Electromagnetic Sum Rules and Low Energy Theorems

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Electromagnetic Sum Rulesand 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)

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The GDH Sum Rule
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D. Drechsel and L. Tiator, Annu. Rev. Nucl. Part.
Sci. 2004, 5469-114
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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 Ip 225 15 mb
  (sr 204.8) In 213 40 mb (sr 233.2)
• ?p 1.88 0.06 (exp 1.793) ?n
  - 1.83 0.35 (exp -1.913)
• (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 2nd resonance region for the neutron very
  large in the 2nd res. region and dominant in
  the 3rd res. region and beyond

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The FFR Sum Rule
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The Low Energy Theorem for neutral pions is
strongly violated in the physical region
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Invariant Amplitudes of Pion Photoproduction
? N ! ? N

• Lorentz invariance, and P, C and T symmetries

4 Lorentz invariant functions of ? (s-u)/4MN
and t
6 functions of ?, t and Q² in electroproduction
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LET for pion photoproduction
in the soft-pion limit for n0, t0 (
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the Born terms in pseudovector coupling have the
correct symmetries and the non-Born terms vanish
in this limit
Born tems with pseudovector coupling
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Dispersion Relations at fixed t
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no sum rule for charged pion
photoproduction
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FFR Sum Rule
Fubini, Furlan, Rossetti, Nuovo Cimento 43 (1966)
161
FFR discrepancy
Heavy Baryon Chiral Perturbation Theory
HBChPT
Dispersion Relation with Im A1 from MAID2003
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soft pion point
pion threshold
physical region of pion photoproduction
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extrapolation to the unphysical region
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FFR Discrepancy ?(n,t) from HBChPT
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at soft-pion point
at physical threshold
FFR ? (?thr, tthr)
? (0, tthr)
FFR (?0,t0)
HBChPT
EXP
MAID03 (t0) (ttthr)
MAID03
?p,-?n
1.792 1.66
2.24
2.29/2.33/2.37
1.793
Proton
1.986 1.82
2.44
2.52/2.56/2.79
1.913
Neutron
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)
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FFR discrepancy ? from MAID03
t tthr
Pasquini, Drechsel, Tiator, Eur. Phys. J A23
(2005)
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Integrands from MAID03
t tthr
third resonance region
D13(1520)
isoscalar
? loops
? (1232)
isovector
? loops
third resonance region
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PROTON
Bernard, et al., ZPC70 (1996) HBChPT at O(p3)
t tthr
Bernard, et al., PLB378 (1996) HBChPT at O(p3)
Bernard, et al. EPJ A11 (2001) HBChPT at O(p4)
DR-MAID
Mainz experiment,Schmidt, et al., PRL 87 (2001)
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NEUTRON
Bernard, et al., ZPC70 (1996) HBChPT at O(p3)
t tthr
Bernard, et al., PLB378 (1996) HBChPT at O(p3)
Bernard, et al. EPJ A11 (2001) HBChPT at O(p4)
DR-MAID
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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?20, ?t0)
• Predictions at ?, t0 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, HBChPT and
  Experiment in the threshold region (?
  ?thr)
• problems with ChPT at low ? lt ?thr because of
  the
• non-relativistic approximation of HBChPT