Measurement of the K + → π + ν ν ¯ Branching Ratio
aa r X i v : . [ h e p - e x ] J a n BNL/79257-2007-JA, KEK-2007-34, TRI-PP-07-18, TUHEP-EX-07-002
Measurement of the K + → π + ν ¯ ν Branching Ratio
S. Adler, V.V. Anisimovsky, M. Aoki, ∗ M. Ardebili, A.V. Artamonov, M. Atiya, B. Bassalleck, A.O. Bazarko, B. Bhuyan, † E.W. Blackmore, D.A. Bryman, S. Chen,
8, 3
I-H. Chiang, I.-A. Christidi, ‡ M.R. Convery, P.S. Cooper, M.V. Diwan, J.S. Frank, T. Fujiwara, J. Haggerty, J. Hu, T. Inagaki, M.M. Ito, A.P. Ivashkin, D.E. Jaffe, S. Kabe, M. Kazumori, § Y. Kuno, ∗ M. Kuriki, ¶ S.H. Kettell, M.M. Khabibullin, A.N. Khotjantsev, P. Kitching, M. Kobayashi, T.K. Komatsubara, A. Konaka, A.P. Kozhevnikov, Yu.G. Kudenko, A. Kushnirenko, ∗∗ L.G. Landsberg, †† B. Lewis, K.K. Li, L.S. Littenberg, J.A. Macdonald, †† D.R. Marlow, R.A. McPherson, P.D. Meyers, J. Mildenberger, O.V. Mineev, M. Miyajima, K. Mizouchi, V.A. Mukhin, N. Muramatsu, T. Nakano, M. Nomachi, T. Nomura, T. Numao, V.F. Obraztsov, K. Omata, D.I. Patalakha, S.V. Petrenko, R. Poutissou, E.J. Ramberg, G. Redlinger, T. Sato, T. Sekiguchi, T. Shinkawa, F.C. Shoemaker, A.J.S. Smith, J.R. Stone, R.C. Strand, S. Sugimoto, Y. Tamagawa, R. Tschirhart, T. Tsunemi, ‡‡ D.V. Vavilov, B. Viren, N.V. Yershov, Y. Yoshimura, and T. Yoshioka §§ Brookhaven National Laboratory, Upton, NY 11973 Institute for Nuclear Research RAS,60 October Revolution Pr. 7a, 117312 Moscow, Russia TRIUMF, 4004 Wesbrook Mall, Vancouver,British Columbia, Canada V6T 2A3 Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544 Institute for High Energy Physics,Protvino, Moscow Region, 142 280, Russia Department of Physics and Astronomy,University of New Mexico, Albuquerque, NM 87131 Department of Physics and Astronomy,University of British Columbia, Vancouver,British Columbia, Canada V6T 1Z1 Department of Engineering Physics, singhua University, Beijing 100084, China Department of Physics and Astronomy,Stony Brook University, Stony Brook, NY 11794 Fermi National Accelerator Laboratory, Batavia, IL 60510 Department of Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan High Energy Accelerator Research Organization (KEK),Oho, Tsukuba, Ibaraki 305-0801, Japan Centre for Subatomic Research, University of Alberta, Edmonton, Canada T6G 2N5 Department of Applied Physics, Fukui University,3-9-1 Bunkyo, Fukui, Fukui 910-8507, Japan Research Center for Nuclear Physics, Osaka University,10-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan Laboratory of Nuclear Studies, Osaka University,1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan Department of Applied Physics, National Defense Academy,Yokosuka, Kanagawa 239-8686, Japan
Abstract
Experiment E949 at Brookhaven National Laboratory studied the rare decay K + → π + ν ¯ ν andother processes with an exposure of 1 . × K + ’s. The data were analyzed using a blindanalysis technique yielding one candidate event with an estimated background of 0 . ± .
03 events.Combining this result with the observation of two candidate events by the predecessor experimentE787 gave the branching ratio B ( K + → π + ν ¯ ν ) = (1 . +1 . − . ) × − , consistent with the StandardModel prediction of (0 . ± . × − . This is a more detailed report of results previouslypublished in Physical Review Letters. PACS numbers: 13.20.Eb, 12.15.Hh, 14.80.Mz ∗ Present address: Department of Physics, Osaka University, Osaka 560-0043, Japan. † Also at Department of Physics, University of Delhi, Delhi 1100007, India ‡ Present address: Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece. § Also at Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan. ¶ Present address: Graduate School of Advanced Sciences of Matter,Hiroshima University, Hiroshima, 739- ∗∗ Present address: Institute for High Energy Physics, Protvino, Moscow Region, 142 280, Russia. †† Deceased. ‡‡ Present address: Department of Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan §§ Present address: International Center for Elementary Particle Physics, University of Tokyo, Tokyo 113-0033, Japan. ontents I. Introduction K + → π + ν ¯ ν K + → π + ν ¯ ν Experiments 12
II. Experimental Method K + Beam Line 15C. Beam Counters 17D. Target 19E. Drift Chamber 20F. Range Stack 201. Scintillation Counters 212. Range Stack Straw Chambers 21G. Photon Veto Counters 22H. Trigger 251. Trigger Architecture 262. K + → π + ν ¯ ν Triggers 273. Monitor Triggers 28I. Data Acquisition 30
III. Data Analysis K + -Decay Background 332. Origins of Beam Background 33B. Analysis Method and Strategy 351. Blind Analysis Method 362. Bifurcation Method for Evaluating Background 363. Analysis Strategy 37C. Track Reconstruction 391. Beam Time Measurements 392. Clustering in the RS 394. Tracking in the UTC 404. Tracking in the Target and B4 405. Track Passage in the IC 426. Tracking in the RS and RSSC 427. Kinematic Measurements of a Track 43D. Monte Carlo Simulation 441. Simulation of K + Propagation 442. Simulation of K + Decay Product 443. Simulation of Trigger 454. Comparison between Data and Simulation 45E. Data Processing and Pre-selection 461. Pass 1 462. Pass 2 46F. Selection Criteria of post Pass 1 and Pass 2 471. Single Beam K + Requirements 482. Decay π + Kinematic Requirements 533. π + → µ + → e + Decay Sequence 564. Photon Veto 64G. Background Evaluation 651. K π Background 692. µ + Background 703. Single Beam Background 734. Double Beam Background 735. Charge Exchange Background 746. Initial Background Evaluated from 1/3 Sample 757. Optimization of Signal Region 758. Correlation and Single Cut Failure Study 769. Final Background Evaluated from 2/3 Sample 7810. Systematic Uncertainty 79H. Acceptance and Sensitivity 791. Acceptance Factors from K µ Events 802. Acceptance Factors from K π Events 825. Kinematic Acceptance from Beam π + Events 834. π + → µ + → e + Decay Acceptance from Beam π + Events 835. Acceptance Factors from Monte Carlo Simulation 846. Correction to T · K µ Branching Ratio 868. Confirmation of the K π Branching Ratio 869. Summary of Acceptance and Sensitivity 87I. Examining the Signal Region 88
IV. Results K + → π + ν ¯ ν B ( K + → π + X ) 93E. Impact on the Unitarity Triangle 94 V. Conclusion Acknowledgments References I. INTRODUCTION
Although the Standard Model (SM) has successfully accounted for all low energy CP-violating phenomena thus far observed, it is insufficient as the source of CP-violation neededto explain the cosmological baryon asymmetry in our universe [1]. According to Sakharov [2],one of the conditions necessary to generate such an asymmetry is that the elementary inter-action violates charge conjugation symmetry (C) and the combined CP symmetry (whereP is the parity symmetry). However, the size of the asymmetry needed for this cannot bederived in model calculations based on the SM [3] and new sources of CP violation havebeen sought for many years in particle physics experiments. Prominent among these are therare decays K → πν ¯ ν which are sensitive to new physics involving both CP-violating and6P-conserving interactions. In this paper, we present a detailed description of the previ-ously reported measurement of the reaction K + → π + ν ¯ ν performed by Brookhaven NationalLaboratory (BNL) experiment BNL-E949 [4]. This paper is arranged as follows. We firstbriefly review CP violation and rare kaon decays, with an emphasis on K + → π + ν ¯ ν de-cays. We then describe previous results on this reaction and discuss the sources of potentialbackground and the methods for suppressing backgrounds. We also discuss the design ofthe K + beamline, the detector and the selection criteria used in data analysis and describethe methods used for estimating background levels and for evaluating the acceptance. Afterexamining the signal region, we present the method used for extracting the branching ratio,making full use of our knowledge of the background in the signal region. In the last section,we show how the measurement of B ( K + → π + ν ¯ ν ) impacts the search for new physics beyondthe SM. A. CP Violation and the Rare Decay K + → π + ν ¯ ν Standard Model CP violation arises from a complex phase in the three-generation quarkmixing matrix [5]. In the Wolfenstein parameterization [6] of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, the parameters can be written in powers of λ = sin θ c ≈ . V CKM = V ud V us V ub V cd V cs V cb V td V ts V tb ≃ − λ / λ Aλ ( ρ − iη ) − λ − λ / Aλ Aλ (1 − ρ − iη ) − Aλ . (1)Where A , λ , ρ and η are real numbers. CP invariance of the Lagrangian for weak interactionsis violated when the CKM matrix is complex. The parameter η quantifies CP violation inthe SM.The unitarity of the CKM matrix implies six unitarity conditions, which can be repre-sented graphically in the form of triangles, all of which must have the same area. The area ofthese triangles is equal to one half of the Jarlskog invariant, J CP [7]. Applying the unitarityproperty V † V = 1 to the CKM matrix in (1) yields V ∗ ub V ud + V ∗ cb V cd + V ∗ tb V td ≃ V ∗ ub − λV ∗ cb + V td = 0 , (2)where the approximations V ud ≃ V ∗ tb ≃ V cd ≃ − λ have been made. This equationcan be represented graphically, as shown in Fig. 1, where we have divided all of the sides7 IG. 1: Unitarity triangles in the ¯ ρ − ¯ η plane. Two sides of the triangle can be expressed by theCKM matrix elements | V td | /Aλ and | V ub /V cb | /λ , respectively, where A and λ are parameters inthe Wolfenstein parameterization. by λV ∗ cb . The apex of the triangle is given by two Wolfenstein parameters, ¯ ρ and ¯ η , where¯ ρ = ρ (1 − λ /
2) and ¯ η = η (1 − λ /
2) [8]. B ’s and K ’s are so far the only two mesons showing evidence of CP violation in theirdecay processes. Whether or not the observed CP violation can be completely explained bythe CKM phase within the SM can be probed by the independent determination of ρ and η ,from B and K decays as shown in Fig. 2. Two sensitive methods for making the comparisonare: • A comparison of angle β from the ratio B ( K L → π ν ¯ ν )/ B ( K + → π + ν ¯ ν ) with thatfrom the CP violating asymmetry ( A CP) in the decay B d → J/ψK s ; and, • A comparison of the magnitude | V td | from K + → π + ν ¯ ν with that from the mixingfrequencies of B s and B d mesons, expressed in terms of the ratio of the mass differences,∆ M B s / ∆ M B d .Although the decay K + → π + ν ¯ ν is a flavor changing neutral current (FCNC) processprohibited at tree level in the SM, it is allowed at the one-loop level. In leading order, itis described by a “Box” diagram and two “ Z -penguin” diagrams, as shown in Fig. 3. Theweak amplitude for this process is represented as M ∼ X i = u,c,t V ∗ is V id γ µ q µ + m i q − m i , (3)8 IG. 2: Unitarity triangle determined by B and K decays. The parameters ¯ ρ and ¯ η can bedetermined in two ways: the angle β from the CP violating asymmetry in the decay B d → J/ψK s ,and from the length of the side from ∆ M B s / ∆ M B d in B − ¯ B mixing; the height of the trianglefrom B ( K L → π ν ¯ ν ) and the radius of a circle centered at (¯ ρ ,
0) from B ( K + → π + ν ¯ ν ). where V ij ’s are the CKM matrix elements, γ µ ’s are the Dirac matrices, q µ is the momentumtransfer, and m i ’s are quark masses. M vanishes if all of the quark masses, m i , are equal,because of the unitarity of the CKM matrix. However, the breaking of flavor symmetry,which results in the variation of quark masses, allows this decay to proceed at a very smallrate. The top quark provides the dominant contribution to the K + → π + ν ¯ ν branching ratiodue to its very large mass in spite of the small coupling of top to down quarks ( V td ) in theCKM matrix.Following Ref. [9], the branching ratio for K + → π + ν ¯ ν is calculated as follows. Theeffective Hamiltonian can be written in the SM as H SMeff = G F √ α π sin Θ W X l = e,µ,τ ( V ∗ cs V cd X l NL + V ∗ ts V td X ( x t ))(¯ sd ) V − A ( ¯ ν l ν l ) V − A , (4)in next-to-leading order (NLO), where X ( x t ) ≡ X ( x t ) + α s ( m t )4 π X ( x t ) ≈ η X · X ( x t ) (5)and X ( x t ) ≡ C ( x t ) − B ( x t ) , η X = 0 . . (6) B ( x j ) and C ( x j ) in (6) are functions of x j ≡ m j /M W , and were derived for the first time9 IG. 3: Second-order weak processes that contribute to the K + → π + ν ¯ ν branching ratio: the“Box” diagram (upper) and two “ Z -penguin” diagrams (bottom). by Inami and Lim in 1981 [10]. The coefficient X l NL and the function X ( x t ) are the charmand top quark contributions, including QCD corrections at NNLO [11, 12, 13, 14, 15].With the top quark mass in the minimal subtraction scheme m t ( m t ) = (162 . ± .
2) GeV [9], X ( x t ) = 1 . ± .
025 (7)is obtained.The perturbative charm contribution gives the largest theoretical uncertainty and can bedescribed in terms of the parameter P c ( X ) ≡ λ [ 23 X e NL + 13 X τ NL ] = 0 . ± . , (8)where the error is obtained by varying the charm mass, m c , the scale factor, µ c = O ( m c )and the coupling constant, α s ( M Z ), by reasonable amounts. One obtains B ( K + → π + ν ¯ ν ) = κ + · "(cid:18) Im λ t λ X ( x t ) (cid:19) + (cid:18) Re λ c λ ( P c ( X ) + δP c,u ) + Re λ t λ X ( x t ) (cid:19) , (9)where κ + ≡ r + α B ( K + → π e + ν )2 π sin Θ W λ = (5 . ± . × − (cid:20) λ . (cid:21) , (10)10 P c,u = 0 . ± .
02 comprises the long-distance contribution calculated in Ref. [16], and the λ j ’s ( ≡ V ∗ js V jd ) are from the CKM matrix elements. The r + (= 0 . K + → π + ν ¯ ν to the well-measured decay K + → π e + ν [17].In obtaining the numerical value in (10), we used [18]sin Θ W = 0 . , α = 1127 . , B ( K + → π e + ν e ) = (4 . ± . × − . (11)Expression (9) describes in the ¯ ρ − ¯ η plane an ellipse with a small eccentricity, namely( σ ¯ η ) + ( ¯ ρ − ¯ ρ ) = σ B ( K + → π + ν ¯ ν )¯ κ + | V cb | X ( x t ) , (12)where ¯ ρ ≡ λ ( P c ( X ) + δP c,u ) | V cb | X ( x t ) , σ ≡ (cid:18) − λ (cid:19) − , κ + ≡ κ + λ . (13)Using (9) and varying m t , | V cb | , P c ( X ) and | V td | , which is constrained by | V ub /V cb | and B − ¯ B mixing in the ¯ ρ − ¯ η plane, the branching ratio of K + → π + ν ¯ ν is predicted to be B ( K + → π + ν ¯ ν ) = (0 . ± . × − (14)within the SM. It should be noted that, of the uncertainty of 27% in (14), the theoreticaluncertainty is ∼
6% at present, mainly due to the uncertainty in the charm quark mass.Theoretically a precise measurement of B ( K + → π + ν ¯ ν ) is one of the cleanest ways toextract | V td | . This is due to the following factors: • the long-distance contributions to the branching ratio are small [19] and under control,the most recent calculation gives a contribution of (+6 ± • the uncertainty from the hadronic matrix element has been reduced to <
1% by recenttheoretical and experimental developments [20], and • the recent NNLO calculation [14, 15] has reduced the total theoretical uncertainties to ∼ K and B decays.If a precise measurement of the neutral analog K L → π ν ¯ ν could also be made, the intrinsictheoretical error on | V td | could be reduced to ∼
1% [15].As determinations of B -system parameters become increasingly precise, the uncertaintyon the SM prediction for K + → π + ν ¯ ν will approach the current theoretical accuracy of11 K + → π + ν ¯ ν branching ratio there-fore provides a stringent test of the SM and probes for new physics. There have beennumerous predictions for K + → π + ν ¯ ν in the models beyond the SM and applications ofthe measured branching ratio to constrain new models. These include the Minimal Su-persymmetric Standard Model with [21, 22] and without [22, 23] new sources of flavor- orCP-violation, generic SUSY with minimal particle content [24], SUSY with non-universalA terms [25], SUSY with broken R-parity [26, 27], topcolor [28], topcolor-assisted techni-color models [29, 30], multiscale walking technicolor [31], four generation models [32], lep-toquarks [33], Left-Right model with right-handed Z ′ [34], extension of the SM to a gaugetheory with J = 0 mesons [35], a multi-Higgs multiplet model [36], light sgoldstinos [37],universal extra dimensions [38], 5-dimensional split fermions [39], a Randell-Sundrum sce-nario [40], a littlest Higgs model [41, 42], non-standard neutrino interactions [43], and aminimal 3-3-1 model [44]. B. History of K + → π + ν ¯ ν Experiments
Searches for this process which began over 35 years ago have used stopped- K + beams. Itwas believed at the time of the first of these that the branching ratio might be as high as afew × − [45]. It was recognized that even at this level, a poor-signature process such as K + → π + ν ¯ ν would need effective particle identification, precise kinematic measurement andthe ability to veto extra charged and neutral tracks to discriminate it from common decaymodes such as K + → µ + ν µ and K + → π + π (referred to as K µ and K π , respectively).The earliest published result was from a heavy liquid bubble chamber experiment [46] at theArgonne Zero Gradient Synchrotron, in which a 90% CL upper limit B ( K + → π + ν ¯ ν ) < − was obtained. In that paper it was recognized that K π decay in flight and hadronic π + interaction in the detector were dangerous sources of potential background.The final analysis of the Argonne experiment improved the limit to 5 . × − [47], butbefore it appeared in print, a subsequent counter/spark-chamber experiment at the BerkeleyBevatron improved the limit to 1 . × − [48]. However this experiment was sensitive to onlythe most energetic of π + , whereas the bubble chamber experiment covered a wide kinematicrange. In addition to the background from common K + decay modes, this experimentconsidered possible background from K + charge exchange in the stopping target followed12y K L → π + e − ¯ ν e , and from beam π + which scattered into the detector. The Chicago-Berkeley group continued their program with a setup sensitive to π + in the kinetic energyrange 60–105 MeV, i.e. below that of the potential background process K π rather thanabove it. This required reconfiguring their photon veto system so that it became nearlyhermetic. Combining results from the two configurations, the branching ratio upper limitwas improved slightly to 5 . × − [49].About a decade later, an experiment at the KEK Proton Synchrotron improved the limitto 1 . × − [50]. The technique of waveform digitization to record the π + → µ + → e + decay chain was introduced for the first time. This experiment was sensitive only to the π + with momenta greater than that from K π (referred to as the “ πν ¯ ν (1)” region) and its setupresembled that of Ref. [48].The BNL series of experiments was initiated in the early 1980’s. They were based on alarge-acceptance solenoidal spectrometer with a hermetic photon veto situated at the endof a highly pure, very intense stopped- K + beam [51] from the BNL Alternating GradientSynchrotron (AGS). The experimental signature of the K + → π + ν ¯ ν decay was a single π + track with π + momentum less than 227 MeV/ c plus no other particle from a K + decay.Fig. 4 shows momentum spectra of major decay modes of K + .The first phase of E787 in 1988-91 achieved a 90% CL upper limit on the branching ratioof 2 . × − [52], using data from the πν ¯ ν (1) region. A separate limit of 1 . × − at 90%CL [53] was extracted from the kinematic region in which the π + is softer than that of the π + from K π (referred to as the “ πν ¯ ν (2)” region). This program completed the identification ofbackgrounds needed to reach the 10 − level of sensitivity and developed methods to reliablymeasure them.A major upgrade of both the beam line and the detector was undertaken between 1992and 1994. The search for K + → π + ν ¯ ν resumed in 1995 and continued through 1998. Thelimit on the branching ratio from the πν ¯ ν (2) region was improved by an order of magnitudeto 2 . × − at 90% CL [54], but the major output of this series of runs was the observation oftwo clean K + → π + ν ¯ ν events [55] in the πν ¯ ν (1) region and a measurement of the branchingratio B ( K + → π + ν ¯ ν ) = (1 . +1 . − . ) × − . The BNL-E787 detector was upgraded againover the period from 1999–2001. The E949 experiment was proposed to use this detector torun for 60 weeks. After the first 12 weeks of running in 2002 no further funds were providedto complete the experiment. Based on the collected BNL-E949 data, the first result was13 IG. 4: Momentum spectra (in MeV/ c ) of charged particles from K + decays in the rest frame.The values in the parentheses represent the branching ratios of the decay modes [18]. The hatchedspectrum shows the π + momentum from K + → π + ν ¯ ν decay assuming the V − A interaction. already published in 2004 [4]. This paper provides an extended and detailed description ofthe detector and data analysis techniques used to produce the E949 result. II. EXPERIMENTAL METHODA. Overview
E949 (BNL-E949) which succeeded BNL-E787 had a sensitivity goal of detecting ten SMsignal events [56]. E949 employed a low momentum beam of K + ’s which were degradedand stopped in the detector. Measurement of the K + → π + ν ¯ ν decay involved observationof the daughter π + in the absence of other coincident activity. The π + was identified by14ts kinematic features obtained from energy, momentum and range measurements, and bythe observation of a π + → µ + → e + decay sequence. Since the signal was expected at the10 − level, the detector was designed to have powerful π + identification for rejection of K µ and K + → µ + ¯ ν µ γ decays ( K µ γ ), 4- π solid angle photon detection coverage for vetoing K π decays, and efficient K + identification system for eliminating beam-related backgrounds.The entire E949 spectrometer was surrounded by a 1 Tesla solenoidal magnetic field alongthe beam direction. The coordinate of detector used a Cartesian coordinate system in whichthe origin was at the center of the target; the + z axis was along the incident beam directionand the + y axis in the vertical up direction as shown in Fig. 5. Under this coordinate system,the azimuthal angle of a track was defined as the arctangent of y/x and, the polar angle θ was defined as the angle with the + z axis. Many detector components have been discussedelsewhere [57, 58, 59, 60, 61, 62]. Fig. 5 shows the E949 detector after upgrades (1999–2001 [56, 63]) with the following improved components: photon veto detection efficiency,tracking and trigger efficiency, and data acquisition (DAQ) live time. E949 was designed torun at the same instantaneous rate as E787, and to achieve a factor of five improvement insensitivity, through the use of a higher duty factor and reduced K + momentum for a higherstopping fraction. The higher duty factor was not achieved in the engineering run in 2001 orthe first physics run in 2002 due to a broken motor generator set that supplied power to theAGS. The regular supply was removed from operation on August 3, 2001 and the backupwas used during the rest of 2001 and 2002. E949 ran at about twice the beam rate of E787. B. Accelerator and K + Beam Line
The K + beam was produced by a high-intensity proton beam from the AGS at BNL: theentire AGS beam of 65 × protons (Tp/spill) at a momentum of 21.5 GeV/ c was deliveredto the E949 K + production target. Prior to 2001 the AGS typically ran at 24 GeV/ c , but atthis momentum the longest spill achievable was 0.5 sec. Combined with the longer cycle time(3.2 s between spills, as compared to 2.3 s), the duty factor at 24 GeV/ c was unacceptablylow. By lowering the proton momentum to 21.5 GeV/ c , the spill length was increased to amaximum of 2.2 s, resulting in a duty factor of 2.2 s/5.4 s. At this lower proton momentumthe production of 710 MeV/ c K + ’s was reduced by 10%. The K + production target was15 IG. 5: Schematic side (a) and end (b) views of the upper half of the E949 detector. Illustratedin this figure, an incoming K + that traverses all the beam instruments, stops in the target andundergoes the decay K + → π + π . The outgoing π + and one photon from π → γγ are also shown.The detector elements and acronyms are described in detail in the text. made of 2/3 of an interaction length of platinum (6 cm along the beam direction), and waslocated on a water-cooled copper base. At the typical AGS running condition 65 Tp onthe production target per 2.2 s spill, the maximum target temperature was measured to be ∼ ◦ C.The Low Energy Separated Beam [64] (LESB III) collected and transported K + ’s emittedat 0 ◦ (along with 500 π + ’s and 500 protons per K + ), and momentum-selected by the firstdipole magnet. Two electro-magneto-static separators swept π + ’s and protons out of the K + beam axis. The resulting beam was further selected by a second dipole magnet. LESB IIIhad a total length of 19.6 m from the production target to the E949 target with an angularacceptance of 12 msr and a momentum acceptance of 4.5% FWHM at a mean momentum of710 MeV/ c . During most of the 2002 running period the first separator voltage was loweredfrom the standard voltage of 600 kV to ∼
250 kV due to high voltage discharges. Underthese conditions a K + : π + ratio in the beam of 3:1 was achieved with a 40% loss in K + flux(typically E787 ran with 4:1). Proton contamination was suppressed to a negligible level bythe separators. At the same typical AGS running condition described previously, 1 . × K + ’s were transported through the beam line.The typical conditions during the 2002 run had 3.5 × K + ’s entering the E949 targetevery spill. This corresponded to a rate of 1.6 × K + ’s/sec during the 2.2 sec spill. Theeffective spill length was actually 8% shorter (2.0 sec) due to some residual modulation of16he beam intensity out of the AGS. The typical instantaneous rate of beam particles at theˇCerenkov counter was 6.3 MHz of K + ’s and 1.5–2.5 MHz of π + ’s. C. Beam Counters
The incoming K + beam traversed a scintillation counter (B0), a ˇCerenkov counter, twobeam wire proportional chambers (BWPC’s), a passive BeO degrader, an active degrader(AD) and a beam hodoscope (B4) as depicted in Fig. 5. The BWPC’s and B0 counter arenot shown in Fig. 5.The B0 counter, which was a 30.5-cm long, 0.6-cm thick and 7.6-cm wide Bicron BC408plastic scintillator, was located just downstream of the last quadrupole magnet and countedall charged particles in the beam. It was read out by an analog-to-digital converter (ADC),a time-to-digital converter (TDC) and a 500 MHz transient digitizers based on gallium-arsenide charge-coupled device (CCD) [57]. The ˇCerenkov counter [51] located just down-stream of the B0 counter identified particles as K + ’s or π + ’s. The ˇCerenkov light from the K + ( π + ) was transmitted (internally reflected) at the downstream surface of the ˇCerenkovradiator and read out with 14 “ K ˇCerenkov” ( C K ) and 14 “ π ˇCerenkov” ( C π ) EMI9954KBPMT’s. The PMT signals were split, with 90% sent to TDC’s via fast LRS3412 discrimina-tors and 10% to a ×
10 amplifier. The amplifier output was sent to CCD’s. The pulse-heightinformation in every 2 ns interval was recorded to reproduce the time development of thepulses and to detect two particles close in time to each other. The multiplicity output of C K ( C π ) PMT’s was discriminated (typical threshold: n >
5) to identify K + ’s and π + ’s inthe trigger ( KB and πB defined in Section II H).The two BWPC’s were located downstream of the ˇCerenkov counter to monitor thebeam profile and identify multiple incoming particles. The first chamber (BWPC1) waslocated 168.5 cm upstream of the target entrance and contained three planes of sense wires:vertical ( x -plane) and ± ◦ to the vertical ( u - and v -planes). The sense wires were 12- µ m-diameter gold-plated tungsten. The x -, u - and v -planes had 144, 60 and 60 readoutchannels, respectively, with a 1.27 mm wire spacing. In the u - and v -planes, pair of wires weremultiplexed in one readout channel. The active area was 17.8 cm (horizontal) by 5.08 cm(vertical). The cathode foils were 25- µ m thick aluminized mylar coated with carbon. Theanode-cathode distance was 3.18 mm, and the total thickness of BWPC1 was approximately176 mm. The second chamber (BWPC2) was located 1.0 m downstream of BWPC1 and alsocontained three planes ( x , u and v ). The direction of the sense wires was vertical ( x -plane)and ± ◦ to the vertical ( u - and v -planes). Each plane had 120 active sense wires with a0.8-mm wire spacing. Among the 120 wires, the central 72 ones were multiplexed by 3 andthe remaining were multiplexed by 6 in the readout channels, yielding a total of 32 readoutchannels for each plane. The cathode foils were 8- µ m single-sided aluminized mylar coatedwith carbon. The anode-cathode distance was 1.6 mm. Both chambers were filled with arecirculated mixture of CF4 (80%) and Isobutane (20%).Downstream of the BWPC’s a degrader slowed the K + ’s so that they stopped in the centerof the scintillator fiber target. The upstream section of the degrader was inactive, consistingof 11.11 cm of beryllium oxide (BeO) and 4.76 mm of Lucite. The high density (3.0 g/cm )and low atomic number of BeO was used to minimize multiple scattering. The AD consistingof 40 layers of 2 mm thick disks of Bicron BC404 scintillator (13.9 cm diameter) alternatingwith 2.2-mm thick copper disks (13.6 cm diameter) was divided into 12 azimuthal segmentswith readout to a single Hamamatsu R1924 PMT through fourteen 1-mm-diameter BicronBCF99-29-AA-MC wave length shifting (WLS) fibers. The PMT outputs were provided toTDC’s, CCD’s and a 4-fold analog sum that was provided to an ADC. These measurementsenabled the AD to identify the beam particles and to detect activity coincident with K + decays.Downstream of the degrader the B4 hodoscope detected the entrance position of theincoming particle in the target and identified the particle type by measuring its energy loss.The B4 hodoscope consisted of two planes, u and v , with about a 11.8-cm diameter orientedat a ± . ◦ angle with respect to the horizontal axis. Each plane had 16 Bicron BC404scintillator fingers with a 7.2-mm pitch. The cross section of each finger had a ‘Z-shape’with a 6.4-mm thick middle part and 3.2-mm thick edges. This shape reduced inactiveregions and improved the spatial resolution. Three Bicron BCF99-29-AA-MC WLS fiberswere embedded in each finger and connected to a single Hamamatsu H3165-10 PMT thatwas read out by TDC’s, ADC’s and CCD’s. At the same position as the B4 hodoscope, butat larger radius was an annular scintillator counter, the ring veto (RV). The RV was designedto veto particles that passed through perimeter of the B4 hodoscope. The RV was composedof two 180 ◦ arcs of 3.3 mm thick Bicron BC404 scintillator with an inner diameter varyingfrom 11.9 cm to 12.0 cm and an outer diameter of 14.6 cm. The two RV elements were18eadout by Hamamatsu H3165-10 PMT’s and the signals were split three ways to ADC’s,TDC’s and CCD’s. D. Target
The target consisted of 413 Bicron BCF10 scintillating fibers of 5-mm square cross sectionand 3.1-m length that were bundled to form a 12-cm-diameter cylinder. A number of 1-mm,2-mm and 3.5-mm square scintillating fibers (called “edge fibers”) filled the gaps near theouter edge of the target. Each of the 5-mm fibers was connected to a Hamamatsu R1635-02PMT, whereas the adjacent edge fibers were grouped onto 16 PMT’s, providing signal readout by ADC’s, TDC’s and CCD’s.The fiducial region of the target was defined by two layers of six plastic-scintillatingcounters surrounding the target. The inner scintillators, called I-Counters (IC’s), helpedto define the fiducial volume and the logical OR of the six IC’s ( IC for trigger condition)used by the trigger for this purpose. The IC’s were 6.4-mm thick (with an inner radius of6.0 cm) and extended 24 cm from the upstream face of the target. The outer scintillators,called V-Counters (VC’s), overlapped the downstream edge of the IC’s by 6 mm, and servedto detect particles that decayed downstream of the fiducial region of the target. The VC’sconsisted of six 5-mm thick and 1.96-m long scintillators, and were staggered azimuthallywith respect to the IC’s. Each IC and VC element was instrumented with an EMI 9954KBPMT which was read out by an ADC, TDC and a 500 MHz transient digitizer (TD) basedon a flash ADC [58].Approximately 27% of the incident K + ’s (typically 3 . × K + ’s/spill) penetrated farenough into the target to satisfy the online target criteria for KB defined in Section II H 2.The remaining K + ’s either underwent decay-in-flight, nuclear interaction in the degraderor scattered in the material of the beam instrumentation and did not reach the target. Itshould be noted that some of the K + ’s ( < KB requirementdidn’t stop in the target, and a factor for the stopping fraction of K + ’s was introducedas described in Section III H 7. The K + ’s deposited an average energy of 100 MeV in thescintillating fiber target when coming to rest in the center of the target fiducial volume. Thelow velocity K + ’s typically lost 5–40 MeV in each fiber, while the nearly minimum ionizing(MIP) π + ’s from K + decays deposited about 1 MeV per fiber, as they passed transversely19hrough the fibers. E. Drift Chamber
The drift chamber, called the “Ultra Thin Chamber” (UTC) [59], was located just outsideof the IC. The primary functions of the UTC were to measure the momenta of chargedparticles and to provide tracking between the target and the Range Stack (RS). The UTChad inner and outer radii of 7.85 cm and 43.31 cm, respectively. Twelve layers of 5–8 mmdrift cells were grouped into three super-layers (each consisting of four layers) with activelengths of 38.8 cm (inner), 44.8 cm (middle) and 50.8 cm (outer). The super-layers werefilled with a 49.8%:49.8%:0.4% mixture of argon, ethane and ethanol. Each anode wire wasinstrumented with an ADC and a TDC. The drift time to the anode wires was used todetermine the ( x, y ) positions for the charged track. At the inner and outer radii of eachsuper-layer were cathode foils, with 7 mm helical cathode strips at a ∼ ◦ pitch angle. Eachcathode strip was instrumented with an ADC and a TDC. The combination of the chargecentroid on the cathode strips and anode hits provided the z hit position. There were twoinactive regions filled with nitrogen gas between the three super-layers. Differential pressuresof ∼ × − radiation lengths. The UTC position resolutions were approximately 175 µ m for x and y and 1 mm for z . F. Range Stack
Located just outside the UTC at an inner radius of 45 cm and an outer radius of 84 cm, theRS consisted of both scintillation counters and embedded straw chambers, providing energyand range measurement of the charged particles, information on the π + → µ + → e + decaysequence and measurement of photon activity.20 . Scintillation Counters The RS consisted of 19 layers of Bicron BC408 plastic scintillator, azimuthally segmentedinto 24 sectors as shown in Fig. 5. Layers 2–18 were 1.9 cm thick and 1.82 m long. Layer19 was 1.0 cm thick and was mainly used to veto longer range muons. Each scintillatorin layers 2–19 was coupled through Lucite light guides to EMI 9954KB PMT’s at bothupstream and downstream ends. The innermost counters were 6.4 mm thick and 52 cm longtrigger-counters (T-counters), defining the fiducial volume for charged K + decay products.The T-counters were thinner than layers 2–19 to suppress rate from photon conversions.Seventeen 1-mm-diameter WLS fibers (Bicron multi-clad BCF-92) with a pitch of 6.9 mmwere embedded in each scintillator and coupled to a single Hamamatsu R1398 PMT at eachend.The signal from each PMT of the RS scintillators was passively split 1:2:2 for ADC’s,discriminators and fan-in modules, respectively. The discriminator output was sent to a TDCand the trigger. Each PMT was read by an ADC and a TDC. The TDC’s (LeCroy 3377)recorded up to 16 hits in a 10.5 µ s time window, and thus allowing for efficient detectionof the µ + → e + decay. The analog fan-in summed the signals from 4 PMT’s on each endin the same hextant (four adjacent sectors) and layer. This analog sum was read out by asingle TD [58] and was provided to mean-timers [63] for good timing on the photon veto inthe trigger and a z -measurement for each layer of the track. The mean-timer output fromeach layer in a hextant was ORed to provide input to the hextant photon veto algorithm inthe trigger (vetoing more than one non-adjacent hextant). The TD’s recorded the chargein 2 ns intervals (500 MHz sampling) in a 2.5 µ s time window with a resolution of 8 bits.The 500 MHz sampling provided sufficient pulse shape information to separate pulses fromdifferent events as close as 5 ns apart, and enabled the detection of the π + → µ + decay asdescribed in Section III F 3. The time window of the TD’s was narrower than that of theTDC’s in order to reduce the data size.
2. Range Stack Straw Chambers
Two range stack straw chambers (RSSC’s) were located outside RS layer 10 and 14 [60].The inner (outer) RSSC consisted of two staggered layers of 24 (28) straws per sector with a21
IG. 6: Schematic end view of the inner RSSC. The tubes run through the beam direction. length 97.8 (113.0) cm. The average density of an RSSC was 0.054 g/cm . Each straw tubewas 3.4 mm in radius with a 50- µ m-diameter gold-coated tungsten anode wire at the center.A schematic drawing is given in Fig. 6. The straw chambers were operated with 67% argonand 33% isobutane mixture with a trace of water in a self-quenching streamer mode at 3,450V. The local x -axis of a chamber was defined to be along the width of the chamber. Therewere a total of 48 chambers, with 2,496 straws installed in the E949 experiment. Becauseof access restriction, a pair of straws from the right and left halves in the same layer wereconnected at the upstream end to allow downstream-only readout.The position of the hit straws provided x − y position information, while the end-to-endtime differences provided the z measurement. The z resolution of the E787 RSSC’s wasdegraded due to a pulse-height dependent time-walk effect. For E949 an amplifier and twodiscriminators were installed on each channel to allow for low threshold timing discriminatorand a high threshold logic discriminator (above the noise level). The z resolution wasimproved from 3.0 cm (RMS) observed in E787 to 1.5 cm (RMS). G. Photon Veto Counters
The detection of activity coincident with the charged track was crucial for suppressingbackground processes that can mimic K + → π + ν ¯ ν . Photons from K π and other radiativedecays were detected by the hermetic photon system as shown in Fig. 5. The photondetectors surrounding the K + decay vertex with a 4 π solid angle coverage were located inthe barrel, upstream and downstream end caps, and near the beam line. The photon systemconsisted of essentially every scintillator detector in experiment: the Barrel Veto (BV), theBarrel Veto Liner (BVL), the RS, the upstream and downstream End Caps (EC’s), the22pstream Photon Veto (UPV), the upstream and downstream Collar detectors (CO), thedownstream Microcollar detector ( µ CO), the Downstream Photon Veto (DPV), the RV, theIC, the VC, the AD and the target. The regions of the target, IC and RS traversed by thecharged track were excluded from the photon veto. The AD and DPV were part of the E949detector upgrade but were only used in the πν ¯ ν (2) analysis, where photon veto near thebeam axis was more important.The 1.9-meter-long, 14.3 radiation length thick BV covering 2/3 of the 4 π sr solid anglewas located in the outermost barrel region with an inner radius of 94.5 cm and an outerradius of 145 cm [51]. The BV was divided into 48 azimuthal sectors. Each sector consistedof four radial layers, in which there were 16 (innermost), 18, 20, 21 (outermost) layers of1-mm thick lead and 5-mm thick Bicron BC408 plastic scintillator. The light collected inthe scintillators accounted for 30% of the total energy deposit in the BV. The azimuthalboundaries of each sector were tilted so that there were no projective cracks for photonsfrom the decay vertex. Both ends of every module were read out by an EMI 9821KB PMTand the signals were recorded by an ADC and a TDC. The time resolution of individual BVcounter was measured to be 1.2 ns.In order to improve the photon veto capability, the BVL, located between the RS andthe BV, replaced the outermost layers 20 and 21 of the RS in E787. Each BVL counter was10 cm wide and 2.2 m long. There were 48 azimuthal sectors, each with 12 layers of 1-mmthick lead and 5-mm thick Bicron BC408 plastic scintillator, for a total thickness of 2.29radiation lengths. Both ends of the BVL modules were read out by EMI 9821KB PMT’sand the signals were recorded by ADC’s and TDC’s. The eight adjacent sectors (hextant) ineach end were grouped and read out by TD’s. The timing resolution of an individual BVLcounter was 0.7 ns. A comparison of radiation length coverage with and without the BVLis shown in Fig.7. A factor of two improvement in the photon veto rejection of K π decayswas expected from the BVL.The EC’s had roughly one-third of the 4 π sr photon coverage [61]. The upstream ECdetector consisted of seventy-five 25 cm long (13.5 radiation lengths) undoped Cesium Iodide(CsI) crystals segmented into four rings, and the downstream EC detector consisted of 68crystals in four rings. To maximize light collection the PMT’s were directly coupled tothe crystals through a Sylgard cookie formed over the PMT and a ultraviolet transmittingoptical filter that selectively passed the fast component of the CsI scintillation light (with a23 IG. 7: Radiation length with (solid curve) and without (dotted curve) the BVL as a function ofthe cosine of the polar angle. These curves account for the contribution from RS, BV, EC andBVL. Other photon veto counters are not accounted for in this plot. decay time of a few tens of nanoseconds at a wavelength of 305 nm) and blocked the slowcomponent. Since the PMT’s were situated in the magnetic field, high-field PMT’s wereused [62] (Hamamatsu R5543 3” PMT’s for the outer three rings and Hamamatsu R55452” PMT’s for the smaller inner ring crystals); the signals were split to ADC’s, constantfraction discriminators (CFD), and CCD’s. The CFD’s outputs were ORed to provide anonline photon veto signal and were also sent to TDC’s. The pulses recorded in the CCD’swere analyzed offline by a pulse-finding algorithm to provide the timing of the EC signalsand to separate two pulses close in time (to reduce the possibility of accidental hits reducingthe efficiency of photon detection). Since the EC was exposed to a high counting-rateenvironment near the beam line, good timing was required to reduce acceptance losses andthe masking of K π photons from early accidental hits.The UPV mounted to the downstream face of the ˇCerenkov counter was 3.1 radiationlengths thick, with an outer dimension of 28.4 cm × × ×
21 WLS fibers coupled to two Hamamatsu R1924 PMT’s. The UPV signals weresent to ADC’s, TDC’s and CCD’s.The upstream (downstream) CO detector was located just upstream (downstream) of theEC’s [65]. Both of the CO’s consisted of twenty-four 2-mm thick lead sheets alternating with25 layers of 5-mm thick Bicron BC404 scintillator sheets, providing about 9 radiation lengthsalong the beam direction. Each scintillator layer was segmented into 12 wedges forming 12identical azimuthal sectors. Light from the wedges was readout by WLS multi-clad fibers(Bicron BCF99-29AA). Each wedge had 16 fibers glued in grooves in the BC404 scintillatorlayer. One end of the fiber was polished and aluminized to provide reflective mirror surface.All fibers for each sector (16 × . < | cos θ | < . µ CO was installed between the inner face of the magnet end-plate and the targetdownstream of the downstream CO to give more photon detection coverage. The µ CO hadan inner (outer) diameter of 15.6 cm (20.0 cm) and a length of 53 cm. The eight azimuthalsectors contained eight layers of 2 mm scintillating fibers (from the original E787 target [51])separated by seven layers of 60 µ m Pb. The 536 fibers from two adjacent sectors were readout by one EMI9954 PMT into ADC’s and TDC’s. H. Trigger
The E949 trigger selected K + → π + ν ¯ ν events from the large number of K + decays andscattered beam particles with requirements on the range of the π + track, the presence of a π + → µ + ν µ decay in the RS, the absence of other activity at the time of the π + and thepresence of a K + at an appropriately earlier time.The trigger was composed of two stages: a fast Level-0 trigger with a decision time of ∼
100 ns and a slower Level-1 trigger with a decision time of 10–100 µ s. The Level-0 triggerwas based on signals from the beam, target, RS and photon veto systems, processed with acombination of commercial and custom-built boards. The Level-1.n trigger was composed of25wo parts running in parallel, the Level-1.1 and Level-1.2 triggers, that involved the partialprocessing of TD and ADC data, and operated only on events that passed the Level-0 trigger.The original E787 trigger has been described previously [51]. The trigger was upgradedfor E949 with the addition of a new programmable Level-0 trigger board and mean-timersfor the photon veto signals [63].E949 collected data with triggers for the K + → π + ν ¯ ν signal and for other physics studiesand calibrations. In this paper, the triggers for calibration and other physics are referred toas the “monitor” samples.
1. Trigger Architecture
For physics and most calibration triggers, Level-0 required a signal from the beam systemand a charged track in the RS. A beam K + or π + was identified in NIM logic by a coincidenceof hits from the ˇCerenkov counter, B4 ( B T arget ) and AGS spill gate (
Spill ).This KB signal served as the Beam Strobe ( BS ) for the trigger. A charged track T · IC and T-counter and second layer (mean-time of the twoends) in a single RS sector. Additional requirements were placed on the crude estimate ofthe track range, which was the track length derived from the deepest layer of the RS hit incoincidence with the T · DC ). The DC was an OR over the six coincidences of theindividual IC’s and the delayed KB signal such that a track in the IC must typically haveoccurred at least 1.5 ns later than a prompt signal. The T · T · DS ) that gated all of theADC’s not associated with the beam system or target and provided a common stop for manyof the TDC’s; the T · T · L .
1) was based on information from the TD system for the hex-tant containing the RS counter in which the charged track was determined to have stopped.26topped π + ’s were preferentially selected (over µ + ’s) by looking for the π → µ decay bycomparing the pulse height to the pulse area (for early decays) or by looking for a seconddetached pulse. This was done by a custom-built ASIC which had access to the TD memo-ries. The ASIC could also reduce the TD data size for readout by discarding the waveformsamples outside the “prompt” time window, keeping instead a calculation of the pulse areaand leading-edge time. The “prompt” window typically extended from 0.5 (2) µ s before(after) the π + track. Level-1.1 typically provided a decision in about 10–20 µ s.The Level-1.2 ( L .
2) used data digitized by the ADC’s and had two components. One ofthem, the “Level-1.1 afterburner” rejected events with an accidental hit near the stoppingcounter. Such hits might defeat the Level-1.1 with an apparent double pulse. The othercomponent, the “
HEX afterburner”, was used for the photon veto, rejecting events withhits in both of the two adjacent hextants when the T · µ s per Level-1.1 trigger. K + → π + ν ¯ ν Triggers
The triggers as described below were valid for the bulk of the 2002 running period. Thetrigger conditions for K + → π + ν ¯ ν were designed differently according to the π + momentum.For high π + momentum, the πν ¯ ν (1) trigger condition was defined as: πν ¯ ν (1) = KB · DC · ( T · · (6 ct + 7 ct ) · (15)19 ct · zf rf · L rr · ( BV + BV L + EC ) · HEX · L . · L . , where KB = C K · B · T arget · Spill. (16)The 6 ct + 7 ct required that the π + reached the 6 th or 7 th layer of the RS and suppressed thecopious 3-body K + → π + π − π + and K + → π + π π backgrounds. The 19 ct signal requiredthat the charged track should not reach the 19 th layer in order to suppress K µ background.The “ ct ” designated the RS sectors that were associated with a T · T · zf rf condition was a fiducial cut on the z -position of the charged track in eachlayer, vetoing tracks that exited the fiducial volume. The L rr z -position of the track from flash TDC’s on layers 3 and 11–13 as well as the deepestlayer of penetration of the track; this rejected events with long range such as the µ + from K µ decay. The photon veto BV + BV L + EC and HEX were from the BV, BVL, EC andRS, respectively, which removed events with photons such as K π , K µ and K µ γ .The number of K + ’s which met the KB trigger requirement when the detector was live( N K ), was recorded at the end of each AGS spill. The total exposure from the πν ¯ ν (1) triggerdata stream was N K = 1 . × . (17)E949 also provided a πν ¯ ν (2) trigger for lower momentum π + ’s. Since the study involvesdifferent background mechanisms and a different kinematic region, the result will be pre-sented in a separate paper.
3. Monitor Triggers
In addition to the K + → π + ν ¯ ν triggers, there were monitor triggers for calibration andnormalization, as well as triggers for other physics modes. All triggers were prescaled toreduce the impact on the total dead time except for the K + → π + ν ¯ ν triggers and the “ γ ”trigger which required the presence of photons in the barrel region for studying K + → π + γγ and K + → π + γ decays [66].To monitor detector performance several processes were employed, including K π and K µ decays, beam particles scattered into the detector fiducial volume, beam K + , chargeexchange events and cosmic rays. These monitor samples were used for calibration andacceptance studies. They were taken simultaneously with the πν ¯ ν (1) and πν ¯ ν (2) triggersto reflect any condition changed into the acceptance and background calculations.The K µ decay has the largest branching ratio. Since the final state does not containphotons or additional tracks and the daughter µ + does not interact strongly, it was a con-venient sample for a variety of acceptance measurements as well as several calibrations.This sample was also used for the normalization of the experiment; the measurement of the K µ branching ratio effectively normalized the counting of K + stops to the well-known K µ Kµ K µ was defined as follows Kµ KB · ( T · · (6 ct + 7 ct ) · (17 ct + 18 ct + 19 ct ) . (18)The final state of the K π decay mode contains one charged π + and two photons from π decay. Since the π + momentum is monochromatic, the K π sample was used to check themeasurement of the charged track momentum, range and energy. Also the π + ’s were usedto study particle identification, while the photons were used to study the photon veto. Two K π triggers were defined, a loose one, Kπ Kπ KB · ( T · · (6 ct + 7 ct ) · ct . (19)and a tighter one, Kπ Kπ Kπ · HEX · L . · L . . (20)The Kπ π + ’s, including some that scatteredinto the fiducial volume of the RS. These π scat events were identified as π + ’s by the C π andhad an in-time track in the RS. The kinematic features of this π + sample were almost thesame as the K + → π + ν ¯ ν signal except that the target pattern was different. This samplewas suitable for calibrating the ionization energy loss ( dE/dx ) of π + ’s and for studying theacceptance. The trigger condition for scattered beam particles was defined as: π scat = πB · DC · ( T · · (6 ct + 7 ct ) · ( BV + BV L + EC ) · HEX. (21)where πB = C π · B · T arget · Spill. (22)The charge exchange process (CEX), K + n → p + K followed by K L → π + l − ¯ ν l can mimic K + → π + ν ¯ ν events when the charged lepton l − from the K L decay had a low momentumand was undetected. The largest uncertainty in this background was the determination ofthe reaction rate. Since K has an equal fraction of K L and K S , the K + n → p + K S processwith a K S decay to π + π − can be used to measure the reaction rate for the determination ofthe background from K + n → p + K L . Since the K S → π + π − could be cleanly identified, a CEX trigger with two charged tracks was defined as:
CEX = KB · DC · T · · (6 ct + 7 ct ) · EC + πB. (23)29 ype Model Standard Resolution SubsystemsADC LRS 4300B CAMAC 10 bits RS,BV,BVL,EC,BeamLRS 1881 Fastbus 13 bits Target,UTCTDC LRS 3377 CAMAC 0.5 ns RS, BVLLRS 1879 Fastbus 2 ns UTC,BV,TargetLRS 1876 Fastbus 1 ns EC,RSSC,BeamWFD TD Fastbus 500 MHz sampling RS,BVL,IC8 bits, up to 10 µ s depthCCD Fastbus 500 MHz sampling Beam,Target,EC8 bits, 256 ns depthTABLE I: Digitizing electronics for E949. In addition, we also defined triggers for beam K + and cosmic ray, which were used intrigger efficiency measurement and detector geometrical alignment. I. Data Acquisition
Analog- and discriminated- signals from the detector were digitized by commercial ADCand TDC, and custom-built waveform digitizer (TD and CCD) systems. When an eventwas accepted by the trigger system, the digitized data for the event were transferred to abuffer module or a local crate controller. At the end of each spill, the data for the spill weretransferred to a host computer. A summary of the digitizing electronics is shown in Table I.For the Fastbus systems, SLAC Scanner Processor (SSP) modules [67] served as cratecontrollers and also to read out, reformat and buffer the data from the front-end after eachtrigger accepted. The CAMAC ADC’s were read out through the FERA bus by a Struck 370QDP DSP (Fastbus) module. The CAMAC TDC’s were read out by custom-built DYC3modules [68] which pushed the data into VME memory boards. The readout time per event(as determined by the slowest crate) was typically 850 µ s.At the end of each spill, the data from the Fastbus buffer memories were read out via thecable segment (12-15 MB/sec) by Struck 340 SFI modules, each controlled by a MVME 260430ingle-board computer (SBC) running VxWorks. The VME memory boards were read outby a separate SBC. Data were transferred from the SBC’s to the host computer (SGI Origin200) via Ethernet (9 MB/sec per link) through a simple network switch. Event fragmentsfrom the readout segments were combined by Event Builder processes running on the hostcomputer. Complete events were distributed to “consumer” processes which included datalogging and online monitoring. The K + → π + ν ¯ ν triggers were written to two DLT-7000drives at 5 MB/sec per drive; a third DLT drive was used to log monitor triggers.A slow control system, based on the MIDAS [69] framework, ran independently of themain DAQ system and was used to monitor a variety of experiment conditions, includingcrate voltages and temperatures.Under typical running conditions, we wrote ∼
300 events per spill with a typical eventsize of ∼
80 kB. This was well within the maximum throughput of the system of about50 MB/spill. The DAQ dead time was due entirely to the speed of the event-by-eventreadout of the front-end electronics at the crate level. The total dead time introduced bythe trigger and DAQ was typically 26%. The E949 experiment collected its physics datafor 12 weeks from March through June of 2002 or about 20% out of the total beam timeapproved by DOE’s Office of High Energy Physics.
III. DATA ANALYSIS
The branching ratio of the K + → π + ν ¯ ν decay in the SM is ∼ − as discussed in Sec-tion I. Unlike the K µ and K π backgrounds, the K + → π + ν ¯ ν signal is continuous with nopeak. To establish that any possible observed candidate event was really from K + → π + ν ¯ ν ,we required that the backgrounds were suppressed to a level substantially below one eventwith small uncertainty. All the detectors were calibrated before the background and accep-tance studies using Kµ Kπ Kπ IG. 8: Range in plastic scintillator (cm) versus the momentum (MeV/ c ) of the charged particlesfor events that passed the πν ¯ ν (1) trigger. These data represent ∼ N K . A. Overview of Background
Data selected by the πν ¯ ν (1) trigger were primarily from background events as shownin Fig. 8. These events were classified into stopped- K + -decay-related and beam-relatedbackgrounds. 32 . Origins of Stopped- K + -Decay Background The stopped- K + -decay backgrounds were categorized into two types: π + -related, µ + -related backgrounds. As can be seen in Fig. 4, multi-body K + decays were suppressed bysetting a signal momentum region higher than the K π peak but lower than the K µ peak.However, since K π and K µ have such large branching ratios (20.92% and 63.44% [18]),migration into the signal region through either resolution or scattering effects was a signif-icant background. Radiative K µ decay and K + → µ + π ν ( K µ ) decays accounted for themajority of the observed µ + band events in Fig. 8. For the K µ decay mode, the backgroundoriginated from a K µ peak event or a K µ low momentum (“tail”) event if the particleidentification was fooled. This was also applicable to the K π decay mode when the pho-tons escaped detection. Because of phase space limits, the πν ¯ ν (1) analysis only needed toconsider the K + -decay-related backgrounds from the K µ , µ + band and K π decay.
2. Origins of Beam Background
The beam-related backgrounds were categorized into three types: single-beam back-ground, double-beam background and CEX background. The first two beam backgroundsaccounted for most of the π + band shown in Fig. 8.The single-beam background consisted of the following components. (1) A K + enteredthe target and decayed in flight to a π + plus a π as illustrated in Fig. 9. The kinematicvalues of the π + were shifted upward to the signal region by the Lorentz boost, faking a K + → π + ν ¯ ν signal event. (2) A π + in the beam was misidentified as a K + , scattered inthe target and entered the fiducial region of the detector as illustrated in Fig. 9. This π + (referred to as scattered π + ) could mimic the target fiber pattern for signal and havekinematic values in the signal region. Rejection of these two background types required bothgood π + /K + beam identification and time resolution for delayed coincidence measurements.The following cases were classified as the double-beam background. A K + came to restin the target accompanied by another K + entering the target that decayed in flight to a π + ,which traversed the fiducial region of the detector (Fig. 10). The second case was similar,except that the beam π + scattered in the target and entered the fiducial region (Fig. 10).33 IG. 9: Schematic diagrams of the single beam background: 1) single beam K + background and2) single beam π + background. The various detector elements and acronyms are described in thetext.FIG. 10: Schematic diagrams of the double beam background: 1) double beam K + − K + back-ground and 2) double beam K + − π + background. The various detector elements and acronymsare described in the text. Both cases could imitate a K + → π + ν ¯ ν signal if the decay products from the first K + weremissed. Rejection of these two backgrounds relied on the ability to observe extra activitythe beam instrumentation, target and RS coincident with the delayed decay.34 IG. 11: Schematic diagram of the charge exchange interaction background. The various detectorelements and acronyms are described in detail in the text.
The CEX background could occur if a K + produced a K in the target and if the K turned into a K L that subsequently underwent semileptonic decay. This process couldproduce background if the charged lepton from a K L decay was undetected and the π + satisfied the kinematics of the signal region (Fig. 11). Rejection of the CEX backgroundwas achieved by using the fact that a K L usually did not deposit energy along the path inthe target, leaving a gap observed between K + and π + fibers in the target. Also exploitedwas matching between the reconstructed z -position of the decay and the energy depositedby the incoming K + and the short flight time of K L in the target. B. Analysis Method and Strategy
Disentangling the K + → π + ν ¯ ν decay from background in this experiment was challengingdue to the poor kinematic signature and very small expected rate of the K + → π + ν ¯ ν signal.These necessitate enormous suppression of background events by rejecting events with verylow levels of extraneous activity. This high level of veto makes it impractical to accuratelysimulate the background rejection power of the detector at the required ∼ − level ofsensitivity. Therefore, an accurate estimate of the background must be obtained from thedata. 35 . Blind Analysis Method A “blind” analysis method was developed to search for the K + → π + ν ¯ ν signal in datasamples. In this method, background sources were identified a priori . The signal regionfor the πν ¯ ν (1), determined so that the sensitivity was optimized, was “blinded” or hiddenuntil the background and acceptance analysis was completed. When possible, selectioncriteria were developed using the monitor samples to avoid examining the signal region. Ifmonitor samples were inadequate and the πν ¯ ν (1) trigger sample were required, at least oneselection criterion distinguishing signal from backgrounds was inverted (i.e. used to select abackground region) to avoid examining the signal region. In addition, the final backgroundestimates were obtained from different samples than that used to determine the selectioncriteria. Each set of three consecutive πν ¯ ν (1) events were selected for “1/3” and “2/3”sample groups. The 1/3 group was used to determine the selection criteria and an unbiasedbackground estimate was obtained from the 2/3 group. The signal region was examinedonly after the background analysis was completed.
2. Bifurcation Method for Evaluating Background
The principal method for background evaluation relied on information from outside thesignal region and involved the application of two complementary but uncorrelated cuts.Fig. 12 illustrates this bifurcation method showing the parameter space of two cuts, “CUT1”and “CUT2”. The number of background events in the signal region (i.e., region “A”) was A events. If the two cuts are uncorrelated, that is, if the rejection of a cut does not dependon the rejection of the other cut, the ratio of the number of background events in region“A” to region “B” must be equal to the ratio in region “C” to region “D”, i.e., A/B = C/D .Background events in the signal region are therefore obtained from the relation A = BC/D .In practice, the present bifurcation analysis was done through two branches. A “nor-malization branch” analysis was used to obtain the number of events in “B” region. A“rejection branch” study was done to get the ratio of
D/C . The rejection was definedas R = ( C + D ) /C . The background level in the signal region was then estimated as N Bkg = B/ ( R − IG. 12: Pictorial explanation of the bifurcation method. Background level in Region “A” canbe estimated from the observed number of events in the other regions, if CUT1 and CUT2 areuncorrelated. cut categories, and B was estimated in the same way as the “first” bifurcation.To check if the two bifurcated cuts were uncorrelated, cuts were loosened simultaneously(as described in Section IV A). The loosening factors were controlled by the predicted back-ground functions, in which the loosening factors were the inputs and the outputs were the cutpositions. By design the functions provided the relative background level and the acceptance.The background level provided by the functions should agree with the observed number ofevents within the newly defined regions if CUT1 and CUT2 were uncorrelated. This methodthus provided input to the evaluation of systematic uncertainties (Section III G 10).
3. Analysis Strategy
The data analysis used the following key steps to determine the selection criteria, evaluatethe background level, investigate the systematic uncertainty, measure the acceptance andfinally obtain the branching ratio. 37
Data were first reconstructed and processed with a number of selection criteria toremove obviously bad events. Then the surviving events were divided into 1/3 and2/3 portions, in which three sub-samples were also skimmed out according to differentbackground features (Section III E). • Blind analysis was adopted in designing, calibrating and tuning the selection criteria.Signal-like and background samples were taken from the monitor trigger samples whenapplicable. When the three skimmed sub-samples were used to have the same exper-imental features of the signal, at least one critical selection criterion was inverted toavoid examining the signal region (Section III F). • The background level was initially evaluated by applying the bifurcation method tothe 1/3 data and controlled to be much less than one event by tuning the selectioncriteria. At least two uncorrelated cuts with large background rejections were chosen.data samples (Section III G 1-III G 5). • The final cut positions were optimized with respect to the signal to background levelestimated from the 1/3 data. This was achieved using the predicted backgroundfunctions estimated in the bifurcation analysis (Section III G 6-III G 7). • Correlations between the bifurcated cuts were checked by conducting a series of re-evaluations of the background levels outside the signal region. A study of single cutfailure was also conducted to investigate a possible flaw in the technique. To avoidthe potential bias of the 1/3 background study, the final background evaluations weretaken from the 2/3 portion (Section III G 8-III G 9). • Acceptances were measured with the monitor trigger samples except for the signalphase space, the trigger efficiency and those that could not be extracted from themonitor trigger data. These were obtained from Monte Carlo. The branching ratioof K π was used to validate the acceptance measurement. Single event sensitivity wasobtained from the total K + exposure and the acceptance. (Section III H). • The signal region was examined. Events observed in this region were all considered asthe signal candidates (Section III I). 38
The branching ratio was obtained from a likelihood analysis incorporating the pre-dicted background rate and acceptance within the signal region (Section IV).
C. Track Reconstruction
Throughout this analysis, the events were reconstructed under the assumption that theywere K + → π + ν ¯ ν events with only a single π + track in the detector.
1. Beam Time Measurements
To fully reconstruct an event, the initial time of the beam particle was required. Thebeam instruments provided several beam times from TDC and CCD measurements on the C K , the C π , the BWPC and the B4. For the TDC measurements, offline analysis treatedall PMT hits coincident with each other as a cluster. The average time of the TDC hits ineach cluster gave t C K , t C π , t BW and t BM , respectively. The CCD measurements were usedto discriminate cases with more than one particle in a beam (referred to as pileup).
2. Clustering in the RS
The track reconstruction routine started by finding clusters in the RS. The hit countersof a positively-charged track (the track counters) were searched for using the TDC timinginformation in the RS. A good T · DS . Fromthis T · T · t rs ) was computedby averaging all the time measurements of the track counters. The stopping counter wasdefined as the one in the outermost layer which was in the most clockwise direction. The T · . Tracking in the UTC When an RS cluster was established, the UTC tracking routine started searching forclusters of hit anode wires in the x − y plane [59]. Hit wires in each super-layer weregrouped into clusters based on their spatial proximity to one another. A straight line fitprovided a crude vector in each super-layer. The solutions due to the left-right ambiguitywere included at this stage. These vectors were then linked to form a track segment inthe UTC. A circle fit was performed with a set of drift distances with left-right ambiguityresolution. The radius of the circle gave the measurement of transverse momentum.If a UTC track in the x − y plane was found, the corresponding track projection on the φ − z plane ( φ was defined as the revolution angle with respect to the closet approach tothe vertex) was then sought in the clusters of hit strips on the UTC cathode foils. A timewindow of ±
15 ns was used to reduce the accidental hits. The calculation of the z positionfor a cluster adopted the ratio method suggested by Ref. [70], which used the three stripswith the highest ADC counts to derive the centroid and to reduce the bias on the z positionmeasurement. A straight line fit was performed in φ − z plane if the z hits were found in atleast 3 foils, thus determining the slope and intercept. The slope was then used to convertthe measured transverse momentum to the total momentum.In case of more than one track pointing to the same T · Kµ
4. Tracking in the Target and B4
Traveling almost parallel to the target fibers, the incident K + deposited significant energyin each fiber (usually > BS . The daughter π + traveled nearly perpendicular to the target fibers, and thus left less energy in each fiber(1 MeV on average) and was in coincidence with the t rs . The BS , t rs and fiber times,positions and energies were the key elements to identify the K + and π + fibers. All the K + π + fibers were linked to form a K + cluster and a π + cluster. A good event should onlyhave one K + cluster and one π + cluster.After a UTC track was found, the target pattern recognition routine started to look forthe fibers belonging to the K + path and the π + path separately on a 1 cm wide strip alongthe UTC track extrapolated into the target. The corresponding energies for K + and π + ( E K and E tg ) and times ( t K and t π ) were calculated from the sum and average in the clusters,respectively. The range of π + in the target ( R tg ) was calculated as the helix traversed bythe π + from the K + decay position to the inner surface of the IC with the polar angle θ correction.Because of ambiguity in the entry and stopping ends in the pattern of the incoming beam,the target reconstruction routine used the B4 measurement on the K + entrance point, whichwas reconstructed by clustering the hits in the two B4 layers to determine the beam time t BM , the energy-weighted beam position and the energy loss in B4. The cluster with t BM closest to the t K was chosen as the one caused by the K + beam. A 0.36 cm positionprecision was obtained by the B4 hodoscope in the x − y plane. With the B4 positionmeasurement, the target reconstruction was repeated to give a better pattern recognition.The K + decay vertex in the x − y plane was determined by the K + fiber closest to the UTCtrack but furthest from the K + entrance point determined by the B4, while the z positionwas calculated from the UTC track extrapolation in the φ − z plane.Target CCD information improved the pattern recognition and π + energy measurementwhen pileup occurred in a K + fiber. When t K and t π were separated by more than 2 ns, theCCD pulses in all of the K + fibers were studied to identify any hidden π + energy.Isolated hit fibers outside the 1 cm search strip and within the time window of t rs ± π + passage and K + decayvertex were determined in the target. The procedure of UTC track fitting was repeatedwith this additional information. This aided the resolution of the left-right ambiguity in theUTC track reconstruction. Iteration of the target reconstruction was also performed withthe improved UTC track. 41 . Track Passage in the IC An allowed IC hit pattern was either one or two adjacent sectors per event. The ICprovided energy loss ( E IC ) and time ( t IC ) measurements. If there was an IC sector crossing,the E IC was from the sum of two IC sectors and the t IC was from the energy-weightedaverage. It was observed that 1% of tracks had extra hits in the IC’s, confounding themeasurements of energy and time in IC’s. This was resolved by using TD information inaddition to the TDC and ADC information. The t IC was always the one closest to the t rs and, the corresponding energy was taken as E IC . The range R IC was computed as thelength of the extrapolated UTC track from the inner to the outer IC radius.
6. Tracking in the RS and RSSC
With a charged track reconstructed in UTC and target, the tracking in RS started fromthe previously found RS cluster. The stopping counter was first analyzed by fitting the TDinformation with a double pulse assumption to find a π + → µ + decay signature. This alsodetermined the µ + energy ( E µ ) from the π + decay at rest.A sector crossing point as illustrated in Fig. 13 was searched for in the RS. A K + → π + ν ¯ ν candidate should not have more than 2 sector crossings. Precise position mea-surement in x − y plane was obtained from the sector crossing points. The z positions weredetermined by using the end-to-end time differences in the hit counters except the T-counter.The average z position resolution was observed to be about 4 to 5 cm.Another precise position measurement was provided by the RSSC’s. All the adjacent hitstraw chambers in the RSSC of the hit sector were grouped to form a cluster. The candidatecluster in each sub-layer of RSSC was defined as the one closest to a series of arcs drawnfrom the UTC track extrapolation to the T counter through the sector crossing point(s).The x − y position was obtained from the average of two sub-layers in the same RSSC. Tominimize the effect of cross-talk, the earliest hit in one sub-layer was chosen as the true hit.The z measurement was given by the time difference, and the precision was about 1.5 cm.An RS track fit in the x − y plane used the entrance point provided by the UTC trackextrapolation, the sector crossing point(s), the RSSC hit position(s) and the expected pathlength predicted by the energy losses in the RS layers, taking into account π + track propa-42 IG. 13: Illustration of the Range Stack track fitting in x − y plane. All the relevant definitionsare given in this plot. The arc represents the extrapolation of the fitted UTC track. gation with energy loss given by the Bethe-Bloch equation in a 1 Tesla magnetic field. The χ of the fitted track was minimized by changing the incident momentum and the angle atthe entrance to the RS (Fig. 13).The total energy loss of the track in the RS ( E rs ) was obtained by summing up all theenergy losses in the track counters, with E µ from the fit to pulse shape subtracted from theenergy in the stopping counter. The range in the RS ( R rs ) was calculated from the pathlength of the fitted track with the polar angle correction. The range in the stopping counterwas estimated from the π + energy loss.
7. Kinematic Measurements of a Track
The total range R and energy E of the track were calculated as R = R tg + R IC + R rs and E = E tg + E IC + E rs , respectively. The total momentum P of the charged track was obtainedfrom the UTC with corrections due to energy loss in the target and IC. Since the momentumreduction in both the target and IC were calculated from the range, some correlation between43 and R was expected, and thus they should not be treated as uncorrelated in the bifurcationanalysis. All of these three kinematic measurements also included tiny contributions from theinactive materials in the UTC. In this analysis, the momentum, energy and range resolutionswere measured to be 1.1%, 2.8% and 2.9% (RMS), respectively, from a study of K π events(Table II). D. Monte Carlo Simulation
Detector responses were modeled by a Monte Carlo simulation package, which was de-veloped for the E787 experiment and maintained or improved for E949. The package usedseveral subroutines from the electromagnetic-shower simulation package EGS4 [71] and anumber of routines written specially for the experiment, including all of the detector ele-ments, except for the beam instrumentation upstream of the target. Simulation sampleswere generated with the same format as the data except for omission of the pulse-shapeinformation and most of the beam counter information.
1. Simulation of K + Propagation
The simulation of K + propagation started from a beam file, which contained K + eventswith a list of measured parameters: the K + stopping position, the t K , the number of K + fiber hits, the number of accidental fiber hits, the B4 hit position, the stopping target fiberelement, the time, energy, fiber element for each K + and accidental hit in the target. Thisfile was obtained from an analysis of the stopping distribution of the Kµ K + propagation hadexactly the same target patten of the data in the simulation. Every K + decay started fromthe stopping fiber at the given K + stopping position. The Kµ
2. Simulation of K + Decay Product
The K + → π + ν ¯ ν decay was generated with the matrix element of semileptonic K + ℓ decay via the V-A interaction, while the K µ decay and the K π decay were generated via44ure phase space. Among the K + decay products, photon and electron interactions andtheir energy deposits were calculated using the routines from EGS4. For charged particles,the energy deposits were calculated by adding the energy losses of each ionization along thesteps taken by the particles. The number of ionization and excitation events was determinedby dividing the total average energy deposited along the step, obtained using the Bethe-Bloch formula, by the minimum energy that a particle lost in a collision. Multiple Coulombscatterings of charged particles with various nuclei in the detector were calculated accordingto the theory of Moliere [72], with corrections for the spin of the scattered particle and theform factor of the nucleus [73]. Hadronic interactions of positively charged π + ’s in the plasticscintillators were calculated using a combination of data and phenomenological models [74].An option in the simulation package allowed users to turn off nuclear absorption reactionsand decays in flight of π + ’s. This was useful in the study of the acceptance.
3. Simulation of Trigger
All the trigger conditions were simulated except for the DC , L . L . Kµ Kµ Kµ x − y plane.
4. Comparison between Data and Simulation
The performance of the Monte Carlo simulation was checked by comparing kinematicresolutions between data and simulation as given in Table II for the K π decay mode. A0.15 cm deviation was observed in the range resolution, which is still not understood. Thisdifference could have affected the acceptance estimate and was investigated when performingthe acceptance study (see Section III H 5). It should also be emphasized that the main role ofMonte Carlo simulation in the E949 experiment was to estimate the acceptance factors thatcould not be obtained from real data (e.g. geometrical and relevant trigger acceptances).45 xperiment σ P (MeV/ c ) σ R (cm) σ E (MeV)Data 2.299 ± ± ± ± ± ± K π peak. E. Data Processing and Pre-selection
Data were stored on the DLT tapes with a total data size of 7 Tera bytes. Two stepsof data processing (“Pass 1” and “Pass 2”) were taken to reduce and skim the data to areasonable volume.
1. Pass 1
Pass 1 involved filtering cuts, which consisted of event reconstruction quality cuts andloose µ + background rejection cuts. Runs with trigger or hardware problems that couldnot be corrected offline were removed from the data analysis. Tracks were required to besuccessfully reconstructed, not to stop in those detector elements which had known hardwareproblems and to have momentum ≤
280 MeV/ c .A π + → µ + double pulse was required to be found by the fit to the TD pulse in thestopping counter and no extra hits were found in the other 3 sectors associated with thestopping counter TD channel. Since photon activity around the stopping counter could causeconfusion in the energy measurement, events with sector crossing in the stopping layer wererejected. Also rejected were the events with a charged track that came to rest in the supportmaterials for the second RSSC layer embedded between the 14 th and 15 th RS layers. Pass 1reduced the data volume by a factor of two.
2. Pass 2
Pass 2 involved cuts that were applied to the sample of events surviving Pass 1 to skimthe data into three categories according to the event features. Each category was uniformly46ivided into a 1/3 portion and 2/3 portion. Pass 2 consisted of five data skimming criteria:quality of the target reconstruction, loose photon veto, quality of the π + → µ + doublepulse fitting result, single beam K + requirement and delayed coincidence cut. These are asdiscussed below and listed in Table III. • The target reconstruction required that the K + decay vertex be inside the targetvolume, | t K − t BM | < | t π − t IC | < E IC be consistent with that expectedfrom the calculated π + range. • The loose photon veto rejected events for | t − t rs | < | t − t rs | < . | t − t rs | < . E RS > . | t − t rs | < E tg > . t was the time measurement in each photon veto counter. • The π + → µ + sequence required a µ + decay pulse in the stopping counter and theabsence of hits within ± . µ + time around the stopping counter. • The beam requirements for signal events were such that the energy loss in the B4 wasgreater than 1 MeV, the t BM differed from t rs by more than 1.5 ns, and the numberof C π hits was less than 4 with | t C π − t rs | < . • The delayed coincidence required t π − t K > K π , µ + and beambackgrounds, respectively, for the 2/3(1/3) portions. These six skims of Pass 2 output datafacilitated the studies to develop the final selection criteria and evaluate the backgrounds.Signal candidates, if any, remained in the six streams. Study of selection criteria using thesesamples was always done with at least one cut inverted to ensure that a blind analysis wasconducted. F. Selection Criteria of post Pass 1 and Pass 2
Further selection criteria were designed and applied in order to gain more backgroundreduction. According to their characteristics, the selection criteria were classified into fourcategories: single beam K + selection criteria, kinematic reconstruction, π + identification47 uts Skim 1 & 4 Skim 2 & 5 Skim 3 & 6Target reconstruction √ √ √ Photon veto √ √ π + → µ + sequence √ √ Beam K + requirement √ √ Delayed coincidence √ TABLE III: The cuts used in Pass 2 for selecting the data streams. Skim 1(4) , 2(5) and 3(6) wasused for studying the K π , µ + and beam background, respectively. See text for more details. and photon veto. All the cuts were selected to optimize the background rejection and thesignal acceptance. Data samples used for this study came from either the 1/3 data (fromSkim 4 to Skim 6) or the monitor trigger events defined in Section II H 3, depending on thenature of the selection criteria that were studied.
1. Single Beam K + Requirements
The beam cuts were used to identify beam particles scattering either in the beam instru-ments or in the target and to ensure a single beam K + particle, making full use of timemeasurements from various sub-detectors, energy loss measurements and pattern recognitioninformation in both the B4 and the target as discussed below. Beam Times:
The t C K , t C π , t BW and t BM cuts were used to reject extra beam particlesscattering in the target when one of them agreed with t rs within ± K + beam intensity was high discriminator dead time was important and the time measure-ments used the CCD information in addition to the TDC information from the C K , C π andB4 hodoscope. Fig. 14 shows the single beam K + signal indicated by the Kµ K µ or K π peak events rejected in the πν ¯ ν (1)momentum distribution. Energy Loss in B4:
The tuning of the energy loss cut in the B4 hodoscope used theacceptance sample from the Kµ π + ’s in the Skim 6 sample. This cut required that the energy loss for an incoming beamparticle should be consistent with a K + ( > . IG. 14: The momentum of charged particles versus t C K − t rs and t C π − t rs . The Kµ K + events. The data plots indicate beam backgroundcontamination at beam time ( t C K or t C π ) close to t rs in the πν ¯ ν (1) trigger sample. The arrowsindicate the rejected timing regions. The statistics in these plots account for about 0.3% of N K . Delayed Coincidence:
The acceptance samples were taken from the K µ peak events inthe Kµ π + ’s in the Skim6 sample. As shown in Fig. 16, the distribution of t π − t K for K + decays at rest was anexponential, as expected, consistent with the known K + lifetime [18]. The distribution forthe scattered π + ’s shows a prompt peak around 0 ns. The delayed coincidence cut required t π − t K > IG. 15: The energy loss in the B4 hodoscope for beam K + (solid) and beam π + (dashed). The K + ’s were the K µ peak events in the Kµ π + ’s were the scattered π + ’sin the Skim 6 sample. The arrow indicates the cut position. resolution was degraded; the delayed coincidence cut was adjusted up to 6 ns delay to takeinto account the resolution. It was expected that this delayed coincidence cut had the samerejection power to the background from K + ’s decaying in flight. Beam Likelihood:
The K + stops in the target were required to have energy loss in theB4 and target consistent with that expected for the measured K + stopping position. Theseconditions helped to eliminate single beam backgrounds with a scattered π + which did nothave a consistent path length in the target. The three quantities were combined into alikelihood function. Fig. 17 shows the likelihood distributions for signal and backgroundusing the K µ peak events in the Kµ π + ’s in the Skim6 sample. Pileup cut:
The pulse shape recorded with the target CCD’s provided pileup information.The time development of the output signal was fitted with both single- and double-pulseassumptions. If the fitted pulse was more likely to be a double pulse and the time of thesecond pulse was coincident with the K + decay time, the event was rejected. Pathology Cuts:
In the target scintillator, charged particles sometimes underwent nuclearinteractions or photons had electromagnetic showers. Complicated target patterns could50
IG. 16: Distribution of the time difference, t π − t K measured by the target, for the K µ peak events(solid) from the Kµ π + ’s (dashed) from the Skim 6 sample.Beam π + ’s can be assigned a time t K by the target fiber reconstruction due to mis-identification.FIG. 17: Beam likelihood for K + decays at rest (solid) from the Kµ π + ’s (dashed) from the Skim 6 sample. The energy loss of K + in the target was required to begreater 25 MeV and the number of K + fibers was required to be greater than 2 before evaluatingthe beam likelihood. • Target π + cluster with an identified kink. This was an indication of a hard scatteringprocess, which might lead to an incorrect measurement of kinematic quantities. • In the B4 the measured energies derived from the ADC and CCD were required to beconsistent within 1.5 MeV and the measured times derived from the TDC and CCDwere required to be consistent within 2 ns. Inconsistency in either the times or energieswas likely due to a second beam particle. • Target π + fiber next to the decay vertex with energy greater than 3.5 MeV. Thismight indicate a K + fiber being mis-identified as a π + fiber, causing an incorrectmeasurement of kinematic quantities. • Target π + fiber with energy greater than 5 MeV. This might imply a K π or a radiative K µ event with a photon hiding in the π + fiber. • If a target edge fiber was identified as a K + fiber with a time within 3 ns of the nearbyIC, the event was rejected. This feature was often an indication of a double beambackground. • Target fibers on the opposite side of the π + track with respect to the vertex havingenergy greater than 2 MeV within ± t π . This usually indicated a K π eventwith a photon conversion opposite to the π + . • Target π + track on the opposite side of the π + track with respect to the K + decayvertex. This usually indicated a double beam or a cosmic ray background in additionto the first K + beam particle. • Target edge fiber greater than 5 MeV within ± t rs . This suggested some photonactivity in the edge fibers, or a double beam event with the second beam hiding in thetrack. • Any activity found in either the UPV or RV within ± t rs .52ost pathology cuts were related to target pattern recognition. These backgrounds couldnot be tagged with the usual procedure of background analysis until the outside-the-boxstudy given in III G 8.
2. Decay π + Kinematic Requirements
Once the π + kinematic reconstruction routines produced the range-, energy- andmomentum-related quantities, cuts on the kinematic values (referred to as KIN cuts), wereidentified and divided into several sub-groups which are described below. Fiducial Cuts:
The π + stopping layer was required to be RS layers 11-18. No chargedtrack was allowed to stop in the RSSC layer embedded between the 14 th and the 15 th RSlayer. Events were also rejected if the stopping layer was 14 with an RSSC hit found inthe same RS sector or one sector clockwise of the stopping counter. The cosine of the polarangle for a charged track was required to be within ± .
5. The z position from the UTCtrack extrapolation to each RS layer was required to be as narrow as ±
30 cm to reject K µ backgrounds with longer path lengths in RS. The effective UTC fiducial volume was definedto be | z | <
25 cm at the UTC outermost layer.
Signal Phase Space:
The phase space cuts required that the momentum, kinetic energyand range of a π + track should be in 211 ≤ P ≤
229 MeV/ c , 115 ≤ E ≤
135 MeV, and33 ≤ R ≤
40 cm. As the resolution of kinematic quantities depended on the azimuthaland polar angles of the π + track, the lower limits of the phase space region were furtherdefined by the requirements (referred to as K π kinematic cuts) that Pdev = ∆ P/σ P ≥ . E/σ E ≥ .
5, and Rdev = ∆
R/σ R ≥ .
75 where the ∆ P , ∆ E and ∆ R were thedeviation from the K π kinematic peak positions, and σ P , σ E and σ R were the correspondingresolutions, which were correlated with the azimuthal and polar angles. Tracking Quality in Target:
Good target tracking relied upon a consistent pattern of hitsin the target. This required the nearest target K + fiber of the track to the B4 hit position tobe no more than 2 cm away. The K + decay vertex was required to be nearest to the extremetip of the K + cluster. No more than one fiber gap was allowed between the K + decay vertexand the closest π + fiber. A target track could include either π + fibers with photons orphoton fibers mis-identified as the π + fibers, leading to an incorrect energy measurementor event classification if the photon veto also failed. The π + fibers were therefore examined53sing likelihoods based on the comparisons of the time, energy, and distance to the trackbetween those values expected from the simulated K + → π + ν ¯ ν sample and the measuredvalues from the Kπ Tracking Quality in UTC and RS:
At least 4 hits out of 6 cathode foils were required toensure a good measurement of a track in the UTC. Further quality checks were developedwith respect to the K µ momentum peak, taking account of all the possible circumstancesthat could lead to an incorrect momentum measurement, such as no hit in the outermostfoil layer, less than 12 layers of anode wire hits, overlapping tracks, or too many wire hitsin a cluster being excluded from the fit. The cuts were adjusted so that the momentumresolution effect did not give a significant contribution to the K π and µ + background es-timates (Section III G). Another good way to ensure a good UTC measurement was torequire consistency among UTC, RS and RSSC’s. In the x − y plane, this could be achievedby requiring consistency between the UTC extrapolation and the positions of the sectorcrossing and the RSSC hits. Similarly, in the φ − z plane consistency was required betweenthe UTC extrapolation and the z positions measured by RSSC and RS. To establish theserequirements, the µ + ’s in Skim 5 were selected with all the other cuts applied, except thatthe maximum momentum cut was not applied and the π + → µ + → e + decay sequence cuts(see Section III F 3) were inverted, and the π + ’s in Skim 6 were selected with all the cutsapplied, except that the cuts on single beam K + were inverted (Section III F 1). These signaland background samples were also used in the relevant studies of energy loss in the RS andrange-momentum consistency in UTC and RS. Energy Loss in RS:
It was noted that a µ + from the K µ might fake a π + due to scattering.In addition a π + from the K π might fake a signal due to either a photon or an accidental hitbeing hidden in the track counters. The resulting background was removed by comparingthe energy measurement in each RS layer with the expected value from the range. Theenergy deviation was required to be within ± σ ’s for each RS counter as shown in Fig. 18.In addition to each layer, a probability for energy loss consistency was also calculated withall track counters except for the T-counter, the stopping counter and the counters withsector crossings. As shown in Fig. 19, the probability cut provided good separation between π + ’s and µ + ’s. Another way to reject the µ + background was to calculate the likelihoodfrom the ratio of the expected ( E iexpt ) and measured ( E imeas ) energy of a track in the i -thlayer, ∆ E ≡ log E iexpt − log E imeas , taking into account the Landau tail of the expected energy54 IG. 18: Distributions of the maximum energy deviation for π + ’s (solid) and µ + ’s (dashed) in RS.The cut positions are indicated by the arrows. distribution. This calculation was done up to and including the layer prior to the stoppinglayer. Fig. 20 shows good π/µ separation using the calculated likelihood value. It was notedthat this cut on the likelihood was correlated with the cuts on the energy deviation and theprobability given above. Range-energy Consistency in IC and Target:
Despite the poor resolution of energy mea-surements in the IC and target, they could still provide a certain level of particle identifi-cation in addition to that from the RS. Signal and background samples used in this studywere from the Kπ Kµ E IC and the expectedvalue from R IC was required to be between − R tg ) in cmand energy ( E tg ) in MeV were used to cut events in which R tg >
12 cm, E tg >
28 MeV,9 . × E tg > × R tg or 10 × E tg < . × ( R tg −
2) to reject background with a photonhiding along the π + track in the target. Range-momentum Consistency in UTC and RS:
This cut was used to check whether therange of the charged track was consistent with that for a π + . The range deviation in RSwas defined as χ rm = ( R rs − R utc ) /σ R , where R utc was the expected range calculated fromthe momentum measured by the UTC with a π + hypothesis, and σ R was the uncertaintyof the measured range as a function of the momentum. The µ + ’s and π + ’s were selected55 IG. 19: Distributions of the probability of energy loss consistency in the RS dE/dx measurementfor π + ’s (solid) and µ + ’s (dashed). The arrow indicates the cut position. from Skim 5 and Skim 6 samples, respectively, as used in the study of energy loss in the RS,except that the maximum momentum requirement was also applied in Skim 5 to removethe K µ range tail events. The distributions of the range deviation for π + ’s and µ + ’s wereshown in Fig. 21. Good π/µ separation was observed. π + → µ + → e + Decay Sequence
All cuts for identifying the π + → µ + → e + decay sequence were put into a special group(referred to as TD cuts). The TD information recorded the pulse shape, providing a tool torecognize the decay sequence. As can be seen below, the TD cuts were independent of thekinematic reconstruction and photon veto and could be used in the bifurcation study. Thesignature for this decay sequence was: • Three energy deposits (pulses) corresponding to the π + kinetic energy, π + → µ + ν µ and µ + → e + ν e ¯ ν µ decays were found in the stopping counter. • The kinetic energy of the µ + from π + → µ + ν µ decay was 4.1 MeV, but due to satura-tion the observed energy was about 3 MeV [75]. Since the path length of the µ + was56 IG. 20: Likelihood distributions of the RS energy measurement for π + ’s (solid) and µ + ’s (dashed).The arrow indicates the cut position. ∼ . µ + exited the stopping counter without depositingmore than 1 MeV was only ∼
1% of π + decays. • The e + from µ + → e + ν e ¯ ν µ decay has a kinetic energy of E e <
53 MeV. Most of the e + ’s exited the stopping counter and deposited energy in the other RS counters.The three energy deposits from the π + → µ + → e + decay sequence should be observed bythe TD’s at both ends of the stopping counter.For the µ + background, only two pulses due to muon kinetic energy and decay wouldbe produced. A µ + could fake a π + when an extra pulse was detected in addition to theexpected two pulses. To suppress the µ + background, two stages of cuts were imposed.At the first stage evidence for the π + → µ + ν µ decay was sought. The pulse developmentin the stopping counter as recorded by the TD’s was fitted with a single- and double-pulsehypothesis in an interval of ∼ π + lifetimes (typically 104 ns). The template shapes usedin the fit were derived from the average of measured pulses from µ + traversal for each endof each RS counter. In addition, a correction was applied to the template shape to take intoaccount the change in shape due to propagation along the counter. The parameters of thesingle-pulse fit were the time, the total area of the pulse and a constant corresponding to apedestal of typically 3 TD counts. The parameters of the double-pulse fit were the time of57 IG. 21: Distributions of the range deviation in the RS for π + ’s (solid) µ + ’s (dashed) tracks. Thearrow indicates the cut position. the first pulse, the time difference of the two pulses, the total pulse area, the fractional areaof the second pulse and the pedestal. A fit to a triple-pulse hypothesis was attempted ifevidence for a third pulse was found based on a rudimentary analysis of the TD informationor if evidence for the π + → µ + ν µ decay from the double-pulse fit was lacking. The twoadditional parameters in the fit were the time difference of the third pulse with respect tothe first and the fractional area of the third pulse. The results of the single- and double-pulsefit hypotheses for π + → µ + ν µ decay were shown in Fig. 22. Loose requirements were firstapplied on the observed µ + energy 1 < E µ <
14 MeV and on the relative quality of theresults of the single- and double-pulse fits, R (1) > R (2) >
2, and R (1) × R (2) > µ + → e + decay could fake the three-pulse π + → µ + → e + decaysequence. • π + time accidental: Accidental activity produced the first pulse, while the µ + → e + decay gave the second and third pulses in the timing sequence. • Early µ + decay: The µ + → e + decay occurred at an early time ( ≤
100 ns), producing58
IG. 22: Results of fits with the single- and double-pulse hypotheses to the TD pulse shapes inthe stopping counter. The solid curve is for the double-pulse hypothesis and the dashed curve isfor the single-pulse hypothesis.Quantity Definition or use E µ Energy of the second pulse T µ Time of the second pulse χ n ( i ) χ for the n -pulse hypothesis for the counterend i ( i = 1 , R ( i ) χ ( i ) /χ ( i )log ( R (1) · R (2)) Neural net inputlog ( χ (1) · χ (2)) Neural net input dz = z π − z µ , neural net input component dt Time difference between both counter endsfor the second pulse, neural net inputTABLE IV: Definitions of quantities determined by the pulse-fitting in the stopping counter. The z positions were determined from the energy ratio between the two ends of the stopping counter.The z position of the nominal π and µ pulse was z π or z µ , respectively. The term “second” pulseidentifies the µ + -candidate pulse. The neural net is described in the second stage of cuts. ut π + time Early µ + µ + time Tailaccidental decay accidental fluctuation π + time consistency Cut √ µ + → e + decay requirement √ Cut on µ + time accidental √ Cut on µ + time accidental √ in the track countersNeural net π + → µ + decay cut √ √ TABLE V: List of the backgrounds targeted by π + → µ + → e + decay sequence cuts. the second pulse, and accidental activity was identified as the third pulse. • µ + time accidental: The µ + → e + decay made the third pulse, while the second pulsewas produced by accidental activity occurring between the µ + stop and decay. • Tail fluctuation: A fluctuation in the falling edge of the first pulse was identified asthe second pulse. The decay positron from the µ + decay made the third pulse. π + Time Consistency Cut:
This cut suppressed the π + time accidental background.When accidental activity made the first pulse and a charged track made the second pulsein the stopping counter, the timing of the first pulse obtained by the TD ( t π,T D ) was notcoincident with t RS obtained from the other RS counters along the track. Events wererejected if | t π,T D − t RS | > . µ + → e + Decay Requirement:
The positron from the µ + → e + decay (the third pulse)generally deposited energy in the stopping counter and other neighboring counters as de-picted in Fig. 23. The positron finding started by looking for a cluster of TDC hits in the RScounters in the region within ± ± ± . z positions of the hits in the cluster, obtained from the end-to-end time differences of thehits, were also required to be consistent with the z position in the stopping counter. If thecandidate was due to a track that passed through the stopping counter, then hits might be60 IG. 23: Schematic view of signal (left) and background (right) of the µ + → e + decay. Eachrectangle represents an RS counter. The central counter represents the stopping counter and theshaded rectangles represent hit counters. The arrow indicates the possible direction of the positronor charged track producing the hits. found on both sides of the stopping counter (Fig. 23). The early µ + decay background wasremoved by requiring that the second pulse from the TD pulse fitting agreed with the timeof the cluster. Cut on Accidental Activity:
Accidental activity in the stopping counter was frequentlyassociated with activity in other RS counters as well as the BV, BVL and EC. Hence,events with activity coincident with the second pulse in the stopping counter were targetedfor rejection. The time windows and energy thresholds for the various subsystems in RS,BV, BVL and EC were optimized in order to have the highest rejection power at a givenacceptance value of 94% for this cut. Events were rejected if the energy sum of the hitswithin a time window in any of the subsystems was greater than the threshold. There wasalso a kind of accidental activity that overlapped the charged track and made a secondpulse in the stopping counter. To reject this accidental background, fits were performedto a double-pulse hypothesis in the two RS counters along the track prior to the stoppingcounter. If the time of the fitted second pulse was within ± χ ratios of the single- to the double-pulse fit hypotheses were greater than 4, then the eventwas rejected. Neural Net π + → µ + Decay Cut:
The tail fluctuation background mimicking the energydeposit for a µ + at the falling edge of the π + -induced pulse was characterized by a smalldecay time and a low pulse area in the second pulse. The variables shown in Fig. 24 anddescribed in Table IV differed for events induced by π + → µ + ν µ and µ + → e + ν e ¯ ν µ decays.61 IG. 24: Distributions of the input variables for the Neural Net function in π + → µ + ν µ induced(solid) and µ + → e + ν e ¯ ν µ induced (dashed) decays. The energy of the second pulse (a), time ofthe second pulse (b), log of the product of the single-pulse fit χ ’s for both ends (c), z positiondifference between the first and second pulses obtained from the energy ratio of both ends (d), logof the product of the χ ratios of single- to double-pulse fits for both ends (e) and time differenceof the second pulses in both ends (f). Application of a fixed cut to each variable would cause a non-negligible acceptance loss. Inorder to achieve a higher acceptance at the same rejection as the fixed cuts, a Neural Network(NN) technique was adopted. The NN function was derived via a Multi-Layer Perceptionprogram incorporated in the library of Physics Analysis Workstation (PAW) [76]. To createthe NN function, the scattered π + ’s and K µ range tail events in the πν ¯ ν (1) trigger, whichpassed all other π + → µ + → e + decay sequence cuts, were used as signal and background62 IG. 25: Distributions of the outputs of the NN function for π + (solid) and µ + (dashed) events.Events with an output of the NN function less than 0.76 were rejected. samples, respectively. A 5-variable NN function was obtained using the six variables shownin Fig. 24 with the differences in z position and time combined to create a single inputvariable, χ ( z, t ) ≡ ( dz/σ dz ) + ( dt/σ dt ) , (24)where the z position and the time were obtained from the energy ratio of both ends, andtime difference of the second pulses in both ends. The resolution of dz ( dt ) was σ dz ( σ dt ).Fig. 25 shows the distributions of the output of the NN function for π + ’s and µ + ’s. Therejection of the NN π + → µ + decay cut as a function of the acceptance is shown in Fig. 2663 IG. 26: Rejection of the NN π + → µ + decay cut as a function of the acceptance. The intersectionof the vertical and horizontal dashed lines shows the rejection and acceptance at the nominal cutposition.
4. Photon Veto
To achieve the background level much less than one event, the total π rejection wasrequired to be of order of 10 . A rejection factor of ∼ was already achieved online, leavinga further ∼ rejection factor to be achieved by the offline analysis. The correspondingphoton veto cuts (referred to as PV cuts) were used to identify the photon activities detectedby all the PV counters.A search for the photons coincident with the track time was performed in the subsystemsof the BV, BVL, RS, EC, target, IC, VC, CO and µ CO. The timing resolution of each photondetection system was a key ingredient in determining the time window for the PV due to the64alse veto rate in the high rate experimental environment. Fig. 27 shows the measurementsof the timing resolution for each system as a function of the visible energy. Since the ECconsisted of 4 rings and the inner ring (ring1) was exposed to high accidental rates fromthe beam, the corresponding time resolution was worse. The time window and energythresholds in each subsystem were optimized by adjusting the cut positions to maximizerejection for a given acceptance. The rejection sample was from Skim 4 with K π events,while the acceptance sample was from the Kµ K π peak, the measured range, energy and momentum of the charged track were requiredto be within three standard deviations of the nominal values. For the K µ , the momentumof the charged track was required to be within three standard deviations of the nominalvalue, and the range was required to be longer than 37 cm to remove any event which couldcontain photon(s). The selected K π and K µ events were also required to pass the KINcuts and the beam cuts.The optimization process [77] started from the initial set of cut parameters. For a newset of parameters, the rejection and acceptance were re-measured. Only one subsystem’scut parameters were varied at a time. If the rejection increased without losing acceptanceor the acceptance increased without losing rejection, the set of parameters was regarded asa good input for the iteration. More preferable cases occurred when both the rejection andacceptance were improved. Fig. 28 illustrates this optimization process for the photon veto.The optimization process continued until no more gain was obtained in rejection withoutlosing acceptance. The boundary point was measured at every given acceptance position.As a result, a profile curve, which gave the maximum achievable rejection, was obtained.Fig. 29 shows the offline rejection of the photon veto cuts against the K π backgroundas a function of acceptance. The time window and energy threshold for each category aretabulated in Table VI. As a reminder, the total offline photon veto is not the simple productof the rejections listed in Table VI due to mutual correlations. G. Background Evaluation
To have an unbiased result, the signal region was always masked until all the backgroundevaluation studies were completed. As described in Section III A, the stopped K + decaybackground and the beam background were subdivided into the following categories:65 IG. 27: The timing resolution as a function of visible energy in various PV counters. • K π background, • K µ range tail background, • µ + band backgrounds, • single beam background, • double beam K + − K + background, • double beam K + − π + background, and66 ategory Time window (ns) Energy threshold (MeV) RejectionBV ± ± ± ± ± ± ± ± ± µ CO ± IG. 29: Offline rejection of the photon veto cuts against the K π background as a function ofthe acceptance. The crossing point of the vertical and horizontal lines shows the rejection andacceptance at the cut position. • CEX background.Except for CEX, all background levels were estimated using the data by means of thebifurcation method. Table VII gives the bifurcation cuts, the data stream categories andthe results for the 1/3 data sample. The background levels given in Table VII must bescaled by a factor of 3 to obtain estimates for the full sample. The 1/3 sample was studiedfirst in order to tune and optimize the cuts. Then the 2/3 sample was used to give the finalbackground estimates. When the normalization branch contained only one or few eventsin region B, a second bifurcation analysis was performed in this branch to improve thestatistics. Details of these procedures are given in the following sections.68 kg. CUT1 CUT2 Category B ( C + D ) /C BC/DK π PV KIN Skim 4 0 . ± .
11 85 . ± . . ± . K µ TD KIN Skim 5 1 . ± .
16 (4 ± × . ± . µ + band TD KIN Skim 5 2 . ± .
74 (4 ± × . ± . ± × . ± . K + − K + BWPC B4 Skim 6 0 . ± .
04 117 ±
37 0 . ± . K + − π + BWPC B4 Skim 6 0 . ± .
11 (7 ± × < . K π Background A K π decay event should have a charged track with a monochromatic momentum, rangeand energy plus two photons. Experimentally, if a K π event appeared in the signal region,the photons from the π decay must have escaped detection and the K π kinematics musthave been distorted by scattering or resolution effects as well.In the background study, the two bifurcation cuts were chosen as the PV cuts (CUT1) andthe signal phase space cuts in the KIN cuts (CUT2), since both of these could independentlygive powerful rejection of the K π background. In order to remove the contamination from µ + and beam events, the bifurcation analysis sample was selected from the Skim 4 sampleby applying the TD cuts, the beam cuts and the KIN cuts other than the phase space cuts.In the rejection branch, K π events were selected by inverting the signal phase space cutin the KIN cuts (CUT2), giving 95,797 events for the region C+D. The PV cuts (CUT1)were then applied to the remaining K π events, leaving 1,124 events for the region C. In thenormalization branch, the K π events with photon activity were selected by inverting thePV cut (CUT1). The signal phase space cuts in the KIN cuts (CUT2) were applied to theabove selected K π sample, resulting in no events ( B = 0) left in the normalization branch.To deal with the above situation and give a non-zero events in the normalization branch,another (second) bifurcation analysis was performed by separating the K π kinematic cuts69nto Edev (CUT1) and Rdev+Pdev cuts (CUT2), since the E measurement was almostindependent of R and P measurements. These two bifurcation cuts were applied sequentiallyto the selected K π sample in the normalization branch. In this second bifurcation study,the lower boundary cuts on E , R and R were removed. Changing the cut positions on theRdev, Pdev and Edev gave the number of events in the normalization branch ( N RP ) and therejection branch ( R E ) in the second bifurcation analysis, which were then used to calculatethe expected number of events in the normalization branch by means of B = N RP / ( R E − K π background with less acceptance loss. Theexpected number of events from the second bifurcation analysis was found to be ∼ R and E when estimating the π + rangein the stopping layer from the measured energy in the stopping layer. The cut positionswere chosen at Rdev > .
75, Edev > . > .
5, in order to reduce the expected K π background level to about 0.01 events level as shown in Table VII. Fig. 30 shows theexpected K π background from the second bifurcation in the 1/3 normalization branch asa function of the Edev cut position. The acceptance was measured using the simulated K + → π + ν ¯ ν sample.The expected number of events from the second bifurcation was used to give the resultfor the normalization branch, which was then used to give the estimated background levelfor the 1/3 sample as shown in Table VII. µ + Background
The µ + background consisted of the K µ range tail events and the µ + band events asindicated in Fig. 8. These µ + ’s lost energy and eventually came to rest in RS and couldmigrate into the signal region through resolution effects if the TD cuts failed.The two bifurcation cuts with the most powerful rejection were the TD cuts (CUT1)and the selected KIN cuts (CUT2), which excluded the fiducial cuts, the lower boundary inthe signal phase space cuts, the cuts on the tracking quality in target and the cuts on therange-energy consistency in IC and target as detailed in Section III F 2. The µ + backgroundsample used for the bifurcation study was selected from the Skim 5 sample by applying the70 IG. 30: The expected kinematic background events in the 1/3 normalization branch (histogramwith left axis) as a function of K π kinematic cut (Edev) position for the K π background (a),maximum momentum cut position for the K µ range tail background (b) and range-momentumcut ( χ rm ) position for the µ + band background (c). Also shown is the relative acceptance changeas the cuts (curve with right axis). PV cuts, the beam cuts and the KIN cuts not used in this bifurcation study, in order toremove the K π and beam backgrounds. In the rejection branch, there were 7,119 µ + eventsin the region C+D when inverting the selected KIN cuts (CUT2). The TD cuts (CUT1) wereapplied to the µ + events, leaving 16 events in the region C. In the normalization branch,the µ + events were obtained by inverting the TD cuts (CUT1). The selected KIN cuts71CUT2) were applied to the above selected µ + sample, leaving only one event ( B = 1) inthe normalization branch.The result from the bifurcation analysis given above had a large statistical uncertainty,since only one events remained in the normalization branch. The K µ range tail and the µ + band events were the only two possible µ + backgrounds in the signal region. The originsof these two backgrounds were due to the momentum and range resolution effects. Oncethese resolution effects were known in the signal region, a better estimate of correspondingevents B in the normalization branch could be obtained. In this analysis, the events with P >
225 MeV/ c were regarded as the K µ range tail background.The momentum resolution effects could be well described by using the momentum dis-tribution from the K µ peak events. In order to enhance the number of events in the nor-malization branch, the RS energy loss cuts were removed from the KIN cuts. Also removedwere the upper boundary cuts on the R , E and P . The K µ peak events were selected byrequiring R >
50 cm. The selected K µ peak events in the πν ¯ ν (1) trigger were found to havelonger range in the target, leading to a bias of 0.5 MeV/ c higher momentum measurementwhen applying a π + hypothesis to the contribution from the energy loss in the target. Aftersubtracting this bias for each K µ peak event, the momentum distribution was seen to bein good agreement with that from the K µ range tail events and with more statistics inthe signal region defined below 229 MeV/ c . Normalizing the number of K µ peak events tothat of the K µ range tail events observed in the region B+D gave the expected number ofevents B in the normalization branch. The expected K µ range tail background is given inTable VII. Fig. 30 shows the number of K µ range tail background events as a function ofthe maximum momentum cut position in the normalization branch.The µ + band background came from the RP resolution effects, which were well describedby the range deviation ( χ rm ) in RS. Enhancing the number of events in the study of the RP resolution effects was achieved by removing the PV cuts and the RS energy loss cutsin the normalization branch sample, since they were not correlated with the range andmomentum measurements. Within the statistical uncertainty, both distributions were seento be consistent except that the distribution without the above requirements gave higherstatistics in the normalization branch (Fig. 30). The estimated background level for the µ + band is given in Table VII. 72 . Single Beam Background In the study of single beam background, the two bifurcation cuts chosen were the offlinedelayed coincidence (DC) cuts (CUT1) and the B4 energy loss cut (CUT2). The offline DCcuts were from the precise offline time measurements from the beam instrumentation, thetarget, the IC and the RS (They were not the ones used in the trigger). The events wereselected from the Skim 6 sample by applying the PV cuts, the KIN cuts, the TD cuts, andthe beam cuts except for the DC cuts and the B4 energy loss cut. In the normalizationbranch, the single beam events were selected by inverting the DC cuts (CUT1). The theB4 energy loss cut (CUT2) was applied to the above selected single beam sample, leaving 8events in the region B for the normalization branch. In the rejection branch, 29,100 singlebeam events in the region C+D were selected by inverting the B4 energy loss cut (CUT2).The DC cuts were applied to these selected events, resulting in 4 events in the region C. Therejection factor was applied for both the π + scattering events and the K + decay-in-flightevents, since there was no reason to have different rejections for these. The backgroundestimate for the single beam background is given in Table VII.
4. Double Beam Background
As already defined in Section III A, the double beam background could be due to a K + − K + event or a K + − π + event. For this double beam background, the DC cuts wereinsufficient to remove the double beam background, but the time difference between thebeam instrumentation ( C K , C π , BWPC’s and B4) and the π + track was a good indicator.Another independent way was to use the target pattern to identify extra particles other thanthe initial K + hit.The two bifurcation cuts chosen were the BWPC timing cuts (CUT1) and the B4 timingcuts (CUT2). The events were selected from the Skim 6 sample by applying the PV cuts,the KIN cuts, the TD cuts and the beam cuts except for the BWPC and the B4 timingcuts. These cuts removed the K π , µ + and single beam backgrounds. In the normalizationbranch, there were no events left if the B4 timing cuts (CUT2) were applied. In order to givea more precise estimate, the second bifurcation was adopted. Time measurements includingthe trailing edge TDC time from the C K and C π were used to select K + − K + events and73 + − π + , separately. The bifurcation analyses were then performed using the B4 timingcuts and the target pattern recognition cuts. Results are given in Table VII. In the rejectionbranch, the double beam background events were selected by inverting the B4 timing cuts(CUT2). The K + − K + and K + − π + events were tagged by the C K and C π , separately;this resulted in 1,170 and 22,150 events for both cases. Applying the BWPC timing cuts(CUT1) resulted in 10 and 3 events observed for the K + − K + and K + − π + backgrounds,respectively. The resulting rejections and background estimates are given in Table VII. The K + − K + background was found to dominate the double beam background.
5. Charge Exchange Background
Since there was no reliable way to isolate the CEX events from the πν ¯ ν (1) trigger data,the background study could only rely on the Monte Carlo simulation. The CEX simulationneeded a number of inputs, such as the CEX re-generation rate as a function of K L energy,the K L decay vertex and the K L momentum, all of which could only be obtained fromthe real data. A special CEX monitor trigger as described in Section II H 3 was used forcollecting data with two charged tracks from the K S decay. The K S ’s were reconstructed inthe π + π − decay mode and used to measure the K S production rate, momentum spectrum,the B4 hit information, decay vertex distribution, and pattern of target K + fibers’ time andenergy. Since a K decays approximately equally to K L and K S states, the measured rateof K S decays can be used to obtain the K L production rate in the target R K L ≡ N K S ǫ K S · A P V · B ( K S → π + π − ) · N K /P S , (25)= 2 . × − , (26)where the quantities used in this calculation are summarized in Table VIII.Both K L → π + µ − ¯ ν µ ( K µ ) and K L → π + e − ¯ ν e ( K e ) decays were simulated to estimatethe CEX background level. Each decay mode was generated with the amount of K L decaysequivalent to 7 . × K L ’s. These Monte Carlo events were passed through all of theselection criteria, except for the π + → µ + → e + decay sequence cuts and the beam cuts thatwere not related to target quantities. There were 56 K e events and 21 K µ events survivingin the signal region. Therefore, the expected CEX events can be estimated by means of N CEX = (cid:16) N K e pass + N K µ pass (cid:17) × N K N MCK × F acc escription Parameter ValuesNumber of selected K S events N K S K S selection efficiency ǫ K S A P V K S → π + π − Branching Ratio B ( K S → π + π − ) 0.686Number of K + Triggers (10 ) N K P S K L Production Rate R K L . × − TABLE VIII: K L production rate and the quantities that are used to estimate it. = (56 + 21) × . × . × × . . ± . stat ) , (27)where N K e pass and N K µ pass are the numbers of K e and K µ events surviving all the cuts, N MCK was the total exposure of generated K + ’s and F acc was the acceptance of the TD and beamcuts that were not applied to in the simulation.
6. Initial Background Evaluated from 1/3 Sample
The initial total background evaluation based on the 1/3 sample was 0 . ± . stat events, which came from the results in Table VII and Equation (27). Given this relativelylow background level, the signal region was expanded to gain acceptance at the cost ofmore background. In addition to the total background level, the bifurcation analyses gavethe corresponding predicted background functions for the TD, the PV and the kinematiccuts, with which the expected background level and relative acceptance change for a givencut position were obtained. These functions were used in optimizing the selection criteria,studying the correlation of two bifurcation cuts and determining the branching ratio as well.
7. Optimization of Signal Region
The distributions of signal and background in the cut space were described well by thepredicted background functions and were used to enlarge the signal region. The cuts to75e loosened were the NN π + → µ + → e + decay cut in the TD cuts (Fig. 26), the PV cuts(Fig. 29) and the K π kinematic cuts (Fig. 30). Loosening the cuts for the beam backgroundsand the other TD and KIN cuts did not provide much acceptance gain. Simultaneouslyloosening the cut positions of the NN π + → µ + → e + decay, PV and K π kinematiccuts could increase the background levels to an unacceptable level. Instead, when one ofthe three cuts was loosened, the cut positions of the other two were kept unchanged. Therevised (extended) signal region consisted of the standard signal region plus three extensions.Hereafter, the revised and standard signal regions were referred to as the extended signalregion and the standard region, respectively. The regions created by loosening the NN π + → µ + → e + decay cut, the PV cuts, and the K π kinematic cuts were referred to as π + → µ + → e + extended, PV extended, and K π kinematic extended regions, respectively.All of which included the standard regions. The loosening factors for these three cuts, f πµe = 4 . f P V = 4 . f K π = 9 .
5, meant the corresponding background increased bythe same factor. The total estimated acceptance gain by enlarging the signal region was31% (12% from the NN π + → µ + → e + decay cut, 7% from the PV cuts, and 12% from the K π kinematic cuts).
8. Correlation and Single Cut Failure Study
The bifurcation procedure described above assumed that the two bifurcation cuts werenot correlated. This assumption was tested by comparing the predicted and observed ratesnear but outside the signal region.A schematic representation of the region near, but outside, the signal region is shown inFig. 31. A near region outside the signal region (A ′ without the black region in Fig. 31)was defined by loosening two bifurcation cuts (CUT1 and CUT2) simultaneously by thefixed factors a and b , respectively. If the predicted background functions associated withthe CUT1 and CUT2 were correct, the expected number of background events in the nearregion ( BG ′ ) was estimated by the bifurcation method as BG ′ = B ′ C ′ /D ′ − BC/D (28)If a deviation was seen between the observed and the predicted numbers in the near region,then a correlation between the bifurcation cuts could be indicated and the background76
IG. 31: Pictorial explanation of the correlation study using the events near, but outside the signalregion. The vertical axis “ a ” and the horizontal axis “ b ” are for the loosening factors, while the1 × estimate might be unreliable. The observation was performed in the same way as thebifurcation method used in the background estimation in the signal region, except that twobifurcation cuts were loosened by factors of a and b .The results of the correlation study for the K π , K µ range tail and µ + band backgroundsare summarized in Table X for the 2/3 data sample. Good agreement was found betweenthe observed and the predicted number of events. The test results were obtained from thecomparisons between the observed and the predicted numbers of events using the predicted77ackground function method.In addition to the study of correlation between the cuts used for the bifurcation method,events that passed all except for a single cut were examined to determine if each cut operatedas designed for the appropriate background mechanism. Such a study provided a way todiscover any new type of background or potential analysis flaw. In the 1/3 sample, six of theeight events that failed a single cut only were far from the cut position, while the other twoevents showed potential analysis flaws. The first flaw would artificially increase the measuredrange and momentum of π + ’s from K π decays that exited the upstream end of the targetthrough the gap between the front face of the target and the B4 hodoscope. Additional cutswith minimal acceptance loss were devised to eliminate such events. The second revealed apossible correlation between the PV cuts and the KIN cuts when using the π + polar angle( θ ) as a reference to exclude the accidental hits in the opposite site of the PV counters.The corresponding calculation used in the PV cuts was subsequently removed. All the cutsdesigned at this stage were referred to as pathology cuts as described in Section III F 1.
9. Final Background Evaluated from 2/3 Sample
Evaluation of the final background levels came from the 2/3 sample. To get the valuesfor the extended box, the corresponding values were scaled by the loosening factors given inSection III G 7. It was noted that loosening the NN π + → µ + → e + decay cut, the PV cuts,and the K π kinematic cuts could lead to a small change in the beam background levels.Therefore, the beam backgrounds were re-estimated in the extended signal region. The TDrejection and PV rejection were measured to be 445 ±
111 and 84 . ± . . ± . stat events, which was dominated by the K π background contribution. As the background distri-bution was not uniform in the signal region, the predicted background functions obtained inthe background study were exploited to interpret any possible candidate events observed andto give a proper branching ratio measurement by using the likelihood technique describedin Section IV. 78 ackground Standard Extended K π ± ± K µ range tail 0.010 ± ± µ + band 0.005 ± ± ± ± ± ± ± ± ± ±
10. Systematic Uncertainty
Systematic uncertainty in the background estimates arose from the possible correlationbetween the two bifurcation cuts. The correlation was investigated using the 2/3 sampleand the results are given in Table X. In addition, the ratios of observations over predictionswere used to quantify the degree of consistency and were found to be consistent with unitywithin a relative uncertainty of 15%, confirming that the background estimations obtainedwith the bifurcation method were reliable.
H. Acceptance and Sensitivity
To reduce the estimated background level to less than one event in the signal region, thisanalysis utilized many selection criteria. The corresponding acceptances for these selectioncriteria were estimated directly from the data when possible, by splitting them into compo-nents that could be measured separately using the monitor trigger data or the Monte Carlosimulation. The latter gave the estimates on the decay phase space, the trigger efficiencyand the nuclear interaction effects. 79 π (PV cuts) × ( K π kinematic cuts)Loosening factor 10 ×
10 20 ×
20 20 ×
50 50 ×
50 50 × ± ± ± ± ± K µ Range Tail ( π + → µ + → e + ) × (Maximum momentum cut)Loosening factor 10 ×
10 20 ×
20 50 ×
50 80 ×
50 120 × ± ± ± ± ± µ + Band ( π + → µ + → e + ) × (Range-momentum cut)Loosening factor 10 ×
10 20 ×
20 50 ×
20 80 ×
20 80 × ± ± ± ± ± K π (top), K µ range tail (middle) and µ + band(bottom) backgrounds in the 2/3 sample. The errors in the predictions are statistical uncertainties.
1. Acceptance Factors from K µ Events
Since the K µ events have the same features as the signal regarding the K + beam, thecharged track and the event topology, the acceptances associated with the relevant cuts aslisted in Table XI were directly measured using the Kµ Tracking in RS:
To measure the acceptance of RS tracking, additional requirements (“set-up cuts”) were applied to Kµ t IC − t C K > K µ events were examined for consistency with RS tracking. The acceptance ofthe RS tracking cuts is given in the second row of Table XI. Tracking in UTC and Target:
Because there should be no photon activity for K µ peakevents, the Kµ ut AcceptanceTracking in RS 0.99996 ± ± ± ± ± ± A K µ ± K + → π + ν ¯ ν selection cuts measured from the K µ monitor triggerdata. The acceptance of beam cuts does not include those using the energy measurement in thetarget. The errors are statistical. also required to meet the timing requirements on the beam instruments. A 5 ns timingconsistency was also required between t IC and t rs . A subset of the PV cuts was applied tosuppress possible K µ γ contamination. The BV and BVL elements of the PV cuts were notapplied to avoid self-vetoing by long range K µ events. Events surviving the set-up cuts werethen checked with the UTC and target requirements except for those involved π + energyand range measurement in the target, giving a measurement of the acceptance of trackingin UTC and target(Table XI). Beam selection criteria:
The K µ events were chosen from those satisfying the require-ments on tracking in the UTC and target described above but without the timing require-ments on the beam instruments. To suppress beam background contamination, the mo-mentum deviation was required to be within two standard deviations of the K µ peak with | cos θ | < .
5. Also required was that there should be no discernible scattering of tracks in theRS. The remaining events then passed through the beam cuts except for the pathology cutsusing π + energy and range measurement information described in Section III F 1, providinga measurement of the corresponding beam selection acceptance. Because K µ events weresimple single tracks, the efficiency of the DC trigger was also measured and included in theacceptance of beam selection criteria (Table XI). Photon veto:
The acceptance of the PV cuts included contributions from both the onlineand offline PV. Ideally, K µ events should not contain any photons, and could thus be used81o measure the acceptance loss due to the PV cuts. In the first step of this procedure,the selection criteria were applied to remove possible beam backgrounds. However, it wasnoted that some µ + ’s could penetrate the whole RS and reach the PV counters, resulting ina time-coincident PV hit and therefore an over-counting in the acceptance loss due to thePV cuts. To avoid this problem, the selected K µ sample was further required to have thestopping layers prior to the 19 th RS layer. The PV cuts were then applied to the surviving K µ events, yielding a measurement of the acceptance loss due to the application of the PVcuts. Since the Kµ Track stop in RSSC:
This cut was classified into the fiducial cuts in Section III F 2 andaimed at vetoing possible associated photon activity detected by the RSSC, even though itwas not included in the PV cuts. Using the Kµ Muon veto in RS:
The πν ¯ ν (1) trigger condition 19 ct was also called as a muon vetoin RS. This trigger requirement could result in acceptance losses when an accidental hithappened in the 19 th layer along with an otherwise good signal candidate event. This losswas measured with the Kµ ct was checked, giving a measurement of the acceptance forthe muon veto in RS (Table XI).
2. Acceptance Factors from K π Events
The Kπ π + energy and range measurement and the KIN cuts involving therange-energy consistency in IC and target. This was a complement to the measurement ofacceptance factors from the K µ events. The K π events were selected with all cuts appliedexcept for those to be measured. To ensure good K π events, the momentum, range andenergy were required to be within two standard deviations of the K π peak positions and82bservation of a π → γγ decay was required. The result was A K π = 0 . ± . stat . (29)
3. Kinematic Acceptance from Beam π + Events
The π scat monitor trigger data provided a pure π + sample to measure the KIN cutsrelated to the particle type: the cut on π + stopping layer in the fiducial cuts, the cuts ontracking quality in UTC and RS, the cuts energy loss in RS and the cut on range-momentumconsistency in UTC and RS. The events were required to pass the Pass 1 cuts and the TDcuts. The K + selection criteria in the beam instruments were inverted to select beam π + ’s.The t IC was required to be within ± t rs . The signal phase space cuts were additionallyapplied to select the events. The acceptance was measured to be A π scat = 0 . ± . stat ± . sys . (30)Since these π + ’s came from the beam π + ’s scattering in the target and not from the K + decays at rest, classification of the K + fibers and π + fibers could be complicated because oftheir nearly coincident times and differences in fiber energy deposits of scattered π + ’s and K + decays at rest. Both of these features would result in more uncertainties in the momentum,energy and range measurements in these π + events. Systematic uncertainties were thereforeinvestigated by loosening or tightening the signal phase space cuts by ± π + → µ + → e + Decay Acceptance from Beam π + Events
The acceptance of π + → µ + → e + decay sequence cuts was measured by using the π scat monitor trigger data. This acceptance measurement included the online and offline (TD) π + identification cuts. The online ones included L . L . A π scat except for the cuts to bemeasured here. It should be noted that this measurement included the acceptance loss dueto the π + absorption and π + decay-in-flight. This loss was estimated to be 1.4% using MonteCarlo in Section III H 5 and should be corrected to remove the double counting problem in83he acceptance. The final acceptance of π + → µ + → e + decay sequence cuts was given below A π → µ → e = 0 . ± . stat ± . sys . (31)Since the π + → µ + → e + decay sequence cuts might be correlated to particle identificationKIN cuts when using information from the RS, the effect on acceptance was investigatedwith and without the RS energy loss cuts. The observed about 2% variation on the relativeacceptance was assigned as the systematic uncertainty.
5. Acceptance Factors from Monte Carlo Simulation
Monte Carlo simulations of the K + → π + ν ¯ ν were used to evaluate the trigger acceptance,the phase space acceptances and the acceptance loss due to π + absorption, decay-in-flight,and nuclear interaction, which could not be measured directly by the monitor trigger data.The K + → π + ν ¯ ν Monte Carlo samples were generated with and without including the nu-clear interaction.
Trigger Requirements:
All the trigger requirements as described in Section II H weresimulated by Monte Carlo except for the DC , L . L .
2, which were already measuredusing the K µ and π scat monitor trigger data. From the Monte Carlo without including thenuclear interaction, the acceptance for the trigger requirements was A trig = 0 . ± . stat ± . sys , (32)where the systematic uncertainty was estimated to be 4.7% from the measurement on thebranching ratio of K π in Section III H 8. It was noted that the trigger acceptance measuredhere was primarily due to geometry. Phase Space:
The phase space acceptance was used to determine the acceptance of offlinecuts on the momentum, range and energy. To measure the phase space acceptances ( A P S ),events were first taken from those surviving from the trigger in the Monte Carlo simulation,and then the phase space cuts were applied. The π + nuclear interaction was not includedin this simulation. The acceptance was measured to be A P S = 0 . ± . stat , (33)which included the loss due to both π + absorption and decay-in-flight.84 orrection for Nuclear Interaction: The nuclear interaction effect was investigated sepa-rately, in order to study the systematic uncertainty associated with it. K + → π + ν ¯ ν eventswere generated with and without nuclear interaction, respectively. The ratio between thetrigger acceptances multiplied by the ratio between the phase space acceptances gave thecorrection for nuclear interaction A nucl. = 0 . ± . stat ± . sys , (34)where the systematic uncertainty took into account the observed 0.15 cm difference on therange resolution (Table II). This difference could affect the acceptance due to the Rdevcut and translated into a 5% uncertainty in the acceptance. It should be pointed that thisdefinition took into account loss associated with nuclear interactions such extra energy in thedetector associated with nuclear interactions that caused the PV counters to fire. This losswas not included in the K µ -based acceptance A K µ given in Table XI because muon-nuclearinteractions are rare.
6. Correction to T · Trigger Inefficiency
The T · T · K µ and K π events in the KB monitor data. UTC track extrapolation was required to give the expected T · T · T · K µ and K π events, separately. Since the energy losses in the T · K µ and K π events were different, simulations were done for these decaymodes to obtain the average energy loss for K µ and K π . Using an energy extrapolationgave the correction to the T · A T · = 0 . ± . stat ) ± . sys ) , (35)where the systematic uncertainty accounted for the fact that there was a 1.5% variationwhen changing the z requirement on the UTC track extrapolation to the T · . Normalization to the K µ Branching Ratio
Since a beam K + could decay after the ˇCerenkov counter with a daughter satisfyingthe B4 and target requirement in KB , or a beam K + could deposit energy in the B4 andtarget but exit the target without stopping, the total number of K + ’s that satisfy the KB trigger requirement should be corrected for the K + stopping fraction ( f s ). This fraction wasobtained by normalizing the total K + exposure to the K µ branching ratio. The K µ eventswere selected from the Kµ π + particle type, the BV and the BVL.The momentum, range and energy cuts for signal were replaced by a minimum 40 cm rangerequirement on the K µ events. The K µ acceptance measurement was also performed inthe same way as that for the signal. This stopping fraction was computed by f s = N K µ N effK ( K µ ) · Acc ( K µ ) · B ( K + → µ + ¯ ν µ ) , = 0 . ± . stat , (36)where the N K µ = 355 ,
119 and was the number of surviving K µ events. N effK ( K µ ) =4 . × and was the total exposure of K + ’s ( N K ) with a prescaling factor for the Kµ Acc ( K µ ) = 17 .
4% and was theacceptance.
8. Confirmation of the K π Branching Ratio
Measurement of the K π branching ratio confirmed the validity of the evaluation of theacceptance of the K + → π + ν ¯ ν selection cuts. Stopped K π events were selected from the Kπ K + → π + ν ¯ ν selection criteriaexcept for those used for the PV cuts and for defining the kinematic signal region. Good K π events should also meet the requirements on the energy, momentum and range, whichwere defined to be within three standard deviations of the K π peaks. Fig. 32 shows the sta-bility of the measured K π branching ratio as a function of run number. The K π branchingratio was measured to be B ( K + → π + π ) = N K π N effK ( K π ) · Acc ( K π ) · f s , = 0 . ± . stat , (37)86 IG. 32: Measurement of the K + → π + π branching ratio as a function of run number in E949. where N K π = 16 ,
405 and was the number of surviving K π events. N effK ( K π ) = 1 . × and was the total exposure of K + ’s ( N K ) with a pre-scaling factor for the Kπ Acc ( K π ) = 7 .
3% and was the acceptance. Thisbranching ratio was in agreement with the world average [18] value of 0.209 ± A trig in Equation (32).
9. Summary of Acceptance and Sensitivity
The acceptances of the K + → π + ν ¯ ν decay were split into several parts as given above.Table XII summarizes all the contributions to the total acceptance in the standard region A standardtotal . To get the acceptance in the extended signal region, the estimated acceptancegains given in Section III G 7 were applied to the acceptances for A π → µ → e , A K π and A P S ,yielding the final acceptance of
Acc. = (2 . ± . stat ± . sys ) × − . (38)This value is 10% higher than that in E787. It is noted that the acceptance of the standardE787 signal region for the E949 data was 84% of the acceptance for the E787 data due tolosses incurred by the higher than expected instantaneous rates (Section II A). Based on87 ontribution Acceptance A K µ . ± . stat A K π . ± . stat A π scat . ± . stat ± . sys A π → µ → e . ± . stat ± . sys A trig . ± . stat ± . sys A P S . ± . stat A nucl. . ± . stat ± . sys A T · . ± . stat ± . sys f s . ± . stat A standardtotal ( × − ) 1 . ± . stat ± . sys TABLE XII: A breakdown of the acceptance for the K + → π + ν ¯ ν selection criteria. the total exposure of K + ’s ( N K ), the single event sensitivity ( SES ) of the E949 2002 runwas given by
SES = (2 . ± . stat ± . sys ) × − . (39) I. Examining the Signal Region
After the background analysis and the acceptance measurement were completed andsatisfactory, all the selection criteria were then be applied to the data. At the stage ofexamining the signal region, no cut could be changed.One candidate event was observed inside the signal region. A close check also foundthat this candidate was located in the π + → µ + → e + extended region as described inSection III G 7. Fig. 33 shows the range and kinetic energy of the events that passed all ofthe selection criteria, except for the phase space cuts on both the range and energy. Thiscandidate together with the events observed in E787 are also shown in Fig. 33. As indicatedin the Figure, the signal box definition in E949 was extended in comparison to that in E787.Fig. 34 is an event display for this candidate. This event had a momentum of 227 . c ,a kinetic energy of 128 . . IG. 33: Range vs kinetic energy of the events satisfying all of the cuts, except for the phase spacecuts on both the range and energy. The plots are shown separately for E949 only (left) and E787plus E949 results. The rectangle represents the signal region defined in E787 (dashed lines) andE949 (solid lines). Events around E = 108 MeV were due to K π , which were not removed by thephoton veto cuts. The light points in the right hand plot represent the expected distribution of K + → π + ν ¯ ν events from simulation. signal. The measured quantities of the observed candidate used in the selection criteriawere compared to the expected distributions for signal to evaluate the signal probabilitydistributions for the candidate. The probabilities for the single beam K + requirements, thedecay π + kinematic requirements and the π + → µ + → e + decay sequence cuts showed afairly flat distribution, which was consistent with the expected signal distribution. Therewas no observed photon activity for this candidate. IV. RESULTS
In this section we describe the method used to obtain the K + → π + ν ¯ ν branching ratioand the impact of the E949 and E787 K + → π + ν ¯ ν candidates on the unitarity triangle.We also describe the implication of the results on the search for the hypothetical decay K + → π + X where X is a stable, massless, non-interacting particle [78].89 IG. 34: Reconstruction of the candidate event (end view). The clusters of squares indicate boththe K + track and the π + track in the target. The hit IC sector is shown next to the π + cluster inthe target. The curve is the result of the UTC track fit. The circles along the track are the hits inthe UTC. The radius of each circle gives the drift distance. The RS and RSSC hit layers are shownoutside the UTC. Also displayed are the TD data in the π + stopping counter, the reconstructionin the target and the CCD data in the K + stopping fiber. In the fit to the TD pulse shape, the π + pulse (dashed) and the µ + pulse (dotted) are shown separately. No obvious π + pulse was observedin the CCD pulse shape for the K + stopping fiber. A. Background Functions
We defined a number of cells in the extended signal region of differing signal/background,and calculated the expected signal/background ( dA/dN ) using individual A typeπνν and N typeπνν functions for each rejection or background type. In total, we had seven types: TD rejection,PV rejection, K π background, K µ range tail background, µ + band background, single-beam background and double-beam background. The variation of each type for different cutposition or cell could be expressed as the change of N typeπνν as a function of the correspondingacceptance A typeπνν , yielding seven functions in total used in this analysis. Two of them were theTD rejection vs. acceptance function (Fig. 26) and the PV rejection vs. acceptance function(Fig. 29). Three of them were the relative background rate in the normalization branch vs.the relative acceptance functions for the kinematic background backgrounds (Fig. 35). The90 IG. 35: The expected relative kinematic background rate in the normalization branch vs. therelative acceptance for the K π background (a), the K µ range tail background (b) and the µ + band background (c). Both background rates and acceptance curves were normalized to one at thecut positions. rest were the relative background rate vs. the relative acceptance functions for the beambackgrounds (Fig. 36). Both relative background rates and the relative acceptance curveswere normalized to one at the cut positions. B. Likelihood Method
The K + → π + ν ¯ ν branching ratio was determined using likelihood analysis incorporatingthe predicted background functions in the signal region. The likelihood ratio X was definedas X ≡ n Y i =1 e − ( s i + b i ) ( s i + b i ) d i d i ! . e − b i b d i i d i ! , (40)where s i and b i were the estimated signal and background in the i th cell, d i was the numberof signal candidates in the i th cell and the product ran over all n cells [79]. In addition thelikelihood estimator X obs was defined as the value of X given the observed candidates. Cellswere defined based on the predicted background functions described in previous section.The predicted background functions showed that there was additional background rejectioncapability within the signal region that could be exploited by sub-dividing the signal region.91 IG. 36: The expected relative beam background rate vs. the relative acceptance for the single-beam and CEX backgrounds (a) and the double-beam background (b). Both background ratesand acceptance curves were normalized to one at the cut positions.
The number of cells to be used for subsequent analysis, 3,781, was established prior to theexamination of the signal region.The total background in the cell containing the signal candidate was estimated to be5 . × − dominated by a contamination of 4 . × − events due to the K µ range tailbackground. The ratio of the acceptance in this cell to the total acceptance in the standardregion ( A standardtotal = 0 . × − . The expected number ofsignal events in this cell was s i = N K · B · A standardtotal · A i = 1 . × × B × . × . × − = 3 . × × B , (41)where B was the K + → π + ν ¯ ν branching ratio. C. Branching Ratio of K + → π + ν ¯ ν The central value of the branching ratio, defined as the value of B that maximized X obs [80], was 0 . × − . Using only the E949 data, B ( K + → π + ν ¯ ν ) = (0 . +4 . − . ) × − where the quoted 68% confidence level (CL) interval was determined from the behavior of92 as described in Ref. [79] and included only the statistical uncertainty. The estimatedprobability that the E949 candidate was due to background alone was 0.074.The results from E787 and E949 were combined to calculate the branching ratio for K + → π + ν ¯ ν . In the E787 K + → π + ν ¯ ν analysis, two K + → π + ν ¯ ν candidate events wereobserved in the signal region [55]. The number of cells describing the E949 signal regionwere augmented by 488 cells that defined the signal region for the E787 analysis to producea likelihood estimator X obs for the combined data.The confidence intervals for the combined E787 and E949 results took into account theestimated systematic uncertainties in the signal acceptance and the background rates. Thesystematic uncertainty of each background source was estimated to be about 15% basedupon the results of the correlation studies. From the study in Section III H, the systematicuncertainty on the acceptance was estimated to be about 8%. The systematic uncertaintyof each background component and the acceptance were assumed to be uncorrelated andto follow a normal distribution with the magnitudes given above regarded as one standarddeviation. With these assumptions, the K + → π + ν ¯ ν branching ratio for the combined E787and E949 result was B ( K + → π + ν ¯ ν ) = (1 . +1 . − . ) × − where the uncertainty denoted the68% CL interval. The corresponding 90% and 95% CL intervals were (0 . , . × − and(0 . , . × − , respectively. The estimated probability that all the K + → π + ν ¯ ν can-didates observed in E787 and E949 were due to background was 0.001. The inclusion ofthe estimated systematic uncertainties had a negligible effect on the CL intervals due to therelatively poor statistical precision inherent in a sample of three candidate events. D. Search for B ( K + → π + X ) The experimental signature of a K + → π + X decay was identical to that of K + → π + ν ¯ ν except that the kinematic signature afforded by the two-body decay ( P π = 227 . c , E π = 127 . R π = 38 . K + → π + ν ¯ ν analysis except that the signal region was defined to be within two standarddeviations of the expected momentum, energy and range of the π + with the upper limits ofthe tightened to P ≤
229 MeV/ c , E ≤
135 MeV, and R ≤
40 cm to suppress K + → µ + X background. The expected background level was small (0.05 events), because the region was93ar from the K π peak. The acceptance studies for K + → π + X decay paralleled those forthe K + → π + ν ¯ ν . The single event sensitivity for the E949 K + → π + X decay analysis wasestimated to be (0 . ± . stat ± . sys ) × − .The candidate event observed in the signal region for K + → π + ν ¯ ν E949 analysis wasalso in the K + → π + X signal region. However, no candidates were observed in the K + → π + X signal region of E787 [55]. The combined E787 and E949 sensitivity was0.196 × − and, using the one observed candidate event without subtraction of theestimated background, the upper limit on the branching ratio was B ( K + → π + X ) < . × − at 90% CL using the Feldman-Cousins method [81]. This limit was largerthan the previous 90% CL limit of 0 . × − of E787 [55] due to the E949 candidateevent. E. Impact on the Unitarity Triangle
As described in Section I, the K + → π + ν ¯ ν branching ratio was directly related to the realand imaginary parts of λ t ≡ V ∗ ts V td (Equation (9)). In Fig. 37 the regions of the complex λ t plane allowed by the K + → π + ν ¯ ν branching ratio determined from the combined E787 andE949 results was compared to the regions allowed by other recent measurements with smalltheoretical uncertainties [82]. The region favored by other CKM-sensitive measurements isat the edge of the 68% CL region allowed by the K + → π + ν ¯ ν measurement.The other CKM-sensitive results [82] used to produce the confidence level intervals inFig. 37 are dominated by measurements of B meson decays. The possible discrepancybetween the λ t regions allowed by the B -decay measurements and by B ( K + → π + ν ¯ ν ) couldbe an indication of physics beyond the SM. As emphasized in Ref. [83], the clean theoreticalinterpretation of K → πν ¯ ν remains valid in most extensions of the SM in distinct contrastto the B -decay measurements currently used to determine the CKM parameters. Thus aprecise measurement of B ( K + → π + ν ¯ ν ) would provide an unambiguous consistency test ofthe flavor sector of the SM. 94 IG. 37: The allowed regions in the λ t plane allowed by the combined E787 and E949 determinationof the K + → π + ν ¯ ν branching ratio (gray), B + → τ + ν (blue) and B -mixing measurements (yellow).The regions outside the lighter (darker) shading have CL > K + → π + ν ¯ ν at CL > V. CONCLUSION
The rare decay K + → π + ν ¯ ν is a flavor-changing-neutral-current process and proceeds via1-loop diagrams mediated mainly by the top quark. Measuring B ( K + → π + ν ¯ ν ) is one of95he cleanest ways to extract | V td | .In this paper we have reported results from the BNL experiment E949, an upgradedversion of the BNL-E787 experiment, designed to improve the sensitivity for measurementof K + → π + ν ¯ ν decay. All the K + decays at rest were analyzed using a blind analysistechnique in which the signal region was masked until the selection criteria were determinedand the background levels were estimated. The development of the cuts and the estimation ofthe background levels were performed using a bifurcation method and, a likelihood analysismethod was developed for interpreting the quality of candidate events. Enlargement ofthe signal region compared to E787 analysis increased the acceptance by 30% with a totalbackground level in the signal region estimated to be 0 . ± .
03 events.An examination of the signal region yielded one event near the upper kinematic limit ofthe decay K + → π + ν ¯ ν . Based on the candidate event, the branching ratio was determinedto be B ( K + → π + ν ¯ ν ) = (0 . +4 . − . ) × − . E787 and E949 results were combined andthe branching ratio was determined to be (1 . +1 . − . ) × − at the 68% CL level basedon three events observed in the momentum region 211 ≤ P ≤
229 MeV/ c . The estimatedprobability that all the K + → π + ν ¯ ν candidates observed in E787 and E949 were due tobackground was 0.001. The measured branching ratio is in agreement with the SM predictionof (0 . ± . × − within the uncertainty. Acknowledgments
We gratefully acknowledge the support and efforts of the BNL Collider-Accelerator Divi-sion for the high quality K + beam delivered. We wish to thank Jose Ocariz of the CKMfitterGroup for producing Fig. 37. This research was supported in part by the U.S. Departmentof Energy, the Ministry of Education, Culture, Sports, Science and Technology of Japanthrough the Japan-U.S. Cooperative Research Program in High Energy Physics and underGrant-in-Aids for Scientific Research, the Natural Sciences and Engineering Research Coun-cil and the National Research Council of Canada, the Russian Federation State ScientificCenter Institute for High Energy Physics, and the Ministry of Science and Education of theRussian Federation. S. Chen was also supported by Program for New Century Excellent96alents in University from the Chinese Ministry of Education. [1] W. Buchmuller, arXiv:hep-ph/0306047.[2] A.D. Sakharov, JETP Lett. , 24 (1967).[3] V.A. Rubakov. M.E. Shaposhnikov, Usp. Fiz Nauk , 493 (1996); Phys. Usp. , 461 (1996);A Riotto and M. Trodden, Annu. Rev. Nucl. Part. Sci. , 35 (1999).[4] V.V. Anisimovsky et al ., Phys. Rev. Lett. , 031801 (2004);[5] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. , 652 (1973).[6] L. Wolfenstein, Phys. Rev. Lett. , 1945 (1983).[7] C. Jarlskog, Phys. Rev. Lett. , 1039 (1985); C. Jarlskog, Z. Phys. C29 , 491 (1985); C.Jarlskog and R. Stora, Phys. Lett.
B208 , 268 (1988).[8] A.J. Buras, M.E. Lautenbacher, and G. Ostermaier, Phys. Rev.
D50 , 3433 (1994).[9] A.J. Buras, F. Schwab, and S. Uhlig, arXiv:hep-ph/0405132 (2004).[10] T. Inami and C.S. Lim, Prog. Theor. Phys. , 297 (1981).[11] G. Buchalla and A.J. Buras, Nucl. Phys. B548 , 309 (1999).[12] G. Buchalla and A.J. Buras, Nucl. Phys.
B398 , 285 (1993).[13] M. Musiak and J. Urban, Phys. Lett.
B451 , 161 (1999).[14] A. J. Buras, M. Gorbahn, U. Haisch and U. Nierste, Phys. Rev. Lett. , 261805 (2005).[15] A. J. Buras et al., JHEP , 002 (2006).[16] G. Isidori, F. Mescia and C. Smith, Nucl. Phys. B , 319 (2005)[17] W.J. Marciano and Z. Parsa, Phys. Rev. D53 , R1 (1996).[18] W. M. Yao et al. [Particle Data Group], J. Phys. G , 1 (2006).[19] D. Rein and L.M. Sehgal, Phys. Rev. D39 , 3325 (1989); J.S. Hagelin and L.S. Littenberg,Prog. Part. Nucl. Phys. , 1 (1989); M. Lu and M.B. Wise, Phys. Lett. B324 , 461 (1994); S.Fajfer, Nuovo Cim.
A110 , 397, (1997); C.Q. Geng, I.J. Hsu, and Y.C. Lin, Phys. Rev.
D54 ,877 (1996).[20] F. Mescia and C. Smith, Phys. Rev.
D76 , 034017 (2007).[21] A.J. Buras et al. , Nucl. Phys. B , 103 (2005).[22] G. Isidori, F. Mescia, P. Paradisi, C. Smith and S. Trine, JHEP , 064 (2006).[23] A.J. Buras et al. , Phys. Lett. B , 161 (2001).
24] A.J. Buras et al. , Nucl. Phys. B , 3 (2000).[25] C. H. Chen, J. Phys. G , L33 (2002).[26] G. Bhattacharyya and A. Raychaudhuri, Phys. Rev. D , R3837 (1998).[27] A. Deandrea, J. Welzel and M. Oertel, JHEP , 038 (2004).[28] G. Buchalla, G. Burdman, C. T. Hill and D. Kominis, Phys. Rev. D , 5185 (1996).[29] Z. J. Xiao, Chin. Phys. Lett. , 712 (1999).[30] Z. J. Xiao, C. S. Li and K. T. Chao, Eur. Phys. J. C , 51 (1999).[31] Z. J. Xiao, L. X. Lu, H. K. Guo and G. R. Lu, Eur. J. Phys. C , 487 (1999).[32] T. Hattori, T. Hasuike and S. Wakaizumi, Phys. Rev. D , 113008 (1999).[33] K. Agashe and M. Graesser, Phys. Rev. D , 4445 (1996).[34] X. G. He and G. Valencia, Phys. Rev. D , 053003 (2004).[35] B. Machet, Mod. Phys. Lett. A , 579 (2000).[36] Y. Grossman, Nucl. Phys. B , 355 (1994).[37] D. S. Gorbunov and V. A. Rubakov, Phys. Rev. D , 054008 (2001).[38] A. J. Buras, M. Spranger and A. Weiler, Nucl. Phys. B , 225 (2003).[39] W. F. Chang and J. N. Ng, JHEP , 077 (2002).[40] G. Burdman, Phys. Rev. D , 076003 (2002).[41] M. Blanke, A. J. Buras, A. Poschenrieder, S. Recksiegel, C. Tarantino, S. Uhlig and A. Weiler,JHEP , 066 (2007).[42] M. Blanke, A. J. Buras, S. Recksiegel, C. Tarantino and S. Uhlig, JHEP , 082 (2007).[43] C. H. Chen, C. Q. Geng and T. C. Yuan, Phys. Rev. D , 077301 (2007).[44] C. Promberger, S. Schatt and F. Schwab, Phys. Rev. D , 115007 (2007).[45] R. J. Oakes, Phys. Rev. , 1520 (1969).[46] U. Camerini et al ., Phys. Rev. Lett. , 326 (1969).[47] D. Ljung and D. Cline, Phys. Rev. D8 , 1307 (1973).[48] J.H. Klems, R.H. Hildebrand, and R. Steining, Phys. Rev. D4 , 66 (1971).[49] G.D.Cable et al ., Phys. Rev. D8 , 3807 (1973).[50] Y. Asano et al ., Phys. Lett., B107 , 159 (1981).[51] M.S. Atiya et al ., Nucl. Instr. Meth.
A321 , 129 (1992).[52] S. Adler et al ., Phys. Rev. Lett. , 1421 (1996).[53] M.S. Atiya et al ., Phys. Rev. D48 , R1 (1993).
54] S. Adler et al ., Phys. Rev.
D70 , 037102 (2004); S. Adler et al ., Phys. Lett.,
B537 , 211 (2002).[55] S. Adler et al ., Phys. Rev. Lett. , 041803 (2002); S. Adler et al ., Phys. Rev. Lett. , 3768(2000); S. Adler et al ., Phys. Rev. Lett. , 2204 (1997).[56] B. Bassalleck et al ., E949 proposal, BNL-67247, TRI-PP-00-06 (1999), .[57] D.A. Bryman et al ., Nucl. Instr. Meth. A396 , 394 (1997).[58] M. Atiya et al ., Nucl. Instr. Meth.
A279 , 180 (1989).[59] E.W. Blackmore et al ., Nucl. Instr. Meth.
A404 , 295 (1998).[60] R.A. McPherson, “Chasing the Rare Decay K + → π + ν ¯ ν ”, Princeton University, Ph.D. Thesis,November, 1995.[61] I.H. Chiang et al ., IEEE Trans. Nucl. Sci. , 394 (1995).[62] T.K. Komatsubara et al ., Nucl. Instr. Meth. A404 , 315 (1998).[63] T. Yoshioka et al ., IEEE Trans. Nucl. Sci. , 334 (2004).[64] J. Doornbos et al ., Nucl. Instr. Meth. A444 , 546 (2000).[65] O. Mineev et al ., Nucl. Instr. Meth.
A494 , 362 (2002).[66] A.V. Artamonov et al ., Phys. Lett.,
B623
192 (2005).[67] H. Brafman et al ., IEEE Trans. Nucl. Sci. , 336 (1985).[68] The reference manual can be found in http://ppd.fnal.gov/elec/dyc3 .[69] The reference manual can be found in either http://midas.triumf.ca or http://midas.psi.ch .[70] N. Khovansky et al ., Nucl. Instr. Meth. A351 , 317 (1994).[71] W.R. Nelson et al ., “The EGS4 Code System”, SLAC 265, SLAC (1985).[72] C. Caso et al ., European Physical Journal C3 , 1 (1998).[73] P. Meyers, “A modified Version of the UMC Multiple Scattering Routine MSCAT1”, E787Technical Note No.77 (1985). Unpublished.[74] A.J. Stevens, “Nuclear Interactions in CH revisited”, E787 Technical Note No.140 (1987).Unpublished.[75] J.B. Birks, Proc. Phys. Soc. A64 , 874 (1951).[76] The reference manual can be found in http://paw.web.cern.ch/paw/mlpfit/pawmlp.html .[77] A.V. Artamonov et al ., Phys. Rev.
D72 , 091102 (2005).
78] F. Wilczek, Phys. Rev. Lett. , 1549 (1982); J.L. Feng, T. Moroi, H. Murayama and E.Schnapka, Phys. Rev. D57 , 5875 (1998).[79] T. Junk, Nucl. Instr. Meth.
A434 , 435 (1999).[80] This definition of the central value of the branching ratio has the desirable feature that inthe limit of N obs observed candidates with high signal-to-background, the central value is thesame as that given by the product of the sensitivity (Section III H) and N obs which is thefundamental definition of the branching fraction.[81] G.J. Feldman and R.D. Cousins, Phys. Rev. D57 , 3873 (1998).[82] The CKMfitter Group, J.Charles et al. , Eur. Phys. J. C , 1, (2005), updated results andplots available at http://ckmfitter.in2p3.fr .[83] D. Bryman et al. , Int. J. Mod. Phys., A , 487 (2006)., 487 (2006).