Nucleon Polarizabilities:Theory and Experiments

Chung-Wen KaoChung-Yuan Christian

University

2007.3 .30. NTU. Lattice QCD Journal Club

What is Polarizability?

Electric Polarizability

Magnetic Polarizability

Polarizability is a measures of rigidity of a system and deeply relates with the excited spectrum.

Excited states

Chiral dynamics and Nucleon Polarizabilities

Real Compton Scattering

﹖

Spin-independent

Spin-dependent

Ragusa Polarizabilities

LO are determined by e, M κ

NLO are determined by 4 spin polarizabilities, first defined by Ragusa

Forward spin polarizability

Backward spin polarizability

Physical meaning of Ragusa Polarizabilities

Forward Compton Scattering

By Optical Theorem :

Dispersion Relation

Relate the real part amplitudes to the imaginary part

Therefore one gets following dispersion relations:

Derivation of Sum rulesExpanded 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

The elastic contribution can be calculated from the Born diagrams with Electromagnetic vertex

Dispersion relation of VVCS

Sum rules for VVCSExpanded by incoming photon energy ν

Combine low energy expansion and dispersion relation one gets 4 sum rulesOn 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)R chiral symmetry.

Therefore SU(2)LXSU(2)R →SU(2)V, ,if mu=md

Quark mass effect

If mq≠0

SU(2)A is broken by the quark mass

QCD Lagrangian is invariant if θR=θL.

Spontaneous symmetry breaking

Mexican hat potential

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.

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).

An analogy: Ferromagnetism

Below TcAbove Tc

＜ M ＞≠0

＜ M ＞ =0

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.

～ t/(4πF)2

Start point of an EFT for pions.

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).

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.

Chiral power D counting

Heavy Baryon Approach

Manifest Lorentz Invariant approach

Theoretical predictions of α and β

LO HBChPT (Bernard, Kaiser and Meissner , 1991)

NLO HBChPT

LO HBChPT including Δ(1232)

Linearly polarized incoming photon+ unpolarized target:

Small energy, small cross section; Large energy, large higher order terms contributes

Extraction of α and β

Extraction of α and β

Theoretical predictions of γ0

MAIDEstimate

Bianchi Estimate

MAID

MAMI(Exp)ELSA(Exp)Bianchi

Total 211±15 -0.94±0.15

GDH sum rule

205

Theoretical predictions of γ0 (Q2) and δ(Q2)

LO+NLO HBChPT (Kao, Vanderhaeghen, 2002)

LO+NLO Manifest Lorentz invariant ChPT (Bernard, Hemmert Meissner2002)

Lo

LO+NLO

Lo Δ

MAID Lo

Data of spin forward polarizabilities

LO+NLO HBChPT

LO+NLO MLI ChPT

MAID

Theoretical predictions of Ragusa polarizabilities

Kumar, Birse, McGovern (2000)

Longitudinal and perpendicularasymmetry

Plan experiments by HIGS, TUNL.

Neutron asymmetry

Proton asymmetry

Polarizabilities on the lattice

Background field method:

Detmold, Tiburzi, Walker-Loud, 2003

Example: Constant electric field at X1 direction

Two-point correlation function

Polarizabilities on the lattice

Summary and Outlook

Polarizabilities are important quantites relating with inner structure of hadron

Tremendous efforts have contributed to Polarizabilities, both theory and experim

ent. We hope our lattice friend can help us to

clarify some issues, in particular, neutron polarizabilities.