Latest results on the 3-loop heavy flavor Wilson coefficients in DIS

1/34 Latest results on the 3-loop heavy flavor Wilson coefficients in DIS A. De Freitas 1, J. Ablinger, A. Behring 1, J. Blümlein 1, A. Hasselhuhn 1,,4, A. von Manteuffel 3, C. Raab 1, C. Schneider, M. Round 1,, F. Wißbrock 1 1 DESY, Zeuthen Johannes Kepler University, Linz 3 J. Gutenberg University, Mainz 4 KIT, Karlsruhe 15.06.015

/34 Introduction Unpolarized Deep Inelastic Scattering DIS): k q k Q := q, x := Q P.q L µν Bjorken x P W µν dσ dq dx WµνLµν W µνq, P, s) = 1 4π 1 g µν qµqν x q d 4 ξ expiqξ) P, s J em µ ξ), J em ν 0) P, s = ) F L x, Q ) + x Q ) qµpν + qνpµ P µp ν + Q x 4x gµν F x, Q ). Structure Functions: F,L contain light and heavy quark contributions.

3/34 Deep Inelastic Scattering DIS): σ cc red Q =.5 GeV 5 GeV 7 GeV m c m c ) GeV) 1.6 1.4 1. NNLO 1 GeV 18 GeV 3 GeV 1. 1.18 1.16 60 GeV 10 GeV 00 GeV 350 GeV x 650 GeV x HERA NNLO fitted NNLO A+B)/ NLO 000 GeV x 1.14 1.1 1.1 NLO 0.11 0.11 0.114 0.116 0.118 0.1 0.1 α s M Z ) NNLO: S. Alekhin, J. Blümlein, K. Daum, K. Lipka, Phys.Lett. B70 013) 17 11.355 m c m c ) = 1.4 ± 0.03exp) +0.03 0.0 scale) +0.00 0.07 thy), α s MZ ) = 0.113 ± 0.011 Yet approximate NNLO treatment Kawamura et al. 105.57.

4/34 α smz ) from NNLO DIS+) analyses from ABM13 α smz ) BBG 0.1134 +0.0019 0.001 valence analysis, NNLO GRS 0.11 valence analysis, NNLO ABKM 0.1135 ± 0.0014 HQ: FFNS N f = 3 JR 0.118 ± 0.0010 dynamical approach JR 0.1140 ± 0.0006 including NLO-jets MSTW 0.1171 ± 0.0014 MSTW 0.1155 0.1175 013) ABM11 J 0.1134 0.1149 ± 0.001 Tevatron jets NLO) incl. ABM13 0.1133 ± 0.0011 ABM13 0.113 ± 0.0011 without jets) CTEQ 0.1159..0.116 CTEQ 0.1140 without jets) NN1 0.1174 ± 0.0006 ± 0.0001 Gehrmann et al. 0.1131 + 0.008 0.00 e + e thrust Abbate et al. 0.1140 ± 0.0015 e + e thrust BBG 0.1141 +0.000 0.00 valence analysis, N 3 LO TH α s = α sn 3 LO) α snnlo) + HQ = +0.0009 ± 0.0006 HQ NNLO accuracy is needed to analyze the world data. = NNLO HQ corrections needed.

5/34 Goals Complete the NNLO heavy flavor Wilson coefficients for twist- in the dynamical safe region Q > 0GeV no higher twist) for F x, Q ) Measure m c and α s as precisely as possible Provide precise CC heavy flavor corrections Consequences for LHC: NNLO VFNS will be provided better constraint on sea quarks and the gluon precise mc and α s on input

6/34 Factorization of the Structure Functions At leading twist the structure functions factorize in terms of a Mellin convolution F,L) x, Q ) = ) C j,,l) x, Q µ, m µ f j x, µ ) }{{} j }{{} nonpert. perturbative into pert.) Wilson coefficients and nonpert.) parton distribution functions PDFs). denotes the Mellin convolution f x) gx) 1 0 dy 1 0 dz δx yz)f y)gz). The subsequent calculations are performed in Mellin space, where reduces to a multiplication, due to the Mellin transformation ˆf N) = 1 0 dx x N 1 f x).

7/34 Wilson coefficients: C j,,l) N, Q µ, m µ ) = C j,,l) N, Q µ ) ) + H j,,l) N, Q µ, m µ At Q m the heavy flavor part ) H j,,l) N, Q µ, m = ) C µ i,,l) N, Q m )A µ ij µ, N i Buza, Matiounine, Smith, van Neerven 1996 Nucl.Phys.B factorizes into the light flavor Wilson coefficients C and the massive operator matrix elements OMEs) of local operators O i between partonic states j ) m A ij µ, N = j O i j. additional Feynman rules with local operator insertions for partonic matrix elements. The unpolarized light flavor Wilson coefficients are known up to NNLO Moch, Vermaseren, Vogt, 005 Nucl.Phys.B. For F x, Q ) : at Q 10m the asymptotic representation holds at the 1% level..

8/34 The Wilson Coefficients at large Q L NS q,,l) NF + 1) = a s A ),NS qq,q NF + 1) δ + Ĉ),NS q,,l) NF ) + a 3 s A 3),NS qq,q NF + 1) δ + A),NS qq,q NF + 1)C,NS q,,l) NF + 1) + Ĉ3),NS q,,l) NF ) L PS q,,l) NF + 1) = a3 s A 3),PS qq,q NF + 1) δ + A) gq,q NF ) NF C g,,l) NF + 1) + NF ˆ C3),PS q,,l) NF ) L S g,,l) NF + 1) = a sa gg,q NF + 1)NF C g,,l) NF + 1) + a3 s A 3) qg,q NF + 1) δ H PS q,,l) NF + 1) = a s H S g,,l) NF + 1) = as +A ) gg,q NF + 1) NF C g,,l) NF + 1) + A) gg,q NF + 1) NF C g,,l) NF + 1) + A ),PS Qg NF + 1) NF C q,,l) NF + 1) + NF ˆ C3) g,,l) NF ), A ),PS Qq N F + 1) δ + C ),PS q,,l) NF + 1) + a 3 s + C 3),PS q,,l) NF + 1) + A) gq,q NF + 1) C g,,l) NF + 1) +A ),PS Qq N F + 1) C,NS q,,l) NF + 1), A Qg NF + 1) δ + C g,,l) NF + 1) + a s A 3),PS Qq N F + 1) δ A ) Qg NF + 1) δ + A Qg NF + 1) C,NS q,,l) NF + 1) + A gg,q NF + 1) C g,,l) NF + 1) ) + C g,,l) NF + 1) + a 3 s A 3) Qg NF + 1) δ + A) Qg NF + 1) C,NS q,,l) NF + 1) { + A ) gg,q NF + 1) C g,,l) NF + 1) + A Qg NF + 1) C ),NS q,,l) NF + 1) } ),PS + C q,,l) NF + 1) + A ) 3) gg,q NF + 1) C g,,l) NF + 1) + C g,,l) NF + 1) J. Ablinger, J. Blümlein, S. Klein, C. Schneider and F. Wißbrock, Nucl. Phys. B 844 011) 6

The Wilson Coefficients at large Q L NS q,,l) NF + 1) = a s A ),NS qq,q NF + 1) δ + Ĉ),NS q,,l) NF ) + a 3 s A 3),NS qq,q NF + 1) δ + A),NS qq,q NF + 1)C,NS q,,l) NF + 1) + Ĉ3),NS q,,l) NF ) L PS q,,l) NF + 1) = a3 s A 3),PS qq,q NF + 1) δ + A) gq,q NF ) NF C g,,l) NF + 1) + NF ˆ C3),PS q,,l) NF ) L S g,,l) NF + 1) = a sa gg,q NF + 1)NF C g,,l) NF + 1) + a3 s A 3) qg,q NF + 1) δ H PS q,,l) NF + 1) = a s H S g,,l) NF + 1) = as +A ) gg,q NF + 1) NF C g,,l) NF + 1) + A) gg,q NF + 1) NF C g,,l) NF + 1) + A ),PS Qg NF + 1) NF C q,,l) NF + 1) + NF ˆ C3) g,,l) NF ), A ),PS Qq N F + 1) δ + C ),PS q,,l) NF + 1) + a 3 s 3),PS + C q,,l) NF + 1) + A) gq,q NF + 1) C g,,l) NF + 1) +A ),PS Qq N F + 1) C,NS q,,l) NF + 1), A Qg NF + 1) δ + C g,,l) NF + 1) + a s A 3),PS Qq N F + 1) δ A ) Qg NF + 1) δ + A Qg NF + 1) C,NS q,,l) NF + 1) + A gg,q NF + 1) C g,,l) NF + 1) ) + C g,,l) NF + 1) + a 3 s A 3) Qg NF + 1) δ + A) Qg NF + 1) C,NS q,,l) NF + 1) { + A ) gg,q NF + 1) C g,,l) NF + 1) + A Qg NF + 1) C ),NS q,,l) NF + 1) } ),PS + C q,,l) NF + 1) + A ) 3) gg,q NF + 1) C g,,l) NF + 1) + C g,,l) NF + 1) J. Ablinger, J. Blümlein, S. Klein, C. Schneider and F. Wißbrock, Nucl. Phys. B 844 011) 6 9/34

Variable Flavor Number Scheme f k n f + 1, µ ) + f k n f + 1, µ ) = A NS qq,q n f, µ ) m + Ã PS qq,q n f, µ ) m f Q+ Qn f + 1, µ ) = Ã PS Qq n f, µ ) m Gn f + 1, µ ) = A S gq,q Σn f + 1, µ ) = = k=1 n f, µ m ) f k n f, µ ) + f k n f, µ ) Σn f, µ ) + Σn f, µ ) + Ã S Qg Ã S qg,q Σn f, µ ) + A S gg,q n f +1 f k n f + 1, µ ) + f k n f + 1, µ ) A NS qq,q Σn f, µ ) + n f Ã S qg,q n f, µ m ) n f, µ m ) + n f Ã PS qq,q n f, µ ) m n f, µ m ) Gn f, µ ) n f, µ m ) Gn f, µ ). n f, µ m ) Gn f, µ ). + ÃPS Qq n f, µ ) m + ÃS Qg n f, µ ) Gn m f, µ ) The choice of matching scales is not free and varies with the process in case of precision observables. Blümlein, van Neerven hep-ph/9811351 = More complicated for masses J. Blümlein, Wißbrock, 013 1/34

13/34 Status of OME calculations Leading Order: Witten 1976, Babcock, Sivers 1978, Shifman, Vainshtein, Zakharov 1978, Leveille, Weiler 1979, Glück, Reya 1979, Glück, Hoffmann, Reya 198 Next-to-Leading Order: Laenen, van Neerven, Riemersma, Smith 1993 Q m : via IBP Buza, Matiounine, Smith, Migneron, van Neerven 1996 Compact results via p F q s Bierenbaum, Blümlein, Klein, 007 Oα s ε) for general N) Bierenbaum, Blümlein, Klein 008, 009 Next-to-Next-to-Leading Order: Q m Moments for F : N =...1014) Bierenbaum, Blümlein, Klein 009 Contributions to transversity: N = 1...13 Blümlein, Klein, Tödtli 009 Terms n f to F general N): Ablinger, Blümlein, Klein, Schneider, Wißbrock 011 At 3-loop order known A PS qq,q, A qg,q, A PS TR qq,q, ANS, qq,q, A gq,q and A PS Qq complete A gg : also complete This talk) A Qg : all terms of On f T F C A/F ) Two masses m 1 m

14/34. Calculation of the 3-loop operator matrix elements The OMEs are calculated using the QCD Feynman rules together with the following operator insertion Feynman rules: p, ν, b p, µ, a 1+ 1) N δ ab p) N gµν p) µpν + νpµ) p + p µ ν, N p, i p1, i µ, a p, j p, j gt a ji µ / γ± δ ij / γ± p) N 1, N 1 N j=0 p1)j p) N j, N p1, µ, a p, ν, b p3, λ, c ig 1+ 1)N f abc νgλµ λgµν) p1 + µp1,ν λ p1,λ ν) p1) N + λ p1p,µ ν + pp1,ν µ p1 pgµν p1 p µ ν N 3 j=0 p1)j p) N 3 j { { p1 p p3 p1 + µ ν λ µ } + } ) p1 p3 p p1, N µ λ ν µ p1, i p, j p3, µ, a p4, ν, b p1, i p, j p3, µ, a p4, ν, b p5, ρ, c g µ ν N 3 N / γ± j=0 l=j+1 p)j p1) N l t a t b )ji p1 + p4) l j 1 + t b t a )ji p1 + p3) l j 1, N 3 g 3 N 4 N 3 N µ ν ρ/ γ± j=0 l=j+1 m=l+1.p)j.p1) N m t a t b t c )ji.p4 +.p5 +.p1) l j 1.p5 +.p1) m l 1 +t a t c t b )ji.p4 +.p5 +.p1) l j 1.p4 +.p1) m l 1 +t b t a t c )ji.p3 +.p5 +.p1) l j 1.p5 +.p1) m l 1 +t b t c t a )ji.p3 +.p5 +.p1) l j 1.p3 +.p1) m l 1 +t c t a t b )ji.p3 +.p4 +.p1) l j 1.p4 +.p1) m l 1 +t c t b t a )ji.p3 +.p4 +.p1) l j 1.p3 +.p1) m l 1, γ+ = 1, γ = γ5. N 4 p1, µ, a p, ν, b p4, σ, d g 1+ 1)N f abe f cde Oµνλσp1, p, p3, p4) p3, λ, c +f ace f bde Oµλνσp1, p3, p, p4) + f ade f bce Oµσνλp1, p4, p, p3), { Oµνλσp1, p, p3, p4) = ν λ gµσ p3 + p4) N +p4,µ σ p4gµσ N 3 i=0 p3 + p4)i p4) N 3 i p1,σ µ p1gµσ N 3 i=0 p1)i p3 + p4) N 3 i + p1 p4gµσ + p1 p4 µ σ p4p1,σ µ p1p4,µ σ } N 4 i i=0 j=0 p1)n 4 i p3 + p4) i j p4) j } { p1 p µ ν { } p3 p4 λ σ { } + p1 p, p3 p4, N µ ν, λ σ )

3) PS Sample of diagrams for A Qq ) 15/34

16/34 The diagrams are generated using QGRAF Nogueira 1993 J. Comput. Phys. A 3),NS qq,q A 3) gq,q A 3),PS Qq A 3) gg,q A 3) Qg No. diagrams 110 86 13 64 133 A Form program was written in order to perform the γ-matrix algebra in the numerator of all diagrams, which are then expressed as a linear combination of scalar integrals. A 3),NS qq,q 746 scalar integrals. A 3) gq,q 159 scalar integrals. A 3),PS Qq 5470 scalar integrals. = Need to use integration by parts identities.

17/34 Integration by parts We use Reduze A. von Manteuffel, C. Studerus, 01 to express all scalar integrals required in the calculation in terms of a smaller) set of master integrals. Reduze is a C++ program based on Laporta s algorithm. It is somewhat difficult to adapt this algorithm to the case where we have operator insertions, due to the dependance on the arbitrary parameter N. For this reason we apply the following trick: k) N x N k) N 1 = 1 x k N=0 This can be then treated as an additional propagator, and Laporta s algorithm can be applied without further modification. If we denote the master integrals by M i, then the reduction algorithm will allow us to express any given integral I as I = i c i x)m i x)

18/34 In fact, any given diagram D will be written this way: D = i c ix)m i x) In general the coefficients c i x) will be rational functions of x, m,.p and the dimension D. We want to obtain each diagram DN) as a function of N. We proceed as follows: 1. Calculate the master integrals M i N) as functions of N.. Evaluate M i x) = N=0 x N M i N). 3. Insert the results in Dx) = i c ix)m i x). 4. Obtain DN) by extracting the Nth term in the Taylor expansion of Dx). Step 1 is done using a variety of techniques to be described shortly. Steps to 4 are done using the Mathematica packages HarmonicSums.m, SumProduction.m and EvaluateMultiSums.m by J. Ablinger and C. Schneider.

19/34 Number of master integrals: A 3),NS qq,q 35 master integrals. A 3) gq,q 41 master integrals. A 3),PS Qq 66 master integrals. A 3) gg,q 05 master integrals. A 3) Qg 334 master integrals. 4 integral families are required and implemented in Reduze.

0/34 Calculation of the master integrals For the calculation of the master integrals we use a wide variety of tools: Hypergeometric functions. Symbolic summation algorithms based on difference fields, implemented in the Mathematica program Sigma C. Schneider, 005. Mellin-Barnes representations. In the case of convergent massive 3-loop Feynman integrals, they can be performed in terms of Hyperlogarithms Generalization of a method by F. Brown, 008, to non-vanishing masses and local operators. Almkvist-Zeilberger algorithm. Differential difference) equations.

1/34 Differential difference) equations In general our N-dependent Feynman master integrals will have the following structure MN) = F N)m ) a+ 3 D p) N Differentiating the master integrals w.r.t. m or p doesn t give any new information, since we already know the functional dependence on these invariants. Differentiating w.r.t. N makes no sense. However, in the x representation of the integrals Mx) = x N F N)m ) a+ 3 D p) N, N=0 we can differentiate w.r.t. x, since as we saw before, this turns the operator insertions into artificial propagators.

/34 Differentiation will raise the powers of the artificial propagators and the resulting integrals can be re-expressed in terms of master integrals. d dx Mx) = i p i x) q i x) M ix) Integrals in a given sector will produce a system of coupled differential equations. Let s consider the following example, M 1 N) = M N) = d D k 1 d D k d D k 3 π) D π) D π) D d D k 1 π) D d D k π) D d D k 3 π) D N j=0 k 3) j k 3 k 1 ) N j, D 1 D D 3 D 4 D 5 N j=0 k 3) j k 3 k 1 ) N j D1 D, D 3 D 4 D 5 where D 1 = k 1 p), D = k p), D 3 = k 3 m, D 4 = k 3 k 1 ) m and D 5 = k 3 k ) m.

3/34 In the x representation, they become M 1 x) = M x) = d D k 1 d D k d D k 3 1 π) D π) D π) D, D 1 D D 3 D 4 D 5 D 6 D 7 d D k 1 d D k d D k 3 1 π) D π) D π) D D1 D, D 3 D 4 D 5 D 6 D 7 with D 6 = 1 x k 3 and D 7 = 1 x k 3 k 1 ). So, d dx M 1x) = 1 1 x d dx M x) = 1 1 x + ɛ 1 ) M 1 x) + x 1 ɛ + 3 ) x ɛ M x) + ɛ 4 + 3ɛ) 1 1 x 1 x 1 x x 1 x M x) + K 1 x) ) M 1 x) + K x) where K 1 x) and K x) are linear combinations of already solved) subsector master integrals.

4/34 K 1 x) and K x) can be turned into the N representation using the RISC Mathematica packages. Then using the fact that M 1 x) = M x) = x N F 1 N)m ) a+ 3 D p) N, N=0 x N F N)m ) a+ 3 D p) N, N=0 we get the following system of coupled difference equations: N + )F 1 N + 1) N + + ɛ)f 1 N) F N 1) = K 1 N), N + ɛ)f N + 1) N + 3 ) ɛ F 1 N) ɛ 4 + 3ɛ) F 1N + ) F 1 N + 1)) = K N), The system can be solved using the RISC Mathematica packages C. Schneider s talk).

5/34 The results are given in terms of standard harmonic sums: F 1 N) = 1 ɛ 3 8 S N) 4 4N S N) + 3 3 3N + 1) + 8 1) N N + ) 3N + 1) + 1 ɛ 1) N 4N 1)S1 N) 3N + 1) 4 5N ) + 1N + 9 43N + 1) 3N + 1) 3 + S N) 8 ) S 1 N) + 3N + 1)S N) 4 S 1 N)S N) + S 3 N) 3N + 1) 3 3N + 1) 3 3 + 16 S,1 N) + 4 5N ) + 10N + 4 S,1 N) 3 3 3N + 1) 3 + 1 9N ) 10N 7 S N) ɛ 3N + 1) + 1) N N 1)S 1 N) 3N + 1) N5N + 9)S 1 N) 3N + 1) 3 + 7N + 5)S N) 19N 3 + 61N ) + 68N + 30 3N + 1) + 3N + 1) 4 +ζ S N) 1 N S N) + N + 1) + 1) N N + ) N + 1) S N)S 1 3N + 1)S N) + N) 10 S 3 N) 3 3N + 1) 3 + 4 S,1 N) + 4 ) S,1 N) S 1 N) 1 ) S N N) + S 3 N) S 1 N) + S N) 4 S 1 N) 3 3 3 N + 1 3 + 43N + 1)S 1 N) + 4 ) 9N + 7) S N) + 9N 1)S 3 N) + S 4 N) S 3,1 N) 43N + )S,1 N) 3N + 1) 3 3N + 1) 3N + 1) 3 3N + 1) 4 S, N) 3N + 1)S,1 N) + 10 S 3,1 N) + 4 S,1,1 N) 4 S,1,1 N) + 19N3 + 63N + 69N + 4 3 3N + 1) 3 3 3 3N + 1) 4 S N)S 1 3 N) S,,1 N) 16 S 3,1,1 N) 14 S,1,1,1 N) + 4 S,1,1,1 N) + 8 S,,1 N) 9 3 3 3 3 + 14 S,1, N) + 3N + 1)S,1,1 N) + 5 S 4,1 N) + 3 3N + 1) 3

6/34 More complicated integrals produce generalized sums, such as the cyclotomic harmonic sums: S {a1,b 1,c 1},...{a l,b l,c l }s 1,..., s l, N) = N s1 k a 1 k 1 + b 1 ) S c1 {a,b,c },...{a l,b l,c l }s,..., s l, N) k 1=1 and sums involving inverse binomials, for example, or N i=1 ) N N N k=0 ) i ) i i i j=1 k 4 k ) S 1 k) k k 1 j j j )S 1 1,, 1; j)

7/34 Emergence of new nested sums : ) N i i ) i 1 i j ) S 1 j 1,, 1; j) i=1 j=1 j 1 = dx x N 1 x H w17, 1,0x) H w 0 x 1 8 + x 18, 1,0x) 1 + ζ dx x)n 1 x 0 x + 1 8 + x H 1x) H 13x) 1 +c 3 dx 8x)N 1 x 0 x + 1 1 x, 8 1 1 w 1 =, w 13 = x8 x) x) x8 x), 1 1 w 17 =, w 18 = x8 + x) + x) x8 + x). 100 associated independent nested sums. The associated iterated integrals request root-valued alphabets with about 30 new letters. J. Ablinger, J. Bümlein, J. Raab, C. Schneider 014.

8/34 Results By now we have computed 6 out of 7 OMEs at Oα 3 s ), namely, A 3),PS qq, A 3) qg Ablinger, Blümlein, Klein, Schneider, Wissbrock, arxiv:1008.3347 3),NS, TR A qq Ablinger, Behring, Blümlein, ADF, von Manteuffel, Schneider, et. al., arxiv:1406.4654 A 3) gq A 3),PS Qq A 3) gg Ablinger, Blümlein, ADF, von Manteuffel, Schneider, et. al., arxiv:140.0359 Ablinger, Behring, Blümlein, ADF, von Manteuffel, Schneider, arxiv:1409.1135 to be published soon and the corresponding Wilson coefficients: L PS q,, L S g,, L NS q,, and Hq, PS We have also partial results for A 3) Qg terms N F T F and Ladder diagrams). The logarithmic contributions to all OMEs and WCs were published recently Behring, Blümlein, ADF, Bierembaum, Klein, Wißbrock, arxiv:1403.6356

9/34 3-Loop OME: A PS Qq { a 3),PS N) = C Qq F T 64 N + N + ) 1 ) 64 N + N + ) 1 ) F N 1)N N + 1) S,, N + ) N 1)N N + 1) S 3,1, N + ) + N 3P 3 S,1 1, 1, N ) 3P 3 S 1 1,1,1, 1, 1, N ) 3P 4 S 1,1 1, 1, N ) + + N 1) N 3 N + 1) N + ) N 1) N 3 N + 1) N + ) N 1) 3 N 4 N + 1) N + ) + N 64 N + N + ) S1,,1, 1, 1, N ) + 64 N + N + ) S1,,1, 1, 1, N ) } + + N 1)N N + 1) N + ) N 1)N N + 1) N + ) +C F T F N F { 16 N + N + ) S1 N) 3 + 16P 9 S 1 N) 7N 1)N N + 1) N + ) 7N 1)N 3 N + 1) 3 N + ) 08 N + N + ) 3P 3 + S S 1 9N 1)N N + 1) N + ) 81N 1)N 4 N + 1) 4 N + ) 3 3P 31 4 N ) + N + } + 43N 1)N 5 N + 1) 5 N + ) 4 + 9N 1)N N + 1) ζ 3 + N + ) { N ) + N + S1 N) 4 4 N + N + ) P 6 S 1 N) 3 +C F C A T F 9N 1)N N + 1) + N + ) 7N 1) N 3 N + 1) 3 N + ) + N 16P S 3, N) N 1)N 3 N + 1) 16P S 1,, 1, N) N 1)N 3 N + 1) + 16P S,1, 1, N) N 1)N 3 N + 1) 16P S 1,1,1, 1, 1, N) N 1)N 3 N + 1) 3 N + N + ) S1,1,, 1, 1, N ) N 1)N N + 1) + 3 N + N + ) S1,1,, 1, 1, N ) N + ) N 1)N N + 1) N + ) + 3 N + N + ) S1,,1, 1, 1, N ) N 1)N N + 1) N + ) 3 N + N + ) S1,1,1,1, 1, 1, 1, N ) N 1)N N + 1) N + ) 3 N + N + ) S1,,1, 1, 1, N ) N 1)N N + 1) N + ) 3 N + N + ) S1,1,1,1, 1, 1, 1, N ) N 1)N N + 1) N + ) + } +

30/34 4000 000 0 x a 3),PS Qq x) 000 4000 exact 6000 Ox 1 ln x) Ox 1 ) + Ox 1 ln x) Ox 1 ) + Ox 1 ln x) + i 8000 Olni x) 10 5 10 4 10 3 10 10 1 1 x Figure: xa 3),PS Qq x) in the low x region solid red line) and leading terms approximating this quantity; dotted line: leading small x approximation Olnx)/x), dashed line: adding the O1/x)-term, dash-dotted line: adding all other logarithmic contributions.

31/34 0-1000 x a 3),PS Qq x) -000-3000 -4000-5000 10 5 10 4 10 3 10 10 1 1 x Approx. A,B exact Ox 1 ln x) Ox 1 ) + Ox 1 ln x) Figure: Comparison of the exact result for xa 3,PS Qq x) solid red line) with an estimate in Kawamura et. al. shaded area). Also shown are the first two leading small x contributions.

0.001 0 0.001 0.00 0.003 0.004 0.005 0.006 0 Oa s) Oa 3 s) Q = 100 GeV Q = 00 GeV 0.005 0.01 0.015 0.0 0 0.01 0.0 0.03 0.04 Q = 500 GeV Q = 1000 GeV 0.05 Q = 5000 GeV Q = 10000 GeV 0.06 10 5 10 4 10 3 10 10 1 10 5 10 4 10 3 10 10 1 1 x Figure: The bottom contribution by the Wilson coefficient H PS Q, to the structure function F x, Q ) as a function of x and Q choosing Q = µ, m b = 4.78GeV on shell scheme). 3/34

33/34 3-Loop OME: A gg,q a 3) gg,q = ) i i i j=1 4 j S 1 j 1) ) j j j 7ζ 3 4 i i + 1 ) 1 + 1) N { C F 16 N ) + N + N T F N N + 1) i=1 ) 18 + γ gq 0) S 4 S 3 S 1 + S 3,1 + S, + 4 5N + 5N ) S 1 S 3NN + 1)N + ) 3NN + 1)N + ) ) i i i j=1 4 j S 1 j 1) ) j j j 7ζ 3 4 i i + 1 ) 4P 69 S 1 3N 1)N 4 N + 1) 4 N + ) + ) + 16P 4 N +C A C F T F 3N 1)N N + 1) + N + ) i=1 3P S, N 1)N N + 1) N + ) 64P 14 S,1,1 16P 3 S 4 4P 63 S 4 + + 3N 1)N N + 1) N + ) 3N 1)N N + 1) N + ) 3N )N 1)N N + 1) N + ) ) i i S 1 j 1) ) i j +C A T 4P 46 N j=1 j j 7ζ 3 56P 5 S, F 3N 1)N N + 1) N + ) i=1 4 i i + 1 ) + 9N 1)N N + 1) N + ) + 3P 30 S,1,1 + 16P 35 S 3,1 + 16P 44 S 4 16P 5 S 8P 36 S 9N 1)N N + 1) + N + ) 7N 1)N N + 1) + N + ) 9N 1)N N + 1) + ) N 16P 48 4 N N 4 i S 1 i 1) ) i=1 i +C F TF i i 7ζ 3 3P 86 S 1 3N 1)NN + 1) N + )N 3)N 1) 81N 1)N 4 N + 1) 4 N + )N 3)N 1) 16P 45 S } + 1 16P 45 S + + 7N 1)N 3 N + 1) 3 N + ) 9N 1)N 3 N + 1) 3 N + ) Also, with this calculation we were able to rederive the three loop anomalous dimension γ gg 3) agreement with the literature. for the terms T F, and obtained

34/34 Conclusions 009: 10-14 Mellin Moments for all massive 3-loop OMEs, WC. 010: Wilson Coefficients L 3),PS q N), L 3),S g N). 013: Reduze has been adapted to the case where we have operator insertions, and used for reductions to master integrals. Complicated master integrals have been computed using a variety of methods. In particular, the differential equations method has allowed us to tackle integrals up until now untreatable with other methods. Emergence of generalized sums and new functions including a large number of root-letters in iterated integrals. 3),S A gq,q, A3),NS,TR qq,q and A 3),PS Qq have been completed. The corresponding 3-loop anomalous dimensions were computed, those for transversity for the first time ab initio. 014: The terms TF in A3) gg was computed. 015: Full A 3) gg is now available. Different new Computer-algebra and mathematical technologies have been and continue to be developed.