε ε ζ ζ, Â 3,NS qq,qq 8 = 1 ε ε ε ζ ζ ζ ζ ζ

27 7 A 3,NS qq,qq = [ 18 T F C F 81 ln3 m µ ln m µ ] 81 ζ 3, ln m µ A 3,NS qq,qq 4 = T FC F [ ln3 m µ ln m µ ln m µ A 3,NS qq,qq 6 = ] 405 ζ 3, T F C F [ ln3 m µ ln m µ ζ 3 ], m ln µ A 3,NS qq,qq 8 = T FC F [ ln3 m µ ln m µ ln m µ ζ 3 ]. All ζ terms vanish after renormalization.

28 8 1 heavy - 1 massless quark A 3,NS qq,qq = n ft FC F [ ln3 m µ ln m µ A 3,NS qq,qq 4 = n ft F C F 81 ζ 3 [ ln3 m µ m ln µ ζ 3 [ A 3,NS qq,qq 6 = n ftfc F ln3 m µ m ln µ A 3,NS qq,qq 8 = n ft F C F [ ln3 m µ ln m µ ζ 3 ], ], ], 835 ζ 3 ],

29 9 7. Conclusions The heavy flavor contributions to F are rather large in the region of lower values of x. QCD precision analyses therefore require the description of the heavy quark contributions to 3 loop order. We calculate the heavy flavor DIS Wilson Coefficients in the asymptotic regime [Q 10m ] using massive operator matrix elements. We newly presented first contributions to these corrections: ā Qg, āps, Qq, ā gg,q, āns, qq ā Qg, āps, Qg = ā NS, qq,q in the unpolarized and polarized case for general values of the Mellin variable. These terms contribute to H 3 ij, H3 ij respectively, through renormalization.

30 30 The calculation is performed in Mellin space, which allows to obtain compact results. The analytic results were obtained using representations in terms of generalized hypergeometric functions. Numerical checks were performed applying Mellin Barnes integrals. Integral techniques and the summation package SIGMA have been used for summation. The results are given in the form of nested harmonic sums. We developed a programme chain to calculate the massive operator matrix elements A 3 ij for fixed Mellin moments based on QGRAF and MATAD First non-logarithmic 3 loop contributions were calculated for a series of Mellin Moments. There are more to come soon for other color structures and higher values of Mellin N. The 3 loop anomalous dimensions are obtained as a by-product from the single pole terms.