Measurement of Decay Amplitudes of B -->(c cbar) Kstar with an Angular Analysis, for (c cbar)=J/psi, psi(2S) and chi_c1

TL;DR

This study measures decay amplitudes of B mesons into charmonium states and K* mesons using angular analysis, revealing polarization differences and no CP violation in the processes.

Contribution

First three-dimensional measurement of B decays to (2S) K* and 1 K* states, updating previous results for J/ K* with a large data sample.

Findings

Longitudinal polarization is larger for 1 than for J/ or (2S).

No evidence of direct CP violation observed.

Measurements improve understanding of decay dynamics.

Abstract

We perform the first three-dimensional measurement of the amplitudes of $B\to \psi(2S) K^*$ and $B\to \chi_{c1} K^*$ decays and update our previous measurement for $B\to J/\psi K^*$. We use a data sample collected with the BaBar detector at the PEP2 storage ring, corresponding to 232 million $B\bar B$ pairs. The longitudinal polarization of decays involving a $J^{PC}=1^{++}$ $\chi_{c1}$ meson is found to be larger than that with a $1^{--}$ $J/\psi$ or $\psi(2S)$ meson. No direct {\it CP}-violating charge asymmetry is observed.

Figures

Tables

Table 1. Table 1: Amplitude coefficients 𝒜 k subscript 𝒜 𝑘 {\cal A}_{k} and angular functions f k ​ ( 𝝎 ) subscript 𝑓 𝑘 𝝎 f_{k}(\boldsymbol{\omega}) that contribute to the differential decay rate. An overall normalization factor 9 / 32 ​ π 9 32 𝜋 9/32\pi (for ψ 𝜓 \psi ) and 9 / 64 ​ π 9 64 𝜋 9/64\pi (for χ c ​ 1 subscript 𝜒 𝑐 1 \chi_{c1} ) has been omitted. In the case of a B ¯ ¯ 𝐵 \kern 1.79993pt\overline{\kern-1.79993ptB}{} decay, the ℑ ⁡ m 𝑚 \Im m terms change sign.

$i$	${𝒜}_k$	$f_k\left(𝝎\right)$ for $\psi$ Aubert:2004cp ; Aubert:2001pe	$f_k\left(𝝎\right)$ for ${\chi }_{c1}$ Stephane
1	${\left|A_0\right|}^2$	$2{\cos }^2{\theta }_{K^{\ast }}\left[1-{\sin }^2{\theta }_{\mathrm{tr}}{\cos }^2{\varphi }_{\mathrm{tr}}\right]$	$2{\cos }^2{\theta }_{K^{\ast }}\left[1+{\sin }^2{\theta }_{\mathrm{tr}}{\cos }^2{\varphi }_{\mathrm{tr}}\right]$
2	${\left|A_{\parallel }\right|}^2$	${\sin }^2{\theta }_{K^{\ast }}\left[1-{\sin }^2{\theta }_{\mathrm{tr}}{\sin }^2{\varphi }_{\mathrm{tr}}\right]$	${\sin }^2{\theta }_{K^{\ast }}\left[1+{\sin }^2{\theta }_{\mathrm{tr}}{\sin }^2{\varphi }_{\mathrm{tr}}\right]$
3	${\left|A_{⟂}\right|}^2$	${\sin }^2{\theta }_{K^{\ast }}{\sin }^2{\theta }_{\mathrm{tr}}$	${\sin }^2{\theta }_{K^{\ast }}\left[2{\cos }^2{\theta }_{\mathrm{tr}}+{\sin }^2{\theta }_{\mathrm{tr}}\right]$
4	$\Im m\left(A_{\parallel }^{\ast }{A}_{⟂}\right)$	${\sin }^2{\theta }_{K^{\ast }}\sin 2{\theta }_{\mathrm{tr}}\sin {\varphi }_{\mathrm{tr}}$	$-{\sin }^2{\theta }_{K^{\ast }}\sin 2{\theta }_{\mathrm{tr}}\sin {\varphi }_{\mathrm{tr}}$
5	$\Re e\left(A_{\parallel }{A}_0^{\ast }\right)$	$-\frac{1}{\sqrt{2}}\sin 2{\theta }_{K^{\ast }}{\sin }^2{\theta }_{\mathrm{tr}}\sin 2{\varphi }_{\mathrm{tr}}$	$\frac{1}{\sqrt{2}}\sin 2{\theta }_{K^{\ast }}{\sin }^2{\theta }_{\mathrm{tr}}\sin 2{\varphi }_{\mathrm{tr}}$
6	$\Im m\left(A_{⟂}{A}_0^{\ast }\right)$	$\frac{1}{\sqrt{2}}\sin 2{\theta }_{K^{\ast }}\sin 2{\theta }_{\mathrm{tr}}\cos {\varphi }_{\mathrm{tr}}$	$-\frac{1}{\sqrt{2}}\sin 2{\theta }_{K^{\ast }}\sin 2{\theta }_{\mathrm{tr}}\cos {\varphi }_{\mathrm{tr}}$

Table 2. Table 2: Summary of the measured amplitudes. For decays to χ c ​ 1 subscript 𝜒 𝑐 1 \chi_{c1} , as A ⟂ subscript 𝐴 perpendicular-to A_{\perp} is compatible with zero, its phase is not defined.

Channel	${\left|A_0\right|}^2$	${\left|A_{\parallel }\right|}^2$	${\left|A_{⟂}\right|}^2$	${\delta }_{\parallel }$	${\delta }_{⟂}$
$J/\psi K^{\ast }$	$0.556\pm 0.009\pm 0.010$	$0.211\pm 0.010\pm 0.006$	$0.233\pm 0.010\pm 0.005$	$-2.93\pm 0.08\pm 0.04$	$2.91\pm 0.05\pm 0.03$
$\psi \left(2S\right)K^{\ast }$	$0.48\pm 0.05\pm 0.02$	$0.22\pm 0.06\pm 0.02$	$0.30\pm 0.06\pm 0.02$	$-2.8\pm 0.4\pm 0.1$	$2.8\pm 0.3\pm 0.1$
${\chi }_{c1}{K}^{\ast }$	$0.77\pm 0.07\pm 0.04$	$0.20\pm 0.07\pm 0.04$	$0.03\pm 0.04\pm 0.02$	$0.0\pm 0.3\pm 0.1$	–

Table 3. Table 3: Difference between the interference terms measured in B 𝐵 B and B ¯ ¯ 𝐵 \kern 1.79993pt\overline{\kern-1.79993ptB}{} decays to J / ψ 𝐽 𝜓 {J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu} .

	$\delta {𝒜}_4$	$\delta {𝒜}_6$
$\left(K^{+}{\pi }^{-}\right)$	$0.002\pm 0.025\pm 0.005$	$-0.011\pm 0.043\pm 0.016$
$\left(K^{+}{\pi }^0\right)$	$-0.017\pm 0.047\pm 0.023$	$-0.051\pm 0.098\pm 0.064$
$\left(K_S^0{\pi }^{+}\right)$	$-0.008\pm 0.049\pm 0.011$	$0.075\pm 0.089\pm 0.009$

Equations

$$g(\mathbf{\omega };\mathbf{A})\equiv \frac{1}{\Gamma }\frac{{\mathrm{d}}^3\Gamma }{\mathrm{d}\cos {\theta }_{K^{\ast }}\mathrm{d}\cos {\theta }_{\mathrm{tr}}\mathrm{d}{\varphi }_{\mathrm{tr}}}=\sum _{k=1}^6{\mathcal{A}}_k{f}_k(\mathbf{\omega }),$$

$$g_{\mathrm{obs}}(\mathbf{\omega };\mathbf{A})=g(\mathbf{\omega };\mathbf{A})\frac{\epsilon (\mathbf{\omega })}{⟨\epsilon ⟩(\mathbf{A})},$$

$$⟨\epsilon ⟩(\mathbf{A})\equiv \int g(\mathbf{\omega };\mathbf{A})\epsilon (\mathbf{\omega })\mathrm{d}\mathbf{\omega }$$

$$g_{\mathrm{obs}}^b(\mathbf{\omega };\mathbf{A})$$

$${\epsilon }^b(\mathbf{\omega })$$

$${\Phi }_k^b$$

$$L^b(\mathbf{A})=\sum _{i=1}^{N_S}\ln \left(g({\mathbf{\omega }}_i;\mathbf{A})\right)-N_S\ln \left(\sum _k{\mathcal{A}}_k(\mathbf{A}){\Phi }_k^b\right),$$

$$L^{\prime b}(\mathbf{A})\equiv \sum _{i=1}^{n_B+N_S}L({\mathbf{\omega }}_i;\mathbf{A})-\frac{{\tilde{n}}_B}{N_B}\sum _{j=1}^{N_B}L({\mathbf{\omega }}_j;\mathbf{A}),$$

Full text

††preprint: BABAR-PUB-07/009††preprint: SLAC-PUB-12430

In the context of measuring the parameters of the Unitarity Triangle of the CKM matrix, $B^{0}$ decays to charmonium-containing final states (${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, $\psi{(2S)}$, $\chi_{c1}$)$K^{*}$, defined collectively here as $B^{0}\rightarrow(c\bar{c})K^{*}$, are of interest for the precise measurement of $\sin 2\beta$, where $\beta\equiv\arg[-V_{cd}V_{cb}^{*}/V_{td}V_{tb}^{*}]$, in a similar way as for $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}$. Furthermore, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*}$ channel allows the measurement of $\cos 2\beta$ Aubert:2004cp .

For the modes considered in this paper, the final state consists of two spin-1 mesons, leading to three possible values of the total angular momentum with different $C\!P$  eigenvalues ($L=1$ is odd, while $L=0,2$ are even). The different contributions must be taken into account in the measurement of $\sin 2\beta$. The amplitude for longitudinal polarization of the two spin-1 mesons is $A_{0}$. There are two amplitudes for polarizations of the mesons transverse to the decay axis, here expressed in the transversity basis Dunietz:1990cj : $A_{\parallel}$ for parallel polarization and $A_{\perp}$ for their perpendicular polarization. Only the relative amplitudes are measured, so that $|A_{0}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2}=1$. Previous measurements by the CLEO Jessop:1997jk , CDF Affolder:2000ec , BABAR Aubert:2004cp and Belle Itoh:2005ks collaborations for the $B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*}$ channels are all compatible with each other, and with a $C\!P$-odd intensity fraction $|A_{\perp}|^{2}$ close to 0.2.

Factorization predicts that the phases of the transversity decay amplitudes are the same. BABAR has observed Aubert:2004cp ; Aubert:2001pe a significant departure from this prediction.

Precise measurements of the branching fractions of $B\rightarrow(c\bar{c})K^{*}$ decays are now available Aubert:2004rz to test the theoretical description of the non-factorizable contributions Chen:2005ht , but polarization measurements are also needed. In particular, measurements for $\psi{(2S)}$ and $\chi_{c1}$, compared to that of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, would discriminate the mass dependence from the quantum number dependence. CLEO has measured the longitudinal polarization of $B$ $\rightarrow$ $\psi{(2S)}$ $K^{*}$ decays to be $|A_{0}|^{2}=0.45\pm 0.11\pm 0.04$ Richichi:2000ca . Belle has studied $B$ $\rightarrow$ $\chi_{c1}$ $K^{*}$ decays and obtained $|A_{0}|^{2}=0.87\pm 0.09\pm 0.07$ Soni:2005fw .

$B\rightarrow(c\bar{c})K^{(*)}$ decays provide a clean environment for the measurement of the CKM angle $\beta$ because one tree amplitude dominates the decay. Very small direct $C\!P$-violating charge asymmetries are expected in these decays, and no such signal has been found Aubert:2004rz . While more than one amplitude with different strong and weak phases are needed to create a charge asymmetry in a simple branching fraction measurement, London et al. have suggested London:2000zi that an angular analysis of vector-vector decays can detect charge asymmetries even in the case of vanishing strong phase difference. Belle has looked for, and not found, such a signal Itoh:2005ks .

In this paper we present the amplitude measurement of charged and neutral $B\rightarrow(c\overline{c})K^{*}$ using a selection similar to that of Ref. Aubert:2004rz , and a fitting method similar to that of Ref. Aubert:2004cp . We use the notation $\psi$ for the $1^{--}$ states ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$. $\psi$ ($\chi_{c1}$) candidates are reconstructed in their decays to $\ell^{+}\ell^{-}$ (${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$), where $\ell$ represents an electron or a muon. Decays to the flavor eigenstates $K^{*0}\rightarrow K^{\pm}\pi^{\mp}$, $K^{*\pm}\rightarrow K^{0}_{\scriptscriptstyle S}\pi^{\pm}$ and $K^{*\pm}\rightarrow K^{\pm}\pi^{0}$ are used. The relative strong phases are known to have a two-fold ambiguity when measured in an angular analysis alone. In contrast to earlier publications Jessop:1997jk ; Affolder:2000ec ; Aubert:2001pe we use here the set of phases predicted in Ref. Suzuki:2001za , with arguments based on the conservation of the $s$-quark helicity in the decay of the $b$ quark. We have confirmed experimentally this prediction through the study of the variation with $K\pi$ invariant mass of the phase difference between the $K^{*}(892)$ amplitude and a non-resonant $K\pi$ $S$-wave amplitude Aubert:2004cp .

The data were collected with the BABAR detector at the PEP-II asymmetric $e^{+}e^{-}$ storage ring, and correspond to an integrated luminosity of about 209 $\mbox{\,fb}^{-1}$ at the center-of-mass energy near the $\mathchar 28935\relax{(4S)}$ mass. The BABAR detector is described in detail elsewhere detector . Charged-particle tracking is provided by a five-layer silicon vertex tracker (SVT) and a 40-layer drift chamber (DCH). For charged-particle identification (PID), ionization energy loss in the DCH and SVT, and Cherenkov radiation detected in a ring-imaging device (DIRC) are used. Photons are identified by the electromagnetic calorimeter (EMC), which comprises 6580 thallium-doped CsI crystals. These systems are mounted inside a 1.5-T solenoidal superconducting magnet. Muons are identified in the instrumented flux return (IFR), composed of resistive plate chambers and layers of iron that return the magnetic flux of the solenoid. We use the GEANT4 geant software to simulate interactions of particles traversing the detector, taking into account the varying accelerator and detector conditions.

${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\rightarrow e^{+}e^{-}$ ($\mu^{+}\mu^{-}$) candidates must have a mass between $2.95-3.14$ ($3.06-3.14$) ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$. $\psi{(2S)}$ candidates are required to have invariant masses $3.44<m_{e^{+}e^{-}}<3.74$ ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ or $3.64<m_{\mu^{+}\mu^{-}}<3.74$ ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$. Electron candidates are combined with photon candidates in order to recover some of the energy lost through Bremsstrahlung. ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates and $\gamma$ candidates with an energy larger than $150\mathrm{\,Me\kern-1.00006ptV}$, are combined to form $\chi_{c1}$ candidates, which must satisfy $350<m_{\ell^{+}\ell^{-}\gamma}-m_{\ell^{+}\ell^{-}}<450$ ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$. $\pi^{0}\rightarrow\gamma\gamma$ candidates must satisfy $113<m_{\gamma\gamma}<153$ ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$. The energy of each photon has to be greater than $50\mathrm{\,Me\kern-1.00006ptV}$. $K^{0}_{\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ candidates are required to satisfy $489<m_{\pi^{+}\pi^{-}}<507$ ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$. In addition, the $K^{0}_{\scriptscriptstyle S}$ flight distance from the $\psi$ vertex must be larger than three times its uncertainty. $K^{*0}$ and $K^{*+}$ candidates are required to satisfy $796<m_{K\pi}<996$ ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$ and $792<m_{K\pi}<992$ ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$, respectively. In addition, due to the presence of a large background of low-energy non-genuine $\pi^{0}$’s, the cosine of the angle $\theta_{K^{*}}$ between the $K$ momentum and the $B$ momentum in the $K^{*}$ rest frame has to be less than 0.8 for $K^{*}\rightarrow K^{\pm}\pi^{0}$. In events where two $B$’s reconstruct to modes with the same $c\bar{c}$ and $K$ candidate, one with a $\pi^{\pm}$ and the other with a $\pi^{0}$, the $B$ candidate with a $\pi^{0}$ is discarded due to the high background induced by fake $\pi^{0}$’s.

$B$ candidates, reconstructed by combining $c\bar{c}$ and $K^{*}$ candidates, are characterized by two kinematic variables: the difference between the reconstructed energy of the $B$ candidate and the beam energy in the center-of-mass frame \mbox{\Delta E}=E_{B}^{*}-\sqrt{s}/2, and the beam-energy substituted mass \mbox{m_{\rm ES}}\equiv\sqrt{(s/2+{\bf p}_{0}\cdot{\bf p}_{B})^{2}/E_{0}^{2}-{\bf p}_{B}^{2}}, where subscript [math] and $B$ correspond to $\mathchar 28935\relax(4S)$ and the $B$ candidate in the laboratory frame. For a correctly reconstructed $B$ meson, $\Delta E$ is expected to peak near zero and $m_{\rm ES}$ near the $B$-meson mass $5.279{\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$. The analysis is performed in a region of the $m_{\rm ES}$ vs $\Delta E$ plane defined by 5.2<\mbox{m_{\rm ES}}<5.3 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ and -120<\mbox{\Delta E}<120 $\mathrm{\,Me\kern-1.00006ptV}$. The signal region is defined as \mbox{m_{\rm ES}}>5.27 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ and |\mbox{\Delta E}| smaller than 40 (30) $\mathrm{\,Me\kern-1.00006ptV}$ for channels with (without) a $\pi^{0}$. For events that have multiple candidates, the candidate having the smallest |\mbox{\Delta E}| is chosen. $m_{\rm ES}$ distributions are available in Ref. Aubert:2006mg .

The $B$ decay amplitudes are measured from the differential decay distribution, expressed in the transversity basis Aubert:2004cp ; Aubert:2001pe , Fig. 1, with conventions detailed in Ref. Stephane .

$\theta_{K^{*}}$ is the helicity angle of the $K^{*}$ decay. It is defined in the rest frame of the $K^{*}$ meson, and is the angle between the kaon and the opposite direction of the $B$ meson in this frame. $\theta_{\rm tr}$ and $\phi_{\rm tr}$ are defined in the $\psi$ ($\chi_{c1}$) rest frame and are the polar and azimutal angle of the positive lepton (${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ daughter of $\chi_{c1}$) , with respect the axis defined by:

• •

$\boldsymbol{x}_{\rm tr}$: opposite direction of the $B$ meson;

• •

$\boldsymbol{y}_{\rm tr}$: perpendicular to $\boldsymbol{x}_{\rm tr}$, in the $(\boldsymbol{x}_{\rm tr},\boldsymbol{p}_{K^{*}})$ plane, with a direction such that $\boldsymbol{p}_{K^{*}}\cdot\boldsymbol{y}_{\rm tr}>0$;

• •

$\boldsymbol{z}_{\rm tr}$: to complete the frame, ie: $\boldsymbol{z}_{\rm tr}=\boldsymbol{x}_{\rm tr}\times\boldsymbol{y}_{\rm tr}$.

In terms of the transversity angular variables $\boldsymbol{\omega}\equiv(\cos{\theta_{K^{*}}},\cos{\theta_{\rm tr}},\phi_{\rm tr})$, the time-integrated differential decay rate for the decay of the $B$ meson is

[TABLE]

where the amplitude coefficients ${\cal A}_{i}$ and the angular functions $f_{k}(\boldsymbol{\omega})$, $k=1\cdots 6$ are listed in Table 1. The $\psi$ decays to two spin-1/2 particles, while the $\chi_{c1}$ decays to two vector particles. The angular dependencies are therefore different Stephane .

The symbol $\boldsymbol{A}\equiv(A_{0},A_{\parallel},A_{\perp})$ denotes the transversity amplitudes for the decay of the $B$ meson, and $\boldsymbol{\overline{A}}$ for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ meson decay. In the absence of direct $C\!P$ violation, we can choose a phase convention in which these amplitudes are related by $\overline{A}_{0}=+A_{0}$, $\overline{A}_{\parallel}=+A_{\parallel}$, $\overline{A}_{\perp}=-A_{\perp}$, so that $A_{\perp}$ is $C\!P$-odd and $A_{0}$ and ${A}_{\parallel}$ are $C\!P$-even. The phases $\delta_{j}$ of the amplitudes, where $j=0,\parallel,\perp$, are defined by $A_{j}=|A_{j}|e^{i\delta_{j}}$. Phases are defined relative to $\delta_{0}=0$.

We perform an unbinned likelihood fit of the three-dimensional angle probability density function (PDF). The acceptance of the detector and the efficiency of the event reconstruction may vary as a function of the transversity angles, in particular as the angle $\theta_{K^{*}}$ is strongly correlated with the momentum of the final kaon and pion. We use the acceptance correction method developped in Ref. Aubert:2004cp . The PDF of the observed events, $g_{\rm obs}$, is :

[TABLE]

where

$\varepsilon(\boldsymbol{\omega})$ is the angle-dependent acceptance and

[TABLE]

is the average acceptance. We take into account the presence of cross-feed from channels with the same $c\bar{c}$ candidate and a different $K^{*}$ candidate that has (due to isospin symmetry) the same $\boldsymbol{A}$ dependence as the signal. The observed PDF for channel ${b}$ $({b}=K^{\pm}\pi^{\mp},K^{0}_{\scriptscriptstyle S}\pi^{\pm},K^{\pm}\pi^{0})$ is then

[TABLE]

where $\varepsilon^{{b}}(\boldsymbol{\omega})$ is the efficiency, defined as the ratio between the reconstructed and generated yield for the process ($B\rightarrow(c\bar{c})K^{*}$, $K^{*}\rightarrow b$), and we do not distinguish between correctly reconstructed signal and cross-feed in the numerator

[TABLE]

$\varepsilon^{{a}\rightarrow{b}}(\boldsymbol{\omega})$ is the probability for an event generated in channel ${a}$ and with angle $\boldsymbol{\omega}$ to be detected as an event in channel ${b}$. $F_{a},{a}=K^{0}_{\scriptscriptstyle S}\pi^{0},K^{\pm}\pi^{\mp},K^{\pm}\pi^{0},K^{0}_{\scriptscriptstyle S}\pi^{\pm}$ denotes the fraction of each channel in the total branching fraction $B\rightarrow c\overline{c}K^{*}$, $\sum_{a}F_{a}=1$. The $\Phi^{{b}}_{k}$ are the $f_{k}(\boldsymbol{\omega})$ moments of the total efficiency $\varepsilon^{{b}}$, including cross-feed :

[TABLE]

Under the approximations of neglecting the angular resolution for signal and cross-feed events, and the possible mis-measurement of the $B$ flavor such as in events where both daughters in $K^{*0}\rightarrow K^{\pm}\pi^{\mp}$ are mis-identified ($K$-$\pi$ swap), the PDF $g_{\rm obs}$ can be expressed as in Eq. (2), and only the coefficients $\Phi_{K}^{b}$ are needed. The biases induced by these approximations have been estimated with Monte Carlo (MC) based studies and found to be negligible.

The coefficients $\Phi^{{b}}_{k}$ are computed with exclusive signal MC samples obtained using a full simulation of the experiment geant ; Evtgen . PID efficiencies measured with data control samples are used to adjust the MC simulation to the observed performance of the detector. Separate coefficients are used for different charges of the final state mesons, in particular to take into account the charge dependence of the interaction of charged kaons with matter, and a possible charge asymmetry of the detector. Writing the expression for the log-likelihood $L^{{b}}(\boldsymbol{A})$ for the PDF $g_{\rm obs}^{{b}}(\boldsymbol{\omega}_{i};\boldsymbol{A})$ for a pure signal sample of $N_{S}$ events, the relevant contribution is

[TABLE]

since the remaining term $\sum_{i=1}^{N_{S}}\ln\left(\varepsilon^{{b}}(\boldsymbol{\omega}_{i})\right)$ does not depend on the amplitudes.

We use a background correction method Aubert:2004cp in which background events from a pure background sample of $N_{B}$ events are added with a negative weight to the log-likelihood that is maximized

[TABLE]

where $L(\boldsymbol{\omega};\boldsymbol{A})=\ln(g_{\rm obs}^{{b}}(\boldsymbol{\omega};\boldsymbol{A}))$. The fit is performed within the $m_{\rm ES}$ signal region. Background events used here for subtraction are from generic ($B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$, $q\overline{q}$) MC samples. $\tilde{n}_{B}$ is an estimate of the unknown number $n_{B}$ of background events that are present in the signal region in the data sample.

As $L^{\prime{b}}$ is not a log-likelihood, the uncertainties yielded by the minimization program Minuit minuit are biased estimates of the actual uncertainties. An unbiased estimation of the uncertainties is described and validated in Appendix A of Ref. Aubert:2004cp . With this pseudo-log-likelihood technique, we avoid parametrizing the acceptance as well as the background angular distributions.

The measurement is affected by several systematic uncertainties. The branching fractions used in the cross-feed part of the acceptance cross section are varied by $\pm 1\sigma$, and the largest variation is retained. The uncertainty induced by the finite size of the MC sample used to compute the coefficients $\Phi_{k}^{b}$ is estimated by the statistical uncertainty of the angular fit on that MC sample Aubert:2001pe . The uncertainty due to our limited understanding of the PID efficiency is estimated by using two different methods to correct for the MC-vs-data differences. The background uncertainty is obtained by comparing MC and data shapes of the $m_{\rm ES}$ distributions for the combinatorial component and by using the corresponding branching errors for the peaking component. The uncertainty due to the presence of a $K\pi$ $S$ wave under the $K^{*}(892)$ peak is estimated by a fit including it. The differential decay rate is described by Eqs. ($6$-$9$) of Ref. Aubert:2004cp .

The results are summarized in Table 2.

The values of $|A_{0}|^{2}$, $|A_{\parallel}|^{2}$, $|A_{\perp}|^{2}$ are negatively correlated due to the constraint $|A_{0}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2}=1$. In particular, $|A_{\parallel}|^{2}$, which would be the least precisely measured parameter in separate one-dimensional fits, is strongly anti-correlated with $|A_{0}|^{2}$, which would be the best measured. The one-dimensional (1D) distributions, acceptance-corrected with an 1D Ansatz and background-subtracted, are overlaid with the fit results and shown on Figure 2. In contrast with the dedicated method used in the fit, for the plots, we simply computed the 1D efficiency maps from the distributions of the accepted events divided by the 1D PDF. As in lower statistics studies, the $\cos\theta_{K^{*}}$ forward backward asymmetry due to the interference with the S wave is clearly visible.

Our measurement of the amplitudes of $B$ decays to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ are compatible with, and of better precision than, previous measurements. A comparison of neutral and charged $B$ decays (not shown) yields results consistent with isospin symmetry. The strong phase difference $\delta_{\parallel}-\delta_{\perp}$ is obtained from a fit in which the phase origin is $\delta_{\perp}\equiv 0$. We confirm our previous observation that the strong phase differences are significantly different from zero, in contrast with what is predicted by factorization. For $B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*}$, it amounts to $\delta_{\parallel}-\delta_{\perp}=0.45\pm 0.05\pm 0.02$. The presence of direct $C\!P$-violating triple-products in the amplitude would produce a $B$ to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ difference in the interference terms ${\cal A}_{4}$ and ${\cal A}_{6}$: $\delta{\cal A}_{4}$ and $\delta{\cal A}_{6}$. Our results (see Table 3), with improved precision relative to Ref. belletp , are consistent with no $C\!P$ violation.

In summary, we have performed the first three-dimensional analysis of the decays to $\psi{(2S)}$ and $\chi_{c1}$. The longitudinal polarization of the decay to $\psi{(2S)}$ is lower than that to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, while the $C\!P$-odd intensity fraction is higher (by 1.4 and 1.0 standard deviations, respectively). This is compatible with the prediction of models of meson decays in the framework of factorization. The longitudinal polarization of the decay to $\chi_{c1}$ is found to be larger than that to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, in contrast with the predictions of Ref. Chen:2005ht , which include non-factorizable contributions. The $C\!P$-odd intensity fraction of this decay is compatible with zero. The parallel and longitudinal amplitudes for $\chi_{c1}$ seem to be aligned ($|\delta_{\|}-\delta_{0}|\sim 0$) while for $\psi$ they are anti-aligned ($|\delta_{\|}-\delta_{0}|\sim\pi$).

We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the US Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the Commissariat à l’Energie Atomique and Institut National de Physique Nucléaire et de Physique des Particules (France), the Bundesministerium für Bildung und Forschung and Deutsche Forschungsgemeinschaft (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), the Foundation for Fundamental Research on Matter (The Netherlands), the Research Council of Norway, the Ministry of Science and Technology of the Russian Federation, Ministerio de Educación y Ciencia (Spain), and the Science and Technology Facilities Council (United Kingdom). Individuals have received support from the Marie-Curie IEF program (European Union) and the A. P. Sloan Foundation.