capbtabboxtable[][]

Novel method for measuring charm-mixing parameters using multibody decays

A. Di Canto1, J. Garra Ticó2, T. Gershon3, N. Jurik4, M. Martinelli1, T.
Pilař3, S. Stahl1, D. Tonelli5

1European Organization for Nuclear Research (CERN), Geneva, Switzerland
2Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
3Department of Physics, University of Warwick, Coventry, United Kingdom
4Department of Physics, University of Oxford, Oxford, United Kingdom
5INFN Sezione di Trieste, Trieste, Italy

We propose a novel method to measure flavor-oscillations and charge-parity
($C\\!P$) violation in charm mixing. The approach applies to multibody charm
decays, such as $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$, and avoids the need for a fit of the decay amplitudes while
suppressing biases due to nonuniform signal-reconstruction efficiencies as
functions of phase space and decay time. Data are partitioned in decay-time
and Dalitz-plot regions (bins). The Dalitz-plot bins are symmetric with
respect to the principal bisector and chosen to ensure nearly constant values
of the strong-interaction phases in each. The ratios of signal yields observed
in each symmetric bin pair are fit as functions of decay time, using
independent auxiliary measurements of the strong-interaction phases as
constraints, to determine the relevant physics parameters. Simulation shows a
35% improvement in sensitivity to the normalized charm-eigenstate mass
difference with respect to existing model-independent methods. In addition, we
introduce a parametrization of oscillation and $C\\!P$-violation effects in
charm mixing that has attractive statistical properties and may find wider
applicability.

Published in Phys. Rev. D99 (2019) 012007

###### Contents

  1. 1 Introduction
  2. 2 The bin-flip method
  3. 3 Dalitz-plot partition and strong-interaction phase inputs
  4. 4 Sensitivity
    1. 4.1 Reach and comparison with other methods
    2. 4.2 Dependence on external inputs
    3. 4.3 Effects of finite resolutions and nonuniform efficiencies
  5. 5 Impact on knowledge of charm-mixing parameters
  6. 6 Conclusions
  7. Acknowledgements
  8. A An alternative parametrization of $C\\!P$ violation in charm mixing

##  1 Introduction

The noncoincidence of mass and flavor eigenstates of neutral flavored mesons
results in flavor oscillations, which are meson-antimeson transitions that
follow an oscillating pattern as a function of time. Flavor oscillations are
sensitive probes for non-standard-model physics since virtual massive
particles can contribute to the amplitude, possibly enhancing the average
oscillation rate or the difference between rates of mesons and those of their
respective antimesons. Indeed, the study of flavor oscillations has long been
established as a powerful instrument to uncover, or constrain, possible
dynamics not described by the standard model.

Oscillations are typically characterized by the dimensionless mixing
parameters $x\equiv\Delta m/\Gamma$ and $y\equiv\Delta\Gamma/2\Gamma$, where
$\Delta m$ ($\Delta\Gamma$) is the difference between the masses (decay
widths) of the neutral-meson eigenstates, and $\Gamma$ is the average decay
width [1]. Oscillations were first observed in the $K^{0}$–$\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0}$ system in 1956 [2], then
established in the $B^{0}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
system in 1987 [3], and in the $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ system in 2006 [4].
Oscillation parameters for all these mesons are known precisely, except for
the width-difference of $B^{0}$ mesons [1]. The first evidence for
$D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillations was
reported in 2007 [5, 6] and the first single-experiment observation in 2012
[7]. However, the underlying charm-mixing parameters still have significant
uncertainties. Recent global combinations yield
$x=(4.6\,^{+\,1.2}_{-\,1.3})\times 10^{-3}$ and $y=(6.2\pm 0.7)\times
10^{-3}$, assuming charge-parity ($C\\!P$) symmetry of doubly Cabibbo-
suppressed decay amplitudes [8]. While the global knowledge of $y$ is rather
precise, less is known about $x$, which has not even been conclusively shown
to differ from zero. Improving the knowledge of $x$ is especially critical
since sensitivity to the small phase $\phi$ that describes $C\\!P$ violation
in the interference between mixing and decay relies predominantly on
observables proportional to $x\sin\phi$.

The most direct experimental access to the charm-mixing parameters is offered
by the analysis of self-conjugate multibody decays, such as $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ (inclusion of charge-conjugate
processes is implied unless stated otherwise). A joint fit of the Dalitz-plot
and decay-time distributions allows for the determination of a $D^{0}$
component growing as a function of decay time in a sample of candidates
produced as $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons, and
vice versa. This approach is challenging as it requires fitting the decay-time
evolution of signal decays across the Dalitz plot with an accurate amplitude
model, accounting for efficiency and resolution effects, and backgrounds
components [9, 10, 11].

With the large samples of $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ decays expected at the LHCb and Belle II experiments [12,
13], the systematic uncertainties due to knowledge of the amplitude model are
likely to limit the final precision on the mixing parameters. Approaches that
obviate the need for an amplitude analysis of the Dalitz-plot distribution
have been proposed to mitigate this issue [14, 15]. These build on ideas
developed to measure the CKM angle $\gamma$ from $B^{-}\rightarrow
D(\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-})K^{-}$ decays,
known as the GGSZ method [16, 17, 18]. By partitioning the Dalitz plot into
bins, the need for an explicit amplitude model is avoided, and the decay-time
distribution depends on a small number of coefficients that encode relevant
information about the decay, in addition to the mixing parameters. At hadron-
collider experiments, however, such model-independent methods still face a
significant challenge. Stringent online event-selection criteria are imposed
on charged-particle momenta and displacements from the primary interaction
space-point to suppress the prevailing backgrounds from light-quark
production. Modeling the resulting biases on signal decay-time and Dalitz-plot
distributions increases the complexity of the analyses introducing further
sources of systematic uncertainty that may offset the intended advantages
[19].

We propose a novel approach for measuring parameters of oscillation and
$C\\!P$ violation in charm mixing using $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$, or other multibody neutral-
charm decays, that requires neither an amplitude analysis of the Dalitz-plot
distribution nor an accurate modeling of the efficiency variations as
functions of decay time and Dalitz-plot position. The sample of
$D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays is
divided into subsamples according to initial $D^{0}$ meson flavor, location on
the Dalitz plot, and decay time. Ratios of decay yields observed in regions
(“bins”) of the Dalitz plot that are symmetric about its bisector are
constructed as functions of decay time. These functions depend on the known
hadronic parameters, dependent on Dalitz-plot bin, that enter the GGSZ method
to determine $\gamma$ [20]. The mixing parameters are obtained from a least-
squares fit of the decay-time-dependent ratios, jointly for mesons produced as
$D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$, in which
external information on the hadronic parameters is used as a constraint. Any
significant $C\\!P$-violating effect in oscillations of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons is observed as a difference
in the ratios between the samples of mesons produced in the $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ states. We dub this approach the
“bin-flip method”.

In Section 2 we develop the formalism of the method; in Section 3 we discuss
the Dalitz-plot partition and external inputs needed; in Section 4 we evaluate
the sensitivity using simulated samples and discuss instrumental effects such
as those due to resolutions and nonuniform reconstruction efficiencies; in
Section 5 we quantify the impact of the method on the knowledge of charm-
mixing phenomenology to finally conclude in Section 6.

##  2 The bin-flip method

Mass eigenstates of neutral charm mesons are expressed as
$|D_{1,2}\rangle=p|D^{0}\rangle\pm q|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$ in terms of flavor
eigenstates, where $p$ and $q$ are complex parameters satisfying
$|q|^{2}+|p|^{2}=1$. In the limit of $C\\!P$ symmetry ($q=p$), we define
$D_{1(2)}$ to be the $C\\!P$-even (odd) eigenstate, and the mixing parameters
as $x=(m_{1}-m_{2})/\Gamma$ and $y=(\Gamma_{1}-\Gamma_{2})/(2\Gamma)$, where
$\Gamma=(\Gamma_{1}+\Gamma_{2})/2$ is the average decay-width, following Refs.
[1, 8]. We specialize the discussion of the method to the $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays, since we anticipate
that it will have a strong impact when used with this mode, but the formalism
can be adapted to other multibody decays.

We parametrize the $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ three-body decay dynamics with two two-body masses following
the Dalitz formalism [21, 22]. We use the following flavor-dependent
definition of squared invariant masses:

$m_{\pm}^{2}\equiv\left\\{\begin{aligned} m^{2}(K^{0}_{\rm\scriptscriptstyle
S}\pi^{\pm})\quad&\text{for}\ D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}\ \text{decays}\\\ m^{2}(K^{0}_{\rm\scriptscriptstyle
S}\pi^{\mp})\quad&\text{for}\ \kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\ \text{decays}\\\
\end{aligned}\right.,$ (1)

which simplifies the simultaneous treatment of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays.

We indicate with $A_{f}(m_{+}^{2},m_{-}^{2})$ and
$\bar{A}_{f}(m_{+}^{2},m_{-}^{2})$ the amplitudes for mesons produced as
$D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$, respectively,
and decaying to the final state $f=K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ at the generic point $(m_{+}^{2},m_{-}^{2})$ of the Dalitz
plane. If $C\\!P$ symmetry is conserved in the decay, the relation
$A_{f}(m_{+}^{2},m_{-}^{2})=\bar{A}_{f}(m_{+}^{2},m_{-}^{2})$ holds. The decay
rates of neutral $D$ mesons tagged in the flavor eigenstates $D^{0}$ and
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ at time $t=0$ evolve in
time as

$\displaystyle\left|{T}_{f}(m_{+}^{2},m_{-}^{2};t)\right|^{2}$
$\displaystyle=\left|{A}_{f}(m_{+}^{2},m_{-}^{2})\,g_{+}(t)+\bar{A}_{f}(m_{-}^{2},m_{+}^{2})\,\frac{q}{p}\,g_{-}(t)\right|^{2}\quad\text{and}$
(2) $\displaystyle\left|\overline{T}_{\\!f}(m_{+}^{2},m_{-}^{2};t)\right|^{2}$
$\displaystyle=\left|\bar{A}_{f}(m_{+}^{2},m_{-}^{2})\,g_{+}(t)+{A}_{f}(m_{-}^{2},m_{+}^{2})\,\frac{p}{q}\,g_{-}(t)\right|^{2},$
(3)

where $g_{\pm}(t)=\theta(t)e^{-imt}e^{-t/2}~{}^{\cosh}_{\sinh}(zt/2)$, $t$ is
the decay time in units of $D^{0}$ lifetime $\tau=1/\Gamma$,
$m=(m_{1}+m_{2})/2$ is the average mass of neutral $D$ mesons, $\theta$ is the
Heaviside function, and $z$ equals $-(y+ix)$.

We divide the Dalitz plane into two sets of $n$ bins each, symmetric about its
principal bisector $m_{+}^{2}=m_{-}^{2}$. Bins are labeled with the index $\pm
b$, where $b=1,...,n$. Positive indices refer to bins in the (lower)
$m_{+}^{2}>m_{-}^{2}$ region, where Cabibbo-favored $D^{0}\rightarrow
K^{*}(892)^{-}\pi^{+}$ decays dominate the amplitude; negative indices refer
to their symmetric counterparts in the (upper) $m_{+}^{2}<m_{-}^{2}$ region.

As oscillations develop as a function of time, the relative variations of
intensities between pairs of bins change depending on the mixing parameters
and relevant charm-decay hadronic parameters. The expressions for the event
yields integrated over each Dalitz-plot bin $b$ are

$\displaystyle N_{b}(t)$
$\displaystyle=\int_{b}dm_{+}^{2}dm_{-}^{2}\left|T_{f}(m_{+}^{2},m_{-}^{2};t)\right|^{2}$
$\displaystyle=F_{b}\left|g_{+}(t)\right|^{2}+\left|\frac{q}{p}\right|^{2}\kern
1.99997pt\overline{\kern-1.99997ptF}{}_{-b}\left|g_{-}(t)\right|^{2}+2\sqrt{\kern
1.99997pt\overline{\kern-1.99997ptF}{}_{-b}F_{b}}{{\rm\,Re}\left[\frac{q}{p}X_{b}\,g_{+}^{\star}(t)g_{-}(t)\right]}\quad\text{and}$
(4) $\displaystyle\kern 1.99997pt\overline{\kern-1.99997ptN}{}_{b}(t)$
$\displaystyle=\int_{b}dm_{+}^{2}dm_{-}^{2}\left|\overline{T}_{\\!f}(m_{+}^{2},m_{-}^{2};t)\right|^{2}$
$\displaystyle=\kern
1.99997pt\overline{\kern-1.99997ptF}{}_{b}\left|g_{+}(t)\right|^{2}+\left|\frac{p}{q}\right|^{2}F_{-b}\left|g_{-}(t)\right|^{2}+2\sqrt{F_{-b}\kern
1.99997pt\overline{\kern-1.99997ptF}{}_{b}}{{\rm\,Re}\left[\frac{p}{q}\bar{X}_{b}\,g_{+}^{\star}(t)g_{-}(t)\right]},$
(5)

where the following definitions are introduced:

$\displaystyle{F}_{b}\equiv\int_{b}dm_{+}^{2}dm_{-}^{2}\left|A_{f}(m_{+}^{2},m_{-}^{2})\right|^{2},\quad\kern
1.99997pt\overline{\kern-1.99997ptF}{}_{b}\equiv\int_{b}dm_{+}^{2}dm_{-}^{2}\left|\bar{A}_{f}(m_{+}^{2},m_{-}^{2})\right|^{2},$
(6) $\displaystyle X_{b}\equiv\frac{1}{\sqrt{F_{b}\kern
1.99997pt\overline{\kern-1.99997ptF}{}_{-b}}}\int_{b}\\!dm_{+}^{2}dm_{-}^{2}\,A_{f}^{\star}(m_{+}^{2},m_{-}^{2})\bar{A}_{f}(m_{-}^{2},m_{+}^{2}),$
(7)

and $\kern 1.99997pt\overline{\kern-1.99997ptX}{}_{b}$ is defined similarly as
in Equation 7 with $A_{f}\leftrightarrow\bar{A}_{f}$ and
$F_{b}\leftrightarrow\kern 1.99997pt\overline{\kern-1.99997ptF}{}_{b}$. Here,
$F_{b}$ and $\kern 1.99997pt\overline{\kern-1.99997ptF}{}_{b}$ are event
yields in the Dalitz bin $b$ at $t=0$. The hadronic parameter of the
interference term $X_{b}$, with (by definition) $X_{-b}=X_{b}^{\star}$ and
$\left|X_{b}\right|\leq 1$, is related to the strong-interaction phase
difference, $\Delta\delta$, and to the weak-interaction phase difference,
$\varphi$, between $A_{f}(m_{+}^{2},m_{-}^{2})$ and
$\bar{A}_{f}(m_{-}^{2},m_{+}^{2})$ averaged over bin $b$. In the limit of
$C\\!P$-conserving decay amplitudes, $A_{f}=\bar{A}_{f}$ so $\kern
1.99997pt\overline{\kern-1.99997ptF}{}_{b}=F_{b}$, $\varphi=0$, and
$X_{b}=\kern 1.99997pt\overline{\kern-1.99997ptX}{}_{b}$ hold. Hence, the real
and imaginary parts of the coefficients $X_{b}\equiv c_{b}-is_{b}$ are

$\displaystyle c_{b}$
$\displaystyle\equiv\frac{1}{\sqrt{F_{b}F_{-b}}}\int_{b}dm_{+}^{2}dm_{-}^{2}\,\left|A_{f}(m^{2}_{+},m^{2}_{-})\right|\,\left|A_{f}(m^{2}_{-},m^{2}_{+})\right|\cos[\Delta\delta(m^{2}_{+},m^{2}_{-})]\quad\text{and}$
(8) $\displaystyle s_{b}$
$\displaystyle\equiv\frac{1}{\sqrt{F_{b}F_{-b}}}\int_{b}dm_{+}^{2}dm_{-}^{2}\,\left|A_{f}(m^{2}_{+},m^{2}_{-})\right|\,\left|A_{f}(m^{2}_{-},m^{2}_{+})\right|\sin[\Delta\delta(m^{2}_{+},m^{2}_{-})],$
(9)

where
$\Delta\delta(m_{+}^{2},m_{-}^{2})=\delta(m_{+}^{2},m_{-}^{2})-\delta(m_{-}^{2},m_{+}^{2})$
and $\delta(m_{+}^{2},m_{-}^{2})$ is the phase of
$A_{f}(m_{+}^{2},m_{-}^{2})$. Constraining the hadronic parameters $c_{b}$ and
$s_{b}$ from independent external measurements offers access to the mixing
parameters.

If the probability $\epsilon(m_{+}^{2},m_{-}^{2})$ to select and reconstruct
the decays is nonuniform across the Dalitz plane, the parameters $F_{b}$ and
$(c_{b},s_{b})$ become

$\displaystyle\tilde{F}_{b}\equiv\int_{b}dm_{+}^{2}dm_{-}^{2}\,\epsilon(m_{+}^{2},m_{-}^{2})\,\left|A_{f}(m_{+}^{2},m_{-}^{2})\right|^{2},\quad\text{and}$
(10)
$\displaystyle\tilde{c}_{b}\equiv\frac{1}{\sqrt{\tilde{F}_{b}\tilde{F}_{-b}}}\\!\int_{b}\\!dm_{+}^{2}dm_{-}^{2}\,\epsilon(m_{+}^{2},m_{-}^{2})\\!\left|A_{f}(m^{2}_{+},m^{2}_{-})\right|\\!\left|A_{f}(m^{2}_{-},m^{2}_{+})\right|\cos[\Delta\delta(m^{2}_{+},m^{2}_{-})],$
(11)
$\displaystyle\tilde{s}_{b}\equiv\frac{1}{\sqrt{\tilde{F}_{b}\tilde{F}_{-b}}}\\!\int_{b}\\!dm_{+}^{2}dm_{-}^{2}\,\epsilon(m_{+}^{2},m_{-}^{2})\\!\left|A_{f}(m^{2}_{+},m^{2}_{-})\right|\\!\left|A_{f}(m^{2}_{-},m^{2}_{+})\right|\sin[\Delta\delta(m^{2}_{+},m^{2}_{-})],$
(12)

respectively. It is important that efficiency-induced biases on
$(c_{b},s_{b})$ are kept small, since these values will be constrained to
externally measured values. This can be achieved by designing selection
strategies aimed at minimizing biases on the Dalitz-plot distribution, when
possible. Otherwise, efficiencies that are nonuniform but still symmetric
across the Dalitz-plot bisector are expected to induce reduced biases on
$(c_{b},s_{b})$. In addition, appropriate choices of binning schemes may also
mitigate the biases on $(c_{b},s_{b})$ induced by efficiency variations. For
example, Dalitz bins defined such that $\Delta\delta(m^{2}_{+},m^{2}_{-})$ is
nearly constant within each bin are expected to reduce the effect of the
nonuniformities of the efficiency on $(c_{b},s_{b})$. We neglect the effect of
efficiency variations as functions of Dalitz-plot position in the discussion
of the method below, and discuss the possible biases in realistic experimental
situations in Section 4.3.

For small mixing parameters ($|z|t\ll 1$), the following approximations hold:

$\displaystyle\left|g_{+}(t)\right|^{2}$ $\displaystyle\approx
e^{-t}+\frac{1}{4}\,e^{-t}\,t^{2}\,{{\rm\,Re}\left(z^{2}\right)}+\mathcal{O}(z^{4}),$
(13) $\displaystyle\left|g_{-}(t)\right|^{2}$
$\displaystyle\approx\frac{1}{4}\,e^{-t}\,t^{2}\,|z|^{2}+\mathcal{O}(z^{4}),\quad\text{and}$
(14) $\displaystyle g_{+}^{\star}(t)g_{-}(t)$
$\displaystyle\approx\frac{1}{2}\,e^{-t}\,t\,z+\mathcal{O}(z^{3}).$ (15)

Terms of $\mathcal{O}(z^{3})$ or higher can be neglected, so that integration
of the above expressions over decay-time bin $j$ yields

$\displaystyle\int_{j}dt\,\left|g_{+}(t)\right|^{2}$ $\displaystyle\approx
n_{j}\left[\displaystyle 1+\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,{{\rm\,Re}\left(z^{2}\right)}\right],$ (16)
$\displaystyle\int_{j}dt\,\left|g_{-}(t)\right|^{2}$ $\displaystyle\approx
n_{j}\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z\right|^{2},\quad\text{and}$ (17)
$\displaystyle\int_{j}dt\,g_{+}^{\star}(t)g_{-}(t)$ $\displaystyle\approx
n_{j}\displaystyle\frac{1}{2}\,\langle t\rangle_{j}\,z,$ (18)

where $\langle...\rangle_{j}$ denotes the average over the exponential
distribution in the decay-time bin $j$, and $n_{j}$ is a normalization
constant that cancels in ratios and is omitted in what follows.

If the probability $\epsilon(t)$ to select and reconstruct the decays is
nonuniform as a function of decay time within bin $j$, the average is
performed over the observed decay-time distribution of mesons that did not
undergo oscillation, $\epsilon(t)e^{-t}$. An advantage of the bin-flip method
is that the dependence of results on $\epsilon(t)$ is minimal.

In the limit of $C\\!P$-conserving decay amplitudes, the decay yields in
Dalitz bin $b$ and decay-time bin $j$ of charm mesons originally produced in
the $D^{0}$ or $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ flavor
states are, respectively,

$\displaystyle N_{bj}$ $\displaystyle=\int_{j}dtN_{b}(t)$
$\displaystyle\approx F_{b}\left[\displaystyle 1+\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,{{\rm\,Re}\left(z^{2}\right)}\right]+\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z\right|^{2}\left|\frac{q}{p}\right|^{2}F_{-b}+\langle
t\rangle_{j}\sqrt{F_{-b}F_{b}}{{\rm\,Re}\left(\frac{q}{p}X_{b}z\right)},$ (19)
$\displaystyle\kern 1.99997pt\overline{\kern-1.99997ptN}{}_{bj}$
$\displaystyle=\int_{j}dt\overline{N}_{b}(t)$ $\displaystyle\approx
F_{b}\left[\displaystyle 1+\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,{{\rm\,Re}\left(z^{2}\right)}\right]+\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z\right|^{2}\left|\frac{p}{q}\right|^{2}F_{-b}+\langle
t\rangle_{j}\sqrt{F_{-b}F_{b}}{{\rm\,Re}\left(\frac{p}{q}X_{b}z\right)}.$ (20)

For each decay-time bin $j$, the ratios between the decay yield in Dalitz bin
$-b$ and Dalitz bin $b$, for mesons originally produced as $D^{0}$ or $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ are, respectively,

$\displaystyle R_{bj}$ $\displaystyle=\frac{N_{-bj}}{N_{bj}}$
$\displaystyle\approx\frac{r_{b}\left[\displaystyle 1+\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,{{\rm\,Re}\left(z^{2}\right)}\right]+\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z\right|^{2}\left|\frac{q}{p}\right|^{2}+\langle
t\rangle_{j}\sqrt{r_{b}}{{\rm\,Re}\left(X_{b}^{\star}\frac{q}{p}\,z\right)}}{\displaystyle
1+\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,{{\rm\,Re}\left(z^{2}\right)}+\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z\right|^{2}r_{b}\left|\frac{q}{p}\right|^{2}+\langle
t\rangle_{j}\sqrt{r_{b}}{{\rm\,Re}\left(X_{b}\frac{q}{p}\,z\right)}},$ (21)
$\displaystyle\kern 1.99997pt\overline{\kern-1.99997ptR}{}_{bj}$
$\displaystyle=\frac{\kern 1.99997pt\overline{\kern-1.99997ptN}{}_{-bj}}{\kern
1.99997pt\overline{\kern-1.99997ptN}{}_{bj}}$
$\displaystyle\approx\frac{r_{b}\left[\displaystyle 1+\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,{{\rm\,Re}\left(z^{2}\right)}\right]+\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z\right|^{2}\left|\frac{p}{q}\right|^{2}+\langle
t\rangle_{j}\sqrt{r_{b}}{{\rm\,Re}\left(X_{b}^{\star}\frac{p}{q}\,z\right)}}{\displaystyle
1+\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,{{\rm\,Re}\left(z^{2}\right)}+\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z\right|^{2}r_{b}\left|\frac{p}{q}\right|^{2}+\langle
t\rangle_{j}\sqrt{r_{b}}{{\rm\,Re}\left(X_{b}\frac{p}{q}\,z\right)}},$ (22)

where $r_{b}=F_{-b}/F_{b}$.

The bin-flip approach consists in performing a joint fit of the $R_{bj}$ and
$\kern 1.99997pt\overline{\kern-1.99997ptR}{}_{bj}$ ratios to determine the
oscillation and $C\\!P$-violation parameters in charm mixing, by constraining
the coefficients $X_{b}$ from external measurements (Section 3). Conceptually,
this is akin to performing the wrong-sign-to-right-sign analysis of
$D^{0}\rightarrow K^{\mp}\pi^{\pm}$ decays [23] simultaneously in specially
chosen subsets of events, the Dalitz-plot bins, for which hadronic parameters
are known. Unlike $D^{0}\rightarrow K^{\mp}\pi^{\pm}$ decays,
$D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays also
proceed through amplitudes with defined $C\\!P$ eigenvalues, with important
consequences on the sensitivity of method, as discussed in Section 4.

In practice, to avoid instabilities of the fit due to $(q/p)^{\pm 1}z$ terms,
where sensitivity to $q/p$ degrades if $\left|z\right|\approx 0$, we
parametrize the ratios using $z_{C\\!P}$ and $\Delta z$, defined by

$z_{C\\!P}\pm\Delta z\equiv\left(q/p\right)^{\pm 1}z.$ (23)

With this definition,

$z^{2}=\left(z_{C\\!P}+\Delta z\right)\left(z_{C\\!P}-\Delta
z\right)=z_{C\\!P}^{2}-\Delta
z^{2},\qquad\left(\frac{q}{p}\right)^{\\!\\!2}=\frac{z_{C\\!P}+\Delta
z}{z_{C\\!P}-\Delta z},$ (24)

and the ratios become

$\displaystyle R_{bj}$ $\displaystyle\approx\frac{r_{b}\left[\displaystyle
1+\frac{1}{4}\,\langle t^{2}\rangle_{j}\,{{\rm\,Re}\left(z_{C\\!P}^{2}-\Delta
z^{2}\right)}\right]+\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z_{C\\!P}+\Delta z\right|^{2}+\sqrt{r_{b}}\langle
t\rangle_{j}{{\rm\,Re}\left[X_{b}^{\star}(z_{C\\!P}+\Delta
z)\right]}}{\displaystyle 1+\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,{{\rm\,Re}\left(z_{C\\!P}^{2}-\Delta
z^{2}\right)}+r_{b}\,\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z_{C\\!P}+\Delta z\right|^{2}+\sqrt{r_{b}}\langle
t\rangle_{j}{{\rm\,Re}\left[X_{b}(z_{C\\!P}+\Delta z)\right]}},$ (25)
$\displaystyle\kern 1.99997pt\overline{\kern-1.99997ptR}{}_{bj}$
$\displaystyle\approx\frac{r_{b}\left[\displaystyle 1+\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,{{\rm\,Re}\left(z_{C\\!P}^{2}-\Delta
z^{2}\right)}\right]+\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z_{C\\!P}-\Delta z\right|^{2}+\sqrt{r_{b}}\langle
t\rangle_{j}{{\rm\,Re}\left[X_{b}^{\star}(z_{C\\!P}-\Delta
z)\right]}}{\displaystyle 1+\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,{{\rm\,Re}\left(z_{C\\!P}^{2}-\Delta
z^{2}\right)}+r_{b}\,\displaystyle\frac{1}{4}\,\langle
t^{2}\rangle_{j}\,\left|z_{C\\!P}-\Delta z\right|^{2}+\sqrt{r_{b}}\langle
t\rangle_{j}{{\rm\,Re}\left[X_{b}(z_{C\\!P}-\Delta z)\right]}}.$ (26)

Using the customary convention for the charm-mixing $C\\!P$-violation phase
$\phi\equiv\arg(q\bar{A}_{f}/pA_{f})\approx\arg(q/p)$, which assumes the
absence of any final-state-dependent weak-interaction phase between decay
amplitudes (consistent with the limit of $C\\!P$-symmetric decay amplitudes),
the interpretation of $z_{C\\!P}$ and $\Delta z$ in terms of the usual mixing
parameters becomes straightforward,

$\displaystyle x_{C\\!P}$ $\displaystyle=-{{\rm\,Im}\left(z_{C\\!P}\right)}$
$\displaystyle=\frac{1}{2}\left[x\cos\phi\left(\left|\frac{q}{p}\right|+\left|\frac{p}{q}\right|\right)+y\sin\phi\left(\left|\frac{q}{p}\right|-\left|\frac{p}{q}\right|\right)\right],$
(27) $\displaystyle\Delta x$ $\displaystyle=-{{\rm\,Im}\left(\Delta z\right)}$
$\displaystyle=\frac{1}{2}\left[x\cos\phi\left(\left|\frac{q}{p}\right|-\left|\frac{p}{q}\right|\right)+y\sin\phi\left(\left|\frac{q}{p}\right|+\left|\frac{p}{q}\right|\right)\right],$
(28) $\displaystyle y_{C\\!P}$
$\displaystyle=-{{\rm\,Re}\left(z_{C\\!P}\right)}$
$\displaystyle=\frac{1}{2}\left[y\cos\phi\left(\left|\frac{q}{p}\right|+\left|\frac{p}{q}\right|\right)-x\sin\phi\left(\left|\frac{q}{p}\right|-\left|\frac{p}{q}\right|\right)\right],$
(29) $\displaystyle\Delta y$ $\displaystyle=-{{\rm\,Re}\left(\Delta z\right)}$
$\displaystyle=\frac{1}{2}\left[y\cos\phi\left(\left|\frac{q}{p}\right|-\left|\frac{p}{q}\right|\right)-x\sin\phi\left(\left|\frac{q}{p}\right|+\left|\frac{p}{q}\right|\right)\right].$
(30)

Conservation of $C\\!P$ symmetry in mixing ($|q/p|=1$) and in the interference
of mixing and decay ($\phi=0$) implies $x_{C\\!P}=x$, $y_{C\\!P}=y$, and
$\Delta x=\Delta y=0$. The observables $\Delta y$, frequently denoted as
$A_{\Gamma}$, and $y_{C\\!P}$ are well known. The introduction of $x_{C\\!P}$
and $\Delta x$ allows for a conveniently symmetric notation and yields
parameters with statistical properties optimally suited for use in
measurements and combinations of results, as discussed in Appendix A.

##  3 Dalitz-plot partition and strong-interaction phase inputs

Various Dalitz-plot binning schemes were developed by the CLEO collaboration
for measuring the coefficients $X_{b}\equiv c_{b}-is_{b}$ [20]. These include
schemes aimed at minimizing the variations of the strong-interaction phase
differences across each bin, as well as alternatives explicitly designed to
optimize the GGSZ sensitivity to $\gamma$. For the bin-flip method we propose
to use the “iso-$\Delta\delta$” scheme with $n=8$ bins defined in each Dalitz
semispace such that

$2\pi(b-3/2)/n<\Delta\delta(m_{+}^{2},m_{-}^{2})<2\pi(b-1/2)/n,\quad
b=1,...,n,$ (31)

where the variation of $\Delta\delta(m_{+}^{2},m_{-}^{2})$ over the Dalitz
plane is evaluated using the “BaBar 2008” amplitude model [24]. Because this
scheme keeps the strong-interaction phase difference approximately constant in
each Dalitz-plot bin, biases due to nonuniform efficiencies are reduced. A
dedicated binning optimization for the bin-flip method may lead to improved
sensitivity, but this is not pursued here, since we intend to rely on existing
measurements of the hadronic parameters to demonstrate quantitatively the
performance of the method.

The iso-$\Delta\delta$ scheme, shown in Figure 1, is available as a look-up
table consisting of a grid of $(m_{+}^{2},m_{-}^{2})$ points spaced
$0.0054{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ apart in both $m_{+}^{2}$
and $m_{-}^{2}$. The corresponding values of $r_{b}$, $c_{b}$, and $s_{b}$ are
reproduced in Table 1, as measured by CLEO in $0.8\mbox{\,fb}^{-1}$ of
$e^{+}e^{-}$ collisions at a center-of-mass energy of
$3.77\mathrm{\,Ge\kern-1.00006ptV}$. The $c_{b}$ and $s_{b}$ correlations are
reported in Table 2.

Figure 1: Iso-$\Delta\delta$ binning of the $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ Dalitz plot, based on the BaBar 2008 amplitude model [20]. The bins are symmetric with respect to the $m_{+}^{2}=m_{-}^{2}$ bisector; positive indices refer to bins in the (lower) $m_{+}^{2}>m_{-}^{2}$ region; negative indices refer to those in the (upper) $m_{+}^{2}<m_{-}^{2}$ region. Colors indicate the absolute value of the bin index $b$. Table 1: Values of $r_{b}$, $c_{b}$, and $s_{b}$ measured by CLEO for the BaBar 2008 iso-$\Delta\delta$ binning scheme [20]. The first contribution to the uncertainty is statistical, the second systematic. $b$ | $r_{b}$ | $c_{b}$ | $s_{b}$
---|---|---|---
1 | $0.488\pm 0.028$ | $0.655\pm 0.036\pm 0.042$ | $-0.025\pm 0.098\pm 0.043$
2 | $0.286\pm 0.027$ | $0.511\pm 0.068\pm 0.063$ | $0.141\pm 0.183\pm 0.066$
3 | $0.319\pm 0.031$ | $0.024\pm 0.140\pm 0.080$ | $1.111\pm 0.131\pm 0.044$
4 | $0.667\pm 0.100$ | $-0.569\pm 0.118\pm 0.098$ | $0.328\pm 0.202\pm 0.072$
5 | $0.632\pm 0.052$ | $-0.903\pm 0.045\pm 0.042$ | $-0.181\pm 0.131\pm 0.026$
6 | $0.220\pm 0.036$ | $-0.616\pm 0.103\pm 0.072$ | $-0.520\pm 0.196\pm 0.059$
7 | $0.125\pm 0.016$ | $0.100\pm 0.106\pm 0.124$ | $-1.129\pm 0.120\pm 0.096$
8 | $0.238\pm 0.018$ | $0.422\pm 0.069\pm 0.075$ | $-0.350\pm 0.151\pm 0.045$
Table 2: Correlation coefficients (in %) between the $c_{b}$ and $s_{b}$ parameters, separately for (top) statistical and (bottom) systematic effects, as measured by CLEO for the BaBar 2008 iso-$\Delta\delta$ binning scheme [20]. | $c_{2}$ | $c_{3}$ | $c_{4}$ | $c_{5}$ | $c_{6}$ | $c_{7}$ | $c_{8}$ | $s_{1}$ | $s_{2}$ | $s_{3}$ | $s_{4}$ | $s_{5}$ | $s_{6}$ | $s_{7}$ | $s_{8}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$c_{1}$ | $-2$ | $-3$ | 5 | 7 | 3 | 1 | $-2$ | 0 | 0 | $-2$ | 0 | 0 | 0 | $-1$ | 0
$c_{2}$ |  | 0 | 4 | 10 | 0 | 0 | 0 | 0 | $-2$ | 0 | 0 | 0 | 0 | 0 | 0
$c_{3}$ |  |  | 0 | 0 | 0 | 2 | $-4$ | 16 | $-4$ | 75 | 7 | $-10$ | 0 | 45 | 4
$c_{4}$ |  |  |  | 1 | 0 | 0 | 5 | 0 | $-1$ | 0 | 7 | $-1$ | 0 | 0 | 0
$c_{5}$ |  |  |  |  | 0 | 1 | 2 | 0 | 3 | 0 | 0 | 3 | 0 | 0 | 0
$c_{6}$ |  |  |  |  |  | $-1$ | $-1$ | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$c_{7}$ |  |  |  |  |  |  | 0 | 2 | 3 | 6 | 0 | 2 | 0 | 1 | 2
$c_{8}$ |  |  |  |  |  |  |  | $-1$ | 0 | $-3$ | 0 | 0 | 0 | $-2$ | 2
$s_{1}$ |  |  |  |  |  |  |  |  | $-8$ | 18 | 11 | $-18$ | $-7$ | 15 | 10
$s_{2}$ |  |  |  |  |  |  |  |  |  | $-3$ | 10 | 31 | $-6$ | $-2$ | 0
$s_{3}$ |  |  |  |  |  |  |  |  |  |  | 11 | $-9$ | $-2$ | 59 | 6
$s_{4}$ |  |  |  |  |  |  |  |  |  |  |  | 0 | $-4$ | 13 | 13
$s_{5}$ |  |  |  |  |  |  |  |  |  |  |  |  | 6 | $-10$ | $-11$
$s_{6}$ |  |  |  |  |  |  |  |  |  |  |  |  |  | $-5$ | $-6$
$s_{7}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 3
$c_{1}$ | $\phantom{-}89$ | $\phantom{-}93$ | $\phantom{-}74$ | $\phantom{-}77$ | $\phantom{-}85$ | $\phantom{-}90$ | $\phantom{-}90$ | $\phantom{-}32$ | $\phantom{-}24$ | $\phantom{-}32$ | $\phantom{-}30$ | $\phantom{-}25$ | $-11$ | $\phantom{-}11$ | $\phantom{-}29$
$c_{2}$ |  | 88 | 70 | 73 | 83 | 87 | 90 | 32 | 25 | 33 | 33 | 25 | $-13$ | 15 | 28
$c_{3}$ |  |  | 73 | 77 | 86 | 91 | 91 | 34 | 22 | 37 | 31 | 23 | $-9$ | 13 | 29
$c_{4}$ |  |  |  | 90 | 80 | 84 | 79 | $-11$ | $-22$ | 13 | $-12$ | 0 | 24 | $-31$ | $-1$
$c_{5}$ |  |  |  |  | 82 | 83 | 81 | $-5$ | $-14$ | 16 | $-6$ | $-1$ | 16 | $-23$ | 2
$c_{6}$ |  |  |  |  |  | 87 | 87 | 12 | 7 | 26 | 15 | 12 | 4 | $-2$ | 17
$c_{7}$ |  |  |  |  |  |  | 91 | 17 | 6 | 24 | 15 | 15 | 3 | $-5$ | 16
$c_{8}$ |  |  |  |  |  |  |  | 24 | 15 | 29 | 24 | 19 | $-4$ | 4 | 20
$s_{1}$ |  |  |  |  |  |  |  |  | 60 | 37 | 57 | 29 | $-43$ | 58 | 48
$s_{2}$ |  |  |  |  |  |  |  |  |  | 31 | 55 | 45 | $-41$ | 67 | 51
$s_{3}$ |  |  |  |  |  |  |  |  |  |  | 31 | 23 | $-9$ | 35 | 40
$s_{4}$ |  |  |  |  |  |  |  |  |  |  |  | 30 | $-42$ | 66 | 49
$s_{5}$ |  |  |  |  |  |  |  |  |  |  |  |  | $-20$ | 27 | 34
$s_{6}$ |  |  |  |  |  |  |  |  |  |  |  |  |  | $-56$ | $-28$
$s_{7}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 40

##  4 Sensitivity

The bin-flip method is validated using simulated experiments. In Section 4.1,
we discuss the tests of the basic assumptions and approximations of the
method, study its properties, and offer an estimate of the best statistical
precision possibly achievable. In Section 4.2, we focus on the dependence of
the method’s sensitivity on external inputs. In Section 4.3, we discuss the
impact of experimental effects, such as finite resolutions and nonuniform
reconstruction efficiencies.

###  4.1 Reach and comparison with other methods

The sensitivity of the bin-flip method to oscillation and $C\\!P$-violation
parameters in charm mixing is determined using ensembles of simulated samples
of $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays,
generated assuming five relevant configurations of the true values of such
parameters:

  1. (i)

No mixing (NM), corresponding to $x=x_{C\\!P}=y=y_{C\\!P}=0$, $|q/p|=1$, and
$\phi=0$ (or $\Delta x=\Delta y=0$);

  2. (ii)

$C\\!P$-conserving world-average mixing (WM), corresponding to
$x=x_{C\\!P}=0.4\%$, $y=y_{C\\!P}=0.6\%$, $|q/p|=1$, and $\phi=0$ (or $\Delta
x=\Delta y=0$);

  3. (iii)

$C\\!P$-conserving large mixing (LM), corresponding to
$x=x_{C\\!P}=y=y_{C\\!P}=1\%$, $|q/p|=1$, and $\phi=0$ (or $\Delta x=\Delta
y=0$);

  4. (iv)

$C\\!P$-violating world-average mixing (WCP), corresponding to $x=0.4\%$,
$y=0.6\%$, $|q/p|=0.93$, and $\phi=-0.15$;

  5. (v)

World-average mixing with $C\\!P$ violation in mixing only (MCP),
corresponding to $x=0.4\%$, $y=0.6\%$, $|q/p|=0.93$, and $\phi=0$.

The $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays
are generated by sampling the decay-time-dependent decay rate of Equations 2
and 3 [25]. The “BaBar 2010” model is used to describe the amplitudes at $t=0$
assuming $C\\!P$-conserving decay amplitudes (Figure 2) [11]. Minor
differences between the amplitude model used in generation and the model used
to define the iso-$\Delta\delta$ Dalitz-plot bins are irrelevant for testing
the method. While the definition of bins requires an amplitude model, the
method remains unbiased against mismodeling [16, 17, 18, 14, 15]. In addition,
both considered models achieve similar descriptions of the variations of the
strong-interaction phases across phase-space, thus keeping the sensitivity of
the method unaltered [20].

Figure 2:  Dalitz-plot distribution for $10^{6}$ simulated $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays in the no-mixing
scenario.

For each scenario we generate an ensemble of $10^{3}$ simulated experiments,
each containing typically $10^{6}$ signal events, corresponding to the size of
the full Belle sample and of that collected by LHCb during 2011–2012. No
background or detector effects are simulated. Each sample is split into ten
equally-populated decay-time bins and eight pairs of iso-$\Delta\delta$
Dalitz-plot bins. The average decay times and squared decay times are
calculated in each bin using decays populating the lower Dalitz subspace
($m^{2}_{-}<m^{2}_{+}$, $b>0$) only, which is enriched in $D^{0}$ mesons that
did not oscillate. The fit minimizes the least-squares function

$\chi^{2}=\sum_{b=1,j=1}^{b=8,j=10}\left[\frac{(N_{-bj}-N_{bj}R_{bj})^{2}}{(\sigma_{-bj})^{2}+(\sigma_{bj}R_{bj})^{2}}+\frac{(\kern
1.99997pt\overline{\kern-1.99997ptN}{}_{-bj}-\kern
1.99997pt\overline{\kern-1.99997ptN}{}_{bj}\kern
1.99997pt\overline{\kern-1.99997ptR}{}_{bj})^{2}}{(\bar{\sigma}_{-bj})^{2}+(\bar{\sigma}_{bj}\kern
1.99997pt\overline{\kern-1.99997ptR}{}_{bj})^{2}}\right].$ (32)

For each decay-time bin $j$ and pair of Dalitz-plot bin $\pm b$, the fit
compares the decay yields $N_{\pm bj}$ ($\kern
1.99997pt\overline{\kern-1.99997ptN}{}_{\pm bj}$) of charm mesons produced as
$D^{0}$ ($\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) flavor states and
observed in the chosen bin with the values expected from Equations 25 and 26
by weighting their squared difference with the variance, which is a function
of the yield’s uncertainties $\sigma_{\pm bj}$ ($\bar{\sigma}_{\pm bj}$).
Signal yields follow Poisson distributions to a good approximation. Hence, we
approximate the uncertainties of the yields as the square roots of the numbers
of decays.

Figure 3: Distributions of (top) fit residuals and (bottom) pulls on (left)
$x_{C\\!P}$ and (right) $y_{C\\!P}$ from $10^{6}$ $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays in the WM scenario, and
fit assuming $C\\!P$ conservation. Only the mixing parameters are determined
by the fits.

Figure 3 shows, as an example, the distribution of the difference between
estimated and generated values (fit residual) and the distribution of the same
difference divided by the estimated standard deviation (fit pull) obtained in
fits assuming $C\\!P$ conservation in the WM scenario, with nuisance
parameters $r_{b}$, $c_{b}$, and $s_{b}$ fixed to their input values (Table
3). The estimated uncertainties on $x_{C\\!P}=x$ and $y_{C\\!P}=y$ are
$0.15\%$ and $0.29\%$, respectively. The difference in sensitivity to $x$ and
$y$ is due to the partial cancellation, in the yield ratio, of mixing terms
proportional to amplitudes with defined $C\\!P$ eigenvalues (such as the
$C\\!P$-odd $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\rho(770)^{0}$
amplitude), which carry sensitivity to $y$. A coarse estimate of the expected
reduction in sensitivity is obtained by further expanding the ratio of
Equation 25 for $\langle
t\rangle_{j}\sqrt{r_{b}}{{\rm\,Re}\left(X_{b}z\right)}\ll 1$ while retaining
only terms linear in decay time, and under the simplifying assumption of
$C\\!P$ conservation,

$\displaystyle R_{bj}$ $\displaystyle\approx\frac{r_{b}+\langle
t\rangle_{j}\sqrt{r_{b}}{{\rm\,Re}\left(X_{b}^{\star}z\right)}}{1+\langle
t\rangle_{j}\sqrt{r_{b}}{{\rm\,Re}\left(X_{b}z\right)}}$ $\displaystyle\approx
r_{b}+\langle
t\rangle_{j}\sqrt{r_{b}}\bigl{[}{{\rm\,Re}\left(X_{b}^{\star}z\right)}-r_{b}{{\rm\,Re}\left(X_{b}z\right)}\bigr{]}$
$\displaystyle=r_{b}-\langle t\rangle_{j}\sqrt{r_{b}}\bigl{[}(1-r_{b})c_{b}\
y-(1+r_{b})s_{b}\ x\bigr{]}.$ (33)

The coefficient multiplying $y$ is typically half of that multiplying $x$
(Table 3) suggesting halved uncertainties on $x_{C\\!P}$ and $\Delta x$ with
respect to those on $y_{C\\!P}$ and $\Delta y$, respectively. In the limit of
a Dalitz-plot bin saturated by $C\\!P$-eigenstate amplitudes, where
$r_{b}\approx 1$ and $s_{b}\approx 0$, sensitivity to the mixing parameters
vanishes.

Table 3:  Values of $r_{b}$, $c_{b}$, and $s_{b}$ resulting from the BaBar 2010 amplitude model [11] used to generate the simulated experiments, with the BaBar 2008 iso-$\Delta\delta$ binning of the $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ Dalitz plot defined by CLEO [20]. $b$ | $r_{b}$ | $c_{b}$ | $s_{b}$
---|---|---|---
1 | $0.479784$ | $0.670828$ | $-0.032140$
2 | $0.221027$ | $0.635411$ | $0.395893$
3 | $0.276147$ | $0.087385$ | $0.850636$
4 | $0.678943$ | $-0.490907$ | $0.783871$
5 | $0.588435$ | $-0.946404$ | $0.113501$
6 | $0.239850$ | $-0.681781$ | $-0.453785$
7 | $0.107627$ | $-0.131118$ | $-0.813580$
8 | $0.208639$ | $0.381420$ | $-0.482809$

Section 4.1 allows an illustration of the bin-flip method through an analogy
with the wrong-sign-to-right-sign analysis of $D^{0}\rightarrow
K^{\mp}\pi^{\pm}$ decays [23, 1]. For the $D^{0}\rightarrow K^{\mp}\pi^{\pm}$
analysis, a similar ratio is obtained, but with parameters that correspond to
a single amplitude ratio rather than their averages over a Dalitz-plot bin.
Thus, $c_{b}$ and $s_{b}$ are replaced by $\cos\delta$ and $\sin\delta$,
respectively, while the replacement for $r_{b}$ is conventionally indicated as
$R_{D}$. Factors $(1\pm R_{D})$ are neglected since $R_{D}\ll 1$. The sign of
the term linear in $\langle t\rangle$ is also flipped, due to a difference in
the conventions to define $\delta$ that amounts to a shift of $\pi$. The
mixing effect in the bin-flip method can therefore be visualized as slopes in
the decay-time ($j$) dependences of $R_{bj}$ that are correlated between
Dalitz-plot bins ($b$), as shown in Figure 4. For bins where $r_{b}$
approaches 1, $s_{b}$ is large, and $c_{b}$ is small (e.g., bin 4 in Table 3),
the effect is mainly due to $x$; for bins where $r_{b}$ and $s_{b}$ are small,
but $c_{b}$ is large (e.g., bin 5), the effect is mainly due to $y$. Hence,
the observed slopes and the known values of $c_{b}$ and $s_{b}$ allow for
determining both $x$ and $y$, unlike in $D^{0}\rightarrow K^{\mp}\pi^{\pm}$
decays, where only a single combination of $x$ and $y$ is accessible (since
$\sin\delta$ is close to zero, the $D^{0}\rightarrow K^{\mp}\pi^{\pm}$
analysis is primarily sensitive to $y$). Contributions from potential
$C\\!P$-violation effects are inferred by comparing the slopes of the ratios
for mesons produced as $D^{0}$ or $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ separately.

Figure 4: Bin-flip ratio in each Dalitz-plot bin as a function of decay time, in the limit of $C\\!P$ symmetry and for two sets of mixing parameters. Ratios are computed from Equation 25 with hadronic parameters from Table 3. Colors identify the various Dalitz-plot bins following Figure 1. Table 4: Expected statistical uncertainties from $10^{6}$ simulated $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays generated in the WM scenario, and analyzed with different approaches assuming $C\\!P$ conservation and allowing only the mixing parameters to float in the fit. Analysis method |  $\sigma(x)$ [%] |  $\sigma(y)$ [%]
---|---|---
Model-dependent | $0.11$ | $0.10$
Standard model-independent | $0.20$ | $0.18$
Bin-flip model-independent | $0.15$ | $0.29$

In addition, we use the simulated samples to compare the performance of the
bin-flip method to those of existing approaches. The customary model-dependent
analysis implies a joint maximum-likelihood fit to the unbinned decay-time and
Dalitz-plot distributions, based on the same amplitude model used in
generation. The established model-independent analysis implies a joint
maximum-likelihood fit to the unbinned decay-time distributions of decays in
the 16 Dalitz-plot bins. While evaluating the performance of both standard
methods, we keep all parameters fixed except $x$ and $y$. Table 4 reports the
results. Predictably, when the underlying amplitude model is exactly known,
the model-dependent analysis offers the best sensitivity to both $x$ and $y$.
The bin-flip method provides better sensitivity to $x$ than the known model-
independent method at the price of reduced sensitivity to $y$. This is
expected because (i) the decay-time binning affects only marginally the
statistical precision with $\mathcal{O}(10^{6})$ signal yields or larger and
(ii) the coefficients of the terms associated with sensitivity to $x$ and $y$
are enhanced or suppressed by $(1+r_{b})$ and $(1-r_{b})$, respectively, in
the bin-flip method compared to the standard model-independent approach, as
shown in Section 4.1. By averaging over the Dalitz-plot bins, the coefficient
multiplying $x$ ($y$) in the bin-flip method becomes approximately $35\%$
larger (smaller) than that from the original model-independent method,
consistent with the sensitivities of Table 4.

Figure 5: Summary of results from simulated experiments of $10^{6}$
$D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays
generated in the five scenarios of $C\\!P$-conserving or $C\\!P$-violating
mixing, and fit (open points) assuming $C\\!P$ conservation or (closed points)
allowing for indirect $C\\!P$ violation. Only the mixing and $C\\!P$-violation
parameters float in the fits.

Figure 5 summarizes the results obtained in the various scenarios and for
either assumption on indirect $C\\!P$ violation. The estimates obtained with
the bin-flip method are unbiased and show proper statistical uncertainties for
all the relevant parameters, regardless of their true values. In addition, the
precision on the $C\\!P$-averaged mixing parameters does not depend on whether
the $C\\!P$-violation parameters are fixed or determined by the fit, which is
expected since $z_{C\\!P}$ and $\Delta z$ are additive, orthogonal parameters.
The customary multiplicative parametrization in terms of $z^{\pm}=z(q/p)^{\pm
1}$ yields larger correlations between mixing and indirect $C\\!P$-violation
parameters, which bias the estimators and induce non-Gaussian uncertainties
(Appendix A). Indeed, the uncertainties on $|q/p|$ and $\phi$ obtained with
the existing model-dependent method depend strongly on the estimated values of
mixing parameters $x$ and $y$ [10], which is undesirable, especially in
combinations of results. The uncertainties on the parameters set out in
Equations 27, 28, 29 and 30 do not depend on the central values of any of the
other parameters, thus showing better statistical properties.

Figure 6: Uncertainties on the mixing and $C\\!P$-violation parameters as
functions of the number of (top) decay-time and (bottom) pairs of Dalitz-plot
bins, as obtained from fitting simulated samples of $10^{6}$ $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays each generated in the WM
scenario. Only mixing and indirect $C\\!P$-violation parameters float in the
fits.

We study the dependence of our findings on bin multiplicity by repeating the
study with various choices for the number of decay-time bins and of pairs of
Dalitz-plot bins. In all tests we consider equipopulated decay-time bins and
iso-$\Delta\delta$ Dalitz-plot bins. Figure 6 shows no significant dependence
on the number of decay-time (pairs of Dalitz-plot) bins if that exceeds
approximately five (ten). Since all schemes investigated experimentally thus
far involve eight pairs of $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ Dalitz-plot bins [20], alternative schemes that are either
optimized for the bin-flip method or simply use more bins could result in
greater sensitivity.

###  4.2 Dependence on external inputs

The sensitivity studies of Section 4.1 assume the ideal case in which the
hadronic parameters $r_{b}$, $c_{b}$, and $s_{b}$ are known exactly. A more
realistic scenario, however, requires these parameters to be either determined
directly from the data, or constrained by external measurements. Since the
samples of $e^{+}e^{-}$ collisions at the $\psi(3770)$ resonance used to
determine the strong-interaction phase parameters are, and will be, smaller
than those used in upcoming measurements of charm-mixing parameters in
$D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays,
independent higher-precision measurements to constrain $r_{b}$ will be
lacking. Moreover, biases due to efficiency variations across the Dalitz plot
are likely to become non-negligible at the precision expected on $r_{b}$. It
is therefore convenient to leave $r_{b}$ as a free parameter in the fit. More
precisely, the free parameter is $\tilde{r}_{b}=\tilde{F}_{-b}/\tilde{F}_{b}$,
following Equation 10, and is treated as an effective, nuisance parameter that
has no straightforward interpretation in terms of the underlying amplitude
models. Conversely, since the mixing and $(c_{b},s_{b})$ parameters cannot
simultaneously be determined precisely without external inputs, an appropriate
binning scheme and care over the efficiency modeling is required to keep the
biases on $c_{b}$ and $s_{b}$ minimal (see Section 4.3). Therefore, the
optimal fit configuration for a realistic analysis corresponds to keeping
$r_{b}$ free to vary and $(c_{b},s_{b})$ constrained.

Table 5: Expected statistical sensitivities from $10^{6}$ simulated $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays generated in the WM scenario, and fit under various assumptions. The fit configuration that best approximates the conditions in a realistic analysis corresponds to allowing $r_{b}$ to float free and keeping $c_{b}$ and $s_{b}$ constrained (last row in each subpanel). Fit configuration |  $\sigma(x_{C\\!P})$ [%] |  $\sigma(y_{C\\!P})$ [%] |  $\sigma(\Delta x)$ [%] |  $\sigma(\Delta y)$ [%]
---|---|---|---|---
No $C\\!P$ violation |  |  |  |
$\quad r_{b}$, $(c_{b},s_{b})$ fixed | $0.15$ | $0.29$ | – | –
$\quad r_{b}$ free, $(c_{b},s_{b})$ fixed | $0.21$ | $0.41$ | – | –
$\quad r_{b}$ fixed, $(c_{b},s_{b})$ constrained | $0.16$ | $0.30$ | – | –
$\quad r_{b}$ free, $(c_{b},s_{b})$ constrained | $0.22$ | $0.43$ | – | –
Indirect $C\\!P$ violation allowed |  |  |  |
$\quad r_{b}$, $(c_{b},s_{b})$ fixed | $0.15$ | $0.29$ | $0.15$ | $0.29$
$\quad r_{b}$ free, $(c_{b},s_{b})$ fixed | $0.21$ | $0.41$ | $0.15$ | $0.29$
$\quad r_{b}$ fixed, $(c_{b},s_{b})$ constrained | $0.16$ | $0.30$ | $0.16$ | $0.31$
$\quad r_{b}$ free, $(c_{b},s_{b})$ constrained | $0.22$ | $0.43$ | $0.16$ | $0.31$

Table 5 shows the sensitivity of the uncertainties to the choice of fit
configuration (unconstrained or constrained) for the nuisance parameters
$r_{b}$, $c_{b}$, and $s_{b}$. The constraints on $(c_{b},s_{b})$ are
implemented by adding to Equation 32 the penalty term

$\chi^{2}_{X}=\sum_{a,b}\left[X^{\rm{gen}}_{a}-X_{a}\right](V_{\rm{CLEO}}^{-1})_{ab}\left[X^{\rm{gen}}_{b}-X_{b}\right],$
(34)

where $X_{b}^{\rm{gen}}$ are the generator-level values of Table 3 and the
covariance matrix $V_{\rm{CLEO}}$ is the sum of the statistical and systematic
covariance matrices from the CLEO measurement of $(c_{b},s_{b})$ derived from
the values reproduced in Tables 1 and 2 [20]. With $10^{6}$ $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays, the impact of the
current precision of measurements of $(c_{b},s_{b})$ is marginal. If $r_{b}$
is unconstrained in the fit, a more significant impact on $\sigma(x_{C\\!P})$
and $\sigma(y_{C\\!P})$ is expected whereas $\sigma(\Delta x)$ and
$\sigma(\Delta y)$ are unaffected.

To assess the impact of the limited precision of external constraints on
future larger samples of $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ decays, such as those expected at the LHCb and Belle II
experiments, the sensitivity is evaluated as a function of sample size. LHCb
is expected to collect about $5\times 10^{7}$ decays by the end of 2018 and at
least an order of magnitude more by 2030, after detector upgrades [12]. Belle
II is expected to collect about $1\times 10^{6}$ decays per 1 ab-1 of
integrated luminosity, for a total of about $5\times 10^{7}$ decays by the end
of 2025 [13]. Table 6 shows uncertainties on the oscillation and
$C\\!P$-violation parameters in charm mixing resulting from fits with
unconstrained $r_{b}$ parameters and $(c_{b},s_{b})$ either constrained or
fixed. The precision of currently available measurements of $(c_{b},s_{b})$
from CLEO will start impacting the precision on $x_{C\\!P}$ and $y_{C\\!P}$
with $10^{7}$ decays, but has negligible impact on the determination of the
$C\\!P$-violation parameters $\Delta x$ and $\Delta y$. However, more precise
inputs are expected owing to $\mathcal{O}(10)$ times larger data sets of
$e^{+}e^{-}$ collisions at center-of-mass energy of
$3.77\mathrm{\,Ge\kern-1.00006ptV}$ that are being collected with the BESIII
detector at the Beijing Electron-Positron Collider. It is therefore plausible
to expect that the uncertainty due to external inputs will reduce, mirroring
the reduction in statistical uncertainty and thus not limiting the precision
of the proposed method.

Table 6: Expected statistical uncertainties as functions of signal yields from fits to simulated $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays generated in the WM scenario, allowing $r_{b}$ to float freely and keeping $(c_{b},s_{b})$ constrained (fixed). The constraints are based on the uncertainties of the CLEO results [20]. Signal yield |  $\sigma(x_{C\\!P})$ [%] |  $\sigma(y_{C\\!P})$ [%] |  $\sigma(\Delta x)$ [%] |  $\sigma(\Delta y)$ [%]
---|---|---|---|---
$1\times 10^{6}$ | $0.22\phantom{0}\,(0.21\phantom{0})$ | $0.43\phantom{0}\,(0.41\phantom{0})$ | $0.16\phantom{0}\,(0.15\phantom{0})$ | $0.31\phantom{0}\,(0.29\phantom{0})$
$5\times 10^{6}$ | $0.10\phantom{0}\,(0.093)$ | $0.24\phantom{0}\,(0.19\phantom{0})$ | $0.068\,(0.065)$ | $0.16\phantom{0}\,(0.13\phantom{0})$
$1\times 10^{7}$ | $0.085\,(0.066)$ | $0.16\phantom{0}\,(0.13\phantom{0})$ | $0.048\,(0.046)$ | $0.095\,(0.091)$
$5\times 10^{7}$ | $0.047\,(0.030)$ | $0.120\,(0.059)$ | $0.021\,(0.021)$ | $0.041\,(0.041)$
$1\times 10^{8}$ | $0.043\,(0.021)$ | $0.091\,(0.042)$ | $0.015\,(0.015)$ | $0.028\,(0.028)$
$5\times 10^{8}$ | $0.034\,(0.009)$ | $0.091\,(0.018)$ | $0.006\,(0.006)$ | $0.013\,(0.013)$

The above analysis is carried out in the limit of $C\\!P$-symmetric $D$ decay
amplitudes. As larger data sets will become available, this assumption might
need to be revisited, possibly resulting in an extension of the method toward
including direct $C\\!P$ asymmetries as has been considered for the GGSZ
method [26]. We expect that doing so will enrich the physics reach of the
method without significantly affecting the sensitivity to oscillation and
indirect $C\\!P$ violation.

###  4.3 Effects of finite resolutions and nonuniform efficiencies

For the bin-flip method to be applicable to experimental data, effects such as
backgrounds, flavor tagging, finite resolutions, and nonuniform efficiency
variations across decay time and Dalitz plane need in principle to be
accounted for. Backgrounds and flavor tagging are not a significant
limitation. Using the $D^{*}(2010)^{+}\rightarrow D^{0}\pi^{+}$ decay chain
provides both very effective background rejection and a highly efficient and
pure identification of the initial $D$ meson flavor. Reconstruction effects
can also be accounted for, by weighting the candidates by the inverse of the
efficiency at a given point in phase space and decay time, for example.
However, the determination of the detector resolution and efficiency
variations often relies on an accurate simulation of the detector response,
which may introduce further unwanted sources of systematic uncertainties and
complexity in the analysis procedures.

The bin-flip method is constructed so as to be insensitive to such effects. To
validate this notion, we incorporate in the simulated samples realistic
resolution and efficiency effects based on publicly available information from
the LHCb and Belle II experiments, which are the environments where this
method is most likely to be considered. In both cases we consider experimental
effects typical of $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}(\rightarrow\pi^{+}\pi^{-})\pi^{+}\pi^{-}$ signal decays reconstructed from
$D^{*}(2010)^{+}\rightarrow D^{0}\pi^{+}$ decays. At LHCb, significant samples
of $D^{0}$ mesons are also obtained from semileptonic $B$-meson decays, with
online selection-requirements that induce less distortion of the kinematic and
decay-time distributions. Such samples can therefore provide results on charm-
mixing parameters complementary to those based on $D^{0}$ mesons produced at
the proton-proton primary interaction [27]. These are not considered in this
work.

Figure 7: Efficiency (normalized to unity at its maximum) as a function of
decay time assumed for $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ decays reconstructed from the $D^{*}(2010)^{+}\rightarrow
D^{0}\pi^{+}$ decay chain with the LHCb detector.

Figure 8: Efficiency (normalized to unity at its maximum) as a function of the
Dalitz plot location assumed for $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays reconstructed from the
$D^{*}(2010)^{+}\rightarrow D^{0}\pi^{+}$ decay chain with the (top) LHCb and
(bottom) Belle II detectors.

For LHCb, we assume a decay-time resolution corresponding to 10% of the
$D^{0}$ lifetime, the decay-time-dependent variation of the reconstruction
efficiency shown in Figure 7, and the efficiency variation over the Dalitz
plane of Figure 8, following Refs. [28, 29]. For Belle II, we assume a decay-
time resolution corresponding to 33% of the $D^{0}$ lifetime following Ref.
[13]. For convenience, we use the model of the efficiency variation over the
Dalitz plot derived for the BaBar detector, in Figure 8 [30]. This is unlikely
to be an accurate model for the Belle II efficiency, but it is sufficient for
the purpose of demonstrating the robustness of the method against
nonuniformities in efficiency, which ought not depend on the details of the
efficiency itself. We also assume that the Belle II reconstruction efficiency
is uniform as a function of decay time. For both LHCb and Belle II, the
resolutions on $m_{+}^{2}$ and $m_{-}^{2}$ are similar to or better than the
$0.0054\,\mbox{$\mathrm{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$}$ spacing used by
CLEO to define the Dalitz bins. Because such spacing is significantly smaller
than the typical size of the Dalitz bins, $m_{+}^{2}$ and $m_{-}^{2}$
resolutions are expected to introduce negligible bin migrations and are
therefore neglected.

Data are generated using the same amplitude model as for the previous studies.
The decay-time resolution is included by smearing the generated decay time
with a Gaussian distribution with a width of $0.1\tau$ ($0.33\tau$) for the
samples simulating LHCb (Belle II) conditions. The effects of selection
requirements on the decay-time and Dalitz-plot distributions are incorporated
by sampling the generated events according to the relevant parametrizations.
The analysis procedure is then repeated as previously described, without
modeling the resolution and efficiency effects in the fits.

Table 7: Biases ($B$), normalized to the statistical uncertainty ($\sigma$), due to neglecting efficiency and resolution effects expected at LHCb and Belle II, as functions of the number of events. We use simulated $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays generated in the WM scenario, and fit allowing $r_{b}$ to float freely and keeping $(c_{b},s_{b})$ constrained. The constraint assumes the current (improved) determination of the external measurements of $(c_{b},s_{b})$ for 1–$10\times 10^{6}$ (5–$50\times 10^{7}$) signal yields. Signal yield | $B/\sigma(x_{C\\!P})$ | $B/\sigma(y_{C\\!P})$ | $B/\sigma(\Delta x)$ | $B/\sigma(\Delta y)$
---|---|---|---|---
LHCb detector |  |  |  |
$\quad 1\times 10^{6}$ | $0.04$ | $0.02$ | $0.05$ | $0.02$
$\quad 5\times 10^{6}$ | $0.07$ | $0.07$ | $0.03$ | $0.10$
$\quad 1\times 10^{7}$ | $0.03$ | $0.09$ | $0.10$ | $0.09$
$\quad 5\times 10^{7}$ | $0.10$ | $0.05$ | $0.27$ | $0.15$
$\quad 1\times 10^{8}$ | $0.12$ | $0.09$ | $0.40$ | $0.16$
$\quad 5\times 10^{8}$ | $0.22$ | $0.10$ | $1.00$ | $0.42$
Belle II detector |  |  |  |
$\quad 1\times 10^{6}$ | $0.03$ | $0.06$ | $0.07$ | $0.05$
$\quad 5\times 10^{6}$ | $0.17$ | $0.06$ | $0.08$ | $0.06$
$\quad 1\times 10^{7}$ | $0.18$ | $0.03$ | $0.05$ | $0.03$
$\quad 5\times 10^{7}$ | $0.40$ | $0.07$ | $0.04$ | $0.03$

The fits are performed with unconstrained $r_{b}$ parameters and
$(c_{b},s_{b})$ parameters constrained. The constraint assumes that the
precision of the external measurements of $(c_{b},s_{b})$ is improved by a
factor two (four) at sample sizes of $5\times 10^{7}$ (1–$5\times 10^{8}$)
signal decays. Table 7 lists the magnitudes of the biases with respect to the
generated values, normalized to the fit uncertainties, as functions of sample
size. In the LHCb case, the observed biases are mostly due to neglecting
efficiency variations across the Dalitz plane. For Belle II, neglecting the
decay-time resolution dominates. As expected, the relative impact of small
constant biases becomes more significant as the statistical precision of the
measurements increases. The largest effect is observed for LHCb, with a
$\Delta x$ bias comparable with the statistical uncertainty in the highest
signal-yield scenario. All other biases do not exceed $40\%$ of the
statistical uncertainty.

These findings show that no accurate knowledge of the decay-time resolution or
efficiency variation as a function of decay time and Dalitz-plane position is
needed to apply the method. This supports the approach as an expedient and
powerful alternative to standard approaches for charm-mixing measurements
using $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and
other multibody decays in current and next generation analyses. Further
refinements will probably be needed to fully exploit the method at the very
high yields expected a decade from now in the final LHCb sample.

##  5 Impact on knowledge of charm-mixing parameters

To assess the impact of a bin-flip analysis on the current global knowledge of
oscillation and $C\\!P$-violation parameters in charm mixing, we compare the
precision of the current world-average determination of $x$, $y$, $\phi$, and
$|q/p|$, with the precision achievable when including a bin-flip analysis of
$1\times 10^{6}$, $5\times 10^{7}$, and $5\times 10^{8}$ $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays. Figure 9 shows the
results assuming unchanged central values, precision of bin-flip results
dominated by statistical uncertainties, and either current or improved
determination of the external measurements of $(c_{b},s_{b})$ parameters.

While the effect on $y$ is relatively minor, the bin-flip method is expected
to have a major impact in the determination of $x$ and of the
$C\\!P$-violation parameters. For instance, the comparison between the current
world-average constraints (blue region), with their update including bin-flip
results based on $10^{6}$ signal decays (orange region), offers a realistic
representation of the impact the bin-flip analysis could have if applied to
typical current LHCb samples. Consistently with Table 6, the precision of the
external inputs has negligible impact on the determination of the
$C\\!P$-violation parameters $|q/p|$ and $\phi$ but will strongly enhance the
reach in $x$ and $y$ when larger samples will be analyzed.

We finally emphasize that the alternative additive parametrization proposed
for the effects of charm mixing offers superior statistical properties to
standard parametrizations and is particularly preferable for combinations, in
which central values cannot be assumed to be known.

Figure 9: Confidence regions at the (inner, darker hatching) 68.3% and (outer,
lighter hatching) 95.5% confidence level in the two-dimensional space of
(left) oscillation parameters ($x$, $y$) and (right) parameters of $C\\!P$
violation in mixing and in the interference between mixing and decay
($|q/p|-1$, $\phi$) corresponding to (blue) current world-average results and
to those results updated to include a bin-flip analysis of (orange) $1\times
10^{6}$, (magenta) $5\times 10^{7}$, and (green) $5\times 10^{8}$ signal
decays. Top panels refer to results based on current CLEO inputs on
$(c_{b},s_{b})$ parameters; bottom panels on improved $(c_{b},s_{b})$ inputs.
The displayed regions assume unchanged central values and precision of bin-
flip results dominated by statistical uncertainties.

##  6 Conclusions

In summary, we propose the bin-flip method, a model-independent approach to
measure parameters of mixing and $C\\!P$ violation in charm from multibody
decays, such as $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$. The method avoids the need for accurate knowledge of either
the decay-time resolution or the signal-reconstruction efficiency as a
function of decay time and position in the Dalitz plot. We also introduce a
novel parametrization of the oscillation and $C\\!P$-violation effects in
charm mixing that has attractive statistical properties and may find wider
applicability.

The bin-flip method offers 35% better statistical sensitivity, compared to
existing model-independent methods, to $C\\!P$-averaged and $C\\!P$-violating
quantities related to the mass difference between the neutral $D$ eigenstates,
while suppressing systematic effects due to nonuniform efficiencies in decay
time and across the Dalitz plane. In addition to the gain in precision, the
demonstrated insensitivity to the details of Dalitz-plot and decay-time
modeling make the application of this method significantly simpler and more
expedient than other model-independent approaches, especially in hadron-
collision experiments.

The bin-flip method is expected to offer good sensitivity in high-yield
multibody decays that receive large contributions from doubly Cabibbo-
suppressed amplitudes. In addition to $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays, it is likely to benefit
the analysis of $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}\pi^{0}$ decays, for which first measurements of the relevant
hadronic parameters have recently become available [31]. The bin-flip method
can also, with straightforward modifications to the formalism, be used with
decays to non-self-conjugate final states such as $D^{0}\rightarrow
K^{\mp}\pi^{\pm}\pi^{0}$ and $D^{0}\rightarrow
K^{\mp}\pi^{\pm}\pi^{+}\pi^{-}$. Conversely, the sensitivity is reduced in
channels where $C\\!P$-eigenstate amplitudes dominate in many of the Dalitz-
plot bins, such as $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}$, $D^{0}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ and $D^{0}\rightarrow
K^{+}K^{-}\pi^{0}$ [32].

A bin-flip analysis of the samples of $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays expected to be collected
at the LHCb or Belle II experiments has the potential to significantly improve
the global knowledge of the charm-mixing parameters and yield more stringent
constraints on $C\\!P$ violation in charm oscillations. The method is expected
to avoid limiting systematic uncertainties even with very large data samples,
when improved knowledge of the hadronic $(c_{b},s_{b})$ parameters from
independent measurements will help to achieve even better precision.

## Acknowledgements

We are grateful to Andrea Contu and Michal Kreps for earlier involvement in
this work, and Jolanta Brodzicka for fruitful discussions and valuable
comments. TG and NJ acknowledge support from the Science and Technology
Facilities Council (United Kingdom). TG and TP acknowledge support from the
European Research Council under FP7.

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##  Appendix A An alternative parametrization of $C\\!P$ violation in charm
mixing

In Section 2, we introduced a new parametrization of charm-mixing effects
expressed as functions of the additive parameters $z_{C\\!P}$ and $\Delta z$,
defined by

$z_{C\\!P}\pm\Delta z\equiv\left(q/p\right)^{\pm 1}z,$ (35)

in terms of the conventional multiplicative parameters $z$ and $q/p$. The
proposed parametrization offers nontrivial advantages in the determination of
parameters from fits to data.

Fits suffer from non-Gaussian estimator distributions when the dimensionality
of the likelihood or least-squares function depends on the estimated value of
one or more parameters. This may happen if all terms sensitive to a parameter
of interest involve products with another parameter, or a function of it, that
can vanish. The likelihood then becomes scarcely sensitive to the parameter of
interest for vanishing values of the multiplication factor, incurring in non-
Gaussian estimator distributions. A multiplicative parametrization as
$(q/p)^{\pm 1}z$ is prone to such effects, as shown using simulated
experiments in the WM scenario in Figure 10: $|q/p|$ pulls are non-Gaussian
and the dispersion of the $\phi$ residual depends on the observed mixing rate.
These issues are avoided when using our parametrization in terms of
$z_{C\\!P}$ and $\Delta z$, as shown in Section 4.

Figure 10: Distributions of (left) fit pull for the $|q/p|$ parameter and
(right) fit residual for the $\phi$ parameter as functions of the observed
mixing rate $x^{2}+y^{2}$.