1. Nucleon Polarizabilities: Theory and Experiments
Chung-Wen Kao
Chung-Yuan Christian University
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)R chiral symmetry.

15. Quark mass effect Therefore SU(2)LXSU(2)R →SU(2)V, ,if mu=md
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
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 is Λ～4πF～1 GeV,
F=93 MeV is the pion decay constant.
Pion coupling must be derivative so
Lagrangian start from L(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

30. MAID MAMI(Exp) ELSA(Exp) Bianchi Total GDH sum rule 205 211±15
-0.94±0.15
GDH sum rule
205

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) Lo
MAID 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 perpendicular asymmetry
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
Example:
Constant electric field at X1 direction

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.