First determination of the CP content of D→ π + π − π + π − and updated determination of the CP contents of D→ π + π − π 0 and D→ K + K − π 0
S. Malde, C. Thomas, G. Wilkinson, P. Naik, C. Prouve, J. Rademacker, J. Libby, M. Nayak, T. Gershon, R.A. Briere
FFirst determination of the CP content of D → π + π − π + π − and updated determination of the CP contents of D → π + π − π and D → K + K − π a , C. Thomas a , G. Wilkinson a,b , P. Naik c , C. Prouve c ,J. Rademacker c , J. Libby d , M. Nayak d , T. Gershon e , R. A. Briere f a University of Oxford, Denys Wilkinson Building, Keble Road, OX1 3RH, UnitedKingdom b European Organisation for Nuclear Research (CERN), CH-1211, Geneva 23,Switzerland c University of Bristol, Bristol, BS8 1TL, United Kingdom d Indian Institute of Technology Madras, Chennai 600036, India e University of Warwick, Coventry, CV4 7AL, United Kingdom f Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
Abstract
Quantum-correlated ψ (3770) → DD decays collected by the CLEO-c experi-ment are used to perform a first measurement of F π + , the fractional CP -evencontent of the self-conjugate decay D → π + π − π + π − , obtaining a value of0 . ± . D → K π + π − and D → K π + π − decays to tag the signal mode. Thissame technique is applied to the channels D → π + π − π and D → K + K − π ,yielding F πππ + = 1 . ± . ± .
022 and F KKπ + = 0 . ± . ± . CP -eigenstate tags, and can be combined to give values of F πππ + = 0 . ± . F KKπ + = 0 . ± . γ using B ∓ → DK ∓ decays, and in time-dependent studies of CP violation and mixing in the D D system. Keywords: charm decay, quantum correlations, CP violation Preprint submitted to Physics Letters B October 3, 2018 a r X i v : . [ h e p - e x ] M a y . Introduction Studies of the process B ∓ → DK ∓ , where D indicates a neutral charmedmeson reconstructed in a state accessible to both D and D decays, givesensitivity to the unitarity triangle angle γ ≡ arg( − V ud V ∗ ub /V cd V ∗ cb ) (also de-noted φ ). Improved knowledge of γ is necessary for testing the StandardModel description of CP violation. In a recent publication [1] it was shownhow inclusive three-body self-conjugate D meson decays can be used for thispurpose, provided their fractional CP -even content is known, a quantity de-noted F + (or F f + when it is necessary to designate the specific decay f ).Measurements of F + for the decays D → π + π − π and D → K + K − π wereperformed, making use of quantum-correlated DD decays coherently pro-duced at the ψ (3770) resonance and collected by the CLEO-c detector. Inthis Letter a first measurement is presented of the CP content of the four-body mode D → π + π − π + π − , again exploiting CLEO-c ψ (3770) data. Thisfully-charged and relatively abundant final state [2] can be reconstructedwith good efficiency by the LHCb detector and hence is a promising modefor improving the determinaton of γ at that experiment, as well as at Belle II.The three-body analysis reported in Ref. [1] exploited events in whichone D meson is reconstructed in the signal mode and the other ‘tagging’ me-son in its decay to a CP eigenstate. The measurement of F π + presentedin this Letter follows the same method, but augments it with other ap-proaches, in particular a complementary strategy in which the tagging modesare D → K , L π + π − , and attention is paid to where on the Dalitz plot this tagdecay occurs. In order to benefit from this strategy for the previously stud-ied decays, this Letter also presents measurements of F πππ + and F KKπ + using D → K , L π + π − tags. Throughout the effects of CP violation in the charmsystem are neglected, which is a good assumption given theoretical expecta-tions and current experimental limits [2–4]. However, as discussed in Ref. [5],knowledge of F + also allows such D decays to be used to study CP -violatingobservables and mixing parameters through time-dependent measurementsat facilities where the mesons are produced incoherently.The remainder of the Letter is structured as follows. Section 2 intro-duces the CP -even fraction F + , derives the relations that are used to mea-sure its value at the ψ (3770) resonance, and reviews how knowledge of F + allows non- CP eigenstates to be cleanly employed in the measurement of γ with B ∓ → DK ∓ decays. Section 3 describes the data set and eventselection. Sections 4, 5 and 6 presents the determination of F + using CP D → K , L π + π − tags and other tags, respectively. In Sect. 7 combi-nations of the individual sets of results are performed for each signal mode;for D → π + π − π and D → K + K − π these combinations include the resultsfrom Ref. [1]. Section 8 gives the conclusions.
2. Measuring the CP content of a self-conjugate D -meson decayand the consequences for the γ determination with B ∓ → DK ∓ Let the amplitude of a D meson decaying to a self-conjugate final state f be written as A ( D → f ( x )) ≡ a x e iθ x , where x indicates a particularpoint in the decay phase space and θ x is a CP -conserving strong phase. Theamplitude is normalised such that (cid:90) x ∈D |A ( D → f ( x )) | d x = B ( f ) , (1)where B ( f ) is the branching fraction of the D decay and D indicates theentire phase space. The D decay amplitude at ¯x is denoted a ¯ x e iθ ¯ x , where ¯x indicates the point in phase space reached by applying a CP transformationto the final-state system at x . CP violation in the charm system is neglected,which implies that the D decay amplitude at ¯x is equal to the D amplitudeat x . It is useful to define the strong phase difference ∆ θ x ≡ θ x − θ ¯ x .It is possible to express the CP -even fraction in terms of the decay am-plitudes introduced above. Let the CP eigenstates be | D CP ± (cid:105) ≡ ( | D (cid:105) ±| D (cid:105) ) / √ D → f in terms of these states. The total CP -even fraction of the inclusive decay is defined as F f + ≡ (cid:82) x ∈D |(cid:104) f ( x ) | D CP + (cid:105)| d x (cid:82) x ∈D |(cid:104) f ( x ) | D CP + (cid:105)| + |(cid:104) f ( x ) | D CP − (cid:105)| d x , (2)and so F f + = (cid:82) x ∈D a x + a x + 2 a x a ¯ x cos ∆ θ x d x (cid:82) x ∈D a x + a x ) d x = 12 (cid:20) B ( f ) (cid:90) x ∈D a x a ¯ x cos ∆ θ x d x (cid:21) . (3)Note also that the following relation is always true in the absence of CP violation: (cid:90) x ∈D a x a ¯ x sin ∆ θ x d x = 0 . (4)3ow consider a quantum-correlated DD system produced in the decayof a ψ (3770) meson. One of the D mesons in the system decays to f at thepoint x , the other to g at y , where in general the phase space of the twodecays is different. The amplitude of the latter decay is denoted b y e iφ y inanalogy with the terminology used above.The amplitude of the ψ (3770) → DD → f g correlated wavefunction canbe written [6] A ( f ( x ) | g ( y )) = 1 √ (cid:2) a x e iθ x b ¯ y e iφ ¯ y − a ¯ x e iθ ¯ x b y e iφ y (cid:3) . (5)The resulting decay probability is then P ( f ( x ) | g ( y )) ∝ (cid:104) a x b y + a x b y − a x b ¯ y a ¯ x b y (cid:16) cos ∆ θ x cos ∆ φ y + sin ∆ θ x sin ∆ φ y (cid:17)(cid:105) . (6)If both D mesons decay to the same final state the probability is dividedby two to avoid double counting. This formula can be used to determinethe population of quantum-correlated decays either integrated over all phasespace or after dividing the phase space into bins.The number of ‘double-tagged’ candidates in which one D meson decaysto f and the other to g , integrating over the phase space of each decay, is M ( f | g ) = ZB ( f ) B ( g ) (cid:104) − (cid:16) F f + − (cid:17) (cid:16) F g + − (cid:17)(cid:105) , (7)where Z is a normalisation constant common to all decay modes. An impor-tant special case, considered in Sect. 4, is where the tagging-mode g is a CP eigenstate, and (2 F g + −
1) reduces to ±
1. Section 6 describes an analysis ofclasses of double-tags where this is not the case.Alternatively, when the tagging-mode g is a multibody decay, its phasespace may be divided into bins. Integrating over the phase space of f resultsin the following decay probability in bin i of the phase space of g : P ( f | g i ) ∝ (cid:90) y ∈D i b y + b y − (cid:16) F f + − (cid:17) b y b ¯ y cos ∆ φ y d y , (8)where D i indicates the phase space encompassed by bin i . In Sect. 5 thisrelation is exploited for the tags D → K , L π + π − .To understand the relevance of the CP -even fraction in the measurementof the unitarity-triangle angle γ consider the decay of a B − meson to DK − ,4ollowing which the D meson decays to a self-conjugate final state f consistingof three or more particles. The amplitude of the B − decay is a superpositionof two decay paths: A ( B − ) = A ( B − → D K − ) A ( D → f ) + A ( B − → D K − ) A ( D → f ) . (9)Following the formalism developed above, the decay amplitude of the D meson at the point x in the phase space is denoted a x e iθ x . The decayamplitude of the B − meson at this point in phase space is therefore A ( B − ( x )) = A ( B − → D K − ) (cid:104) a x e iθ x + r B e i ( δ B − γ ) a ¯ x e iθ ¯ x (cid:105) , (10)where r B and δ B are respectively the ratio of moduli and the strong phasedifference between the suppressed and favoured B − decay amplitudes. Theresulting decay probability is P ( B − ( x )) ∝ a x + r B a x + 2 r B a x a ¯ x cos ( δ B − γ + θ x − θ ¯ x ) (11)= a x + r B a x + 2 r B a x a ¯ x (cid:104) cos( δ B − γ ) cos ∆ θ x − sin( δ B − γ ) sin ∆ θ x (cid:105) . The expression for B + ( x ) is identical except that the sign in front of γ is re-versed and x ↔ ¯x . The total yield of B ∓ decays is determined by integratingover the entire D phase space: Y ∓ = h ∓ (cid:90) x ∈D P ( B ∓ ( x )) d x = h ∓ (cid:104) r B + (cid:16) F f + − (cid:17) r B cos( δ B ∓ γ ) (cid:105) , (12)where h ∓ is a normalisation constant and Eqs. 3 and 4 have been employed.This expression is very similar to that derived in Ref. [7] for the case whenthe D meson decays to a CP eigenstate and is indeed identical in the event F f + = 0 or 1. Hence measurements of Y ∓ , and observables built from theseyields [1], can be used to obtain information on the angle γ and the otherparameters of the B ∓ decay, provided that F f + is known. In Ref. [1] it isdemonstrated how the effects of D D mixing, neglected in Eq. 12, may alsobe accommodated.
3. Data set and event selection
The data set analysed consists of e + e − collisions produced by the Cor-nell Electron Storage Ring (CESR) at √ s = 3 .
77 GeV corresponding to an5 able 1: D -meson final states reconstructed in this analysis. Type Final statesMixed
CP π + π − π + π − , π + π − π , K + K − π , K , L π + π − CP -even K + K − , π + π − , K π π , K π , K ωCP -odd K π , K ω , K η , K η (cid:48) integrated luminosity of 818 pb − and collected with the CLEO-c detector.The CLEO-c detector is described in detail elsewhere [8]. Monte Carlo sim-ulated samples of signal decays are used to estimate selection efficiencies.Possible background contributions are determined from a generic D D sim-ulated sample corresponding to approximately fifteen times the integratedluminosity of the data set. The EVTGEN generator [9] is used to simulatethe decays. The detector response is modelled using the GEANT softwarepackage [10].Table 1 lists the D -meson final states considered in the analysis. Double-tag candidates are reconstructed in which one D meson decays into π + π − π + π − and the other into a CP eigenstate, or where one D meson decays into π + π − π + π − , π + π − π or K + K − π and the other into one of the mixed- CP modes K π + π − or K π + π − . The combinations π + π − π + π − vs. π + π − π + π − and π + π − π + π − vs. π + π − π are also reconstructed.The unstable final state particles are reconstructed in the following decaymodes: π → γγ , K → π + π − , ω → π + π − π , η → γγ , η → π + π − π and η (cid:48) → η ( γγ ) π + π − . The π , K , ω , η and η (cid:48) reconstruction procedure isidentical to that used in Ref. [11].Final states that do not contain a K are fully reconstructed via the beam-constrained candidate mass, m bc ≡ (cid:112) s/ (4 c ) − p D /c , where p D is the D -candidate momentum, and ∆ E ≡ E D − √ s/
2, where E D is the D -candidateenergy. The m bc and ∆ E distributions of correctly reconstructed D -mesoncandidates peak at the nominal D mass and zero, respectively. Neither∆ E nor m bc distributions exhibit any peaking structure for combinatoricbackground. The double-tag yield is determined from counting events insignal and sideband regions of m bc after placing requirements on ∆ E [1, 11–13]. The selection criteria of candidates involving the modes D → K + K − and D → π + π − do not include the cosmic ray muon and radiative Bhabha vetoesthat are described in Ref. [1]. This is because these sources of background do6ot contaminate the double-tag sample, and the vetoes are found to perturbthe selection efficiency of the other D meson in the event. When selecting D → K π + π − candidates it is demanded that the K decay products form avertex that is significantly displaced from the e + e − collision point; in contrast,for D → π + π − π + π − and D → π + π − π candidates the π + π − vertex mustbe consistent with originating from the collision point in order to suppresscontamination from D → K π + π − and D → K π decays, respectively.The double-tag yield determination procedure is identical to that pre-sented in Refs. [11, 12] except for the selections where the signal decay is π + π − π + π − and the tag decay is K + K − , π + π − , π + π − π or π + π − π + π − , whichare all dominated by a background from continuum production of light quark-antiquark pairs. For these modes an unbinned maximum likelihood fit is per-formed to the distribution of the average m bc of the two D -meson candidates.The background is modelled with an ARGUS function [14] and the signal ismodelled with the sum of two Crystal Ball functions [15] with power-lawtails on opposite sides. The parameters of the Crystal Ball functions arefixed from fits to large samples of simulated data.Figures 1 (a) and (b) show the average m bc distributions for CP -tagged D → π + π − π + π − candidates, summed over all tag modes that are CP -evenand CP -odd eigenstates, respectively, where the CP -tag final state does notcontain a K meson. Figure 2 shows the average m bc distributions for D → π + π − π + π − , D → π + π − π and D → K + K − π candidates tagged with D → K π + π − decays, while Figs. 3 (a)–(c) show the Dalitz-plot distributions ofthe tag decay for these three signal modes.Many K mesons do not deposit any reconstructible signal in the detector.However, double-tag candidates can be fully reconstructed using a missing-mass squared ( m ) technique [16] for tags containing a single K meson.Yields are determined from the signal and sideband regions of the m distribution. Figure 1 (c) shows the m distributions for D → π + π − π + π − candidates tagged with either a K π or K ω decay. Figure 4 shows the m distributions for D → π + π − π + π − , D → π + π − π and D → K + K − π candidates tagged with D → K π + π − decays, and Figs. 3 (d)–(f) show thecorresponding tag-side Dalitz-plot distributions.In events where more than one pair of decays is reconstructed an algo-rithm is applied to select a single double-tag candidate based on the informa-tion provided by the m bc and ∆ E variables. The particular choice of metricvaries depending on the category of double tag and is optimised throughsimulation studies. 7 c [GeV/ bc m ) c E v e n t s / ( M e V / c [GeV/ bc m ) c E v e n t s / ( M e V / c / [GeV m -0.2 0.2 0.6 1 ) c / E v e n t s / ( M e V Figure 1: Distributions of D → π + π − π + π − candidates tagged by CP -eigenstates. Sub-figures (a) and (b) show average m bc distributions for CP -even tags and CP -odd tagsnot involving K mesons, respectively. Sub-figure (c) shows the m distribution forcandidates tagged by CP eigenstates that contain a K meson. The shaded histogram isthe estimated peaking background and the vertical dotted lines indicate the signal region. ] c [GeV/ bc m ) c E v e n t s / ( M e V / c [GeV/ bc m ) c E v e n t s / ( M e V / c [GeV/ bc m ) c E v e n t s / ( M e V / Figure 2: Average m bc distributions for (a) D → π + π − π + π − , (b) D → π + π − π and (c) D → K + K − π candidates tagged by a D → K π + π − decay. The shaded histogram isthe estimated peaking background and the vertical dotted lines indicate the signal region. The peaking background estimates are determined from the generic MonteCarlo sample of D D events. For double tags involving a CP mode withouta K meson the peaking backgrounds are found to constitute 5-10% of theselected events, and are predominantly from residual D → K π + π − contam-ination. The peaking backgrounds for final states with a K are generallylarger; for K π and K ω this contamination amounts to 15–20% of the sig-nal yield, whereas for K π + π − it is ∼
10% of the signal yield. The dominantsource of peaking background in each case is the equivalent decay containing8 c / [GeV +2 m ] c / [ G e V - m c / [GeV +2 m ] c / [ G e V - m c / [GeV +2 m ] c / [ G e V - m c / [GeV +2 m ] c / [ G e V - m c / [GeV +2 m ] c / [ G e V - m c / [GeV +2 m ] c / [ G e V - m Figure 3: Dalitz-plot distributions for D → K π + π − reconstructed against (a) D → π + π − π + π − , (b) D → π + π − π and (c) D → K + K − π , and D → K π + π − reconstructedagainst (d) D → π + π − π + π − , (e) D → π + π − π and (f) D → K + K − π . The axis labels m ± are the invariant-mass squared of the π ± K , L pair. ] c / [GeV m -0.2 0.2 0.6 1 ) c / E v e n t s / ( M e V c / [GeV m -0.2 0.2 0.6 1 ) c / E v e n t s / ( M e V c / [GeV m -0.2 0.2 0.6 1 ) c / E v e n t s / ( M e V Figure 4: m distributions for (a) D → π + π − π + π − , (b) D → π + π − π and (c) D → K + K − π candidates tagged by a D → K π + π − decay. The shaded histogram is theestimated peaking background and the vertical dotted lines indicate the signal region. K instead of a K meson. The contamination from specific modes in theother categories of double tags is typically 10% or less. The statistical un-certainties on these background estimates arising from the finite size of thesimulated samples are included in the total statistical uncertainties on thesignal yields.The measured double-tag event yields after background subtraction aregiven in Table 2. Table 2: Double-tagged signal yields after background subtraction. Information on theentries marked ‘ † ’, not studied in the current analysis, can be found in Ref. [1]. π + π − π + π − π + π − π K + K − π K + K − . ± . † † π + π − . ± . † † K π π . ± . † † K π . ± . † † K ω . ± . † † K π . ± . † † K ω . ± . † † K η ( γγ ) 18 . ± . † † K η ( π + π − π ) 3 . ± . † † K η (cid:48) . ± . † † K π + π − . ± . . ± . . ± . K π + π − . ± . . ± . . ± . π + π − π + π − . ± . . ± . CP -eigenstate modes is requiredfor normalisation purposes. Since the single-tag reconstruction criteria ap-plied are identical to those employed in Ref. [1], all information on theseyields is taken from the earlier publication. It is also necessary to know thesingle-tag yield for the decay D → π + π − π . A fit to the m bc distributionreturns a result of 29998 ±
320 signal candidates, after the subtraction ofsmall peaking-background contributions.
4. Analysis with the CP tags The yields of the single and double tags containing a CP eigenstate areused as inputs to determine the CP -even fraction, F π + . Following on from10q. 7, the expected number of observed events, M , where one D meson decaysto the π + π − π + π − final state, and the other decays to X , a CP eigenstatewith eigenvalue η CP , is given by M (4 π | X ) = 2 N DD B (4 π ) B ( X ) ε (4 π | X ) (cid:104) − η CP (cid:16) F π + − (cid:17)(cid:105) , (13)where N DD is the number of DD pairs, B (4 π ) and B ( X ) are the branchingfractions for the two reconstructed final states and ε (4 π | X ) is the efficiencyof reconstructing such a double tag. The double tag yield is denoted by M − ( M + ) for CP -even ( CP -odd) tags. Experimentally it is advantageous toeliminate dependence on N DD , the branching fractions and the reconstructionefficiency, which can be achieved by normalising by the single-tag yields. Theyield of single tags, S + ( S − ) decaying to a CP -odd ( CP -even) eigenstate X ,is given by S ( X ) = 2 N DD B ( X ) ε ( X ) , (14)where ε ( X ) is the reconstruction efficiency of the single tag. The smalleffects of D D mixing are eliminated from the measurement by correctingthe measured single-tag yields S ± meas such that S ± = S ± meas / (1 − η CP y D ) where y D = (0 . ± . D -mixing parameter [17]. A furthercorrection is applied in the case of the tags K + K − and π + π − because ofthe differing selection requirements for the single and double-tag case, asdescribed in Sect. 3. This correction factor is determined by taking the ratioof the selection efficiency of the single tag from simulation with the twodiffering selections. It is determined to be 1.15 and 1.10 for the D → K + K − and D → π + π − modes, respectively, with an uncertainty of ± .
05. Theother uncertainties on these single-tag yields are assigned following the sameprocedure described in Ref. [1].For the case of the two CP tags involving a K meson a different treatmentis required, since it is not possible to measure the single-tag yield directlyfor these modes. Following the procedure described in Ref. [1], the effectivesingle-tag yield is evaluated using Eq. 14, where the effective single-tag ef-ficiency ε ( K X ) is calculated from the ratio of ε (4 π | K X ) /ε (4 π ), and theleading systematic uncertainties are associated with the branching fractionsand the value used for the effective single-tag efficiency. The effective single-tag yields are determined to be 21726 ± ± K π and K ω , respectively.Assuming that the reconstruction efficiencies of each D meson are inde-pendent, then the ratio of the double-tagged and single-tagged yields are11 N p K w K ) gg ( h K ) ppp ( h K ' h K (a) - N -0.001 0 0.001 0.002 0.003 0.004 0.005KK pp p p K w K p K (b) Figure 5: D → π + π − π + π − results for (a) N + and (b) N − . In each plot the vertical yellowband indicates the value obtained from the combination of all tags. The black portionof the uncertainty represents the statistical uncertainty only while the red represents thetotal. independent of the branching fraction and reconstruction efficiency of the CP tag and N DD . This ratio is defined as N + ≡ M + /S + , with an analogousexpression for N − . The CP -even fraction F π + is then given by F π + = N + N + + N − . (15)The measured values for N + and N − for each CP tag are displayed in Fig. 5.It can be seen that there is consistency between the individual tags for eachmeasurement. The mean value < N + > = (5 . ± . × − is significantlylarger than < N − > = (1 . ± . × − , indicating that the π + π − π + π − final state is predominantly CP even.If the acceptance across the phase space of the D → π + π − π + π − decayis not uniform it has the potential to bias the measurement of F π + . Usingsimulated data the selection efficiency of individual pions in D → π + π − π + π − decays is determined in bins of momentum and polar angle with respect to thebeam direction. The candidates in data are then weighted by the normalisedefficiency. Each pion is treated independently and an overall weight, typicallylying within 5–10% of unity, is found by multiplying the individual weights.The scaled signal yields are used to re-determine F π + and the difference12etween this and the value found without efficiency correction is 0.008, whichis taken as the systematic uncertainty due to non-uniform acceptance.Using the CP tags only, and accounting for the correlations between thesystematic uncertainties, yields F π + = 0 . ± . ± .
5. Analysis with the D → K , L π + π − tags For each of the signal samples that are tagged by D → K π + π − or D → K π + π − decays the Dalitz plot of the tag mode is divided into eightpairs of symmetric bins by the line m = m − , where m ± is the invariant-masssquared of the π ± K , L pair. The bins lying on one side of this line ( m > m − )are labelled − → −
8, and those on the other side 1 →
8. The binningdefinition follows the ‘Equal ∆ δ D BABAR 2008’ scheme of Ref. [18], in whichthe boundaries are chosen according to the strong-phase prediction of a modeldeveloped by the BaBar collaboration [19]. The expected distribution ofentries is symmetric and so the analysis considers the absolute bin number | i | , which contains the contents of the pair of bins − i and i .Following Eq. 8, the expected population of bin | i | for signal decays with K π + π − tags is M | i | = h (cid:104) K i + K − i − (cid:16) F + − (cid:17) c i (cid:112) K i K − i (cid:105) , (16)where h is a normalisation factor specific to the signal category, K i is theflavour-tagged fraction, being the proportion of K π + π − decays to fall in bin i in the case that the mother particle is known to be a D meson, and c i isthe cosine of the strong-phase difference between D and ¯ D decays averagedin bin i and weighted by the absolute decay rate (a precise definition may befound in Ref. [6]). The only difference between the form of this expressionand the case when the signal decays into a pure CP -even eigenstate [6] is theprefactor of (2 F + −
1) in the final term.Similarly, when the tagging meson decays to K π + π − then the numberof double-tag decays produced in bin | i | is M (cid:48)| i | = h (cid:48) (cid:104) K (cid:48) i + K (cid:48)− i + (cid:16) F + − (cid:17) c (cid:48) i (cid:113) K (cid:48) i K (cid:48)− i (cid:105) , (17)where the primed quantities are now specific to this case. The reversed signin front of the final term reflects the fact that the K meson is almost entirelya CP -odd eigenstate. 13he values of c i and c (cid:48) i within these bins have been measured by theCLEO collaboration [18]. The values of the K i parameters are taken froman analysis of the predictions of various B -factory models [19–22] presentedin Ref. [13], and those of the K (cid:48) i parameters from measurements performedwith CLEO-c data [23].The double-tagged samples are analysed to determine the background-subtracted signal yield in each Dalitz-plot bin. The distribution of back-ground between the different bins is assigned according to its category. Flatbackground is assumed to contribute proportionally to the bin area. Peakingbackgrounds that occur on the signal side affect the distribution of tag decaysin K , L π + π − phase space according to their nature. For example, in the caseof D → K π decays that are wrongly reconstructed as D → π + π − π , the tagdecay will be in a CP -even state and distributed accordingly. Similarly, thedistribution of K ( π π ) π + π − decays that are misreconstructed as K π + π − tags is well understood and modelled appropriately. The distribution of theresidual K π + π − vs. K π + π − events that contaminate the π + π − π + π − vs. K π + π − selection is determined from data by inverting the K veto on thesignal decay.It is also necessary to account for relative bin-to-bin efficiency variationsin the background-subtracted signal yields. The correction factors are de-termined from simulation and typically differ (cid:46)
5% from unity. The signalyields in each bin after background subtraction and relative efficiency cor-rection are shown in Table 3 for K π + π − tags and in Table 4 for K π + π − tags.A log-likelihood fit is performed to the efficiency-corrected signal yieldsof each sample, assuming the expected distributions given by Eqs. 16 and 17.The fit parameters are the CP -even fraction and the overall normalisation.The values of K i , K (cid:48) i , c i and c (cid:48) i are also fitted, but with their measurementuncertainties and correlations imposed with Gaussian constraints. Separatefits are performed for the D → K π + π − tags, the D → K π + π − tags, andfor both samples combined. Fits to large ensembles of simulated experimentsdemonstrate that the returned uncertainties are reliable and that there is nosignificant bias in the procedure. All data fits are found to be of good quality.The results are plotted in Fig. 6 for the D → K π + π − tags and in Fig. 7 forthe D → K π + π − tags. The numerical results for the CP -even fraction aregiven in Table 5 for D → π + π − π + π − and in Table 6 for D → π + π − π and D → K + K − π .The dominant systematic uncertainty is associated with the distribution14 able 3: Double-tagged signal yields vs. K π + π − after background subtraction in absolutebin numbers of the D → K π + π − Dalitz plot. The yields are corrected for relative bin-to-bin efficiency variations and then scaled so that the totals match the values in Table 2. | i | π + π − π + π − π + π − π K + K − π . ± . . ± . . ± .
12 19 . ± . . ± . . ± .
63 16 . ± . . ± . . ± .
54 10 . ± . . ± . . ± .
55 55 . ± . . ± . . ± .
16 21 . ± . . ± . . ± .
47 27 . ± . . ± . . ± .
88 36 . ± . . ± . . ± . . .
019 and 0 . F π + , F πππ + and F KKπ + , respectively. The uncertainty as-sociated with the measurement errors on K i , K (cid:48) i , c i and c (cid:48) i is estimated byre-running the fit with these quantities set as fixed parameters and subtract-ing in quadrature the new fit uncertainty from that obtained with the orig-inal procedure. This component is found to be 0 .
013 for D → π + π − π + π − ,0 .
010 for D → π + π − π and 0 .
025 for D → K + K − π , and is accountedas a systematic uncertainty in the final results. An uncertainty is evalu-ated to account for non-uniformities in acceptance across phase space. For D → π + π − π + π − this contribution is calculated with the same procedure asin Sect. 4, and found to be 0 .
002 for the joint K , L π + π − fit. For D → π + π − π and D → K + K − π the acceptance uncertainties are taken to be 0.001 and0 .
6. Other tags
The double-tagged yield of π + π − π + π − vs. π + π − π can be used to de-termine F π + , benefiting from the well-measured value of F πππ + . The ratio15 able 4: Double-tagged signal yields vs. K π + π − after background subtraction in absolutebin numbers of the D → K π + π − Dalitz plot. The yields are corrected for relative bin-to-bin efficiency variations. | i | π + π − π + π − π + π − π K + K − π . ± . . ± . . ± .
12 59 . ± . . ± . . ± .
03 55 . ± . . ± . . ± .
14 20 . ± . . ± . . ± .
95 46 . ± . . ± . . ± .
16 24 . ± . . ± . . ± .
77 61 . ± . . ± . . ± .
78 84 . ± . . ± . . ± . Table 5: The F π + fit results for the D → K π + π − tags, where the first uncertainty isstatistical and the second systematic. The row K , L π + π − indicates the configurationwhere the CP -even fraction is a common fit parameter shared between the D → K π + π − and D → K π + π − samples. Tag F π + K π + π − ± ± K π + π − ± ± K , L π + π − ± ± D → π + π − π single-tag yields is defined as N πππ ≡ M (4 π | πππ ) /S ( πππ ), where a very small correction is applied to the mea-sured single-tag yield to account for mixing effects. Following from Eq. 7,the ratio N πππ /N + removes dependence on the signal branching fractionand reconstruction efficiency and is given by N πππ N + = (cid:104) − (cid:16) F π + − (cid:17) (cid:16) F πππ + − (cid:17)(cid:105) F π + , (18)which can be rearranged to yield F π + = N + F πππ + N πππ − N + + 2 N + F πππ + . (19)16 bsolute bin number1 2 3 4 5 6 7 8 E v e n t s / b i n E v e n t s / b i n E v e n t s / b i n Figure 6: Data (points) and fit results (solid line) in absolute bin numbers for K π + π − tags vs. (a) D → π + π − π + π − , (b) D → π + π − π and (c) D → K + K − π . Also shown ineach case is the expectation if F + = 0 (dotted line) or F + = 1 (dashed line). Absolute bin number1 2 3 4 5 6 7 8 E v e n t s / b i n E v e n t s / b i n E v e n t s / b i n Figure 7: Data (points) and fit results (solid line) in absolute bin numbers for K π + π − tags vs. (a) D → π + π − π + π − , (b) D → π + π − π and (c) D → K + K − π . Also shown ineach case is the expectation if F + = 0 (dotted line) or F + = 1 (dashed line). The choice of N + in the denominator of Eq. 18 is preferred to N − as it ismeasured with better relative precision.Taking as input the yields given in Sect. 3, the value of N + reportedin Sect. 4 and the final result for F πππ + presented in Sect. 7 implies F π + =0 . ± . ± . D → πππ single-tag yield and small violationsof the efficiency-factorisation ansatz assumed in Eq. 18; the understandingof the peaking background component in the sample; and the possible effectsof non-uniform acceptance. 17 able 6: The F πππ + and F KKπ + fit results for the D → K π + π − tags, where the firstuncertainty is statistical and the second systematic. The row K , L π + π − indicates theconfiguration where the CP -even fraction is a common fit parameter shared between the D → K π + π − and D → K π + π − samples. Tag F πππ + F KKπ + K π + π − ± ± ± ± K π + π − ± ± ± ± K , L π + π − ± ± ± ± π + π − π + π − vs. π + π − π + π − also carries informa-tion on the value of F π + . This sample is however only used for a consistencycheck, as there are large backgrounds from both the continuum and frommisidentification of D → K ππ decays that are a potential source of signif-icant systematic bias. Furthermore, the predicted yield and measurementuncertainty means that the result from analysis of these double tags wouldhave low weight in the combined measurement of F π + . Using Eq. 7 thenumber of observed self-tagged events is given by M (4 π | π ) = 4 R F π + (cid:16) − F π + (cid:17) , (20)where R = N DD B (4 π ) ε (4 π | π ). The predicted double-tagged yield usingthe value of F π + obtained from the CP tags is 17 ±
2, which is consistent withthe measured yield reported in Table 2.
7. Combination of results
The results for F π + from the CP tags, the K , L π + π − tags and the π + π − π tag are summarised in Table 7. They are compatible and are therefore com-bined, taking account of correlated uncertainties. Correlations arise fromthe non-flat Dalitz plot acceptance between all three measurements and theuse of N + as an input to both the CP tags and π + π − π tag measurements.There is a further small correlation between the results obtained with the CP and π + π − π tags, associated with the uncertainty on the value of themixing parameter y D . The final result is F π + = 0 . ± . F πππ + and F KKπ + obtained with K , L π + π − tags, together with those determined from CP tags. The K , L π + π − mea-18 able 7: Results for F π + for each tag category, and combined. When two uncertaintiesare shown, the first is statistical and the second systematic. For the combined result thetotal uncertainty is given. Tag F π + CP eigenstates 0.754 ± ± K , L π + π − ± ± π + π − π ± ± ± F πππ + = 0 . ± .
017 and F KKπ + = 0 . ± .
055 are obtained. The K , L π + π − tags improve the relative precision on F πππ + by 6% and on F KKπ + by 10%. Table 8: Results for F πππ + and F KKπ + for each tag category, and combined. The CP -eigenstate tag results are from Ref. [1]. When two uncertainties are shown, the first isstatistical and the second systematic. For the combined result the total uncertainty isgiven. Tag F πππ + F KKπ + CP eigenstates 0.968 ± ± ± ± K , L π + π − ± ± ± ± ± ±
8. Conclusions
A first measurement has been made of the CP -even fraction of the de-cay D → π + π − π + π − , exploiting quantum-correlated double-tags involving CP -eigenstates, a binned Dalitz-plot analysis of the modes D → K , L π + π − ,and D → π + π − π decays. The result, F π + = 0 . ± . D decays that can be harnessed for the mea-surement of the unitarity-triangle angle γ through the process B ∓ → DK ± .19he decays D → K , L π + π − have also been employed as a tag to measure the CP contents of the modes D → π + π − π and D → K + K − π . The resultsconfirm the conclusion of a previous analysis [1], based on CP -eigenstate tags,and also suggested by earlier amplitude-model studies [24–26], that the CP -even content of the π + π − π final state is very high, and therefore this decaytoo is a powerful mode for the measurement of γ . Combining the two setsof measurements yields F πππ + = 0 . ± .
017 and F KKπ + = 0 . ± . CP -even fractions have been measured, all three decay modesmay also be used for studies of indirect CP violation and mixing in the D D system [5]. Acknowledgments
This analysis was performed using CLEO-c data, and as members of theformer CLEO collaboration we thank it for this privilege. We are gratefulfor support from the UK Science and Technology Facilities Council, the UK-India Education and Research Initiative, and the European Research Councilunder FP7.
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