# Precise tests of low energy QCD from decay properties

**ABSTRACT** We report results from the analysis of the $\mbox {$\mbox { ($\mbox {$\mbox {) 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 $\mbox {$\mbox { and $\mbox {$\mbox {, the I = 0 and I = 2 S-wave ππ scattering lengths: $\mbox {$\mbox {. Combination of this result with the other NA48/2 measurement obtained in the study of $\mbox {$\mbox { decays brings an improved determination of $\mbox {$\mbox { and the first precise experimental measurement of $\mbox {$\mbox {, providing a 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: $\mbox {$\mbox { and $\mbox {$\mbox {.

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**ABSTRACT:**The first two non-trivial moments of the distribution of the topological charge (or gluonic winding number), i.e., the topological susceptibility and the fourth cumulant, can be computed in lattice QCD simulations and exploited to constrain the pattern of chiral symmetry breaking. We compute these two topological observables at next-to-leading order in three-flavour Chiral Perturbation Theory, and we discuss the role played by the eta propagation in these expressions. For hierarchies of light-quark masses close to the physical situation, we show that the fourth cumulant has a much better sensitivity than the topological susceptibility to the three-flavour quark condensate, and thus constitutes a relevant tool to determine the pattern of chiral symmetry breaking in the limit of three massless flavours. We provide the complete formulae for the two topological observables in the isospin limit, and predict their values in the particular setting of the recent analysis of the RBC/UKQCD collaboration. We show that a combination of the topological susceptibility and the fourth cumulant is able to pin down the three-flavour condensate in a particularly clean way in the case of three degenerate quarks.Journal of High Energy Physics 09/2012; 2012(12). · 5.62 Impact Factor

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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

1

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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

3

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1Introduction

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

2Beam 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

4

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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.

3Event 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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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

4Theoretical 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

-20000 200040006000

Zv(cm)

8000

5000

10000

15000

20000

25000

30000

35000

40000

0.8

1

1.2

5052 5456586062 64 666870

10000

20000

30000

40000

50000

60000

0.8

1

1.2

1020 3040 506070

10000

20000

30000

40000

50000

60000

70000

80000

90000

0.8

1

1.2

051015 20253035 40

(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

00.050.10.150.20.25

-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

00.025 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 -0.200.20.40.60.81

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

6Analysis 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