Papers in Physics, vol. 9, art. 090009 (2017)

Received: 27 June 2017, Accepted: 9 October 2017
Edited by: D. Restrepo
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.090009

www.papersinphysics.org

ISSN 1852-4249

An improvement to the measurement of Dalitz plot parameters

S. Ghosh,1∗ A. Roy1†

Precise measurement of the Dalitz plot parameters of the η′ → η π+ π− decay gives a
better insight of the dynamics of the heavy pseudo-scalar meson. Various measurements
of the Dalitz plot parameters of the η′ meson have been performed with the detection of
all the final state particles including the neutral decay of η → 2γ. In many experiments,
reconstruction of the η meson from the neutral decay modes comes with the disadvantage of
poor resolution and low efficiency compared to that of the charged particles. In this article,
a study of the Dalitz plot parameters keeping the η meson as a missing particle is presented.
The method is found to be advantageous in the case of poor photon resolution. Effect of
the charged particle resolution on the Dalitz variable Y is also examined. This work may
provide guidance to select a suitable method for the Dalitz plot analysis, depending on the
detector resolution.

I. Introduction

Low energy quantum chromodynamics (QCD) is
effectively studied in the framework of the chiral
perturbation theory (ChPT). The Lagrangians for
the hadronic processes are derived from ChPT and
the effective degrees of freedom are, usually, the
octet of pseudo-scalar mesons (π, K, η) [1]. The
observed mass of the η′ meson in nature is much
higher than the other pseudo-scalar mesons, which
gives the axial U(1) anomaly [2, 3]. This suggests
the possible existence of many unsolved problems
that could provide inputs to the Lagrangian. Study
of the production and decay of the η′ meson is,
therefore, important to both theory and experi-
ment [4]. Some possible dynamics that could be
studied in the η′ decay model are the new sym-

∗E-mail: phd12115113@iiti.ac.in
†E-mail: ankhi@iiti.ac.in

1 Discipline of Physics, School of Basic Sciences, Indian
Institute of Technology Indore, Khandwa Road, Simrol,
MP-453552, India.

metry and the symmetry breaking during interac-
tions [5], gluonic contributions, gauge field config-
urations with non-zero winding number, and the
quark instantons [6, 7]. This has motivated many
groups to study both charged and uncharged de-
cays of the η′ meson [1, 8].

The Dalitz plot plays an important role as a use-
ful tool to study decay dynamics of a meson decay-
ing into three bodies. As the three-body decay has
two degrees of freedom, one can define two linearly
independent variables to represent the decay in the
phase space. In this article, we shall study the de-
cay of η′ → η π+ π− and for that we define the
Dalitz plot variables [9] as

X =

√
3(Tπ+ −Tπ−)

Q
, (1)

Y =
(mη + 2mπ)

mπ
·
Tη
Q
− 1.

Here T and m are, respectively, the kinetic energy
(in the rest frame of η′) and the mass of the par-
ticles indicated by the subscripts. Then, Q = Tπ+

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Papers in Physics, vol. 9, art. 090009 (2017) / S. Ghosh et al.

+ Tπ+ + Tπ+ = mη′ - mη - 2mπ, is the available
energy of the reaction. The decay dynamics of the
η′ mesons is studied in the form of Dalitz plot pa-
rameters which are obtained by fitting the Dalitz
plot with the following general parameterization:

f(X,Y ) = A
(
1 + aY + bY 2 + cX + dX2

)
. (2)

Here, a, b, c, and d are the Dalitz plot parame-
ters of the decay and A is the overall normalization
constant. The measurement of Dalitz plot param-
eters are specifically important to understand and
crosscheck the correct inputs to the theoretical dis-
tribution of the Lagrangian [10]. It is, therefore,
important to measure the Dalitz plot parameters
precisely with good statistics and resolution.

The Dalitz plot parameters of the η′ → η π+
π− decay have been measured in the experiments
of the VES [11] and the BESIII [12] collaborations.
GAMS-4π [13] and IHEP-IISN-LANL-LAPP col-
laborations [7] have reported the Dalitz plot pa-
rameters for the neutral decay mode (η π0 π0) of
the η′ meson. The values of the Dalitz plot pa-
rameters for both of these decay modes of the η′

meson should be the same under the isospin limit.
However, this is not observed in the above experi-
ments. This discrepancy may be reconciled by pre-
cise measurements of the small Dalitz plot parame-
ters with high statistics. The previous experiments
have identified the final state particles (π+, π− and
γ) of the η′ → (η) π+π− → (2γ) π+π− decay and
used the invariant mass method for the Dalitz plot
analysis. If the photon resolution is poor, the re-
construction efficiency of the η meson will be signif-
icantly low and will also affect the Dalitz variable
Y and the parameters related to this variable.

Here, we will be reporting a method that im-
proves the Dalitz plot parameters in the measure-
ments of the η′ decay modes and found particu-
larly advantageous in the case of poor photon res-
olution. We shall consider the production of η′

meson through photo-production reaction first and
then introduce the detector resolution to all the
final state particles π+, π−, p and γ of the chan-
nel γ + p → η′p → (η) π+π−p → (2γ) π+π− p
to simulate the detector environment. The Dalitz
plot parameters are then calculated using the fol-
lowing methods: (a) exclusive measurement or in-
variant mass method and (b) missing measurement
method. The exclusive measurement method is

commonly used, where all the final state parti-
cles (π+, π− and γ) are considered and η is re-
constructed as the invariant mass (η → 2γ). In
the missing measurement method, we have recon-
structed the η′ meson as a missing particle using
the information of the recoiled proton, the beam
photon and the target proton (γbeam + ptarget →
η′ precoil). The η

′ information along with π+ and
π− is used to reconstruct the η meson as a missing
particle. The calculations of the Dalitz plot param-
eters using both these methods are described and
compared. In addition to that, a bin width study
is also performed with missing analysis to optimize
the bin size for the extraction of the Dalitz plot
parameters. In the missing measurement method,
a dependence of the Dalitz variable Y on the de-
tector resolutions of the charged particles has also
been examined. Further, to study the effect of the
background, method (b) was subjected to a com-
binatoric background channel and the Dalitz plot
parameters are then systematically studied with
varying background component in the mixture of
the signal and the background. The background is
then subtracted from every Dalitz plot bin and the
parameters are reported.

II. Model to fold the detector reso-
lutions

The Pluto simulation framework (version 5.42) de-
veloped by the HADES collaboration was used to
generate hadronic physics reactions for this analy-
sis [14]. Dalitz plot parameters of the η′ → η π+
π− decay from the BESIII [12] experiment were
used as input parameters for the generated events.
A total of 105 γ + p → η′ p events, each with
a photon beam energy of 2.5 GeV were generated
in the phase space model for this analysis. How-
ever, the energy of the η′ meson is arbitrary as the
Dalitz plot parameters are independent of the ini-
tial energy of the η′ meson. The detector effects
are absent in the events generated by the Pluto
event generator. However, in reality, the momen-
tum of the particle passing through the detector is
modified according to the detector resolution. Gen-
erally, the detector response is distributed with a
resolution which is dependent on the magnitude of
the particle’s momentum. To incorporate the de-
tector response, each component of the momentum

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Papers in Physics, vol. 9, art. 090009 (2017) / S. Ghosh et al.

Figure 1: (a) Number of events as a function of the momentum, where the solid histogram is for the
events with unit momentum before folding and the dashed histogram is for the events after the folding
with a Gaussian distribution whose σ is 3% of the mean. (b) Number of generated vs. reconstructed
events in the Dalitz variable bins X. (c) Number of generated vs. reconstructed events in the Dalitz
variable bins Y. (d) The Dalitz variable X vs. Y showing the bins completely inside the Dalitz plot
boundary.

of the particles are convoluted with a random num-
ber sampled from a Gaussian distribution of mean 1
and σ = 3% of the mean, which introduces the mo-
mentum dependent resolution to the particle. This
effect transforms particles with a single momentum
into a particle with a momentum distribution as
shown in Fig. 1(a) [15].

Since we are studying a case where the photon
resolution is poor compared to the charged parti-
cle resolution, the final state charged particles (π+,
π− and p) are folded with 1% resolution [16, 17] in
the momentum components to simulate a detector-
like effect, whereas 6% resolution is used for the
photons. There are experiments that suffer from
poor photon resolution for which 6% is rather an
underestimation [18]. Dalitz plot variables X and
Y, after the addition of resolution to the final state
particles in momentum components, are compared

with the generated Dalitz plot variables as shown
in Fig. 1(b) and (c). It is observed in Fig. 1(c) that
at higher values of Y, the bin migration is more sig-
nificant. The reason for this is that a higher value
of Y corresponds to a high energy η meson, which
decays to energetic photons with poor resolution.
This leads to a enhanced bin migration at higher
value of Y.

Folding has the effect of migrating the events
from one bin to the neighboring bins of the Dalitz
plot. It distributes the events in the bins away from
the diagonal, creating a homogeneous migration of
the events and thus allows to calculate the param-
eters from a properly binned Dalitz plot. Both
Dalitz plot variables, shown in Fig. 1, are divided
into 18 bins from −1.5 to 1.5. Though the migra-
tion is observed for the individual Dalitz variables
shown in Fig 1(b) and (c), the maximum number

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Papers in Physics, vol. 9, art. 090009 (2017) / S. Ghosh et al.

Figure 2: (a) Number of events vs. difference of the reconstructed (after folding) and the true value of
the Dalitz variable X and (b) the Dalitz variable Y for the exclusive measurement method. (c) Number
of events vs. difference of the reconstructed and the true value of Dalitz variable X and (d) the Dalitz
variable Y for the missing measurement method (b).

of events, however, lie on the diagonal and confined
well within the resolution of the bins.

The general parameterization of Eq. (2) was used
to parametrize the decay. A least square fitting
procedure MINUIT [19] was used to minimize the
χ2 given in Eq. (3) to fit each bin of the Dalitz plot.

χ2 =
∑(Nj −f(Xj,Yj)

∆Nj

)2
. (3)

Where Nj and ∆Nj denotes the number of events
and their statistical uncertainties for the Dalitz plot
bin j (j = 1, 2, ...,n). Xj and Yj are the central co-
ordinates of each bin, and f(Xj,Yj) denotes the
fitted form of the polynomial. The bins on the
boundary of the Dalitz plot are removed and only
bins which are completely inside the boundary are
considered as shown in Fig. 1(d).

i. Method (a): Exclusive measurement

In this method, measurement of the Dalitz plot pa-
rameters are performed with all the detected final
state particles π+, π− and γ. To calculate the reso-
lution of Dalitz variables X and Y, first a difference
of the reconstructed (after folding) and the true
value of the Dalitz variable is considered. Then,
the standard deviation of X and Y for a large num-
ber of events is calculated using

σ =

√√√√(∑(x−µ)2
N

)
, (4)

where x represents each value of the Dalitz vari-
ables and µ is the mean, which should be zero in
this case, and N is the total number of events. As
seen from Fig. 2(a) and (b), the resolutions of the
Dalitz variables X and Y are respectively 0.019 and
0.099, after the folding in the momentum compo-

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Papers in Physics, vol. 9, art. 090009 (2017) / S. Ghosh et al.

Table 1: Dalitz plot parameters with varying number of bins in the Dalitz variables X and Y for methods
(a) and (b).

Method (a) Method (b)

Par 17 × 17 18 × 18 19 × 19 17 × 17 18 × 18 19 × 19
a −0.067 ±0.008 −0.062 ±0.008 -0.066 ± 0.008 -0.045 ± 0.008 -0.044 ± 0.008 -0.044 ± 0.008
b −0.134 ±0.015 −0.142 ±0.015 -0.145 ± 0.014 -0.058 ± 0.016 -0.063 ± 0.015 -0.081 ± 0.015
c 0.014 ± 0.005 0.014 ± 0.005 0.009 ± 0.005 0.014 ± 0.006 0.014 ± 0.005 0.009 ± 0.006
d −0.098 ±0.009 −0.096 ±0.009 -0.093 ± 0.009 -0.085 ± 0.010 -0.081 ± 0.009 -0.092 ± 0.009
χ2

ndf
1.25 1.04 1.09 1.08 1.14 0.95

Figure 3: Comparison of the Dalitz plot parameters
in an exclusive measurement method with vary-
ing photon resolution and missing measurement
method along with the generated BESIII param-
eters.

nents. The resolution of the variables X and Y
are independent in this method as the former was
calculated from π+ and π− but the later was calcu-
lated from the photons and, therefore, causing the
resolution to be poor.

ii. Method (b): Missing measurement

In the missing measurement method, the Dalitz
variables X and Y are calculated without using the
information of the final photons from the decay of
the η meson. Only π+ and π− are folded with the
resolutions and the produced η′ information is used
to calculate the Dalitz variables X and Y. The cal-
culated resolutions of the Dalitz variables X and Y
are both 0.016 and are shown in Fig. 2(c) and (d).
In contrast to the exclusive measurement method,
in this method, the resolutions of X and Y are same.

Figure 4: Variation of the Y resolution with the
charged particle resolution.

This is due to the fact that both X and Y use the
information of the same particles (π+ and π−) with
added resolution [20].

iii. Comparison of the method (a) and (b)

To compare the methods described above, we have
extracted the Dalitz parameters by fitting the
Dalitz plot with the parameterization defined in
Eq. (2). All the bins are fitted to the general pa-
rameterization with the χ2 minimization. To find
an optimized binning, the Dalitz plot parameters
are calculated using the method (a) and (b) for
varying number of bins in X and Y axis as shown
in Table 1. It is found that the best χ2/ndf is ob-
tained for 105 events with a binning of 18 × 18 for
both methods. It can be seen that a systematics
arises due to varying bin size on the parameters a
and b in the invariant mass method. In both the
methods a binning of 18 × 18 is thus used to cal-
culate the Dalitz plot parameters, which is higher

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Papers in Physics, vol. 9, art. 090009 (2017) / S. Ghosh et al.

than the resolution of the Dalitz variables X and
Y. A comparison of the parameters obtained using
the methods (a), with different photon resolution,
and (b), with the generated BESIII parameters, is
shown in Fig. 3 along with their errors and χ2/ndf.
It can be seen from the figure that with poorer
photon resolution the parameters a and b system-
atically deviate from the generated value. Though
the values of the parameters c and d are consistent
as shown in Table 1, it is observed that due to poor
photon resolution, deviation of the a and b param-
eters from the input in the exclusive measurement
method is significant compared to the missing 2γ
measurement method. Even the deviation of pa-
rameter b is higher than that of a, as b is the coeffi-
cient of a quadratic Y. The poor photon resolution
directly affects the η momentum resolution, which
is reflected in the poor Y resolution. The missing
measurement method is, therefore, preferable for
the Dalitz plot analysis in this case of poor photon
resolution.

Since in the missing measurement method the
Dalitz variable Y depends on the charge parti-
cle resolution, we have carried out another study
to examine the variation of the Y resolution with
the charged particle resolution. We found that Y
varies linearly with the charged particle resolution
as shown in Fig. 4. It can be concluded that if
the charge particle resolution is equally poor as
the photon resolution, the missing analysis method
is still a better choice compared to the exclusive
measurement method. In the present experimen-
tal condition of 1% charged particle resolution and
6% neutral particle resolution, the exclusive mea-
surement method introduces a poorer Y resolution
compared to the missing measurement method.

III. Background study

The missing measurement method comes with a
cost of additional combinatoric background for the
channel η′ → η π+ π−. This combinatoric back-
ground arises from π+ and π− produced in the de-
cays η′ → η π+ π− and η → π+ π− π0, which
leaves the π+ and π− in both decays with a similar
available energy and hence similar kinematics. We
consider a background η′ → η π0 π0 generated with
the same Dalitz plot parameters because of isospin
symmetry [1] as η′ → η π+ π− and η is further

Figure 5: The number of events vs. the miss-
ing mass Mx(η

′, π+π−) from the signal and back-
ground channel.

decayed into η → π+ π− π0. We mixed the signal
channel with 5%, 10% and 15% background and
then folded it to add resolution. The Dalitz plot pa-
rameters from these three set are given in Table 2.
The missing mass Mx(η

′, π+π−) in Fig. 5 clearly
shows the contribution of the background channel
in the η meson mass region due to misidentifica-
tion. The Dalitz plot parameters a and b related
to the Dalitz variable Y are systematically shifted
from their generated values due to the presence of
this background. Meanwhile, the Dalitz parame-
ters c and d do not change since π+ and π− from
both channels have similar kinematics, as shown in
Fig. 6. However, this combinatoric background can
be subtracted by determining the number of back-
ground events in each bin of the Dalitz plot. The
contribution of background in each bin is calculated
from the generated Monte-Carlo samples before im-
plementing the resolution to the particles, which in-
troduces a systematics from the migration of back-
ground events to the nearby bins in the Dalitz plot.
The general parameterization of Eq. (2) is fitted
to the background subtracted Dalitz plot and the
parameters are given in Table 3. The systematics
from the migration of background events is small
in the missing measurement method, and the ex-
tracted parameters after background subtraction
show that method (b) is a better choice than the
exclusive measurement [method (a)] when charge
particle resolution is 1% and photon resolution is
6%.

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Papers in Physics, vol. 9, art. 090009 (2017) / S. Ghosh et al.

Table 2: Comparison of the Dalitz plot parameters extracted from method (b) with different background
contributions.

input pars method (b) (No bkg) method (b) (5% bkg) method (b) (10% bkg) method (b) (15% bkg)

a (−0.047) -0.044 ± 0.008 −0.074 ± 0.008 −0.087 ± 0.008 −0.125 ± 0.008
b (−0.069) -0.063 ± 0.015 −0.078 ± 0.015 −0.130 ± 0.015 −0.119 ± 0.015
c ( 0.019) 0.014 ± 0.005 0.016 ± 0.005 0.020 ± 0.005 0.015 ± 0.005
d (−0.073) -0.081 ± 0.009 −0.070 ± 0.009 −0.064 ± 0.009 −0.062 ± 0.009

χ2

ndf
1.14 0.86 0.94 1.16

Table 3: Comparison of the Dalitz plot parameters extracted from method (a) and the background
subtracted Dalitz plot obtained from method (b).

input pars method (a) method (b) (5% bkg) method (b) (10% bkg) method (b) (15% bkg)

a (−0.047) −0.062 ± 0.008 −0.045 ± 0.008 −0.045 ± 0.008 −0.042 ± 0.008
b (−0.069) −0.142 ± 0.015 −0.066 ± 0.016 −0.086 ± 0.016 −0.084 ± 0.017
c (+0.019) 0.014 ± 0.005 +0.018 ± 0.006 +0.022 ± 0.006 +0.019 ± 0.006
d (−0.073) −0.096 ± 0.009 −0.073 ± 0.010 −0.060 ± 0.010 −0.076 ± 0.010

χ2

ndf
1.04 0.93 1.15 1.02

Figure 6: Comparison of the Dalitz plot parameters
from generated and missing mass method without
any background and with the background added
in a proportion of 5%, 10% and 15% of the total
number of events.

IV. Conclusions

In this article, we have described two methods for
measuring the Dalitz variables X and Y in the con-
text of the decay η′ → η (2γ) π+ π−. The exclu-
sive measurement method and the invariant mass

method, though commonly used for measuring the
Dalitz variables, are shown here to be unsuitable in
the case of poor photon resolution. In contrast, the
missing (2γ) analysis method is found to be more
suitable when the photon resolution is poor. The
resolution in Y versus charge particle resolution, in
the missing analysis method, suggests that, even in
the case of equally poor charged particle resolution
and photon resolution the missing analysis method
is a better choice compared to the exclusive mea-
surement method. The missing analysis method,
however, comes with the disadvantage of the com-
binatoric background and the Dalitz plot parame-
ters deviate systematically from their central values
depending on the fraction of background present.
Once the background subtraction is implemented,
the missing analysis method performs even better
and becomes more attractive. In conclusion, this
work provides guidance to select a suitable method
for the extraction of the Dalitz plot parameters de-
pending on the detector resolution as well as the
background present in the signal region and may
be extended to the three body decay channels of
other mesons when the detector resolution is poor.

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