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Dynamics of the decay j ! 3ú
0
S. Teige a , A. R. Dzierba a , J. Gunter a , J. J. Manak d ,
J. Napolitano c , D. R. Rust a , E. Scott a , P. T. Smith a ,
T. Sulanke a , A. Szczepaniak b , and D. P. Weygand d
a Department of Physics, Indiana University,Bloomington IN 47405, USA
b Nuclear Theory Center, Indiana University,Bloomington IN 47405, USA
c Department of Physics, Rensselaer Polytechnic Institute, Troy NY 12180,USA
d Thomas Jefferson National Accelerator Facility, Newport News Virginia
Abstract
The parameter ff = \Gamma0:005 \Sigma 0:007 (stat) \Sigma 0:002 (syst) describing the density of
events on the j ! 3ú 0 Dalitz plot has been measured. The result is compared to
previous values and theoretical predictions.
1 Introduction
Chiral perturbation theory offers a consistent description of low energy QCD
particularly applicable to meson decays. An outstanding difficulty is the large
discrepancy between the predicted and observed rate of the j ! 3ú de­
cay. (1). We report here a measurement of the transition matrix element
j M(j ! 3ú 0 ) j 2 . The density of events on the Dalitz plot has been calculated
theoretically to be uniform by Gasser and Leutwyler (1) and by Di Vecchia (2).
More recently a calculation of Kambor, Wiesendanger and Wyler (3) found a
density which decreases as the distance from the center of the plot increases.
The Dalitz plot density is specified by a single parameter, conventionally de­
noted as ff. Kambor et al. show that both ff and the rate \Gamma(j ! 3ú) are
sensitive to a parameter of the theory ( c in their notation ). The dependence
of ff and \Gamma on c is such that a more negative value of ff reduces the value
of \Gamma thereby increasing the disagreement with experiment. Two values of ff
are determined by (3), ­0.014 and ­0.007, the latter containing higher order
corrections and leading to a prediction for \Gamma(j ! 3ú) more consistent with
experiment.
Preprint submitted to Elsevier Preprint 3 August 1999

To second order in the center of mass pion energy (4) the Dalitz plot density
for the decay j ! 3ú 0 is parameterized as
j Mj 2 / 1 + 2ffz (1)
with
z = 2
3
3
X
i=1
/
3E i \Gamma m j
m j \Gamma 3m ú 0
! 2
=
/
ae
ae max
! 2
(2)
where E i is the energy of the i'th pion in the j rest frame and ae is the distance
from the center of the Dalitz plot.
A uniformly populated Dalitz plot ( as in (2) and (1)) gives ff = 0. ff has been
measured previously by Baglin et al. (5) , Alde et al. (6) , and Abele et al.
(7). Recent results are shown in figure 1 along with the result of this work.
Reference (7) finds a value significantly different than zero and in probable
disagreement (2oe) with theoretical expectations. (5) and (6) are consistent
with both zero and the non­zero expectation of (3).
Fig. 1. Previous measurements of ff and the result of this work. Also shown are the
results obtained by varying the analysis. The dashed lines indicate our estimate of
the systematic error in the measurement. See the text for a discussion of systematic
errors.
2

In this work we report on a determination of ff from the decay j ! 3ú 0 where
the j was produced by 18.2 GeV negative pions incident on liquid hydrogen.
The data was collected in 1995 by the Brookhaven National Laboratory E852
collaboration. The apparatus has been described previously (8),(9) and con­
sisted of a large, segmented lead glass calorimeter (LGD), a charged particle
tracking and veto system and a trigger system capable of determining the
charged particle multiplicity and the total energy deposited in the lead glass.
The data used for this analysis was collected with the ``all­neutral'' trigger
which required no charged particles, an energy deposition greater than 12
GeV in the lead glass and no photons in the downstream photon veto.
j ! 3ú 0 decays were selected by requiring exactly 6 reconstructed clusters
in the LGD and a visible total energy of greater than 16.5 GeV. All combi­
nations of assignments to the hypothesis 3ú 0 were tested and if the best ü 2
corresponded to a probability greater than 1% the event was considered a can­
didate. A preliminary fit using only the ú 0 mass constraint was performed to
allow evaluation of the 3ú 0 effective mass. Figure 2 shows the resulting distri­
bution. Events with a mass less than 0.65 GeV=c 2 (corresponding to 3oe from
the j mass ) were subjected to a full kinematic fit to the hypothesis ú \Gamma p ! jn
; j ! 3ú 0 . A confidence level (CL) cut selected decays for the final analysis.
It also was required that no photon have an energy less than 0.25 GeV and
that the minimum separation between photons be larger than 9 centimeters.
The effect and purpose of these requirements is discussed below. The final
resulting data set contained 87,500 events.
To determine ff eqn. 1 is fitted to the observed distribution of z corrected
for acceptance and phase space dependence. The acceptance correction was
based a monte­carlo taking into account the apertures of the apparatus and a
large sample of GEANT generated electromagnetic showers. The requirement
on the minimum photon energy and the minimum allowable photon separa­
tion mentioned above removed any uncertainties in the monte­carlo associated
the transverse development of electromagnetic showers and the behavior of
low energy photons. The separation requirement also selected events where
the nearest two electromagnetic showers were well resolved. The phase space
dependence of z was removed by dividing the observed distribution by the
distribution due to a uniformly populated Dalitz plot.
The distribution resulting from removing the phase space dependence and
correcting for the acceptance is proportional to M 2 and is shown in figure 3
along with the result of the fit. The distribution has been numerically scaled
to give a proportionality constant of one. Our result is ff = \Gamma0:0047 \Sigma 0:0074
where the error is statistical only.
Systematic effects have been considered and the important contributors are
3

3p 0 effective mass, GeV/c 2
Events
per
0.010
GeV/c
2
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0.45 0.5 0.55 0.6 0.65 0.7
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0.45 0.5 0.55 0.6 0.65 0.7
Fig. 2. The observed 3ú 0 effective mass distribution. The j signal dominates the
data with a secondary peak due to the decay K \Lambda0 ! K 0
s ú; K 0
s ! 2ú 0 visible. The
inset shows the j region, there are 161,000 events under the peak with an estimated
signal to noise ratio of 100 at the mass of the j.
shown in figure 1. The analysis required that no two photons have recon­
structed impact positions closer than 9 cm, a distance denoted by \Deltar. This
value was chosen by varying \Deltar and determining ff. It was found that ff does
not depend on the value of \Deltar provided \Deltar ? 9 cm. The contribution to the
systematic error is taken as the difference between our reported value of ff and
the extrema of values obtained with 8:5 Ÿ \Deltar Ÿ 11 cm.
A similar analysis was performed to determine the value used for the confi­
dence level cut. It was found that ff does not depend on the value used for
the confidence level cut provided CL – 0:3. A contribution to the systematic
error of magnitude similar that of the \Deltar study was also determined.
The effect of requiring the lowest energy photon to have an energy larger than
0.25 GeV was investigated by removing this requirement and by replacing it
with a requirement of 0.5 GeV. A contribution of \Sigma0:001 to ff was found.
Finally, the data set was divided into two statistically independent halves by
experimental run number and, separately, by momentum transfer. In both
4

0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
0 0.2 0.4 0.6 0.8 1
Z
|M|
2
Fig. 3. z­distribution of j ! 3ú 0 events on the Dalitz plot. The solid line is the
result of a fit corresponding to eqn. 1
cases, the values determined for ff were not statistically significantly different
leading us to conclude these selections do not contribute to our systematic
error. Adding the effects considered in quadrature give our final estimate of
the systematic error, \Sigma0:002
2 Conclusion
We have measured the ff parameter of the transition matrix element M(j !
3ú 0 ) to be ff = \Gamma0:005 \Sigma 0:007 (stat) \Sigma 0:002 (syst). Our result differs from
one previous measurement (7) by slightly more than 2oe but is consistent with
several other (6), (5) measurements. When compared to theoretical expecta­
tions, the value of zero favored by Di Vecchia, Gasser and Leutwyler (2), (1)
cannot be ruled out. The result favors the predicted value of ­0.007 of Kam­
bor, Wiesendanger and Wyler (3) but their other predicted value, ­0.014, is
not definitively ruled out.
5

References
[1] J. Gasser and H. Leutwyler, Nucl. Phys. B250 539 (1985)
[2] P. Di Vecchia et al., Nucl. Phys. B181 318 (1981)
[3] J. Kambor, C. Wiesendanger and D. Wyler, Nucl. Phys. B465 215
(1996)
[4] C. Zemach, Phys. Rev. D24 1325 (1981)
[5] C. Baglin et al., Nucl. Phys. B22 66 (1970)
[6] D. Alde et al., Z. Phys. C25 225 (1984)
[7] A. Abele et al., Phys. Lett. B417 193 (1998)
[8] R. R. Crittenden et al., Nucl. Instrum. Methods A387 (1997) 377
[9] S. Teige et al., Phys. Rev. D59 012001 (1999)
6