Ke4 decays and Wigner cusp
Ke4 decays and Wigner cusp
Contribution to the proceedings of HQL06,
Munich, October 16th-20th 2006
Lucia Masetti1
Institut für Physik
Universität Mainz
D-55099 Mainz, GERMANY
1 Introduction
The single-flavour quark condensate 〈0 |qq| 0〉 is a fundamental parameter of χPT ,
determining the relative size of mass and momentum terms in the expansion. Since
it can not be predicted theoretically, its value must be determined experimentally,
e.g. by measuring the ππ scattering lengths, whose values are predicted very precisely
within the framework of χPT , assuming a big quark condensate [1], or of generalised
χPT , where the quark condensate is a free parameter [2].
The K+−e4 decay is a very clean environment for the measurement of ππ scattering
lengths, since the two pions are the only hadrons and they are produced close to
threshold. The only theoretical uncertainty enters through the constraint [3] between
the scattering lengths a20 and a
0. In the K
± → π0π0π± decay a cusp-like structure
can be observed at M200 = 4m
π+ , due to re-scattering from K
± → π+π−π±. The
scattering lengths can be extracted from a fit of the M200 distribution around the
discontinuity.
2 Experimental setup
Simultaneous K+ and K− beams were produced by 400 GeV energy protons from the
CERN SPS, impinging on a beryllium target. The kaons were deflected in a front-
end achromat in order to select the momentum band of (60± 3) GeV/c and focused
at the beginning of the detector, about 200 m downstream. For the measurements
presented here, the most important detector components are the magnet spectrom-
eter, consisting of two drift chambers before and two after a dipole magnet and the
quasi-homogeneous liquid krypton electromagnetic calorimeter. The momentum of
1Present address: Physikalisches Institut, Universität Bonn, D-53012 Bonn, GERMANY
the charged particles and the energy of the photons are measured with a relative
uncertainty of 1% at 20 GeV. A detailed description of the NA48/2 detector can be
found in Ref. [4].
3 K± → π+π−e±νe
The K+−e4 selection consisted of geometrical criteria, like the requirement of having
three tracks within the detector acceptance and building a good vertex; particle iden-
tification requirements, based mainly on the different fraction of energy deposited
by pions and electrons in the electromagnetic calorimeter; kinematical cuts for back-
ground rejection, like an elliptical cut in the (pT ,M3π) plane centered at (0,MK). In
order to improve the pion rejection, the electron identification also included a Linear
Discriminant Analysis combining the three quantities with the highest discriminating
power. Two reconstruction strategies can be applied to the K+−e4 events: either im-
posing the kaon mass and extracting the kaon momentum from a quadratic equation,
or imposing the kaon momentum to be the mean beam momentum (60 GeV/c along
the beam axis) and extracting the kaon mass from a linear equation (see Fig. 1).
50 52 54 56 58 60 62 64 66 68 70
pK(GeV/c)
)2 (GeV/c
0.3 0.4 0.5 0.6 0.7 0.8
510  datae4K
 MCe4K
Background
Figure 1: Kaon momentum (left) and mass (right) of the K+−e4 events reconstructed
with a quadratic or a linear equation, respectively. The data (crosses) are compared
to signal MC (open histogram) plus background (yellow).
Analysing part of the 2003 data, 3.7× 105 K+−e4 events were selected with a back-
ground contamination below 1%. The background level was estimated from data,
using the so-called “wrong sign” events, i.e. with the signature π±π±e∓νe, that, at
the present statistical level, can only be background, since the corresponding kaon
decay violates the ∆S = ∆Q rule and is therefore strongly suppressed [5]. The main
background contributions are due to K± → π+π−π± events with π → eν or a pion
mis-identified as an electron. The background estimate from data was cross-checked
using Monte Carlo simulation (MC).
3.1 Form factors
Figure 2: Topology of the Ke4 decay.
The form factors of the K+−e4 decay are parametrised as a function of five kinematic
variables [6] (see Fig. 2): the invariant masses Mππ and Meν and the angles θπ, θe
and φ. The matrix element
V ∗usu(pν)γµ(1− γ5)v(pe)(V
µ − Aµ)
contains a hadronic part, that can be described using two axial (F and G) and one
vector (H) form factors [7]. After expanding them into partial waves and into a Taylor
series in q2 = M2ππ/4m
π+ − 1, the following parametrisation was used to determine
the form factors from the experimental data [8, 9]:
F = (fs + f
2 + f ′′s q
4)eiδ
2) + fp cos θπe
iδ11(q
G = (gp + g
2)eiδ
H = hpe
iδ11(q
In a first step, ten independent five-parameter fits were performed for each bin in
Mππ, comparing data and MC in four-dimensional histograms in Meν , cos θπ, cos θe
and φ, with 1500 equal population bins each. The second step consisted in a fit of the
distributions in Mππ (see Figs. 3,4), to extract the (constant) form factor parameters.
The polynomial expansion in q2 was truncated according to the experimental
sensitivity. The dependence on Meν and the D-wave were found to be negligible
within the total uncertainty and the corresponding parameters were therefore set to
zero. The δ = δ00 − δ11 distribution was fitted with a one-parameter function given by
the numerical solution of the Roy equations [3], in order to determine a00, while a
was constrained to lie on the centre of the universal band. The following preliminary
result was obtained:
f ′s/fs = 0.169± 0.009stat ± 0.034syst
0.28 0.3 0.32 0.34 0.36 0.38 0.4
NA48/2 Ke4 PRELIMINARY
 statistical errors only
’ ’ ’
0.28 0.3 0.32 0.34 0.36 0.38 0.4
NA48/2 Ke4 PRELIMINARY
 statistical errors only
-0.15
-0.05
0.28 0.3 0.32 0.34 0.36 0.38 0.4
NA48/2 Ke4 PRELIMINARY
 statistical errors only
0.28 0.3 0.32 0.34 0.36 0.38 0.4
NA48/2 Ke4 PRELIMINARY
 statistical errors only
Figure 3: F , G and H dependence on Mππ. The points represent the results of the
first-step fits, the lines are fitted in the second step.
f ′′s /fs = −0.091± 0.009stat ± 0.031syst
fp/fs = −0.047± 0.006stat ± 0.008syst
gp/fs = 0.891± 0.019stat ± 0.020syst
g′p/fs = 0.111± 0.031stat ± 0.032syst
hp/fs = −0.411± 0.027stat ± 0.038syst
a00 = 0.256± 0.008stat ± 0.007syst ± 0.018theor,
where the systematic uncertainty was determined by comparing two independent
analyses and taking into account the effect of reconstruction method, acceptance, fit
method, uncertainty on background estimate, electron-ID efficiency, radiative correc-
tions and bias due to the neglected Meν dependence. The form factors are measured
relative to fs, which is related to the decay rate. The obtained value for a
0 is compat-
ible with the χPT prediction a00 = 0.220±0.005 [10] and with previous measurements
[11, 12].
4 K± → π0π0e±νe
About 10,000 K00e4 events were selected from the 2003 data and about 30,000 from
the 2004 data with a background contamination of 3% and 2%, respectively. The
0.28 0.3 0.32 0.34 0.36 0.38 0.4
NA48/2 Ke4 PRELIMINARY
 Universal Band fit
 statistical errors only
Figure 4: δ = δ00 − δ11 distribution as a function of Mππ. The points represent the
results of the first-step fits, the line is fitted in the second step.
selection criteria were similar to the ones used for the K+−e4 events, apart from the
requirement of containing one track and 4 photons compatible with two π0s at the
same vertex. The electron identification was based on the fraction of energy deposited
in the electromagnetic calorimeter and on the width of the corresponding shower. The
background level was estimated from data by reversing some of the selection criteria
and was found to be mainly due to K± → π0π0π± events with a pion mis-identified
as an electron (see Fig. 5).
2 in GeV/cKm
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
Signal
 MC00e4K
)γe3(K
Figure 5: Invariant mass distribution in logarithmic scale of the K00e4 events selected
from the 2003 data (crosses) compared to the signal MC (red) plus physical (yellow)
and accidental (blue) background.
The branching fraction was measured, as a preliminary result from the 2003 data
only, normalised to K± → π0π0π±:
BR(K00e4 ) = (2.587± 0.026stat ± 0.019syst ± 0.029ext)× 10
where the systematic uncertainty takes into account the effect of acceptance, trigger
efficiency and energy measurement of the calorimeter, while the external uncertainty
is due to the uncertainty on the K± → π0π0π± branching fraction. This result is
about eight times more precise than the best previous measurement [13].
For the form factors the same formalism is used as in K+−e4 , but, due to the
symmetry of the π0π0 system, the P -wave is missing and only two parameters are
left: f ′s/fs and f
s /fs. Using the full data sample, the following preliminary result
was obtained:
f ′s/fs = 0.129± 0.036stat ± 0.020syst
f ′′s /fs = −0.040± 0.034stat ± 0.020syst,
which is compatible with the K+−e4 result (see Fig. 6).
s/fsf’
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
-0.14
-0.12
-0.08
-0.06
-0.04
-0.02
σ 1e4
 68%e4
σ 1e4
 68%e4
Figure 6: Comparison of the f ′s/fs and f
s /fs measurements in K
e4 and K
5 K± → π0π0π±
From 2003 data, about 23 million K± → π0π0π± events were selected, with negligible
background. The squared invariant mass of the π0π0 system (M200) was computed
imposing the mean vertex of the π0s, in order to improve its resolution close to
threshold. At M200 = 4m
π+ , the distribution shows evidence for a cusp-like structure
(see Fig. 7, left) due to ππ re-scattering.
x 10 2
0.08 0.09 0.1 0.11 0.12 0.13
25000
30000
35000
40000
45000
50000
0.076 0.077 0.078 0.079 0.08
x 10 2
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Figure 7: Left: M200 of the selection K
± → π0π0π± data events. The arrow indicates
the position of the cusp. Right: angle between the π± and the π0 in the π0π0 centre
of mass system. The points represent the data, the three curves, the MC distribution
for different values of k′
Fitting the distribution with the theoretical model presented in Ref. [14] and using
the unperturbed matrix element
M0 = A0(1 +
h′u2 + 1
k′v2),
the following result was obtained [15], assuming k′ = 0 [16]:
g0 = 0.645± 0.004stat ± 0.009syst
h′ = −0.047± 0.012stat ± 0.011syst
a2 = −0.041± 0.022stat ± 0.014syst
a0 − a2 = 0.268± 0.010stat ± 0.004syst ± 0.013theor,
where the a0 − a2 measurement is dominated by the uncertainty on the theoretical
model.
In a further analysis, the value of k′ was obtained from a fit above the cusp in the
plane cos θ vs M200, where θ is the angle between the π
+ and the π0 in the π0π0 centre
of mass system. Evidence was found for a non-zero value of k′ (see Fig. 7, right):
k′ = 0.0097± 0.0003stat ± 0.0008syst,
where the systematic uncertainty takes into account the effect of acceptance and
trigger efficiency. Reweighting the MC with the obtained value of k′, the standard fit
of the M200 distribution with the Cabibbo-Isidori model was performed to obtain the
cusp parameters, that were found to be compatible with the published values.
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	Introduction
	Experimental setup
	K + - e e
	Form factors
	K 0 0 e e
	K 0 0
