Electromagnetic Sum Rules and Low Energy Theorems Barbara Pasquini, Dieter Drechsel, L. T., Sabit Kamalov Pavia, Mainz, Dubna - the Gerasimov-Drell-Hearn sum rule (GDH 1966) - the Fubini-Furlan-Rossetti sum rule (FFR 1965) - the Nambu-Shrauner-Lurié sum rule (NSL 1962)

The GDH Sum Rule

D. Drechsel and L. Tiator, Annu. Rev. Nucl. Part. Sci. 2004, 54:69-114

Summary on GDH  GDH sum rule: relation between the anomalous magnetic moment and the helicity difference of photoabsorption on the nucleon  Within sizeable uncertainties (especially for the neutron) we can see agreement with the GDH sum rule I p = 225 ± 15  b (sr: 204.8) I n = 213 ± 40  b (sr: 233.2)  p = 1.88 ± 0.06 (exp: 1.793)  n = ± 0.35 (exp: )  (with Regge models a perfect agreement can be achieved)  2-pion contributions may be very different for proton and neutron for the proton: mainly in the 2 nd resonance region for the neutron: very large in the 2 nd res. region and dominant in the 3 rd res. region and beyond

The FFR Sum Rule

The Low Energy Theorem for neutral pions is strongly violated in the physical region

Invariant Amplitudes of Pion Photoproduction    (k) i 4 Lorentz invariant functions of = (s-u)/4M N and t  + N !  + N Lorentz invariance, and P, C and T symmetries: 6 functions of, t and Q² in electroproduction

in the soft-pion limit for =0, t=0 ( ) LET for pion photoproduction Born tems with pseudovector coupling the Born terms in pseudovector coupling have the correct symmetries and the non-Born terms vanish in this limit

Dispersion Relations at fixed t LET soft pion limit ( = 0, t =0 )

no sum rule for charged pion photoproduction

FFR Sum Rule Fubini, Furlan, Rossetti, Nuovo Cimento 43 (1966) 161 Dispersion Relation with Im A 1 from MAID2003 HBChPT Corrections from physical pion mass as function of at fixed FFR discrepancy Heavy Baryon Chiral Perturbation Theory

pion threshold soft pion point physical region of pion photoproduction

extrapolation to the unphysical region

FFR Discrepancy  t  from HBChPT

FFR ( = 0,t=0 ) FFR +  ( thr, t thr ) HBChPT EXP t =0) (t=t thr ) MAID03 ( t =0) (t=t thr ) /2.56/ Neutron /2.33/ Proton MAID03 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(p 4 ) P waves at O(p 3 ) S and P waves at O(p 4 )  p, -  n  (0, t thr ) at soft-pion pointat physical threshold

neutron proton FFR discrepancy  from MAID03 Pasquini, Drechsel, Tiator, Eur. Phys. J A23 (2005) t = t thr

Integrands from MAID03 third resonance region D 13 (1520)  1232)  loops t = t thr  S +  S   S +  S =   V +  V =   V +  V =  isoscalar isovector

PROTON Bernard, et al., PLB378 (1996) HBChPT at O(p 3 ) Bernard, et al., ZPC70 (1996) HBChPT at O(p 3 ) DR-MAID Bernard, et al. EPJ A11 (2001) HBChPT at O(p 4 ) Mainz experiment, Schmidt, et al., PRL 87 (2001) t = t thr

NEUTRON Bernard, et al., ZPC70 (1996) HBChPT at O(p 3 ) DR-MAID Bernard, et al. EPJ A11 (2001) HBChPT at O(p 4 ) Bernard, et al., PLB378 (1996) HBChPT at O(p 3 ) t = t thr

Summary on FFR  FFR sum rule: linear relation between the anomalous magnetic moment and single-pion photoproduction on the nucleon in the soft-pion limit (m  2 =0, =t=0)  Predictions at =, t=0 : extrapolation of MAID amplitudes in the unphysical region give very good results  p = (exp: 1.793)  n = (exp: )  Corrections to the sum rule from the physical pion mass:  (, t thr ) good agreement between MAID, HBChPT and Experiment in the threshold region ( = thr ) problems with ChPT at low  thr because of the non-relativistic approximation of HBChPT