Measurement of violating asymmetries
in decays with a time-dependent Dalitz approach
We report a measurement of violating asymmetries in decays with a time-dependent Dalitz approach. This analysis is based on a data sample of pairs accumulated at the resonance with the Belle detector at the KEKB asymmetric-energy collider. As the result of an unbinned maximum likelihood fit to the selected candidates, the mixing-induced and direct violation parameters, and are obtained for , and other decays. We find four solutions that describe the data. There are
The values for the violating phase in are similar but other properties of the Dalitz plot are quite different for the four solutions. These four solutions have consistent values for all three meson decay channels and none of them deviates significantly from the values measured in decays with the currently available statistics. In addition, we find no significant direct violation.
The Belle Collaboration
violation in the quark sector is described in the standard model (SM) by the Kobayashi-Maskawa (KM) theory KM (). In this theory, the existence of a single irreducible phase gives rise to violating asymmetries in the time-dependent rates of and decays into a common eigenstate carter (). Specifically, for neutral meson decays dominated by transitions such as , we can measure violating quantities that determine the phi1_beta () angle of the Unitarity Triangle. The measurements have been performed by Belle jpsiks_Belle () and BaBar jpsiks_BABAR () collaborations, and provide a precise reference value for because of the very small theoretical uncertainty.
Recently, measurements of time-dependent violation of penguin-mediated decays have become interesting because these decay modes proceed via loop diagrams and are, therefore, expected to be sensitive probes of the physics beyond the SM. In these decay modes, searches for new physics effects are carried out by investigating deviations of violating parameters from those determined by processes b2sTheory ().
Among these decays, is one of the most promising modes because of its very small Cabibbo-suppressed tree diagram contribution. Previous Belle measurements of the violating asymmetries have been performed separately in the mass region around the mass ref:belle_phiks_cp () and at higher masses ref:belle_kskk_cp (), while neglecting interference between intermediate states. It is, however, expected that the sensitivity to violating parameters would improve in a measurement using the time-dependent Dalitz plot distribution because of the correct treatment of interferences between various resonant and nonresonant processes.
In the decay chain (4S) , where one of the mesons decays at time to the final state and the other decays at time to a final state that distinguishes between and , the decay rate has a time dependence given by
where is the neutral meson lifetime, is the mass difference between the two neutral mass eigenstates, , is the total amplitude of , and the -flavor charge when the tagged meson is a . The Dalitz plot variables , , and are defined as , and , where , , and are the four-momenta of the , , and , respectively. These variables satisfy by energy-momentum conservation. In the isobar approximation ref:isobar (), the total amplitude for is given by the sum of the decay channels with that final state,
where is a complex coefficient describing the relative magnitude and phase for the -th decay channel, including the weak phase dependence. The Dalitz-dependent amplitudes, , contain only strong dynamics and, thus, . The amplitudes of the contributions considered in the decay are summarized in Table 1. We use the same formalism as references belle_pipiks_dalitz (); babar_kkks_dalitz (). We utilize Flatté ref:Flatte () and relativistic Breit Wigner (RBW) ref:pdg () lineshapes to describe the resonances.
|Resonances||Fixed parameters (MeV)||Resonance shape|
|no fixed parameters|
|no fixed parameters|
|no fixed parameters|
In the Dalitz-dependent amplitudes, (Eq. 2), we choose a convention in which the mixing phase () is absorbed into the decay amplitude, . These complex coefficients, and , can be redefined as
in which case a resonance, , has a direct violating asymmetry given by
where the ’s are restricted by definition to lie between and .
For cases where the contribution is a eigenstate, the mixing-induced violating parameter, , equals the fitted parameter ,
and is related to the mixing-induced violating asymmetry as
where is the eigenvalue of the final state. Note that and are restricted by these definitions to lie in the physical region.
This time-dependent Dalitz measurement of violating parameters in decays is based on a large data sample that contains pairs, collected with the Belle detector at the KEKB asymmetric-energy (3.5 on 8 GeV) collider KEKB () operating at the resonance. The is produced with a Lorentz boost factor of along the -axis, which is antiparallel to the positron beam direction. Since the pairs are produced nearly at rest in the center-of-mass system (cms), is determined from , the distance between the two meson decay vertices along the -direction: , where is the speed of light.
The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside the coil is instrumented to detect mesons and to identify muons (KLM). The detector is described in detail elsewhere Belle (). Two inner detector configurations were used. A 2.0 cm radius beam pipe and a 3-layer silicon vertex detector were used for the first sample of pairs, while a 1.5 cm radius beampipe, a 4-layer silicon detector and a small-cell inner drift chamber were used to record the remaining pairs svd2 ().
We reconstruct candidates from an oppositely-charged kaon pair and a candidate. The charged kaons are selected from the charged tracks having their impact parameters consistent with coming from the interaction point (IP). To suppress background from particle misidentification, charged tracks that are positively identified as pions, protons, or electrons are excluded. The particle species are identified by using particle information (PID) from the CDC, ACC, TOF, and ECL systems. We reconstruct candidates from pairs of oppositely charged tracks having invariant mass within 12 MeV/ of the mass. The direction of the momentum is required to be consistent with the direction of vertex displacement with respect to IP ref:belle_b2sqq_2005 ().
We combine the pair and to form a neutral meson. Signal candidates are identified by two kinematic variables defined in the cms: the beam-energy constrained mass and the energy difference , where is the cms beam energy, and and are the cms three momentum and energy of the reconstructed meson candidate, respectively. We use candidates in a signal region defined as a ellipse around the and mean values: , where is the nominal neutral meson mass ref:pdg (). A larger region in and , GeV/ and GeV GeV, is used to determine the signal and background fractions. The sideband regions used for the continuum background study are defined as GeV/ GeV/ and GeV GeV for the distribution, GeV/ GeV/ and GeV GeV excluding the rectangular region of GeV/ GeV/ and GeV GeV for the Dalitz distribution.
The dominant source of background is continuum and production. To reduce it, we require that , where is the angle between the thrust axis of the candidate and that of the rest of the event. This requirement retains 83% of the signal while 79% of the continuum events are removed. The background is found to be mostly originating from -decays, which peaks in the signal region with an estimated yield of events: charge_conj () and decays. There are also potential backgrounds from and . Backgrounds due to – misidentification are also found. All these peaking background decays are suppressed to a negligible level by applying and (other modes) vetoes on the invariant masses; these vetoes are summarized in Table 2. For backgrounds that arise from misidentified particles, the invariant masses are recalculated by assuming an alternate mass hypothesis for the charged kaon. The remaining contribution is included in the nominal fit as the background component. Yields for signal, continuum and backgrounds as well as PDFs for those are described in more detail later.
|Vetoed mode||Vetoed region|
We identify the flavor of the accompanying meson from inclusive properties of particles that are not associated with the reconstructed candidate. The algorithm for flavor tagging is described in detail elsewhere tagging (). To represent the tagging information, we use two parameters, defined in Eq.( Measurement of violating asymmetries in decays with a time-dependent Dalitz approach) and . The parameter is an event-by-event Monte Carlo (MC) determined flavor-tagging quality factor that ranges from for no flavor discrimination to for unambiguous flavor assignment. It is used only for sorting data into seven intervals. The wrong tag fractions for the seven intervals, , and the difference in between and decays, , are determined from data tagging (). The vertex position for the decay is reconstructed using the charged kaon pair and the transverse components of IP. The vertex position of is obtained using tracks that are not assigned to the candidate and IP.
We find that 1.5 % of the selected events have more than one candidate. In these events, we choose the candidate that is formed from the most kaon-like charged kaon candidate and the candidate closest to the nominal mass.
After all the selections are applied, we obtain 98982 candidates in the - fit region, of which 2333 are in the signal region. We extract the signal yield using a three-dimensional extended unbinned maximum likelihood fit to the distributions of , and the flavor-tag quality () interval, , for the selected events. For the probability density function (PDF) of the signal component, we use a sum of two Gaussians (a single Gaussian) for the () shape. All parameters of the PDFs are free in the fit, except the ratio of the area of the broader Gaussian component to that of the core Gaussian, and the width of the broader Gaussian in . These additional parameters are fixed from the results of a fit to a data control sample. For the continuum background component, the () shape is modeled by a first-order polynomial (an ARGUS argus ()) function, with shape parameters floated in the fit. The background component is parameterized by two-dimensional binned histograms from MC. In the fit, the total signal, continuum and background yields are also free parameters. The fit yields signal events in the signal region. The projections of the , and distributions for the candidate events are shown in Fig. 1. The average signal, continuum and fractions in the signal ellipse are calculated to be 50 %, 49 % and 1 %, respectively. The event-by-event signal probabilities as a function of , and obtained with this fit are used in the unbinned maximum likelihood fit with a time-dependent Dalitz approach that is used to extract violation parameters.
In a Dalitz plot as a function of (, ), signal and continuum events densely populate the kinematic boundaries with low , which correspond to the and resonances. Large variations in a small area of the Dalitz plane make it difficult to use histograms to describe the background. Therefore, we apply the transformation,
where is the Jacobian of this transformation. The parameters and are given by the transformation,
where is invariant mass, and are kinematic limits of , is the helicity angle, defined as the angle between the and the in the rest frame. With this transformation, the Dalitz plot turns into a “square Dalitz plot” with a smooth density variation. Figure 2 (a) and (b) show the Dalitz distributions based on our signal model with the usual Dalitz parameterization, and , the square Dalitz parameterization, and , respectively. As can be seen, the highlighted region where most of the signal and background events are located, is magnified in the square Dalitz parameterization. The square Dalitz plot is described in detail elsewhere belle_pipiks_dalitz (); babar_sdl_pipipi ().
The PDF expected for the signal distribution, , is given by
which accounts for dilution from the incorrect flavor tagging. This function is convolved with the resolution function ref:belle_phiks_cp (); the impact of detector resolution on the Dalitz plot is ignored because the intrinsic widths of the dominant resonances are larger than the mass resolution. We determine the variations of the signal detection efficiency across the Dalitz plane due to detector acceptance, , by using a large MC sample.
The PDF for continuum background is
where , , and are the Dalitz distribution PDF, the Dalitz-plot-dependent flavor asymmetry and the PDF, respectively. The function is modeled as a sum of exponential and prompt components, and is convolved with a double Gaussian that represents the resolution. All parameters of are determined by a fit to the distribution in the sideband region that is defined above. The Dalitz distribution PDF, , is a two-dimensional binned histogram PDF. To determine the PDF, we use the sideband region around the signal region with a less restrictive requirement, , to increase statistics. We have checked that the Dalitz distribution for this sideband region is similar to that for the signal region, using a MC sample. There is a flavor asymmetry due to the jet-like topology of continuum because a high momentum in is accompanied by a high momentum in ; to account for this, we extract the Dalitz plot asymmetry using almost the same region as the region used in extraction: since we find no correlation between and , we enlarge the lower limit of the sideband region in this fit from GeV/ to GeV/ in .
Using high-statistics MC sample, we find no violating asymmetry in the background coming from charmless and charmed decays. Therefore, the PDFs for and backgrounds are given by
respectively. Dalitz distribution PDFs as well as , are modeled with two-dimensional histograms from MC. The PDF for both models, , are described by exponential functions with effective lifetimes while are the resolution functions. The effective lifetimes are obtained from fits to the MC sample.
To account for a small fraction of events with large values not yet described by either signal or background PDFs, an outlier PDF is introduced, , where is a Gaussian and is the two-dimensional binned histogram PDF of the Dalitz plot of data itself.
For the -th event, the following likelihood function is evaluated:
where runs over a total of four components including signal and backgrounds. The probability of each component () is calculated using the result of the -- fit on an event-by-event basis.
As there is only sensitivity to the relative amplitudes and phases between decay modes, we fix and . In addition, and non-resonant contributions are combined and have a single common violating parameters. The combined component is referred to as “others” throughout this paper. The parameters and are fixed to the world average values of 21.5 and 0, respectively. We determine 19 parameters of the Dalitz plot and asymmetries by maximizing the likelihood function , where the product is over all events.
We find four preferred solutions with consistent parameters but significantly different amplitudes for and . The fitted results are summarized in Table 3. These are obtained by performing a large number of fits with random input parameters. For each resonance, , the relative fractions can be calculated as
where the sum of fractions over all decay channels may not be 100% due to interference. Table 4 summarizes the relative fractions for all solutions.
|Parameter||Solution 1||Solution 2||Solution 3||Solution 4|
|0.16||0.10||-0.01 0.11||0.09 0.07|
|-0.02 0.10||-0.04 0.09||0.01 0.10||-0.10 0.09|
|0.07 0.06||0.03 0.08||0.01||-0.02|
|0.12||0.06 0.04||0.10 0.04||0.18 0.03|
|Parameter||Solution 1||Solution 2||Solution 3||Solution 4|
|26.0 7.4||54.0 9.6||26.4 7.8||68.1 12.3|
|14.2 1.2||14.5 1.2||14.2 1.2||14.4 1.2|
|5.10 1.39||5.89 1.86||39.6 2.6||59.0 3.0|
|3.73 0.74||3.71 0.73||3.68 0.73||4.15 0.79|
|138.4 44.8||175.0 52.6||157.4 29.5||48.1 11.7|
|1.65 4.17||21.0 17.3||4.63 6.76||7.87 4.78|
|26.0 12.9||78.0 36.2||38.6 18.1||6.27 3.81|
|215.2 47.5||352.0 66.8||284.5 36.3||207.9 18.4|
By translating the fit results using Eqs. 4 and 5, we determine the time-dependent violating parameters of and decays and other decays with the final state. Table 5 summarizes the violating parameters for all solutions.
|Solution 1||Solution 2||Solution 3||Solution 4|