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NNLO QCD Analysis of the Virtual Photon Structure Functions

NNLO QCD Analysis of the Virtual Photon Structure Functions. Ken Sasaki (Yokohama National Univ.). Takahiro Ueda (KEK). Yoshio Kitadono (Hiroshima/Kyoto Univ.). & Tsuneo Uematsu (Kyoto Univ. ). Talk at Radcor07 at GGI October 1-5, 2007. 1. Introduction and Motivation.

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NNLO QCD Analysis of the Virtual Photon Structure Functions

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  1. NNLO QCD Analysis of the Virtual Photon Structure Functions Ken Sasaki (Yokohama National Univ.) Takahiro Ueda (KEK) Yoshio Kitadono (Hiroshima/Kyoto Univ.) & Tsuneo Uematsu (Kyoto Univ. ) Talk at Radcor07 at GGI October 1-5, 2007

  2. 1. Introduction and Motivation Two-photon processin e+e－collision in the future linear collider ILC Photon structure functions and for the case of virtual photon ( ) target as in the case of nucleon structure functions Two-photon process and

  3. Virtual-Photon Kinematics : Bjorken variable : Mass squared of probe photon : Mass squared of target photon In the kinematic region: e+e－collision & QCD interactions structure fns. and perturbatively calculable !

  4. in Perturbative QCD Simple parton model • For real photon target ( ) Point-like contribution dominates Hadronic piece (LO) (NLO) Walsh-Zerwas (1973) Witten (1977) OPE+RGE QCD improved parton model lowest order in Bardeen-Buras (1979) Gluck-Reya (1983),… • For highly virtual photon target ( ) : QCD scale parameter (LO) (NLO) Walsh-TU (1981,1982) Hadronic piece can also be dealt with perturbatively Definite prediction of , its shape and magnitude, is possible

  5. Present Motivation NNLOextension Moch-Vermaseren-Vogt (2002, 2006) Motivated by the calculation of 3-loop anomalous dimensions Vogt-Moch-Vermaseren (2004,2006) We extend the analysis of virtual photon structure functions toNNLO( ) K. Sasaki, T. Ueda and T.U., Phys. Rev.D75 (2007) 114009 Here we also consider target mass effects to NNLO Y. Kitadono, K. Sasaki, T. Ueda and T.U., in preparation For parton distributions inside virtual photon in theNNLO, see Ken Sasaki’s next talk

  6. Structure Functions Spin-averaged Structure Tensor Forward photon-photon scattering cf. Structure tensor no spin averaged Symmetric part unpolarized photon structure functions Anti-symmetric part polarized photon structure functions and

  7. The OPE near the light-cone Spin-twist-2 operators 2. Theoretical framework : OPE and RGE light-cone : over relevant ops. (hadronic ops. + photon op.) ( -th moment) (dominant) quark : gluon : (flavor singlet) photon : (flavor non-singlet) QCD with massless quarks with flavors

  8. Moment sum rule of Inverse Mellin transformation: -space -space for even we take Photon matrix element RG improved coefficients Perturbatively calculable when Solution of RG eq. for coefficient functions : QCD beta function : anomalous dimensions Expand up to NNLO ( ) numerically inverted

  9. Anomalous dimension matrix 4 x 4 matrix Anomalous dimensions of hadronic operators 1-loop anom. dim. Mixing anomalous dimensions of photon-hadron ops.

  10. Summarized as LO ( ) NLO ( ) for even NNLO ( ) Moment Sum Rule for

  11. QCD -function Anomalous dimensions of hadronic operators Hadronic coefficient functionsvan Neerven-Zijlstra (1991,1992) Renomalization Group parameters Tarasov-Vladimirov-Zarkov (1980) Larin-Vermaseren (1993) 1-loop 2-loop 3-loop Moch-Vermaseren-Vogt (2004) 1-loop 2-loop 3-loop tree 1-loop 2-loop

  12. LO : Coefficients of the moment sum rule tree 1-loop 1-loop tree NLO : 1-loop 2-loop 2-loop 1-loop New evaluation needed NNLO : Mixing anomalous dimensions 2-loop 3-loop 3-loop 2-loop known Photon coefficient function Photon matrix elements

  13. Mixing anomalous dimensions No exact expressions for 3-loop yet But the compact parametrization of 3-loop photon-parton splitting functions (and exact ) available (the deviation < 0.1 %) 1-loop 2-loop 3-loop published A.Vogt, S.Moch and J.Vermaseren Acta Phys.Polon. B37,683(2006) We adopt this parametrization

  14. Photon coefficient function , are obtained by the replacement of group factors [example] 1-loop 2-loop

  15. Unrenormalizedgluon matrix elements of hadronic operators were calculated in -space We obtain , after taking moments, renormalization and the replacement of group factors Photon matrix elements Perturbatively calculable when 1-loop 2-loop Matiounine-Smith-van Neerven (1998)

  16. 2 2 2 2 L n =4 Q =30GeV P =1GeV =200MeV LO LO LO LO 0.7 f NLO NLO NLO NLO NNLO NNLO NNLO NNLO 0.6 ) 2 /P 0.5 2 ln(Q 0.4 a p ñ NLO LO NNLO 4 e á f 0.3 3n ¤ 2 g F 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 x 3. Numerical Analysis Numerical Plot

  17. 2 2 2 2 L n =4 Q =30GeV P =1GeV =200MeV LO LO LO 0.7 f NLO NLO NLO 0.6 LO NLO 0.5 p a ñ 4 e á 0.4 f 3n ¤ 0.3 L g F 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 x Numerical Plot

  18. Sum rule : first moment By taking the limit we get a sum rule Corrections 8-10 e.g. Note: and develop singularities at which are cancelled by a factor vanishing of anom. dim of energy-mom tensor

  19. For the virtual photon target, the maximal value of the Bjorken variable due to the constraint In the nucleon case We study the Target Mass Effects (TME) based on Operator Product Expansion (OPE) taking into account the trace terms of the matrix elements of the operators. This amounts to consider Nachtmann moments. By using the orthogonality in the Gegenbauer polynomial we project out the definite spin contributions. 4. Target mass effects for virtual photon structures Y.Kitadono, K.Sasaki, T.Ueda and T.U.

  20. Nachtmann moments for and where Inverting the moments where

  21. where the four functions F, G, H and FLare defined as Mellin-inverted moments : with and defined as ordinary n-th moment given as obtained to NNLO

  22. Numerical Results (1) F, G, H, FL

  23. (2) at larger x TME No TME

  24. (3) at larger x No TME TME Note with TME

  25. Comparison with existing experimental data : Effective photon structure fn. PLB483 (2000) 373

  26. The virtual photon structure function investigated in the kinematical region Definite predictions made up toNNLO ( ) NNLO corrections appear sizable at large x We also analyzed the virtual photon structure function to NLO ( ) The target mass effects (TME) investigated TME becomes sizable at larger x. TME reduces and enlarges at larger x. 5. Summary

  27. Power corrections of the form due to higher-twist effects should also be studied Here we have treated active flavors as massless quarks Off course, surely we have to take into account mass effects of heavy quark flavors So far we have studied the NNLO QCD correction to the 1st moment of We should also interested in full NNLO analysis of and Experimental confrontation in the future is anticipated Future subjects

  28. Extra slides

  29. Moment Sum Rule for LO ( ) NLO ( )

  30. LO : Coefficients of the moment sum rule NLO :

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