Post on 06-Jan-2016
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Spin polarizabilities in Heavy Baryon Chiral Perturbation Theory
Chung-Wen KaoChung-Yuan Christian
University
2008.10 .6. University of Virginia, Charlottesville 18th international Symposium on Spin Physics
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
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
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
Spin Structure functions
Moment of structure functions
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.
Theorists can easily calculate forward Compton amplitudes and compare with data!
Brief introduction to HBChPT
This would be a little bit boring for
experts and absolutely boring
for everyone else…..
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 γ0
Convergence is very poor!
MAIDEstimate
Bianchi Estimate
MAID
MAMI(Exp)
ELSA(Exp)
Bianchi
Total 211±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
p
n
p
n
Data of spin forward polarizabilities
LO+NLO HBChPT
LO+NLO MLI ChPT
MAID
M. Amarian et al, PRL 93, 152301(2004)
neutron
Data of Generalized GDH sum rule
A. Deur et al. PRL 93, 212001 (2004)
More and more data…..
When theorists are taking a nap……..
Experimentalists are working very hard to get more and more data……
The Good Data……
arXiv 0802.2232 by CLAS collaboration (Y. Pork et. Al.), Submitted to PRL
LO+NLO HB
LO+NLO MLI
The excellent dataA. Duer et. al. PRD78,032001 (2008)
Very low Q2 !
HBChPT does a very good job, even better than MLI at medium Q^2!
The embarrassing ones…
arXiv 0802.2232 by CLAS collaboration (Y. Pork et. Al.), Submitted to PRL
Proton
The one I shouldn’t have shown you…..
A. Duer et. al. PRD78,032001 (2008)isovector
isoscalar
Melancholia………
Sum rule cannot be wrong because generalized GDH sum rule looks very good.
So, what goes wrong?
Including NLO Δ and/or NNLO
HB expansion is not responsible for it becauseMLI doesn’t work well, either.
Δ contribution is important but it should be isoscalar at tree level.
From the analytical forms one may need calculate up to NNLO
NLO Δ
In progress………Contribute to isovector channel
Knight: Hard working Physicists
Death of HBChPT?
Devil : Spin polarizabilities !