Sensitivity of the deuteron form factor to nucleon resonances Page: 4 of 25
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5 j
(p+q)M ei PM
FIG. 2. The basic electromagnetic coupling between tower members
are then fit to the Nijmegen 1993 phase shifts below 350 MeV [21]. Excellent
fits are obtained using towers with either one or three members. Fits using a
two member tower were attempted, but proved to be less successful than those
obtained from one- and three-member towers. The reason for these poor fits
is under investigation; it may be related to the need for a certain symmetry
between the coupling constants. Due to the larger number of free parameters,
X2s for three-member tower models are slightly improved over those using only
the proton. These results will be presented in Sec. IC.
In order to explore the effect of the resonances in a simple NN system where
they can can only make virtual contributions, we study the deuteron form factor.
The electromagnetic coupling between tower members, illustrated in Fig. 2, is
chosen to be proportional to ei = tj e, where e is the electric charge. The full
currents with electromagnetic form factors are constructed using the methods of
Gross and Riska [22] (see Eq. (2.29) for more detail). In this simple first treatment
we choose (4y = bij so the photon itself cannot excite or de-excite members of
the tower. In order to satisfy gauge invariance an interaction current must be
included, and the inclusion and study of this current is one of the novel features
of this work. This current is derived from the kernel V by minimal substitution
using the methods of Ito et. al. [231. However, because the structure of the kernel
is different from the one assumed by Ito et. al., a slightly different argument is
needed, and because the derivation is somewhat lengthly the details are presented
in the Appendix.TABLE 1. Parameters in the tower of states models. Numbers in bold face were
varied during the fitting procedure. The form factor parameters and all masses are in
GeV; all the coupling constants have dimensions GeV4.
Model 1 Model 3 Model 31
Parameter 3So 3s, 1S 3s, 1S 3S
a 0.198125 0.155094 0.197611 0.157080 0.197750 0.150054
p 1.09772 1.27872 1.09795 1.29825 1.10168 1.28994
y 0.171623 0.136404 0.172138 0.139047 0.172141 0.129887
Pc 0.3492 0.412689 0.3492 0.412689 0.3492 0.412689
911 -1516.96 -2553.31 -1500.89 -2615.86 -1504.81 -2554.04
912 155.0 406.731 190.0 404.563
913 155.0 -105.0 190.0 -105.0
922 10.0 -105.0 15.0 -105.0
923 272.175 597.206 253.928 591.489
933 10.0 -105.0 15.0 -105.0
x2 6.97 7.60 5.06 6.83 3.66 6.04
mi 0.93825 0.93825 0.93825
m2 1.44 1.17
M31.52 1.52
C. Numerical results
From the real and imaginary parts of M we calculate the phase shifts and
perform a least-squares fit to fix the free parameters in the models. The resulting
parameters for three models are shown in Table I. Model 1 consists of a tower
with the proton only, Model 3 employs a three-member tower consisting of the
proton, Roper, and D13, and Model 31 uses the same three-member tower, except
that the mass of the Roper has been lowered to increase its inelastic contributions.
The inelastic region has no effect on the fitting procedure, as only data below
350 MeV are used in the fit.
In order to best understand the nature of the restrictions placed on these
parameters, look at the expression for tan 6. For a one-member tower (Model 1),
this expression reduces to
TABLE II. Binding energy of the deuteron as calculated in each of the three models.
The percent error shown is the error between the calculated value and the expected
value of 2.22 MeV. The energies are given in MeV.
Model 1 Model 3 Model 31
Binding Energy 2.15 2.02 2.47
Percent Error 3.2 9.0 11.37
6
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Herbst, Kelly Ann & Gross, Franz. Sensitivity of the deuteron form factor to nucleon resonances, article, October 1, 1997; Newport News, Virginia. (https://digital.library.unt.edu/ark:/67531/metadc710625/m1/4/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.