Johannes Blümlein, DESY [in collaboration with I. Bierenbaum and S. Klein]

Transcription

1 Oα 3 1 s Heavy Flavor Wilson Coefficients in Q m Johannes Blümlein, DESY [in collaboration with I. Bierenbaum and S. Klein Introduction Theory Status The Method Asymptotic Loop Results all N Asymptotic 3 Loop Results fixed moments Conclusions Johannes Blümlein RADCOR09, Ascona October 009

2 References: J. Blümlein, A. De Freitas, W.L. van Neerven, and S. Klein: Nucl. Phys. B I. Bierenbaum, J. Blümlein and S. Klein: Phys. Lett. B , Nucl. Phys. B , Phys. Lett. B , Nucl. Phys. B I. Bierenbaum, J. Blümlein, S. Klein and C. Schneider: Nucl. Phys. B J. Blümlein, S. Klein and B. Tödtli: arxiv: , Phys. Ref. D 009 in print.

3 Deep Inelastic Scattering DIS: k q k P W µν 1. Introduction L µν Q := q, x := Q pq ν := Pq M, dσ dq dx W µνl µν Bjorken x 3 W µν q, P, s = 1 4π { unpol. = 1 x { pol. d 4 ξ expiqξ P, s [Jµ em ξ, Jem ν 0 P, s g µν q µq ν q F L x, Q + x Q M Pq ε µναβq α [ s β g 1 x, Q + P µ P ν + q µp ν + q ν P µ x s β sq Pq pβ g x, Q. Q 4x g µν F x, Q Structure Functions: F,L, g 1, contain light and heavy quark contributions

4 4 Let us compare the scaling violations in the massless and massive case. F x,q F,c x,q LO charm contributions : PDFs from [Alekhin, Melnikov, Petriello, 006. = different scaling violations!

5 Light and Heavy Quark Contributions 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 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 MRST004 MRST NNLO MRST004FF3 CTEQ6.5 CTEQ5F x = 0.000, i = 13 x = 0.003, i = 1 x = , i = 11 x = , i = 10 x = 0.013, i = 9 x = 0.00, i = 8 x = 0.03, i = x= , i=15 x=0.0005, i=14 x=0.0006, i=13 x=0.0008, i=1 x=0.0010, i=11 x=0.0015, i= x = 0.050, i = 6 x = 0.080, i = 5 x = 0.13, i = 4 x = 0.18, i = 3 x = 0.5, i = x=0.00, i=9 x=0.003, i=8 x= , i=7 x=0.005, i=6 x=0.006, i= H1 PDF 000 extrapolation x = 0.40, i = 1 x = 0.65, i = Q / GeV H1 Collaboration ZEUS D H1 D H1 Displaced Track High statistics for F and F c c = Accuracy will increase in the future. x=0.01, i=4 x=0.013, i=3 x=0.030, i= x=0.03, i=1 Q /GeV [Thompson, 007

6 6 HERA F cc H1+ZEUS BEAUTY CROSS SECTION in DIS F cc Q = GeV H1 HERA I: D* VTX ZEUS HERA II: D* D +, D 0, D s + H1 prel. HERA II: D* VTX ZEUS prel. HERA II: D* D + µ 4 GeV 7 GeV NLO QCD: CTEQ5F3 MRST004FF3 σ bb _ Q = 5 GeV Q = 1 GeV Q = 5 GeV 0 11 GeV 18 GeV 30 GeV 0 Q = 60 GeV Q = 130 GeV Q = 00 GeV GeV 130 GeV 500 GeV [Krüger H1 and Z. Coll., x Q = 650 GeV x x H1 Prel. HERA I+II VTX ZEUS Prel. HERA II 39 pb -1 03/04 µ ZEUS Prel. HERA II 15 pb µ MSTW08 Prel. CTEQ CTEQ5F3 x [Krüger H1 and Z. Coll., 008.

7 Evolution of Light Quark Distributions The scaling violations are described by the splitting functions P ij x, a s. They describe the probability to find parton i radiated from parton j and carrying its momentum fraction x. They are related to the anomalous dimensions via a Mellin Transform: M[fN := 1 0 dzz N 1 fz, γ ij N, a s := M[P ij N, a s. The splitting functions govern the scale evolution of the parton densities. d ΣN, Q d lnq = γ qq γ qg ΣN, Q, GN, Q GN, Q γ gq γ gg d d lnq q NSN, Q = γ qq,ns q NS. The singlet light flavor density is defined by Σn f, µ = n f f i n f, µ + f i n f, µ. i=1 The anomalous dimensions are presently known at NNLO [Moch, Vermaseren, Vogt, 004 and N 3 LO for NS +, N= [Baikov & Chetyrkin, 006 7

8 . Theory Status 8 Leading Order : F,L x, Q [Witten, 1976; Babcock, Sivers, 1978; Shifman, Vainshtein, Zakharov, 1978; Leveille, Weiler, 1979; Glück, Reya, 1979; Glück, Hoffmann, Reya, 198. Leading Order : g 1 x, Q [Watson, 198; Glück, Reya, Vogelsang, 1991; Vogelsang, 1991 Leading Order : g x, Q [J.B., Ravindran, van Neerven, 003 = holds to all orders Wandzura-Wilczek relation. Soft resummation: F,L x, Q [ Laenen & Moch, 1998; Alekhin & Moch, 008 Next-to-Leading Order : F,L x, Q [Laenen, Riemersma, Smith, van Neerven, 1993, 1995 asymptotic: [Buza, Matiounine, Smith, Migneron, van Neerven, 1996; Bierenbaum, J.B., Klein, 007 Next-to-Leading Order : g 1 x, Q asymptotic: [Buza, Matiounine, Smith, Migneron, van Neerven, 1997; Bierenbaum, J.B., Klein, 009 Fast semi-analytic representation in Mellin space: [Alekhin & J.B., 003 This includes power corrections. = 3-Loop corrections needed to line up with the accuracy reached for light flavor contributions.

9 3. The Method 9 massless RGE and light cone expansion in Bjorken limit {Q, ν}, x fixed: [ lim Jξ, J0 c N ξ i,τ 0 ξ, µ ξ µ1...ξ µn O µ 1...µ m i,τ 0, µ. 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 Oa 3 s Q C,L;i fl = δ i,q + a l s Cfl,l,L,i, µ l=1 : [Moch, Vermaseren, Vogt, 005. i = q, g Heavy quark contributions given by heavy quark Wilson coefficients, H S,NS Q,L,i µ, m Q.

10 10 In the limit Q m h [Q 10 m for F, g 1 : 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 k i = δ k,i + a l sa S,NS,l m k,i, i = q, g [Buza, Matiounine, Migneron, Smith, van Neerven, 1996; Buza, Matiounine, Smith, van Neerven, Heavy OMEs occur as well as transition functions to define the VFNS starting from a fixed flavor number schemeffns. [Buza, Matiounine, Smith, van Neerven, 1998; Chuvakin, Smith, van Neerven, µ

11 11 Comparison for LO: R ξ Q m H 1,g H 1,g,asym Comparison to exact order Oa s result: asymptotic formulae valid for Q > 0 GeV/c in case of F cc x, Q and Q > 800 GeV/c for F cc L x, Q Drawbacks: - Power corrections m /Q k cannot be calculated using this method. - The case of two different heavy quark masses is still too complicated = scale problem to be treated semianalytically. - Only inclusive quantities can be calculated = structure functions.. R ξ=q /m c

12 1 Renormalization  ij = δ ij + Mass renormalization on mass shell scheme Charge renormalization k=0 â k sâ k ij MOM scheme MS scheme D = 4 + ε: NEW! Terms were forgotten even at NLO. Renormalization of ultraviolet singularities = are absorbed into Z factors given in terms of anomalous dimensions γ ij. Factorization of collinear singularities = are factored into Γ factors Γ NS, Γ ij,s and Γ qq,ps. For massless quarks it would hold: Γ = Z 1. Here: Γ matrices apply to parts of the diagrams with massless lines only. Generic formula for operator renormalization and mass factorization: A ij = Z 1 il  lm Γ 1 mj = Oε terms of the loop OMEs are needed for renormalization at 3 loops.

13 4. Oa s Contributions to Oε 13 Refs.: [I. Bierenbaum, J.B., S. Klein, Phys. Lett. B ; B ; Nucl. Phys. B ; I. Bierenbaum, J.B., S. Klein, C. Schneider, Nucl. Phys. B Use of generalized hypergeometric functions for general analytic results = allows feasible computation of higher orders in ε & automated check for fixed values of N. use of Mellin-Barnes integrals for numerical checks MB, [Czakon, 006. 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 1 S + S S 1ζ 3 + ζ S1 + S use of integral techniques and the Mathematica package SIGMA [Schneider, 007., [Bierenbaum, J.B., Klein, Schneider, 007, 008. Partial checks for fixed values of N using SUMMER, [Vermaseren, Computation within general R ξ gauge..

14 14 We calculated all loop Oε terms in the unpolarized case and several loop Oε terms in the polarized case: a Qg, a,ps Qq, a gg,q, a gq,q, a,ns qq,q. a Qg, a,ps Qq, a,ns qq,q. We verified all corresponding loop Oε 0 results by van Neerven et. al. A remark on the appearing functions: 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 harmonic sums with index { 1} cancel holds even for each diagram J.B., 004; J.B., Ravindran, 005,006; J.B., Klein, 007; J.B., Moch in preparation; J.B [hep-ph; [math-ph Expectation for 3 loops: weight 5 6 harmonic sums

15 ¼ Ö 15 Diagrams contain two scales: the mass m and the Mellin parameter N. point functions with on shell external momentum, p = 0. reduce for N = 0 to massive tadpoles. E.g. diagrams contributing to the gluonic OME Â Qg = Ñ Ó Ð Ô Õ Ò Ö Ø

16 16 Example: 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 5. Fixed moments at 3 Loop: F QQ 17 F QQ L x, 3-loop: [J.B., De Freitas, Klein, van Neerven, Nucl. Phys. B Refs.: [I. Bierenbaum, J.B., S. Klein, Nucl.Phys.Proc.Suppl ; arxiv:081.47; PoS Contributing OMEs: Singlet A Qg A qg,q A gg,q A gq,q Pure Singlet A PS Qq A PS qq,q } mixing Non Singlet A NS,+ qq,q All loop Oε terms in the unpolarized case are known. 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. = The calculation will provide first independent checks on γ qg, γ qq,ps and on respective color projections of γ qq,ns+, γ gg and γ gq. The calculation proceeds in the same way in the polarized case. Independent checks provided by pole terms anomalous dimensions, lower order OMEs and sum rules for N =.

18 18 Fixed moments using MATAD 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 3 loop OMEs are generated with QGRAF [Nogueira, Color factors are calculated with [van Ritbergen, Schellekens, Vermaseren, Translation to suitable input for MATAD [Steinhauser, 001. Tests performed: Various loop calculations for N =, 4, 6,... were repeated agreement with our previous calculation. Several non trivial scalar 3 loop diagrams were calculated using Feynman parameters for all N agreement with MATAD.

19 19 A 3,PS Qq + A 3,PS General structure of the result: the PS case qq,q = ˆγ0 qg γ gq ˆγ 1,PS qq ˆγ,PS qq + n f ζ 16 γ 0 gg γ 0 qq + 4n f + 1β 0,Q + 6β 0 ln 3 m µ n f + 1β 0,Q β 0 + ˆγ0 qg 8 ˆγ 0 qg a n f + 1ˆγ gq 1 γ gq 1 γ0 gq ˆγ qg 1 8 ˆγ qg 0 γ 0 gq ζ γ gg 0 γ qq 0 + 4n f + 1β 0,Q + 6β 0 a,ps Qq β 0 16 gq,q γ0 gq a Qg 4n f β 0,Qˆγ qq 1,PS + ˆγ qg 0 γ gq 1 m ln µ +C F ζ ˆγ qg 0 γ gq 0 4ˆγ qq 1,PS ln m µ γ gq 0 ˆγ 0 qg + ζ 3 γ gg 0 γ qq 0 + 4n f β 0,Q + 6β β 0 + β 0,Q a,ps Qq + γ gq 0 a Qg n f + 1ˆγ qg 0 a gq,q + 1a,PS Qq + a 3,PS Qq + a 3,PS qq,q. n f dependence non trivial. Take all quantities at n f flavors and adopt notation ˆγ ij γ ij n f + 1 γ ij n f, β 0,Q β 0 n f + 1 β 0 n f. There are similar formulas for the remaining OMEs.

20 0 Number of Diagrams to be calculated: A 3,PS Qqq : 13, A3,NS qq : 18, A 3 gq : 89, A 3 Qg We calculated the terms A 3,PS Qq + A 3,PS qq,q, A3,NS qq,q, A3 gq,q : 1498, A3 gg,q : 1; A3 Qg, A3 qg,q, A3 gg,q : days CT : 10; for N =, 4, 6, 8, 10, 1 using MATAD and find agreement of the pole terms with the prediction obtained from renormalization. An additional check is provided by the sum rule +A 3,PS +A 3,PS +A 3 N= N= N= A 3,NS qq,q which is fulfilled by our result. qq,q Qq gq,q = 0, N= All terms proportional to ζ cancel in the renormalized result for A 3,NSPS Qqq, A 3 gq,q We observe the number The term B4 appears as B4 = 4ζ ln + 3 ln4 13 ζ Li 4 1 C F C A B4. = 8σ 3, ζ 4.

21 1 Result for the renormalized PS term for N = 4. A 3,PS Qq + + A 3,PS qq,q N=4 = C FT F n f C FC A T F C F T F C FT F C FC A T F C F T F C FT F ζ ζ C F T F n f ζ C F C A T F ln m µ C F C A T F + 75 ζ ζ C F T F C FT F n f ln 3 m µ C m F T F ln µ B4 + 5 ζ ζ B4 5 ζ 4 C F T F.

22 The constant terms: N = 1 â 3 gq,q : â 3 gq,q N=1 m 3ε/ = T F {C µ F T F n f +C F C A [ B ζ ζ ζ ζ ζ [ + CF B ζ ζ ζ ζ }. + C F T F [ ζ 3

23 3 The constant terms: N = 1 â 3,NS qq,q N=1 = T F m µ 3ε/ â 3,NS qq,q : { C F T F n f ζ ζ [ C F C A B ζ ζ ζ C F [ B ζ ζ ζ [ } +C F T F ζ ζ

24 4 The constant terms: N = 1 â 3,PS Qq â 3,PS qq,q N=1 N=1 = T F m = µ 3ε/ â 3,PS Qq { + â 3,PS qq,q : C F T F n f ζ ζ [ 148 +C F C A B ζ ζ ζ C F ζ ζ [ B ζ + C F T F [ ζ 3 } ζ m { 3ε/ CF µ TF 4964 n f ζ ζ }.

25 5 The constant terms: N = 10 â 3 gg,q : â 3 gg,q N=10 m 3ε/ = T F {n µ f T F [ 1518 C A 4455 ζ ζ [ +C F ζ ζ ζ ζ ζ ζ C A[ B 4 + C A C F [ B ζ ζ [ +CF B ζ ζ ζ T F C A [ ζ ζ T F C F [ ζ ζ } ζ 3T F.

26 6 The constant terms: N = 10 â 3 Qg + â3 qg,q : â 3 Qg â 3 qg,q N=10 N=10 = T F m µ 3ε/ [ {n f T F C A ζ ζ [ C F ζ ζ ζ ζ ζ ζ C F [ B C A C F [ B ζ 4 + C A[ B ζ ζ ζ ζ + T F C A [ ζ ζ T F C F [ ζ ζ } T F ζ 3. = n f TF m 3ε/ [ {C µ A ζ ζ [ C F ζ ζ }

27 7 We obtain for the moments of the PS, and ˆγ gq anomalous dimensions N ˆγ,PS qq /T F /C F T F 1 + n f + 56 C F C A ζ C A C F T F 1 + n f C F C A ζ C A C F T F 1 + n f T F 1 + n f T F 1 + n f T F 1 + n f C F C F C A ζ C A C F C F C A ζ C A C F C F C A ζ C A C F C F C A ζ C A N ˆγ gq /T F /C F 7 81 T F 1 + n f + 51 C A C F ζ C A C F T F 1 + n f T F 1 + n f T F 1 + n f T F 1 + n f T F 1 + n f C A C F ζ C A C F C A C F ζ C A C F C A C F ζ C A C F C A C F ζ C A C F C A C F ζ C A C F

28 8 The moments of the ˆγ qg and ˆγ gg anomalous dimensions N ˆγ qg /T F + ζ n f T F C A C F n f T F C A C F 416C A C F + 88C A + 18C F 3 + ζ C A C AC F C F n f T F C A C F 5 83C A 3876C AC F C F + ζ C A C AC F C F n f T F C A C F C A C AC F C F 69864C A 94664C AC F C F + ζ C A C AC F C A C AC F C F n f T F C A C F 7 C F + ζ C A C AC F C F C A C AC F C F N ˆγ gg /T F + ζ n f T F C A C F n f T F C A C F 88C A + 416C AC F 18C F 3 + ζ C A C AC F C F n f T F C A C F C A C AC F C F n f T F C A C F 664C A + 658C AC F 64C F + ζ C A C AC F C A C AC F C F 147 C F + ζ C A C AC F C A C AC F C F n f T F C A C F 7 C F + ζ C A C AC F C F C A C F C A C F

29 The moments of the NS + ˆγ qq anomalous dimensions 9 N ˆγ,NS,+ qq /T F /C F T F 1 + n f + 56 C F C A ζ C A C F T F 1 + n f + 51 C F C A ζ C A C F T F 1 + n f T F 1 + n f T F 1 + n f T F 1 + n f C F C F C A ζ C A C F C F C A ζ C A C F C F C A ζ C A C F C F C A ζ C A = Agreement for the terms T F with [Larin, Nogueira, Ritbergen, Vermaseren, 1997; Moch, Vermaseren, Vogt, 004. How far can we go? N = 14 in some cases; generally: N = 10 achieved. CPU time: various months; up to 64 Gb processors needed. = Phenomenology Unfortunately not enough to perform the automatic fixed moments all moments turn. [J.B., Kauers, Klein, Schneider, [hep-ph. Transversity: 13 moments calculated at Oa 3 s ; agreement with Gracey s anomalous dimensions to N = 8 [J.B., Klein, Toedtli 009

30 30 Fixed moments using Feynman parameters ν 3 ν Consider e.g the 3 loop tadpole diagram 5 ν 4 ν ν 1 Using Feynman parameters, one obtains a representation in terms of a double sum I = CΓ [ + ε/ ν1, + ε/ ν 5, ν 1 ε/, ν 45 ε/, ν ε, ν /ε m,n=0 ν 1, ν, ν 4, + ε/, ν 345 ε/, ν ε ν 345 ε/ n+m ν /ε m + ε/ ν 1 m + ε/ ν 5 n ν 45 ε/ n m!n!ν ε n+m ν 345 ε/ m ν 345 ε/ n, which derives from a Appell function of the first kind, F 1.

31 First all N results As in the loop case, for any diagram deriving from the tadpole ladder topology, one obtains for fixed values of N a finite sum over double sums of the same type. Consider e.g. the scalar diagram 31 For the above diagram, we obtained a result for arbitrary N using similar techniques as in the loop case and the package SIGMA. S,1,1 N + N N + 5S 3,1 S 1 4 N + 1N + N S,1 + N + 3S 1 + 5N + 6 S N S 5N + N N + 1 S 7N N + 1N + 5 S 1 S + N N + 1 S N + 5S 1 N + 1N + + 4N + 3S 1 N + 1 N + N + 3S 4 N Oε. N N + I 1 = 4N + 1S N + 1 N + ζ N + 1S 1 4N N + 1 For fixed N, this formula agrees with the result we obtain using MATAD.

32 6. Conclusions The heavy flavor contributions to the structure function F are rather large in the region of lower values of x. Competitive QCD precision analyzes therefore require the description of the heavy quark contributions to 3 loop order. The inclusive heavy flavor contributions to F,L x, Q are known to NLO in semi-analytic form with fast numeric implementations. Complete analytic results are known in the region Q m at NLO for F,L x, Q, g 1, x, Q. They are expressed in terms of massive operator matrix elements and the corresponding massless Wilson coefficients. Threshold resummations were performed. F L x, Q is known to NNLO for Q m. The calculation of fixed moments of the unpolarized massive operator matrix elements and for transversity at Oa 3 s has been finished for N = 10, 1, 13, 14, which will be followed by some first phenomenological parameterization. We also calculate the matrix elements necessary to transform from the FFNS to the VFNS. First steps towards the calculation of the massive operator matrix elements at Oa 3 s for general values of N have been performed already in the same way we solved the -loop problem. 3