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Precise Measurements of Direct CP Violation, CPT Symmetry, and Other

Parameters in the Neutral Kaon System

E. Abouzaid,4 M. Arenton,11 A.R. Barker,5, ∗ M. Barrio,4 L. Bellantoni,7 E. Blucher,4 G.J. Bock,7

C. Bown,4 E. Cheu,1 R. Coleman,7 M.D. Corcoran,9 B. Cox,11 A.R. Erwin,12 C.O. Escobar,3 A. Glazov,4, †

A. Golossanov,11 R.A. Gomes,3 P. Gouffon,10 J. Graham,4 J. Hamm,12 Y.B. Hsiung,7 D.A. Jensen,7 R. Kessler,4

K. Kotera,8 J. LaDue,5 A. Ledovskoy,11 P.L. McBride,7 E. Monnier,4, ‡ H. Nguyen,7 R. Niclasen,5

D.G. Phillips II,11 V. Prasad,4 X.R. Qi,7 E.J. Ramberg,7 R.E. Ray,7 M. Ronquest,11 A. Roodman,4

E. Santos,10 P. Shanahan,7 P.S. Shawhan,4 W. Slater,2 D. Smith,11 N. Solomey,4 E.C. Swallow,4, 6

S.A. Taegar,1 P.A. Toale,5 R. Tschirhart,7 Y.W. Wah,4 J. Wang,1 H.B. White,7 J. Whitmore,7 M. J. Wilking,5

B. Winstein,4 R. Winston,4 E.T. Worcester,4 T. Yamanaka,8 E. D. Zimmerman,5 and R.F. Zukanovich10

1University of Arizona, Tucson, Arizona 857212University of California at Los Angeles, Los Angeles, California 90095

3Universidade Estadual de Campinas, Campinas, Brazil 13083-9704The Enrico Fermi Institute, The University of Chicago, Chicago, Illinois 60637

5University of Colorado, Boulder, Colorado 803096Elmhurst College, Elmhurst, Illinois 60126

7Fermi National Accelerator Laboratory, Batavia, Illinois 605108Osaka University, Toyonaka, Osaka 560-0043 Japan

9Rice University, Houston, Texas 7700510Universidade de São Paulo, São Paulo, Brazil 05315-970

11The Department of Physics and Institute of Nuclear and ParticlePhysics, University of Virginia, Charlottesville, Virginia 22901

12University of Wisconsin, Madison, Wisconsin 53706(Dated: November 2, 2010)

We present precise tests of CP and CPT symmetry based on the full dataset of K → ππ decayscollected by the KTeV experiment at Fermi National Accelerator Laboratory during 1996, 1997, and1999. This dataset contains 16 million K → π0π0 and 69 million K → π+π− decays. We measurethe direct CP violation parameter Re(ǫ′/ǫ) = (19.2 ±2.1)×10−4. We find the KL-KS mass difference∆m = (5270 ± 12)×106 ~s−1 and the KS lifetime τS = (89.62 ± 0.05)×10

−12 s. We also measureseveral parameters that test CPT invariance. We find the difference between the phase of the indirectCP violation parameter, ǫ, and the superweak phase, φǫ − φSW = (0.40 ± 0.56)

◦. We measure thedifference of the relative phases between the CP violating and CP conserving decay amplitudes forK → π+π− (φ+−) and for K → π

0π0 (φ00), ∆φ = (0.30± 0.35)◦. From these phase measurements,

we place a limit on the mass difference betweenK0 andK0, ∆M < 4.8 × 10−19 GeV/c2 at 95% C.L.These results are consistent with those of other experiments, our own earlier measurements, andCPT symmetry.

PACS numbers: 11.30.Er, 13.25.Es, 14.40.Df

I. INTRODUCTION

Since the 1964 discovery of CP violation inKL → π+π− decay[1], significant experimental effort hasbeen devoted to understanding the mechanism of CP vi-olation. Early experiments showed that the observed ef-fect was due mostly to a small asymmetry between theK0 → K0 and K0 −→ K0 transition rates, which isreferred to as indirect CP violation. Decades of addi-tional effort were required to demonstrate the existenceof direct CP violation in a decay amplitude. This paperreports the final measurement of direct CP violation by

∗Deceased.†Permanent address DESY, Hamburg, Germany‡Permanent address C.P.P. Marseille/C.N.R.S., France

the KTeV Experiment (E832) at Fermilab.Direct CP violation can be detected by comparing the

level of CP violation for different decay modes. The pa-rameters ǫ and ǫ′ are related to the ratio of CP violatingto CP conserving decay amplitudes for K → π+π− andK → π0π0:

η+− ≡ A (KL → π+π−)

A(KS → π+π−)≈ ǫ+ ǫ′,

η00 ≡A(

KL → π0π0)

A(

KS → π0π0) ≈ ǫ− 2ǫ′,

(1)

where ǫ is a measure of indirect CP violation, which iscommon to all decay modes. The relation among thecomplex parameters η+−, η00, ǫ, and ǫ

′ is illustrated inFig. 1.If CPT symmetry holds, the phase of ǫ is equal to the

“superweak” phase:

φSW ≡ tan−1 (2∆m/∆Γ) , (2)

FERMILAB-PUB-10-440-E

Operated by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy.

http://arxiv.org/submit/0138525/pdf

2

0

0.5

1

1.5

0 0.5 1 1.5Re x10-3

Im x

10-3

φε

φ00

φ+-

ε'-2ε'

εη 00 η +-

FIG. 1: Diagram of CP violating kaon parameters. For thisillustration, the ǫ parameter has the central value measured byKTeV and the value of ǫ′ is scaled by a factor of 50. Althoughthey appear distinct in this diagram, note that φ+− and φ00are consistent with each other within experimental errors.

where ∆m ≡ mL−mS is the KL-KS mass difference and∆Γ = ΓS − ΓL is the difference in the decay widths.The quantity ǫ′ is a measure of direct CP violation,

which contributes differently to the π+π− and π0π0 de-cay modes, and is proportional to the difference be-tween the decay amplitudes for K0 → π+π−(π0π0) andK0 → π+π−(π0π0). Measurements of ππ phase shifts [2]show that, in the absence of CPT violation, the phase ofǫ′ is approximately equal to that of ǫ. Therefore, Re(ǫ′/ǫ)is a measure of direct CP violation and Im(ǫ′/ǫ) is a mea-sure of CPT violation.Experimentally, Re(ǫ′/ǫ) is determined from the double

ratio of the two pion decay rates of KL and KS:

Γ(KL → π+π−) /Γ(KS → π+π−)Γ(

KL → π0π0)

/Γ(

KS → π0π0)

=

∣

∣

∣

∣

η+−η00

∣

∣

∣

∣

2

≈ 1 + 6Re(ǫ′/ǫ). (3)

For small |ǫ′/ǫ|, Im(ǫ′/ǫ) is related to the phases of η+−and η00 by

φ+− ≈ φǫ + Im(ǫ′/ǫ),φ00 ≈ φǫ − 2Im(ǫ′/ǫ),∆φ ≡ φ00 − φ+− ≈ −3Im(ǫ′/ǫ) .

(4)

The Standard Model accommodates both direct andindirect CP violation [3–5]. Most recent Standard Modelpredictions for Re(ǫ′/ǫ) are less than 30 × 10−4 [6–15];however, there are large hadronic uncertainties in thesecalculations. Experimental results have established that

Re(ǫ′/ǫ) is non-zero [16–20]. The previous result fromKTeV, which was based on about half of the KTeVdataset, is Re(ǫ′/ǫ) = (20.7± 2.8)× 10−4[20]. This resultwas published in 2003 and will be referred to in this textas “KTeV03.” The result based on all data from NA48at CERN is Re(ǫ′/ǫ) = (14.7± 2.2)× 10−4[19].This paper reports the final measurement of Re(ǫ′/ǫ)

by KTeV. The measurement is based on 85 million re-constructed K → ππ decays collected in 1996, 1997, and1999. This full sample is two times larger than, andcontains, the sample on which the KTeV03 results arebased. We also present measurements of the kaon pa-rameters ∆m and τS , and tests of CPT symmetry basedon measurements of ∆φ and φǫ − φSW . Using our phasemeasurements, we place a limit on the mass differencebetween K0 and K0.For this analysis we have made significant improve-

ments to the data analysis and the Monte Carlo sim-ulation. The full dataset, including the data used inKTeV03, has been reanalyzed using the improved re-construction and simulation. These results supersedethe previously published KTeV03 results[20], which werebased on data from 1996 and 1997.This paper describes the KTeV experiment in Sec. II,

the analysis technique in Sec. III, and the extractionof physics results in Sec. IV. We emphasize changesand improvements since the KTeV03 publication. Wewill refer to [20] for some details that have not changedsince KTeV03. Section V presents the final KTeV re-sults, including correlations between the parameters andcrosschecks of the results. Section VI is a summary anddiscussion of the results. Appendix A contains a discus-sion of the dependence of our measurements on details ofkaon regeneration.

II. MEASUREMENT TECHNIQUE AND

APPARATUS

A. Overview

The measurement of Re(ǫ′/ǫ) requires a source of KLandKS decays, and a detector to reconstruct the charged(π+π−) and neutral (π0π0) final states. The strategyof the KTeV experiment is to produce two identical KLbeams, and then to pass one of the beams through a “re-generator” that is about two hadronic interaction lengthslong. The beam that passes through the regenerator iscalled the regenerator beam, and the other beam is calledthe vacuum beam. The regenerator creates a coherent|KL〉+ρ |KS〉 state, where ρ, the regeneration amplitude,is a physical property of the regenerator. The regener-ator is designed such that most of the K → ππ decaysdownstream of the regenerator are from the KS compo-nent. The charged spectrometer is the primary detectorfor reconstructing K → π+π− decays and the pure Ce-sium Iodide (CsI) calorimeter is used to reconstruct thefour photons from K → π0π0 decays. A Monte Carlo

3

simulation is used to correct for the average acceptancedifference between K → ππ decays in the two beams,which results from the very different KL and KS life-times. The decay-vertex distributions provide a criticalcheck of the simulation. The measured quantities arethe vacuum-to-regenerator “single ratios” forK → π+π−and K → π0π0 decay rates. These single ratios are pro-portional to |η+−/ρ|2 and |η00/ρ|2, respectively, and theratio of these two quantities gives Re(ǫ′/ǫ) via Eq. 3.

B. KTeV Experiment

The KTeV kaon beams are produced by a beamlineof magnets, absorbers, and collimators that act on theproducts of a proton beam incident on a fixed target.The 800 GeV/c proton beam, provided by the FermilabTevatron, has a 53 MHz RF structure so that the pro-tons arrive in ∼1 ns wide “buckets” at 19 ns intervals.This beam is incident on a beryllium oxide (BeO) targetthat is about one proton interaction length long. Im-mediately downstream of the target, the beam consistsof protons, muons, and other charged particles, neutralkaons, neutrons, photons, and hyperons. This beam iscollimated into two beams and the non-kaon componentis reduced by magnets and absorbers in a 100 meter longbeamline. At the start of the fiducial decay region, 120 mdownstream of the target, the average kaon momentumis about 70 GeV/c. The neutron-to-kaon ratio is 1.3 inthe vacuum beam and 0.8 in the regenerator beam. TheKTeV beams and the beamline elements that producethem are described in detail in [20].KTeV reconstructs kaon decays that occur in an evac-

uated decay region 90-160 m downstream of the target.Figure 2 is a schematic of the detector. In the KTeV coor-dinate system, the positive x-axis points to the left if theobserver is facing downstream, the positive y-axis pointsup, and the positive z-axis points downstream from thetarget. At the upstream end of the decay region, the re-generator alternates between the two beams to minimizeacceptance differences between decays in the vacuum andregenerator beams. The charged spectrometer and CsIcalorimeter are located downstream of the vacuum win-dow at the end of the decay region. The decay region andprimary detectors are surrounded by a system of photonveto detectors to detect particles with trajectories thatmiss the CsI calorimeter. The major detector elementsare described in more detail in the following paragraphs.The regenerator consists mainly of 84 10× 10× 2 cm3

scintillator modules as seen in Fig. 3a. Its primary pur-pose is to provide KS regeneration, but it is also usedas part of the trigger and veto systems. Each module isviewed by two photomultiplier tubes (PMTs), one fromabove and one from below. The downstream end of theregenerator has a lead-scintillator sandwich called the“regenerator Pb module” (Fig. 3b), which is also viewedby two PMTs. This last module of the regenerator isused to define a sharp upstream edge for the kaon decay

25 cm

120 140 160 180Z = Distance from kaon production target (meters)

Beams

Regenerator BeamVacuum Beam

AnalysisMagnet

Regenerator

DriftChambers

TriggerHodoscope

MuonVeto

Steel

CsI

Mask Anti

Photon VetoesPhotonVetoes

CA

FIG. 2: Schematic of the KTeV detector. Note that the ver-tical and horizontal scales are different.

region in the regenerator beam.The charged spectrometer consists of four drift cham-

bers (DCs) and a dipole magnet. Each drift chambermeasures charged-particle positions in both the x andy views. A chamber consists of two planes of horizontalwires to measure y hit coordinates, and two planes of ver-tical wires to measure x hit coordinates; the two x-planesand the two y-planes are offset to resolve position ambi-guities. The DC planes have a hexagonal cell geometryformed by six field-shaping wires surrounding one sensewire (Fig. 4). There are a total of 1972 sense wires in thefour drift chambers. The cells are 6.35 mm wide, and thedrift velocity is about 50µm/ns. The analyzing magnetimparts a kick of 412 MeV/c in the horizonal plane. Thewell-known kaon mass is used to set the momentum scalewith 10−4 precision.The CsI calorimeter consists of 3100 pure CsI crys-

tals viewed by photomultiplier tubes. The layout of the1.9 × 1.9 m2 calorimeter is shown in Fig. 5. There are2232 2.5 × 2.5 cm2 crystals in the central region, and868 5 × 5 cm2 crystals surrounding the smaller crys-tals. The crystals are all 50 cm (27 radiation lengths)long. Each crystal is wrapped in 12 µm, partially-blackened, aluminized mylar in a manner designed tomake the longitudinal response of each crystal as uniformas possible. The calorimeter is read out by custom dig-itizing electronics (DPMTs) placed directly behind thePMTs[21]. Momentum-analyzed electrons and positronsfrom KL → π±e∓ν (Ke3) decays are used to calibratethe CsI energy scale to 0.02%.An extensive veto system is used to reject events com-

ing from interactions in the regenerator, and to reducebackground from kaon decays into non-ππ final statessuch as KL → π±µ∓ν and KL → π0π0π0. The veto sys-tem consists of a number of lead-scintillator detectors inand around the primary detectors.KTeV uses a three-level trigger to select events. Level 1

uses fast signals from the detector and introduces no

4

1 2 74 75 76 77 78 79 80 81 82 83 84

20 mm

(a)

K→π+π−Edge

K→π0π0Edge

Lead Scint Lead Scint

4 mm5.6 mm

100

mm

(b)

FIG. 3: Diagram of the regenerator. (a) Layout of the 85 re-generator modules, including the lead-scintillator module. (b)Zoomed diagram of the lead-scintillator regenerator module.The PMTs above and below are not shown. The thickness ofeach lead (scintillator) piece is 5.6 (4.0) mm. The transversedimension is 100 mm, and is not drawn with the same scaleas the z-axis. The kaon beam enters from the left. The ar-rows indicate the location and ±1σ uncertainty of the effectiveupstream edges for reconstructed K → π0π0 and K → π+π−

decays for 1999 data.

deadtime. Level 2 is based on more sophisticated pro-cessing from custom electronics and introduces a dead-time of 2-3 µs; when an event passes Level 2 the entiredetector is read out with an average deadtime of 15 µs.Level 3 is a software filter; the processors have enoughmemory that no further deadtime is introduced.

Individual triggers are defined to select K → π+π−and K → π0π0 decays; the trigger efficiencies are studiedusing decays collected in separate minimum-bias triggers.Additional triggers select decays such as KL → π±e∓νand KL → π0π0π0 which are used for calibration and ac-ceptance studies. The “accidental” trigger uses a set ofcounters near the target to collect events based on pri-mary beam activity; these events are uncorrelated withdetector signals that come from the beam particles andare used to model the effects of intensity-dependent ac-cidental activity.

Several changes were made to the KTeV experiment to

Track 1 Track 2

6.35mm

FIG. 4: Diagram of drift chamber geometry showing six fieldwires (open circles) around each sense wire (solid dots). Thesolid lines illustrate the hexagonal cell geometry; they do notrepresent any physical detector element. The vertical dashedlines are separated by 6.35 mm and are used to define thetrack separation cut described in Sec. III B 2.

1.9 mFIG. 5: Beamline view of the KTeV CsI calorimeter, showingthe 868 larger outer crystals and the 2232 smaller inner crys-tals. Each beam hole size is 15 × 15 cm2 and the two beamhole centers are separated by 0.3 m. The positive z directionis into the page.

5

improve data collection efficiency for the 1999 run.

1. Neutral Beams. The proton extraction cycle of theTevatron was improved from 20 second extractions(or “spills”) every 60 seconds in 1996 and 1997 to 40second extractions every 80 seconds in 1999. Themaximum available intensity was∼2 ×1011 protonsper second. In 1999, KTeV chose to take about halfof the data at an average intensity of ∼ 1.6× 1011protons/s and half at a lower average intensity of∼ 1× 1011 protons/s as a systematic cross-check.

2. CsI Calorimeter Electronics. During 1996 and1997 data taking, individual channels of the customreadout electronics for the CsI calorimeter failedoccasionally. These failures account for half of the20% data-taking inefficiency during 1996 and 1997.They also affect the data quality and complicatethe calibration of the calorimeter. All of the cus-tom electronics were re-fabricated and installed inthe CsI calorimeter in preparation for the 1999 run.The re-fabrication of the chips was successful; noCsI calorimeter electronics had to be replaced dur-ing the 1999 run.

3. Drift Chambers. The drift chambers required somerepair due to radiation damage sustained duringdata taking in 1996 and 1997. About half of onedrift chamber was restrung and a second chamberwas cleaned. The drift chamber readout electronicswere modified to allow the system to run at highergain without causing the system to oscillate or trig-ger on noise.

4. Helium Bags. Helium bags are placed between thedrift chambers to minimize the matter seen by theneutral beams after leaving the vacuum decay re-gion and to reduce multiple scattering of chargedparticles. In 1996 and 1997, one of the small he-lium bags was leaky and contained mostly air bythe end of the 1997 run, so it was necessary in theanalysis to correct for the increased multiple scat-tering resulting from this increased material in thedetector. The bags were replaced for 1999. Thisreduction in material traversed by the beams wasoffset by a change in the buffer gas used in the driftchambers; the total ionization loss upstream of theCsI calorimeter was less than 5 MeV in each year.This energy loss occurs mostly in the scintillatorhodoscope just upstream of the CsI calorimeter.

5. Trigger. In 1999, the trigger was adjusted to selectmore KL → π+π−π0 decays for the measurementof kaon flux attenuation in the regenerator beam,called “regenerator transmission.” The improve-ment of this measurement reduces several system-atic uncertainties associated with the fitting proce-dure as described in Sec. IVB.

C. Monte Carlo Simulation

KTeV uses a Monte Carlo (MC) simulation to calculatethe detector acceptance and to model background to thesignal modes. The different KL and KS lifetimes lead todifferent z-vertex distributions in the vacuum and regen-erator beams. We determine the detector acceptance asa function of kaon decay z-vertex and energy, includingthe effects of geometry, detector response, and resolu-tion. To help verify the accuracy of the MC simulation,we collect and study decay modes with approximately tentimes higher statistics than the K → ππ signal samples:KL → π±e∓ν, KL → π0π0π0, and KL → π+π−π0.The Monte Carlo simulates K0/K0 generation at the

BeO target following the parameterization in [22], prop-

agates the coherent K0/K0 state through the absorbersand collimators along the beamline to the decay point,simulates the decay including decays inside the regen-erator, traces the decay products through the detector,and simulates the detector response including the digi-tization of the detector signals and the trigger selection.The parameters of the detector geometry are based bothon data and survey measurements. Many aspects of thetracing and detector response are based on samples ofdetector responses, called “libraries,” that are generatedwith GEANT[23] simulations; the use of libraries keepsthe MC relatively fast. The effects of accidental activityare included in the simulation by overlaying data eventsfrom the accidental trigger onto the simulated events.The Monte Carlo event format is identical to data andthe events are reconstructed and analyzed in the samemanner as data. More details of the simulation are avail-able in [20].Many improvements have been made to the MC simu-

lation since KTeV03[20]. We have improved the simula-tion to include finer details of electromagnetic showeringin the CsI calorimeter and charged particle propagationthrough the detector. These changes are described indetail below.

1. Shower library. For this analysis, the GEANT-based library used to simulate photon and electronshowers in the CsI calorimeter has been improvedto simulate the effects of incident particle angle.The library used for KTeV03 was binned in en-ergy and incident position. There were 325 posi-tion bins and six logarithmic energy bins (2 GeV,4 GeV, 8 GeV, 16 GeV, 32 GeV, and 64 GeV).The effect of angles was approximated by shiftingthe incident position based on the angle of inci-dence. The shower library has now been expandedto include nine angles in x and y (-35 mrad to 35mrad) for photons and 15 angles in x and y (-85mrad to 85 mrad) for electrons. Electron anglesmay be larger than photon angles because of themomentum kick imparted by the analyzing magnet.The position and energy binning is unchanged fromKTeV03. Differences between the library angle and

6

10-3

10-2

10-1

(a)

0.850.9

0.951

1.051.1

1.15

(b)

0.850.9

0.951

1.051.1

1.15

(c)

FIG. 6: Data-MC comparison of fraction of energy in each ofthe 49 CsI crystals in an electron shower. (a) The fractionof energy in each of the 49 CsI crystals in electron showersfor data. (b) KTeV03 data/MC ratio. (c) Current data/MCratio.

the desired angle are accounted for by shifting theincident position. The particle energy cutoff ap-plied in the GEANT shower library generation hasbeen lowered from 600 keV to 50 keV for electrons;the photon cutoff of 50 keV is unchanged. Sixteenshowers per bin have been generated. Energy de-posits are corrected for energy lost in the 12 µmmylar wrapping around the CsI crystals.

The current Monte Carlo produces a significantlybetter simulation of electromagnetic showers in theCsI calorimeter. Figure 6 shows the data-MC com-parison of the fraction of energy in each of the 49CsI crystals in electron showers relative to the to-tal reconstructed shower energies for electrons fromKL → π±e∓ν decays. The majority of the energy isdeposited in the central crystal since the Moliere ra-dius of CsI is 3.8 cm. These plots are made for 16-32GeV electrons with incident angles of 20-30 mrad;the quality of agreement is similar for other ener-gies and angles. The data-MC agreement improvesdramatically as seen in Fig. 6. This improvementin the modeling of electromagnetic shower shapesleads to important reductions in the systematic un-certainties associcated with the reconstruction ofphoton showers from K → π0π0 decays (see Sec.III C).

2. Ionization Energy Loss. In KTeV03, we did notinclude the effect of ionization energy losses forcharged particles in the simulation. In the currentsimulation, we include the ionization loss for eachvolume of material in the detector. The total lossup to the surface of the CsI is less than 5 MeV.This is a very small effect for K → π+π− decays

but it is important for low-energy electrons used inthe calibration of the CsI calorimeter and affectsconverted photons from K → π0π0 decays.

3. Bremsstrahlung. In KTeV03, the MC included elec-tron Bremsstrahlung in materials upstream of theanalyzing magnet only. In the current analysis, theBremsstrahlung rate and photon angle in each vol-ume of material in the detector are included in thesimulation.

4. Delta Rays. In the KTeV03 simulation, delta raysproduced in a drift chamber cell deposited all oftheir energy in that cell. The MC now has a morecomplete treatment in which delta rays may scat-ter into adjacent cells of the drift chamber. Highmomentum delta rays are traced through the de-tector like any other particle and low momentumdelta rays are simulated using a library createdwith GEANT4[24]. This treatment of delta raysimproves our simulation of the distribution of ex-tra in-time hits in the drift chambers.

5. Pion Interactions. The probability for pions to in-teract hadronically with material in the spectrom-eter is 0.7%; hadronic interactions in the spectrom-eter were not simulated in KTeV03. These eventsare now simulated using a GEANT-based librarywhich contains a list of secondary particles pro-duced by each hadronic interaction. An averageof nine secondary particles are produced per inter-action; these secondary particles are read in fromthe shower library and traced through the rest ofthe detector like any other particle. Events withpion interactions typically trigger the photon vetosystem and so do not pass selection criteria in theanalysis.

6. Fringe Fields. The simulation of fringe fields fromthe analysis magnet has been refined. Fringe fieldsfrom the analysis magnet inside the vacuum tankand between all four drift chambers have beenmeasured and are now simulated. The maximumstrength of the fringe field in these regions is com-parable to the earth’s magnetic field. The fringefield simulation improves the MC description of theazimuthal dependance of the π+π− invariant massfor K → π+π− data.

7. Position Resolution. The position resolution of thedrift chambers is dependent upon position withinthe cell as shown in Fig. 7. In KTeV03, the reso-lution was treated as flat across the cell; the posi-tion dependence of the resolution is now simulated.The position dependence of the resolution is alsoconsidered in the analysis of K → π+π− data; thisis described in Sec. III B.

The position resolution of CsI calorimeter clustersin the MC is slightly worse than in data. To bettermatch the resolutions, we artificially improve the

7

0

50

100

150

200

0 2 4 6D, mm

σ, µ

m

FIG. 7: Dependence of drift chamber position resolution onposition within the cell. D is the distance from the sense wire.Crosses represent the measured central values and uncertain-ties of the resolution in bins of D and the line represents apolynomial fit to the data. This fit is used in the simula-tion to parameterize the position dependence of the positionresolution.

resolution of the MC by 9%. This is done by mov-ing the reconstructed position toward the generatedparticle position.

III. DATA ANALYSIS

The analysis is designed to identify K → ππ decayswhile removing poorly reconstructed events that are dif-ficult to simulate, and to reject background. For eachdecay mode, the same requirements are applied to de-cays in the vacuum and regenerator beams, so that mostsystematic uncertainties cancel in the single ratios usedto measure |η+−/ρ|2 and |η00/ρ|2. The following sectionsdescribe the analysis and the associated systematic un-certainties in Re(ǫ′/ǫ).

A. Common Features

Although many details of the charged and neutral de-cay mode analyses are different, several features are com-mon to reduce systematic uncertainties. We select anidentical 40-160 GeV/c kaon momentum range for boththe charged and neutral decay modes. We also use thesame z-vertex range of 110-158 m from the target for eachdecay mode. To simplify the treatment of backgroundfrom kaons that scatter in the regenerator, the veto re-quirements for the charged and neutral mode analysesare made as similar as possible.When discussing systematic uncertainties, we typically

estimate a potential shift s ± σs, where s is the shift inRe(ǫ′/ǫ) and σs is the statistical uncertainty on s. Weassign a symmetric systematic error, ∆s, such that the

range [−∆s,+∆s] includes 68.3% of the area of a Gaus-sian with mean s and width σs.

B. Charged Reconstruction and Systematics

The K → π+π− analysis consists primarily of the re-construction of tracks in the spectrometer; the verticesand momenta of the tracks are used to calculate kine-matic quantities describing the decay. The CsI calorime-ter is used for particle identification by comparing thereconstructed cluster energy to the measured track mo-mentum. The analysis requirements provide clean identi-fication of well-reconstructed K → π+π− events with lit-tle background contamination. The cuts are sufficientlyloose to reduce systematics from modeling of resolutiontails. The K → π+π− reconstruction and event selectionare described in the following sections; more details ofthe analysis are found in [20].

1. K → π+π− Reconstruction

The spectrometer reconstruction begins by findingtracks separately in the x- and y-views. Track segmentsare found in the two drift chambers upstream of the mag-net and the two drift chambers downstream of the mag-net; these segments are then extrapolated to the centerof the magnet. We require the extrapolated segments tomatch within 6 mm at the magnet mid-plane to form acombined track; they typically match to within 0.5 mm.Each particle momentum is determined from the trackbend-angle in the magnet and a map of the magneticfield.The process of finding track segments depends on the

alignment and calibration of the drift chambers. For thecurrent analysis, we made new measurements of the driftchamber sizes and rotations. The survey of the wire posi-tions used a large coordinate measurement machine witha camera and magnifying lens mounted on the end of amovable arm. The measured drift chamber size is about0.02% larger than the nominal value found by scalingthe 6.35 mm “cell” size. The relative non-orthogonalitybetween DC1 and DC2 is limited to ±30 µrad. The un-certainty in Re(ǫ′/ǫ) associated with the drift chamberalignment and calibration is 0.20×10−4. The momen-tum measurement uses the known kaon mass as a con-straint; the 0.022 MeV/c2 uncertainty in the kaon masscorresponds to an uncertainty in Re(ǫ′/ǫ) of 0.10×10−4.If two x-tracks and two y-tracks are found, the recon-

struction continues by extrapolating both sets of tracksupstream to define vertices projected on the x-z and y-z planes. The difference between these two projections,∆zvtx, is used to define a vertex-χ

2,

χ2vtx ≡ (∆zvtx/σ∆z)2 , (5)

where σ∆z is the resolution of ∆zvtx. This resolutiondepends on momentum and opening angle, and accounts

8

for multiple scattering effects. The two x-tracks and twoy-tracks are assumed to originate from a common vertexif χ2vtx < 100.To determine the full particle trajectory, the x and

y tracks are matched to each other based on their pro-jections to the CsI; the projected track positions mustmatch CsI cluster positions to within 7 cm.An event is assigned to the regenerator beam if the

regenerator x-position has the same sign as the x-coordinate of the kaon trajectory at the downstream faceof the regenerator; otherwise, the event is assigned to thevacuum beam.In KTeV03 [20], the track segments were reconstructed

assuming that the position resolution of the drift cham-bers does not depend on the hit position within a cham-ber cell. To check this assumption, a special data samplewas collected with the magnetic field turned off. Threechambers are used to reconstruct straight tracks andthese tracks are compared to the hits reconstructed inthe fourth chamber. The resulting position resolution(see Fig. 7) shows a significant dependence on the dis-tance between the track and the sense wire. Tracks pass-ing close to a sense wire have worse resolution because ofthe time separation of drift electrons reaching the sensewire. The variation of the resolution for tracks close tothe boundary of a drift cell can be partly explained bythe electric field configuration. The drift-time depen-dent resolution is included in the MC and is used for thetrack segment reconstruction in the current analysis. Thenew resolution measurement improves reconstruction ofz-vertex and track momentum. For example, the widthof the π+π− invariant mass is reduced by ∼ 14%; thisimprovement is more significant for higher momentumkaons where it reaches ∼ 25%.The kaon decay vertex position and the momenta of

the two tracks forming the vertex are used to calculatethe π+π− invariant mass, their energy, and p2T , the sumof their momenta transverse to the beam direction.

2. K → π+π− Selection

The K → π+π− event selection begins with the three-level trigger during data taking. Level 1 uses hits inthe trigger hodoscopes and the drift chambers to selectevents consistent with two charged particles coming fromthe decay of a kaon that did not undergo large angle scat-tering in the defining collimator or regenerator prior tothe decay. Level 2 uses custom hit counting electron-ics and a track finding system to select events with twotracks from a common vertex. The vertex requirementat trigger level is loose compared to the selection crite-ria in the offline analysis. The inefficiencies of the Level1 and Level 2 triggers are studied using KL → π±e∓νdecays from minimum-bias triggers. The uncertainty inRe(ǫ′/ǫ) associated with Level 1 and Level 2 event selec-tion is 0.2×10−4. The Level 3 software filter reconstructstwo charged tracks and makes loose cuts on reconstructed

mass and particle indentification. To measure the Level 3inefficiency of the K → π+π− trigger, we perform the fulloffline analysis on “random accepts,” a prescaled subsetof the K → π+π− trigger that has no Level 3 require-ment. We find that the bias in Re(ǫ′/ǫ) from the Level3 trigger inefficiency is (0.30 ± 0.12)×10−4. We correctRe(ǫ′/ǫ) for this bias and assign an uncertainty in Re(ǫ′/ǫ)of 0.12×10−4.The offline selection criteria for K → π+π− decays

are tighter than those imposed by the trigger. TheK → π+π− analysis requirements and any associatedsystematic uncertainties in Re(ǫ′/ǫ) are described in thefollowing paragraphs.

We make a number of cuts on energy deposits in theveto detectors. The most important veto requirementare the muon veto cuts, which suppress background fromKL → π±µ∓νµ decays, and the regenerator cuts, whichreduce background from scattered kaons. Additional vetocuts are made for consistency with the K → π0π0 anal-ysis.

We also use the spectrometer and the calorimeter as“veto detectors.” We reject events with any tracks otherthan those from the vertex. We require the ratio of re-constructed cluster energy to track momentum, E/p, tobe less than 0.85 to identify the tracks as pions. We re-quire that the track momentum be greater than 8 GeV/cto ensure 100% efficiency for the muon veto detectors.These cuts suppress background from Ke3 and Kµ3 de-cay modes.

We remove events with 1.112 GeV/c2 < mpπ <1.119 GeV/c2, where mpπ is the reconstructed invariantmass assuming the higher momentum particle is a proton.This removes background from Λ → pπ− and Λ̄ → p̄π+decays where the proton is mis-identified as a pion.

Figure 8 shows the π+π− invariant mass dis-tributions for the two beams; we require 488MeV/c2 < mπ+π− < 508 MeV/c

2. Figure 9 shows thep2T distributions; we require p

2T < 250 MeV

2/c2. The p2Trequirement is the only K → π+π− selection criterionthat results in a systematic uncertainty in Re(ǫ′/ǫ). Wevary the p2T cut from 125 MeV

2/c2 to 1000 MeV2/c2 andassign a systematic uncertainty in Re(ǫ′/ǫ) of 0.10×10−4based on the change in Re(ǫ′/ǫ).

To reduce our sensitivity to details of the Monte Carlosimulation, we require track trajectories to be clear ofa number of physical apertures. We require that trackspoint at least 2 mm into the CsI calorimeter away fromthe edges of the Collar Anti detector that surrounds thebeamholes and at least 2.9 cm inside the outer edge ofthe CsI calorimeter. If the vertex position is upstreamof the Mask Anti (MA, see Fig. 2), we require that thetrack position at the MA be less than 4 cm in x and yfrom the nominal beam center. We cut away from wiresat the edges of the drift chambers. To reduce the possi-bility of x and y track candidate mismatches, we requirethat the projections of the tracks at the CsI calorimeterbe separated by 6 cm in x and 3 cm in y. We require thatdecays originate from within one of the beams by requir-

9

Vac beam π+π- mass

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FIG. 8: π+π− invariant mass distribution for K → π+π−

candidate events in the vacuum (left) and regenerator (right)beams. The data distribution is shown as dots, the K →π+π−(γ) signal MC (MC Sig) is shown as dotted histogramand the sum of signal and background MC is shown as a solidhistogram.

Vac beam π+π- PT2

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FIG. 9: p2T distribution for K → π+π− candidate events in

the vacuum (left) and regenerator (right) beams. The datadistribution is shown as dots, the K → π+π−(γ) signal MC(MC Sig) is shown as dotted histogram and the sum of signaland background MC is shown as a solid histogram.

ing that the projection of the vertex (x,y) position alongthe kaon direction reconstructs inside a 75 cm2 square atthe z of the downstream edge of the regenerator.

We require a minimum separation between the tracksin the x and y views at each drift chamber. This cut isdefined in terms of the DC cell through which the trackpasses; we require that the tracks be separated by at least3 cells at each chamber. This track separation cut formsa limiting inner aperture and depends on the position ofeach wire within the drift chambers. The wire spacing isknown with an uncertainty of 20 µm. There are varia-tions in the actual wire spacing, which are measured indata, but are not simulated in the Monte Carlo. To deter-

mine the effect of these variations, we convolve the trackillumination with the wire-cell size to determine the num-ber of events that migrate across the track separation cutin data but not in MC. We find that the bias in Re(ǫ′/ǫ)is (-0.16 ± 0.12)×10−4; the corresponding uncertainty inRe(ǫ′/ǫ) is ±0.22×10−4.The effective regenerator edge, shown in Fig. 3, defines

the upstream edge of acceptance for K → π+π− decaysin the regenerator beam. We find the effective regenera-tor edge by calculating the probability for two minimumionizing pions to escape the last piece of scintillator with-out depositing enough energy to be vetoed. This calcu-lation depends on the measured average energy depositof a muon passing through the regenerator Pb module,the fraction of energy coming from the last piece of scin-tillator due to the geometry of the phototube placementon the Pb module, the value of the trigger threshold,and the value of the offline cut on the energy deposit inthe Pb module. We find the effective regenerator edgeto be (1.65 ± 0.4) mm upstream of the physical edgein 1997 and (0.7 ± 0.4) mm upstream of the physicaledge in 1999. The difference in effective edges is due todifferent offline cuts on the energy deposit in the Pb mod-ule. In 1997, the edge is defined by the trigger threshold.In 1999, a tight offline cut is applied. We evaluate theuncertainty in this measurement by varying the triggerthreshold and the fraction of energy coming from the lastpiece of scintillator by ∼15% each. The 0.4 mm uncer-tainty in the position of the effective regenerator edgeleads to an uncertainty in Re(ǫ′/ǫ) of 0.20×10−4.We estimate the systematic uncertainty in Re(ǫ′/ǫ) as-

sociated with the Monte Carlo simulation of drift cham-ber efficiencies by generating separate sets of MC inwhich scattering, DC efficiency maps, and accidental ac-tivity are turned off. We take 10% of the resulting vari-ation in Re(ǫ′/ǫ), 0.15×10−4, to be the systematic un-certainty associated with the simulation of drift chamberefficiencies. We vary the simulated drift chamber resolu-tions by 5% and, from the resulting variation in Re(ǫ′/ǫ),we assign a systematic error of 0.15×10−4.The systematic uncertainties in Re(ǫ′/ǫ) associated

with the K → π+π− analysis are summarized in Ta-ble I. The total systematic uncertainty associated withthe K → π+π− analysis is 0.81×10−4; this is reduced by∼ 35% from KTeV03.

C. Neutral Reconstruction and Systematics

To reconstruct K → π0π0 decays, we first identify fourclusters of energy in the calorimeter and reconstruct theenergies and positions of the photons associated witheach cluster. A number of corrections are then madeto the measured cluster energies based on our knowledgeof the CsI calorimeter performance and the reconstruc-tion algorithm. We use the cluster positions and ener-gies along with the well-known pion mass to determinewhich pair of photons is associated with which neutral

10

Source Error on Re(ǫ′/ǫ) (×10−4)KTeV03 Result Current Result

1997 1999 Total

L1 and L2 Trigger 0.20 0.20 0.20 0.20L3 Trigger 0.54 0.20 0.14 0.12Alignment and Calibration 0.28 0.20 0.20Momentum scale 0.16 0.10 0.10p2T 0.25 0.10 0.10DC efficiency modeling 0.37 0.15 0.15DC resolution modeling 0.15 0.15 0.15Background 0.20 0.20 0.20Wire Spacing 0.22 0.22 0.22Reg Edge 0.20 0.20 0.20 0.20Acceptance 0.79 0.87 0.25 0.41Upstream z — 0.33 0.48 0.40Monte Carlo Statistics 0.41 0.28 0.28 0.20Total 1.26 1.12 0.82 0.81

TABLE I: Summary of systematic uncertainties in Re(ǫ′/ǫ)from the K → π+π− analysis. For errors which are evaluatedindividually for each year, the individual errors are listed incolumns and the total is the weighted average of the individ-ual errors. For those errors which are evaluated for the fulldataset or taken to be the same for both years, only one num-ber is listed. The value of each systematic uncertainty fromKTeV03 is provided for reference.

pion from the kaon decay and to calculate the decay ver-tex, the center-of-energy, and the π0π0 invariant mass.The precision of the CsI calorimeter energy and positionreconstruction is crucial to the K → π0π0 analysis andhas been improved significantly since KTeV03. SectionIII C 1 gives details of the CsI calorimeter reconstruction,Sec. III C 2 describes the reconstruction of K → π0π0decays, Sec. III C 3 describes the selection criteria forK → π0π0 decays, and Sec. III C 4 describes the sys-tematic uncertainties associated with the CsI calorimeterenergy reconstruction.

1. CsI Calorimeter Energy and Position Reconstruction

The first step in reconstructing clusters is to determinethe energy deposited in each crystal of the CsI calorime-ter. We convert the digitized information to an energy us-ing constants for each channel that are determined fromthe electron calibration. An in-situ laser, which deliv-ers light at known intensities via quartz fibers to eachCsI crystal, is used to calibrate the DPMTs and to mea-sure the less than 1% spill-to-spill drifts in each channel’sgain. The “laser correction” removes these spill-to-spillchanges and is applied before any clustering is performed.

We define a “cluster” as a 7×7 array of small crystalsor a 3×3 array of large crystals. Clusters near the bound-ary between the small and large crystals (see Fig. 5) maycontain both sizes of crystals; in this case the cluster isdefined as a 3× 3 array of “large” crystals where the en-ergy deposit in four small crystals is summed to form a

“large” crystal as needed. Each cluster is centered on a“seed crystal,” containing the maximum energy depositamong the crystals in the cluster. An initial approxi-mation of the cluster energy is found by summing theenergies of the crystals in the cluster.The x, y position of a cluster is reconstructed by calcu-

lating the fraction of energy in neighboring columns androws of crystals in the cluster. The x, y position algo-rithm uses a map that is based on assuming a uniformphoton illumination across each crystal to convert theseratios to a position within the seed crystal. The positionmaps are made using isolated clusters from K → π0π0data; no corrections to the position are applied based onincident particle angle. The final position is evaluatedafter all energy corrections are applied.The raw cluster energy must be corrected for a num-

ber of geometric and detector effects. We apply “crystal-level” corrections that adjust the energy in each crys-tal that makes up the cluster and “cluster-level” correc-tions, which are multiplicative corrections to the totalcluster energy. Many of the crystal-level corrections relyon “transverse energy maps”; as a function of positionwithin the seed crystal, these maps predict the distribu-tion of energy among the crystals within a cluster . Theyare made using isolated photon clusters from K → π0π0data. The crystal-level and cluster-level energy correc-tions are enumerated below.

1. Partial Clusters. We correct for energy that is miss-ing from the cluster because of crystal energies thatare below the readout threshold or because portionsof the 3×3 or 7×7 cluster are located in the beamholes or outside the calorimeter. The energy inmissing crystals is estimated using the transverseenergy maps. The energy in crystals that were be-low threshold is estimated by a parameterization,which was determined from data, of the ratio ofenergy in a crystal to the readout threshold. Thefraction of the readout threshold energy predictedto be present in a crystal decreases with distancefrom the seed crystal and increases logarithmicallywith cluster energy.

2. Out of Cone. The ∼5% “out-of-cone correction” isapplied because an electromagnetic shower is notfully contained by the 7×7 small-crystal or 3×3large-crystal clusters. We determine the out-of-cone correction using the same GEANT simulationused to generate the Monte Carlo shower library(see Sec. II C). The correction is parameterized bya quadratic function of the reconstructed distanceof the cluster position from the center of the seedcrystal and a linear function of the reconstructedenergy. The size of the correction varies by about1% across the face of a crystal and by about 0.2%per 100 GeV. There is no explicit dependence of theout-of-cone correction on incident angle; becausethe reconstructed positions are not corrected forincident particle angle, the angle effect is included

11

implicitly in our parameterization as a function ofreconstructed position. The correction is generatedseparately for photons and electrons, and for smalland large crystals. In KTeV03, the out-of-cone cor-rection was determined for small and large crystalsusing 8 GeV GEANT showers, but there was noadjustment for the energy, position, or type of theincident particle.

3. Longitudinal Response. We correct the energy ineach crystal for the ∼5% non-uniformity of re-sponse along the length of each CsI crystal. Thelongitudinal response of each CsI crystal is mea-sured in ten 5-cm z bins using cosmic ray muonsthat pass vertically through the CsI calorimeter.These muons are detected by a cosmic ray ho-doscope consisting of three sets of 3 m-long, over-lapping plastic scintillation counters placed aboveand below the CsI calorimeter. Typically the crys-tal response increases with z as the shower nearsthe PMT. The measured CsI response is convolvedwith a GEANT prediction of the shower’s longi-tudinal distribution to correct the energy in eachcrystal. The GEANT shower profiles are generatedseparately for photons and electrons. There are in-dividual profiles for each crystal position within thecluster; they are binned in local position relative tothe center of the seed crystal and in the same sixlogarithmic cluster energy bins used in the MonteCarlo (see Sec. II C). The mean shower depth forphotons and electrons varies logarithmically withenergy. These crystal-by-crystal shower profiles area significant improvement to the longitudinal uni-formity correction; in the KTeV03 analysis, the uni-formity correction was applied at cluster level basedonly on a predicted average longitudinal energy dis-tribution for a whole shower.

4. Shared Energy. For clusters that overlap, we mustpartition the energy in the shared crystals. The“overlap correction” separates the energy depositedin two or more clusters that share crystals by usingthe transverse energy maps to predict how muchenergy each particle contributed to the shared crys-tals.

The “neighbor correction” estimates the amount ofunderlying energy in each crystal that comes fromnearby clusters that are less than 50 cm away butoutside the 3×3 or 7×7 cluster boundary. The cor-rection uses a 13×13 map to predict the energycontribution from neighboring clusters. This mapis similar to the transverse energy maps but doesnot depend on position within the CsI crystal andis generated using GEANT rather than data.

We correct clusters near the beam holes for ex-tra energy that comes from nearby clusters thatdo not share crystals but which leak energy acrossthe beam holes. This correction uses maps madeusing electrons from KL → π±e∓ν data.

5. Detector Effects. We correct for a number of detec-tor effects including the observed transverse non-uniformity of energies across each crystal that re-mains after the out-of-cone correction, the non-linearity of each channel with energy which re-mains after the longitudinal uniformity correction,and global time variations in the CsI calorimeterresponse. These corrections are measured usingE/p of electrons from KL → π±e∓ν decays, andare applied multiplicatively to the total cluster en-ergy. The “transverse non-uniformity correction”is made by dividing each cluster seed crystal into a5×5 grid and measuring the cluster energies of elec-trons in each of these position bins. A multiplica-tive correction is applied to the total cluster energybased on the cluster’s reconstructed position withinthe seed crystal. The correction is normalized suchthat the average correction over each crystal (25bins) is 1.0. The “channel-by-channel linearity cor-rection” removes the residual energy non-linearity.It is measured separately for each CsI calorimeterchannel in data and Monte Carlo. The “spill-by-spill correction” is applied to correct for time vari-ations in the response of the calorimeter as a whole;it is measured and applied separately for each spill.Each of these corrections has a maximum magni-tude of less than 1%.

6. Photon-Electron Differences. For the K → π0π0analysis we apply a “photon correction” that is de-signed to remove any residual differences betweenphotons and the electrons that are used to calibratethe calorimeter. The correction is based on pho-tons from K → π0π0 and KL → π0π0π0 decays.It is measured separately for 1996, 1997, and 1999in nine regions of the calorimeter by fitting eachevent for the photon energies applying six (four)kinematic constraints for π0π0π0 (π0π0). The de-tails of the kinematic fit are described in [25]. Thiscorrection is most important for photons with ener-gies below 20 GeV; its magnitude is less than 0.2%.The photon correction is new for the current analy-sis; no correction of photon-electron differences wasapplied in KTeV03.

The quality of the calibration and the CsI calorime-ter performance is evaluated by analyzing electrons fromthe KL → π±e∓ν calibration sample with all correctionsapplied. The electron calibration for 1996, 1997, and1999 is based on 1.5 billion total electrons. Figure 10shows the E/p distribution and the energy resolution asa function of momentum of these electrons after all cor-rections. The final energy resolution of the calorimeteris σE/E ≃ 2%/

√E ⊕ 0.4%, where E is in GeV.

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FIG. 10: Ke3 electrons after all corrections. (a) E/p for 1.5× 109 electrons. (b) Energy resolution. The fine curve showsthe momentum resolution function that has been subtractedfrom the E/p resolution to find the energy resolution.

2. K → π0π0 Reconstruction

K → π0π0 and KL → π0π0π0 events are fully recon-structed using the positions and energies of the fouror six photon clusters in the CsI calorimeter. TheKL → π0π0π0 reconstruction is almost identical to theK → π0π0 reconstruction, but for simplicity this discus-sion will be in terms of the π0π0 reconstruction. Usingcluster energies and positions, we are able to reconstructthe z vertex of the kaon decay, the (x,y) components ofthe center-of-energy of the kaon, the kaon energy, andthe π0π0 invariant mass.We must first determine which pairs of photons are as-

sociated in the K → π0π0 decay. For four photons, thereare three possible pairings. For each pairing we calculated12, the distance in z between the π

0 decay vertex andZCsI , the mean shower depth in the CsI crystals. Usingthe pion mass as a constraint, in the small angle approx-imation, we find the distance for each pair of photons tobe

d12 ≈√E1E2mπ0

r12, (6)

where r12 is the transverse distance between the two pho-tons at the CsI calorimeter.For each pairing, we compare the calculated distance

for each candidate pion. In most cases, only the correctpairing will give a consistent distance for both pions. Theconsistency of the measured distance is quantified usingthe pairing chi-squared variable:

χ2π0 ≡(

d12 − davgσ12

)2

+

(

d34 − davgσ34

)2

. (7)

In Eq. 7, dij is the calculated distance for each pion, davgis the weighted average of the distance dij for both pions,and σij is the energy dependent vertex resolution for eachpion. We choose the pairing that gives the minimumvalue of χ2π0 . Using Monte Carlo events, we find that thisprocedure selects the wrong pairing for less than 0.01%

of K → π0π0 decays in the final event sample. The zvertex of the kaon decay is taken to be ZCsI - davg forthe best pairing.We find the center-of-energy of the kaon decay at the

CsI calorimeter plane by weighting the position of eachphoton with its energy. The x and y components of thecenter-of-energy are

xcoe ≡∑

xiEi∑

Ei, ycoe ≡

∑

yiEi∑

Ei, (8)

where the sums are over all four photons. The center-of-energy is the point at which the kaon would have in-tercepted the plane of the CsI calorimeter if it had notdecayed, so we can calculate the (x,y) position of the de-cay vertex by assuming it lies on the line between thetarget and the center-of-energy. The x coordinate of thekaon decay vertex is used to determine whether the kaoncame from the regenerator or the vacuum beam.The π0π0 invariant mass is calculated from the coor-

dinates of the kaon decay vertex and the four photonpositions and energies. The kaon energy is calculated asthe sum of the four photon energies.

3. K → π0π0 Selection

The K → π0π0 event selection begins with three lev-els of trigger requirements during data taking. TheK → π0π0 Level 1 trigger requires that the total en-ergy in the CsI calorimeter be greater than 30 GeV. Theinefficiency in this trigger is studied using K → π+π−π0decays from the K → π+π− trigger; the inefficiency ata given energy, Etotal, is the ratio of events with energygreater than Etotal for which the Level 1 trigger bit is notset to the total number of events with energy greater thanEtotal. We find inefficiencies ranging from (0.5-1.6)×10−4in the 40-160 GeV energy range used for the analysis.The impact of this inefficiency is slightly different in thevacuum and regenerator beams because of small differ-ences in the energy distributions. The resulting bias inRe(ǫ′/ǫ) is less than 0.02×10−4, which we assign as asystematic error.The Level 2 trigger requirement is based on the “hard-

ware cluster counter” (HCC)[26]. The inefficiency in thistrigger is measured using KL → π0π0π0 decays from atrigger that has no Level 2 requirement. We reconstructtheKL → π0π0π0 decays without any requirement on theHCC; the Level 2 inefficiency is the ratio of the numberof events that do not meet the HCC requirement to thetotal number of events found in the offline reconstruc-tion. The bias in Re(ǫ′/ǫ) produced by this inefficiencyis determined using K → π0π0 MC. The inefficiency issimulated to within 10% by the Monte Carlo, so we take10% of the measured bias as the systematic uncertaintyin Re(ǫ′/ǫ). The total uncertainty in Re(ǫ′/ǫ) associatedwith the Level 2 trigger is 0.19×10−4.The inefficiency of the Level 3 K → π0π0 trigger is

studied using “random accepts,” a prescaled subset of

13

the K → π0π0 trigger that has no Level 3 requirement.We find no statistically signficant bias in Re(ǫ′/ǫ) andquote an uncertainty of 0.07×10−4 in Re(ǫ′/ǫ) based onthe statistical precision of the bias measurement.

The offline selection criteria for the K → π0π0 sam-ple are designed to select events that are cleanly recon-structed, to suppress background, and to select kinematicand fiducial regions appropriate for the KTeV detector.We evaluate the systematic uncertainty associated witheach of the following requirements by loosening or remov-ing the cut and evaluating the change in Re(ǫ′/ǫ).

The energy of each CsI calorimeter cluster is requiredto be greater than 3 GeV because the clustering correc-tions and MC simulation are not reliable at very low en-ergies. The minimum distance between the reconstructedpositions of CsI calorimeter clusters is required to begreater than 7.5 cm because it is difficult to separatethe energy deposits in two very close clusters. Clustersvery near the beam holes are not as well reconstructedbecause of energy leakage across the beam holes and mul-tiple overlapping or nearby clusters. In KTeV03, the in-ner CsI aperture was defined by the Collar Anti (CA)detector; we now remove events with clusters having aseed crystal in the first ring of crystals around the beamholes. We do not find any systematic variation of Re(ǫ′/ǫ)with these requirements.

The variable describing the quality of the photon pair-ing, χ2

π0(Eq. 7), is required to be less than 50. This is a

rather loose cut since more than 99% ofK → π0π0 eventspassing all other cuts have χ2

π0values below 10. The pri-

mary purpose of this cut is to reduce background fromKL → π0π0π0 events in which two of the photons escapethe detector; in this case it is likely that the missingphotons come from different pions causing the remainingphotons to be paired incorrectly. The systematic un-certainty in Re(ǫ′/ǫ) associated with this requirement is0.14×10−4.The “shape chi-squared” variable, χ2γ , is a measure of

how well the transverse energy distribution of each CsIcalorimeter cluster matches the expected distribution fora photon. This variable, which is not a true chi-squaredbecause of correlations that are not considered, is calcu-lated by comparing the transverse energy distribution ofeach cluster to the transverse energy maps described inSec. III C 1. The maximum value of χ2γ for each eventis required to be less than 48. The purpose of this cutis to remove background from KL → π0π0π0 events inwhich two or more photons overlap in the CsI calorimeterand are reconstructed as a single cluster. In these casesthe transverse distribution of energy would tend to bedifferent from that of a single photon cluster. The sys-tematic uncertainty in Re(ǫ′/ǫ) associated with the shapechi-squared requirement is 0.15×10−4.We make a number of cuts on the veto detectors to re-

duce background. We also use the calorimeter, spectrom-eter, and trigger hodoscope as “veto detectors” by cut-ting on extra clusters, tracks, and hits. There is no sys-tematic uncertainty associated with these requirements.

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DataMCBG

FIG. 11: K → π0π0 RING distributions for data and signalMC in the vacuum (left) and regenerator (right) beams. Thedashed line indicates our cut.

In the K → π+π− analysis we use p2T to remove eventsin which the kaon scatters in the collimator or the re-generator. This variable is not available for K → π0π0decays since we do not measure the photon angles, so weuse the “ring number” variable to reject scattered kaondecays. Ring number is calculated using the center-of-energy of the reconstructed clusters, and is defined as

RING = 40000×Max(∆x2coe,∆y2coe), (9)

where ∆xcoe and ∆ycoe are the distances from the center-of-energy to the center of the closest beam hole. A changeof ∆RING= 1 corresponds to an incremental area of 1cm2 centered on the beam hole. Events with ring numberless than 81 cm2 should be from kaons decaying insideone of the two beams. Figure 11 shows the ring numberdistributions for both beams for data and Monte Carlo.The ring number is required to be less than 110 cm2; thesystematic uncertainty in Re(ǫ′/ǫ) associated with thisrequirement is 0.27×10−4.The limiting apertures for K → π0π0 events are the

CsI calorimeter inner aperture at the beamholes, the CsIcalorimeter outer aperture, the upstream edge in eachbeam, and an effective inner aperture resulting fromthe 7.5 cm photon separation requirement at the CsIcalorimeter. The CsI calorimeter inner and outer aper-tures are defined by rejecting events in which a photonhits the innermost or outermost ring of CsI crystals. Theupstream aperture in the vacuum beam is defined by theMask Anti and the upstream aperture in the regeneratorbeam is defined by the lead module at the downstreamedge of the regenerator. The systematic errors associatedwith the precision of these apertures are discussed in theKTeV03 paper[20] and have not changed; the individualvalues are listed in Table II. The total systematic uncer-tainty in Re(ǫ′/ǫ) associated with limiting apertures inthe K → π0π0 analysis is 0.48×10−4.Figure 12 shows the reconstructed kaon mass distribu-

tions for both beams for data and Monte Carlo. The massis required to be 490 MeV/c2 < mπ0π0 < 505 MeV/c

2.The sidebands of the mπ0π0 distribution are almost ex-clusively KL → π0π0π0 background, with a small con-

14

10 2

10 3

10 4

10 5

10 6

10 7

0.45 0.475 0.5 0.525 0.55Vacuum beam invariant mass (GeV/c2)

Eve

nts

per

5 M

eV

DataMCBG

10 2

10 3

10 4

10 5

10 6

10 7

0.45 0.475 0.5 0.525 0.55Regenerator beam invariant mass (GeV/c2)

Eve

nts

per

5 M

eV

DataMCBG

FIG. 12: K → π0π0 mπ0π0 distributions for data and signalMC in the vacuum (left) and regenerator (right) beams. Thedashed lines indicate our cuts.

tribution from events in which the photons have beenmispaired. The peaking background at the kaon mass isfrom decays of kaons which scattered with non-zero an-gle in the regenerator and the defining collimators. Moredetails on the background are given in Sect. III D.

4. Energy Systematics

The reconstruction of K → π0π0 decays depends en-tirely on the reconstruction of energies and positions ofphoton showers in the CsI calorimeter. Reconstructedquantities may depend upon the absolute energy scaleor the energy linearity of the CsI calorimeter. We ap-ply corrections that match the energy scale between dataand Monte Carlo, and we assign systematic uncertaintiesbased on any disagreement in either absolute energy scaleor energy linearity between data and Monte Carlo. Theprocedures for matching the energy scale and evaluatingthe energy systematics are described in this section.The energy scale of the CsI calorimeter is set by the

electron calibration, but there is a small, residual differ-ence in energy scale between data and Monte Carlo forK → π0π0 events. This difference is removed by adjust-ing the energy scale in data such that the sharp edgein the z vertex distribution at the regenerator matchesbetween data and Monte Carlo, as shown in Fig. 13.The correction is determined by sliding finely binnedK → π0π0 data and Monte Carlo z vertex distributionsin the regenerator beam past each other and using theKolmogorov-Smirnov (KS) test to determine how muchthe data must be adjusted to best match the MC. Thecorrection is binned in kaon energy in the same 10 GeVenergy bins that are used to extract our results (see Sec.IVB). The same correction is applied to each cluster inan event.The final energy scale adjustment is shown as a func-

tion of kaon energy in Fig. 14. The average size of thez-vertex shift is ∼2.5 cm. This corresponds to an aver-age energy correction of ∼0.04%, compared to ∼0.1% inKTeV03. As a result of improvements to the simulation

0

20

40

60

80

100

124 124.5 125 125.5 126 126.5 127z vertex (m)

Th

ou

sa

nd

s p

er

10

cm

Uncorrected Data

MC

0

20

40

60

80

100

124 124.5 125 125.5 126 126.5 127z vertex (m)

Th

ou

sa

nd

s p

er

10

cm

Corrected Data

MC

FIG. 13: Regenerator beam K → π0π0 z vertex distributionnear the regenerator for 1999 data and Monte Carlo. (a) Un-corrected data. (b) Data with energy scale correction applied.

0.998

0.9985

0.999

0.9995

1

1.0005

1.001

1.0015

1.002

40 60 80 100 120 140 160EK (GeV)

En

erg

y S

cale

Co

rrec

tio

n

KTeV03

current analysis

-0.1

-0.05

0

0.05

0.1

Ver

tex

Sh

ift

(m)

FIG. 14: Change in the final energy scale adjustment rel-ative to KTeV03. The dashed line represents no data-MCmismatch. The y axis on the right side of the plot shows thedata-MC z vertex shift in meters.

and reconstruction of clusters, the required energy scaleadjustment in the current analysis is smaller and less de-pendent on kaon energy than in the KTeV03 analysis.

This final energy scale adjustment ensures that the en-ergy scale matches between data and MC at the regener-ator edge, but we must check whether the data and MCenergy scales remain matched for the full length of thedecay volume. Any non-linearity would result in differenteffective energy scales at different decay points because ofthe correlation between the z vertex and kaon energy dis-tributions. We check the energy scale at the downstreamend of the decay region by studying the z-vertex distri-bution of π0π0 pairs produced by hadronic interactions

15

in the vacuum window and other downstream detectorelements in data and MC. To verify that this type ofproduction has a comparable energy scale to K → π0π0,we also study the z-vertex distribution of hadronic π0π0

pairs produced in the regenerator. The z-vertex distri-bution of regenerator hadronic events is Gaussian whilethe distribution of downstream events is more compli-cated, as described below. The methods for making thedata-MC comparison in each case are described in thefollowing paragraphs.

We compare the Gaussian z-vertex distributions ofhadronically produced regenerator events between dataand MC by sliding the distributions past each other andusing the chi-squared test. The average data-MC differ-ence is plotted in Fig. 15; we find no significant data-MCmismatch in this sample.

For the downstream hadronic events, we consider in-teractions in four separate detector volumes: the vacuumwindow, the upstream drift chamber, and the two heliumbags surrounding the drift chamber. The production ofπ0π0 pairs in each of these volumes is simulated sepa-rately; a fit is used to determine the relative contributionof each material, and to find the difference between thedata and MC z-vertex distributions. The fit is performedseparately for the 1996, 1997, and 1999 data samples.Figure 16 shows the z-vertex distributions of downstreamhadronic π0π0 pairs for 1999 data and MC, before andafter the Monte Carlo data are shifted by the measured1.06 cm data-MC difference. The z shifts measured foreach year are plotted in Fig. 15.

To convert these shifts to an uncertainty in Re(ǫ′/ǫ),we consider a linearly varying energy scale distortion suchthat no adjustment is made at the regenerator edge andthe z shift at the vacuum window is that measured by thehadronic downstream sample. This distortion is shownby the shaded region in Fig. 15. We rule out energyscale distortions that vary non-linearly as a function ofz vertex because they introduce data-MC discrepanciesin other distributions. The systematic error on Re(ǫ′/ǫ)due to uncertainties in the K → π0π0 energy scale is0.65×10−4.To evaluate the effect of energy non-linearities on the

reconstruction, we study the way the reconstructed kaonmass, which does not depend on the absolute energyscale, varies with reconstructed kaon energy, kaon z ver-tex, minimum cluster separation, and incident photonangle. Data-MC comparisons for these distributions forthe 1999 data sample are shown in Fig. 17. To mea-sure any bias resulting from the nonlinearities that causethe small data-MC differences seen in these distribu-tions, we investigate adjustments to the cluster energiesthat improve the agreement between data and MC inthe plot of reconstructed kaon mass vs kaon energy. Wefind that a 0.1%/100 GeV distortion produces the bestdata-MC agreement for the 1997 and 1999 datasets. Fig-ure 18 shows the improvement in data-MC agreementwith this distortion applied to 1999 data. The 1996dataset has slightly larger non-linearities; we find that

-2

-1

0

1

2

3

125 130 135 140 145 150 155 160

Hadronic π0π096 hadronic π0π0

97 hadronic π0π0

99 hadronic π0π0

KTeV03

K→π0π0

z vertex (meters)

ZD

ata−

ZM

C (

cm)

FIG. 15: Energy scale tests at the regenerator and vacuumwindow. The difference between the reconstructed z posi-tions for data and MC is plotted for K → π0π0 events, andfor hadronically produced π0π0 pairs at the regenerator andthe downstream detector elements. The solid point at the re-generator edge is the K → π0π0 sample; there is no differencebetween data and MC by construction. The open point at theregenerator edge is the average shift of the hadronic regener-ator samples for all three datasets. The points at the vacuumwindow are the shifts for the downstream hadronic events foreach dataset separately. The shaded region shows the rangeof data-MC shifts covered by the total systematic uncertaintyfrom the energy scale. For reference, the data-MC shift at thevacuum window from KTeV03 is also plotted.

0

500

1000

1500

2000

2500

3000

3500

4000

Ev

en

ts p

er

8 c

m

Data

Default MC

0

500

1000

1500

2000

2500

3000

3500

4000

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

z(π0π

0) − z(vacwin) (m)

Data

MC shifted1.06 cmdownstream

(a)

(b)

FIG. 16: z-vertex distributions of π0π0 pairs producedhadronically in downstream detector elements for 1999 dataand MC. (a) Data (dots) and nominal MC (histogram). (b)Data (dots) and MC that is shifted 1.06 cm downstream tomatch the data (histogram).

16

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

110 120 130 140 150 160

z vertex (m)

≠≠ -

MK (

Me

V/c

2M

)

Data

MC

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

50 75 100 125 150

EK (GeV)

M≠≠ -

MK (

Me

V/c

2)

Data

MC

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2 0.4 0.6 0.8

Minimum cluster separation (m)

M≠≠ -

MK (

Me

V/c

2)

Data

MC

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 10 20 30

Photon angle (rad)

M≠≠ -

MK (

Me

V/c

2)

Data

MC

(a) (b)

(c) (d)

FIG. 17: Comparisons of the reconstructed kaon mass vs(a) z-vertex, (b) kaon energy, (c) minimum cluster separa-tion, and (d) photon angle for 1999 data (circles) and MC(stars). The values plotted are the difference between the re-constructed kaon mass for each bin and the PDG kaon mass.

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

40 60 80 100 120 140 160

EK (GeV)

Mπ

π - M

K (

MeV

/c2 )

Data

Data with distortion (0.1%/100 GeV)

MC

FIG. 18: Effect of 0.1%/100 GeV distortion on MK vs EKfor 1999 data. The values plotted are the difference betweenthe reconstructed kaon mass for each bin and the PDG kaonmass.

a 0.3%/100 GeV distortion produces the best data-MCagreement for this dataset. The data-MC agreement inthe reconstructed kaon mass as a function of kaon energyhas been significantly improved compared to KTeV03,where a 0.7%/100 GeV distortion was required.To evaluate the systematic error associated with these

non-linearities, we apply the distortions to the data andfind that Re(ǫ′/ǫ) changes by less than 0.2×10−4 for allthree datasets. Properly weighting the three datasets, wefind that the systematic uncertainty on Re(ǫ′/ǫ) due toenergy non-linearities is 0.15×10−4.The systematic uncertainties in Re(ǫ′/ǫ) from the K →

π0π0 analysis are summarized in Table II. The K →

π0π0 analysis contributes an uncertainty in Re(ǫ′/ǫ) of1.55×10−4, which is reduced by ∼23% from KTeV03.

Source Error on Re(ǫ′/ǫ) (×10−4)KTeV03 Result Current Result

1996 1997 1999 Total

L1 Trigger 0.10 0.01 0.01 0.03 0.02L2 Trigger 0.13 0.20 0.12 0.23 0.19L3 Trigger 0.08 0.20 0.04 0.05 0.07Ring Number 0.24 0.27 0.27Pairing χ2 0.20 0.14 0.14Shape χ2 0.20 0.15 0.15Energy Nonlinearity 0.66 0.10 0.10 0.20 0.15Energy Scale 1.27 0.45 0.82 0.59 0.65Position Reconstruction 0.35 0.35 0.35Background 1.07 1.14 1.06 1.07CsI Inner Aperture 0.42 0.42 0.42MA Aperture 0.18 0.18 0.18Reg Edge 0.04 0.04 0.04CsI Size 0.15 0.15 0.15Acceptance 0.39 0.48 0.48MC Statistics 0.40 0.75 0.37 0.41 0.25

Total 2.01 1.69 1.63 1.56 1.55

TABLE II: Summary of systematic uncertainties in Re(ǫ′/ǫ)from the K → π0π0 analysis. For errors which are evaluatedindividually for each year, the individual errors are listed incolumns and the total is the weighted average of the individ-ual errors. For those errors which are evaluated for the fulldataset or taken to be the same for all years, only one num-ber is listed. The value of each systematic uncertainty fromKTeV03 is provided for reference.

D. Background and Systematics

Background to the K → ππ signal modes is simulatedusing the Monte Carlo, normalized to data outside thesignal region, and subtracted. There are two categoriesof background in this analysis: scattered K → ππ eventsand non-ππ background. We use decays from coher-ently regenerated kaons only; any kaons that scatterwith non-zero angle in the regenerator are treated asbackground. This regenerator scattering background andbackground from kaons that scatter in the defining col-limators have the same momentum and p2T distributionsfor both K → π+π− and K → π0π0 decays. This back-ground can be identified using the reconstructed trans-verse momentum of the decay products in the charged de-cay mode. Therefore, the scattering background is smallin the charged mode, and we may use K → π+π− decaysto tune the simulation of scattering background on whichwe must rely in the neutral mode.Non-ππ background is present because of misidentifi-

cation of high branching-ratio decay modes. The back-ground to K → π+π− decays comes from KL → π±e∓νand KL → π±µ∓ν decay modes. The background toK → π0π0 decays comes from KL → π0π0π0 decays

17

and hadronic interactions in the regenerator. The back-ground estimation procedure and the associated system-atic uncertainties are described in detail in [20].

There is only one significant change to the backgroundestimation procedure since KTeV03. Hadronic produc-tion of K∗ and ∆ resonances via KL+N → K∗S +X andn+N → ∆+X are now included in the K → π+π− re-generator beam background analysis; these backgroundsources were not considered in the KTeV03 analysis. Theincident neutron spectrum is assumed to be the same asthat of the Λ baryon, which is measured in data. For K∗Sdecays, both K±π∓ and π0KS ,KS → π+π− modes aresimulated. The K∗S → π0KS background is normalizedusing the transverse momentum side band in the regen-erator beam. The K∗S → K±π∓ and ∆ → p±π∓ decaysare normalized using mass sidebands in the regeneratorbeam reconstructed assuming the vertex is located at theregenerator edge. These two modes are seperated usingthe momentum asymmetry distribution of the secondaryparticles. The hadronic K∗ and ∆ background sampleshave negligible contributions to the signal region after allselection cuts, but including them improves the descrip-tion of mass and pT sidebands.

The background levels in K → π+π− are illustrated inFig. 8 and Fig. 9, and the background to K → π0π0 maybe seen in Fig. 11 and Fig. 12. Background contributesless than 0.1% of K → π+π− events and about 1% ofK → π0π0 events. Tables III and IV contain summariesof all the background fractions for each dataset. Thereare some variations in background levels among the yearsdue to differences in trigger and veto requirements. Thesystematic uncertainty in Re(ǫ′/ǫ) due to background is0.20×10−4 from K → π+π− and 1.07×10−4 from K →π0π0.

Vacuum Beam Regenerator BeamSource 1997 1999 1997 1999

Regenerator Scattering — — 0.073% 0.075%Collimator Scattering 0.009% 0.008% 0.009% 0.008%KL → π

±e∓ν 0.032% 0.032% 0.001% 0.001%KL → π

±µ∓ν 0.034% 0.030% 0.001% 0.001%Total Background 0.074% 0.070% 0.083% 0.085%

TABLE III: Summary of K → π+π− background levels.

E. Data Summary

The numbers of events collected in each beam are sum-marized in Table V. After all event selection require-ments are applied and background is subtracted, we havea total of 25 million vacuum beamK → π+π− decays and6 million vacuum beam K → π0π0 decays.

IV. ACCEPTANCE AND FITTING

A. Acceptance Correction and Systematics

We use the Monte Carlo simulation to estimate the ac-ceptance of the detector in momentum and z-vertex binsin each beam. We evaluate the quality of this simulationby comparing z-vertex distributions in the vacuum beambetween data and Monte Carlo. To account for small dif-ferences in the energy spectrum between data and MonteCarlo, we reweight the distributions, using the same 10GeV/c momentum bins used by the fitter (see SectionIVB), by adjusting the number of MC events in each binso that the data and MC kaon momentum distributionsagree. We fit the data-MC ratio of z-vertex distributionsto a line, and call the slope of this line, s, the acceptance“z-slope.” We use this z-slope to evaluate the systematicerror on Re(ǫ′/ǫ).A z-slope affects the value of Re(ǫ′/ǫ) by producing

a bias between the regenerator and vacuum beams be-cause of the different z vertex distributions in the twobeams. A good approximation of the bias on Re(ǫ′/ǫ) iss∆z/6 where ∆z is the difference of the mean z valuesfor the vacuum and regenerator beam z vertex distribu-tions. The factor of 6 converts the bias on the vacuum-regenerator beam ratio to a bias on Re(ǫ′/ǫ). The valuesof ∆z are 5.6 m for the K → π+π− sample and 7.2 mfor the K → π0π0 sample. We use the measured bias onRe(ǫ′/ǫ) and the statistical error on that measurement toassign a systematic uncertainty in Re(ǫ′/ǫ).The z-slopes for the full dataset are shown in Fig. 19.

We use the 25 million vacuum beam K → π+π− de-cays to measure the z-slope in the charged decay mode.The uncertainty in Re(ǫ′/ǫ) associated with this z-slope is0.41 ×10−4. We assign an additional uncertainty of 0.40×10−4 based on a Re(ǫ′/ǫ) fit which excludes K → π+π−decays from the region upstream of the MA. This regionis very sensitive to the value of the MA apperture cut(see Sec. III B), and, since it lies upstream of the re-generator edge, there are no regenerator beam decays tocompensate for this dependence. Combining these twouncertainties, the systematic uncertainty in Re(ǫ′/ǫ) as-sociated with the acceptance correction is 0.57×10−4.We also measure the data-MC z-slope in the high

statistics KL → π±e∓ν decay mode and find a slopethat is similar in magnitude to the systematic uncertaintyfrom K → π+π−. We do not use the KL → π±e∓ν z-slope to set the systematic error because it is sensitive todifferent detector effects and has different particle typesin the final state than K → π+π−.We use 88 million KL → π0π0π0 decays to measure

the z-slope in the neutral decay mode. This mode has thesame type of particles in the final state asK → π0π0, andit is more sensitive than π0π0 to potential problems inthe reconstruction due to close clusters, energy leakage atthe CsI calorimeter edges, and low photon energies. Weassign an uncertainty on Re(ǫ′/ǫ) from the neutral modeacceptance of 0.48 ×10−4 based on the KL → π0π0π0 z-

18

Vacuum Beam Regenerator BeamSource 1996 1997 1999 1996 1997 1999

Regenerator Scattering 0.288% 0.260% 0.258% 1.107% 1.092% 1.081%Collimator Scattering 0.102% 0.122% 0.120% 0.081% 0.093% 0.091%KL → π

0π0π0 0.444% 0.220% 0.301% 0.015% 0.006% 0.012%Photon Mispairing 0.007% 0.007% 0.008% 0.007% 0.008% 0.007%Hadronic Production 0.002% 0.001% — 0.007% 0.007% 0.007%Total Background 0.835% 0.603% 0.678% 1.209% 1.197% 1.190%

TABLE IV: Summary of K → π0π0 background levels. Note that photon mispairing is not subtracted from the data and isnot included in the total background sum.

Vacuum Beam Regenerator BeamK → π+π− 25107242 43674208K → π0π0 5968198 10180175

TABLE V: Summary of event totals after all selection criteriaand background subtraction.

slope. We also measure the z-slope in K → π0π0 decaysand find that the results are consistent with those fromKL → π0π0π0 decays. There is no significant change inRe(ǫ′/ǫ) when the region upstream of the MA is excludedin the neutral mode, so no additional systematic uncer-tainty is required.

B. Fitting and Systematics

The value of Re(ǫ′/ǫ) and other kaon parameters ∆m,τS , φǫ, and Im(ǫ

′/ǫ) are determined using a fitting pro-gram. The fitting procedure is to minimize χ2 betweenbackground subtracted data and a prediction function.The prediction function uses the detector acceptance de-termined with the Monte Carlo simulation. The fits areperformed in 10 GeV/c kaon momentum bins. There isno z binning to determine Re(ǫ′/ǫ), while a z-binned fit isperformed to measure the other kaon parameters. Uncer-tainties from the fitting procedure are mainly related toregenerator properties and the dependence of the resulton external parameters.Neglecting the contribution from KS produced at the

target (called target-KS), the number of K → ππ eventsin the vacuum beam for a given p, z is

Nππ(p, z) ∼ F(p)|η|2 exp(

− tτL

)

, (10)

where t = (z − zreg)mK/p is the measured proper timerelative to decays at the regenerator edge, η = η+− =ǫ+ ǫ′ (η = η00 = ǫ−2ǫ′) for charged (neutral) decays andF(p) is the kaon flux. The fitting program includes thecontribution of target-KS by using a phenomenologicalmodel for K0/K0 production at the target and propagat-ing the kaon states up to the decay volume. The model oftarget-KS production is checked by floating the K

0/K0

flux ratio in the fit. The fitted fraction of target-KS de-

viates from the model by (2.5 ± 1.6)%. The associatedsystematic uncertainty in Re(ǫ′/ǫ) is ±0.12×10−4.The number of events in the regenerator beam is

Nππ(p, z) ∼ F(p)Treg(p)×[

|ρ(p)|2 exp(

− tτS)

+ |η|2 exp(

− tτL)

+ 2|ρ(p)||η| cos (∆mt+ φρ(p)− φη) exp(

− tτave)]

,

(11)where ρ(p) is the momentum-dependent coherent regen-eration amplitude, φρ(p) = arg(ρ), 1/τave = (1/τS +1/τL)/2 and Treg(p) is the relative kaon flux transmis-sion in the regenerator beam. The prediction functionaccounts for decays inside the regenerator by using theeffective regenerator edge (Fig. 3b) as the start of thedecay region.The parameters from Eqs. 10,11 are determined as dis-

cussed below. The kaon flux, F(p), is a free parameter foreach of the twelve 10 GeV/c momentum bins. Separatekaon fluxes are allowed for charged and neutral decays toaccount for slight differences in the data samples, so thereare a total of 12× 2 = 24 free fit parameters to describethe kaon flux. The flux ratio of decays in the vacuum andregenerator beams is, however, the same in both chargedand neutral decay modes. The 1996 K → π0π0 data hasno corresponding K → π+π− data, so it is possible thatthere could be small differences in the flux ratio betweenthe two years which do not cancel in the fit. We assignan uncertainty in Re(ǫ′/ǫ) of ±0.03×10−4 from this pos-sibility, following [18].The relative kaon flux attenuation in the regenera-

tor beam, Treg, results from the shadow absorber andthe regenerator itself. The attenuation is measured di-rectly from data by comparing the rate ofKL → π+π−π0decays in the vacuum (N+−0vac ) and regenerator (N

+−0reg )

beams. As mentioned earlier, a dedicated trigger was in-troduced in 1999 to improve the statistical precision ofthis measurement; the improved measurement