Nucleon Polarizabilities: Theory and Experiments

1. Nucleon Polarizabilities:Theory and Experiments Chung-Wen Kao Chung-Yuan Christian University 2007.3 .30. NTU. Lattice QCD Journal Club

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. Chiral dynamics and Nucleon Polarizabilities

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

5. 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

6. Physical meaning of Ragusa Polarizabilities

7. Forward Compton Scattering

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

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

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

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

12. 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

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.

14. 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.

15. 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

16. 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).

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

18. 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 π interacts each other weakly because they must couple derivatively! • Actually if their momenta go to zero, π must decouple with any particles, including itself. Start point of an EFT for pions. ～t/(4πF)2

19. 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).

20. 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.

21. Chiral power D counting

22. Heavy Baryon Approach

23. Manifest Lorentz Invariant approach

24. Theoretical predictions of α and β LO HBChPT (Bernard, Kaiser and Meissner , 1991) NLO HBChPT LO HBChPT including Δ(1232)

25. Extraction of α and β Linearly polarized incoming photon+ unpolarized target: Small energy, small cross section; Large energy, large higher order terms contributes

26. Extraction of α and β

28. Theoretical predictions of γ0

29. MAID Estimate Bianchi Estimate

31. Theoretical predictions of γ0(Q2) and δ(Q2) LO+NLO HBChPT (Kao, Vanderhaeghen, 2002) LO+NLO Manifest Lorentz invariant ChPT (Bernard, Hemmert Meissner 2002) MAID Lo Lo Lo Δ LO+NLO

34. Data of spin forward polarizabilities LO+NLO HBChPT LO+NLO MLI ChPT MAID

35. Theoretical predictions of Ragusa polarizabilities Kumar, Birse, McGovern (2000)

36. Longitudinal and perpendicularasymmetry Plan experiments by HIGS, TUNL.

37. Neutron asymmetry

38. Proton asymmetry

39. Polarizabilities on the lattice Detmold, Tiburzi, Walker-Loud, 2003 Background field method:

40. Polarizabilities on the lattice Two-point correlation function Constant electric field at X1 direction Example:

41. Summary and Outlook • Polarizabilities are important quantites relating with inner structure of hadron • Tremendous efforts have contributed to Polarizabilities, both theory and experiment. • We hope our lattice friend can help us to clarify some issues, in particular, neutron polarizabilities.