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Nucleon Polarizabilities:Theory and Experiments
Chung-Wen KaoChung-Yuan Christian
University
2007.3 .30. NTU. Lattice QCD Journal Club
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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
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Chiral dynamics and Nucleon Polarizabilities
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Real Compton Scattering
﹖
Spin-independent
Spin-dependent
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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
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Physical meaning of Ragusa Polarizabilities
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Forward Compton Scattering
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By Optical Theorem :
Dispersion Relation
Relate the real part amplitudes to the imaginary part
Therefore one gets following dispersion relations:
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Derivation of Sum rulesExpanded by incoming photon energy ν:
Comparing with the low energy expansion of forward amplitudes:
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Generalize to virtual photon
Forward virtual virtual Compton scattering (VVCS) amplitudes
h=±1/2 helicity of electron
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The elastic contribution can be calculated from the Born diagrams with Electromagnetic vertex
Dispersion relation of VVCS
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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
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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.
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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.
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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.
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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).
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An analogy: Ferromagnetism
Below TcAbove Tc
< M >≠0
< M > =0
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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.
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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).
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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.
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Chiral power D counting
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Heavy Baryon Approach
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Manifest Lorentz Invariant approach
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Theoretical predictions of α and β
LO HBChPT (Bernard, Kaiser and Meissner , 1991)
NLO HBChPT
LO HBChPT including Δ(1232)
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Linearly polarized incoming photon+ unpolarized target:
Small energy, small cross section; Large energy, large higher order terms contributes
Extraction of α and β
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Extraction of α and β
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Theoretical predictions of γ0
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MAIDEstimate
Bianchi Estimate
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MAID
MAMI(Exp)ELSA(Exp)Bianchi
Total 211±15 -0.94±0.15
GDH sum rule
205
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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
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Data of spin forward polarizabilities
LO+NLO HBChPT
LO+NLO MLI ChPT
MAID
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Theoretical predictions of Ragusa polarizabilities
Kumar, Birse, McGovern (2000)
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Longitudinal and perpendicularasymmetry
Plan experiments by HIGS, TUNL.
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Neutron asymmetry
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Proton asymmetry
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Polarizabilities on the lattice
Background field method:
Detmold, Tiburzi, Walker-Loud, 2003
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Example: Constant electric field at X1 direction
Two-point correlation function
Polarizabilities on the lattice
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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.