How to extract the P{sub 33}(1232) resonance contributions from the amplitudes M{sub 1{sup +}}{sup 3/2},E{sub 1{sup +}}{sup 3/2},S{sub 1{sup +}}{sup 3/2} of pion electroproduction on nucleons Page: 4 of 27
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(Sec. 3) shows that the particular solutions of integral equations generated by the Born
term should be considered as nonresonance background to the contribution of the P33
resonance. These solutions are produced by rescattering effects in the diagrams 1(a-c)
corresponding to the Born term. For the first time the presence of such nonresonance
contribution in the multipoles M1+2, E3/2 was mentioned in Ref. [13] within dynamical
model describing pion photoproduction in the P33 region in terms of the diagrams of Fig.
1 taking into account rescattering effects.
In Sec. 4 taking into account nonresonance contributions in the multipoles Mi2, E3/2
the amplitudes Ai/2, A3/2 for the yN - P33(1232) transition are extracted from experi-
ment. The magnitudes of these amplitudes turned out to be smaller than the magnitudes
which are traditionally extracted from experiment without taking into account nonres-
onance background contributions in the amplitude M12. As a result, the traditionally
mentioned disagreement between quark model predictions and experiment for the ampli-
tudes A1/2, A3/2 turned out to be removed.
2 Dispersion relations for invariant and multipole
amplitudes
In this Section we will present very briefly main formulas which are necessary for our
calculations. Following the work [11] we choose invariant amplitudes in accordance with
the following definition of the hadron current:
{ = 2,(p2)5 [7y(7yk) - (yk)yP] + 2PNB2 + 2q/B3 + 2kB4
-,y B5 + (yk)P"B6 + (yk)k"B7 + (yk)q"Bsu(pi), (5)
where k, q, pl, p2 are the 4-momenta of virtual photon, pion, initial and final nucleons,
respectively, P = 2(p1 + p2), Q2 = -k2, B1, B2, ...B8 are invariant amplitudes which are
functions of the invariant variables s = (k + pi)2, t = (k - q)2, Q2.
The conservation of the hadron current leads to the relations:
4Q2B4 = (s - u)B2 - 2(t + Q2 _ 2)B3, (6)4
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Aznauryan, I. G. How to extract the P{sub 33}(1232) resonance contributions from the amplitudes M{sub 1{sup +}}{sup 3/2},E{sub 1{sup +}}{sup 3/2},S{sub 1{sup +}}{sup 3/2} of pion electroproduction on nucleons, article, August 1997; Newport News, Virginia. (https://digital.library.unt.edu/ark:/67531/metadc706849/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.