Non-model determination of the ∆(1232) parameters
The ∆(1232) resonance and pole parameters are determined from the data of pN elastic scattering analysis in the framework of a non-model approach.
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| Date: | 2001 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Series: | Вопросы атомной науки и техники |
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| Cite this: | Non-model determination of the ∆(1232) parameters / A.S. Omelaenko // Вопросы атомной науки и техники. — 2001. — № 1. — С. 48-49. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-784422015-03-18T03:01:51Z Non-model determination of the ∆(1232) parameters Omelaenko, A.S. Nuclear reactions The ∆(1232) resonance and pole parameters are determined from the data of pN elastic scattering analysis in the framework of a non-model approach. 2001 Article Non-model determination of the ∆(1232) parameters / A.S. Omelaenko // Вопросы атомной науки и техники. — 2001. — № 1. — С. 48-49. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 13.75Gx. http://dspace.nbuv.gov.ua/handle/123456789/78442 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Nuclear reactions Nuclear reactions Omelaenko, A.S. Non-model determination of the ∆(1232) parameters Вопросы атомной науки и техники |
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The ∆(1232) resonance and pole parameters are determined from the data of pN elastic scattering analysis in the framework of a non-model approach. |
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Article |
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Omelaenko, A.S. |
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Omelaenko, A.S. |
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Omelaenko, A.S. |
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Non-model determination of the ∆(1232) parameters |
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Non-model determination of the ∆(1232) parameters |
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Non-model determination of the ∆(1232) parameters |
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Non-model determination of the ∆(1232) parameters |
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Non-model determination of the ∆(1232) parameters |
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non-model determination of the ∆(1232) parameters |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2001 |
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Nuclear reactions |
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Non-model determination of the ∆(1232) parameters / A.S. Omelaenko // Вопросы атомной науки и техники. — 2001. — № 1. — С. 48-49. — Бібліогр.: 8 назв. — англ. |
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Вопросы атомной науки и техники |
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AT omelaenkoas nonmodeldeterminationofthe1232parameters |
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2025-07-06T02:32:24Z |
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N U C L EAR R EA C TI O N S
NON-MODEL DETERMINATION OF THE ∆(1232) PARAMETERS
A.S. Omelaenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
The ∆(1232) resonance and pole parameters are determined from the data of πN elastic scattering analysis in the
framework of a non-model approach.
PACS: 13.75Gx.
In [1] the significant discrepancies in the pole pa-
rameters of the P33 πN scattering amplitude, namely in
the absolute value and phase of the corresponding to the
∆(1232) resonance residue, were discussed using a real-
istic resonance model. Here we continue the discussion
trying to perform model-independent evaluation of the
resonance and pole parameters.
The P33 amplitude of the elastic πN scattering sup-
posed to be purely elastic in the ∆(1232) excitation re-
gion. Corresponding element of S matrix depends on the
total energy W and can be written in the following gen-
eral form through real K matrix:
)(1
)(1)(
WiK
WiKWS
−
+= , (1)
for one-channel case the K matrix element can be writ-
ten in the terms of the phase shift δ33 as
)(tan)( 33 WWK δ= . (2)
It is well known that the right-hand side of Eq. (2)
has a pole at W0≅1232 MeV and decreases as ∼q3→0 if
W goes to its value at the πN threshold (W0 is the point
where the phase shift δ33 passes through value 90° and q
is the c. m. momentum). These feathers have such a sol-
id experimental and theoretical basement that introduc-
ing them explicitly in parameterization of the K matrix
element cannot be treated as some kind of a real model
restriction:
)(
)(
)()(
0
0
0
3
3
WF
WWWq
WqWK
−
Γ
=
. (3)
In Eq. (3) Γ0 is the experimental width of the ∆
(1232) resonance, and function F(W) contains all dy-
namics of the P33 amplitude aside from the threshold be-
havior and the pole property. In the framework of any
specific resonance description F(W) presents the energy
dependence of the experimental width. For example, in
[2,3] one can find many model variants of F(W). In any
phenomenological model with explicit background on
level with the resonance interaction the total K matrix
element also can be presented in form (3) with F(W) de-
pending on the background parameters. Quite similar
situation takes place in more complicated dynamical
models ([4], for example).
In searching the non-model description of the P33
amplitude we use power series for F(W) up to some
maximal degree n:
...)()(1)( 2
0201 +−+−+= WWcWWcWF . (4)
Actually, this approach is a model-independent base
for determination of the resonance and pole parameters
of the P33 amplitudes in region of the ∆(1232) excitation,
if the used series has sufficient converging near the point
W0. Using the K matrix gives an advantage of treating
the most simple series expansion with real coefficients.
In addition, in the complex W plane a circle of a fixed
radius covers the maximal number of experimental
points when its center is situated on the real axe. An in-
terval of real axes from ∼(W0 − Γ0/2) up to ∼(W0 + Γ0/2)
seems be the most preferable in the role of the corre-
sponding mathematical vicinity, as in this case the re-
gion of convergence in the complex plane W reaches the
pole position on the second Riemann’s sheet.
1.
2.
obtained from the SAID system (http://said.-
phys.vt.edu). Data from different energy intervals
W1...W2 were fitted by χ2 method with using Eqs. (2),
(3) and (4) for calculation of the phase shift δ33. As all
but SM99s solutions are given without errors we have
used an arbitrary error 0.25° for each point. So, W0, Γ0
and coefficients c1...n are free parameters (n≤4). Parame-
ters W0, Γ0, coordinates of the pole in the complex W
plate Re Wp, Im Wp, absolute value res and phase ϕ of
the residue are presented in the table. n in (4) was re-
stricted by the maximal value at which the fit is mean-
ingful. The numbers of used points N and the χ2 per
number of degree of freedom are indicated, too.
As it follows from the first five lines of the table, all
parameters, mentioned above, are practically the same
for n equal 3 and 4. This situation is illustrated by the
48 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 1.
Series: Nuclear Physics Investigations (37), p. 48-49.
figure, too. Results of fitting on the narrower energy in-
terval confirm this observation. The small shifts in val-
ues of being the most sensitive residue parameters give
some measure of real errors. It is interesting to note that
for SM99 fits on the full energy interval W from
1100 MeV to 1350 MeV give practically the same re-
sults as the fits with the data in vicinity of W0. These an-
swers are in accordance with the resonance model ([1]).
The most plausible rounded values of obtained parame-
ters are Γ0=115.3 MeV; Re Wp=1211.5 MeV; Im Wp=-
50.8 MeV; |res|=53.1 MeV; ϕ=46.7°.
Ana-
lyses
W1,
MeV
W2,
MeV
N n χ2/d.f. W0,
MeV
Γ0, MeV Re Wp,
MeV
Im Wp,
MeV
|res|,
MeV
ϕ, deg.
SP99 1180 1260 17 0 .83D+02 1230.88 128.15 1204.29 -40.93 35.84 -61.32
17 1 .40D+00 1232.66 116.72 1208.73 -51.89 55.44 -55.42
17 2 .17D-02 1232.53 115.19 1211.65 -51.33 54.92 -46.50
17 3 .21D-03 1232.53 115.32 1211.56 -50.84 53.11 -46.73
17 4 .23D-03 1232.53 115.32 1211.53 -50.84 53.11 -46.87
SP99 1160 1280 25 0 .21D+03 1229.14 119.77 1204.57 -39.52 34.85 -59.45
25 1 .21D+01 1232.68 118.64 1208.08 -51.66 54.40 -56.21
25 2 .13D-01 1232.55 115.17 1211.64 -51.50 55.30 -46.67
25 3 .27D-03 1232.53 115.28 1211.60 -50.92 53.40 -46.63
25 4 .22D-03 1232.53 115.30 1211.51 -50.92 53.37 -46.98
SP99s 1180 1260 5 0 .15D+03 1230.51 132.19 1202.92 -41.41 36.11 -62.40
5 1 .28D+01 1232.49 114.82 1209.21 -52.32 56.92 -54.72
5 2 .54D+00 1231.83 112.21 1214.34 -48.64 49.39 -36.62
SP99s 1160 1280 7 0 .30D+03 1230.01 135.91 1201.51 -41.79 36.32 -63.40
7 1 .59D+01 1232.89 116.12 1209.03 -53.59 59.06 -55.67
7 2 .11D+01 1232.14 112.18 1213.60 -50.72 54.42 -40.64
7 3 .53D+00 1232.04 113.87 1211.87 -47.85 44.98 -45.38
SP99 1100 1350 51 2 .33D+00 1232.63 116.33 1210.78 -51.84 55.65 -48.97
51 3 .63D-02 1232.55 115.22 1211.66 -51.09 53.89 -46.52
51 4 .20D-03 1232.53 115.30 1211.53 -50.91 53.34 -46.89
KA84 1180 1260 17 2 .50D+00 1231.27 116.35 1212.18 -51.89 55.56 -39.55
1160 1280 25 2 .88D+00 1231.32 118.15 1208.68 -52.15 55.89 -50.23
25 3 .80D+00 1231.40 117.86 1208.41 -54.10 62.73 -51.82
KP80 1180 1260 10 1 .79D+00 1230.94 116.62 1207.00 -49.78 51.51 -55.53
10 2 .68D+00 1230.91 115.12 1210.53 -50.04 52.29 -44.68
KP80 1160 1280 13 1 .20D+01 1231.00 118.21 1206.46 -50.50 52.44 -56.36
13 2 .46D+00 1230.92 115.36 1209.81 -50.14 52.52 -46.99
13 3 .52D+00 1230.93 115.31 1209.84 -50.32 53.15 -46.93
KP80 1100 1350 23 2 .16D+01 1231.01 116.90 1208.45 -50.93 53.76 -50.80
23 3 .67D+00 1230.92 115.31 1209.79 -49.63 50.68 -47.17
23 4 .69D+00 1230.94 115.10 1210.07 -49.95 51.63 -46.29
The resonance and pole parameters for solution SM99
vs total number of free parameters n+2
Good convergence of the procedure discussed for
solution SM99 can be partially conditioned by a form of
the energy-dependent parameterization. Nevertheless
such convergence for SM99s as for solutions from pre-
vious analyses is not reached. This can be considered as
an argument in favor of additional measurements of
elastic scattering in the first resonance excitation region.
REFERENCES
3. A.S. Omelaenko. Determining the ∆(1232) pole pa-
rameters // VANT, 2000, № 2, p. 3-6.
4. S.S. Vasan. Determination of the position and
residues of the ∆++ and ∆0 poles // Nucl. Phys. 1976,
v. B106, p. 535-545.
5. A.S. Omelaenko. Determination of the mass and
width of ∆(1232) P33 resonance in the phenomeno-
logical approach inspired by the Lippman-Schwinger
49
equation // Yad. Fiz. 1992, v. 55, p. 1050-1060 (Sov.
J. Nucl. Phys. 1992, v. 55, p. 591-597).
6. A.S. Omelaenko. Determination of the mass and
width of ∆(1232) resonance within the framework of
a nonrelativistic model with a separable potential //
Ukr. Fiz. J. 1996, v. 41, p. 524-529.
7. A. Arndt, R.L. Workman, I.I. Strakovsky, M. Pavan.
Partial-wave analysis of πN-scattering. Eprint nucl-
th/9807087 (submitted to Phys. Rev. C).
8. R. Koch, E. Pietarinen. Low-energy πN partial wave
analysis // Nucl. Phys. 1980, v. A336, p. 331-346.
??????? ??????? ????? ? ???????. 2000, ?2.
?????: ??????-?????????? ???????????? (36), ?. 3-6.
50
A.S. Omelaenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
PACS: 13.75Gx.
obtained from the SAID system (http://said.phys.vt.edu). Data from different energy intervals W1...W2 were fitted by 2 method with using Eqs. (2), (3) and (4) for calculation of the phase shift 33. As all but SM99s solutions are given without errors we have used an arbitrary error 0.25 for each point. So, W0, 0 and coefficients c1...n are free parameters (n4). Parameters W0, 0, coordinates of the pole in the complex W plate Re Wp, Im Wp, absolute value res and phase of the residue are presented in the table. n in (4) was restricted by the maximal value at which the fit is meaningful. The numbers of used points N and the 2 per number of degree of freedom are indicated, too.
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