Determination of the S-wave pi pi scattering lengths from a study of K+- -> pi+- pi0 pi0 decays
aa r X i v : . [ h e p - e x ] D ec Determination of the S-wave ππ scattering lengths from a studyof K ± → π ± π π decays J.R. Batley , A.J. Culling , G. Kalmus , C. Lazzeroni , D.J. Munday , M.W. Slater , S.A. Wotton ,R. Arcidiacono , G. Bocquet , N. Cabibbo , A. Ceccucci , D. Cundy , V. Falaleev , M. Fidecaro , L. Gatignon ,A. Gonidec , W. Kubischta , A. Norton , A. Maier , M. Patel , A. Peters , S. Balev , P.L. Frabetti ,E. Goudzovski , P. Hristov , V. Kekelidze , V. Kozhuharov , L. Litov , D. Madigozhin , E. Marinova ,N. Molokanova , I. Polenkevich , Yu. Potrebenikov , S. Stoynev , A. Zinchenko , E. Monnier , E. Swallow ,R. Winston , P. Rubin , A. Walker , W. Baldini , A. Cotta Ramusino , P. Dalpiaz , C. Damiani , M. Fiorini ,A. Gianoli , M. Martini , F. Petrucci , M. Savri´e , M. Scarpa , H. Wahl , M. Calvetti , E. Iacopini , G. Ruggiero ,A. Bizzeti , M. Lenti , M. Veltri , M. Behler , K. Eppard , K. Kleinknecht , P. Marouelli , L. Masetti ,U. Moosbrugger , C. Morales Morales , B. Renk , M. Wache , R. Wanke , A. Winhart , D. Coward ,A. Dabrowski , T. Fonseca Martin , M. Shieh , M. Szleper , M. Velasco , M.D. Wood , G. Anzivino ,E. Imbergamo , A. Nappi , M. Piccini , M. Raggi , M. Valdata-Nappi , P. Cenci , M. Pepe , M.C. Petrucci ,C. Cerri , R. Fantechi , G. Collazuol , L. DiLella , G. Lamanna , I. Mannelli , A. Michetti , F. Costantini ,N. Doble , L. Fiorini , S. Giudici , G. Pierazzini , M. Sozzi , S. Venditti , B. Bloch-Devaux , C. Cheshkov ,J.B. Ch`eze , M. De Beer , J. Derr´e , G. Marel , E. Mazzucato , B. Peyaud , B. Vallage , M. Holder ,M. Ziolkowski , C. Biino , N. Cartiglia , F. Marchetto , S. Bifani , M. Clemencic , S. Goy Lopez ,H. Dibon , M. Jeitler , M. Markytan , I. Mikulec , G. Neuhofer , and L. Widhalm Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, UK b2 CERN, CH-1211 Gen`eve 23, Switzerland Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia The Enrico Fermi Institute, The University of Chicago, Chicago, IL 60126, USA Department of Physics and Astronomy, University of Edinburgh, JCMB King’s Buildings, Mayfield Road, Edinburgh, EH93JZ, UK Dipartimento di Fisica dell’Universit`a e Sezione dell’INFN di Ferrara, I-44100 Ferrara, Italy Dipartimento di Fisica dell’Universit`a e Sezione dell’INFN di Firenze, I-50019 Sesto Fiorentino, Italy Sezione dell’INFN di Firenze, I-50019 Sesto Fiorentino, Italy Institut f¨ur Physik, Universit¨at Mainz, D-55099 Mainz, Germany q10
Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA Dipartimento di Fisica dell’Universit`a e Sezione dell’INFN di Perugia, I-06100 Perugia, Italy Sezione dell’INFN di Perugia, I-06100 Perugia, Italy Sezione dell’INFN di Pisa, I-56100 Pisa, Italy Scuola Normale Superiore e Sezione dell’INFN di Pisa, I-56100 Pisa, Italy Dipartimento di Fisica dell’Universit`a e Sezione dell’INFN di Pisa, I-56100 Pisa Italy DSM/IRFU - CEA Saclay, F-91191 Gif-sur-Yvette, France Fachbereich Physik, Universit¨at Siegen, D-57068 Siegen, Germany w18
Sezione dell’INFN di Torino, I-10125 Torino, Italy Dipartimento di Fisica Sperimentale dell’Universit`a e Sezione dell’INFN di Torino, I-10125 Torino, Italy ¨Osterreichische Akademie der Wissenschaften, Institut f¨ur Hochenergiephysik, A-10560 Wien, Austria z Published in The European Physical Journal C: Volume 64, Issue 4 (2009), Page 589
Abstract.
We report the results from a study of the full sample of ∼ . × K ± → π ± π π decays recorded by the NA48/2 experiment at the CERN SPS. As first observed in this ex-periment, the π π invariant mass ( M ) distribution shows a cusp-like anomaly in the regionaround M = 2 m + , where m + is the charged pion mass. This anomaly has been interpretedas an effect due mainly to the final state charge exchange scattering process π + π − → π π in K ± → π ± π + π − decay. Fits to the M distribution using two different theoretical formulationsprovide the presently most precise determination of a − a , the difference between the ππ S-wave scattering lengths in the isospin I = 0 and I = 2 states. Higher-order ππ rescatter-ing terms, included in the two formulations, allow also an independent, though less precise,determination of a . PACS. π , K and η mesons J.R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays Introduction
The main purpose of the NA48/2 experiment at the CERNSPS was to search for direct CP violation in K ± decay tothree pions [1,2,3]. The experiment used simultaneous K + and K − beams with momenta of 60 GeV/ c propagatingthrough the detector along the same beam line. Data werecollected in 2003-2004, providing large samples of fully re-constructed K ± → π ± π + π − and K ± → π ± π π decays.From the analysis of the data collected in 2003, we havealready reported the observation of a cusp-like anomaly a University of Birmingham, Edgbaston, Birmingham, B152TT, UK b Funded by the UK Particle Physics and Astronomy Re-search Council c Dipartimento di Fisica Sperimentale dell’Universit`a eSezione dell’INFN di Torino, I-10125 Torino, Italy d Universit`a di Roma “La Sapienza” e Sezione dell’INFN diRoma, I-00185 Roma, Italy e Istituto di Cosmogeofisica del CNR di Torino, I-10133Torino, Italy f Dipartimento di Fisica dell’Universit`a e Sezione dell’INFNdi Ferrara, I-44100 Ferrara, Italy g Scuola Normale Superiore, I-56100 Pisa, Italy h CERN, CH-1211 Gen`eve 23, Switzerland i Faculty of Physics, University of Sofia “St. Kl. Ohridski”,5 J. Bourchier Blvd., 1164 Sofia, Bulgaria j Sezione dell’INFN di Perugia, I-06100 Perugia, Italy k Northwestern University, 2145 Sheridan Road, Evanston,IL 60208, USA l Centre de Physique des Particules de Marseille, IN2P3-CNRS, Universit´e de la M´editerran´ee, Marseille, France m Department of Physics and Astronomy, George Mason Uni-versity, Fairfax, VA 22030, USA n Dipartimento di Fisica, Universit`a di Modena e ReggioEmilia, I-41100 Modena, Italy o Istituto di Fisica, Universit`a di Urbino, I-61029 Urbino,Italy p Physikalisches Institut, Universit¨at Bonn, D-53115 Bonn,Germany q Funded by the German Federal Minister for Education andresearch under contract 05HK1UM1/1 r SLAC, Stanford University, Menlo Park, CA 94025, USA s Royal Holloway, University of London, Egham Hill, Egham,TW20 0EX, UK t UCLA, Los Angeles, CA 90024, USA u Laboratori Nazionali di Frascati, I-00044 Frascati (Rome),Italy v Institut de F´ısica d’Altes Energies, UAB, E-08193 Bel-laterra (Barcelona), Spain w Funded by the German Federal Minister for Research andTechnology (BMBF) under contract 056SI74 x University of Bern, Institute for Theoretical Physics, Si-dlerstrasse 5, CH-3012 Bern, Switzerland y Centro de Investigaciones Energeticas Medioambientales yTecnologicas, E-28040 Madrid, Spain z Funded by the Austrian Ministry for Traffic and Re-search under the contract GZ 616.360/2-IV GZ 616.363/2-VIII, and by the Fonds f¨ur Wissenschaft und Forschung FWFNr. P08929-PHY in the π π invariant mass ( M ) distribution of K ± → π ± π π decays in the region around M = 2 m + , where m + is the charged pion mass [4]. The existence of thisthreshold anomaly had been first predicted in 1961 byBudini and Fonda [5], as a result of the charge exchangescattering process π + π − → π π in K ± → π ± π + π − decay. These authors had also suggested that the studyof this anomaly, once found experimentally, would allowthe determination of the cross-section for π + π − → π π at energies very close to threshold. However, samples of K ± → π ± π π decay events available in those years werenot sufficient to observe the effect, nor was the M resolu-tion. As a consequence, in the absence of any experimentalverification, the article by Budini and Fonda [5] was for-gotten.More recently, Cabibbo [6] has proposed an interpreta-tion of the cusp-like anomaly along the lines proposed byBudini and Fonda [5], but expressing the K ± → π ± π π decay amplitude in terms of the π + π − → π π amplitudeat threshold, a x . In the limit of exact isospin symmetry a x can be written as ( a − a ) /
3, where a and a arethe S-wave ππ scattering lengths in the isospin I = 0 and I = 2 states, respectively.Here we report the results from a study of the finalsample of ∼ . × K ± → π ± π π decays. Best fits totwo independent theoretical formulations of rescatteringeffects in K ± → π ± π π and K ± → π ± π + π − decays ([7]and [8,9]) provide a precise determination of a − a , andan independent, though less precise, determination of a . The layout of the beams and detectors is shown schemati-cally in Fig. 1. The two simultaneous beams are producedby 400 GeV/ c protons impinging on a 40 cm long Be tar-get. Particles of opposite charge with a central momen-tum of 60 GeV/ c and a momentum band of ± .
8% ( rms )produced at zero angle are selected by two systems ofdipole magnets forming “achromats” with null total de-flection, focusing quadrupoles, muon sweepers and colli-mators. With 7 × protons per pulse of ∼ . . × (2 . × )particles per pulse, of which ∼ .
7% ( ∼ . K + ( K − ). The decay volume is a 114 m long vacuum tankwith a diameter of 1.92 m for the first 66 m, and 2.40 mfor the rest.A detailed description of the detector elements is avail-able in [10]. Charged particles from K ± decays are mea-sured by a magnetic spectrometer consisting of four driftchambers (DCH1–DCH4, denoted collectively as DCH)and a large-aperture dipole magnet located between DCH2and DCH3 [10]. Each chamber has eight planes of sensewires, two horizontal, two vertical and two along each oftwo orthogonal 45 ◦ directions. The spectrometer is locatedin a tank filled with helium at atmospheric pressure andseparated from the decay volume by a thin Kevlar R (cid:13) win-dow with a thickess of 0.0031 radiation lengths ( X ). A16 cm diameter aluminium vacuum tube centred on the .R. 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K + K − K + K − (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) KABES 3 K e v l a r w i nd o w FRONT−END ACHROMAT
50 200 10 cm
He tank + HO D L K r H AC M UV DCH 1
F DFD
Protectingcollimator
DFFD
Quadrupole ACHROMATTarget
Defining collimator Finalcollimator Cleaning
250 m
SpectrometerQuadruplet tankVacuum focused beams
Decay volume
Magnet
DCH 4
KABES 2KABES 1 Z collimator TAX 17, 18
Fig. 1.
Schematic side view of the NA48/2 beam line, decay volume and detectors (TAX 17, 18: motorised collimators;FDFD/DFDF: focusing quadrupoles; KABES 1-3: beam spectrometer stations (not used in this analysis); DCH1-4: drift cham-bers; HOD: scintillator hodoscope; LKr: liquid Krypton calorimeter; HAC: hadron calorimeter; MUV: muon veto). Thick linesindicate beam axes, narrow lines indicate the projections of the beam envelopes. Note that the vertical scales are different inthe left and right part of the figure. beam axis runs the length of the spectrometer throughcentral holes in the Kevlar window, drift chambers andcalorimeters. Charged particles are magnetically deflectedin the horizontal plane by an angle corresponding to atransverse momentum kick of 120 MeV/ c . The momentumresolution of the spectrometer is σ ( p ) /p = 1 . ⊕ . p ( p in GeV/ c ), as derived from the known properties ofthe spectrometer and checked with the measured invari-ant mass resolution of K ± → π ± π + π − decays. The mag-netic spectrometer is followed by a scintillator hodoscopeconsisting of two planes segmented into horizontal andvertical strips and arranged in four quadrants.A liquid Krypton calorimeter (LKr) [11] is used to re-construct π → γγ decays. It is an almost homogeneousionization chamber with an active volume of ∼
10 m ofliquid krypton, segmented transversally into 13248 2 cm × X thick and has an energy resolution σ ( E ) /E = 0 . / √ E ⊕ . /E ⊕ . σ x = σ y = 0 . / √ E ⊕ .
06 cm for eachtransverse coordinate x, y .An additional hodoscope consisting of a plane of scin-tillating fibers is installed in the LKr calorimeter at adepth of ∼ . X with the purpose of sampling electro-magnetic showers. It is divided into four quadrants, eachconsisting of eight bundles of vertical fibers optically con-nected to photomultiplier tubes. The K ± → π ± π π decays are selected by a two leveltrigger. The first level requires a signal in at least onequadrant of the scintillator hodoscope (Q1) in coincidencewith the presence of energy depositions in LKr consis-tent with at least two photons (NUT). At the second level(MBX), an on-line processor receiving the drift chamberinformation reconstructs the momentum of charged parti-cles and calculates the missing mass under the assumptionthat the particle is a π ± originating from the decay of a60 GeV/ c K ± travelling along the nominal beam axis.The requirement that the missing mass is not consistentwith the π mass rejects most of the main K ± → π ± π background. The typical rate of this trigger is ∼ , c , measured with a maximumerror of 6% (much larger than the magnetic spectrometerresolution), and at least four energy clusters in the LKr,each consistent, in terms of size and energy, with the elec-tromagnetic shower produced by a photon of energy above3 GeV, are selected for further analysis. In addition, therelative track and photon timings must be consistent withthe same event within 10 ns, and the clusters must be intime between each other within 5 ns.The distance between any two photons in the LKr isrequired to be larger than 10 cm, and the distance betweeneach photon and the impact point of any track on theLKr front face must exceed 15 cm. Fiducial cuts on thedistance of each photon from the LKr edges and centreare also applied in order to ensure full containment of J.R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays the electromagnetic showers. In addition, because of thepresence of ∼
100 LKr cells affected by readout problems(“dead cells”), the minimum distance between the photonand the nearest LKr dead cell is required to be at least2 cm.At the following step of the analysis we check the con-sistency of the surviving events with the K ± → π ± π π decay hypothesis. We assume that each possible pair ofphotons originates from a π → γγ decay and we calcu-late the distance D ij between the π decay vertex and theLKr front face: D ij = p E i E j R ij m where E i , E j are the energies of the i -th and j -th pho-ton, respectively, R ij is the distance between their impactpoints on LKr, and m is the π mass.Among all possible π pairs, only those with D ij valuesdiffering by less than 500 cm are retained further, andthe distance D of the K ± decay vertex from the LKr istaken as the arithmetic average of the two D ij values. Thischoice gives the best π π invariant mass resolution nearthreshold: at M = 2 m + it is ∼ .
56 MeV/ c , increasingmonotonically to ∼ . c at the upper edge of thephysical region. The reconstructed distance of the decayvertex from the LKr is further required to be at least 2 mdownstream of the final beam collimator to exclude π -mesons produced from beam particles interacting in thecollimator material (the downstream end of the final beamcollimator is at Z = −
18 m).Because of the long decay volume, a photon emittedat small angle to the beam axis may cross the aluminiumvacuum tube in the spectrometer or the DCH1 centralflange, and convert to e + e − before reaching the LKr. Insuch a case the photon must be rejected because its energycannot be measured precisely. To this purpose, for eachphoton detected in LKr we require that its distance fromthe nominal beam axis at the DCH1 plane must be >
11 cm, assuming an origin on axis at D −
400 cm. In thisrequirement we take into account the resolution of the D measurement (the rms of the difference between D valuesfor the two photon pairs distribution is about 180 cm).Each surviving π pair is then combined with a chargedparticle track, assumed to be a π ± . Only those combina-tions with a total π ± π π energy between 54 and 66 GeV,consistent with the beam energy distribution, are retained,and the π ± π π invariant mass M is calculated, after cor-recting the charged track momentum vector for the effectof the small measured residual magnetic field in the decayvolume (this correction uses the decay vertex position, D,as obtained from LKr information).For each π ± π π combination, the energy-weighed av-erage coordinates (center-of-gravity, COG) X COG , Y
COG are calculated at each DCH plane using the photon im-pact points on LKr and the track parameters measuredbefore the magnet (so the event COG is a projection ofthe initial kaon line of flight). Acceptance cuts are thenapplied on the COG radial position on each DCH planein order to select only K ± → π ± π π decays originating from the beam axis. In addition, we require a minimalseparation between the COG and the charged track coor-dinates X t , Y t , as measured in each DCH plane: q X COG + Y COG < R
COGmax , p ( X COG − X t ) + ( Y COG − Y t ) > R COG − trackmin , where the limits depend on the COG and track impactpoint distributions at each drift chamber (see Table 1). Table 1.
Acceptance cuts on event COG and charged trackcoordinates.Drift chamber R COGmax (cm) R COG − trackmin (cm)DCH1 2.0 17.0DCH2 2.0 19.0DCH3 2.0 19.0DCH4 3.0 15.5 The values of R COG − trackmin take into account both thebeam width (the cut is made with respect to each eventCOG rather than to the nominal beam center) and thearea where the track impact point distribution is still sen-sitive to the detailed features of the beam shape. In thisway the effect of these cuts does not depend strongly onthe beam shape and on the precise knowledge of the beamposition in space (during data taking, the average beamtransverse position was observed to move slightly by up to2 mm). This cut removes about 28% of events, mainly atlarge M , but the statistical precision of the final resultson the ππ scattering lengths is not affected.For events with more than one accepted track-clustercombination ( ∼ .
8% of the total), the K ± → π ± π π decay is selected as the π ± π π combination minimizinga quality estimator based on two variables: the difference ∆D of the two D ij values and the difference ∆M betweenthe π ± π π invariant mass and the nominal K ± mass [12]: (cid:18) ∆Drms D ( D ) (cid:19) + (cid:18) ∆Mrms M ( D ) (cid:19) , where the space and mass resolutions rms D , rms M arefunctions of D , as obtained from the measured ∆D and ∆M distributions.Fig. 2 shows the distribution of ∆M , the differencebetween the π ± π π invariant mass and the nominal K ± mass for the selected K ± → π ± π π decays (a total of6 . × events). This distribution is dominated bythe gaussian K ± peak, with a resolution σ = 1 . c .There are small non Gaussian tails originating from uniden-tified π ± → µ ± decay in flight or wrong photon pairing.The fraction of events with wrong photon pairing in thissample is 0 . The beam is focused at the DCH1 plane, where its widthis ∼ .
45 cm..R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays 5 -0.01 -0.005 0 0.005 0.01 ∆M (GeV/ c ) e v e n t s /0 . G e V / c Fig. 2.
Distribution of the difference between the π ± π π invariant mass and the nominal K ± mass for the selected K ± → π ± π π decays. Fig. 3 shows the distribution of the square of the π π invariant mass, M , for the final event sample. This distri-bution is displayed with a bin width of 0.00015 (GeV/ c ) ,with the 51 st bin centred at M = (2 m + ) (for most ofthe physical region the bin width is smaller than the M resolution, which is 0.00031 (GeV/ c ) at M = (2 m + ) ).The cusp at M = (2 m + ) = 0 . c ) is clearlyvisible. Samples of simulated K ± → π ± π π events ∼
10 timeslarger than the data have been generated using a full de-tector simulation based on the GEANT-3 package [13].This Monte Carlo (MC) program takes into account alldetector effects, including the trigger efficiency and thepresence of a small number ( < K ± → π ± π + π − events, which provide pre-cise information on the average beam angles and positionswith respect to the nominal beam axis. Furthermore, therequirement that the average reconstructed π ± π + π − in-variant mass is equal to the nominal K ± mass for both K + and K − fixes the absolute momentum scale of themagnetic spectrometer for each charge sign and magnetpolarity, and monitors continuously the beam momentumdistributions during data taking.The Dalitz plot distribution of K ± → π ± π π decayshas been generated according to a series expansion in the ab M (GeV/ c ) e v e n t s /0 . ( G e V / c ) Fig. 3. a : distribution of M , the square of the π π invariantmass; b : enlargement of a narrow region centred at M =(2 m + ) (this point is indicated by the arrow). The statisticalerror bars are also shown in these plots. Lorentz-invariant variable u = ( s − s ) /m , where s i =( P K − P i ) ( i =1,2,3), s = ( s + s + s ) / P K ( P i ) is the K ( π ) four-momentum, and i = 3 corresponds to the π ± [12]. In our case s = M , and s = ( m K + 2 m + m ) / π π invariant massthe simulation provides the detection probability and thedistribution function for the reconstructed value of M .This allows the transformation of any theoretical distribu-tion into an expected distribution which can be compareddirectly with the measured one. ππ scatteringlengths a and a The sudden change of slope (“cusp” ) observed in the M distribution at M = (2 m + ) (see Fig. 3) can be inter-preted [5] [6] as a threshold effect from the decay K ± → π ± π + π − contributing to the K ± → π ± π π amplitudethrough the charge exchange reaction π + π − → π π . Inthe formulation by Cabibbo [6] the K ± → π ± π π decayamplitude is described as the sum of two terms: M ( K ± → π ± π π ) = M + M , (1)where M is the tree level K ± → π ± π π weak decayamplitude, and M is the contribution from the K ± → π ± π + π − decay amplitude through π + π − → π π charge J.R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays exchange, with the normalization condition M = 0 at M = (2 m + ) . The contribution M is given by M = − a x m + M + s(cid:18) m + M (cid:19) − , (2)where a x is the S-wave π + π − charge exchange scatter-ing length (threshold amplitude), and M + is the K ± → π ± π + π − decay amplitude at M = 2 m + . M changesfrom real to imaginary at M = 2 m + with the conse-quence that M interferes destructively with M in theregion M < m + , while it adds quadratically above it.In the limit of exact isospin symmetry a x = ( a − a ) / a and a are the S-wave ππ scattering lengths inthe I = 0 and I = 2 states, respectively.However, it was shown in ref. [4] that a fit of this simpleformulation to the NA48/2 M distribution in the inter-val 0 . < M < .
097 (GeV/ c ) using a x m + as a freeparameter gave only a qualitative description of the data,with all data points lying systematically above the fit inthe region near M = (2 m + ) . It was also shown in ref. [4]that a good fit could be obtained using a more completeformulation of ππ final state interaction [7] which tookinto account all rescattering processes at the one-loop andtwo-loop level.In the following sections we present the determina-tion of the ππ scattering lengths a and a by fits of thefull data set described in Section 2 to two theoretical ap-proaches: the Cabibbo-Isidori (CI) formulation [7], andthe more recent Bern-Bonn (BB) formulation [8].In the CI approach, the structure of the cusp singu-larity is treated using unitarity, analiticity and cluster de-composition properties of the S -matrix. The decay ampli-tude is expanded in powers of ππ scattering lengths upto order ( scattering length ) , and electromagnetic effectsare omitted.The BB approach uses a non-relativistic Lagrangianframework, which automatically satisfies unitarity andanaliticity constraints, and allows one to include electro-magnetic contributions in a standard way [9].In all fits we also need information on the K ± → π ± π + π − decay amplitude. To this purpose, we use a sam-ple of 4 . × K ± → π ± π + π − decays which are alsomeasured in this experiment [14]. In the Cabibbo-Isidori (CI) formulation [7] the weak am-plitudes for K ± → π ± π π and K ± → π ± π + π − decay attree level are written as M = 1 + 12 g u + 12 h u + 12 k v , (3) M + = A + (1 + 12 gu + 12 hu + 12 kv ) , (4)respectively. In Eq. (3) u = ( s − s ) /m , where s =( m K + 2 m + m ) /
3, while in Eq. (4) u = ( s − s + ) /m + , where s + = m K / m ; for both amplitudes s i = ( P K − P i ) , where P K ( P i ) is the K ( π ) four-momentum and i =3 corresponds to the odd pion ( π ± from K ± → π ± π π , π ∓ from K ± → π ± π + π − decay), and v = ( s − s ) /m . Itmust be noted that in ref. [7] the v dependence of both am-plitudes had been ignored because the coefficients k and k were consistent with zero from previous experiments.Within the very high statistical precision of the presentexperiment this assumption is no longer valid.Pion-pion rescattering effects are evaluated by meansof an expansion in powers of the ππ scattering lengthsaround the cusp point, M = (2 m + ) . The terms addedto the tree-level decay matrix elements depend on five S-wave scattering lengths which are denoted by a x , a ++ , a + − , a +0 , a , and describe π + π − → π π , π + π + → π + π + , π + π − → π + π − , π + π → π + π , π π → π π scat-tering, respectively. In the limit of exact isospin symmetrythese scattering lengths can all be expressed as linear com-binations of a and a .At tree level, omitting one-photon exchange diagrams,isospin symmetry breaking contributions to the elastic ππ scattering amplitude can be expressed as a function ofone parameter η = ( m − m ) /m = 0 .
065 [15,16,17]. Inparticular, the ratio between the threshold amplitudes a x , a ++ , a + − , a +0 , a and the corresponding isospin sym-metric amplitudes – evaluated at the π ± mass – is equalto 1 − η for π + π + → π + π + , π + π → π + π , π π → π π ,1 + η for π + π − → π + π − , and 1 + η/ π + π − → π π .These corrections have been applied in order to extract a and a from the fit to the M distribution.The CI formulation [7] includes all one-loop and two-loop rescattering diagrams and can be used to fit both K ± → π ± π π and K ± → π ± π + π − decay distributions.However, rescattering effects are much smaller in K ± → π ± π + π − than in the K ± → π ± π π decay because theinvariant mass of any two-pion pair is always ≥ m + . In-deed, a good fit to the K ± → π ± π + π − Dalitz plot [14] canbe obtained with or without the addition of rescatteringterms to the tree-level weak amplitude of K ± → π ± π + π − decay. We have checked that both the values of the best fitparameters and their statistical errors, as obtained fromfits to the M distribution of K ± → π ± π π decay, un-dergo negligible changes whether or not rescattering ef-fects are included in the K ± → π ± π + π − decay amplitude.This can be understood from the fact that the K ± → π ± π + π − decay amplitude enters into the CI formulationof rescattering effects in K ± → π ± π π decays as thecomplete expression given by Eq. (4). Thus Eq. (4), withparameters extracted from a fit to the K ± → π ± π + π − data, provides an adequate phenomenological descriptionof K ± → π ± π + π − decay which can be used in calculatingrescattering effects in K ± → π ± π π decay.In the fits to the M distribution from K ± → π ± π π decay, the free parameters are ( a − a ) m + , a m + , g , h ,and an overall normalization constant. The coefficient k cannot be directly obtained from a fit to the M distribu-tion. Its value is determined independently from the Dalitzplot distribution of K ± → π ± π π decays, as described .R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays 7 in the Appendix. The value k = 0 . M + parameters are fixed from data: the coeffi-cients g , h , k are obtained from a separate fit to the K ± → π ± π + π − decay Dalitz plot [14], using M + asgiven by Eq. (4), and taking into account Coulomb ef-fects; and A + is obtained from the measured ratio, R ,of the K ± → π ± π + π − and K ± → π ± π π decay rates, R = 3 . ± .
050 [12], which is proportional to A . Thefit gives g = − . ± . h = 0 . ± . k = − . ± . A + = 1 . ± . M distri-bution from K ± → π ± π π decay.As explained in Section 6 all fits are performed over the M interval from 0 . . c ) (bin26 to 226). The CI formulation [7] does not include ra-diative corrections, which are particularly important near M = 2 m + , and contribute to the formation of π + π − atoms (“pionium”). For this reason we first exclude fromthe fit a group of seven consecutive bins centred at M =4 m (an interval of ± .
94 MeV/ c in M ). The qualityof this fit is illustrated in Fig. 4a, which displays the quan-tity ∆ ≡ (data – fit)/data as a function of M . The smallexcess of events from pionium formation is clearly visible. -0.02-0.015-0.01-0.00500.0050.010.0150.02 0.075 0.08 0.085 0.09 0.095 0.1 0.105-0.02-0.015-0.01-0.00500.0050.010.0150.02 0.075 0.08 0.085 0.09 0.095 0.1 0.105 ∆∆ ab M (GeV/ c ) Fig. 4. ∆ = (data – fit)/data versus M for the rescatteringformulation of ref. [7]: a – fit with no pionium formation andexcluding seven consecutive bins centred at M = (2 m + ) (theexcluded region is shown by the two vertical dotted lines; b –fit with pionium CI (see text). The two vertical dashed linesshow the M interval used in the fit. The point M = (2 m + ) is indicated by the arrow. Pionium formation and its dominating decay to π π are taken into account in the fit by multiplying the contentof the bin centred at M = 4 m (bin 51) by 1 + f atom ,where 1 + f atom describes the contribution from pioniumformation and decay. The pionium width is much narrowerthan the bin width, since its mean lifetime is measured tobe ∼ × − s [18]; however, the M resolution is takeninto account in the fits as described in the last paragraphof Section 3. The results of a fit with f atom as a free param-eter and with no excluded bins near M = 4 m are givenin Tables 2 and 3 (fit CI ): the quality of this fit is shown inFig. 4b. The best fit value f atom = 0 . ± . K ± → π ± + pionium decay, normalizedto the K ± → π ± π + π − decay rate, of (1 . ± . × − ,which is larger than the predicted value ∼ . × − [19,20]. As discussed in Section 5, this difference is due to ad-ditional radiative effects, which are not taken into accountin the CI formulation [7] and, contrary to pionium forma-tion and decay, affect more than one bin. For this reasonfor the fits without the radiative effects taken into accountwe prefer to fix f atom = 0 . M = 4 m . Theresults of this fit are listed as Fit CI A in Tables 2 and 3.We have also performed fits using the constraint be-tween a and a predicted by analyticity and chiral sym-metry [21] (we refer to this constraint as the ChPT con-straint): a m + = ( − . ± . . a m + − . − . a m + − . − . a m + − . (5)The results of these fits are shown in Tables 2 and3 (fits CI χ and CI χA ). For fit CI χ no bins near the cusppoint are excluded and f atom is a free parameter, while forfit CI χA the seven bins centred at M = 4 m are excludedand f atom is kept fixed at the value obtained from fit CI χ . The Bern-Bonn (BB) formulation [8] describes the K → π decay amplitudes using two expansion parameters: a ,the generic ππ scattering amplitude at threshold; and aformal parameter ǫ such that in the K -meson rest framethe pion momentum is of order ǫ , and its kinetic energy T is of order ǫ . In the formulation of ref. [8] the K → π decay amplitudes include terms up to O ( ǫ , aǫ , a ǫ ).However, in the formulae used in the fits described belowthese amplitudes include terms up to O ( ǫ , aǫ , a ǫ ). Inthe BB formulation the description of the K → π decayamplitudes is valid over the full physical region .At tree level the K → π decay amplitudes are ex-pressed as polynomials containing terms in T , T , and( T − T ) , where T is the kinetic energy of the “odd”pion ( π ± from K ± → π ± π π , π ∓ from K ± → π ± π + π − decay) in the K ± rest frame, while T and T are the We thank the Bern-Bonn group for providing the computercode which calculates the K → π decay amplitudes. J.R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays kinetic energies of the two same-sign pions. Since thesevariables can be expressed as functions of the relativisticinvariants u and v defined previously, for consistency withthe fits described in the previous subsection we prefer touse the same forms as given in Eqs. (3) and (4). It mustbe noted, however, that the best fit polynomial coefficientsare not expected to be equal to those obtained from thefits to the CI formulation [7] because the loop diagramcontributions are different in the two formulations.As for CI, also in the BB formulation rescattering ef-fects are much smaller in K ± → π ± π + π − than in the K ± → π ± π π decay, and a good fit to the M ±± distri-bution alone can be obtained with or without the additionof rescattering terms to the tree-level weak amplitude of K ± → π ± π + π − decay. However, contrary to CI, the coef-ficients of the tree-level K ± → π ± π + π − amplitudes enterinto the K ± → π ± π π rescattering terms in differentcombinations. Therefore, the use of a phenomenologicaldescription of the K ± → π ± π + π − decay amplitude ex-tracted from a fit to K ± → π ± π + π − data alone is not jus-tified in this case. Thus, in order to obtain a precision onthe fit parameters which matches the BB approximationlevel, the value of each coefficient of the K ± → π ± π + π − tree-level amplitude is obtained from the fit. We perform simultaneous fits to two distributions: the M distribution described in Section 2 and the M ±± dis-tribution from K ± → π ± π + π − decay, obtained as a pro-jection of the Dalitz plot described in ref. [14]. This latterdistribution is made with the same binning as for the M distribution from K ± → π ± π π decay and consists of4 . × events.All fits are performed over the M interval from0 . . c ) (bin 26 to 226), and from0 . . c ) (bin 70 to 330) for the M ±± distribution from K ± → π ± π + π − decay. As for the M distribution from K ± → π ± π π decay, a very largesample of simulated K ± → π ± π + π − decays (see ref. [14])is used to obtain the detection probability and the distri-bution function for the reconstructed value M ±± for anygenerated value of M ±± .In all fits the free parameters are ( a − a ) m + and a m + (or only a m + for the fit using the ChPT constraintgiven by Eq. (5)), the coefficients of the tree-level weakamplitudes g , h , g , h , k (see Eqs. (3, 4)), and two over-all normalization constants (one for each distribution).The coefficient k (see Eq. (3)) is determined indepen-dently from a separate fit to the Dalitz plot distributionof K ± → π ± π π decays (see the Appendix). The fixedvalue k = 0 . f atom ,is also a free parameter.Since the detection of K ± → π ± π π and K ± → π ± π + π − decays involves different detector componentsand different triggers (no use of LKr information is made Nevertheless, if one fixes the coefficients g, h, k in the fitto the values obtained from fits to K ± → π ± π + π − data onlywith or without rescattering terms, the corresponding varia-tions of the best fit a , a values are much smaller than the a , a statistical errors. to select K ± → π ± π + π − decays), the ratio of the de-tection efficiencies for the two decay modes is not knownwith the precision needed to extract the value of A + (seeEq. (4)) from the fit. Therefore, as for the CI fits, also forthe BB fits A + is obtained from the ratio of the K ± → π ± π + π − and K ± → π ± π π decay rates, measured byother experiments, R = 3 . ± .
050 [12].Tables 2 and 3 show the results of a fit (fit BB ) using f atom as a free parameter and including all bins aroundthe cusp point in the fit; for fit BB A the value of f atom isfixed and seven bins centred at M = 4 m are excluded.A comparison with the results of the corresponding CI fits(fits CI and CI A , respectively) shows that the differencebetween the best fit values of ( a − a ) m + is rather small(about 3%), while the difference between the two a m + values is much larger. We note that in the BB fits a m + has a stronger correlation with other fit parameters thanin the CI fits (see Tables 4 and 5). Table 4.
Parameter correlations for the CI fits (fit CI A inTable 2). g h a − a a g . h − .
701 1 . a − a . − .
793 1 . a − .
902 0 . − .
869 1 . Table 5.
Parameter correlations for the BB fits (fit BB A inTable 2). g h g h k a − a a g . h .
996 1 . g − . − .
960 1 . h .
206 0 . − .
247 1 . k − . − .
423 0 .
359 0 .
803 1 . a − a − . − .
817 0 . − .
402 0 .
141 1 . a .
976 0 . − .
958 0 . − . − .
794 1 . Fits BB χ and BB χA (see Tables 2 and 3) are similarto BB and BB A , respectively, but the ChPT constraintgiven by Eq. (5) is used. Here the best fit value of a m + agrees well with the value obtained from the CI fit (fit CI χA ). Radiative corrections to both K ± → π ± π π and K ± → π ± π + π − decay channels have been recently studied by ex-tending the BB formulation [8] to include real and virtualphotons [9]. In the K ± rest frame the emission of realphotons is allowed only for photon energies E < E cut .We have performed simultaneous fits to the M dis-tribution from K ± → π ± π π and to the M ±± distribu-tion from K ± → π ± π + π − decays using the formulation .R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays 9 Table 2.
Fit results without radiative corrections: ππ scattering parameters. Parameter values without errors have been keptfixed in the fit or calculated using the constraint between a and a given by Eq. (5).Fit χ /NDF a m + a m + ( a − a ) m + f atom CI . − . . . CI A . − . . . CI χ . − . . . CI χA . − . . . BB . − . . . BB A . − . . . BB χ . − . . . BB χA . − . . . Table 3.
Fit results without radiative corrections: coefficients of the tree-level K → π weak decay amplitudes. Parametervalues without errors have been kept fixed in the fit.Fit g h k g h kCI . − . . − . . − . CI A . − . . − . . − . CI χ . − . . − . . − . CI χA . − . . − . . − . BB . − . . − . − . − . BB A . − . . − . − . − . BB χ . − . . − . − . − . BB χA . − . . − . − . − . of ref. [9]. Our event selection does not exclude the pres-ence of additional photons; however, energetic photonsemitted in K ± decays result in a reconstructed π ± π π invariant mass lower than the K mass. We set E cut =0.010 GeV in order to be consistent with the measured π ± π π invariant mass distribution shown in Fig. 2 (thesame is true for the π ± π + π − invariant mass distributionfrom K ± → π ± π + π − decay measured in this experiment[14]). For each fit we adjust the value of A + (see Eq. (4)) sothat the ratio of the K ± → π ± π + π − and K ± → π ± π π decay rates is consistent with the measured one [12].The formulation of ref. [9] does not include pioniumformation, and the K ± → π ± π π amplitude, A rad , hasa non-physical singularity at M = (2 m + ) . To avoidproblems in the fits, the square of decay amplitude at thecenter of bin 51, where the singularity occurs, is replacedby | A | (1 + f atom ), where A is the decay amplitudeof the BB formulation without radiative corrections [8],and f atom is again a free parameter.The results of simultaneous fits to the M distributionfrom K ± → π ± π π decays, and to the M ±± distributionfrom K ± → π ± π + π − decay are shown in Tables 6 and 7.In all these fits the M and M ±± intervals are equal tothose of the fits described in Sections 4.1 and 4.2 (see Ta-bles 2 and 3). In fit BB all bins around the cusp point areincluded and f atom is a free parameter, while in fit BB A seven consecutive bins centred at M = (2 m + ) are ex-cluded and f atom is fixed to the value given by fit BB .A comparison of fit BB or BB A with radiative correc-tions taken into account (Table 6) with the correspondingfits without radiative corrections (fits BB , BB A of Table2) shows that radiative corrections reduce ( a − a ) m + by ∼ a m + is much larger, possibly suggesting again that thedetermination of this scattering length is affected by largetheoretical uncertainties.Fits BB χ and BB χA in Tables 6 and 7 are similar to BB and BB A , respectively, but the constraint between a and a predicted by analyticity and chiral symmetry[21] (see Eq. (5)) is used. A comparison of fits BB χ and BB χA with the corresponding fits obtained without radia-tive corrections (fits BB χ , BB χA of Table 2) shows thatradiative corrections reduce a m + by ∼ BB χ to BB χA in Tables 6 and 7 the effect ofchanging the maximum allowed photon energy E cut from0.005 to 0.020 GeV is found to be negligible.No study of radiative corrections has been performedin the framework of the CI approach [7]. However, thedominating radiative effects (Coulomb interaction andphoton emission) are independent of the specific approxi-mation. Therefore, extracting the relative effect of radia-tive corrections from the BB calculation and using it forthe fit to the CI formula is justified. In order to obtain anapproximate estimate of radiative effects in this case, wehave corrected the fit procedure by multiplying the abso-lute value of the K ± → π ± π π decay amplitude given inref. [7] by | A rad /A | [22], as obtained in the frameworkof the BB formulation [8,9]. Because of the non-physicalsingularity of A rad at M = (2 m + ) in the BB formula-tion, in the calculation of the K ± → π ± π π decay am-plitude for the 51 st bin we also multiply the squared am-plitude of ref. [7] by 1 + f atom .The results of these radiative-corrected fits to the M distribution from K ± → π ± π π decay performed using ππ scattering lengths from a study of K ± → π ± π π decays Table 6.
Fit results with electromagnetic corrections: ππ scattering parameters. Parameter values without errors have beenkept fixed in the fit or calculated using the constraint between a and a given by Eq. (5).Fit χ /NDF a m + a m + ( a − a ) m + f atom CI . − . . . CI A . − . . . CI χ . − . . . CI χA . − . . . BB . − . . . BB A . − . . . BB χ . − . . . BB χA . − . . . Table 7.
Fit results with electromagnetic corrections: coefficients of the tree-level K → π weak decay amplitudes. Parametervalues without errors have been kept fixed in the fit.Fit g h k g h kCI . − . . − . . − . CI A . − . . − . . − . CI χ . − . . − . . − . CI χA . − . . − . . − . BB . − . . − . − . − . BB A . − . . − . . − . BB χ . − . . − . − . − . BB χA . − . . − . − . − . Table 8.
Fit parameter correlations for the CI formulationwith radiative correction (fit CI in Table 6). g h a − a a f atom g . h − .
629 1 . a − a . − .
719 1 . a − .
913 0 . − .
873 1 . f atom − .
516 0 . − .
650 0 .
542 1 . Table 9.
Fit parameter correlations for the BB formulationwith radiative correction (fit BB in Table 6). g h g h k f atom a − a a g . h .
997 1 . g − . − .
965 1 . h .
234 0 . − .
255 1 . k − . − .
225 0 .
194 0 .
889 1 . f atom .
597 0 . − .
652 0 . − .
111 1 . a − a − . − .
843 0 . − . − . − .
682 1 . a .
977 0 . − .
976 0 . − .
310 0 . − .
839 1 . the CI formula are listed in Tables 6 and 7 (Fits CI to CI χA ). The parameter correlations for two fits which in-clude electromagnetic effects are shown in Tables 8 and9. Fig. 5 illustrates the fit results for the fits CI and BB with and without radiative corrections. All the fits areperformed using the same K ± → π ± π π data sample. -0.08-0.06-0.04-0.0200.02 0.24 0.25 0.26 0.27 0.28 0.29 a m + ( a − a ) m + CICI BB BB
Fig. 5.
68% confidence level ellipses taking into account thestatistical uncertainties only. Dashed line ellipses: fits CI and BB without radiative corrections. Solid line ellipses: fits CI and BB with radiative corrections. The theoretical band al-lowed by the ChPT constraint (see Eq. (5)) is shown by thedotted curves..R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays 11 Pionium formation in particle decay and in charged parti-cle scattering was studied in early theoretical work [20,23],but a unified description of its production together withother electromagnetic effects near threshold was missing.In a more recent approach [24], electromagnetic ef-fects in K ± → π ± π π decay have been studied in theframework of nonrelativistic quantum mechanics using apotential model to describe the electromagnetic interac-tion between the π + π − pair in loop diagrams. This modelis equivalent to a perturbative one, in which all simplesequential π + π − loops with electromagnetic interactionsbetween the two charged pions are taken into account toall orders (including the formation of electromagneticallybound final states), but there is no emission of real pho-tons and the electromagnetic interaction with the other π ± from the K ± → π ± π + π − decay is ignored. Because ofthese limitations, the model of ref. [24] cannot be directlyapplied to the full physical region of the K ± → π ± π π decay; however, contrary to the BB formulation [9], its in-tegral effect over a narrow region which includes the cusppoint ( M = 4 m ) can be calculated.We have implemented the electromagnetic effects pre-dicted by the model of ref. [24] in the parameterization ofthe CI formulation [7] (the detailed procedure is describedin Eqs. (6, 7, 8) of ref. [25]). In the theoretical M distri-bution the electromagnetic correction for the bin centredat 4 m (bin 51), averaged over the bin, depends on thebin width, as it includes contributions from both pioniumbound states with negligible widths and a very narrowpeak of unbound π + π − states annihilating to π π . Forthe bin width of 0.00015 (GeV/ c ) used in the fits, theseeffects increase the content of bin 51 by 5.8%, in agree-ment with the results of the fits performed using f atom asa free parameter (see Tables 2, 6). Thus the model of ref.[24] explains why the typical fit result for f atom is nearlytwice as large as the prediction for pionium contributiononly, as calculated in refs. [19,20].Near the cusp point the two calculations of electromag-netic effects [9] and [24,25] are very similar numerically,thus increasing the confidence in the central cusp bin ra-diative effect calculated using Eq. (8) of ref. [25]. However,at larger distances from the cusp the approach of refs. [24,25] leads to deviations from the electromagnetic correc-tions of ref. [9]. This can be explained by the fact that themodel of ref. [24] takes into account only processes thatdominate near the cusp point. For this reason we do notuse this model in the fits, but we consider it as a comple-mentary calculation limited to a region very close to thecusp point, providing a finite result for the bin centredat M = 4 m which the formulation of ref. [9] does notprovide. As shown below, all systematic corrections affecting thebest fit values of the coefficients describing the K ± → π ± π π weak amplitude at tree level, g and h (see Eq. (3)),are found to be much smaller than the statistical errors.We use these corrections as additional contributions to thesystematic uncertainties instead of correcting the centralvalues of these parameters.For a given fit, we find that the systematic uncertain-ties affecting the best fit parameters do not change ap-preciably if the fit is performed with or without electro-magnetic corrections. In addition, we find that, with theexception of f atom , the systematic uncertainties affectingall other parameters are practically the same if in the fitthe seven consecutive bins centred at M = 4 m are in-cluded (and f atom is used as a free parameter), or if theyare excluded (and the value of f atom is fixed).For these reasons, we give detailed estimates of thesystematic uncertainties only for fits CI , CI χ , BB , BB χ performed with the decay amplitude corrected for electro-magnetic effects.The parameters g, h, k which describe the K ± → π ± π + π − weak amplitude at tree level are used as freeparameters when fitting the data to the BB formulation[8,9]. However, they enter into the K ± → π ± π π de-cay amplitude only through rescattering terms, thus wedo not consider the best fit values of these parametersas a measurement of physically important values. Herewe do not estimate the systematic uncertainties affect-ing them and we discuss the uncertainties associated with K ± → π ± π + π − decay in Section 7. In the study of thesystematic uncertainties affecting the K ± → π ± π π de-cay parameters we fix the values of the K ± → π ± π + π − decay parameters g, h, k in the BB formulation to theirbest fit values shown in Table 7.The fit interval for the presentation of the final re-sults (bins 26–226 of width 0.00015 (GeV/ c ) , with bin51 centred at 4 m π + ) has been chosen to minimize the totalexperimental error of the measured a − a . If the upperlimit of the fit region, s max , is increased, the statisticalerror decreases. All our fits give good χ up to rather high s max values where the acceptance is small . However, thesystematic error increases with s max , especially the contri-butions from trigger inefficiency and non-linearity of theLKr response. The total experimental error on a − a ,obtained by adding quadratically the statistical and sys-tematic error, has a minimum when the upper limit of thefit interval corresponds to bin 226. The detector acceptance to K ± → π ± π π decays de-pends strongly on the position of the K ± decay vertexalong the nominal beam axis, Z , so the Z distribution At the maximum kinematically allowed s value the π ± isat rest in the K ± decay frame. In this case, it moves along the K ± flight path inside the beam vacuum tube and cannot bedetected. Near this maximum s value the acceptance is verysensitive to the precise beam shape and position due to the π ± narrow angular distribution, and it is difficult to reproduce itin the Monte-Carlo simulation.2 J.R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays provides a sensitive tool to control the quality of the ac-ceptance simulation.Fig. 6 shows the comparison between the data andMonte-Carlo simulated Z distributions. The small differ-ence between the shapes of the two distributions in theregion Z < Z region in thearea where the acceptance drops because of the increasingprobability for the charged pion track to cross the spec-trometer too close to the event COG. The effect of thisacceptance difference has been checked by introducing asmall mismatch in the track radius cuts between real andsimulated data, and also by applying small changes to theLKr energy scale (equivalent to shifts of the event Z posi-tion similar to the effect observed in the acceptance). Thecorresponding small changes of the fit results are consid-ered as the acceptance related contribution to the sys-tematic uncertainties (quoted as Acceptance(Z) in Tables11–14). Z (cm) ab e v e n t s /180 c m Fig. 6. K ± → π ± π π decay Z distributions for data andMonte-Carlo simulation. a : Experimental (solid circles) andsimulated (histogram) distributions, normalized to experimen-tal statistics. b : Ratio between the experimental and simu-lated distributions. The nominal position of LKr front face isat Z = 12108 . The Monte Carlo sample from which the acceptanceand resolution effects used in the fits are derived, is gen-erated under the assumption that the K ± → π ± π π matrix element, M , depends only on u . We have stud- ied the sensitivity of the fit results to the presence of a v -dependent term by adding to |M| a term of the form k v or k ′ Re ( M ) v , consistent with the observed v depen-dence in the data. The largest variations of the fit resultsare shown in Tables 11–14 as the contributions to the sys-tematic uncertainties arising from the simplified matrixelement used in the Monte Carlo (they are quoted as Ac-ceptance(V)). During data taking in 2003 and 2004 some changes tothe trigger conditions were introduced following improve-ments in detector and electronics performance. In addi-tion, different minimum bias triggers with different down-scaling factors were used. As a consequence, trigger effectshave been studied separately for the data samples takenduring seven periods of uniform trigger conditions. De-tails of the trigger efficiency for the K ± → π ± π π decayevents are given in [1,3].As described in Section 2, K ± → π ± π π events wererecorded by a first level trigger using signals from the scin-tillator hodoscope (Q1) and LKr (NUT), followed by a sec-ond level trigger using drift chamber information (MBX).Events were also recorded using other triggers with differ-ent downscaling factors for different periods: a minimumbias NUT trigger (ignoring both Q1 and MBX); and aminimum bias Q1*MBX trigger (ignoring LKr informa-tion). Using the event samples recorded with these down-scaled triggers, and selecting K ± → π ± π π decays asdescribed in section 2, it was possible to measure sepa-rately two efficiencies:1. the efficiency of the minimum bias Q1*MBX triggerusing the event sample recorded by the minimum biasNUT trigger;2. the efficiency of the minimum bias NUT trigger usingthe events recorded by the minimum bias Q1*MBXtrigger.These two efficiencies were multiplied together to ob-tain the full trigger efficiency.The measured efficiencies for seven different periodsare shown in Fig. 7 as a function of the reconstructed M .In the initial data taking periods the samples of minimumbias events were rather small, resulting in relatively largestatistical errors. However, we can improve the estimateof the trigger efficiency for these periods under the ad-ditional assumption that it is a smooth function of M (this assumption is justified by the fact that no anomalyis expected nor observed in its behaviour). We find that a2-nd degree polynomial p + p ∗ ( M − m ) + p ∗ ( M − m ) (6)describes well the trigger efficiency over the M fit inter-val. Moreover, over this interval the dependence is almostlinear, so we expect a negligible effect on the determina-tion of the scattering lengths. .R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays 13 abcdefg M (GeV/ c ) ǫ Fig. 7.
Trigger efficiency ǫ as a function of M for the differ-ent time periods with different trigger conditions ( a – c : 2003, d – g : 2004). The errors are defined by the available statistics ofthe event samples recorded by the two minimum bias triggers. Table 10.
Trigger efficiency corrections for the best fit param-eters of fits CI and CI χ of Table 6.fit CI fit CI χ g . . h . . a − a ) m + − . a m + - 0 . a m + . f atom . − . Fits are made separately for each of the data takingperiods shown in Fig. 7. In a first fit, the M distributionfrom the data and the corresponding trigger efficiency arefitted simultaneously, and the theoretical M distribu-tion, distorted by the acceptance and resolution effects, ismultiplied by the corresponding trigger efficiency, as pa-rameterized using Eq. (6). The fit to the M distributionalone is then repeated under the assumption of a fully ef-ficient trigger, and the results of the two fits are comparedto obtain the trigger efficiency correction and its effectiveerror. As an example, Table 10 lists the trigger correctionsto the best fit parameters of fits CI and CI χ (see Table 6).The trigger corrections are all in agreement with zerowithin their statistical uncertainties. For a conservativeestimate, we combine in quadrature the corrections andtheir errors to obtain the trigger efficiency contributionto the systematic uncertainties of the best fit results (seeTables 11–14). As described in Section 2, the π π invariant mass M isdetermined using only information from the LKr calorime-ter (photon energies and coordinates of their impact points).The measurement of the scattering lengths relies, there-fore, on the correct description of the M resolution inthe Monte Carlo simulation.In order to check the quality of the LKr energy reso-lution we cannot use the π mass peak in the two-photoninvariant mass distribution, because the nominal π mass[12] is used in the reconstruction of the two-photon decayvertex (see Section 2). We find that a convenient variablewhich is sensitive to all random fluctuations of the LKrresponse, and hence to its energy resolution, is the ra-tio m π /m π , where m π and m π are the measured two-photon invariant masses for the more and less energetic π , respectively, in the same K ± → π ± π π decay. Thedistributions of this ratio for real and simulated events areshown in Fig. 8. One can see that the width of the dis-tribution for simulated events is slightly larger than thatof the data: the rms value of the simulated distribution is0.0216, while it is 0.0211 for the data.In order to check the sensitivity of the fit results toa resolution mismatch of this size, we have smeared themeasured photon energies in the data by adding a ran-dom energy with a Gaussian distribution centred at zeroand with σ = 0 .
06 GeV (see Fig. 8). Such a change in-creases the rms value of the m π /m π distribution from0.0211 to 0.0224. A fit is then performed for the data sam-ple so modified, and the values of the fit parameters arecompared with those obtained using no energy smearing.The artificial smearing of the photon energies describedabove introduces random shifts of the fit parameters withintheir statistical errors. In order to determine these shiftsmore precisely than allowed by the statistics of a singlefit, we have repeated the fit eleven times using for eachfit a data sample obtained by smearing the original pho-ton energies with a different series of random numbers,as described in the previous paragraph. The shifts of thefit parameters, averaged over the eleven fits, represent thesystematic effects, while the errors on those average val-ues are the corresponding uncertainties. Conservatively,the quadratic sum of the shifts and their errors is quotedas “LKr resolution” in Tables 11–14. In order to study possible non-linearity effects of the LKrcalorimeter response to low energy photons, we select π pairs from K ± → π ± π π events using the following cri-teria:1. both π → γγ decays must be close to symmetrical(0 . < E γ E π < . π (denoted as π ) must fulfil therequirement22 GeV < E π <
26 GeV. ππ scattering lengths from a study of K ± → π ± π π decays m π /m π ab e v e n t s /0 . Fig. 8.
Distributions of the measured ratio m π /m π (seetext) for the data of 2004. a : solid circles - data events; opencircles - data events with the LKr cluster energies artificiallysmeared as described in the text; histogram - simulated distri-bution, normalized to data statistics. b : corresponding ratiosof data and simulated distributions. For the π pairs selected in such way we define the ratioof the two-photon invariant masses, r = m π /m π , where π is the lower energy π . Fig. 9 shows the average ratio h r i as a function of E π / π → γγ decays E π / , h r i depends on the lowest pion energyeven in the case of perfect LKr linearity. However, asshown in Fig. 9, for E π / . h r i for simulated events are systematically above those of thedata, providing evidence for the presence of non-linearityeffects of the LKr response at low energies.To study the importance of these effects, we modify allsimulated events to account for the observed non-linearitymultiplying each photon energy by the ratio h r Data ih r MC i , where h r Data i and h r MC i are the average ratios for data and sim-ulated events, respectively. As shown in Fig. 9, the valuesof h r i for the sample of simulated events so modified arevery close to those of the data. The small shifts of the bestfit parameters obtained using these non-linearity correc-tions are taken as contributions to the systematic uncer- The small resolution mismatch between data and simulatedevents introduces a negligible effect here. tainties in Tables 11–14, where they are quoted as “LKrnon-linearity”. E π / h r i Fig. 9.
Average r = m π /m π versus E π / π pairs from K ± → π ± π π decays selected as described in the text. Solidcircles: data; crosses: simulated events; open circles: simulatedevents corrected for non-linearity (see text). The π energy isdivided by 2 to compare with the γ energy for symmetric π decays. The π ± interaction in the LKr may produce multiple en-ergy clusters which are located, in general, near the impactpoint of the π ± track and in some cases may be identifiedas photons. To reject such “fake” photons a cut on thedistance d between each photon and the impact point ofany charged particle track at the LKr front face is imple-mented in the event selection, as described in Section 2.In order to study the effect of these “fake” photons on thebest fit parameters we have repeated the fits by varyingthe cut on the distance d between 10 and 25 cm in the se-lection of both data and simulated K ± → π ± π π events.The largest deviations from the results obtained with thedefault cut value ( d =15 cm) are taken as contributions tothe systematic uncertainties (see Tables 11–14). The Monte Carlo program includes a complete simula-tion of the beam magnet system and collimators with the .R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays 15 P K (GeV/ c ) ab e v e n t s /0 . G e V / c Fig. 10.
Distributions of the reconstructed K ± momentum P K from the data and from Monte-Carlo simulation (2003data). a : solid circles – experimental data; dashed line his-togram – simulation; solid line histogram – simulation withthe corrected K ± spectrum width. b : corresponding ratios ofdata and simulated spectra. purpose of reproducing the correlation between the in-cident K ± momenta and trajectories. However, the ab-solute beam momentum scale cannot be modelled withthe required precision, hence we tune the average value tothe measured ones for each continuous data taking period(“run”) using K ± → π ± π + π − events which are recordedduring data taking, and also simulated by the Monte Carloprogram.After this adjustment, a residual systematic differencestill exists between the measured and simulated K ± mo-mentum distributions, as shown in Fig. 10. In order tostudy the sensitivity of the best fit parameters to this dis-tribution, we have corrected the width of the simulated K ± momentum distribution to reproduce the measureddistribution (see Fig. 10) using a method based on the re-jection of simulated events. To minimize the random effectof this rejection, a fraction of events has also been removedfrom the uncorrected MC sample in such a way that thecorrected and uncorrected MC samples have a maximumoverlap of events and the same statistics. The correspond-ing changes of the best fit parameters are included in thecontributions to the systematic uncertainties and quotedas “ P K spectrum” in Tables 11–14.In order to take into account changes of running condi-tions during data taking, the number of simulated K ± → π ± π π events for each run should be proportional to thecorresponding number of events in the data. However, be-cause of changes in the trigger efficiency and in acceptance related to minor hardware problems, the ratio between thenumber of simulated and real events varies by a few per-cent during the whole data taking period. In order to studythe effect of the small mismatch between the two sampleson the best fit parameters, we have made them equal runby run by a random rejection of selected events. The corre-sponding shifts of the best fit parameters are considered asa Monte Carlo time dependent systematic error, and arelisted in Tables 11–14, where they are quoted as “MC(T)”. Table 11.
Fit parameter systematic uncertainties in units of10 − for the CI formulation with electromagnetic corrections(fit CI in Table 6). The factor m + which should multiply thescattering lengths is omitted for simplicity.Source g h a a a − a f atom Acceptance(Z) 22 17 11 14 3 1Acceptance(V) 9 3 5 6 1 3Trigger efficiency 10 17 22 30 8 11LKr resolution 4 2 11 17 7 56LKr nonlinearity 2 21 39 49 11 5 P K spectrum 5 3 11 23 12 8MC(T) 3 2 4 1 5 25 k error 8 6 3 4 1 1Hadronic showers 9 3 3 13 9 20Total systematic 29 33 49 67 22 66Statistical 22 18 56 92 45 93 Table 12.
Fit parameter systematic uncertainties in units of10 − for the CI formulation with electromagnetic correctionsand with the ChPT constraint (fit CI χ in Table 6). The factor m + which should multiply the scattering lengths is omitted forsimplicity.Source g h a a a − a f atom Acceptance(Z) 24 14 4 1 3 9Acceptance(V) 8 4 2 0 2 0Trigger efficiency 13 15 8 2 6 10LKr resolution 0 2 2 0 1 46LKr nonlinearity 12 13 13 3 10 31 P K spectrum 0 0 2 1 2 5MC(T) 2 2 6 1 4 24 k error 7 7 1 0 0 2Hadronic showers 5 3 4 1 3 19Total systematic 33 26 18 4 14 65Statistical 9 8 28 6 21 77 The most important source of external error is the valueof | A + | , obtained from the measured ratio of the K ± → π ± π + π − and K ± → π ± π π decay rates, R = 3 . ± .
050 [12]. This ratio is proportional to | A + | , so δ | A + | / | A + | = 0 . δR ) /R. ππ scattering lengths from a study of K ± → π ± π π decays Table 13.
Fit parameter systematic uncertainties in units of10 − for the BB formulation with electromagnetic corrections(fit BB in Table 6). The factor m + which should multiply thescattering lengths is omitted for simplicity.Source g h a a a − a f atom Acceptance(Z) 31 21 16 20 4 0Acceptance(V) 6 1 7 8 1 4Trigger efficiency 26 22 29 39 10 13LKr resolution 10 9 21 29 9 60LKr nonlinearity 34 36 56 67 12 1 P K spectrum 12 11 18 32 13 10MC(T) 2 1 4 1 5 25 k error 5 5 4 6 2 1Hadronic showers 2 4 8 18 10 20Total systematic 56 50 72 94 25 70Statistical 47 46 92 129 48 97 Table 14.
Fit parameter systematic uncertainties in units of10 − for the BB formulation with electromagnetic correctionsand with the ChPT constraint (fit BB χ in Table 6). The factor m + which should multiply the scattering lengths is omitted forsimplicity.Source g h a a a − a f atom Acceptance(Z) 24 14 4 1 3 9Acceptance(V) 8 4 2 1 2 0Trigger efficiency 14 16 9 2 7 8LKr resolution 0 1 2 1 2 46LKr nonlinearity 12 13 13 3 10 31 P K spectrum 0 0 2 1 2 5MC(T) 2 2 6 1 4 24 k error 7 7 0 0 0 2Hadronic showers 5 3 4 1 3 17Total systematic 33 26 18 4 14 64Statistical 9 9 32 8 24 77 The typical | A + | uncertainty is, therefore, δ | A + | ≈ . | A + | within its uncertainty. Each fit is redonetwice changing the | A + | value by + δ | A + | and − δ | A + | . Onehalf of the variation of the fit parameters correspondingto these two fits is listed in Table 15, and is taken as theexternal contribution to the full parameter uncertainty. Table 15.
Contributions to the fit parameter uncertainties (inunits of 10 − ) due to the external error δ | A + | .Fit g h a m + a m + ( a − a ) m + f atom CI CI χ BB BB χ ππ scattering lengths: final results The BB formulation with radiative corrections [9] providespresently the most complete description of rescatteringeffects in K → π decay. For this reason we use the resultsfrom the fits to this formulation to present our final resultson the ππ scattering lengths:( a − a ) m + = 0 . ± . stat. ) ± . syst. ) ± . ext. ); (7) a m + = − . ± . stat. ) ± . syst. ) ± . ext. ) . (8)The values of the ππ scattering lengths, ( a − a ) m + and a m + , are obtained from fit BB of Table 6. In additionto the statistical, systematic and external errors discussedin the previous sections, these values are affected by atheoretical uncertainty. We note that, at the level of ap-proximation of the BB and CI amplitude expression usedin the fits, a difference of 0.0088(3.4%)is found betweenthe values of ( a − a ) m + and of 0.015(62%) for a m + .For the sake of comparison with other independent resultson the ππ scattering lengths we take into account thesedifferences as theoretical uncertainty.From the measurement of the lifetime of pionium bythe DIRAC experiment at the CERN PS [18] a value of | a − a | m + = 0 . +0 . − . was deduced which agrees,within its quoted uncertainty, with our result (it should benoted that this measurement provides only a determina-tion of | a − a | , while our measurement of K ± → π ± π π decay is also sensitive to the sign).Previous determinations of the ππ scattering lengthshave also relied on the measurement of K ± → π + π − e ± ν e ( K e ) decay. Fig. 11 compares our results (Eqs. (7, 8))with the results from the most recent analysis of a largesample of K e decays, also collected by the NA48/2 col-laboration [26].If we use the ChPT constraint (see Eq. (5)), we obtain(see fit BB χ of Table 6)( a − a ) m + = 0 . ± . stat. ) ± . syst. ) ± . ext. ) . (9)For this fit the theoretical uncertainty affecting the valueof a − a is estimated to be ±
2% ( ± . K ± → π ± π π decay amplitude [27] in the frame of theCI formulation [7] (the goals of this study included a moreprecise estimate of the theoretical uncertainties affectingthe ππ scattering lengths). This theoretical uncertaintyis smaller than that affecting the result of the fit with a − a and a as free parameters, because the theoreti-cal uncertainty on a becomes negligible when using theChPT constraint.The 68% confidence level ellipse corresponding to theresult given by Eq. (9) is also shown in Fig. 11, togetherwith a fit to the K e data which uses the same ChPTconstraint. The a − a vs a correlation coefficient for this .R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays 17 -0.06-0.05-0.04-0.03-0.02-0.0100.01 0.24 0.25 0.26 0.27 0.28 0.29 0.3 a m + ( a − a ) m + NA48/2 K e NA48/2 cusp DIRAC ✲✛ Fig. 11.
68% confidence level ellipses corresponding to thefinal results of the present paper (small solid line ellipse: fitwith the ChPT constraint (see Eq. (5)); large solid line ellipse:fit using a − a and a as independent parameters), and from K e decay [26] (small dashed line ellipse: fit with the ChPTconstraint; large dashed line ellipse: fit using a and a as in-dependent parameters). Vertical lines: central value from theDIRAC experiment [18] (dotted line) and error limits (dashedlines). The 1-sigma theoretical band allowed by the ChPT con-straint (see Eq. (5)) is shown by the dotted curves. figure has been calculated taking into account statistical,systematic and external covariances. Its value is − . − .
839 (see Table9).
Summary and conclusions
We have studied the π π invariant mass distribution mea-sured from the final sample of 6 . × K ± → π ± π π fully reconstructed decays collected by the NA48/2 exper-iment at the CERN SPS. As first observed in this exper-iment [4], this distribution shows a cusp-like anomaly at M = 2 m + which is interpreted as an effect due mainly tothe final state charge-exchange scattering process π + π − → π π in K ± → π ± π + π − decay [5,6].Good fits to the M distribution have been obtainedusing two different theoretical formulations [7] and [8,9],all including next-to-leading order rescattering terms. Weuse the results of the fit to the formulation which includesradiative corrections [9] to determine the difference a − a ,which enters in the leading-order rescattering term, and a , which enters in the higher-order rescattering terms,where a and a are the I = 0 and I = 2 S-wave ππ scattering lengths, respectively. These values are given inEqs. (7) and (8), while Eq. (9) gives the result from a fitthat uses the constraint between a and a predicted byanalyticity and chiral symmetry [21] (see Eq. (5)).As discussed in Section 8, our results agree with thevalues of the ππ scattering lengths obtained from the studyof K e decay [26], which have errors of comparable magni-tude. The value of a − a as quoted in Eqs. (7) and (9) arealso in agreement with theoretical calculation performedin the framework of Chiral Perturbation Theory [28,29],which predict ( a − a ) m + = 0 . ± . K ± → π ± π + π − and K ± → π ± π π decays. In the case of K ± → π ± π + π − decay there is no cusp singularity in the physicalregion because the invariant mass of any pion pair is al-ways ≥ m + . As a consequence, rescattering effects can bereabsorbed in the values of the Dalitz plot parameters g , h , k obtained from fits without rescattering, such as thosediscussed in ref. [14]. On the contrary, a correct descriptionof the K ± → π ± π π Dalitz plot is only possible if rescat-tering effects are taken into account to the next-to-leadingorder. Furthermore, the values of the parameters g , h , k which describe the weak K ± → π ± π π amplitude attree level depend on the specific theoretical formulation ofrescattering effects used to fit the data.In a forthcoming paper we propose an empirical pa-rameterization capable of giving a description of the K ± → π ± π π Dalitz plot, which does not rely on any ππ rescattering mechanisms, but nevertheless reproducesthe cusp anomaly at M = 2 m + . This parameterizationis useful for computer simulations of K ± → π ± π π decayrequiring a precise description of all Dalitz plot details. Acknowledgements
We gratefully acknowledge the CERN SPS accelerator andbeam-line staff for the excellent performance of the beam.We thank the technical staff of the participating laborato-ries and universities for their effort in the maintenance andoperation of the detectors, and in data processing. We aregrateful to G. Isidori for valuable discussions on the fittingprocedure. It is also a pleasure to thank G. Colangelo, J.Gasser, B. Kubis and A. Rusetsky for illuminating discus-sions and for providing the computer code to calculate the K ± → π ± π + π − and K ± → π ± π π decay amplitudes inthe framework of the Bern-Bonn formulation. Appendix: Measurement of the k parameter In order to measure the k parameter which describes the v dependence of the weak amplitude for K ± → π ± π π decay at tree level (see Eq.(3)), we have performed fitsto the π ± π π Dalitz plot. Because of technical complica-tions associated with two-dimensional fits, we do not usethe results of these fits to determine the scattering lengths,but focus mainly on the measurement of k .We use two independent methods. In the first method,the Dalitz plot is described by two independent variables: ππ scattering lengths from a study of K ± → π ± π π decays -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 cos( θ ) e v e n t s /0 . Fig. 12.
Projections of the K ± → π ± π π Dalitz plot ontothe cos( θ ) axis (see text). Full circles: data. Dashed (full) line:best fit to the CI formulation [7] with k = 0 ( k = 0 . M and cos( θ ), where θ is the angle between the mo-mentum vectors of the π ± and one of the two π in therest frame of the π pair (with this choice of variablesthe Dalitz plot has a rectangular physical boundary). The M fit interval is identical to the one used for the one-dimensional fits described in Sections 4.1, 4.2, but the binwidth is increased from 0.00015 to 0.0003 (GeV/ c ) , andfour consecutive bins around M = 4 m are excluded.The cos( θ ) variable is divided into 21 equal bins from − .
05 to 1 .
05, but only the interval − . < cos( θ ) < . × × ×
21) is obtained from the Monte Carlo simulationdescribed in Section 3. This matrix is used to transformthe true simulated Dalitz plot into an expected one whichcan be directly compared with the measured Dalitz plotat each step of the χ minimization.Fits to the CI formulation [7] are performed with afixed value a = − . k parameter is kept fixedat zero, the fit quality is very poor ( χ = 4784 . k is used as a free param-eter in the fit, the best fit value is k = 0 . ± . χ = 1223 . θ ) axis.A simultaneous fit to the Dalitz plot from K ± → π ± π π decay and to the M ±± distribution from K ± → π ± π + π − decay is performed in the frame of the BB formu-lation [8] using the constraint between a and a predictedby analyticity and chiral symmetry (see Eq.(5)). The best fit gives k = 0 . ± . χ = 1975 . k value soobtained and that obtained from a fit to the CI formula-tion [7] is due to the rescattering contributions which aredifferent in the two formulations. When radiative correc-tions are included in the fit [9], k is practically unchanged(its best fit value is 0.008495), demonstrating that electro-magnetic corrections have a negligible effect on its deter-mination.The second fitting method is based on the event weight-ing technique. In order to study the size of the trigger ef-fect on the fit parameters, we use a fraction of the datataken with uniform trigger conditions and associated witha large minimum bias event sample which allows a preciseevaluation of the trigger efficiency.The Dalitz plot is described by the u and | v | vari-ables (see Eq.(3)), and the intervals − . < u < . | v | < . < | v | < . v max , u < . K ± → π ± π π decay probability by the factor 1.055 in the in-terval | M − m | < . c ) . The fits areperformed with a fixed value a = − . ∼ . × simulated events generated with a simple matrix element M sim without rescattering effects and with fixed values of g , h and k . At every iteration in the χ minimization,each simulated event is reweighted by the ratio |M| |M sim | ,where M is the matrix element which includes rescat-tering and is calculated with the new fitting parameters,and both M and M sim are calculated at the generated u , | v | values. The simulated events so weighted are thenrebinned, and their two-dimensional u, | v | distribution iscompared with that of the data.A good fit ( χ = 1166 for 1257 degrees of freedom) isobtained when the trigger efficiency is taken into account,giving k = 0 . ± . χ value is somewhat worse ( χ = 1276) and weobtain k = 0 . ± . k by ≈ . k = (0 . . / . k value, and conservatively take one half of thedifference between them as the contribution to the sys-tematic error due to the different fitting techniques. Asmentioned above, the trigger correction shifts the k cen-tral value by − . .R. Batley et al.: Determination of the S-wave ππ scattering lengths from a study of K ± → π ± π π decays 19 with a partial data sample, we also add it in quadratureto the systematic error. So our measurement of k in theframe of the CI rescattering formulation [7] gives k = 0 . ± . stat. ) ± . syst. )= 0 . ± . . For most of the one-dimensional fits discussed in thepresent paper we do not apply any trigger correction, sohere we use the effective value k = 0 . k = 0 . k is kept fixed in thosefits, we check the variations of all the best fit parametersby varying k within the limits defined by its full error.These variations are listed in Tables 11–14, where they aredenoted as “ k error”. References
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