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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.

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Nucleon polarizabilities theory and experiments

Nucleon Polarizabilities:Theory and Experiments

Chung-Wen Kao

Chung-Yuan Christian University

2007.3 .30. NTU. Lattice QCD Journal Club

What is polarizability
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.

Chiral dynamics and nucleon polarizabilities
Chiral dynamics and Nucleon Polarizabilities

Real compton scattering
Real Compton Scattering



Ragusa polarizabilities
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

Physical meaning of ragusa polarizabilities
Physical meaning of Ragusa Polarizabilities

Dispersion relation
Dispersion Relation

Relate the real part amplitudes to the imaginary part

By Optical Theorem :

Therefore one gets following dispersion relations:

Derivation of sum rules
Derivation of Sum rules

Expanded by incoming photon energy ν:

Comparing with the low energy expansion of forward amplitudes:

Generalize to virtual photon
Generalize to virtual photon

Forward virtual virtual Compton scattering (VVCS) amplitudes

h=±1/2 helicity of electron

Dispersion relation of vvcs
Dispersion relation of VVCS

The elastic contribution can be calculated from

the Born diagrams with Electromagnetic vertex

Sum rules for vvcs
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
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 m q 0
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.

Nucleon polarizabilities theory and experiments

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

Spontaneous symmetry breaking
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.


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).

An analogy ferromagnetism
An analogy: Ferromagnetism

Above Tc

Below Tc



Pion as 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.


Chiral perturbation theory
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
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 and
Theoretical predictions of α and β

LO HBChPT (Bernard, Kaiser and Meissner , 1991)


LO HBChPT including Δ(1232)

Extraction of and
Extraction of α and β

Linearly polarized incoming photon+ unpolarized target:

Small energy, small cross section;

Large energy, large higher order terms contributes

Extraction of and1
Extraction of α and β

Nucleon polarizabilities theory and experiments



Bianchi Estimate

Theoretical predictions of 0 q 2 and q 2
Theoretical predictions of γ0(Q2) and δ(Q2)

LO+NLO HBChPT (Kao, Vanderhaeghen, 2002)

LO+NLO Manifest Lorentz invariant ChPT (Bernard, Hemmert Meissner





Lo Δ


Data of spin forward polarizabilities
Data of spin forward polarizabilities




Longitudinal and perpendicular asymmetry
Longitudinal and perpendicularasymmetry

Plan experiments by HIGS, TUNL.

Polarizabilities on the lattice
Polarizabilities on the lattice

Detmold, Tiburzi, Walker-Loud, 2003

Background field method:

Nucleon polarizabilities theory and experiments

Polarizabilities on the lattice

Two-point correlation function

Constant electric field at X1 direction


Summary and outlook
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.