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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
CERN-PH-EP-2010-036
23 September 2010
CERN-PH-EP-2010-036
23 September 2010
Precise tests of low energy QCD from Ke4decay properties
The NA48/2 Collaboration1
Abstract
We report results from the analysis of the K±→ π+π−e±ν (Ke4) decay by the NA48/2
collaboration at the CERN SPS, based on the total statistics of 1.13 million decays collected
in 2003 − 2004. The hadronic form factors in the S- and P-wave and their variation with
energy are obtained. The phase difference between the S- and P-wave states of the ππ
system is accurately measured and allows a precise determination of a0
I=2 S-wave ππ scattering lengths: a0
−0.0432 ± 0.0086stat± 0.0034syst± 0.0028th . Combination of this result with the other
NA48/2 measurement obtained in the study of K±→ π0π0π±decays brings an improved
determination of a0
stringent test of Chiral Perturbation Theory predictions and lattice QCD calculations. Using
constraints based on analyticity and chiral symmetry, even more precise values are obtained:
a0
0= −0.0444±0.0007stat±0.0005syst±0.0008ChPT.
Submitted for publication in European Physical Journal C
0and a2
0, the I=0 and
0= 0.2220 ± 0.0128stat± 0.0050syst± 0.0037th,a2
0=
0and the first precise experimental measurement of a2
0, providing a
0= 0.2196±0.0028stat±0.0020systand a2
1See next pages for the list of authors
Copyright CERN for the benefit of the Collaboration
contact: brigitte.bloch-devaux@cern.ch
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The NA48/2 Collaboration
J.R. Batley, G. Kalmus, C. Lazzeroni1, D.J. Munday, M.W. Slater1, S.A. Wotton
Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, UK2
R. Arcidiacono3, G. Bocquet, N. Cabibbo4, A. Ceccucci, D. Cundy5, V. Falaleev,
M. Fidecaro, L. Gatignon, A. Gonidec, W. Kubischta, A. Norton6, A. Maier,
M. Patel7, A. Peters
CERN, CH-1211 Gen` eve 23, Switzerland
S. Balev8, P.L. Frabetti, E. Goudzovski1, P. Hristov8, V. Kekelidze, V. Kozhuharov9,
L. Litov, D. Madigozhin, E. Marinova10, N. Molokanova, I. Polenkevich,
Yu. Potrebenikov, S. Stoynev11, A. Zinchenko
Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
E. Monnier12, E. Swallow, R. Winston
The Enrico Fermi Institute, The University of Chicago, Chicago, IL 60126, USA
P. Rubin13, A. Walker
Department of Physics and Astronomy, University of Edinburgh, JCMB King’s Buildings, Mayfield
Road, Edinburgh, EH9 3JZ, UK
W. Baldini, A. Cotta Ramusino, P. Dalpiaz, C. Damiani, M. Fiorini8, A. Gianoli, M. Martini,
F. Petrucci, M. Savri´ e, M. Scarpa, H. Wahl
Dipartimento di Fisica dell’Universit` a e Sezione dell’INFN di Ferrara, I-44100 Ferrara, Italy
A. Bizzeti14, M. Lenti, M. Veltri15
Sezione dell’INFN di Firenze, I-50125 Firenze, Italy
M. Calvetti, E. Celeghini, E. Iacopini, G. Ruggiero16
Dipartimento di Fisica dell’Universit` a e Sezione dell’INFN di Firenze, I-50125 Firenze, Italy
M. Behler, K. Eppard, K. Kleinknecht, P. Marouelli, L. Masetti, U. Moosbrugger, C. Morales
Morales, B. Renk, M. Wache, R. Wanke, A. Winhart
Institut f¨ ur Physik, Universit¨ at Mainz, D-55099 Mainz, Germany17
D. Coward18, A. Dabrowski8, T. Fonseca Martin19, M. Shieh, M. Szleper,
M. Velasco, M.D. Wood20
Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
P. Cenci, M. Pepe, M.C. Petrucci
Sezione dell’INFN di Perugia, I-06100 Perugia, Italy
G. Anzivino, E. Imbergamo, A. Nappi, M. Piccini, M. Raggi21, M. Valdata-Nappi
Dipartimento di Fisica dell’Universit` a e Sezione dell’INFN di Perugia, I-06100 Perugia, Italy
C. Cerri, R. Fantechi
Sezione dell’INFN di Pisa, I-56100 Pisa, Italy
G. Collazuol, L. DiLella, G. Lamanna8, I. Mannelli, A. Michetti
Scuola Normale Superiore e Sezione dell’INFN di Pisa, I-56100 Pisa, Italy
F. Costantini, N. Doble, L. Fiorini22, S. Giudici, G. Pierazzini, M. Sozzi, S. Venditti
Dipartimento di Fisica dell’Universit` a e Sezione dell’INFN di Pisa, I-56100 Pisa, Italy
B. Bloch-Devaux23, C. Cheshkov24, J.B. Ch` eze, M. De Beer, J. Derr´ e, G. Marel,
E. Mazzucato, B. Peyaud, B. Vallage
DSM/IRFU – CEA Saclay, F-91191 Gif-sur-Yvette, France
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M. Holder, M. Ziolkowski
Fachbereich Physik, Universit¨ at Siegen, D-57068 Siegen, Germany25
S. Bifani26, C. Biino, N. Cartiglia, M. Clemencic8, S. Goy Lopez27, F. Marchetto
Dipartimento di Fisica Sperimentale dell’Universit` a e Sezione dell’INFN di Torino,
I-10125 Torino, Italy
H. Dibon, M. Jeitler, M. Markytan, I. Mikulec, G. Neuhofer, L. Widhalm
¨Osterreichische Akademie der Wissenschaften, Institut f¨ ur Hochenergiephysik,
A-10560 Wien, Austria28
Submitted for publication in European Physical Journal C
1University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
2Funded by the UK Particle Physics and Astronomy Research Council
3Dipartimento di Fisica Sperimentale dell’Universit` a e Sezione dell’INFN di Torino, I-10125 Torino, Italy
4Universit` a di Roma “La Sapienza” e Sezione dell’INFN di Roma, I-00185 Roma, Italy
5Istituto di Cosmogeofisica del CNR di Torino, I-10133 Torino, Italy
6Dipartimento di Fisica dell’Universit` a e Sezione dell’INFN di Ferrara, I-44100 Ferrara, Italy
7Department of Physics, Imperial College, London, SW7 2BW, UK
8CERN, CH-1211 Gen` eve 23, Switzerland
9Faculty of Physics, University of Sofia “St. Kl. Ohridski”, 1164 Sofia, Bulgaria
10Sezione dell’INFN di Perugia, I-06100 Perugia, Italy
11Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
12Centre de Physique des Particules de Marseille, IN2P3-CNRS, Universit´ e de la M´ editerran´ ee, F-13288
Marseille, France
13Department of Physics and Astronomy, George Mason University, Fairfax, VA 22030, USA
14Dipartimento di Fisica, Universit` a di Modena e Reggio Emilia, I-41100 Modena, Italy
15Istituto di Fisica, Universit` a di Urbino, I-61029 Urbino, Italy
16Scuola Normale Superiore, I-56100 Pisa, Italy
17Funded by the German Federal Minister for Education and research under contract 05HK1UM1/1
18SLAC, Stanford University, Menlo Park, CA 94025, USA
19Laboratory for High Energy Physics, CH-3012 Bern, Switzerland
20UCLA, Los Angeles, CA 90024, USA
21Laboratori Nazionali di Frascati, via E. Fermi, 40, I-00044 Frascati (Rome), Italy
22Institut de F´ ısica d’Altes Energies, UAB, E-08193 Bellaterra (Barcelona), Spain
23Dipartimento di Fisica Sperimentale dell’Universit` a di Torino, I-10125 Torino, Italy
24Institut de Physique Nucl´ eaire de Lyon, IN2P3-CNRS, Universit´ e Lyon I, F-69622 Villeurbanne, France
25Funded by the German Federal Minister for Research and Technology (BMBF) under contract 056SI74
26University College Dublin School of Physics, Belfield, Dublin 4, Ireland
27Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain
28Funded by the Austrian Ministry for Traffic and Research under the contract GZ 616.360/2-IV GZ 616.363/2-
VIII, and by the Fonds f¨ ur Wissenschaft und Forschung FWF Nr. P08929-PHY
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1 Introduction
At high energy, strong interactions between elementary particles are described by Quantum
Chromo Dynamics (QCD) whose Lagrangian can be expanded in power series of the strong
coupling constant. At low energy (below ∼ 1 GeV) the strong coupling becomes large and the
perturbative description is no longer possible. Another approach, Chiral Perturbation Theory
(ChPT), has been considered: it introduces an effective Lagrangian [1] where the elementary
constituents are light pseudo-scalar mesons instead of quarks. The physical observables are
then described by an expansion in terms of external momenta and light quark masses. At
the cost of a number of free parameters (determined from experimental measurements), ChPT
can quantitatively describe meson structure and form factors but can also compute hadronic
contributions to some low energy observables like the g − 2 of the muon which is very precisely
measured [2]. Testing the predictions of ChPT and its underlying assumptions is then of prime
interest.
ChPT has been particularly powerful in describing ππ scattering at low energy and over the
past 40 years, calculations at Leading Order (LO) and at the two subsequent Orders (NLO,
NNLO) have converged towards very precise values of the underlying constants of the theory,
the S-wave ππ scattering lengths in the isospin 0 and 2 states, denoted a0
Experimental determinations of the scattering lengths have been pursued over more than
four decades, but more recently, precise measurements have been obtained in several channels:
- The study of Ke4decays is of particular interest as it gives access to the final state interaction
of two pions in absence of any other hadron. The asymmetry of the dilepton system with respect
to the dipion system is related to the difference between the S- and P- wave ππ scattering phases
for isospin states 0 and 1 (δ0
1). Under the assumption of isospin symmetry, values of the
scattering lengths a0
0have been reported by NA48/2 [3] at the CERN SPS, based on a
partial sample of 670 000 K±decays collected in 2003, E865 [4] at the BNL AGS, based on 400
000 K+decays and S118 [5] (often referred to as Geneva-Saclay Collaboration) at the CERN
PS, based on 30 000 K+decays. The results from the analysis of the full available statistics of
NA48/2 (1.13 million decays) will be given here and discussed in detail.
- The study of K±→ π0π0π±decays (K3π) has shown evidence for a cusp-like structure in
the Mπ0π0 distribution, explained by re-scattering effects in the ππ system below and above the
2mπ± threshold. This has been published by NA48/2 for partial (2.287 · 107decays) [6] and
total (6.031·107decays) [7] statistics. The combination of the independent NA48/2 final results
from the two channels, Ke4and K3πcusp, will be reported here and compared to the currently
most precise theoretical predictions.
- Another challenging approach is the formation of ππ atoms as studied by the DIRAC
collaboration [8] at the CERN PS from 6 500 observed π+π−pairs. The lifetime measurement
of such pionium atoms is directly related to the underlying charge-exchange scattering process
π+π−→ π0π0. The result of a larger sample analysis is also expected.
Isospin symmetry breaking effects have been fully considered in the last two processes which
would not occur otherwise. With the achieved experimental precision from Ke4decays, mass
effects (mπ+ ?= mπ0,mu?= md), neglected in previous studies, should be included when relating
phase measurements to scattering length values. The impact of these effects on the low energy
QCD stringent tests performed will also be discussed.
0and a2
0, respectively.
0− δ1
0and a2
2 Beam and detector
A sketch of the beam geometry and detector layout is shown in Figure 1. The two simultaneous
K+and K−beams are produced by 400 GeV primary protons from the CERN SPS impinging on
a 40 cm long beryllium target. Opposite charge particles, with a central momentum of 60 GeV/c
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and a momentum band of ±3.8% (rms), are selected by two systems of dipole magnets (each
forming an ”achromat”), focusing quadrupoles, muon sweepers and collimators. At the entrance
of the decay volume, a 114 m long evacuated vacuum tank, the beams contain ∼ 2.3 × 106K+
and ∼ 1.3 × 106K−per pulse of about 4.5 s duration with a flux ratio K+/K−close to 1.8.
The two beams are focused ∼ 200 m downstream of the production target in front of the first
spectrometer chamber [9]. The NA48 detector and its performances are described in full detail
elsewhere [10]. The components used in the Ke4analysis are listed here:
- Charged particle momenta from K±decays are measured in a magnetic spectrometer
consisting of four drift chambers (DCH1 through DCH4) and a large aperture dipole magnet
located between the second and third chamber. Each chamber consists of four staggered double
planes of sense wires along the horizontal, vertical and ±45◦directions. The spectrometer is
located in a tank filled with 95% purity helium at atmospheric pressure and separated from
the decay volume by a thin (0.0031 radiation length thick) Kevlarr ?window to reduce multiple
scattering. The spectrometer magnet gives a transverse momentum kick of 120 MeV/c to
charged particles in the horizontal plane. The momentum resolution of the spectrometer is
σ(p)/p = (1.02 ⊕ 0.044 p)% (p in GeV/c).
- A hodoscope (HOD) consisting of two planes of scintillators segmented into horizontal and
vertical strips is used to trigger the detector readout on charged track topologies. The hodoscope
surface is logically subdivided into 16 non-overlapping square regions. Its time resolution is ∼
150 ps.
- A liquid-krypton calorimeter (LKr) measures the energy of electrons and photons. The
transverse segmentation into 13248 2 cm × 2 cm projective cells and the 27 radiation length
thickness result in an energy resolution σ(E)/E = (3.2/√E ⊕ 9.0/E ⊕ 0.42)% (E in GeV) and
a space resolution for transverse position of isolated showers σx= σy= (0.42/√E ⊕ 0.06) cm.
This allows to separate electrons (E/p ∼ 1) from pions (E/p < 1).
- The muon veto counters (MUV) consist of one horizontal and one vertical plane of plastic
scintillator slabs read out by photo-multipliers and preceded each by 0.8 m thick iron absorbers.
The MUV itself is also preceded by the hadron calorimeter (HAC, not used in this analysis)
with a total thickness of 1.2 m of iron.
- A beam spectrometer (KABES), based on Micromegas amplification in a TPC [11], allows
to measure the incident kaon momentum with a relative precision better than 1%.
- A two-level trigger logic selects and flags events. At the first level (L1), charged track
topologies are selected by requiring coincidences of hits in the two HOD planes in at least two of
the 16 square regions. At the second level (L2), a farm of asynchronous microprocessors performs
a fast reconstruction of tracks and runs a decision-taking algorithm. Three complementary
configurations are used: a) 2VTX, selecting events with at least three tracks forming consistent
two-track vertices with the beam line; b) 1VTX, selecting events with at least two tracks forming
a vertex consistent with a beam particle decay; and c) 1TRKP, which selects tracks originating
from the beam line and kinematically inconsistent with K±→ π±π0decay. This trigger logic
ensures a very high trigger efficiency for such topologies.
3 Event selection
Events from the whole data sample recorded in 2003 and 2004 were selected using criteria similar
to those applied to the 2003 sample [3]. These criteria are recalled here for completeness, and
complemented by the additional requirements applied in the final analysis.
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KABES 3
Kevlar window
1000
FRONT−END ACHROMAT
1 cm
0.36
mrad
50 200
10 cm
He tank +
Spectrometer
HOD
LKr
HAC
MUV
DCH 1
FDFD
Protecting
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focused beams
Decay volume
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DCH 4
KABES 2
KABES 1
Z
TAX 17, 18
Figure 1: Sketch of the NA48/2 beam line, decay volume and detectors. Note the different
vertical scales in the left and right parts of the figure
3.1Signal topology selection
The signal topology is characterized by three charged tracks consistent with a common decay
vertex, with two opposite charge pions and one electron or positron. In addition, missing energy
and transverse momentum should be allowed for the undetected neutrino.
In the 2004 run, the LKr calorimeter information was only recorded for a fraction of the
2VTX and 1VTX topologies, while events flagged as 1TRKP were fully recorded. An additional
downscaling was applied off-line to the latter trigger configuration to ensure similar trigger
conditions throughout the two years.
The whole data sample was then selected for three well reconstructed charged tracks. The
timings of the three tracks, as measured from the DCH information, must agree within 6 ns,
while the timings of the corresponding hodoscope signals must agree within 2 ns. The three-
track reconstructed vertex position had to lie within a 5 cm radius transverse to the beam
line and within 2 to 95 meters downstream of the final collimator. Two opposite sign pions
(E/p < 0.8) and one electron or positron (0.9 < E/p < 1.1) were required.
momentum requirement of 3 GeV/c (5 GeV/c) for the electron (pion) was applied while the
maximum momentum sum was set at 70 GeV/c. The distance between any two tracks at DCH1
was required to be larger than 2 cm and the distance between any track and the beam line
larger than 12 cm. The track impact at the LKr front face was required to fall within the
active fiducial region and away from any dead cell by at least 2 cm to ensure reliable energy
measurement. The track-to-track distance at the LKr front face had to be larger than 20 cm to
prevent shower overlaps. No more than 3 GeV energy deposits in the calorimeter, not associated
to tracks but in-time with the considered track combination, were allowed to eliminate events
possibly biased by emission of hard photon(s). No track-associated signal in the MUV detector
was allowed to reject possible π → µν decays in flight. The reconstructed three-track invariant
mass (assigning the pion mass to each track) and the transverse momentum ptrelative to the
beam axis had to be outside an ellipse centered on the kaon mass and zero pt, with semi-axes
20 MeV/c2and 35 MeV/c, respectively, thus requiring a non-zero ptvalue for the undetected
neutrino and excluding K±→ π+π−π±three-body decays.
The reconstruction of the kaon momentum under the assumption of a four-body decay with
an undetected massless neutrino provides a more precise estimate than the 60 GeV/c average
beam momentum. Imposing energy-momentum conservation in the decay and fixing the kaon
A minimum
6
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mass and the beam direction to their nominal value, a quadratic equation in pK, the kaon
momentum, is obtained. If solutions exist in the range between 50 and 70 GeV/c, the event is
kept and the solution closer to 60 GeV/c is assigned to pK.
3.2Background rejection
There are two main background sources: K±→ π+π−π±decays with subsequent π → eν decay
or a pion mis-identified as an electron; and K±→ π±π0(π0) decays with subsequent Dalitz
decay of a π0(π0
Additional selection criteria are applied against background events: the elliptic cut in the
plane (M3π,pt) rejects K±→ π+π−π±decays with one pion mis-identified as an electron but
no missing mass in the K3π hypothesis and low pt value. By varying the ellipse semi-axes
one can change the amount of accepted contamination. Requiring the square invariant mass
M2
decays. An invariant mass of the e+e−system (assigning an electron mass to the opposite
charge pion) larger than 0.03 GeV/c2ensures rejection of converted photons and of some
multi-π0events. Additional rejection against pions mis-identified as electrons is achieved by
using a dedicated linear discriminant variable (LDA) based on shower properties (E/p, radial
shower width and energy weighted track-cluster distance). The training of this variable has been
performed on pion tracks from well reconstructed K3π events having E/p > 0.9, and electron
tracks from Ke3(K±→ π0e±ν) decays selected on the basis of kinematics only (missing mass
of the (K±− π0− e±) system compatible with the neutrino mass). It provides a high, almost
momentum independent, efficiency for electron tracks and additional rejection of pion tracks.
The precise rejection level can be adjusted according to the discriminant variable value. To
ensure a low level of contamination, the kaon momentum, reconstructed under the four-body
assumption, was required to be within the range of 54 to 66 GeV/c. This momentum cut
removes ∼ 40% of the remaining background along with a ∼ 2% loss of signal events (illustrated
in section 5).
The background contamination to signal “right sign” (RS) events (π+π−e±ν) is estimated
from the observed “wrong sign” (WS) events (π±π±e±ν), which, assuming the validity of the
∆S = ∆Q rule, can only be background. Such events are selected with the same criteria as the
signal events apart from the requirement of two opposite sign pions which is changed to two same
sign pions. The background contribution to RS signal events has the same magnitude as that
measured from WS events if originating from K±→ π±π0(π0) decays but has to be multiplied
by a factor of 2 if originating from K3π decays because of the two equal charge pions. This
factor has been cross-checked using Monte Carlo simulated events from the various background
topologies.
A total of 1 130 703 Ke4 candidates (726 367 K+and 404 336 K−) were selected from a
sample of ∼ 2.5 1010triggers recorded in 2003-2004. The subtracted background was estimated
to 2×3 386 (2×2 109 for K+and 2×1 277 for K−) events according to twice the observed numbers
of WS events. The ∼ 0.6% relative background level was found to be constant throughout the
two-year data taking.
D→ e+e−γ) with an electron mis-identified as a pion and photon(s) undetected.
Xin the decay K±→ π±X to be larger than 0.04 (GeV/c2)2further rejects K±→ π±π0
4 Theoretical formulation
4.1Kinematics
The decay K±→ π+π−e±ν is conveniently described using three different rest frames: the K±
rest frame, the dipion rest frame and the dilepton rest frame. The kinematics is then fully
described by the five Cabibbo-Maksymowicz variables [12] as shown in the sketch of Figure 2:
7
Page 8
- Sπ= Mππ2, the square of the dipion invariant mass,
- Se= Meν2, the square of the dilepton invariant mass,
- θπ, the angle of the π±in the dipion rest frame with respect to the flight direction of the
dipion in the K±rest frame,
- θe, the angle of the e±in the dilepton rest frame with respect to the flight direction of the
dilepton in the K±rest frame,
- φ, the angle between the dipion and dilepton rest frames.
dipion dilepton
θθ
Figure 2: Topology of the charged K+
e4decay showing the angle definitions.
4.2Decay probability
We recall the expression of the decay amplitude which is the product of the weak current of the
leptonic part and the (V - A) current of the hadronic part:
Gw
√2V∗
us¯ uνγλ(1 − γ5)ve?π+π−|Vλ− Aλ|K+?, where
?π+π−|Aλ|K+? =
−i
mK(F(pπ+ + pπ−)λ+ G(pπ+ − pπ−)λ+ R(pe+ pν)λ) and
?π+π−|Vλ|K+? =−H
m3
K
?λµρσ(pπ+ + pπ− + pe+ pν)µ(pπ+ + pπ−)ρ(pπ+ − pπ−)σ
In the above expressions, p is the four-momentum of each particle, F,G,R are three axial-
vector and H one vector complex form factors with the convention ?0123= 1.
The decay probability summed over lepton spins can be written as:
d5Γ =
G2
2(4π)6m5
F|Vus|2
K
ρ(Sπ,Se) I(Sπ,Se,cosθπ,cosθe,φ) dSπdSedcosθπdcosθedφ,
where
ρ(Sπ,Se) is the phase space factor Xσπ(1 − ze), with
X =1
2λ1/2(m2
λ(a,b,c) = a2+ b2+ c2− 2(ab + ac + bc).
The function I, using four combinations of F,G,R,H complex hadronic form factors
(Fi,i = 1,4), reads [13, 14, 15]:
K,Sπ,Se), σπ= (1 − 4m2
π/Sπ)1/2, ze=m2
e
Se, and
I = 2(1 − ze)(I1+ I2cos2θe+ I3sin2θe· cos2φ + I4sin2θe· cosφ + I5sinθe· cosφ
+ I6cosθe+ I7sinθe· sinφ + I8sin2θe· sinφ + I9sin2θe· sin2φ)
8
Page 9
where
I1
=1
= −1
= −1
=1
= − (Re(F∗
= −
= − (Im(F∗
=1
= −1
4
?
4(1 − ze)
4(1 − ze)?|F2|2− |F3|2?
1F3) + zeRe(F∗
?Re(F∗
2(1 − ze) Im(F∗
2(1 − ze) Im(F∗
(1 + ze)|F1|2+1
2(3 + ze)(|F2|2+ |F3|2) sin2θπ+ 2ze|F4|2?
|F1|2−1
sin2θπ,
2(1 − ze)Re(F∗
4F2)) sinθπ,
2F3) sin2θπ− zeRe(F∗
1F2) + zeIm(F∗
1F3) sinθπ,
2F3) sin2θπ.
,
I2
I3
I4
I5
I6
I7
I8
I9
?
2(|F2|2+ |F3|2) sin2θπ
?
,
1F2) sinθπ,
1F4)?,
4F3)) sinθπ,
In Ke4decays, the electron mass can be neglected (ze= 0) and the terms (1 ± ze) become
unity. One should also note that the form factor F4is always multiplied by zeand thus does
not contribute to the full expression.
With this simplification, the complex hadronic form factors Fireduce to:
F1= m2
K(γ F + α G cosθπ),F2= m2
K(β G),F3= m2
K(βγ H),
where one uses the three dimensionless complex form factors F,G (axial), H (vector), and three
dimensionless combinations of the Sπand Seinvariants:
α = σπ(m2
K,β = σπ(SπSe)1/2/m2
K− Sπ− Se)/2m2
?α2− β2.
K,γ = X/m2
K,
related by σπγ = 2
If T-invariance holds, the Watson theorem [16] tells us that a partial-wave amplitude of
definite angular momentum l and isospin I must have the phase of the corresponding ππ
amplitude δI
l.
Developing further F1,F2,F3in a partial wave expansion with respect to the variable cosθπ
using Legendre functions Pl(cosθπ) and their derivative P?
l(cosθπ),
∞
?
F1/m2
K
=
∞
?
l=0
Pl(cosθπ) F1,leiδl,F2(3)/m2
K
=
l=1
P?
l(cosθπ) F2(3),leiδl,
one can now express the form factors F,G,H using explicitly the modulus and phase of each
complex contribution. A D-wave contribution would appear as a cos2θπterm for F and cosθπ
terms for G,H with its own phase.
F=Fseiδfs+ Fpeiδfpcosθπ+ Fdeiδfdcos2θπ
Gpeiδgp+ Gdeiδgdcosθπ
Hpeiδhp+ Hdeiδhdcosθπ
G=
H= (1)
Limiting the expansion to S- and P-waves and considering a unique phase δpfor all P-wave form
factors in absence of CP violating weak phases, the function I is then expressed as the sum of
12 terms, each of them being the product of two factors, Ai, which depends only on the form
factor magnitudes and one single phase δ (= δs−δp), and Biwhich is function of the kinematical
variables only (see Table 1):
I =
12
?
i=1
Ai(Fs,Fp,Gp,Hp,δ) × Bi(Sπ,Se,cosθπ,cosθe,φ).(2)
Going from K+to K−under CPT conservation, θeshould be replaced by π − θe, φ should
be replaced by π + φ and Hpby −Hp[17]. Under the assumption of CP conservation, this is
equivalent to obtaining the φ distribution of K−decays from the φ distribution of K+decays
with the same Hpvalue by changing φ to −φ. This property can be verified in the expressions
given in Table 1.
9
Page 10
Table 1: Contributions to the Ke4decay probability from S- and P-wave terms in absence of CP
violating weak phases.
Ai
F2
F2
G2
Bi
1
2
3
s
γ2sin2θe
γ2cos2θπsin2θe
α2cos2θπsin2θe
p
p
+2αβsinθπcosθπsinθecosθecosφ
+β2sin2θπ(1 − sin2θecos2φ)
β2γ2sin2θπ(1 − sin2θesin2φ)
2γ2cosθπsin2θe
2γsinθe(βsinθπcosθecosφ + αcosθπsinθe)
2βγsinθπsinθesinφ
−2βγ2sinθπsinθecosφ
−2βγ2sinθπsinθecosθesinφ
−2βγ2sinθπcosθπsinθecosφ
−2βγsinθπ(βsinθπcosθe+ αcosθπsinθecosφ)
2γcosθπsinθe(βsinθπcosθecosφ + αcosθπsinθe)
4
5
6
7
8
9
10
11
12
H2
p
FsFpcosδ
FsGpcosδ
FsGpsinδ
FsHpcosδ
FsHpsinδ
FpHp
GpHp
FpGp
5Monte Carlo simulation
Signal events were generated in the kaon rest frame according to the decay matrix element as
given in section 4.2 and with values of form factors as measured in [4, 5], and then boosted to
the laboratory frame. The incident kaon trajectory and momentum were generated taking into
account the time variations of the beam properties for each kaon charge, and the decay vertex
position according to the exponential decay law. As a precise description of the acceptance and
resolution in the five-dimensional space of the kinematic variables is necessary, a detailed GEANT3-
based [18] Monte Carlo (MC) simulation was used, including full detector geometry and material
implementation, DCH alignment and local inefficiencies. A large time-weighted MC production
was achieved, providing an event sample about 25 times larger than the data and reproducing
the observed ratio (K+/K−) = 1.8. The same reconstruction and selection codes as for data were
used, except for the timing cuts. The LDA cut was applied to the simulated electron candidates
as a momentum-dependent efficiency. This represents the optimal implementation of the cut
effect as it avoids reliance on the details of the shower developments, including fluctuations and
limited statistics of the simulation. Two independent codes were used for the decay matrix
element according to the Pais-Treiman formulation, one with a smooth phase shift variation
[19] and constant form factors, the other with a more elaborated phase shift variation following
ChPT prediction [20, 21, 22] and form factors depending on invariant masses (as published
in [4]). They were used in independent analyses of a subset of the data.
The quality of the simulation can be seen from the plots of Figure 3 where distributions
of simulated variables in the laboratory frame are compared to data distributions. Not only
acceptance but also resolutions are well described in the simulation. Residual discrepancies
will be studied in section 7.Acceptances in the five-dimensional space are shown as two-
and one-dimensional projections in Figure 4 emphasizing their correlations. The experimental
resolutions, projected on each of the five variables, vary smoothly across each spectrum. They
are respectively (the mean value corresponds to a mixture of K±in the same ratio as in the
data):
10
Page 11
rms
σ(Mππ)
σ(Meν)
σ(cosθπ)
σ(cosθe)
σ(φ)
mean value
1.5 MeV/c2
9.6 MeV/c2
0.052
0.052
307 mrad
variation across spectrum
increasing from 0.5 to 2.5 MeV/c2,
decreasing from 13 to 6 MeV/c2,
decreasing from 0.058 (cosθπ= 0) to 0.040 (cosθπ= ±1),
increasing from 0.025 (cosθe= −1) to 0.070 (cosθe= 1),
decreasing from 370 mrad (φ = 0) to 240 mrad (φ = ±π).
5000
10000
15000
20000
25000
30000
35000
0.8
1
1.2
-2000020004000 6000
Zv(cm)
8000
5000
10000
15000
20000
25000
30000
35000
40000
0.8
1
1.2
50525456586062 64666870
10000
20000
30000
40000
50000
60000
0.8
1
1.2
10 20 30405060 70
10000
20000
30000
40000
50000
60000
70000
80000
90000
0.8
1
1.2
0510152025303540
(a)
WS x 10
(b)
WS x 10
Rmin(cm)
(c)
WS x 10
pK(GeV/c)
(d)
WS x 10
pe(GeV/c)
Figure 3: Distributions of (a) the reconstructed vertex longitudinal position; (b) the minimum
track radius at DCH1; (c) the reconstructed K±momentum; (d) the reconstructed electron
momentum. Data (background subtracted) are shown as full circles with error bars, simulations
as histograms and background (wrong sign events increased by a factor of 10 to be visible) as
shaded areas. The inserts show the ratio of data to simulated distributions. The arrows on plot
(c) show the reconstructed kaon momentum range selected in the final analysis. Errors shown are
statistical only; residual discrepancies will be discussed in section 7 (Systematic uncertainties).
11
Page 12
-1
-0.5
0
0.5
1
00.05 0.10.15 0.20.25
-1
-0.5
0
0.5
1
0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46
0
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
0
5
10
15
20
25
0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25
0
0.05
0.1
0.15
0.2
0.25
0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46
0
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
0
5
10
15
20
25
0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46
0
5
10
15
20
25
-1-0.8 -0.6 -0.4 -0.20 0.20.40.60.81
0
5
10
15
20
25
-1 -0.8 -0.6 -0.4 -0.20 0.20.40.60.81
0
5
10
15
20
25
-3-2 -10123
%
%
%
%
%
Mππ(GeV/c2)Meν(GeV/c2)
cosθe
cosθe
%
%Meν(GeV/c2)
Meν(GeV/c2)Mππ(GeV/c2)
Mππ(GeV/c2)cosθπ
cosθe
φ
Figure 4: Distribution of the acceptance (in %) as obtained from the simulation in the five-
dimensional space and projected onto the (Meν,cosθe), (Mππ,cosθe) and (Mππ,Meν) planes and
along the five kinematical variables. The smooth acceptance variation along single kinematical
variables results from contributions of very different acceptance regions in the five-dimensional
space.
12
Page 13
Radiative corrections were implemented in the simulation in two successive steps:
- first, virtual photon exchange between charged particles is described by the classical
Coulomb attraction/repulsion between two opposite/same charge particles (i,j = π+,π−,e±)
and applied as a weight to the Ke4decay probability according to the Gamow function:
C(Sij) =
?
i?=j
ωij
eωij− 1
with ωij= 2παQiQj/βij, where α is the fine structure constant, QiQj= −1 for opposite charge
particles (+1 for same charge particles) and βijis the relative velocity (in unit of c) expressed
?
(Sij− m2
effect comes from the attraction between the two pions at low relative velocity, which distorts
the Mππspectrum near threshold. The electron (positron) being always relativistic, its relative
velocity is very close to 1 and the corresponding weight is a constant.
- second, real photons are generated by the program PHOTOS version 2.15 [23] interfaced
to the simulation. Only 10% of the events have photons adding up to more than 1 GeV in
the laboratory frame. Furthermore the event selection reduces the acceptance for events with
energetic photons. For these events, the resulting effect is a bias of the measured Meν and θe
variables as hard photon emission mostly affects the electron kinematics.
as βij=
?
?
?1 −
4m2
im2
i− m2
j
j)2using the invariant mass Sij of the (ij) system. The largest
6 Analysis method
As an extension of the method proposed originally in [13], based on partially integrated
distributions of the Cabibbo-Maksymovicz variables in an ideal detector, we have chosen to
work in the five-dimensional space to take into account the precise knowledge of the experimental
acceptance and resolution. The high statistics now available allows a definition of a grid of equal
population boxes adapted to both detector acceptance and resolution, and to the form factor
variations to be studied. There are of course many possible choices and we stick to the grid
used in [3] for simplicity: the data sample is first distributed over ten Mππslices to follow the
variation of physical parameters along this variable; each sub-sample is then distributed over
five Meνequi-populated slices, then over five cosθπslices, five cosθeslices and twelve φ slices to
ensure that correlations in the plane (cosθe,φ) are precisely described. This procedure results in
a total of 15 000 five-dimensional boxes (Nbox) of unequal sizes and achieves equal populations
of 48 data events per box in the K+sample and 27 in the K−sample, which are analyzed
separately since the simultaneous K±beam geometries are not identical. A dedicated estimator
(suited to account for Poisson fluctuations of the small number of events per box and limited
simulation statistics) is used in the minimization procedure (as in previous analyses [3, 4, 5]).
It is defined as:
T2=2
Nbox
?
j=1
njln
?
nj
rj(1 −
1
mj+ 1)
?
+ (nj+ mj+ 1)ln
?
1 + rj/mj
1 + nj/(mj+ 1)
?
,
where nj is the number of data events in box j, mj is the number of observed simulated
events in the same box and rj is the number of expected simulated events (rj
mj· NData/Nfit
(section 4.2) and computed for each event using the generated values of the kinematic variables
and the current values of the fitted parameters (F,G,H,δ)fit, while Nfit
Nbox
?
=
MC· I(F,G,H,δ)fit/I(F,G,H,δ)gen). The expression I is defined by Eq. 2
MCis the corresponding
total number of simulated events
j=1
mj· I(F,G,H,δ)fit/I(F,G,H,δ)gen. This takes into
13
Page 14
account resolutions in the five-dimensional space and is independent of the particular set of form
factors (F,G,H,δ)genused at generation step provided that the simulated sample populated all
regions of the five-dimensional space accessible to the data.
We note that the more “classical” Log-likelihood L and least squares χ2estimators
L =
Nbox
?
Nbox
?
j=1
2njln(nj/rj) + 2(rj− nj) for large values of mj,
χ2=
j=1
(rj− nj)2
nj
mj
mj+ nj
for large values of mjand nj,
are almost equivalent to T2within the available statistics.
In this analysis, the branching fraction is not measured, so only relative form factors are
accessible: Fp/Fs, Gp/Fs, Hp/Fsand the phase shift δ. Neglecting a possible Meνdependence
and without prior assumption on the shape of their variation with Mππ, the form factors and
phase shift are measured in independent Mππbins. Fits are performed in the four-dimensional
space, separately for the K+and K−samples but using the same Mππ bin definitions. The
results are found consistent for both charge signs and then combined in each bin according
to their statistical weight.Identical results are obtained by fitting simultaneously the two
independent samples to a single set of form factors and phase in the same Mππ bins. The
relative normalizations (NData/Nfit
equal to unity at the ππ threshold. Last, values of Fp/Fs, Gp/Fs, Hp/Fsare deconvoluted of
the observed Fsvariation in each bin and plotted against q2= (Sπ/4m2
possible further dependence. Potential variations with Meν are then explored and quantified
when found significant.
In a second stage of the analysis, the observed variations of the form factors and phase shift
with Mππand Meνare used to determine other parameter values through specific models. Series
expansions of the variables q2= (Sπ/4m2
factor variations (section 8.1). More elaborated models related to the physical parameters (a0
and a2
0) will be used when studying the phase variation (section 8.2).
MC) are proportional to F2
sand are rescaled to have a value
π) − 1 to investigate a
π) − 1 and Se/4m2
πwill be used to quantify the form
0
7Systematic uncertainties
Two independent analyses were performed on a large fraction of the 2003 data sample. They were
based on different event selection and reconstruction, different detector corrections and different
binning and fitting procedures. Consistent results were obtained, ensuring the robustness of the
analysis. The final analysis was performed on the full statistics recorded over two years and
follows one of the two validated analyses.
The studies reported in [3] have been repeated and extended to the whole data sample.
Several systematic errors were limited by the available statistics and are now reduced. With
respect to the analysis described in [3], the additional cut on the reconstructed kaon momentum
ensures a lower relative background contamination (WS/RS = 0.0030 instead of 0.0046) and
helps decreasing the impact of background related systematics. For each investigated item,
the analysis was repeated varying one condition at a time and a systematic uncertainty was
quoted for each fitted parameter in each Mππbin. A particular attention was given to possible
bin-to-bin correlations, which are indeed observed in some cases.
- Fitting procedure: the number of boxes used in the fitting procedure was varied within a
factor of 2, keeping, however, the same definition for the 10 Mππ bins. This last number was
also extended to 12 and 15 bins. The grid definition was also varied as well as the estimator
minimized in the fit. No visible bias was observed.
14
Page 15
- Trigger efficiency: two independent methods to measure the high (∼ 99.3%) trigger
efficiency were used. The first one considers Ke4selected candidates satisfying the Level 1 trigger
condition (downscaled by 100 and thus based on small statistics) and measures the efficiency
from events which satisfy the Level 2 trigger. The second approach focuses on K±→ π±π0
events satisfying the Level 1 trigger condition, kinematic cuts and loose particle identification.
Assigning a pion mass to both π±and opposite charge electron tracks allows coverage of the full
Mππrange with sufficient statistics. Both methods have been used to apply the trigger efficiency
to the simulation in the five-dimensional space. As the efficiency is practically uniform and very
stable over the two years, the overall effect is almost negligible.
- MUV efficiency: imperfect modeling of the MUV response to pion punch-through has
been studied with pion tracks from fully reconstructed decays (K3π,Kππ0
function of the pion momentum. An additional inefficiency per pion track of 0.5 to 1.5% has
been introduced in the simulation resulting in an average inefficiency of 1.7% varying between
1.3 and 2.3% over the five-dimensional space. The observed change in the fit parameters has
been quoted as systematic uncertainty.
- Acceptance, resolution and beam geometry: the analysis method does not rely on the
detailed matrix element assumptions, provided that the whole phase space is covered. Particular
care was taken in controlling the geometrical acceptance and in following the time-dependence
of the beam geometry. The cut values on the longitudinal vertex position were varied in steps
of few meters. The cut value on the minimum track-beam axis distance at DCH1 was varied in
steps of one cm. Both variables are sensitive to the acceptance, trigger composition and beam
geometry. The maximum effect observed for each variable was quoted as systematic uncertainty.
A reweighting of the kaon simulated spectrum was considered in order to reproduce the data
distribution (Figure 3c) and the difference observed in the result was quoted as systematic error.
It accounts for residual imperfections in the beam geometry and detector resolution modeling.
These three effects of similar size have been added in quadrature under the same label in Table 2.
- Background contamination: the analysis was repeated subtracting the WS events according
to their five-dimensional distributions, and scaled by a factor one, two or three. The dependence
of each fitted parameter with the WS events scale factor was measured in each Mππbin. The
scale factor for the background subtraction was cross-checked using a detailed simulation of
contributing processes and found to be 2.0±0.3. The effect of the 0.3 uncertainty is propagated
to each point according to the measured slopes and quoted as systematic uncertainty (labeled
background level in Table 2). The effect is bin-to-bin correlated, as expected.
The background measured from wrong sign (WS) events is observed at low Sπ values as
expected from K3π decays where Sπ cannot exceed (MK− Mπ)2, and shows a component
clustering at Se= m2
πfrom π → eν decays (Figure 5). Varying the semi-axes of the elliptic cut
in the plane (M3π,pt) accepts different fractions and shapes of the K3π background. Results
were found to be stable with respect to this cut without bin-to-bin correlation. Residual effects
were quoted as systematic uncertainty (labeled background shape in Table 2).
- Electron identification: the final rejection against pions mis-identified as electrons (E/p >
0.9) is achieved by a cut on an LDA variable. In the simulation, the cut effect is applied as a
momentum dependent efficiency. The cut value was varied from 0.85 to 0.90 (nominal cut) and
0.95. The analysis was repeated in the three conditions and the residual variation quoted as a
systematic uncertainty.
- Radiative corrections: no systematic uncertainty was assigned to the Coulomb correction
as its formulation is well established. The PHOTOS photon emission was switched off in the
simulation to evaluate its effect on the fitted parameters. One tenth of the full effect was quoted
as theoretical uncertainty on the radiative corrections. This is based on detailed comparisons
between the PHOTOS and KLOR codes available for the KL→ π±e∓ν mode [24], and on more
recent evaluations for the Ke4mode [25]. As expected, the effect comes mostly from removing
D
D) and quantified as a
15
Page 16
events with hard photon emission.
- Dependence on Se: in the first stage of the analysis, the form factors were assumed to
be independent of Meν. The effect of this assumption was explored by analyzing again the
data with a simulation reweighted for a linear dependence of Fs on Se/4m2
0.068, as measured. The observed deviation between the two analyses was quoted as systematic
uncertainty.
Many checks were performed to test the stability of the results, splitting the data in
statistically independent sub-samples according to the kaon charge, achromat polarity, dipole
magnet polarity, decay vertex longitudinal position, transverse impact position of the electron
on the calorimeter front face and data taking time. Results were compared in each bin and
found to be consistent within the statistical errors.
In addition, a different reconstruction of the Cabibbo-Maksymowicz variables, based on the
information of the KABES detector to measure precisely the kaon momentum and incident
direction, improves the resolutions by ∼ 50% for the cosθπ,cosθe,φ variables. However, as this
information was only available for 65.6% of the event sample, and also affected by different
systematic uncertainties (such as a mis-tagging rate of few percent), this alternative analysis
was only used as a cross-check of the standard procedure. The results were found to be in
good agreement and the statistical errors on the fitted parameters were reduced by 5 to 10%
with respect to the standard analysis of the same subsample, yet this was not enough of an
improvement to compensate for the 20% increase from the reduced statistics.
πwith a slope of
8 Results and interpretation
The detailed numerical results obtained in the ten independent slices of Mππ are given in
the Appendix (Table 7 to Table 10).As explained in the previous section, the systematic
uncertainties do have a bin-to-bin correlated component, albeit much smaller than the
uncorrelated one. In the tables, only the diagonal term of the matrix is quoted. The agreement
between data and simulation distributions can be seen in Figure 5 where K+and K−data are
added and compared to the sum of the simulated distributions using the common set of fitted
parameters.
8.1 S- and P-wave form factors
Under the assumption of isospin symmetry, the form factors can be developed in a series
expansion of the dimensionless invariants q2= (Sπ/4m2
Two slope and one curvature terms are sufficient to describe the Fsform factor variation
within the available statistics (the overall scale factor fsis to be determined from the branching
fraction, not reported here):
π) − 1 and Se/4m2
π[26].
Fs
= fs(1 + f?
s/fsq2+ f??
s/fsq4+ f?
e/fsSe/4m2
π), (3)
while two terms (offset and slope) are enough to describe the Gpform factor:
Gp/fs
= gp/fs+ g?
p/fsq2,(4)
and two constants to describe the Fpand Hpform factors. The χ2of the fit to Fsis 111.5 for
81 degrees of freedom and blows up to 230.1 for 82 degrees of freedom if the Sedependence is
set to zero. The numerical results for all terms are given in Table 3 and displayed in Figure 6.
It has been checked that potential D-wave contributions (Eq. 1) are indeed consistent with zero
and do not affect the S- and P-wave measured values.
16
Page 17
Table 2: Systematic uncertainties (in units of 10−4) affecting each of the dimensionless fitted
parameters. The background level and Se dependence contributions are 100% bin-to-bin
correlated. Form factor description follows Eq. 3,4 in section 8.1. Scattering lengths (expressed
in units of 1/mπ+) are given for Models B and C according to Eq. 6,7 in section 8.2.
f?
s/fs
f??
f?
Source
fit procedure43 16
trigger efficiency25 231
MUV efficiency215
acceptance/resolution597
background shape16154
electron identification5 108
radiative corrections 11 1427
background level20 1651
Sedependence 2652 42
total systematic 466474
statistical error 707060
s/fs
e/fs
fp/fs
gp/fs
g?p/fs
hp/fs
4
4
13
9
18
13
12
22
44
43
69
72
45
129
170
11
1
< 131
515
39
28
35
65
44
101
100
23
27
54
21
12
7
9
22
24
37
30
8
26
75
150
a0
0
a2
0
a0
0
two-parameter fit
Model B
one-parameter fit
Model C
Source
fit procedure
trigger efficiency
MUV efficiency
acceptance/resolution
background shape
electron identification
radiative corrections
background level
Sedependence
total systematic
statistical error
9
6
6
4
7
4
2
10
20
31
14
19
19
13
50
128
< 1
13
18
10
12
10
7
10
7
5
7
57
34
86
18
49
Table 3: Results of the form factor measurements. When relevant, the correlations between
fitted parameters are given.
value
0.152
−0.073
0.068
−0.048
0.868
0.089
−0.398
errorscorrelations
f??
s/fs
s/fs−0.954
f??
s/fs
f?
f??
s/fs=
f?
e/fs=
fp/fs=
gp/fs=
g?p/fs=
hp/fs=
s/fs=± 0.007stat
± 0.007stat
± 0.006stat
± 0.003stat
± 0.010stat
± 0.017stat
± 0.015stat
± 0.005syst
± 0.006syst
± 0.007syst
± 0.004syst
± 0.010syst
± 0.013syst
± 0.008syst
f?
0.080
0.019
e/fs
f?
gp/fs
g?p/fs−0.914
17
Page 18
Sπ(GeV/c2)2
Se(GeV/c2)2
cosθπ
cosθe
φ(K+)(rad)φ(K−)(rad)
WS x 10WS x 10
WS x 10WS x 10
WS x 10WS x 10
10000
20000
30000
40000
50000
60000
70000
0.8
1
1.2
0.08 0.10.12 0.140.16 0.180.2 0.22
2000
1.2
4000
6000
8000
10000
x 10
0.8
1
00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
5000
1.2
10000
15000
20000
25000
30000
35000
0.8
1
-1-0.8 -0.6 -0.4 -0.200.20.40.60.81
5000
1.2
10000
15000
20000
25000
30000
35000
40000
0.8
1
-1-0.8 -0.6 -0.4 -0.200.20.40.60.81
2000
1.2
4000
6000
8000
10000
12000
14000
16000
18000
0.8
1
-3-2 -10123
2000
1.2
4000
6000
8000
10000
0.8
1
-3 -2-10123
Figure 5:
dimensional space. The full circles are the K±summed data after background subtraction,
the histograms are the simulation with the best fit parameters, and the shaded areas correspond
to the WS events multiplied by a factor of 10 to be visible. The inserts show the Data/Simulation
ratios. The φ distributions are shown separately for K+and K−. The errors shown are statistical
only.
Distribution of the Cabibbo-Maksymovicz variables projected from the five-
18
Page 19
F2
s/f2
s
0.95
1
1.05
1.1
1.15
1.2
00.10.20.30.40.5 0.60.70.80.9
q2
1
Fp/fs
-0.15
-0.1
-0.05
0
0.05
00.10.20.30.40.50.60.70.80.9
q2
1
F2
s/f2
s
0.95
1
1.05
1.1
1.15
00.050.1 0.150.20.250.30.350.40.45
Se/4m2
π
Gp/fs
0.7
0.8
0.9
1
1.1
00.10.20.30.40.50.60.70.80.9
q2
1
F2
s(Data)/F2
s(MC)
0.9
0.95
1
1.05
1.1
010203040506070 8090100
bin number in (q2,Se) plane
Hp/fs
-0.9
-0.6
-0.3
0
0.3
00.10.20.30.4 0.50.60.7 0.8 0.9
q2
1
Figure 6: Variation of the fitted form factors with q2and Se. Left column: F2
the q2axis assuming no Sedependence and residual variation of the projection on the Seaxis
when the q2dependence is accounted for. The bottom plot shows the ratio (Data/MC) after fit in
each bin of the plane (q2,Se) displayed on a linear scale where Sebins run in each q2bin. Right
column: Fp/fs (χ2/ndf = 16.6/9),Gp/fs (χ2/ndf = 17.5/8) and Hp/fs (χ2/ndf = 18.2/9)
versus q2. The errors displayed in these figures are statistical only.
sprojected on to
19
Page 20
Low energy constants (LEC Li) which are parameters of ChPT can also be extracted from
combined fits of meson masses, decay constants and form factor measurements. This has been
done in refs [15, 27, 28, 29] including successively S118, E865 and NA48/2 results [3] but such a
study is beyond the scope of this article. It is not yet clear if the most recent NNLO calculations
support the energy dependence of the now precisely measured form factors. Isospin breaking
effects may have to be taken into account as suggested in some preliminary work [30]. However,
we can compare NA48/2 results with previous experimental results in terms of slopes and
relative form factors using the absolute fsvalue of each experiment as a normalization factor
and propagating errors as uncorrelated in absence of any published correlation information.
The available measurements are summarized in Table 4. While S118 results were limited by
statistics and E865 errors were dominated by systematics, the NA48/2 values are now precise
in both respects. The three sets of results are compatible within the experimental errors.
Table 4: Results of the form factor measurements for all Ke4experiments. The errors quoted
in parentheses are statistical, then systematic (when available). Fits with one slope only and
one slope and curvature for Fshave also been performed for direct comparison with previous
experimental results. Due to lack of sensitivity, the values labeled by (∗) have been given by
[5, 4] only as a test. The value labeled by (†) is not exactly the same quantity as that measured
by S118 and NA48/2.
S118 [5]
5.59(14)
E865 [4]
5.75(2)(8)
NA48/2
fs
one term (q2)
f?
s/fs
two terms (q2,q4)
f?
s/fs
f??
s/fs
three terms (q2,q4,Se/4m2
f?
s/fs
f??
s/fs
f?
e/fs
fp/fs
gp/fs
g?p/fs
hp/fs
n.a.
0.080(20)0.079(15)0.073(2)(2)
−
−
0.184(17)(70)
−0.104(21)(70)
0.147(7)(5)
−0.076(7)(6)
π)
−
−
−
n.a.
n.a.
0.152(7)(5)
−0.073(7)(6)
0.068(6)(7)
−0.048(3)(4)
0.868(10)(10)
0.089(17)(13)
−0.398(15)(8)
−0.056(18)(42)∗
−0.059(17)(47)†
0.809(9)(12)
0.120(19)(7)
−0.513(33)(35)
0.009(32)∗
0.855(41)
0.070(20)
−0.480(122)
8.2Phase shift and scattering lengths in the ππ system
Theoretical framework
To extract the ππ scattering lengths from the measurements of the phase shift δ = δs−δp, more
theoretical ingredients are needed. To perform a fair comparison of experimental results, we
must take into account the evolution of the theoretical predictions over the last 30-40 years.
The Roy equations [31] were at the origin of many theoretical developments. These equations
are based on the fundamental principles of analyticity, unitarity and crossing symmetries and
allow a prediction of the ππ phase values close to threshold using experimental measurements
above the matching point (√s = 0.8 GeV), and the subtraction constants a0
0 and 2 S-wave scattering lengths (in units of 1/mπ+). One should note, however, that the two-
pion system of the Ke4decay is never in the I=2 state, but the combination (2a0
as a subtraction constant and brings some sensitivity to a2
0and a2
0, the isospin
0− 5a2
0) enters
0when solving the Roy equations.
20
Page 21
Conversely, from measurements of the phases and using the Roy equations, one can determine
the corresponding values of the subtraction constants.
We will consider three successive implementations of the solutions:
- Model A: in the mid 70’s, several authors had given solutions of these equations [19, 32] and
proposed a parametrization δ0
is explicitly given in [5] for δ = δ0
1as
0= f(a0
0− δ1
0,q2). We will consider the parametrization of [19] which
sin2δ =2 σπ(a0
0+ b q2),(5)
where
scattering length. In the plane (a0
b = 0.19−(a0
within an uncertainty of ±0.04 which reflects the input data precision. There is no dependence
on a2
0in this formulation.
- Model B: numerical solutions of the Roy equations were published 25 years later by two
groups [20, 33] with a parametrization of the phases δI
b = b0− a1
1is the difference between the S-wave slope with q2and the P-wave
0,b) the two parameters are related by an empirical formula:
0−0.15)2, where the slope b is a quadratic function of the S-wave scattering length
lwith energy (s = Sπ):
tanδI
l(s) =σπq2l{AI
l+ BI
lq2+ CI
lq4+ DI
lq6}
?
4m2
s − sI
π− sI
l
l
?
, (6)
where the Schenk coefficients XI
expansion of the variables (a0
the same boundary conditions at the matching point√s = 0.8 GeV: δ0
δ1
The authors of ref. [33] have in addition parameterized the coefficients as a linear expansion
around the values of the phases at the matching point.
In the plane (a0
0), the values are constrained to lie within a band (called “Universal Band”,
UB) fixed by the input data above 0.8 GeV and the Roy equations, defined by the equation of
the centre line:
a2
l(X = A,B,C,D,s) are written as a third degree polynomial
0− 0.225) and (a2
0+ 0.03706). Both predictions agree when using
0= 82.3◦(±3.4◦) and
1= 108.9◦(±2.0◦).
0,a2
0= −0.0849 + 0.232 a0
0− 0.0865 (a0
0)2and by a width ±0.0088.
- Model C: in the framework of ChPT, an additional constraint has been established [21, 22, 34]
which can be used together with the Roy equations solutions discussed above to give more precise
predictions lying within a ChPT band defined by the equation of the centre line:
a2
0=−0.0444 + 0.236 (a0
0− 0.220) − 0.61 (a0
0− 0.220)2− 9.9 (a0
0− 0.220)3
(7)
and by a reduced width ±0.0008.
Including more phenomenological ingredients like the scalar radius of the pion, very precise
predictions at NNLO have been made by the same authors:
a0
0(ChPT) = 0.220 ± 0.005th,a2
0(ChPT) = −0.0444 ± 0.0010th.(8)
Because of the different formulations, a given set of phase measurements will translate to
different values of the scattering lengths.
More recently, triggered by the early NA48/2 precise results [3], new theoretical work [35] has
shown that isospin symmetry breaking may also alter the phases measured in Ke4decay when
all mass effects (mπ+ ?= mπ0,mu?= md), neglected so far in previous analyses, are considered.
The measured phase of the I=0 S-wave is no longer δ0
0but ψ0
0:
ψ0
0=
1
32πF2
π
?
(4∆π+ s) σ±+ (s − m2
π0)
?
1 +
3
2R
?
σ0
?
+ O(p4),(9)
21
Page 22
where Fπ is the pion decay constant, s = Sπ, ∆π = m2
π± − m2
π0, R =
ms− ˆ m
md− mu
and
σx=
?
1 −4m2
Even if the difference between the mass-symmetric δ0
modest in terms of absolute magnitude (10 to 15 mrad) over the whole energy range accessible in
Ke4decays, the coherent shift toward higher values of the phases has non negligible implications
when extracting scattering lengths from such measurements as shown in Figure 7. These effects
were of course present but neglected in the results of the S118 [5] and E865 [4] experiments.
πx
s
, with x = [±,0] .
0(∆π= 0, 1/R = 0, σ±= σ0) and ψ0
0is
Other models, based on analyticity and unitarity but not using Roy equations have also been
developed [36, 37]. They exploit the Ke4phase measurements associated or not with other ππ
scattering results to extract a value for a0
0through a conformal transformation and an effective
range function developed in a series of the variable w(s) =
√s −
√s +
?
?
4m2
K− s
K− s
4m2
with coefficients
(B0,B1,..). Results from such fits, using only the NA48/2 phase shift measurements will be
reported as well.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.280.30.320.340.360.380.4
NA48/2 Ke4 (2003-2004)
no isospin corrections
with isospin corrections
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.2 0.210.220.230.240.25 0.260.27
NA48/2 Ke4 (2003-2004)
UB
ChPT
no isospin corrections
with isospin corrections
Figure 7: Left: Phase shift (δ) measurements without (open circles) and with (full circles)
isospin mass effects correction from NA48/2 Ke4data. The lines correspond to the two-parameter
fit within model B. Errors are statistical only. Right: Fits of the NA48/2 Ke4data in the (a0
plane without (black) and with (red) isospin mass effects. Errors are statistical only. Ellipses
are 68% CL contours (model B) and circles are the result of the one-parameter fit imposing the
ChPT constraint (Model C). The small (green) ellipse corresponds to the best prediction from
ChPT.
0,a2
0)
NA48/2 Results
We first focus on the most elaborated models B and C. The NA48/2 phase measurements are
used as input to a two-parameter fit (Eq. 6, Model B) leading to:
a0
a2
0=
0=
0.2220
−0.0432
± 0.0128stat
± 0.0086stat
± 0.0050syst
± 0.0034syst
± 0.0037th,
± 0.0028th.
22
Page 23
with a 97% correlation coefficient and a χ2of 8.84 for 8 degrees of freedom for statistical
errors only. The theoretical errors in the two-parameter fit have been estimated following the
prescription described in [35] and are dominated by the experimental precision of the inputs
to the Roy equation (for a2
0) and the neglected higher order terms when introducing the mass
effects (for a0
0). The breakdown of the theoretical errors is given in Table 5. Using the additional
ChPT constraint (Eq. 7, Model C) , the one-parameter fit gives a χ2/ndf of 8.85/9 for statistical
errors only and the best fit value:
a0
0= 0.2206 ± 0.0049stat± 0.0018syst± 0.0064th
corresponding to (a2
0= −0.0442) from the ChPT constraint (Eq. 7). This result can be compared
to the most precise prediction of ChPT (Eq. 8).
Table 5: Contributions to the theoretical uncertainty on the scattering length values obtained
in the two-parameter fit (Eq. 6) and the constrained fit (Eq. 6 and 7). The Bern solutions
correspond to [20], the Orsay solutions to [33].
two-parameter fit
a0
0
one-parameter fit
a0
a2
00
Roy equation solutions
|(Bern) − (Orsay)|
δ0
δ1
Isospin corrections
R = 37 ± 5.
Fπ= (86.2 ± 0.5) MeV
Higher Orders
quadratic sum without CHPT
ChPT constraint ±0.0008
quadratic sum
0.0000
0.0010
0.0000
0.0006
0.0027
0.0002
0.0013
0.0043
0.0003
0± 3.4◦at matching point
1± 2.0◦at matching point
0.0005
0.0003
0.0035
0.0037
−
−
0.0000
0.0001
0.0005
0.0028
−
−
0.0008
0.0003
0.0042
0.0062
0.0017
0.0064
The result of a two-parameter fit based on analyticity only [36] from isospin corrected phase
measurements leads to:
B0 = 10.229 ± 2.433,B1 = −8.768 ± 5.560,
0= 0.2255+0.0125
χ2/ndf = 8.87/8
with a 99.7% correlation, corresponding to a0
predictions for a0
−0.0140stat, in agreement also with ChPT
0, while no value can be given for a2
0.
Discussion
We have repeated the S118 analysis within Model A (Eq. 5) using the published phase
measurements and found results consistent with the published scattering length value [5]. Then
we have extended the analysis to Models B and C (Eq. 6 and 7) with and without the latest
isospin mass effect corrections. The same exercise has been also performed using the E865
published phase values, after taking into account the recently published errata [38] which solved
most of the inconsistencies between the E865 global fit (which cannot be repeated by an external
analysis) and the model independent fit (which can be repeated) results . Table 6 summarizes all
fit results for existing Ke4data, as originally published, and also refitted under various conditions
using the same model formulations.
A comparison of the two-parameter fit results of the three experiments under various model
assumptions is shown in Figure 8. The effect of the isospin corrections is marginal for S118 due
to limited statistics but brings a significant shift for the E865 and NA48/2 results.
23
Page 24
Table 6: Fits of experimental phase values within three models based on the Roy equations. In
case of Model C, the value of a2
0is fixed by the constraint and given within parentheses.
Model A (Eq. 5)Model B (Eq. 6)Model C (Eq. 6,7)
S118 [5] published Model A: a0
no isospin correction
a0
0= 0.310 ± 0.109
b = 0.110 ± 0.190
including isospin correction (Eq. 9)
a0
0= 0.282 ± 0.110
b = 0.122 ± 0.192
0= 0.31 ± 0.11, b = 0.11 ± 0.16
a0
a2
0= 0.309 ± 0.125
0= 0.013 ± 0.105
a0
0= 0.245 ± 0.037
(a2
0= −0.0390)
a0
a2
0= 0.280 ± 0.124
0= 0.003 ± 0.104
a0
0= 0.224 ± 0.040
(a2
0= −0.0435)
E865 [4, 38] published global fit results for two models:
Model B: a0
0= −0.055 ± 0.023
Model C: a0
no isospin correction
a0
0= 0.213 ± 0.035
b = 0.269 ± 0.059
including isospin correction (Eq. 9)
a0
0= 0.184 ± 0.036
b = 0.284 ± 0.060
0= 0.203 ± 0.033, a2
0= 0.216 ± 0.013, (a2
0= −0.0454 ± 0.0031)
a0
0= −0.063 ± 0.023
0= 0.206 ± 0.033a0
0= 0.235 ± 0.013
(a2
0= −0.0409)a2
a0
0= −0.072 ± 0.023
0= 0.179 ± 0.033a0
0= 0.213 ± 0.013
(a2
0= −0.0461)a2
NA48/2 final result
no isospin correction
a0
0= 0.263 ± 0.012
b = 0.191 ± 0.018
including isospin correction (Eq. 9)
a0
0= 0.236 ± 0.013
b = 0.202 ± 0.018
a0
0= −0.036 ± 0.009
0= 0.247 ± 0.013a0
0= 0.242 ± 0.005
(a2
0= −0.0395)
a2
a0
0= −0.043 ± 0.009
0= 0.222 ± 0.013a0
0= 0.221 ± 0.005
(a2
0= −0.0442)a2
Figure 9-left shows all experimental phase measurements from Ke4data after correction for
the isospin mass effects. Another interesting feature of Model C is the possibility to measure
a0
band [34]:
0from each phase value by solving a polynomial expansion in q2along the ChPT constraint
δ =
q
?1 + q2
0.2527 + 0.151(a0
?
a0
0+ b q2+ c q4+ d q6?
± e,with
b =
0− 0.22) + 1.14(a0
0− 0.22)2+ 35.5(a0
0− 0.22)3,
c =0.0063 − 0.145(a0
0− 0.22), d = −0.0096 and e = 0.0035 q3+ 0.0015 q5.
Using such a method to extract a single value of a0
correlations are negligible. This assumption is only approximately true, in particular because the
systematic uncertainty from background subtraction and the isospin corrections are examples
of point-to-point correlated effects. However, it is a meaningful check of the consistency of the
experimental measurements within the model. Figure 9-right shows the values of a0
the 21 individual measurements of δ from the three experiments. The theoretical uncertainty e
0implies that possible point-to-point
0obtained for
24
Page 25
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.10.15 0.20.250.3 0.350.40.450.5
a0
Model A
68% CL contours
S118
E865
NA48/2NA48/2
S118
E865
0
b
a0
0
a2
0
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0.140.160.18 0.20.220.240.26
68% CL contours
UB
ChPT
E865E865
NA48/2NA48/2
Figure 8: Left: Results from all Ke4experiments in the (a0
slope b and a0
0in Model A [19] is represented by the large band. Contours for two-parameter
fits results at 68% CL are drawn as dotted before isospin corrections are applied and solid when
applied. Right: Results from Ke4 experiments in the (a0
experiment has little sensitivity to a2
0and is not shown here. The UB (Model B) and ChPT
(Model C) bands show the region allowed by the Roy equation solutions and the additional
ChPT constraint, respectively.
0,b) plane. The relation between the
0,a2
0) plane for Model B. The S118
on the relation inverted in the fit has been added in quadrature to the statistical error to obtain
the error on each point.
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.280.3 0.320.340.360.380.4
Ke4 Data with isospin corrections
S118
E865
NA48/2 (fit)
a0
0
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.280.3 0.320.340.360.380.4
Ke4 Data with isospin corrections
S118
E865
NA48/2
Figure 9: Left: Phase shift (δ) measurements corrected for isospin mass effects for all Ke4
available results. The line corresponds to the two-parameter fit of the NA48/2 data alone.
Right: values obtained for each individual measurement from the inverted ChPT constraint.
The band corresponds to the global fit over the NA48/2 data, including point to point
correlations and is in agreement with the individual values. It means that the model gives
a good description of the data points over the whole range.
25
Page 26
Going further in the framework of ChPT, the low energy constant (LEC)¯l3can be extracted
from the scattering length values determined from the NA48/2 phase measurements through
the relations [34]:
a0
a2
0=
0=
0.225
−0.0434
−1.6 10−3 ¯l3− 1.3 10−5 ¯l3
−3.6 10−4 ¯l3− 4.3 10−6 ¯l3
2
2
From the above equations, it should be noted that a2
0. From the most precise value obtained within Model C, a0
one can deduce a range for¯l3between −0.55 and 5.75, in other words:¯l3= 2.6 ± 3.2 which is
in very good agreement with the preferred value of ChPT [39]¯l3= 2.9±2.4 and those obtained
by lattice calculations [40] clustering around¯l3= 3 ± 0.5. The NA48/2 result excludes large
negative values of¯l3 allowed in Generalized ChPT [33]. In this framework, it means that¯l3
brings only a few percent correction to the leading order term in the pion mass expression,
product of the quark masses (mu+ md) and the quark condensate |?0|¯ uu|0?| in the chiral limit,
normalized to the pion decay constant Fπ[34]:
¯l3
32π2F2M4+ O(M6) with M2= (mu+ md) |?0|¯ uu|0?| /F2
0is five times less sensitive to¯l3than
0= 0.2206±0.0049stat±0.0018syst,
a0
M2
π= M2−
π.
8.3Combination with other NA48/2 results
The analysis of the decay K±→ π0π0π±by NA48/2 has enlightened another effect of the
scattering lengths through the cusp observed in the Mπ0π0 distribution as a consequence of
re-scattering effects in the ππ system below and above the 2mπ± threshold [6, 7].
The cusp and Ke4 results are obviously statistically independent. They have systematic
uncertainties of different origins (control of calorimetry and neutral trigger in one case,
background and particle identification in the other) and show different correlations between
the fitted scattering lengths:
K3πcuspKe4
a0
0=
0=
a2
0=
0.2220 ± 0.0128stat± 0.0050syst
a0
0− a2
0.2571 ± 0.0048stat± 0.0029syst
−0.0241 ± 0.0129stat± 0.0096syst
−0.0432 ± 0.0086stat± 0.0034syst
correlation−0.839 (stat. only),−0.774 (all)0.967 (stat. only),0.969 (all)
The systematic errors quoted for the cusp results includes internal and external uncertainties, but
no uncertainty associated to theory. In the Ke4result, the theoretical uncertainty contribution
(Table 5) is even smaller than the experimental systematic error and thus has very little impact
on the overall precision.
Neglecting potential (but small) common systematic contribution to the experimental errors,
it is possible to combine the two measurements and to get a more precise result (χ2/ndf =
1.84/2):
a0
0= 0.2210 ± 0.0047stat± 0.0040syst, a2
a0
0= −0.0429 ± 0.0044stat± 0.0028syst
0= 0.2639 ± 0.0020stat± 0.0015syst.
0− a2
The two input sets of values and their combination are displayed in Figure 10. This last
result, which does not require any additional theoretical ingredient, is in very good agreement
with the most precise ChPT predictions (Eq. 8) given with a similar precision and recalled here:
a0
0(ChPT) = −0.0444 ± 0.0010th,
a0
0= 0.265 ± 0.004th.
An alternative picture of the same results can be seen in Figure 11a for the variables a0
and a2
0measured by NA48/2 and DIRAC experiments through three different processes.
0(ChPT) = 0.220 ± 0.005th, a2
0− a2
0− a2
0
26
Page 27
It is worth comparing the now precise a2
theoretical lattice QCD calculations involving also ChPT formulations:
0experimental measurement with very precise
a2
a2
0=
0=
−0.04330 ± 0.00042stat
−0.04385 ± 0.00028stat± 0.00038syst
When using the ChPT constraint, the combined NA48/2 results become (χ2/ndf = 1.87/1):
by the NPLQCD collaboration [41],
by the ETM collaboration [42].
a0
0= 0.2196 ± 0.0028stat± 0.0020syst, a0
a2
0= −0.0444 ± 0.0007stat± 0.0005syst± 0.0008ChPT
where the last error on a2
0comes from the ChPT constraint uncertainty.
0− a2
0= 0.2640 ± 0.0021stat± 0.0015syst,
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.2 0.210.220.230.240.250.26
(stat. + syst.) errors
68% CL contour
NA48/2 combined Ke4 + Cusp
Ke4
Cusp
ChPT
UB
DIRAC
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.240.250.260.270.280.29 0.3
(stat. + syst.) errors
68% CL contour
NA48/2 combined Ke4 + Cusp
Ke4
Cusp
ChPT
DIRAC
Figure 10: NA48/2 Ke4 (black) and cusp (blue) results from the two-parameter fits in the
(a0
0) (right) planes. In each plane the smallest (red) contour corresponds
to the combination of the NA48/2 results. The correlation coefficient is then 0.21 in the left
plane and 0.92 in the right plane. The dashed lines visualize the ChPT constraint band and the
solid (black) lines the Universal Band. The other (green) lines correspond to the DIRAC result
band [8].
0−a2
0,a2
0) (left) and (a0
0,a2
Such precise values of a0
violating amplitude in the process KL→ ππ through interference between amplitudes in the
isospin states 0 and 2. This phase is given by the value of (δ2
the neutral kaon mass. Propagating the NA48/2 scattering length values and their correlated
experimental errors to the phase values using the numerical solutions of Roy equations [20],
we obtain δ0
0= (47.67 ± 0.06exp) degrees at MK0 where the uncertainty corresponds to
statistical and systematic errors added in quadrature. This result is fully consistent with the
expectations of [22, 20] but with a much reduced experimental uncertainty. The width of the
ChPT constraint translates to an additional uncertainty of ± 0.3 degree to be added linearly to
the experimental one. Using these values, we obtain the phase of ε?, φε? = (42.3 ± 0.4) degrees.
0and a2
0can be used to evaluate the phase of ε?, the direct CP
0− δ0
0+π
2) at the energy of
0− δ2
9Summary
NA48/2 results
From the study of 1.13 millions Ke4 decays with both charge signs and with a low relative
background of ∼ 0.6%, the S- and P-wave form factors and their variation with energy have
been measured (Table 3). Evidence for a ∼ 5% contribution from Fphas been established; a
constant Hpand linear Gpvariation with Sπhave been measured. The Fsform factor variation
27
Page 28
0.20.250.3
NA48/2
Ke4
NA48/2
Cusp
NA48/2
Combined
DIRAC
pionium
-0.1-0.050
NA48/2
Ke4
NA48/2
Cusp
NA48/2
Combined
(a)
0.15 0.20.250.3
S118
Ke4
E865
Ke4
NA48/2
Ke4
All Ke4
combined
-0.1 -0.050
S118
Ke4
E865
Ke4
NA48/2
Ke4
All Ke4
combined
(b)
Figure 11: (a): Two-parameter best fit values for a0
combined result. The DIRAC result is shown as well. (b): Two-parameter best fit values for
a0
0from each Ke4experiment and combined result (dominated by the NA48 precision).
The right part of the large S118 error bar is truncated. Vertical bands correspond to the best
predictions from ChPT: a0
0−a2
0and a2
0from both NA48/2 channels and
0and a2
0= 0.220 ± 0.005, a0
0−a2
0= 0.265 ± 0.004, a2
0= −0.0444 ± 0.0010.
in the plane (Mππ,Meν) can be described by a slope and curvature in Sπand a slope in Se. The
precise measurement of the phase shift of the ππ system has allowed to extract the scattering
lengths a0
0using the Roy equations after correction for isospin breaking mass effects:
0and a2
a0
a2
0=
0=
0.2220
−0.0432
± 0.0128stat
± 0.0086stat
± 0.0050syst
± 0.0034syst
± 0.0037th,
± 0.0028th.
This very sensitive test strongly confirms the predictions of Chiral Perturbation Theory and the
underlying assumption of a large quark condensate contributing to the pion mass. Combining
both NA48/2 Ke4 and cusp results from independent analyses with different sensitivities, an
even more precise set of values is obtained :
a0
a0
0= 0.2639 ± 0.0020stat± 0.0015syst.
which brings the first experimental determination of a2
very precise calculations of lattice QCD.
Using the additional constraint from ChPT (Eq. 7), the results from the Ke4analysis alone
are:
a0
0= 0.2206 ± 0.0049stat± 0.0018syst± 0.0064th,
and combined with the cusp results:
0= 0.2210 ± 0.0047stat± 0.0040syst, a2
0− a2
0= −0.0429 ± 0.0044stat± 0.0028syst
0in perfect agreement with the currently
a0
0= 0.2196 ± 0.0028stat± 0.0020syst, a0
corresponding to a2
0− a2
0= 0.2640 ± 0.0021stat± 0.0015syst,
0= −0.0444 ± 0.0007stat± 0.0005syst± 0.0008ChPT.
These last values can be used to estimate the phase of the direct CP violating amplitude ε?,
giving φε? = (42.3 ± 0.4) degrees at the MK0 energy.
28
Page 29
Combination with other Ke4results
Combining Ke4NA48/2 results with previous experimental results [5, 38] and applying isospin
breaking corrections to all phase shift values, we obtain for the two-parameter fit (Eq. 6, Model
B): a0
0.0028th, where all experimental errors are considered as independent between experiments
and theoretical errors common to all experiments.
(Eq. 7, Model C), we obtain: a0
0= 0.2198 ± 0.0046stat± 0.0016syst± 0.0064thcorresponding to
a2
0= −0.0445±0.0011stat±0.0004syst±0.0008ChPT. The new world average result is dominated
by the NA48/2 experimental precision and illustrated in Figure 11b.
0= 0.2173 ± 0.0118stat± 0.0043syst± 0.0037th, a2
0= −0.0462 ± 0.0079stat± 0.0030syst±
Using the additional ChPT constraint
Acknowledgments
We gratefully acknowledge the CERN SPS accelerator and beam-line staff for the excellent
performance of the beam and the technical staff of the participating institutes for their effort
in the maintenance and operation of the detectors. We enjoyed constructive exchanges with
S. Pislak and P. Tru¨ ol from the E865 collaboration and we also would like to thank all theory
groups who expressed their interest in this work by fruitful discussions which triggered the latest
developments of the analysis, in particular G. Colangelo, J. Gasser and the late J. Stern for their
constant support and contribution.
29
Page 30
Appendix: Fit results for independent Mππbins.
The following tables give the definition of the Mππbins (Table 7) and the fit results for the four
form factors (Table 8, 9) and δ phase shift (Table 10) in each individual bin.
Table 7: Definition of the ten bins in Mππ: bin range, event numbers (K++ K−), barycenter
and χ2of the fits for (2 × 1496) degrees of freedom in each bin.
binMππrange Number of events
number (MeV/c2) (K++ K−)
1 279.00 − 291.29
2291.29 − 300.50
3300.50 − 309.22
4309.22 − 317.73
5 317.73 − 326.35
6 326.35 − 335.33
7335.33 − 345.25
8 345.25 − 357.03
9357.03 − 373.27
10> 373.27 74336 + 41560
Mππbarycenter
(MeV/c2)
286.06
295.95
304.88
313.48
322.02
330.80
340.17
350.94
364.57
389.95
χ2
ndf = 2992
3087.66
2955.93
3092.96
2977.36
2954.31
2962.53
3010.69
3082.64
3113.97
2929.37
71940 + 39572
72197 + 40354
71671 + 40177
71558 + 40164
72725 + 40181
72618 + 40290
72817 + 39995
73273 + 40751
73232 + 41292
Table 8: Result of the fits for F2
An arbitrary scale has been applied to set F2
an indication of the variation as the analysis has been done in the plane (q2,Se). The quoted
systematic errors correspond to the bin-to-bin uncorrelated part.
s/F2
s(0) (neglecting a possible Meνdependence) and Fp/Fs(0).
s/F2
s(0) = 1 at threshold. Values are given only as
bin
F2
s/F2
value
1.0050
1.0379
1.0567
1.0743
1.0875
1.0975
1.1104
1.1191
1.1257
1.1550
s(0)statistical
error
0.0030
0.0032
0.0032
0.0033
0.0033
0.0034
0.0034
0.0034
0.0034
0.0035
systematic
error
0.0031
0.0016
0.0018
0.0018
0.0011
0.0016
0.0013
0.0010
0.0012
0.0060
Fp/Fs(0)
value
−0.0318
−0.0569
−0.0367
−0.0273
−0.0641
−0.0672
−0.0381
−0.0530
−0.0542
−0.0462
statistical
error
0.0117
0.0109
0.0104
0.0101
0.0097
0.0095
0.0096
0.0094
0.0095
0.0103
systematic
error
0.0027
0.0060
0.0057
0.0056
0.0072
0.0066
0.0080
0.0057
0.0055
0.0057
1
2
3
4
5
6
7
8
9
10
30
Page 31
Table 9: Result of the fits for Gp/Fs(0) and Hp/Fs(0). The quoted systematic errors correspond
to the bin-to-bin uncorrelated part only.
bin
Gp/Fs(0)
value
0.8856
0.9091
0.8661
0.8581
0.9193
0.9261
0.8859
0.9227
0.9389
0.9497
statistical
error
0.0453
0.0249
0.0189
0.0160
0.0140
0.0128
0.0120
0.0109
0.0100
0.0096
systematic
error
0.0116
0.0147
0.0104
0.0089
0.0102
0.0099
0.0103
0.0065
0.0052
0.0057
Hp/Fs(0)
value
−0.3147
−0.3659
−0.3498
−0.4820
−0.3585
−0.4741
−0.3861
−0.3253
−0.3673
−0.5022
statistical
error
0.0908
0.0560
0.0468
0.0432
0.0415
0.0410
0.0412
0.0419
0.0441
0.0503
systematic
error
0.0339
0.0135
0.0171
0.0126
0.0171
0.0154
0.0273
0.0193
0.0380
0.0310
1
2
3
4
5
6
7
8
9
10
Table 10: Result of the fits for the phase shift δ. The quoted systematic errors correspond to
the bin-to-bin uncorrelated part only. The isospin correction (later subtracted) is given in the
last column.
binδ valuestatistical
(mrad)error (mrad)
1 52.77525.102
2
91.56414.267
3116.819 11.772
4 139.29510.399
5157.3578.927
6178.2378.354
7 218.9018.329
8 238.0247.714
9284.913 7.425
10325.7787.011
systematic
error (mrad)
isospin corr.
(mrad)
12.542
11.555
11.543
11.760
12.071
12.441
12.866
13.377
14.044
15.322
7.934
7.787
4.743
3.908
2.327
1.723
4.821
2.261
2.168
1.769
31
Page 32
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