Measurements of CKM angles at Belle

TL;DR

This paper reviews recent measurements of CP violation and hadronic B decays at Belle, focusing on determining the CKM angles φ₁ and φ₂ to test the Standard Model's description of CP violation.

Contribution

It provides the latest experimental results on CKM angles φ₁ and φ₂ from Belle, enhancing understanding of CP violation in B meson decays.

Findings

Precise measurements of CKM angles φ₁ and φ₂.

Confirmation of CP violation effects consistent with the Standard Model.

Improved constraints on the unitarity triangle parameters.

Abstract

In this review recent studies on $CP$ violation and related hadronic $B$ decays by the Belle experiment, in particular measurements of CKM angles $\phi_1$ and $\phi_2$ are reported.

Equations

$$\ln \mathcal{L}=\sum _i\ln {\mathcal{P}}_i^{\mathrm{Belle}}+{\mathcal{P}}_i^{\mathrm{BaBar}}.$$

$${\mathcal{P}}^{\mathrm{ExP}}=\sum _k{f}_k\int \left[P_k(\Delta t^{\prime })R_k^{\mathrm{Exp}}(\Delta t-\Delta t^{\prime })\right]d\Delta t^{\prime },$$

$$P_{\mathrm{sig}}(\Delta t,q)=\frac{1}{4{\tau }_{B^0}}{e}^{\frac{-|\Delta t|}{{\tau }_{B^0}}}\left[1+q\left(S\sin (\Delta m\Delta t)-A\cos (\Delta m\Delta t)\right)\right],$$

$$-{\eta }_{f}S=+0.66\pm 0.10(stat.)\pm 0.06(syst.),A=-0.02\pm 0.07(stat.)\pm 0.03(syst.).$$

$$P_i(\Delta t,{\varphi }_1)$$

$$\sin (2{\varphi }_1)$$

$$\cos (2{\varphi }_1)$$

$${\varphi }_1$$

$$\mathcal{B}(B^0\to \psi (2S){\pi }^0)=[1.17\pm 0.17(stat.)\pm 0.08(syst.)]\times 10^{-5}.$$

$$\mathcal{B}(B^0\to {\rho }^{+}{\rho }^{-})=[28.3\pm 1.5(stat.)\pm 1.5(syst.)]\times 10^{-6},$$

$$f_L=0.988\pm 0.012(stat.)\pm 0.023(syst.),$$

$$S=-0.13\pm 0.15(stat.)\pm 0.05(syst.),A=0.00\pm 0.10(stat.)\pm 0.06(syst.).$$

Full text

1 Introduction

In the standard model (SM) of electroweak interaction, charge-parity $(CP)$ violation arises from an irreducible complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix [1]. The Belle and BaBar experiments have established $CP$ violating effects in the $B$ meson system. Both experiments use their measurements of the mixing-induced $CP$ violation in $b\to c\bar{c}s$ transitions  to precisely determine the parameter $\sin(2\phi_{1})$, where $\phi_{1}$ is defined as $\arg[-V_{cd}V^{*}_{cb}/V_{td}V^{*}_{tb}]$, with $V_{ij}$ is the CKM matrix element of quarks $i,~{}j$. In this proceeding an overview of recent measurements of the CKM angles $\phi_{1}$ and $\phi_{2}$ \big{(}\arg[-V_{td}V^{*}_{tb}/V_{ud}V^{*}_{ub}]\big{)} is presented. Unless stated otherwise, all measurements presented here are based on Belle’s final dataset of $772\times 10^{6}$ $B\bar{B}$ pairs.

2 First observation of $CP$ violation in $\bar{B}^{0}\to D_{CP}^{(*)}h^{0}$ decays with Belle + BaBar data

The decay $\bar{B}^{0}\to D^{(*)}h^{0}$, where $h^{0}$ is a light, unflavored neutral meson ($h^{0}\in{\pi^{0},~{}\eta,~{}\omega}$), is dominated by a $b\to c\bar{u}d$ color-suppressed tree diagram in the SM. The final state $D^{(*)}h^{0}$ is a $CP$ eigenstate if the neutral $D$ meson decays to a $CP$ eigenstate as well ($i.e.$, $D^{0}_{CP}\to K_{S}^{0}\pi^{0}$, $D^{0}_{CP}\to K_{S}^{0}\omega$ ($CP=-1$) or $D^{0}_{CP}\to K^{+}K^{-}$ ($CP=+1$) and $D^{*}_{CP}\to D^{0}_{CP}\pi^{0}$). Therefore, a time-dependent $CP$ asymmetry measurement is applicable in the same way as used in the $b\to c\bar{c}s$ decays, but with a small correction from the $b\to u\bar{c}d$ process. This $b\to u\bar{c}d$ amplitude is suppressed by $V_{ub}V^{*}_{cd}/V_{cb}V^{*}_{ud}\approx 0.02$ relative to the leading amplitude. Neglecting the suppressed amplitude, the time evolution of $\bar{B}^{0}\to D_{CP}^{(*)}h^{0}$ decays is governed by $\phi_{1}$[2].

Due to the limited available statistics, previous measurements performed separately by the BaBar and Belle collaborations were not able to establish $CP$ violation in these or related decays [3]. This motivated a joint analysis using the combined full dataset of Belle and BaBar experiments [4]. Using a fit to the beam-energy constrained mass $M_{\rm bc}=\sqrt{E^{2}_{\rm beam}-p^{2}_{B}}$, where $E_{\rm beam}$ is the beam energy and $p_{B}$ is the reconstructed $B$ meson momentum in the center-of-mass system, we extract $508\pm 31$ signal events in the BaBar data of 431 million $B\bar{B}$ events and $757\pm 44$ signal events in the Belle data of 772 million $B\bar{B}$ events, as shown in Fig. 1. The dominant source of background originates from $e^{+}e^{-}\to p\bar{q}~{}(q\in{u,~{}d,~{}s,~{}c})$ continuum events. To suppress this background, we use a multivariate analyzer based on a neural network. The neural network uses the so-called event shape variables to discriminate continuum events, which tend to be jetlike, from spherical $B\bar{B}$ events.

The time-dependent $CP$ violation measurement is performed using established Belle and BaBar techniques for the vertex reconstruction, the flavor-tagging, and the modeling of $\Delta t$ resolution effects, where $\Delta t$ is the proper time interval between the decays of the two $B$ mesons produced in an $\Upsilon(4S)$ decay. Combined analysis is performed by maximizing a joint log-likelihood function

[TABLE]

The experiment-dependent probality density function (PDF) $\mathcal{P}^{\rm Exp}$ is defined as

[TABLE]

where the index $k$ represents the signal and background PDF components. The symbol $P_{k}$ denotes the PDF describing the proper time interval of the particular physical process and $R_{k}^{\rm Exp}$ refers to the corresponding resolution function. The fractions $f_{k}$ are evaluated on an event-by-event basis as a function of $M_{\rm bc}$. While the background model is determined from the $M_{\rm bc}$ sideband and hence is experiment-dependent, the signal model is expressed as

[TABLE]

where the $B^{0}$ meson lifetime is represented by $\tau_{B^{0}}$, $B^{0}-\bar{B}^{0}$ mixing frequency by $\Delta m$ and $q$ is event- and experiment-dependent tagging quality parameter. In the SM, the coefficients, $S=-\eta_{f}\sin(2\phi_{1})$ and $A=0$, where $\eta_{f}$ is the $CP$ eigenvalue of the final state. $S$ and $A$ quantify mixing-induced and direct $CP$ violation, respectively. The combined fit gives

[TABLE]

These results correspond to the first observation of $CP$ violation in $\bar{B}^{0}\to D_{CP}^{(*)}h^{0}$ decays with a significance of 5.4 standard deviations and are in agreement with the value of $\phi_{1}$ measured from $b\to c\bar{c}s$ transitions.

3 Measurement of $\phi_{1}$ in $B^{0}\to\bar{D}^{(*)0}h^{0}$ with time-dependent binned Dalitz plot analysis

In this analysis, we present a model-independent measurement of the angle $\phi_{1}$ in $b\to c\bar{u}d$ transitions governing $B^{0}\to\bar{D}^{(*)0}h^{0}$ decays, with subsequent decay $\bar{D}^{0}\to K_{S}^{0}\pi^{+}\pi^{-}$ is not a $CP$ eigenstate [5]. From a fit to $M_{\rm bc}$ and $\Delta E=E_{B}-E_{\rm beam}$, where $E_{B}$ is the reconstructed $B$ mesons energy in the center-of-mass system, we extract total $962\pm 41$ signal events, of which $464\pm 26$ events are from $B^{0}\to\bar{D}^{0}\pi^{0}$ mode (Fig. 2), with a signal fraction $(72.1\pm 4.1)\%$ and $182\pm 18$ events from $B^{0}\to\bar{D}^{0}\omega$ with a fraction of $(58.4\pm 5.7)\%$. The signal fraction of other decay modes ranges between 44% and 70%.

Our measurement of $\phi_{1}$ is based on the binned Dalitz distribution approach. This idea was proposed in Ref. [6] to measure the angle $\phi_{3}$. Events are divided into 16 bins on the Dalitz plot plane and the number of events in bin $i~{}(i=-8,...,-1,+1,...,+8)$ is modeled as

[TABLE]

where $h$ is the normalization constant, $\xi_{h^{0}}$ is the $CP$ eigenvalue of $h^{0}$ meson, $L$ is the relative angular momentum in the $D^{(*)0}h^{0}$ system, $K_{i}$ is the integrated squared amplitude, and $S_{i}$ and $C_{i}$ represent the weighted averages of the sine and cosine of the phase difference between $\bar{D}^{0}$ and $D^{0}$ decay amplitudes over the $i$th Dalitz plot bin. The parameters $K_{i}$ can be measured with a set of flavor-tagged neutral $D$ mesons such as $D^{*+}\to D^{0}\pi^{+}$ or $B^{+}\to\bar{D}^{0}\pi^{+}$ decays, by measuring signal yield in each Dalitz plot bin. The measurement of the phase parameters $S_{i}$ and $C_{i}$ is more complicated and can be done with coherent decays of $D^{0}\bar{D}^{0}$ pairs [7]. We obtain

[TABLE]

The value $\sin(2\phi_{1})=0.691\pm 0.017$ measured in $b\to c\bar{c}s$ transitions determines the absolute value of $\cos(2\phi_{1})$, leading two possible solutions in the $0^{\circ}\leq\phi_{1}\leq 180^{\circ}$ range. Our measurement is inconsistent with the negative solution, corresponding to the value $\phi_{1}=68.1^{\circ}$ at the level of 5.1 standard deviations, but in agreement with the positive solution, corresponding to the value $\phi_{1}=21.9^{\circ}$ at 1.3 standard deviations.

4 First observation of the decay $B^{0}\to\psi(2S)\pi^{0}$

Although decays mediated via $b\to c\bar{c}s$ transitions allow us to access the $\phi_{1}$ at first order (tree), its value is prone to distortion from suppressed higher-order loop-induced (penguin) amplitudes containing different weak phases. The related $b\to c\bar{c}d$ induced decays can be used to quantify the shift in $\phi_{1}$ caused by these loop contributions and may provide useful information about the penguin pollution [8]. Since the dominant $b\to c\bar{c}d$ tree amplitude is also suppressed, $B^{0}\to J/\psi\pi^{0}$ is the only mode measured so far, providing $\sin(2\phi_{1})=0.65\pm 0.21~{}{(\rm stat.)}\pm 0.05~{}{(\rm syst.)}$ by Belle [9], which is consistent with $\sin(2\phi_{1})$ from $b\to c\bar{c}s$. The possible next mode, $B^{0}\to\psi(2S)\pi^{0}$ was not observed previously.

The decay mode $B^{0}\to\psi(2S)\pi^{0}$ is reconstructed with $\psi(2S)\to\ell^{+}\ell^{-}~{}(\ell=e,~{}\mu)$ or $\psi(2S)\to J/\psi(\to\ell^{+}\ell^{-})\pi^{+}\pi^{-}$ [10]. The major background contribution originates from $b\to c\bar{c}q$ decays other than the signal. The background arises from $e^{+}e^{-}\to q\bar{q}~{}(q=u,~{}d,~{}s,~{}c)$ continuum events is not so problematic and is suppressed by applying a loose requirement on the ratio of second- to zeroth-order Fox-Wolfram moments. The signal is extracted from a fit to $M^{\prime}_{\rm bc}$ and $\Delta E$, where $M^{\prime}_{\rm bc}$ is the modified beam-constrained mass to take into account the worse energy resolution of $\pi^{0}$ than rest of the particles. The fit gives $85\pm 12$ signal events with a significance of 7.2 standard deviations. The branching fraction is measured to be

[TABLE]

This measurement constitutes the first observation of this decay and it will contribute to the future time-dependent $CP$ asymmetry measurement of the $b\to c\bar{c}d$ process.

5 Study of $B^{0}\to\rho^{+}\rho^{-}$ decays

In order to access $\phi_{2}$, charmless decay modes that are mediated via $b\to u\bar{u}d$ transitions are necessary. Examples are the decays $B\to\pi\pi,~{}\rho\pi,~{}\rho\rho$. At tree level, one expects $A=0$ and $S=\sin(2\phi_{2})$. Possible penguin contributions can give rise of direct $CP$ violation, $A\neq 0$ and also pollute the measurement of $\phi_{2}$, $S=\sqrt{1-A^{2}}\sin(2\phi_{2}^{\rm eff})$, where the observed $\phi_{2}^{\rm eff}=\phi_{2}-\Delta\phi_{2}$ is shifted by $\Delta\phi_{2}$ due to different weak and strong phases from additional non-leading contributions. This inconvience can be overcome by estimating $\Delta\phi_{2}$ using either an isospin analysis [11] or $SU(3)$ flavor symmetry [12]. In this analysis, we present a measurement of the branching fraction and the longitudinal polarization fraction of $B^{0}\to\rho^{+}\rho^{-}$ decays, as well as the time-dependent $CP$ violating parameters [13].

In addition to combinatorial background, the presence of multiple background components with the same four-pion final state as $B^{0}\to\rho^{+}\rho^{-}$ make this decay quite difficult to isolate and interferences between the various four-pion modes need to be considered. A multi-dimensional maximum likelihood fit is performed. The fit uses the variables $\Delta E$, $M_{\rm bc}$, the masses and helicity angles (angle between one of the daughter of $\rho^{\pm}$ meson and the $B$ flight direction in the corresponding rest frame of the $\rho^{\pm}$.) of the two reconstructed $\rho^{\pm}$ mesons to separate longitudinally polarized states from transversely polarized states, a fisher discriminant to separate the jet-like continuum events from the spherical $B\bar{B}$ decays and the $\Delta t$ distribution for the two flavors of $B_{\rm tag}$. We obtain the branching fraction

[TABLE]

the fraction of longitudinal polarization

[TABLE]

and the $CP$ violating parameters

[TABLE]

These results together with the other Belle measurements [14] are used to perform an isospin analysis to constrain the CKM angle $\phi_{2}$ and obtain two solutions with $\phi_{2}=(93.7\pm 10.6)^{\circ}$ being most compatible with other SM based fits to the data. The size of the penguin pollution is consistent with zero: $\Delta\phi_{2}=(0.0\pm 9.6)^{\circ}$. Figure 3 shows the $\phi_{2}$ can from the isospin analysis.

6 Summary

The first observation of $CP$ violation in $\bar{B}^{0}\to D_{CP}^{(*)}h^{0}$ decays from a combined analysis of Belle and BaBar dataset is presented. The result is consistent with the value of $\sin(2\phi_{1})$ measured in the $b\to c\bar{c}s$ process. Using a similar process, $B^{0}\to\bar{D}^{(*)0}h^{0}$ with $\bar{D}^{0}\to K_{S}^{0}\pi^{+}\pi^{-}$, a model-independent time-dependent Dalitz plot analysis is performed and excludes the second $\phi_{1}$ solution by 5.1 standard deviations. Observation of $B^{0}\to\psi(2S)\pi^{0}$ is presented, which will contribute to the $\phi_{1}$ measurement in future. And finally $\phi_{2}$ measurement from $B\to\rho\rho$ decays is presented.

Acknowledgements

The author thanks the workshop organizers for hosting a fruitful and stimulating workshop and providing excellent hospitality. This research is supported by the U.S. Department of Energy.