You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We present measurements of the time-dependent -violation parameters  and  in $\B^0\ra {\ensuremath{\eta^{\prime}}\xspace}\Kz$ decays. The data sample corresponds to 384 million  pairs produced by  annihilation at the . The results are ${\mbox{$S$}}= {\ensuremath{0.58\pm 0.10\pm 0.03}}$, and ${\mbox{$C$}}= {\ensuremath{-0.16\pm 0.07\pm 0.03}}$. We observe mixing-induced violation with a significance of 5.5 standard deviations in this $b\ra s$ penguin dominated mode.'
title: ' **Observation of  violation in   Decays** '
---

-PUB-[06]{}/[063]{}\
SLAC-PUB-[12127]{}\

authors\_sep2006.tex

Measurements of time-dependent   asymmetries in $B^0$ meson decays through Cabibbo-Kobayashi-Maskawa (CKM) favored $b \rightarrow c \bar{c} s$ amplitudes [@babar] have provided crucial tests of the mechanism of  violation in the Standard Model (SM) [@SM]. Decays of $B^0$ mesons to charmless hadronic final states such as ${\ensuremath{\eta^{\prime}}\xspace}K^0$ proceed mostly via a single loop (penguin) amplitude. In the SM the penguin amplitude has approximately the same weak phase as the $b \to c \bar{c} s$ transition, but it is sensitive to the possible presence of new heavy particles in the loop [@Penguin]. The measurement of   asymmetries in  thus provides an important test for such effects.

Within the SM, CKM-suppressed amplitudes and multiple particles in the loop introduce additional weak phases whose contribution may not be negligible [@Gross; @Gronau; @BN; @london]. The time-dependent -violation parameter $S$ (defined in Eq. \[eq:FCPdef\] below) measured in the decay  is compared with the value of  from measurements of time-dependent  violation in $B$ decays to states containing charmonium and a neutral kaon. The deviation $\deltaS=S - \stwob$ has been estimated in several theoretical approaches: QCD factorization (QCDF) [@BN; @BN2], QCDF with modeled rescattering [@Cheng], Soft Collinear Effective Theory [@Zupan], and SU(3) symmetry [@Gross; @Gronau; @Jonat]. These models estimate $|\deltaS|$ to be of the order $0.01$, and with uncertainties give bounds $|\deltaS | \lesssim 0.05$.

The time-dependent  asymmetry in the decay  has been measured previously by the  [@Previous] and Belle [@BELLE] Collaborations. In this Letter we update our previous measurements using an integrated luminosity of 349 fb$^{-1}$, corresponding to $384 \pm 4$ million pairs, recorded at the $\Upsilon (4S)$ resonance (center-of-mass energy $\sqrt{s}=10.58\ {\ensuremath{\mathrm{\,Ge\kern -0.1em V}}\xspace}$). Belle has since updated their results [@NewBelle]. Our data were collected with the   detector [@BABARNIM] at the PEP-II asymmetric-energy  collider. In addition to the  decays used previously, we now also include the decay .

Charged particles from  interactions are detected, and their momenta measured, by a combination of five layers of double-sided silicon microstrip detectors and a 40-layer drift chamber, both operating in the 1.5 T magnetic field of a superconducting solenoid. Photons and electrons are identified with a CsI(Tl) electromagnetic calorimeter (EMC). Charged particle identification is provided by the average energy loss in the tracking devices and by an internally reflecting ring imaging Cherenkov detector covering the central region. The instrumented flux return (IFR) of the magnet allows the identification of muons and  mesons.

We reconstruct a decaying into the  eigenstate ${\mbox{$\eta^{\prime} \KS$}}$ or ${\mbox{$\eta^{\prime} \KL$}}$ ($B_{\CP}$). From the remaining particles in the event we also reconstruct the decay vertex of the other $B$ meson ($B_{\rm tag}$) and identify its flavor. The difference $\deltat \equiv {\mbox{$t_{\CP}$}}- {\ensuremath{t_{\rm tag}}}$ of the proper decay times ${\mbox{$t_{\CP}$}}$ and ${\ensuremath{t_{\rm tag}}}$ of the  and tag $B$ mesons, respectively, is obtained from the measured distance between the $B_{\CP}$ and $B_{\rm
tag}$ decay vertices and from the boost ($\beta \gamma =0.56$) of the system. The  distribution is given by: $$\begin{aligned}
  F({\deltat}) &=&
        \frac{e^{-\left|\deltat\right|/\tau}}{4\tau} [1 \mp\Delta w \pm
                                                   \label{eq:FCPdef}\\
   &&\hspace{-1em}(1-2w)\left(-\eta S\sin(\deltamd\deltat) -
C\cos(\deltamd\deltat)\right)]\,\nonumber\end{aligned}$$ where $\eta$ is the  eigenvalue of the final state ($-1$ for , $+1$ for ). The upper (lower) sign denotes a decay accompanied by a () tag, $\tau$ is the mean $\Bz$ lifetime, $\deltamd$ is the mixing frequency, and the mistag parameters $w$ and $\Delta w$ are the average and difference, respectively, of the probabilities that a true $\Bz$ is incorrectly tagged as a $\Bzb$ or vice versa. The tagging algorithm has six mutually exclusive tagging categories and a measured analyzing power of $( 30.4\pm 0.3)\%$ [@s2b]. A non-zero value of the parameter $C$ would indicate direct  violation.

We establish the event selection criteria with the aid of a detailed Monte Carlo (MC) simulation of the  production and decay sequences, and of the detector response [@geant]. These criteria are designed to retain signal events with high efficiency while removing most of the background.

The -daughter candidates are reconstructed through their decays $\piz\ra\gaga$, $\eta\ra\gaga$ ([$\eta_{\gaga}$]{}), $\eta\ra\pip\pim\piz$ ([$\eta_{3\pi}$]{}), ${\ensuremath{\eta^{\prime}}\xspace}\ra{\ensuremath{\eta_{\gaga}}}\pip\pim$ ([${\ensuremath{\eta^{\prime}}\xspace}_{\eta(\gamma\gamma)\pi\pi}$]{}), ${\ensuremath{\eta^{\prime}}\xspace}\ra{\ensuremath{\eta_{3\pi}}}\pip\pim$ ([${\ensuremath{\eta^{\prime}}\xspace}_{\eta(3\pi)\pi\pi}$]{}), ${\ensuremath{\eta^{\prime}}\xspace}\ra{\ensuremath{\rho^0}}\gamma$ ([${\ensuremath{\eta^{\prime}}\xspace}_{\rho\gamma}$]{}), where ${\ensuremath{\rho^0}}\ra\pip\pim$, $\KS\ra\pip\pim$ ($K^0_{\pi^+\pi^-}$) or $\piz\piz$ ($K^0_{\pi^0\pi^0}$). Only the [${\ensuremath{\eta^{\prime}}\xspace}_{\eta(\gamma\gamma)\pi\pi}$]{} mode is used for the  sample. The requirements on the invariant masses of these particle combinations are the same as in our previous analysis [@Previous]. The list of all decay modes used in the current analysis can be seen in Table \[tab:Results\]. Signal  candidates are reconstructed from clusters of energy deposited in the EMC or from hits in the IFR not associated with any charged track in the event [@Stefan]. From the cluster centroid and the decay vertex we determine the direction (but not the magnitude) of the $\KL$ momentum ${{\bf p}}_{\KL}$.

For  decays we reconstruct the -meson candidate by combining the four-momenta of the  and [$\eta^{\prime}$]{} with a vertex constraint. We also constrain the $\eta$, [$\eta^{\prime}$]{}, and  masses to world-average values [@PDG2006]. From the kinematics of  decays we determine the energy-substituted mass $\mes \equiv \sqrt{({\mbox{${1\over2}$}}s +
{{\bf p}}_0\cdot{{\bf p}}_B)^2/E_0^2 - {{\bf p}}_B^2}$ and the energy difference ${\ensuremath{\Delta E}}\equiv E_B^*-{\mbox{${1\over2}$}}\sqrt{s}$, where $(E_0,{{\bf p}}_0)$ and $(E_B,{{\bf p}}_B)$ are the laboratory four-momenta of the  and the $B$ candidate, respectively, and the asterisk denotes the  rest frame. The resolution is $3\ {\ensuremath{\mathrm{\,Me\kern -0.1em V}}\xspace}$ in  and $20-50\ {\ensuremath{\mathrm{\,Me\kern -0.1em V}}\xspace}$ in [$\Delta E$]{}, depending on the decay mode.

For  candidates we obtain [$\Delta E$]{} and ${{\bf p}}_{\KL}$ from a fit with the $B^0$ and  masses constrained to world-average values [@PDG2006]. To make a match with the measured  direction we construct the missing momentum ${{\bf p}}_{\rm
miss}$ from ${{\bf p}}_0$ and all charged tracks and neutral clusters other than the  candidate. We then project ${{\bf p}}_{\rm miss}$ onto ${{\bf p}}_{\KL}$, and require the component perpendicular to the beam line, $p_{{\rm miss}\perp}^{\rm proj}$, to satisfy $p_{{\rm miss}\perp}^{\rm proj}-p_{\KL\perp} > -0.5\ {\ensuremath{\mathrm{\,Ge\kern -0.1em V}}\xspace}$. This value was chosen to minimize the yield uncertainty in the presence of background.

For   we require $5.25<\mes<5.29\ {\ensuremath{\mathrm{\,Ge\kern -0.1em V}}\xspace}$ and $|{\ensuremath{\Delta E}}|<0.2$ [$\mathrm{\,Ge\kern -0.1em V}$]{}, for  we require $-0.01<{\ensuremath{\Delta E}}<0.04$ [$\mathrm{\,Ge\kern -0.1em V}$]{}, and for all decays $|{\deltat}|<20$ ${\rm ps}$, and, for the error on , ${\ensuremath{\sigma_{\deltat}}}<2.5$ ${\rm ps}$.

Background events arise primarily from random combinations of particles in continuum $\epem\ra\qqbar$ events ($q=u,d,s,c$). We reduce these with requirements on the angle [$\theta_{\rm T}$]{} between the thrust axis of the $B$ candidate in the  frame and that of the rest of the charged tracks and neutral calorimeter clusters in the event. In the fit we discriminate further against  background with a Fisher discriminant  that combines several variables that characterize the production dynamics and energy flow in the event [@AngMom]. For the [${\ensuremath{\eta^{\prime}}\xspace}_{\rho\gamma}$]{} decays we require $|\cos\theta^{\rho}_{\rm dec}|< 0.9$ to reduce the combinatorial background. Here $\theta_{\rm dec}^\rho$ is the angle between the momenta of the [$\rho^0$]{}daughter  and of the [$\eta^{\prime}$]{}, measured in the [$\rho^0$]{} rest frame.

For  candidates we require that the cosine of the polar angle of the total missing momentum in the laboratory system be less than 0.95, to reject very forward  jets. The purity of the  candidates reconstructed in the EMC is further improved by a requirement on the output of a neural network (NN) that takes cluster-shape variables as inputs. The NN was trained on MC signal events and data events in the region . We check the performance of the NN on data with  candidates in the larger $\Bz \to J/\psi K^0_L$ data sample.

The average number of candidates found per selected event is between 1.08 and 1.32, depending on the final state. In the case of events with multiple candidates we choose the candidate with the smallest value of a $\chi^2$ constructed from the deviations from expected values of one or more of the daughter resonance masses, or with the best decay vertex probability for the $B$, depending on the decay channel. Furthermore, in the sample, if several $B$ candidates have the same vertex probability, we choose the candidate with the  information taken from, in order, EMC and IFR, EMC only, or IFR only. From the simulation we find that this algorithm selects the correct-combination candidate in about two thirds of the events containing multiple candidates.

We obtain the common -violation parameters and signal yields for each channel from a maximum likelihood fit with the input observables [$\Delta E$]{}, , , and . The selected sample sizes are given in the first column of Table \[tab:Results\]. We estimate from the simulation a contribution to the input sample of less than 1.1 % of background from other charmless $B$ decay modes. These events have final states different from the signal, but similar kinematics, and exhibit broad peaks in the signal regions of some observables. We find that the  background component is needed only for the channels with [${\ensuremath{\eta^{\prime}}\xspace}_{\rho\gamma}$]{}. We account for these with a separate component in the probability density function (PDF). For each component $j$ (signal, combinatorial background, or  background) and tagging category $c$, we define a total probability density function for event $i$ as: $${\cal P}_{j,c}^i \equiv {\cal P}_j ( \mes^i ) \cdot {\cal
P}_j ( {\ensuremath{\Delta E}}^i ) \cdot { \cal P}_j( {\mbox{${\cal F}$}}^i ) \cdot 
{ \cal  P}_j (\deltat^i, \sigma_{\deltat}^i;c)\,,$$ except for  for which ${\cal P}_j ( \mes^i )$ is omitted. The factored form of the PDF is a good approximation since linear correlations are small.

We write the extended likelihood function for all events of the decay mode $d$ as $${{\cal L}_d} =  \prod_c  \exp{(-n_c)} \prod_i^{N_c} \left[ \sum_{j}n_j
f_{j,c}  {\cal P}_{j,c}^i  \right]\,,$$ where $n_j$ is the yield of events of component $j$, $f_{j,c}$ is the fraction of events of component $j$ for each category $c$, $n_c =  n_{sig}f_{sig,c}+n_{q\bar{q}}f_{q\bar{q},c}+n_{B\bar{B}}f_{B\bar{B},c}$ is the number of events found by the fitter for category $c$, and $N_c$ is the number of events of category $c$ in the sample. When combining decay modes we form the grand likelihood ${\cal L}=\prod{\cal L}_d$. We fix both $f_{{\rm sig},c}$ and $f_{B\bar{B},c}$ to $f_{{\ensuremath{B_{\rm flav}}},c}$, the values measured with the large sample of fully reconstructed $B^0$ decays into flavor eigenstates ([$B_{\rm flav}$]{} sample) [@Stefan].

The PDF ${ \cal P}_{\rm sig} ({\deltat},\, {\ensuremath{\sigma_{\deltat}}}; c)$, for each category $c$, is the convolution of $F({\deltat};\, c)$ (Eq. \[eq:FCPdef\]) with the signal resolution function (sum of three Gaussians) determined from the [$B_{\rm flav}$]{} sample. The other PDF forms are: the sum of two Gaussians for ${\cal P}_{\rm
sig}(\mes)$ and ${\cal P}_{\rm sig}({\ensuremath{\Delta E}})$; the sum of three Gaussians for ${\cal P}_{\qqbar}({\deltat}; c)$ and ${\cal P}_{\BB}({\deltat}; c)$; an asymmetric Gaussian with different widths below and above the peak for ${\cal P}_j({\mbox{${\cal F}$}})$ (a small “tail" Gaussian is added for ${\cal P}_{\qqbar}({\mbox{${\cal F}$}})$); a linear dependence for ${\cal P}_{\qqbar}({\ensuremath{\Delta E}})$ and a fourth-order polynomial for ${\cal P}_{\BB}({\ensuremath{\Delta E}})$; for ${\cal
P}_{\qqbar}(\mes)$ and ${\cal
P}_{\BB}(\mes)$ the function $x\sqrt{1-x^2}\exp{\left[-\xi(1-x^2)\right]}$, with $x\equiv2\mes/\sqrt{s}$ and $\xi$ a free parameter [@Argus] and the same function plus a Gaussian, respectively.

For the signal and  background components we determine the PDF parameters from simulation. We study large control samples of $B$ decays to charm final states of similar topology to verify the simulated resolutions in [$\Delta E$]{} and , adjusting the PDFs to account for any differences found. The  background parameters are free to vary in the final fit. Thus, for the six channels listed in Table \[tab:Results\], we perform a single fit with 93 free parameters: $S$, $C$, signal yields (6), ${\ensuremath{\eta^{\prime}}\xspace}_{\rho\gamma} K^0$  background yields (2), continuum background yields (6) and fractions (30), background , , [$\Delta E$]{},  PDF parameters (47). The parameters $\tau$ and $\deltamd$ are fixed to world-average values [@PDG2006].

We test and calibrate the fitting procedure by applying it to ensembles of simulated experiments with  events drawn from the PDF into which we have embedded the expected number of signal and  background events randomly extracted from the fully simulated MC samples. We find negligible bias for $C$. For $S$ we find and apply multiplicative correction factors for bias from dilution due to cross-feed from  background to signal events equal to 1.03 in the final states ${\ensuremath{{\ensuremath{\eta^{\prime}}\xspace}_{\rho\gamma} K^0}}_{\pi^+\pi^-}$, [${\ensuremath{\eta^{\prime}}\xspace}_{\eta(\gamma\gamma)\pi\pi} \KL $]{}, and ${\ensuremath{{\ensuremath{\eta^{\prime}}\xspace}_{\rho\gamma} K^0}}_{\pi^{0}\pi^{0}}$.

  Mode                                                        \# events   Signal yield                    $ S$                                       $C$
  ---------------------------------------------------------- ----------- -------------- ----------------------------------------- -----------------------------------------
  ${\ensuremath{\eta^{\prime}}\xspace}_1 K^0_{\pi^+\pi^-}$       664       $224\pm16$    ${\ensuremath{\phantom{-}}}0.61\pm0.23$               $-0.26\pm0.14$
  ${\ensuremath{\eta^{\prime}}\xspace}_2 K^0_{\pi^+\pi^-}$      11943      $566\pm30$    ${\ensuremath{\phantom{-}}}0.56\pm0.14$               $-0.24\pm0.10$
  ${\ensuremath{\eta^{\prime}}\xspace}_3 K^0_{\pi^+\pi^-}$       177        $73\pm9$     ${\ensuremath{\phantom{-}}}0.89\pm0.35$   ${\ensuremath{\phantom{-}}}0.14\pm0.25$
  ${\ensuremath{\eta^{\prime}}\xspace}_1 K^0_{\pi^0\pi^0}$       490        $52\pm9$     ${\ensuremath{\phantom{-}}}0.84\pm0.42$               $-0.26\pm0.36$
  ${\ensuremath{\eta^{\prime}}\xspace}_2 K^0_{\pi^0\pi^0}$      13915      $133\pm24$    ${\ensuremath{\phantom{-}}}0.56\pm0.41$   ${\ensuremath{\phantom{-}}}0.15\pm0.27$
  ${\mbox{$\eta^{\prime} \KS$}}$                                                         ${\ensuremath{\phantom{-}}}0.62\pm0.11$               $-0.18\pm0.07$
  ${\ensuremath{\eta^{\prime}}\xspace}_1\KL$                    4199       $204\pm24$    ${\ensuremath{\phantom{-}}}0.32\pm0.28$   ${\ensuremath{\phantom{-}}}0.08\pm0.23$
  ${\ensuremath{{\ensuremath{\eta^{\prime}}\xspace}K^0}}$                                ${\ensuremath{\phantom{-}}}0.58\pm0.10$               $-0.16\pm0.07$

  : Results of the fits. Subscripts for [$\eta^{\prime}$]{} decay modes denote [${\ensuremath{\eta^{\prime}}\xspace}_{\eta(\gamma\gamma)\pi\pi}$]{} (1), [${\ensuremath{\eta^{\prime}}\xspace}_{\rho\gamma}$]{} (2), and [${\ensuremath{\eta^{\prime}}\xspace}_{\eta(3\pi)\pi\pi}$]{} (3). []{data-label="tab:Results"}

![\[fig:projMbDE\] Distributions projected (see text) onto (a)  and (b) [$\Delta E$]{} for candidates, and (c) [$\Delta E$]{} for  candidates. The solid lines shows the full fit result and the dashed lines show the background contributions.](projMbDE.eps "fig:"){width="50.00000%"}\

Results from the fit for the signal yields and the parameters $S$ and $C$ are presented in Table \[tab:Results\]. In Fig. \[fig:projMbDE\] we show the projections onto  and [$\Delta E$]{} for a subset of the data for which the ratio between the likelihood of signal events and the sum of likelihoods of signal and background events (computed without the variable plotted) exceeds a mode-dependent threshold that optimizes the sensitivity. In Fig. \[fig:DeltaTProj\] we give the $\Delta t$ and asymmetry projections of the events selected as for Fig. \[fig:projMbDE\]. We measure a correlation of $3.2\%$ between $S$ and $C$ in the fit.

We perform several crosschecks of our analysis technique including time-dependent fits for $B^+$ decays to the charged final states ${\ensuremath{{\ensuremath{\eta^{\prime}}\xspace}_{\eta(\gamma\gamma)\pi\pi} K^{+} }}$, ${\ensuremath{{\ensuremath{\eta^{\prime}}\xspace}_{\rho\gamma} K^+}}$, and ${\ensuremath{{\ensuremath{\eta^{\prime}}\xspace}_{\eta(3\pi)\pi\pi} K^{+}}}$; fits removing one fit variable at a time; fits without  PDFs; fits with multiple  components; fits allowing for non-zero information in events; fits with $C
= 0$. In all cases, we find results consistent with expectation. The value $S=0.62\pm0.11$ for  differs from our previous measurement $S=0.30\pm0.14$ [@Previous] due to the improved event reconstruction (with a contribution of +0.08) and selection (+0.12) and to the additional data collected (+0.12). With a model of the data sample changes introduced by our revised event reconstruction and new data, we find that our current result has a statistical probability of 35% (50%) for an assumed true value of $S$ of 0.61 (0.70).

![Projections (see text) onto $\Delta t$ for (a)  and (c)  of the data (points with error bars for  tags ($N_{\Bz}$) in red empty rectangles and  tags ($N_{\Bzb}$) in in blue solid circles), fit function (red dashed and blue solid lines for  and  tagged events, respectively), and background function (black shaded regions). We show the raw asymmetry, $(N_{\Bz}-N_{\Bzb})/(N_{\Bz}+N_{\Bzb})$, for (b)  and (d) ; the lines represent the fit functions.[]{data-label="fig:DeltaTProj"}](projDeltaTKs.eps "fig:"){width="23.00000%"} ![Projections (see text) onto $\Delta t$ for (a)  and (c)  of the data (points with error bars for  tags ($N_{\Bz}$) in red empty rectangles and  tags ($N_{\Bzb}$) in in blue solid circles), fit function (red dashed and blue solid lines for  and  tagged events, respectively), and background function (black shaded regions). We show the raw asymmetry, $(N_{\Bz}-N_{\Bzb})/(N_{\Bz}+N_{\Bzb})$, for (b)  and (d) ; the lines represent the fit functions.[]{data-label="fig:DeltaTProj"}](projDeltaTKl.eps "fig:"){width="23.00000%"}

We have studied the systematic uncertainties arising from several sources (in decreasing order of magnitude): variation of the signal PDF shape parameters within their errors, modeling of the signal distribution, use of  signal parameters from the [$B_{\rm flav}$]{} sample, interference between the CKM-suppressed $\bar{b}\to\bar{u} c\bar{d}$ amplitude and the favored $b\to c\bar{u}d$ amplitude for some tag-side $B$ decays [@dcsd],  background, SVT alignment, and position and size of the beam spot. The [$B_{\rm flav}$]{} sample is used to determine the errors associated with the signal  resolutions, tagging efficiencies, and mistag rates. We take the uncertainties in $\tau_B$ and  from published measurements [@PDG2006]. Summing all systematic errors in quadrature, we obtain 0.03 for $S$ and 0.03 for $C$.

In conclusion, we have used a sample containing $1252\pm50$ flavor-tagged   events to measure the time-dependent violation parameters, ${\mbox{$S$}}= {\ensuremath{0.58\pm 0.10\pm 0.03}}$ and ${\mbox{$C$}}= {\ensuremath{-0.16\pm 0.07\pm 0.03}}$. This sample is 2.1 times as large as that of our previous measurement [@Previous]. Our result for  is consistent with the world average of  measured in $\Bz\ra
J/\psi\KS$ [@PDG2006]. We observe mixing-induced  violation in $B^0$ decays to   with a significance (systematic uncertainties included) of 5.5 standard deviations. Our result for direct-violation is 2.1 standard deviations from zero.

acknow\_PRL.tex

[99]{}

 Collaboration, B. Aubert [[*et al.*]{}]{}, [[Phys. Rev. Lett.]{} [**89**]{}]{}, 201802 (2002); Belle Collaboration, K. Abe [[*et al.*]{}]{}, [[Phys. Rev.]{} D [**66**]{}]{}, 071102(R) (2002).

N. Cabibbo, [[Phys. Rev. Lett.]{} [**10**]{}]{}, 531 (1963); M. Kobayashi and T. Maskawa, [[Prog. Theor. Phys. [**49**]{}]{}]{}, 652 (1973).

Y. Grossman and M. P. Worah, [[Phys. Lett.]{} B [**395**]{}]{}, 241 (1997); D. Atwood and A. Soni, [[Phys. Lett.]{} B [**405**]{}]{}, 150 (1997); M. Ciuchini [[*et al.*]{}]{}, [[Phys. Rev. Lett.]{} [**79**]{}]{}, 978 (1997).

Y. Grossman [[*et al.*]{}]{}, [[Phys. Rev.]{} D [**68**]{}]{}, 015004 (2003). C.-W. Chiang [[*et al.*]{}]{}, [[Phys. Rev.]{} D [**68**]{}]{}, 074012 (2003); M. Gronau, J. L. Rosner and J. Zupan, [[Phys. Lett.]{} B [**596**]{}]{}, 107 (2004). M. Beneke and M. Neubert, [[Nucl. Phys.]{} B [**675**]{}]{}, 333 (2003). D. London and A. Soni, [[Phys. Lett.]{} B [**407**]{}]{}, 61 (1997).

M. Beneke, [[Phys. Lett.]{} B [**620**]{}]{}, 143 (2005); G. Buchalla [[*et al.*]{}]{}, JHEP [**0509**]{}, 074 (2005). H. Y. Cheng [[*et al.*]{}]{}, [[Phys. Rev.]{} D [**72**]{}]{}, 014006 (2005), [[Phys. Rev.]{} D [**71**]{}]{}, 014030 (2005); S. Fajfer [[*et al.*]{}]{}, [[Phys. Rev.]{} D [**72**]{}]{}, 114001 (2005).

A. R. Williamson and J. Zupan, [[Phys. Rev.]{} D [**74**]{}]{}, 014003 (2006).

M. Gronau, J. L. Rosner and J. Zupan, arXiv:hep-ph/0608085, 2006.

 Collaboration, B. Aubert [[*et al.*]{}]{}, [[Phys. Rev. Lett.]{} [**94**]{}]{}, 191802 (2005). Belle Collaboration, K. Abe [[*et al.*]{}]{}, [[Phys. Rev.]{} D [**72**]{}]{}, 012004 (2005).

Subsequent to this submission, Belle updated their results: Belle Collaboration, K. F. Chen [[*et al.*]{}]{}, Following letter

 Collaboration, B. Aubert [[*et al.*]{}]{}, [[Nucl. Instr. Meth.]{} A [**479**]{}]{}, 1 (2002).

 Collaboration, B. Aubert [[*et al.*]{}]{}, [[Phys. Rev. Lett.]{} [**94**]{}]{}, 161803 (2005).

S. Agostinelli [[*et al.*]{}]{}, [[Nucl. Instr. Meth.]{} A [**506**]{}]{}, 250 (2003).

 Collaboration, B. Aubert [[*et al.*]{}]{}, [[Phys. Rev.]{} D [**66**]{}]{}, 032003 (2002).

Particle Data Group, Y.-M. Yao [[*et al.*]{}]{}, [[J. Phys. [**G[**33**]{}**]{}]{}]{}, 1 (2006).

 Collaboration, B. Aubert [[*et al.*]{}]{}, [[Phys. Rev. Lett.]{} [**92**]{}]{}, 061801 (2004).

ARGUS Collaboration, H. Albrecht [[*et al.*]{}]{}, [[Phys. Lett.]{} B [**241**]{}]{}, 278 (1990).

O. Long [[*et al.*]{}]{}, [[Phys. Rev.]{} D [**68**]{}]{}, 034010 (2003).