Spin polarizabilities in Heavy Baryon Chiral Perturbation Theory

1. Spin polarizabilities in Heavy Baryon Chiral Perturbation Theory Chung-Wen Kao Chung-Yuan Christian University 2008.10 .6. University of Virginia, Charlottesville 18th international Symposium on Spin Physics

2. What is Polarizability? Excited states Electric Polarizability Magnetic Polarizability Polarizability is a measures of rigidity of a system and deeply relates with the excited spectrum.

3. Real Compton Scattering Spin-independent Spin-dependent ﹖

4. Ragusa Polarizabilities Forward spin polarizability Backward spin polarizability LO are determined by e, M κ NLO are determined by 4 spin polarizabilities, first defined by Ragusa

5. Forward Compton Scattering

6. Dispersion Relation Relate the real part amplitudes to the imaginary part By Optical Theorem : Therefore one gets following dispersion relations:

7. Derivation of Sum rules Expanded by incoming photon energy ν: Comparing with the low energy expansion of forward amplitudes:

8. Generalize to virtual photon Forward virtual virtual Compton scattering (VVCS) amplitudes h=±1/2 helicity of electron

9. Dispersion relation of VVCS The elastic contribution can be calculated from the Born diagrams with Electromagnetic vertex

10. Sum rules for VVCS Expanded by incoming photon energy ν Combine low energy expansion and dispersion relation one gets 4 sum rules On spin-dependent vvcs amplitudes: Generalized GDH sum rule Generalized spin polarizability sum rule

11. Spin Structure functions

12. Moment of structure functions

13. Theory vs Experiment • Theorists can calculate Compton scattering amplitudes and extract polarizabilities. • On the other hand, experimentalists have to measure the cross sections of Compton scattering to extract polarizabilities. • Experimentalists can also use sum rules to get the values of certain combinations of polarizabilities. • Theorists can easily calculate forward Compton amplitudes and compare with data!

14. Brief introduction to HBChPT This would be a little bit boring for experts and absolutely boring for everyone else…..

15. Chiral Symmetry of QCD if mq=0 Left-hand and right-hand quark: QCD Lagrangian is invariant if Massless QCD Lagrangian has SU(2)LxSU(2)Rchiral symmetry.

16. Quark mass effect If mq≠0 QCD Lagrangian is invariant if θR=θL. Therefore SU(2)LXSU(2)R →SU(2)V, ,if mu=md SU(2)A is broken by the quark mass

17. Spontaneous symmetry breaking Spontaneous symmetry breaking: a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. The system no longer appears to behave in a symmetric manner. Example: V(φ)=aφ2+bφ4, a<0, b>0. Spontaneous symmetry Mexican hat potential U(1) symmetry is lost if one expands around the degenerated vacuum! Furthermore it costs no energy to rum around the orbit →massless mode exists!! (Goldstone boson).

18. An analogy: Ferromagnetism Above Tc Below Tc ＜M＞=0 ＜M＞≠0

19. Pion as Goldstone boson • π is the lightest hadron. Therefore it plays a dominant the long-distance physics. • More important is the fact that soft πinteracts each other weakly because they must couple derivatively! • Actually if their momenta go to zero, πmust decouple with any particles, including itself. Start point of an EFT for pions. ～t/(4πF)2

20. Chiral Perturbation Theory • Chiral perturbation theory (ChPT) is an EFT for pions. • The light scale is p and mπ. • The heavy scale isΛ～4πF～1 GeV, F=93 MeVisthe pion decay constant. • Pion coupling must be derivative so Lagrangian start fromL(2).

21. Set up a power counting scheme • kn for a vertex with n powers of p or mπ. • k-2 for each pion propagator: • k4 for each loop:∫d4k • The chiral power :ν=2L+1+Σ(d-1) Nd • Since d≧2 therefore νincreases with the number of loop.

22. Chiral power D counting

23. Heavy Baryon Approach

24. Manifest Lorentz Invariant approach

25. Theoretical predictions of γ0 Convergence is very poor!

26. MAID Estimate Bianchi Estimate

28. Theoretical predictions of γ0(Q2) andδ(Q2) LO+NLO HBChPT (Kao, Vanderhaeghen, 2002) LO+NLO Manifest Lorentz invariant ChPT (Bernard, Hemmert Meissner 2002) MAID Lo Lo Lo Δ p p LO+NLO n n

31. Data of spin forward polarizabilities M. Amarian et al, PRL 93, 152301(2004) neutron LO+NLO HBChPT LO+NLO MLI ChPT MAID

32. Data of Generalized GDH sum rule A. Deur et al. PRL 93, 212001 (2004)

33. More and more data….. When theorists are taking a nap…….. Experimentalists are working very hard to get more and more data……

34. The Good Data…… arXiv 0802.2232 by CLAS collaboration (Y. Pork et. Al.), Submitted to PRL LO+NLO HB LO+NLO MLI

35. The excellent data A. Duer et. al. PRD78,032001 (2008) HBChPT does a very good job, even better than MLI at medium Q^2! Very low Q2 !

36. The embarrassing ones… arXiv 0802.2232 by CLAS collaboration (Y. Pork et. Al.), Submitted to PRL Proton

37. The one I shouldn’t have shown you….. A. Duer et. al. PRD78,032001 (2008) isovector isoscalar

38. Melancholia……… So, what goes wrong? Sum rule cannot be wrong because generalized GDH sum rule looks very good. HB expansion is not responsible for it because MLI doesn’t work well, either. Δ contribution is important but it should be isoscalar at tree level. From the analytical forms one may need calculate up to NNLO Including NLO Δ and/or NNLO

39. NLO Δ In progress……… Contribute to isovector channel

40. Knight: Hard working Physicists Death of HBChPT? Devil : Spin polarizabilities!