1. Adler-Bardeen Theorem for the Axial Anomaly and the First Moment of the Polarized Virtual Photon Structure Function Takahiro Ueda (Yokohama National Univ.) Tsuneo Uematsu (Kyoto Univ.) Ken Sasaki (Yokohama National Univ.) Phys.Rev.D73,094024(2006), [arXiv:hep-ph/0604130] SPIN 2006 Kyoto 2006.10.6 Dedicated to the memory of Jiro Kodaira

2. 1 / 18 Introduction Future linear collider experiment (e.g. ILC) Polarized collision (virtual) photon spin structure function and Just as analogy with nucleon case and We focus on sum rule Two-photon process

3. 2 / 18 Definition (virtual) photon structure : Antisymmetric part is with (polarized)

4. 3 / 18 Spin Structure Function Photon spin structure function is… Bjorken variable Mass squared of probe photon Mass squared of target photon target probe more easily measured than for kinematical reason The 1st moment related to triangle anomaly

5. 4 / 18 For perturbatively calculable The1st Moment Sum Rule For real photon target ( ) For virtual photon target ( ) Efremov-Teryaev (1990)Bass (1992) Narison-Shore-Veneziano (1993) Bass-Brodsky-Schmidt (1998) Narison-Shore-Veneziano (1993) Uematsu-Sasaki (1999) Shore (2005) NLO result correction ? : QCD scale parameter : # of flavor LO in all orders in QED and QCD (pert.)+(non-pert.) QED-anomaly QCD-anomaly

6. 5 / 18 Transition of from on-shell ( ) to virtual state ( ) T. Ueda, T.Uematsu and K. Sasaki, Phys. Lett. B640 (2006)188

7. 6 / 18 The 1st moment of is related to triangle anomaly in OPE : OPE QED-anomaly : OPE QCD-anomaly

8. 7 / 18 Framework For OPE of two EM currents, only gauge-invariant operators need be included No gauge-invariant twist-2 and gluon nor photon Operators. Only quark operators considered (axial current) 1st. moment sum rule J.C.Collins : Flavor singlet : Flavor non-singlet We choose : charge matrix

9. 8 / 18 Go To NNLO RGE for coefficient function For, we need… where : up to (3-loop) (2-loop) (3-loop) perturbatively calculable for

10. 9 / 18 Anomalous Dimension Expand as… For non-singlet axial current For singlet axial current Kodaira (1980) Larin (1993) NS-axial current is conserved in massless limit triangle anomaly

11. 10 / 18 Coefficient Function Expand as… where Larin (1994) Larin-Vermaseren (1991) Kodaira-Matsuda-Muta-Sasaki -Uematsu (1979)

12. 11 / 18 Adler-Bardeen Theorem Adler-Bardeen Theorem for axial anomaly Anomaly comes from one-loop triangle diagram Anomaly term is not affected by higher-order radiative corrections (Non-renormalization theorem) QED anomary QCD anomary Color :

13. 12 / 18 Photon Matrix Element Expand as : : : : : : Vanishes due to non-renormalization theorem for triangle anomaly (singlet only)

14. 13 / 18 Calculation of Photon Matrix Element Adler-Bardeen theorem for QCD Consider Green functions like (as operator identity)

15. 14 / 18 Calculation of Photon Matrix Element Operate and set we obtain Leading is Contribute to

16. 15 / 18 Calculation of Photon Matrix Element + + + + + + ( ) counter diagram for sub-divergence we can extract Multiply and set

17. 16 / 18 Final Answer NLO LO NNLO

18. 17 / 18 Numerical Evaluation For For example, taking 3 3 4 4 5 5 1 1 1 1 1 1 LO -0.0816 -0.0913 -0.0986 NLO -0.0250 -0.0288 -0.0309 NNLO -0.0272 -0.0317 -0.0343 NNLO / (LO+NLO) 3%

19. 18 / 18 Summary We study sum rule up to NNLO To evaluate photon matrix elements of axial currents, we resort to Adler-Bardeen theorem for the axial anomaly. The three-loop-level calculation for reduces to the one in two-loop level. Numerical evaluation shows typical NNLO contribution is for 3%

20. Appendix

21. Counter Diagram Appendix - A Corresponding counter diagram is with

22. Treatment of Reading point method Appendix - B Put at the right-hand side of the string of -matrices Do not assume any property of, except anticommutativity of After all calculation (integration and contraction) (dimensional regularization)

23. Backup

24. 23 / 18 NLO Result In kinematic region structure functions : QCD scale parameter pQCD gives whole information on

25. 24 / 18 Photon Matrix Element Adler-Bardeen theorem QED anomary QCD anomary Color :

26. 25 / 18 Calculation of Photon Matrix Element Operate and set we obtain

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