The Fubini-Furlan-Rossetti sum rule
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The The Fubini-Furlan-RossettiFubini-Furlan-Rossetti
sum rulesum rule
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LET
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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:
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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
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Dispersion Relations at fixedDispersion Relations at fixed tt
LET soft pion limit ( = 0, t =0 )
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POLE
BORN-PV
obtained from tree-level Feynman diagrams:
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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
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pion threshold
soft pion point
physical regionof pion photoproduction
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extrapolationto the unphysical region
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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
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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))
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neutron
proton
FFR discrepancy FFR discrepancy from from MAID03MAID03
Pasquini, Drechsel, Tiator, Eur. Phys. J A23 (2005)
t = tthr
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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 =
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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
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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
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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