The The Fubini-Furlan-RossettiFubini-Furlan-Rossetti

sum rulesum rule

LET

Pion PhotoproductionPion Photoproduction

(k)

i

4 Lorentz invariant functions of = (s-u)/4MN and

t

+ N + N !! + N + N

Lorentz invariance, and P, C and T symmetries:

PCAC hypothesis

current algebraPion Low Energy Theorems

soft pion limit

LET for pion photoproductionLET for pion photoproduction

Born with pseudovector coupling

the contribution from is zero

Dispersion Relations at fixedDispersion Relations at fixed tt

LET soft pion limit ( = 0, t =0 )

POLE

BORN-PV

obtained from tree-level Feynman diagrams:

FFR Sum RuleFFR Sum RuleFubini, Furlan, Rossetti, Nuovo Cimento 43 (1966) 161

Dispersion Relation with Im A1 from

MAID2003

HBChPT

Corrections from physical pion mass as function of at fixed

FFR discrepancy

Heavy BaryonChiral Perturbation Theory

pion threshold

soft pion point

physical regionof pion photoproduction

extrapolationto the unphysical region

FFR Discrepancy FFR Discrepancy from HBChPT from HBChPT (heavy baryon chiral perturbation theory )(heavy baryon chiral perturbation theory )

Dispersion integral at fixed Dispersion integral at fixed t=tt=tthrthr with with ImIm A A11 from MAID03 from MAID03

EXP MAID03 (tt=0)=0) (t=t(t=tthrthr))

HBChPT

MAID03

Proton 1.793

1.792 1.66 2.29/2.33/2.37 2.24

Neutron

1.913

1.986 1.822.52/2.56/2.79 2.44

HBChPT:

Bernard, Kaiser, Meissner, Z. Phys. C70 (1996)

Bernard, Kaiser, Meissner, Phys. Lett. B378 (1996)

Bernard, Kaiser, Meissner, Eur. Phys. J. A11 (2001)

S wave at O (p4)

P waves at O(p3)

S and P waves at O (p4)

FFR + FFR + ((thrthr, t, tthrthr))FFR FFR ((==0,t=00,t=0))

p, -n

(0, t(0, tthrthr))

neutron

proton

FFR discrepancy FFR discrepancy from from MAID03MAID03

Pasquini, Drechsel, Tiator, Eur. Phys. J A23 (2005)

t = tthr

Integrands from MAID03Integrands from MAID03

third resonance region

third resonance region

D13(1520)

1232)

loops

loops

t = tthr

S + S

S + S =

V + V =

V + V =

PROTONPROTON

Bernard, et al., PLB378 (1996)

HBChPT at O(p3)

Bernard, et al., ZPC70 (1996)

HBChPT at O(p3)

DR-MAID

Bernard, et al. EPJ A11 (2001)

HBChPT at O(p4)

Mainz experiment,Schmidt, et al., PRL 87 (2001)

t = tthr

NEUTRONNEUTRON

Bernard, et al., ZPC70 (1996)

HBChPT at O(p3)

DR-MAID

Bernard, et al. EPJ A11 (2001)

HBChPT at O(p4)

Bernard, et al., PLB378 (1996)

HBChPT at O(p3)

t = tthr

SummarySummary FFR sum rule: relation between the anomalous magnetic moment and single pion-photoproduction on the nucleon

in the soft pion limit (m2=0, =t=0)

Predictions at =, t=0 : extrapolation of MAID amplitudes in the unphysical region give very good results

p = 1.792 (exp: 1.793) n = - 1.986 (exp: -1.913)

Corrections to the sum rule from the physical pion mass: (, tthr)

good agreement between MAID and HBChPT in the threshold region ( = thr)

problems at low thr because of the non-relativistic approximation of HBChPT