Measurement of CKM Angle $\phi _{3}$ Using $B^{\pm } \rightarrow D (K_\mathrm{S}^{0}\pi ^{+}\pi ^{-}\pi ^{0})K^{\pm }$ Decays at Belle

Abstract

The CKM angle $\phi _3$ can be determined in a theoretically clean way as it is accessible via the tree-level decays, $B^{\pm }\rightarrow DK^{\pm }$. The current uncertainty on $\phi _3$ is significantly larger than that of the standard model (SM) prediction. A more precise measurement of $\phi _3$ is crucial for testing the SM description of CP violation and probing for new physics effects. The statistical uncertainty can be reduced if information from additional D meson final states is included, which in practice means new three and four-body decay modes. Here, we measure $\phi _3$ with $B^{\pm }\rightarrow D(K_\mathrm{S}^0\pi ^+\pi ^-\pi ^0)K^{\pm }$ decays using a data sample corresponding to an integrated luminosity of 711 fb$^{-1}$ collected with the Belle detector at KEKB asymmetric $e^+e^-$ collider. This four-body D final state has a large branching fraction of 5.2% and the phase space is rich with different resonance substructures. We adopt a model-independent method to estimate $\phi _3$ by studying various regions of the phase space. The D decay strong-phase information is obtained from the quantum-correlated D meson pairs produced at CLEO-c.

P. K. Resmi, J. F. Libby, and K. Trabelsi on behalf of Belle Collaboration.

18.1 Introduction

The current best measurement of the CKM [1, 2] angle $\phi _3$, combining all the results from different experiments, is ($73.5^{+4.2}_{-5.1}$)$^{\circ }$ [3]. This large uncertainty is due to the small branching fractions of the decays sensitive to $\phi _3$. The value of $\phi _3$ estimated indirectly from other parameters of the unitarity triangle is ($65.3^{+1.0}_{-2.5}$)$^{\circ }$ [3]. Any disagreement between these results could imply that there is new physics beyond the standard model (SM). But a comparison would be meaningful only if the associated uncertainties are comparable. Thus, an improved measurement of $\phi _3$ is essential for testing the SM description of $CP$ violation. The color-favored $B^{-} \rightarrow D^{0} K^{-}$ and color-suppressed $B^{-} \rightarrow \overline{D^{0}}K^{-}$ decays, where D indicates a neutral charm meson reconstructed in a final state common to both $D^{0}$ and $\overline{D^{0}}$, provide $CP$-violating observables, that are sensitive to $\phi _3$. Here and elsewhere in this paper, charge conjugation of final states is implied unless explicitly stated otherwise. The Feynman diagrams are shown in Fig. 18.1. These are tree-level decays and hence the theoretical uncertainty is negligible ($\mathcal {O}(10^{-7})$) [4].

If the amplitude for the color-favored decay is $A_\mathrm{fav} = A$, then the color-suppressed one can be written as $A_\mathrm{sup} = Ar_{B}e^{i(\delta _{B} - \phi _{3})}$, where $\delta _{B}$ is the strong-phase difference between the decay processes, and

$$\begin{aligned} r_{B} = \frac{\mid A_\mathrm{{sup}} \mid }{\mid A_\mathrm{{fav}}\mid }. \end{aligned}$$

(18.1)

The statistical uncertainty on $\phi _3$ is proportional to $r_B$. For $B^{+} \rightarrow DK^{+}$ decays, $r_{B} \sim 0.1$, whereas for $B^{+}\rightarrow D\pi ^{+}$, it is 0.005. Though $B^{+} \rightarrow D \pi ^{+}$ decays are not very sensitive to $r_{B}$ and $\phi _{3}$, they serve as excellent control sample modes for signal extraction procedure in $B^{+} \rightarrow DK^{+}$ due to their similar kinematics. This also helps in determining the cross-feed background due to the misidentification of kaons and pions from data.

Fig. 18.1

Color-favored (left) and color-suppressed (right) $B^{-} \rightarrow DK^{-}$ processes

The limitations on the current $\phi _3$ measurements due to statistical precision can be reduced by exploring more and more D final states. Here, we study the four-body self-conjugate state, $D\rightarrow K_\mathrm{S}^0\pi ^+\pi ^-\pi ^0$. This decay mode has a branching fraction of 5.2% [5], which is almost twice that of $D \rightarrow K_\mathrm{S}^0\pi ^+\pi ^-$, the dominant multibody D final state used to determine $\phi _3$ [6, 7]. This decay proceeds via interesting resonance substructures like $K_\mathrm{S}^0\omega$, $K^{*}\rho$, etc., thus facilitating a model-independent extraction of $\phi _3$ by studying the D phase space regions. We present the expected results from $B^+\rightarrow DK^+$ decays by analyzing simulated samples and preliminary results obtained from the $B^{+} \rightarrow D\pi ^{+}$ data sample, i.e., the calibration mode.

18.2 Formalism to Measure $\phi _{3}$ Sensitive Parameters

The methods to determine $\phi _3$ vary according to the D meson final state under consideration. When it is a multibody self-conjugate state, there are two methods: model-dependent and model-independent. In the model-dependent method, the D amplitudes are fitted to a model corresponding to the intermediate resonances. The model assumptions cause large uncertainties that could limit the precision of the $\phi _3$ measurement. The model-independent approach provides measurements of CP violating asymmetries made in independent regions of the D phase space [8, 9]. This binning reduces the statistical precision, but the uncertainty due to model assumptions are no longer present as the average strong-phase measurements are used. This analysis follows the model-independent method.

The D phase space is binned into regions with differing strong phases, which allows $\phi _{3}$ to be determined from a single channel in a model-independent manner. The signal yield for $B^{\pm }\rightarrow DK^{\pm }$ decays in each bin is given as

$$\begin{aligned} \Gamma _{i}^{\pm } \propto K_{i} + r_{B}^{2}\overline{K_{i}} +2\sqrt{K_{i}\overline{K_{i}}}(c_{i}x_{\pm } \mp s_{i}y_{\pm }), \end{aligned}$$

(18.2)

where $x_{\pm } = r_{B}\cos (\delta _{B} \pm \phi _{3})$ and $y_{\pm } = r_{B}\sin (\delta _{B} \pm \phi _{3})$. The $x_{\pm }$ and $y_{\pm }$ parameters, that are sensitive to $\phi _3$, can be obtained when the phase space is divided into three or more bins. Here, $K_{i}$ and $\overline{K_{i}}$ are the fraction of flavor-tagged $D^{0}$ and $\overline{D^{0}}$ events in the $i^\mathrm{th}$ bin, respectively, which can be estimated from $D^{*+} \rightarrow D^0 \pi ^{+}$ decays with good precision due to their large sample size. The parameters $c_{i}$ and $s_{i}$ are the amplitude-weighted average of the cosine and sine of the strong-phase difference between $D^{0}$ and $\overline{D^{0}}$ over the $i^\mathrm{th}$ bin; these parameters need to be determined at a charm factory experiment like CLEO-c or BESIII, where the quantum-entangled $D^{0}\overline{D^{0}}$ pairs are produced via $e^{+}e^{-} \rightarrow \psi (3770) \rightarrow D^{0}\bar{D^{0}}$ [10]. The values of $c_i$ and $s_i$ parameters for $D\rightarrow K_\mathrm{S}^0\pi ^+\pi ^-\pi ^0$ decays as well as the binning scheme to divide the D phase space reported in [11] are used in this analysis.

18.3 Data Samples and Event Selection

The $e^+e^-$ collision data sample at a center-of-mass energy corresponding to the pole of the $\Upsilon (4S)$ resonance collected by the Belle detector [12, 13] is used in this analysis. It corresponds to an integrated luminosity of 711 fb$^{-1}$ and contains 772 $\times$ 10$^6$ $B\overline{B}$ pairs. The Belle detector is located at the interaction point (IP) of KEKB asymmetric $e^+e^-$ collider [14]. A detailed description of the Belle detector is given in [12, 13]. Monte Carlo (MC) samples are used to optimize the selection criteria, determine the efficiencies, and identify various sources of background.

We reconstruct $B^{+}\rightarrow DK^{+}$ and $B^{+} \rightarrow D\pi ^{+}$ decays in which the D decays to the four-body final state of $K_\mathrm{S}^0\pi ^+\pi ^-\pi ^0$. The decays $D^{*+}\rightarrow D\pi ^{+}$ produced via the $e^+e^-\rightarrow c\bar{c}$ continuum process are also selected to measure the $K_i$ and $\overline{K_i}$ parameters. We select the charged particle candidates produced within 0.5 cm and ± 3.0 cm of the IP in perpendicular and parallel directions to the z-axis, respectively, where the z-axis is defined to be opposite to the $e^+$ beam direction. These tracks are then identified as kaons or pions with the help of the particle identification system at Belle [12]. We reconstruct the $K_\mathrm{S}^0$ candidates from two oppositely charged pion tracks. The invariant mass of these pion candidates is required to be within ±3$\sigma$ of the nominal $K_\mathrm{S}^0$ mass [5], where $\sigma$ is the mass resolution. The background due to random combinations of pions is reduced with the help of a neural network [15] based selection with 87% efficiency [16].

We reconstruct $\pi ^0$ candidates from a pair of photons detected in the electromagnetic calorimeter (ECL). The $\pi ^0$ candidates within the diphoton invariant mass range 0.119–0.148 GeV/$c^2$ are retained. The photon energy thresholds are optimized separately for candidates detected in the barrel, forward endcap, and backward endcap regions of the ECL. Furthermore, kinematic constraints are applied to $K_\mathrm{S}^0$, $\pi ^0$, and D invariant masses and decay vertices. This improves the energy and momentum resolution of the B candidates and the invariant masses used to divide the D phase space into bins.

While reconstructing $D^{*+}\rightarrow D\pi ^{+}$ decays, it is required that the accompanying pion has at least one hit in the silicon vertex detector. This pion carries a small fraction of the momentum due to the limited phase space of the decay and hence is known as a slow pion. The D meson momentum in the laboratory frame is chosen to be between 1–4 GeV/c so that it matches to that in $B^+\rightarrow Dh^+ (h= K, \pi )$ sample. The signal candidates are identified by the kinematic variables $M_D$, the invariant mass of D candidate and $\Delta M$, the difference in the invariant masses of $D^*$ and D candidates. We retain events that satisfy the criteria, $1.80< M_D < 1.95$ GeV/$c^2$ and $\Delta M < 0.15$ GeV/$c^2$. A kinematic constraint is applied so that the D and $\pi$ candidates come from the common vertex position. When there are more than one candidate in an event, the one with the smallest $\chi ^2$ value from the $D^*$ vertex fit is retained for further analysis. The overall selection efficiency is 3.7%.

The B meson candidates are reconstructed by combining a D candidate with a charged kaon or pion track. Events with D meson invariant mass in the range 1.835–1.890 GeV/$c^2$ are selected. The kinematic variables energy difference $\Delta E$ and beam constrained mass $M_\mathrm{bc}$ are used to identify the signal candidates. They are defined as $\Delta E~=~ E_{B} - E_\mathrm{beam}$ and $M_\mathrm{bc}~=~ c^{-2}\sqrt{E_\mathrm{beam}^{2}-|\mathbf {\mathbf {p}}_{B}|^2c^{2}}$, where $E_{B}$ and $\mathbf {\mathbf {p}}_{B}$ are the energy and momentum of the B candidate and $E_\mathrm{beam}$ is the beam energy in the center-of-mass frame. The candidates that satisfy the criteria $M_\mathrm{bc} >$ 5.27 GeV/$c^2$ and $-0.13< \Delta E < 0.30$ GeV are selected. In events with more than one candidate, the candidate with the smallest value of $(\frac{M_\mathrm{bc}-M_{B}^{PDG}}{\sigma _{M_\mathrm{bc}}})^{2} + (\frac{M_{D}-M_{D}^{PDG}}{\sigma _{M_{D}}})^{2} + (\frac{M_{\pi ^{0}}-M_{\pi ^{0}}^{PDG}}{\sigma _{M_{\pi ^{0}}}})^{2}$ is retained. Here, the masses $M_i^\mathrm{PDG}$ are those reported by the Particle Data Group in [5] and the resolutions $\sigma _{M_\mathrm{bc}},~\sigma _{M_{D}}$ and $\sigma _{M_{\pi ^{0}}}$ are obtained from MC simulated samples of signal events.

The main source of background is from $e^+e^-\rightarrow q\bar{q},~q = u, d, s, c$ continuum processes, and these are suppressed by exploiting the difference in their event topology to that of $B\overline{B}$ events. The continuum events are jet-like in nature and $B\overline{B}$ events have a spherical topology. These events are separated with the help of a neural-network-based algorithm [15]. We require the neural network output to be greater than −0.6, which reduces the continuum background by 67% at the cost of 5% signal loss. The overall selection efficiency is 4.7 and 5.3% for $B^+\rightarrow DK^+$ and $B^+\rightarrow D\pi ^+$ modes, respectively.

18.4 Determination of $K_i$ and $\overline{K_i}$

The $K_i$ and $\overline{K_i}$ parameters indicate the fraction of $D^0$ and $\overline{D^0}$ events in each D phase space bin. They are measured from the $D^{*+}\rightarrow D\pi ^+$ sample; the charge of the pion determines the flavor of the D meson. The signal yield is obtained from a two-dimensional extended maximum-likelihood fit to $M_D$ and $\Delta M$ distributions independently in each bin. Appropriate probability density functions (PDF) are used to model the distributions. A quadratic correlation between $M_D$ and $\Delta M$ is taken into account for the signal component. The yields along with $K_i$ and $\overline{K_i}$ values are given in Table 18.1.

Table 18.1 $D^0$ and $\overline{D^0}$ yield in each bin of D phase space along with $K_i$ and $\overline{K_i}$ values measured in $D^{*}$ tagged data sample

18.5 Signal Extraction in $B^+\rightarrow Dh^+$ Sample

The signal yield in each D phase space bin is determined from a two-dimensional extended maximum-likelihood fit to $\Delta E$ and neural network output (NB). The latter is transformed as

$$\begin{aligned} NB' = \log \left( \frac{NB - NB_\mathrm{low}}{NB_\mathrm{high} - NB} \right) , \end{aligned}$$

(18.3)

where $NB_\mathrm{low}$ = $-0.6$ and $NB_\mathrm{high}\approx 1.0$ are the minimum and maximum values of NB in the sample, respectively. The three background components are continuum background, combinatorial $B\overline{B}$ background due to final state particles from both the B mesons and cross-feed peaking background due to the misidentification of a kaon as a pion or vice versa.

The sum of a Crystal Ball (CB) [17] function and two Gaussian functions with a common mean is used as the PDF to model the $\Delta E$ signal component in both the B samples. The sum of a Gaussian and an asymmetric Gaussian with different mean values is used to parametrize the PDF that describes the $NB'$ signal component. The continuum background distribution in $\Delta E$ is modeled with a first-order Chebyshev polynomial and that in $NB'$ is described by the sum of two Gaussian PDFs with different mean values. The $\Delta E$ distribution of random $B\overline{B}$ background in $B^{+} \rightarrow D\pi ^{+}$ is described by an exponential function. There is a small peaking structure due to misreconstructed $\pi ^0$ events and this is modeled by a CB function. A first-order Chebyshev polynomial is added to the above two PDFs in the case of $B^{+}\rightarrow DK^{+}$ decays. The $NB'$ distribution for both the samples are modeled by an asymmetric Gaussian function. The cross-feed peaking background in $\Delta E$ is modeled with the sum of three Gaussian functions, whereas the signal PDF itself is used for the $NB'$ distribution. The fit projections in $B^+\rightarrow DK^+$ MC sample are shown in Fig. 18.2. These are signal-enhanced projections with events in the signal region of the other variable, where the signal regions are defined as $|\Delta E| < 0.05$ GeV and $0 {<} NB' {<} 12$.

Fig. 18.2

Signal-enhanced fit projections of $\Delta E$ (left) and $NB'$ (right) for $B^{\pm }\rightarrow DK^{\pm }$ MC sample having equivalent luminosity as that of full data sample collected by Belle. The black points with the error bar are the data and the solid blue curve is the total fit. The dotted red, blue, magenta, and green curves represent the signal, continuum, random $B\overline{B}$ backgrounds, and cross-feed peaking background components, respectively. The pull between the data and the fit are shown for both the projections

The $\phi _3$ sensitive parameters are determined directly from the fit by expressing the signal yield as in (18.2). The $K_i$ and $\overline{K_i}$ values along with the $c_i$ and $s_i$ measurements reported in [11] are used as input parameters. Efficiency corrections are applied and the effect of migration of events between the bins due to finite momentum resolution is also taken into account. The preliminary results obtained from $B^+\rightarrow D\pi ^+$ data sample are summarized in Table 18.2. The dominant source of systematic uncertainty is the size of the signal MC sample used for estimating the efficiency and the extent of migration between the bins. The statistical likelihood contour is given in Fig. 18.3.

Table 18.2 Preliminary results of $x_{\pm }$and $y_{\pm }$ parameters from $B^{\pm }\rightarrow D\pi ^{\pm }$ data sample. The first uncertainty is statistical, second is systematic and the third one is due to the uncertainty on the $c_i$, $s_i$ measurements

Fig. 18.3

One (solid line), two (dashed line), and three (dotted line) standard deviation likelihood contours for the $(x_{\pm },y_{\pm })$ parameters for $B^{\pm }\rightarrow D\pi ^{\pm }$ data sample

18.6 Summary

A precise measurement of the CKM angle $\phi _3$ is essential to establish the SM description of CP violation. Here, we present the feasibility of $D\rightarrow K_\mathrm{S}^0\pi ^+\pi ^-\pi ^0$ final state to do so in $B^+\rightarrow DK^+$ decays. This is the first attempt to analyze this particular decay mode. The signal extraction procedure is established in an MC sample, as well as $B^+\rightarrow D\pi ^+$ data sample, the calibration mode. MC predictions estimate the statistical uncertainty on $x_{\pm },~y_{\pm }$ in $B^+\rightarrow DK^+$ to be 0.08 and 0.17, respectively.

An improved measurement is possible once an amplitude model for $D^{0}\rightarrow K_\mathrm{S}^0\pi ^{+}\pi ^{-}\pi ^{0}$ is available to guide the binning of the phase space such that maximum sensitivity to $\phi _3$ is obtained. Furthermore, a more precise measurement of $c_i,~s_i$ parameters could be performed with a larger sample of $e^{+}e^{-}\rightarrow \psi (3770)$ data that has been collected by BESIII, thus reducing the systematic uncertainty. The Belle II detector is expected to collect about 50 times larger B sample. Thus, the improved binning combined with the larger B sample make $B^{+}~\rightarrow ~D(K_\mathrm{S}^0\pi ^+\pi ^-\pi ^0)K^{+}$ a promising addition to the set of modes to be used to determine $\phi _3$ to a precision of 1–2$^\circ$ [18].