Moments of the 3-Loop Corrections to the Heavy Flavor Contribution to - PowerPoint PPT Presentation

1 Moments of the 3-Loop Corrections to the Heavy Flavor Contribution to F 2 ( x, Q 2 ) for Q 2 ≫ m 2 Sebastian Klein, DESY in collaboration with I. Bierenbaum and J. Bl¨ umlein • Introduction and Theory Status • The Method • 2 Loop Results • Asymptotic 3 Loop Results (Fixed Moments) & Anomalous Dimensions • Towards an all– N Result at 3 Loops. • Conclusions Sebastian Klein Fermilab, Theory Seminar 18.06.2009

2 References : - I. Bierenbaum, J. Bl¨ umlein, and S. K., Phys. Lett. B648 (2007) 195, 0702265 [hep-ph] ; Nucl. Phys. B780 (2007) 40, 0703285 [hep-ph] ; Phys. Lett. B672 (2009) 401, 0901.0669 [hep-ph] ; Nucl. Phys. B (2009), 0904.3563 [hep-ph] ; in print . - I. Bierenbaum, J. Bl¨ umlein, S. K. and C. Schneider, Nucl. Phys. B803 (2008) 1, 0803.0273 [hep-ph] ; 0707.4659 [math-ph] . - J. Bl¨ umlein, M. Kauers, S. K. and C. Schneider, Comput. Phys. Commun. (2009), 0902.4091 [hep-ph] ; in print . - J. Bl¨ umlein, A. De Freitas, W.L. van Neerven and S. K., Nucl. Phys. B755 (2006) 272, 0605310 [hep-ph] .

3 1. Introduction Deep–Inelastic Scattering: k ′ k Q 2 Q 2 := − q 2 , x := 2 P · q , Bjorken–x L µν q ν := P · q , M dσ dQ 2 dx ∼ L µν W µν P W µν The picture of the proton at short distances [Feynman, 1969; Bjorken, Paschos, 1969.] • The proton mainly consists of light partons. • There are three valence partons: two up quarks and one down quark. • The sea–partons are: u, u, d, d, s, s and the gluon g .

4 The hadronic tensor cannot be calculated perturbatively. It can be decomposed into several scalar structure functions. For DIS via single photon exchange, it is given by: � W µν ( q, P, s ) = 1 d 4 ξ exp( iqξ ) � P, s | [ J em µ ( ξ ) , J em (0)] | P, s � ν 4 π � � � � � − Q 2 = 1 g µν − q µ q ν F L ( x, Q 2 ) + 2 x P µ P ν + q µ P ν + q ν P µ F 2 ( x, Q 2 ) unpol. 4 x 2 g µν q 2 Q 2 2 x 2 x � � � � � − M s β − sq s β g 1 ( x, Q 2 ) + g 2 ( x, Q 2 ) 2 Pq ε µναβ q α Pq p β pol. . In Bjorken limit, { Q 2 , ν } → ∞ , x fixed, at twist τ = 2–level: � � x, Q 2 µ 2 , m 2 � F i ( x, Q 2 ) f j ( x, µ 2 ) , k = C i,j ⊗ µ 2 � �� � � �� � j � �� � structure functions parton densities, Wilson coefficients, non-perturbative perturbative = ⇒ Wilson coefficients contain both light and heavy flavor contributions: � � � � � � x, Q 2 µ 2 , m 2 x, Q 2 x, Q 2 µ 2 , m 2 = C light k k C i,j + H i,j , k = c, b . i,j µ 2 µ 2 µ 2

5 The Discovery of Heavy Quarks J [Aubert et al. , 1974] @ BNL Υ [Herb et al. , 1977] @ FERMILAB Ψ [Augustin et al. , 1974] @ SLAC • Masses of charm and bottom [PDG, 2008.] : m c ≈ 1 . 3 GeV, m b ≈ 4 . 2 GeV

6 Heavy Quarks in DIS • Assume only light partons in the proton. Light quarks may directly scatter off the exchanged vector boson, the gluon via quark–pair production. • Heavy quarks (c or b) emerge in final states through hard scattering processes (top outside the HERA region). • LO contribution to F (2 ,L ) by heavy quark production: photon-gluon fusion � 1 � x � z , m 2 dz a = 1 + 4 m 2 /Q 2 . z H (1) F QQ (2 ,L ) ( x, Q 2 ) = 4 e 2 G ( z, Q 2 ) , c a s (2 ,L ) ,g Q 2 ax � • open c(b) • heavy quark J / Ψ = cc Q Υ = bb production: resonances: D u = uc , ... cc = J / Ψ Q B u = ub , ... bb = Υ . [Witten, 1976; Gl¨ uck, Reya, 1979, ...] [Berger, Jones, 1981.] • Observation of charmonium in DIS [Aubert et al. , 1983.]

7 The Gluon Distribution • Gluon carries roughly 50% of the proton momentum. • Heavy quark production is an excellent way to extract the gluon density via a measurement of - scaling–violations of F 2 , - F QQ . L • First extraction of the gluon density including heavy quark effects by uck, Hoffmann, Reya, 1982.] : [Gl¨ - Unfold the gluon density via � f ( x, Q 2 ) G ( x, Q 2 ) = P − 1 qg ⊗ x � − 2 3 P cg ⊗ G ( x, Q 2 ) .

8 – cc H1+ZEUS BEAUTY CROSS SECTION in DIS HERA F 2 _ ∼ bb – cc Q 2 = 5 GeV 2 Q 2 = 12 GeV 2 Q 2 = 25 GeV 2 F 2 Q 2 = 2 GeV 2 4 GeV 2 7 GeV 2 σ H1 HERA I: NLO QCD: D* VTX CTEQ5F3 0.02 ZEUS HERA II: 0.4 MRST2004FF3 D + , D 0 , D s + D* H1 (prel.) HERA II: D* VTX ZEUS (prel.) HERA II: 0.01 D + 0.2 D* µ 0 0 Q 2 = 60 GeV 2 Q 2 = 130 GeV 2 Q 2 = 200 GeV 2 11 GeV 2 18 GeV 2 30 GeV 2 0.1 0.4 0.05 0.2 0 0 -4 -3 -2 -4 -3 -2 Q 2 = 650 GeV 2 10 10 10 x 10 10 10 60 GeV 2 130 GeV 2 500 GeV 2 x 0.04 H1 (Prel.) HERA I+II VTX 0.4 0.03 ZEUS (Prel.) HERA II 39 pb -1 (03/04) µ ZEUS (Prel.) HERA II 125 pb -1 (05) µ 0.02 0.2 MSTW08 (Prel.) 0.01 CTEQ6.6 0 CTEQ5F3 0 -4 -3 -2 -5 -3 -5 -3 -5 -3 -1 10 10 10 10 10 10 10 10 10 10 x x [Kr¨ uger (H1 and Z. Coll.), 2008.] [Kr¨ uger (H1 and Z. Coll.), 2008.]

9 10 12 × 4 i x=0.00003, i=22 F 2 ⋅ 2 i H1 e + p x = 0.000050, i = 21 MRST2004 _ F 2cc x=0.00005, i=21 x = 0.000080, i = 20 10 11 10 6 ZEUS e + p x = 0.00013, i = 19 x=0.00007, i=20 MRST NNLO x = 0.00020, i = 18 x=0.00013, i=19 BCDMS MRST2004FF3 x = 0.00032, i = 17 10 10 x=0.00018, i=18 10 5 x = 0.00050, i = 16 NMC CTEQ6.5 x = 0.00080, i = 15 x=0.000197, i=17 10 9 x = 0.0013, i = 14 CTEQ5F3 x=0.00020, i=16 x = 0.0020, i = 13 10 4 10 8 x=0.00035, i=15 x = 0.0032, i = 12 x = 0.0050, i = 11 x=0.0005, i=14 10 7 10 3 x = 0.0080, i = 10 x=0.0006, i=13 x=0.0008, i=12 x = 0.013, i = 9 10 6 x = 0.020, i = 8 x=0.0010, i=11 10 2 x = 0.032, i = 7 10 5 x=0.0015, i=10 x = 0.050, i = 6 x=0.002, i=9 x = 0.080, i = 5 10 10 4 x=0.003, i=8 x = 0.13, i = 4 x=0.00316, i=7 x = 0.18, i = 3 10 3 1 x = 0.25, i = 2 x=0.005, i=6 10 2 x=0.006, i=5 x = 0.40, i = 1 -1 ZEUS D ✽ 10 x=0.012, 10 i=4 H1 D ✽ H1 Collaboration x=0.013, i=3 -2 x = 0.65, i = 0 H1 PDF 2000 H1 Displaced Track 1 10 x=0.030, i=2 extrapolation -1 x=0.032, 10 -3 i=1 10 2 3 1 10 10 10 2 3 4 5 Q 2 /GeV 2 1 10 10 10 10 10 Q 2 / GeV 2 [Thompson, 2007.] • High statistics for F 2 and F c ¯ c 2 . Accuracy will increase in the future. • F c ¯ 2 ( x, Q 2 ) ∼ 20 − 40 % of F 2 ( x, Q 2 ) for small values of x , but different scaling violations. c

10 Splitting Functions • The scaling violations are described by the splitting functions P ij ( x, a s ). • They describe the probability to find a parton i being radiated from parton j and carrying its momentum fraction x . • They are related to the anomalous dimensions via a Mellin–Transform: � 1 dzz N − 1 f ( z ) , γ ij ( N, a s ) := − M [ P ij ]( N, a s ) . M [ f ]( N ) := 0 • The splitting functions govern the scale–evolution of the parton densities.        Σ( N, Q 2 )  Σ( N, Q 2 )  γ qq γ qg d  ⊗  , = −  d ln Q 2 G ( N, Q 2 ) G ( N, Q 2 ) γ gq γ gg d d ln Q 2 q NS ( N, Q 2 ) − γ NS = ⊗ q NS . qq • The singlet light flavor density is defined by n f � ( f i ( n f , µ 2 ) + ¯ Σ( n f , µ 2 ) f i ( n f , µ 2 )) . = i =1 • The anomalous dimensions are presently known at NNLO [Moch, Vermaseren, Vogt, 2004.] )

11 Theory Status of Heavy Quark Corrections Leading Order : F 2 ,L ( x, Q 2 ) [Witten, 1976; Babcock, Sivers, 1978; Shifman, Vainshtein, Zakharov, 1978; Leveille, Weiler, 1979; Gl¨ uck, Reya, 1979; Gl¨ uck, Hoffmann, Reya, 1982.] Leading Order : g 1 ( x, Q 2 ) [Watson, 1982; Gl¨ uck, Reya, Vogelsang, 1991; Vogelsang, 1991] Leading Order : g 2 ( x, Q 2 ) [Bl¨ umlein, Ravindran, van Neerven, 2003] Soft Resummation: F 2 ,L ( x, Q 2 ) [ Laenen & Moch, 1998; Alekhin & Moch, 2008] Next-to-Leading Order : F 2 ,L ( x, Q 2 ) [Laenen, Riemersma, Smith, van Neerven, 1993, 1995] asymptotic: [Buza, Matiounine, Smith, Migneron, van Neerven, 1996; Bierenbaum, Bl¨ umlein, S.K., 2007] Mellin–space expressions: [Alekhin, Bl¨ umlein, 2003.] . Next-to-Leading Order : g 1 ( x, Q 2 ) asymptotic: [Buza, Matiounine, Smith, Migneron, van Neerven, 1997; Bierenbaum, Bl¨ umlein, S.K., 2009] Next-to-Next-to-Leading Order : F L ( x, Q 2 ) asymptotic: [Bl¨ umlein, De Freitas, S.K., van Neerven, 2006.] O ( α 3 s ) : Light flavor Wilson coefficients: [Moch, Vermaseren, Vogt, 2005.] = ⇒ 3-loop heavy quark corrections needed to reach the same accuracy as for the light flavor contributions.

12 Need for the Calculation: • Heavy flavor (charm) contributions to DIS structure functions are rather large [20–40 % 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 ( M 2 ⇐ ⇒ Z ), from DIS data: δα s /α s < 1 %. Z ) = 0 . 1141 +0 . 0020 (Recent NS N 3 LO analysis: α s ( M 2 − 0 . 0022 = ⇒ δα s /α s ≈ 2% [Bl¨ ottcher, Guffanti, 2007] .) umlein, B¨ ⇐ ⇒ Precise determination of the gluon and sea quark distributions. Derivation of variable flavor number scheme for heavy quark production to O ( a 3 ⇐ ⇒ s ). • Calculation of the heavy flavor Wilson coefficients to higher orders for Q 2 ≥ 25 GeV 2 [sufficient in many applications]. Goal: • First recalculation of the fermionic contributions to the NNLO anomalous dimensions.