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### Nucleon Polarizabilities:Theory and Experiments

Chung-Wen Kao

Chung-Yuan Christian University

2007.3 .30. NTU. Lattice QCD Journal Club

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.

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

Dispersion Relation

Relate the real part amplitudes to the imaginary part

By Optical Theorem :

Therefore one gets following dispersion relations:

Derivation of Sum rules

Expanded by incoming photon energy ν:

Comparing with the low energy expansion of forward amplitudes:

Generalize to virtual photon

Forward virtual virtual Compton scattering (VVCS) amplitudes

h=±1/2 helicity of electron

Dispersion relation of VVCS

The elastic contribution can be calculated from

the Born diagrams with Electromagnetic vertex

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

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.

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.

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

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).

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

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).

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.

Theoretical predictions of α and β

LO HBChPT (Bernard, Kaiser and Meissner , 1991)

NLO HBChPT

LO HBChPT including Δ(1232)

Extraction of α and β

Linearly polarized incoming photon+ unpolarized target:

Small energy, small cross section;

Large energy, large higher order terms contributes

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

Theoretical predictions of Ragusa polarizabilities

Kumar, Birse, McGovern (2000)

Longitudinal and perpendicularasymmetry

Plan experiments by HIGS, TUNL.

Polarizabilities on the lattice

Two-point correlation function

Constant electric field at X1 direction

Example:

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.

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