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 !