arX

iv:1

604.

0085

7v2

[he

p-ex

] 2

0 Ju

n 20

16

BABAR-PUB-15/009SLAC-PUB-16505

Measurement of the neutral D meson mixing parameters in a time-dependent

amplitude analysis of the D0→ π

+π

−

π0 decay

J. P. Lees, V. Poireau, and V. TisserandLaboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP),

Universite de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France

E. GraugesUniversitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain

A. PalanoINFN Sezione di Bari and Dipartimento di Fisica, Universita di Bari, I-70126 Bari, Italy

G. EigenUniversity of Bergen, Institute of Physics, N-5007 Bergen, Norway

D. N. Brown and Yu. G. KolomenskyLawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA

H. Koch and T. SchroederRuhr Universitat Bochum, Institut fur Experimentalphysik 1, D-44780 Bochum, Germany

C. Hearty, T. S. Mattison, J. A. McKenna, and R. Y. SoUniversity of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1

V. E. Blinovabc, A. R. Buzykaeva, V. P. Druzhininab, V. B. Golubevab, E. A. Kravchenkoab,

A. P. Onuchinabc, S. I. Serednyakovab, Yu. I. Skovpenab, E. P. Solodovab, and K. Yu. Todyshevab

Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090a,Novosibirsk State University, Novosibirsk 630090b,

Novosibirsk State Technical University, Novosibirsk 630092c, Russia

A. J. LankfordUniversity of California at Irvine, Irvine, California 92697, USA

J. W. Gary and O. LongUniversity of California at Riverside, Riverside, California 92521, USA

A. M. Eisner, W. S. Lockman, and W. Panduro VazquezUniversity of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA

D. S. Chao, C. H. Cheng, B. Echenard, K. T. Flood, D. G. Hitlin, J. Kim,

T. S. Miyashita, P. Ongmongkolkul, F. C. Porter, and M. RohrkenCalifornia Institute of Technology, Pasadena, California 91125, USA

Z. Huard, B. T. Meadows, B. G. Pushpawela, M. D. Sokoloff, and L. Sun∗

University of Cincinnati, Cincinnati, Ohio 45221, USA

J. G. Smith and S. R. WagnerUniversity of Colorado, Boulder, Colorado 80309, USA

D. Bernard and M. VerderiLaboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France

D. Bettonia, C. Bozzia, R. Calabreseab, G. Cibinettoab, E. Fioravantiab, I. Garziaab, E. Luppiab, and V. Santoroa

2

INFN Sezione di Ferraraa; Dipartimento di Fisica e Scienze della Terra, Universita di Ferrarab, I-44122 Ferrara, Italy

A. Calcaterra, R. de Sangro, G. Finocchiaro, S. Martellotti, P. Patteri, I. M. Peruzzi, M. Piccolo, and A. ZalloINFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy

S. Passaggio and C. Patrignani†

INFN Sezione di Genova, I-16146 Genova, Italy

B. BhuyanIndian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India

U. MallikUniversity of Iowa, Iowa City, Iowa 52242, USA

C. Chen, J. Cochran, and S. PrellIowa State University, Ames, Iowa 50011, USA

H. AhmedPhysics Department, Jazan University, Jazan 22822, Kingdom of Saudi Arabia

A. V. GritsanJohns Hopkins University, Baltimore, Maryland 21218, USA

N. Arnaud, M. Davier, F. Le Diberder, A. M. Lutz, and G. WormserLaboratoire de l’Accelerateur Lineaire, IN2P3/CNRS et Universite Paris-Sud 11,

Centre Scientifique d’Orsay, F-91898 Orsay Cedex, France

D. J. Lange and D. M. WrightLawrence Livermore National Laboratory, Livermore, California 94550, USA

J. P. Coleman, E. Gabathuler, D. E. Hutchcroft, D. J. Payne, and C. TouramanisUniversity of Liverpool, Liverpool L69 7ZE, United Kingdom

A. J. Bevan, F. Di Lodovico, and R. SaccoQueen Mary, University of London, London, E1 4NS, United Kingdom

G. CowanUniversity of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom

Sw. Banerjee, D. N. Brown, and C. L. DavisUniversity of Louisville, Louisville, Kentucky 40292, USA

A. G. Denig, M. Fritsch, W. Gradl, K. Griessinger, A. Hafner, and K. R. SchubertJohannes Gutenberg-Universitat Mainz, Institut fur Kernphysik, D-55099 Mainz, Germany

R. J. Barlow‡ and G. D. LaffertyUniversity of Manchester, Manchester M13 9PL, United Kingdom

R. Cenci, A. Jawahery, and D. A. RobertsUniversity of Maryland, College Park, Maryland 20742, USA

R. CowanMassachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA

R. Cheaib and S. H. RobertsonMcGill University, Montreal, Quebec, Canada H3A 2T8

B. Deya, N. Neria, and F. Palomboab

3

INFN Sezione di Milanoa; Dipartimento di Fisica, Universita di Milanob, I-20133 Milano, Italy

L. Cremaldi, R. Godang,§ and D. J. SummersUniversity of Mississippi, University, Mississippi 38677, USA

P. TarasUniversite de Montreal, Physique des Particules, Montreal, Quebec, Canada H3C 3J7

G. De Nardo and C. SciaccaINFN Sezione di Napoli and Dipartimento di Scienze Fisiche,

Universita di Napoli Federico II, I-80126 Napoli, Italy

G. RavenNIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands

C. P. Jessop and J. M. LoSeccoUniversity of Notre Dame, Notre Dame, Indiana 46556, USA

K. Honscheid and R. KassOhio State University, Columbus, Ohio 43210, USA

A. Gaza, M. Margoniab, M. Posoccoa, M. Rotondoa, G. Simiab, F. Simonettoab, and R. Stroiliab

INFN Sezione di Padovaa; Dipartimento di Fisica, Universita di Padovab, I-35131 Padova, Italy

S. Akar, E. Ben-Haim, M. Bomben, G. R. Bonneaud, G. Calderini, J. Chauveau, G. Marchiori, and J. OcarizLaboratoire de Physique Nucleaire et de Hautes Energies,IN2P3/CNRS, Universite Pierre et Marie Curie-Paris6,Universite Denis Diderot-Paris7, F-75252 Paris, France

M. Biasiniab, E. Manonia, and A. Rossia

INFN Sezione di Perugiaa; Dipartimento di Fisica, Universita di Perugiab, I-06123 Perugia, Italy

G. Batignaniab, S. Bettariniab, M. Carpinelliab,¶ G. Casarosaab, M. Chrzaszcza, F. Fortiab,M. A. Giorgiab, A. Lusianiac, B. Oberhofab, E. Paoloniab, M. Ramaa, G. Rizzoab, and J. J. Walsha

INFN Sezione di Pisaa; Dipartimento di Fisica, Universita di Pisab; Scuola Normale Superiore di Pisac, I-56127 Pisa, Italy

A. J. S. SmithPrinceton University, Princeton, New Jersey 08544, USA

F. Anullia, R. Facciniab, F. Ferrarottoa, F. Ferroniab, A. Pilloniab, and G. Pireddaa

INFN Sezione di Romaa; Dipartimento di Fisica,Universita di Roma La Sapienzab, I-00185 Roma, Italy

C. Bunger, S. Dittrich, O. Grunberg, M. Heß, T. Leddig, C. Voß, and R. WaldiUniversitat Rostock, D-18051 Rostock, Germany

T. Adye and F. F. WilsonRutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom

S. Emery and G. VasseurCEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France

D. Aston, C. Cartaro, M. R. Convery, J. Dorfan, W. Dunwoodie, M. Ebert, R. C. Field, B. G. Fulsom,

M. T. Graham, C. Hast, W. R. Innes, P. Kim, D. W. G. S. Leith, S. Luitz, V. Luth, D. B. MacFarlane,

D. R. Muller, H. Neal, B. N. Ratcliff, A. Roodman, M. K. Sullivan, J. Va’vra, and W. J. WisniewskiSLAC National Accelerator Laboratory, Stanford, California 94309 USA

M. V. Purohit and J. R. Wilson

4

University of South Carolina, Columbia, South Carolina 29208, USA

A. Randle-Conde and S. J. SekulaSouthern Methodist University, Dallas, Texas 75275, USA

M. Bellis, P. R. Burchat, and E. M. T. PuccioStanford University, Stanford, California 94305, USA

M. S. Alam and J. A. ErnstState University of New York, Albany, New York 12222, USA

R. Gorodeisky, N. Guttman, D. R. Peimer, and A. SofferTel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel

S. M. SpanierUniversity of Tennessee, Knoxville, Tennessee 37996, USA

J. L. Ritchie and R. F. SchwittersUniversity of Texas at Austin, Austin, Texas 78712, USA

J. M. Izen and X. C. LouUniversity of Texas at Dallas, Richardson, Texas 75083, USA

F. Bianchiab, F. De Moriab, A. Filippia, and D. Gambaab

INFN Sezione di Torinoa; Dipartimento di Fisica, Universita di Torinob, I-10125 Torino, Italy

L. Lanceri and L. VitaleINFN Sezione di Trieste and Dipartimento di Fisica, Universita di Trieste, I-34127 Trieste, Italy

F. Martinez-Vidal and A. OyangurenIFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain

J. Albert, A. Beaulieu, F. U. Bernlochner, G. J. King, R. Kowalewski,

T. Lueck, I. M. Nugent, J. M. Roney, and N. TasneemUniversity of Victoria, Victoria, British Columbia, Canada V8W 3P6

T. J. Gershon, P. F. Harrison, and T. E. LathamDepartment of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom

R. Prepost and S. L. WuUniversity of Wisconsin, Madison, Wisconsin 53706, USA

We perform the first measurement on the D0 − D0 mixing parameters using a time-dependentamplitude analysis of the decay D0 → π+π−π0. The data were recorded with the BABAR detector atcenter-of-mass energies at and near the Υ (4S) resonance, and correspond to an integrated luminosityof approximately 468.1 fb−1. The neutralD meson candidates are selected from D∗(2010)+ → D0π+

s

decays where the flavor at the production is identified by the charge of the low momentum pion,π+s . The measured mixing parameters are x = (1.5± 1.2± 0.6)% and y = (0.2± 0.9± 0.5)%, where

the quoted uncertainties are statistical and systematic, respectively.

PACS numbers: 13.25.Ft, 11.30.Er, 12.15.Ff, 14.40.Lb

∗Now at: Wuhan University, Wuhan 43072, China†Now at: Universita di Bologna and INFN Sezione di Bologna,

I-47921 Rimini, Italy‡Now at: University of Huddersfield, Huddersfield HD1 3DH, UK

§Now at: University of South Alabama, Mobile, Alabama 36688,

USA¶Also at: Universita di Sassari, I-07100 Sassari, Italy

5

I. INTRODUCTION

The first evidence for D0−D0 mixing, which had beensought for more than two decades since it was first pre-dicted [1], was obtained by BABAR [2] and Belle [3] in2007. These results were rapidly confirmed by CDF [4].The techniques utilized in those analyses and more re-cent, much higher statistics LHCb analyses [5–7] do not

directly measure the normalized mass and the widthdifferences of the neutral D eigenstates, x and y. Incontrast, a time-dependent amplitude analysis of theDalitz-plot (DP) of neutral D mesons decaying into self-conjugate final states provides direct measurements ofboth these parameters. This technique was introducedusing D0 → K0

Sπ−π+ decays by the CLEO collabora-

tion [8], and the first measurement by the Belle Col-laboration [9] provided stringent constraints on the mix-ing parameters. More recent measurements with this fi-nal state by the BABAR and Belle collaborations [10, 11]contribute significantly to the Heavy Flavor AveragingGroup (HFAG) global fits that determine world averagemixing and CP violation parameter values [12].This paper reports the first measurement of mixing

parameters from a time-dependent amplitude analysisof the singly Cabibbo-suppressed decay D0 → π+π−π0.The inclusion of charge conjugate reactions is impliedthroughout this paper. No measurement of CP violationis attempted as the data set lacks sufficient sensitivityto be interesting. The D0 candidates are selected fromD∗(2010)+ → D0π+

s decays where the D0 flavor at pro-duction is identified by the charge of the slow pion, π+

s .The D0 and D0 meson flavor eigenstates evolve and

decay as mixtures of the weak Hamiltonian eigenstatesD1 and D2 with masses and widths m1,Γ1 and m2,Γ2,respectively. The mass eigenstates can be expressed assuperpositions of the flavor eigenstates, |D1,2〉 = p

∣

∣D0⟩

±q∣

∣D0⟩

where the complex coefficients p and q satisfy

|p|2+|q|2 = 1. The mixing parameters are defined as nor-malized mass and width differences, x ≡ (m1 −m2)/ΓD

and y ≡ (Γ1 − Γ2)/2ΓD. Here, ΓD is the average decaywidth, ΓD ≡ (Γ1 + Γ2)/2. These mixing parameters ap-pear in the expression for the decay rate at each point(s+, s−) in the D0 decay Dalitz-plot at the decay timet, where s± ≡ m2(π±π0). For a charm meson tagged att = 0 as a D0, the decay rate is proportional to

∣

∣M(D0)∣

∣

2 ∝ 1

2e−ΓDt

{

|Af |2 [cosh (yΓDt) + cos (xΓDt)]

+

∣

∣

∣

∣

q

pAf

∣

∣

∣

∣

2

[cosh(yΓDt)− cos(xΓDt)]

− 2

[

Re

(

q

pA∗

fAf

)

sinh(yΓDt)

−Im

(

q

pA∗

fAf

)

sin(xΓDt)

]}

, (1)

where f represents the π+π−π0 final state that is com-

monly accessible to decays of both flavor eigenstates, andAf and Af are the decay amplitudes for D0 and D0 tofinal state f . The amplitudes are functions of positionin the DP and are defined in our description of the fit-ting model in Sec. IVA Eq. (3). In Eq. (1), the firstterm is the direct decay rate to the final state f andis always the dominant term for sufficiently small decaytimes. The second term corresponds to mixing. Initially,the cosh(yΓDt) and cos(xΓDt) contributions to this termcancel, but over time the cosh(yΓDt) contribution can be-come dominant. The third term is the interference term.It depends explicitly on the real and imaginary parts ofA∗

fAf and on the real and imaginary parts of q/p. Asfor the mixing rate, the interference rate is intially zero,but it can become important at later decay times. Thevariation of the total decay rate from purely exponentialdepends on the relative strengths of the direct and mixingamplitudes, their relative phases, the mixing parametersx and y, and on the magnitude and phase of q/p. HFAGreports the world averages to be x = (0.49+0.14

−0.15)% andy = (0.61± 0.08)% assuming no CP violation [12].

In this time-dependent amplitude analysis of the DP,we measure x, y, τD ≡ 1/ΓD, and resonance parame-ters of the decay model. At the level of precision of thismeasurement, CP violation can be neglected. Direct CPviolation in this channel is well constrained [13], and in-direct CP violation due to q/p 6= 1 is also very small, asreported by HFAG [12]. We assume no CP violation, i.e.,q/p = 1, and Af (s+, s−) = Af (s−, s+).

This paper is organized as follows: Section II discussesthe BABAR detector and the data used in this analysis.Section III describes the event selection. Section IVpresents the model used to describe the amplitudes inthe DP and the fit to the data. Section V discusses andquantifies the sources of systematic uncertainty. Finally,the results are summarized in Sec. VI.

II. THE BABAR DETECTOR AND DATA

This analysis is based on a data sample correspondingto an integrated luminosity of approximately 468.1 fb−1

recorded at, and 40MeV below, the Υ (4S) resonanceby the BABAR detector at the PEP-II asymmetric en-ergy e+e− collider [14]. The BABAR detector is describedin detail elsewhere [15, 16]. Charged particles are mea-sured with a combination of a 40-layer cylindrical driftchamber (DCH) and a 5-layer double-sided silicon vertextracker (SVT), both operating within the 1.5 T mag-netic field of a superconducting solenoid. Informationfrom a ring-imaging Cherenkov detector is combined withspecific ionization (dE/dx) measurements from the SVTand DCH to identify charged kaon and pion candidates.Electrons are identified, and photons measured, with aCsI(Tl) electromagnetic calorimeter. The return yoke ofthe superconducting coil is instrumented with trackingchambers for the identification of muons.

6

III. EVENT SELECTION

We reconstruct D∗+ → D0π+s decays coming from

e+e− → cc in the channel D0 → π+π−π0. D∗+ can-didates from B-meson decays are disregarded due tohigh background level. The pion from the D∗+ de-cay is called the “slow pion” (denoted π+

s ) because ofthe limited phase space available. The mass differenceof the reconstructed D∗+ and D0 is defined as ∆m ≡m

(

π+π−π0π+s

)

− m(

π+π−π0)

. Many of the selectioncriteria and background veto algorithms discussed beloware based upon previous BABAR analyses [17, 18].To select well-measured slow pions, we require that the

π+s tracks have at least 10 hits measured in the DCH;

and we reduce backgrounds from other non-pion tracksby requiring that the dE/dx values reported by the SVTand DCH be consistent with the pion hypothesis. TheDalitz decay π0 → γe+e− produces background when wemisidentify the e+ as a π+

s . We reduce such backgroundby trying to reconstruct an e+e− pair using the candidateπ+s track as the e+ and combine it with a γ. If the e+e−

vertex is within the SVT volume and the invariant massis in the range 115 < m (γe+e−) < 155MeV, then theevent is rejected. Real photon conversions in the detec-tor material are another source of background in whichelectrons can be misidentified as slow pions. To identifysuch conversions, we first create a candidate e+e− pairusing the slow pion candidate and an identified electron,and perform a least-squares fit. The event is rejectedif the invariant mass of the putative pair is less than60MeV and the constrained vertex position is within theSVT tracking volume.We require that the D0 and π+

s candidates originatefrom a common vertex, and that the D∗+ candidate orig-inates from the e+e− interaction region (beam spot). Akinematic fit to the entire decay chain is performed withgeometric constraints at each decay vertex. In addition,the γγ and π+π−π0 invariant masses are constrained tobe the nominal π0 and D0 masses, respectively [13]. Theχ2 probability of the D∗+ fit must be at least 0.1%.About 15% of events with at least one candidate satis-fying all selection criteria (other than the final D0 massand Deltam cuts described below) have at least two suchcandidates. In these events, we select the candidate withthe smallest χ2 value.To suppress misidentifications from low-momentum

neutral pions, we require the laboratory momentum ofthe π0 candidate to be greater than 350 MeV. The recon-structed D0 proper decay time t, obtained from our kine-matic fit, must be within the time window −2 < t < 3 psand have an uncertainty σt < 0.8 ps. Combinatorialand B meson decay background is removed by requir-ing p∗(D0) > 2.8GeV, where p∗ is the momentum mea-sured in the e+e− center-of-mass frame for the event.The reconstructed D0 mass must be within 15 MeV ofthe nominal D0 mass [13] and the reconstructed ∆mmust be within 0.6 MeV of the nominal D∗+–D0 massdifference [13]. After imposing all other event selec-

tion requirements as mentioned earlier, these p∗(D0), σt,m(π+π−π0), and ∆m criteria were chosen to maximizethe significance of the signal yield obtained from a 2D-fitto the m,∆m plane of data, where the significance wascalculated as S/

√S +B with S and B as the numbers

of signal and background events, respectively.The signal probability density functions (PDFs) in

both m and ∆m are each defined as the sum of two Gaus-sian functions. The m(π+π−π0) background distributionis parameterized by the sum of a linear function and a sin-gle Gaussian, which is used to model the D0 → K−π+π0

contribution when we misidentify the kaon track as apion. We use a threshold-like function [19] to model the∆m background as a combination of realD0 mesons withrandom slow pion candidates near kinematic threshold.For many purposes, we use “full” Monte Carlo (MC)

simulations in which each data set is roughly the samesize as that observed in the real data and the backgroundis a mixture of bb, cc, τ+τ− and uu/dd/ss events scaled tothe data luminosity. The signal MC component is gener-ated with four combinations of x = ±1%, y = ±1%. Wecreate four samples for each set of mixing values exceptx = y = +1% which has ten samples.Based upon detailed study of full MC events, we have

identified four specific misreconstructions of the D0 can-didate that we can safely remove from the signal regionwithout biasing the measured parameters. The first mis-reconstruction creates a peaking background in the cor-ner of the DP when the K− daughter of a D0 → K−π+

decay is misidentified as a pion. To veto these events, weassign the kaon mass hypothesis for the π+π− candidatesand calculate the m(K−π+) invariant mass. We removemore than 95% of these mis-reconstructions by requiring∣

∣m(K−π+)−m(D0)∣

∣ > 20 MeV.

The second mis-reconstruction occurs when theD0 sig-nal candidate shares one or more tracks with a D0 →K−π+π0 decay. To veto these decays, we create a list ofall D0 → K−π+π0 candidates in the event that satisfy∣

∣m(K−π+π0)−m(D0)∣

∣ < 20 MeV, |∆m−∆mPDG| < 3

MeV, and χ2veto < 1000, where

χ2veto(m,∆m) =

(

m(K−π+π0)−mPDG(D0)

σm

)2

+

(

∆m−∆mPDG

σ∆m

)2

, (2)

where mPDG denotes the nominal value for the masstaken from Ref. [13] and σm (σ∆m) is the m (∆m) un-certainty reported by the fit. Such an additional vetois applied for the specific case when the π+π0 from aD0 → K−π+π0 decay is paired with a random π− toform a signal candidate. We can eliminate more than95% of these mis-reconstructions by finding the K− can-didate in the event that yields a m(K−π+π0) invari-ant mass closest to the nominal D0 mass and requir-ing

∣

∣m(K−π+π0)−m(D0)∣

∣ > 40MeV. The background

fromD0 → K−π+π0 due to misidentifying the kaon track

7

as a pion falls outside the signal region mass window andis negligible.The third mis-reconstruction is the peaking back-

ground when the π+π− pair from aD0 → K0Sπ+π− decay

is combined with a random π0 to form a signal candi-date. To veto these events, we combine the π+π− from aD0 → π+π−π0 candidate with K0

S→ π+π− candidates

in the same event and require∣

∣m(K0Sπ+π−)−m(D0)

∣

∣ >20 MeV for each.The fourth mis-reconstruction is pollution from D0 →

K0Sπ0 → (π+π−)π0 decay. Although a real D0 decay,

its amplitude does not interfere with those for “prompt”D0 → π+π−π0. We eliminate ∼ 99% of these events byremoving candidates with 475 < m(π+π−) < 505MeV.The K0

Sveto also removes other potential backgrounds

associated with K0Sdecays.

Figure 1 shows the m(π+π−π0) and ∆m distributionsof D0 candidates passing all the above requirements ex-cept for the requirement on the shown variable. We relaxthe requirements on ∆m and m(π+π−π0) to perform a2D-fit in the m(π+π−π0)–∆m plane, whose projectionsare also shown in Fig. 1. The fit determines that about91% of the ∼ 138,000 candidates satisfying all selectionrequirements (those between the dashed lines in Fig. 1),including those for m(π+π−π0) and ∆m cuts, are signal.

IV. MEASUREMENT OF THE MIXING

PARAMETERS

A. Fit Model

The mixing parameters are extracted through a fit tothe DP distribution of the selected events as a function oftime t. The data is fit with a total PDF which is the sumof three component PDFs describing the signal, “broken-charm” backgrounds, and combinatorial background.The signal DP distribution is parametrized in terms

of an isobar model [20–22]. The total amplitude is acoherent sum of partial waves Wk with complex weightsck,

Af (s−, s+) = Af (s+, s−) =∑

k

ckWk(s+, s−) , (3)

where Af and Af are the final state amplitudes in-troduced in Eq. (1). Our model uses relativisticBreit-Wigner functions each multiplied by a real spin-dependent angular factor using the same formalism withthe Zemach variation as described in Ref. [23] for Wk,and constant WNR = 1 for the non-resonant term. As inRef. [23], Wk also includes the Blatt-Weisskopf form fac-tors with the radii of D0 and intermediate resonances setat 5 GeV−1 and 1.5 GeV−1, respectively. The CLEO col-laboration modeled the decay as a coherent combinationof four amplitudes: those with intermediate ρ+, ρ0, ρ−

resonances and a uniform non-resonant term [24]. Thisform works well to describe lower statistics samples. In

) [GeV]0π

-π

+πm(

1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98

Eve

nts

/ 2

.5 M

eV

0

2000

4000

6000

8000

10000

Background types:

Combinatorial

Broken-charm

(a)

m [GeV]∆

0.14 0.142 0.144 0.146 0.148 0.15 0.152

Eve

nts

/ 0

.1 M

eV

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

22000

Background types:

Combinatorial

Broken-charm

(b)

FIG. 1: (Color online). (a) The reconstructed D0 mass dis-tribution of data (dots) with its fit projection (blue line), re-quiring |∆m−∆mPDG| < 0.6 MeV; (b) The ∆m distributionof data (dots) with its fit projection (blue line), requiring∣

∣m(π+π−π0)−mD0

∣

∣ < 15 MeV. The underlying histogramsshown in shaded bands represent contributions from differ-ent background categories defined in Section IV. The verticaldashed lines mark the actual m(π+π−π0) or ∆m requirementfor the DP analysis.

this analysis we use the model we developed for ourhigher statistics search for time-integrated CP viola-tion [18], which also includes other resonances as listed inTable I. The partial wave with a ρ+ resonance is the refer-ence amplitude. The true decay time distribution at anypoint in the DP depends on the amplitude model and themixing parameters. We model the observed decay timedistribution at each point in the DP as an exponentialwith average decay time coming from the mixing formal-ism (Eq. (1)) convolved with the decay time resolution,modeled as the sum of three Gaussians with widths pro-portional to σt and determined from simulation. As theability to reconstruct t varies with the position in theDP, our parameterization of the signal PDF includes σt

functions that depend on m2(π+π−), defined separatelyin six ranges, each as an exponential convolved with aGaussian. Efficiency variations across the Dalitz-plot aremodeled by a histogram obtained from simulated decays

8

generated with a uniformly populated phase space.In addition to correctly reconstructed signal decay

chains, a small fraction of the events, < 1%, containD0 → π+π−π0 (D0 → π+π−π0) decays which are cor-rectly reconstructed, but then paired with false slow pioncandidates to create fake D∗+ (D∗−) candidates. Asthese are real D0 decays, their DP and decay time dis-tributions are described in the fit assuming a randomlytagged flavor. The total amplitude for this contributionis A′

f (s+, s−) = fRSAf (s+, s−) + (1 − fRS)Af (s−, s+),where fRS is the “lucky fraction” that we have a fake slowpion with the correct charge. As roughly half of theseevents are assigned the wrongD flavor, we set fRS = 50%in the nominal fit. We later vary this fraction to deter-mine a corresponding systematic uncertainty.Backgrounds from mis-reconstructed signal decays and

other D0 decays are referred to as “broken-charm”. Inthe fit, the Dalitz-plot distribution for this category isdescribed by histograms taken from the simulations. Thedecay time distributions are described by the sum of twoexponentials convolved with Gaussians whose parametersare taken from fits to the simulations.We use sideband data to estimate combinatorial back-

ground. The data are taken from the sidebands withm

(

π+π−π0)

< 1.80GeV or m(

π+π−π0)

> 1.92GeV,and outside of the region 0.144 < ∆m < 0.147 GeV,where most of the broken-charm background events re-side. The weighted sum of the two sideband regions isused to describe the combinatorial background in the sig-nal region. The sideband weights and their uncertaintiesare determined from full MC simulation. We model theseevents in t similarly to the broken-charm category. Thedecay time is described by the sum of two exponentialsconvolved with Gaussians. As an ad hoc description ofσt between 0 and 0.8 ps, the σt function for the combi-natorial background is an exponential convolved with aGaussian, but we use different values in six ranges of |t|.The best-fit parameters are determined by an unbinned

maximum-likelihood fit. The central values for x andy were blinded until the systematic uncertainties wereestimated. Because of the high statistics and the com-plexity of the model, the fit is computationally intensive.We have therefore developed an open-source frameworkcalled GooFit [25] to exploit the parallel processing powerof graphical processing units. Both the framework andthe specific analysis code used in this analysis are pub-licly available [26].

B. Fit Results

The time-integrated Dalitz-plot for the signal regiondata is shown in Fig. 2(a). The amplitude parameters de-termined by the fit described above are listed in Table I.Our amplitude parameters and the associated fractionsare generally consistent with the previous BABAR resultsbased on a subset of our data [18]. The normalized dif-ference between the signal DP and the model is shown

in Fig. 2(b). The m2(π±π0) and m2(π+π−) projectionsof the data and model are shown in Fig. 2(c)–(e). Differ-ences between the data and the fit model are apparent inboth the Dalitz-plot itself and the projections. Large pullvalues are observed predominantly near low and high val-ues of m2 in all projections. However, we understand theorigin of these discrepancies, and the systematic uncer-tainties induced on the mixing parameters are small, asdiscussed below. Our fit reports the raw mixing param-eters as x = (2.08± 1.17)% and y = (0.14± 0.89)%. Thecorrelation coefficient between x and y is −0.6%. Themeasured D0 lifetime is τD = (410.2± 3.8) fs, and agreeswith the world average of (410.1± 1.5) fs [13]. The cen-tral values of x and y are later corrected by the estimatedfit biases as discussed in Sec. V.

V. SYSTEMATIC UNCERTAINTIES

Most sources of systematic uncertainty are studied byvarying some aspect of the fit, measuring the resultingx and y values, and taking the full differences betweenthe nominal and the varied results as the correspondingsystematic uncertainty.To study instrumental effects that may not be well-

simulated and are not covered in other studies, we dividethe data into four groups of disjoint bins and calculateχ2 with respect to the overall average for each group forboth x and y. Within a group, each bin has roughly thesame statistics. Four bins of m(π+π−π0) give χ2 = 3.9(0.2) for x (y); five bins of each of D0 laboratory mo-mentum plab, cos θ, and φ give χ2 values of 1.5, 1.2, and3.2 (5.9, 5.1, and 6.9) for x (y), respectively. Altogether,the summed χ2 is 27.9 for ν = 37 degrees of freedom. Ig-noring possible correlations, the p-value for the hypoth-esis that the variations are consistent with being purelystatistical fluctuations around a common mean value is≈ 85%. Therefore, we assign no additional systematicuncertainties.Table II summarizes the systematic uncertainties de-

scribed in detail below. Combining them in quadrature,we find total systematic uncertainties of 0.56% for x and0.46% for y.As mentioned earlier, one source of background comes

from events in which the D0 is correctly reconstructed,but is paired with a random slow pion. We assume thelucky fraction fRS to be exactly 50% in the nominal fit.To estimate the uncertainty associated with this assump-tion, we vary the fraction from 40% to 60% and take thelargest variations as an estimate of the uncertainty.The detector resolution leads to correlations between

reconstructed D0 mass and the decay time, t. We dividethe sample into four ranges of D0 mass with approxi-mately equal statistics and fit them separately; we findthe variations consistent with statistical fluctuations. Be-cause the average decay time is correlated with the re-constructed D0 mass, we refit the data by introducingseparate time resolution functions for each range, allow-

9

]2) [GeV0π

+π(2m

0 0.5 1 1.5 2 2.5 3

]2

) [G

eV

0π

-π(

2m

0

0.5

1

1.5

2

2.5

3

0

5

10

15

20

25

30

35

40

45

(a)

]2) [GeV0π

+π(2m

0 0.5 1 1.5 2 2.5 3

]2

) [G

eV

0π

-π(

2m

0

0.5

1

1.5

2

2.5

3

-5

-4

-3

-2

-1

0

1

2

3

4

5

(b)

]2

) [GeV0π

+π(2m

0 0.5 1 1.5 2 2.5 3

2E

ve

nts

/ 0

.01

25

Ge

V

0

500

1000

1500

2000

2500

Norm

aliz

ed r

esid

ual

-4

-2

0

2

4

(c)

]2

) [GeV0π

-π(2m

0 0.5 1 1.5 2 2.5 3

2E

ve

nts

/ 0

.01

25

Ge

V

0

200

400

600

800

1000

1200

1400

Norm

aliz

ed r

esid

ual

-4

-2

0

2

4

(d)

]2

) [GeV-π

+π(2m

0 0.5 1 1.5 2 2.5 32

Eve

nts

/ 0

.01

25

Ge

V0

200

400

600

800

1000

1200

1400

1600

Norm

aliz

ed r

esid

ual

-4

-2

0

2

4

(e)

FIG. 2: (Color online). The (a) Dalitz-plot and (b) difference between the Dalitz-plot and fit model prediction normalized bythe associated statistical uncertainty in each bin, both time-integrated for the data. Also shown underneath are the projectionsof (c) m2

π+π0 , (d) m2

π−π0 , and (e) m2

π+π−for our data (points) and fit model (blue solid lines), together with the fit residuals

normalized by the associated statistical uncertainties. The PDF components for signal (red dotted) and background (greendashed) events are shown. Note the narrow gap in (e) due to the K0

S veto.

ing the sets of parameters to vary independently. Theassociated systematic uncertainties are taken as the dif-ferences from the nominal values.

The DP distribution of the signal is modeled as a co-herent sum of quasi-two-body decays, involving severalresonances. To study the sensitivity to the choice ofthe model, we remove some resonances from the coher-ent sum. To decide if removing a resonance provides a“reasonable” description of the data, we calculate the χ2

of a fit using an adaptive binning process where eachbin contains at least a reasonable number of events sothat its statistical uncertainty is well determined. With1762 bins, the nominal fit has χ2 = 2794. We separatelydrop the four partial waves that individually increase χ2

by less than 80 units: f0(1370), f0(1500), f0(1710), andρ(1700). We take the largest variations as the systematicuncertainties. The other partial waves individually whenremoved produce ∆χ2 > 165. Additional uncertaintiesfrom our amplitude model due to poor knowledge of themass and width of f0(500) are accounted for by float-ing the mass and width of f0(500) in the fit to data andtaking the variations in x and y. The default resonanceradius used in the Breit-Wigner resonances in the isobarcomponents is 1.5 GeV−1, as mentioned earlier. We varyit in steps of 0.5 GeV−1 from a radius of 0 to 2.5 GeV−1

and again take the largest variations.

The efficiency as a function of position in the DP inthe nominal fit is modeled using a histogram taken from

10

TABLE I: Results of the fit to the D0 → π+π−π0 sample showing each resonance amplitude magnitude, phase, and fit fractionfr ≡

∫

|ckAk (s+, s−)|2ds−ds+. The uncertainties are statistical only. We take the mass (width) of the f0(500) to be 500 (400)

MeV. In the fit, all resonance masses and widths are fixed to the listed values, which are taken from earlier world averagesproduced by the Particle Data Group [13].

Resonance parameters Fit to data resultsState JPC Mass (MeV) Width (MeV) Magnitude Phase (◦) Fraction fr (%)ρ(770)+ 1−− 775.8 150.3 1 0 66.4±0.5ρ(770)0 1−− 775.8 150.3 0.55±0.01 16.1±0.4 23.9±0.3ρ(770)− 1−− 775.8 150.3 0.73±0.01 −1.6±0.5 35.6±0.4ρ(1450)+ 1−− 1465 400 0.55±0.07 −7.7±8.2 1.1±0.3ρ(1450)0 1−− 1465 400 0.19±0.07 −70.4±15.9 0.1±0.1ρ(1450)− 1−− 1465 400 0.53±0.06 8.2±6.7 1.0±0.2ρ(1700)+ 1−− 1720 250 0.91±0.15 −23.3±10.3 1.5±0.5ρ(1700)0 1−− 1720 250 0.60±0.13 −56.3±16.0 0.7±0.3ρ(1700)− 1−− 1720 250 0.98±0.17 78.9±8.5 1.7±0.6f0(980) 0++ 980 44 0.06±0.01 −58.8±2.9 0.3±0.1f0(1370) 0++ 1434 173 0.20±0.03 −19.6±9.5 0.3±0.1f0(1500) 0++ 1507 109 0.18±0.02 7.4±7.4 0.3±0.1f0(1710) 0++ 1714 140 0.40±0.08 42.9±8.8 0.3±0.1f2(1270) 2++ 1275.4 185.1 0.25±0.01 8.8±2.6 0.9±0.1f0(500) 0++ 500 400 0.26±0.01 −4.1±3.7 0.9±0.1NR 0.43±0.07 −22.1±11.7 0.4±0.1

TABLE II: Summary of systematic uncertainties. The varioussources are added in quadrature to find the total systematicuncertainty.

Source x [%] y [%]

“Lucky” false slow pion fraction 0.01 0.01Time resolution dependence

0.03 0.02on reconstructed D0 massAmplitude-model variations 0.31 0.12Resonance radius 0.02 0.10DP efficiency parametrization 0.03 0.03DP normalization granularity 0.03 0.04Background DP distribution 0.21 0.11Decay time window 0.18 0.19σt cutoff 0.01 0.01Number of σt ranges 0.11 0.26σt parametrization 0.05 0.03Background-model MC time

0.06 0.11distribution parametersFit bias correction 0.29 0.02SVT misalignment 0.20 0.23

Total 0.56 0.46

events generated with a uniform phase space distribution.As a variation, we parameterize the efficiency using athird-degree polynomial in s+, s− and take the differencein mixing parameters as the uncertainty in the efficiencymodel. Normalization over the DP is done numericallyby evaluating the total PDF on a 120× 120 grid. To findthe sensitivity to the accuracy of the normalization inte-gral, we vary the granularity of the grid from 120× 120to 240 × 240 and take the largest variations as system-atic uncertainties. The combinatorial background in theDP is modeled by sideband data summed according to

weights taken from simulation. We repeat the fit using ahistogram taken from simulation and vary the weights by±1 standard deviation. Additionally, we vary the num-ber of bins used in the “broken-charm” histograms.

In the nominal fit, we consider events in the decay timewindow between −2 and +3 ps, i.e. about −5 to +7 τD0 .To test our sensitivity to high-|t| events, the window isvaried, with the low end ranging from −3.0 ps to −1.5 psand the high end ranging from 2.0 ps to 3.0 ps. We assignan uncertainty of 0.18% to x and 0.19% to y, the largestvariations from this source. We vary the maximum al-lowed uncertainty on the reconstructed decay time σt tostudy the effect of poorly measured events. The nomi-nal cutoff at 0.8 ps is relaxed to 1.2 ps in steps of 0.1ps and we use the largest variations as the uncertaintiesfrom this source. To account for the variation of σt acrossthe DP, the nominal fit has six different σt distributions,one for each range of m2(π+π−). We reduce the num-ber of ranges to two and increase it to eight, and use thelargest difference as the uncertainty associated with thenumber of ranges. Additionally, instead of using a func-tional form to describe the σt distribution in each range,we repeat our nominal fit using a histogram taken fromsimulation. This produces extremely small changes in themeasured mixing parameters; we take the full differenceas an estimate of the uncertainty.

In the nominal fit, the background components havetheir decay time dependences modeled by the sums of twoexponentials convolved with Gaussians whose parametersare fixed to values found from fits to simulated data. Wevary each parameter in sequence by ±1 standard devia-tion and take the largest variations as estimates of thesystematic uncertainty.

Our fits combine two effects: detector resolution and

11

efficiency. We ignore the migration of events which areproduced at one point in the DP and reconstructed atanother point; we parameterize detection efficiency fromsimulated events, generated with a uniformly populatedDP using the observed positions, in the numerator. Asnoted earlier, this leads to discrepancies between fit pro-jections and data for simulated data which are very sim-ilar to those observed for real data as observed in Fig. 2.We believe this is due to ignoring the systematic migra-tion of events away from the boundaries of phase spaceinduced by misreconstruction followed by constrained fit-ting. We have further checked the migration effect byfitting the data in a smaller DP phase space with allthe boundaries shifted 0.05 GeV2 inwards. In addition,detector resolution leads to a correlation between recon-structed D0 mass and t, also noted earlier. To estimatethe level of bias and systematic uncertainty introduced bythese factors, we studied the full MC samples describedin Section III. The fit results display small biases in xand y. From the fit to each sample, we determine thepull values for x and y, defined as the differences of fit-ted and input values. We then correct for fit biases bysubtracting +0.58% from x and −0.05% from y where thenumerical values are the mean deviations from the gener-ated values. The assigned systematic uncertainties arehalf the shifts in each variable.To test the sensitivity of our results to small un-

certainties in our knowledge of the precise positions ofthe SVT wafers, we reconstruct some of our MC sam-ples with deliberately wrong alignment files that producemuch greater pathologies than are evident in the data.We again create background mixtures and fit these mis-aligned samples. Four samples are generated, all withx = y = +1%. Each sample has roughly the same mag-nitude of effect caused by the five different misalignmentsconsidered. As the misalignments used in this study areextreme, we estimate the systematic uncertainties as halfof the averages of the absolute values of the shifts in xand y.

VI. SUMMARY AND CONCLUSIONS

We have presented the first measurement of D0–D0

mixing parameters from a time-dependent amplitudeanalysis of the decay D0 → π+π−π0. We find x =

(1.5 ± 1.2 ± 0.6)% and y = (0.2 ± 0.9 ± 0.5)%, wherethe quoted uncertainties are statistical and systematic,respectively. The dominant sources of systematic uncer-tainty can be reduced in analyses with larger data sets.Major sources of systematic uncertainty in this measure-ment include those originating in how we determine shiftsfor detector misalignment and the choice of decay timewindow. We estimated conservatively the former as it isalready small compared to the statistical uncertainty ofthis measurement. The latter can be reduced by morecarefully determining the signal-to-background ratio asa function of decay time. However, since the systematicuncertainties are already small compared to the statisti-cal uncertainties, we choose not to do so in this analysis.Similar considerations suggest that systematic uncertain-ties will remain smaller than statistical uncertainties evenwhen data sets grow to be 10 to 100 times larger in ex-periments such as LHCb and Belle II.

VII. ACKNOWLEDGMENTS

We are grateful for the extraordinary contributions ofour PEP-II colleagues in achieving the excellent luminos-ity and machine conditions that have made this work pos-sible. The success of this project also relies critically onthe expertise and dedication of the computing organiza-tions that support BABAR. The collaborating institutionswish to thank SLAC for its support and the kind hospi-tality extended to them. This work is supported by theUS Department of Energy and National Science Foun-dation, the Natural Sciences and Engineering ResearchCouncil (Canada), the Commissariat a l’Energie Atom-ique and Institut National de Physique Nucleaire et dePhysique des Particules (France), the Bundesministeriumfur Bildung und Forschung and Deutsche Forschungsge-meinschaft (Germany), the Istituto Nazionale di FisicaNucleare (Italy), the Foundation for Fundamental Re-search on Matter (The Netherlands), the Research Coun-cil of Norway, the Ministry of Education and Science ofthe Russian Federation, Ministerio de Economıa y Com-petitividad (Spain), the Science and Technology Facili-ties Council (United Kingdom), and the Binational Sci-ence Foundation (U.S.-Israel). Individuals have receivedsupport from the Marie-Curie IEF program (EuropeanUnion) and the A. P. Sloan Foundation (USA).

[1] A. Pais and S. B. Treiman, Phys. Rev. D 12, 2744 (1975).[2] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett.

98, 211802 (2007).[3] M. Staric et al. (Belle Collaboration), Phys. Rev. Lett.

98, 211803 (2007).[4] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett.

100, 121802 (2008).[5] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett.

110, 101802 (2013).

[6] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett.111, 251801 (2013).

[7] R. Aaij et al. (LHCb Collaboration), arXiv:1602.07224[hep-ex] (2016).

[8] D. M. Asner et al. (CLEO Collaboration), Phys. Rev. D72, 012001 (2005).

[9] K. Abe et al. (Belle Collaboration), Phys. Rev. Lett. 99,131803 (2007).

[10] P. del Amo Sanchez et al. (BABAR Collaboration), Phys.

12

Rev. Lett. 105, 081803 (2010).[11] T. Peng et al. (Belle Collaboration), Phys. Rev. D 89,

091103 (2014).[12] Y. Amhis et al. (Heavy Flavor Averaging Group

(HFAG)), arXiv:1412.7515 [hep-ex] (2014), (May 2015updated values taken from HFAG website.).

[13] K. Olive et al. (Particle Data Group), Chin.Phys. C38,090001 (2014).

[14] J. P. Lees et al. (BABAR Collaboration), Nucl. Instr.Meth. Phys. Res., Sect. A 726, 203 (2013).

[15] B. Aubert et al. (BABAR Collaboration), Nucl. Instr.Meth. Phys. Res., Sect. A 479, 1 (2002).

[16] B. Aubert et al. (BABAR Collaboration), Nucl. Instr.Meth. Phys. Res., Sect. A 729, 615 (2013).

[17] B. Aubert et al. (BABAR Collaboration), Phys. Rev. D

74, 091102 (2006).[18] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett.

99, 251801 (2007).[19] H. Albrecht et al. (ARGUS), Z. Phys. C48, 543 (1990).[20] G. N. Fleming, Phys. Rev. 135, B551 (1964).[21] D. Morgan, Phys. Rev. 166, 1731 (1968).[22] D. J. Herndon et al., Phys. Rev. D 11, 3165 (1975).[23] S. Kopp et al. (CLEO), Phys. Rev. D 63, 092001 (2001).[24] D. Cronin-Hennessy et al. (CLEO Collaboration), Phys.

Rev. D 72, 031102 (2005).[25] R. Andreassen et al., IEEE Access 2, 160 (2014).[26] The code is published on GitHub at

http://github.com/GooFit/GooFit