11institutetext:  Computer Science and Engineering, New York University,
Brooklyn, NY, USA. 11email: {shijie.li<EMAIL_ADDRESS>22institutetext:
Ophthalmology, University Hospital Bonn, Bonn, Germany

# Improved Counting and Localization from Density Maps for Object Detection in
2D and 3D Microscopy Imaging

Shijie Li 11 Thomas Ach 22 Guido Gerig 11

###### Abstract

Object counting and localization are key steps for quantitative analysis in
large-scale microscopy applications. This procedure becomes challenging when
target objects are overlapping, are densely clustered, and/or present fuzzy
boundaries. Previous methods producing density maps based on deep learning
have reached a high level of accuracy for object counting by assuming that
object counting is equivalent to the integration of the density map. However,
this model fails when objects show significant overlap regarding accurate
localization. We propose an alternative method to count and localize objects
from the density map to overcome this limitation. Our procedure includes the
following three key aspects: 1) Proposing a new counting method based on the
statistical properties of the density map, 2) optimizing the counting results
for those objects which are well-detected based on the proposed counting
method, and 3) improving localization of poorly detected objects using the
proposed counting method as prior information. Validation includes processing
of microscopy data with known ground truth and comparison with other models
that use conventional processing of the density map. Our results show improved
performance in counting and localization of objects in 2D and 3D microscopy
data. Furthermore, the proposed method is generic, considering various
applications that rely on the density map approach. Our code will be released
post-review.

Figure 1: Left to right: the input image, its ”ground truth” density map, and
the predicted map produced by [3]. The areas highlighted by the  red, green
and  blue boxes contain overlapping objects which typically cause localization
problems.

##  1 Introduction

Object counting and localization techniques are important processing steps for
highly automated quantitative analysis of large-scale microscopy images driven
by biological and medical applications. The main aim is the prediction of the
number of target objects such as cells, nuclei, granules and also accurate
localization of these objects for subsequent semantic segmentation. Such tasks
are often based on simple point annotations, which take significantly less
effort compared to precise outlining of sought objects. The use of point
annotation becomes particularly important when objects are densely clustered,
but also for object analysis from 3D imagery where manual outlining may not be
feasible.

Supervised object counting and localization methods fall into two categories:
counting [3, 6, 9, 10, 13, 15, 17] and detection emphasized [2, 11, 16]. The
former faces limitations with overlapping objects in regard to counting
performance. Modern crowd counting methods are based on the training of a
regression method that maps images to density maps, the integration of which
is shown to be equivalent to the number of crowd objects. Placing a Gaussian
kernel for each annotated dot is the most common approach to generate the
density map. Lempitsky et al. [6] train a linear transform from the feature
space to the density map space. Others [17, 7, 15] proposed convolutional
neural networks as feature extractors. Cohen et al. [10] generate the density
map with a square kernel instead of a Gaussian kernel. Improving the objective
function for optimization was proposed by [9, 13]. Recent work by Guo et al.
[3] shows superior counting performance on various cell datasets. To localize
the objects, they use the connected components after thresholding or take
local maxima. However, the evaluation of localization performance in [3]
indicates that these postprocessing steps fail to predict the center of the
object and tend to produce false positives when occlusion occurs, and do not
seem to be robust enough to background noise. Furthermore, those methods are
sensitive to hyperparameters such as threshold values.

In this paper, we propose a counting method that automatically determines the
threshold to estimate the number of objects in a connected component. We use
Gaussian mixture fitting to decode this information from the predicted density
map. These two methods result in a postprocessing approach of the density map
without the need for any additional hyperparameters. We validate our approach
on multiple 2D and 3D datasets and demonstrate that the proposed method
provides improved density maps and increases counting and localization
performance in comparison to previous baselines.

##  2 Density Map Analysis

Overview In this section, we briefly summarize the basic idea of the density
map (DM) approach. We then provide a mathematical analysis of the issues of
density maps generated with Gaussian kernels. Density map methods assume that
$N$ images
$\mathbf{I}_{1},\mathbf{I}_{2},\dots,\mathbf{I}_{N}\in\mathbb{R}^{H\times
W\times C}$ and their corresponding annotations which are N sets of 2D/3D
points $P_{1},P_{2},\dots,P_{N}$ are provided. For each point annotation set
$P_{i}$ we generate a density map $\mathbf{D}_{i}\in\mathbb{R}^{H\times
W}_{>0}$ as follows,

$\mathbf{D}_{i}=\sum_{j=1}^{C_{i}}\mathbf{D}_{i}^{j},\quad\mathbf{D}_{i}^{j}(\mathbf{x})=\mathcal{N}(\mathbf{x};\mathbf{p}_{i}^{j},\mathbf{\Sigma})$
(1)

where $C_{i}\in\mathbb{Z}$ is the number of objects in the image
$\mathbf{I}_{i}$, $\mathbf{p}_{i}^{j}\in P_{i}$, $\mathbf{x}$ is the pixel
location, and $\mathbf{\Sigma}$ is the covariance matrix often defined related
to the object size and the image isotropy. The learning step of the DM method
is a training of a regressor mapping from the image $\mathbf{I}_{i}$ to the
density map $\mathbf{D}_{i}$. During testing, the counting prediction
$\hat{C}_{i}=\sum_{\mathbf{x}}\mathbf{\hat{D}}_{i}$ is computed, where
$\mathbf{\hat{D}}_{i}$ is the predicted density.

Decomposition of the Density Map Localizing of objects from the density map is
based on a decomposition of the predicted density map by solving,

$\min\quad\|\hat{\mathbf{D}}_{i}-\sum_{j=1}^{\hat{C}_{i}}\hat{\mathbf{D}}_{i}^{j}\|,\quad\textrm{s.t.}\quad\sum_{\mathbf{x}}\hat{\mathbf{D}}_{i}^{j}=1.$
(2)

Applying connected component analysis after thresholding the density map is
one simple but effective way to solve this problem. However, it fails to take
additional constraints into consideration, which might have the risk of
undercounting several objects merged into one or overcount when assigning
background noise to an object. Furthermore, a particular threshold value has
to be tuned manually during the training stage individually for different
datasets. Therefore, we are looking for a way to determine the threshold
automatically and to estimate the number of objects in connected components.
We make the following assumptions:

  1. 1.

A thresholding perfectly decomposes the density map. That is to say,
$\mathbf{1}_{\hat{\mathbf{D}}_{i}>T}=\sum_{j=1}^{\hat{C}_{i}}\mathbf{1}_{\hat{\mathbf{D}}_{i}^{j}>T},$
(3) $\hat{\mathbf{D}}_{i}=\sum_{j=1}^{\hat{C}_{i}}\hat{\mathbf{D}}_{i}^{j}.$
(4)

  2. 2.

Each decomposed component is the same as Eq. 1 with the same
$\mathbf{\Sigma}$.

Multiplying Eq. 3 with Eq. 4 , we then get
$\mathbf{1}_{\hat{\mathbf{D}}_{i}>T}\hat{\mathbf{D}}_{i}=\sum_{j=1}^{\hat{C}_{i}}\mathbf{1}_{\hat{\mathbf{D}}_{i}^{j}>T}\sum_{j=1}^{\hat{C}_{i}}\hat{\mathbf{D}}_{i}^{j}$.
According to Assumption 1, we can have
$\mathbf{1}_{\hat{\mathbf{D}}_{i}^{j}>T}\hat{\mathbf{D}}_{i}^{k}\approx\mathbf{0}$
if $j\neq k$. Eq. 2 can then be written as
$\mathbf{1}_{\hat{\mathbf{D}}_{i}>T}\hat{\mathbf{D}}_{i}=\sum_{j=1}^{\hat{C}_{i}}\mathbf{1}_{\hat{\mathbf{D}}_{i}^{j}>T}\hat{\mathbf{D}}_{i}^{j}$.
Following Assumption 2, we can get
$\sum_{\mathbf{x}}\mathbf{1}_{\hat{\mathbf{D}}_{i}^{j}>T}\hat{\mathbf{D}}_{i}^{j}=F(r_{T})$,
where $r_{T}$ is the Mahalanobis distance satisfying
$\mathcal{N}(r_{T},\mathbf{p_{i}^{j}},\mathbf{\Sigma})=T$. Then the
optimization problem Eq. 2 is equivalent to

$\displaystyle\min\quad$
$\displaystyle\|\mathbf{1}_{\hat{\mathbf{D}}_{i}>T}\hat{\mathbf{D}}_{i}-\sum_{j=1}^{\hat{C}_{i}}\mathbf{1}_{\hat{\mathbf{D}}_{i}^{j}>T}\hat{\mathbf{D}}_{i}^{j}\|,$
(5) s.t.
$\displaystyle\sum_{\mathbf{x}}\mathbf{1}_{\hat{\mathbf{D}}_{i}^{j}>T}\hat{\mathbf{D}}_{i}^{j}/F(r_{T})=1.$
(6)

This indicates that if we choose the right threshold, Eq. 6 should hold.

Figure 2: Overall pipeline: Our method contains two modules to be applied at
the testing stage: 1) a counting method, and 2) center estimation. The
threshold value will be determined automatically to identify the number of
objects inside a connected component. The center estimation will localize the
object center according to the number of objects in the connected component.
Section 3 will discuss the proposed counting and localization procedures in
detail.

##  3 Methodology

In the previous section, we reframed the localization problem into a modified
density map decomposition problem when two assumptions hold. In real
applications, there are two situations that may break these assumptions. For
the convenience of illustration, we name the value on the left side of Eq. 6
the count of the connected component.

###  3.1 Regression Noise

Problem Statement While training the density map regressor, there is no
constraint to assure Assumption 2 to hold. This means that even though we may
choose the right threshold and the decomposition satisfies Assumption 1, the
count of a connected component might be slightly greater or less than one. We
call this ”regression noise”. In addition, density maps may depict small peaks
due to background noise.
Solution We observe that the connected component count will be around 1 if a
single object exists and 0 for no objects. A straightforward remedy would be
to round the count of connected component between 0.5 and 1.5 to 1, for
example. The connected components in green from Figure 2 are identified as
containing one object, while those in the red box no object.

###  3.2 Imperfect Object Partition

Problem Statement In real cases, due to clustering, overlapping objects and
fuzzy edges, individual components in Eq. 4 are so close that it is hard to
automatically find an optimal threshold, or a threshold for object separation
may even not exist. In these situations, the Assumption 1 does not hold.

Solution There are two clues that will be used to approach problems with
localization. For areas with multiple nearby objects, counting based on
connected component will give us a good counting estimate, and second, centers
of objects will most likely show peaks in the density map. With these two
observations, we a apply similar method as in [14] where we first normalize
the density within the connected component area such that we can treat it as a
probability distribution. We then perform Monte Carlo Sampling to generate
random samples. Finally, we apply a Gaussian mixture model to fit the
distribution. Centers of objects are then the fitted means of the Gaussian
mixture.

Dataset | Method | MAE$(\downarrow)$ | Precision$(\uparrow)$ | Recall$(\uparrow)$ | F-measure$(\uparrow)$
---|---|---|---|---|---
VGG | IoDM | $\mathbf{2.05(\pm 0.09)}$ | - | - | -
CCA-T | $17.18(\pm 1.37)$ | $\mathbf{99.99(\pm 0.00)}$ | $91.20(\pm 0.47)$ | $95.36(\pm 0.26)$
DMA (ours) | $2.94(\pm 0.47)$ | $98.32(\pm 0.21)$ | $\mathbf{96.63(\pm 0.24)}$ | $\mathbf{97.46(\pm 0.18)}$
Ellipse | IoDM | $5.02(\pm 0.44)$ | $-$ | $-$ | $-$
CCA-T | $6.48(\pm 0.96)$ | $\mathbf{98.74(\pm 0.16)}$ | $95.77(\pm 0.5)$ | $97.20(\pm 0.27)$
DMA (ours) | $\mathbf{4.68(\pm 0.59)}$ | $98.15(\pm 0.43)$ | $\mathbf{96.48(\pm 0.48)}$ | $\mathbf{97.29(\pm 0.30)}$
MBM | IoDM | $6.56(\pm 1.42)$ | - | - | -
CCA-T | $\mathbf{6.02(\pm 1.47)}$ | $\mathbf{90.18(\pm 0.86)}$ | $\mathbf{90.97(\pm 1.30)}$ | $\mathbf{90.49(\pm 0.69)}$
DMA (ours) | $6.23(\pm 1.12)$ | $89.55(\pm 1.19)$ | $89.87(\pm 1.63)$ | $89.63(\pm 0.86)$
ADI | IoDM | $\mathbf{12.72(\pm 1.75)}$ | - | - | -
CCA-T | $13.90(\pm 1.43)$ | $87.60(\pm 1.02)$ | $90.47(\pm 1.37)$ | $88.62(\pm 1.11)$
DMA (ours) | $13.45(\pm 2.05)$ | $\mathbf{90.76(\pm 1.30)}$ | $\mathbf{88.56(\pm 1.69)}$ | $\mathbf{89.32(\pm 1.19)}$
DCC | IoDM | $3.12(\pm 0.39)$ | - | - | -
CCA-T | $3.17(\pm 0.53)$ | $\mathbf{90.75(\pm 1.11)}$ | $90.37(\pm 3.60)$ | $89.19(\pm 3.5)$
DMA (ours) | $\mathbf{2.83(\pm 0.30)}$ | $89.53(\pm 2.05)$ | $\mathbf{90.53(\pm 2.11)}$ | $\mathbf{89.28(\pm 1.75)}$
BBBC027 (3D) | IoDM | $\mathbf{13.58(\pm 4.54)}$ | - | - | -
CCA-T | $87.18(\pm 55.10)$ | $\mathbf{97.25(\pm 2.55)}$ | $79.25(\pm 12.80)$ | $\mathbf{86.80(\pm 10.00)}$
DMA (ours) | $13.63(\pm 4.91)$ | $85.99(\pm 7.20)$ | $\mathbf{85.35(\pm 6.23)}$ | $85.63(\pm 6.70)$
SIM (3D) | IoDM | $67.00(\pm 45.33)$ | - | - | -
CCA-T | $161.36(\pm 61.90)$ | - | - | -
DMA | $\mathbf{64.58(\pm 48.11)}$ | - | - | -

Table 1: Quantitative results of different methods on various test datasets.
($\uparrow$) means larger is better. $(\downarrow)$ means smaller is better.

Figure 3: Illustration of the model’s capability to localize objects while
also coping with strongly overlapping structures, displayed for five different
datasets (columns). Circles represent the ground truth of object locations,
and red stars mark the predicted object locations. Boxes mark locations where
method’s differ. Histograms (bottom row) show posterior probability
distributions of predicted versus ground truth counts calculated from the
testing split of the datasets.

##  4 Experiments

Here, we introduce datasets used for verification, describe experimental
configuration and evaluation metrics, and show qualitative and quantitative
results. As we mentioned in sections 1 and 2, commonly used methods to count
and localize are integration of density map (IoDM) and connected component
analysis with threshold (CCA-T). To evaluate counting performance, we compare
results of our proposed density map analysis (DMA) with those of the other
methods when applied to different synthetic and real datasets.

###  4.1 Datasets

For a full comparison, all the 2D datasets as also used in [3] are included
for evaluation. Furthermore, we generate and use synthetic data where we can
control occlusions, applying the tool from [5]. We will also apply procedures
to a volumetric Structural Illumination Microscopy (SIM) dataset to evaluate
the method’s performance in a clinical 3D application.

VGG Simulated bacterial cells in fluorescence-light microscopy image using
[5]. There are 200 $256\times 256$ images with $176(\pm 61)$ cells in each
image. Synthetic Ellipse Dataset By adding an eccentricity parameter to [5],
we generate a dataset containing object shape with eccentricities from $0$ to
$0.85$. For each image, we allow a proportion $p$ of objects to overlap, where
$p\sim U[0,0.5]$. MBM The dataset was first introduced by [4] containing
eleven $1200\times 1200$ images of healthy human bone marrow from eight
different patients. The center point of the cell nuclei has been annotated.
[10] further crop it into 44 $600\times 600$ images with $126\pm 33$ cell
nuclei in each image. DCC The Dublin Cell Counting datasets is quite
challenging. It contains 177 images of stem cells from various tissues and
species shot by different devices. Data therefore shows large variability in
regard to the density, morphology, size and artifacts. The number of cells
varies from 0 to 101. ADI We use images of human subcutaneous adipose tissue
from [8]. [10] cropped 200 regions of size $1700\times 1700$ and downsampled
them to $150\times 150$. There are $165\pm 44$ cells on average in each image.
BBBC027 We used image set with low SNR in the BBBC027[12] dataset for 3D
validation. It synthesized 3D digital phantoms in human colon tissue. The
weight center of the instance label is calculated as the point annotation. SIM
RPE cells were imaged with super-resolution structured illumination microscopy
(SIM) as part of a recent ophthalmological study [1]. Our biomedical partners
localized each organelle per cell with point annotations. There are 420 images
in total. There are 424 objects on average in each image.

###  4.2 Experimental setup

We set up four experiments for each dataset. We first train with similar steps
as proposed in [3]111The original tensorflow version is available at
https://github.com/mzlr/sau-net.git. We reproduce results with pytorch. and
select the best models with the smallest counting mean square error based on
IoDM and DMA separately. We then compare the counting performance with no
postprocessing (IoDM), thresholding and connected-component analysis (CCA-T)
and our density map analysis method (DMA). We find the best threshold of CCA-T
during the validation step for each dataset. The image size, batch size, data
augmentation, selection of the $\mathbf{\Sigma}$ in Eq. 1 and the scaling
coefficient of the loss function are the same as [3]. To rule out randomness
of results, we run 10 different data splits for each dataset and average the
evaluation metrics. All experiments are carried out on a desktop computer with
an AMD Ryzen 9 5950X desktop processor and a single NVIDIA GeForce RTX 3080
GPU. Each experiment takes about one hour to complete.

###  4.3 Evaluation Metrics

We use Precision, Recall and F-measure to evaluate the model performance of
localization, and MAE for the counting ability. A proposed object center is
True Positive (TP) if and only if its distance to the nearest annotation
points is less than a threshold (8 for MBM and 5 for the rest following [2]).
Each proposed point can only be assigned to one annotation point. Those
annotation points without a corresponding proposal will be counted as False
Negatives (FN). Proposals that are not True Positives are labeled as False
Positives (FP).

###  4.4 Results and Analysis

Figure 3 shows results for qualitative evaluation of our method on
localization performance, highlighting that our method can distinguish
strongly overlapping objects while also eliminating regression noise in those
areas where the CCA-T method fails. Table 3.2 lists quantitative results. All
methods present relatively high precision regarding localization among
different datasets. But the proposed method generally has lower mean absolute
counting error and higher recall and $F_{1}$ score especially in those
datasets with severe object overlaps (VGG, Ellipse). This indicates that our
method has better performance in counting and is producing less false
positives of object localization in areas of clustered objects.

##  5 Conclusion

Crowd object counting and localization are important steps for quantitative
analysis in large-scale microscopy applications. The consistency between
counting and localization is also important for manual correction in certain
applications where no ground-truth is available. In this work, we propose a
counting method applied to deep network predicted density maps leveraging the
property of multivariate normal distributions. Validation and comparison to
previously proposed methods demonstrate improved performance in localization -
an aspect which may also be important for subsequent semantic segmentation.
Analysis also shows improvements in 3D data where there is rapidly growing
interest for automated analysis. Furthermore, our comparisons also reveal
improvement of object counting results. The proposed processing is generic in
view of crowd object counting applications based on density maps. Future work
will tackle the still existing problem of incorrect detection produced by the
regression model.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2021-099 LHCb-PAPER-2021-009 September 14, 2021

Observation of the mass difference between neutral charm-meson eigenstates

LHCb collaboration†††Authors are listed at the end of this paper.

A measurement of mixing and $C\\!P$ violation in neutral charm mesons is
performed using data reconstructed in proton–proton collisions collected by
the LHCb experiment from 2016 to 2018, corresponding to an integrated
luminosity of 5.4$\text{\,fb}^{-1}$. A total of $30.6$ million
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays are
analyzed using a method optimized for the measurement of the mass difference
between neutral charm-meson eigenstates. Allowing for $C\\!P$ violation in
mixing and in the interference between mixing and decay, the mass and decay-
width differences are measured to be $x_{{C\\!P}}=\left[3.97\pm
0.46\text{\,(stat)}\pm 0.29\text{\,(syst)}\right]\times 10^{-3}$ and
$y_{{C\\!P}}=\left[4.59\pm 1.20\text{\,(stat)}\pm
0.85\text{\,(syst)}\right]\times 10^{-3}$, respectively. The $C\\!P$-violating
parameters are measured as $\Delta x=\left[-0.27\pm 0.18\text{\,(stat)}\pm
0.01\text{\,(syst)}\right]\times 10^{-3}$ and $\Delta y=\left[0.20\pm
0.36\text{\,(stat)}\pm 0.13\text{\,(syst)}\right]\times 10^{-3}$. This is the
first observation of a nonzero mass difference in the ${D}^{0}$ meson system,
with a significance exceeding seven standard deviations. The data are
consistent with $C\\!P$ symmetry, and improve existing constraints on the
associated parameters.

Published in Phys. Rev. Lett. 127 (2021) 111801

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

Neutral charm mesons propagating freely can change (oscillate) into their own
antiparticles, as the mass eigenstates are linear combinations of the flavor
eigenstates. These flavor-changing neutral currents do not occur at tree level
in the Standard Model (SM) and allow for hypothetical particles of arbitrarily
high mass to contribute significantly to the process. This can affect the
mixing of mesons and antimesons such that measurements of these processes can
probe physics beyond the SM [1, 2, 3, 4].

The mass eigenstates of charm mesons can be written as $|D_{1,2}\rangle\equiv
p|{{D}^{0}}\rangle\pm q|{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rangle$, where $p$ and $q$ are
complex parameters and, in the limit of charge-parity ($C\\!P$) symmetry,
$|D_{1}\rangle$ ($|D_{2}\rangle$) is defined as the $C\\!P$ even (odd)
eigenstate. Mixing of flavor eigenstates is described by the dimensionless
parameters $x\equiv(m_{1}-m_{2})c^{2}/\Gamma$ and
$y\equiv(\Gamma_{1}-\Gamma_{2})/(2\Gamma)$, where $m_{1(2)}$ and
$\Gamma_{1(2)}$ are the mass and decay width of the $D_{1(2)}$ state,
respectively, and $\Gamma$ is the average decay width [5]. In ${D}^{0}$ and
${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ decays to a common final
state, $f$, $C\\!P$ violation in mixing manifests itself if $|q/p|\neq 1$ or
in the interference between mixing and decay if
$\phi_{f}\equiv\arg(q\bar{A}_{f}/pA_{f})\neq 0$. Here $A_{f}$ ($\bar{A}_{f}$)
denotes the amplitude of the decay process ${{D}^{0}}\rightarrow f$ (${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow f$). In the
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decay
studied in this Letter, $C\\!P$ violation in the decay
($|A_{f}|^{2}\neq|\bar{A}_{f}|^{2}$) is not considered, as in the SM it is
negligible for the doubly Cabibbo-suppressed (DCS) and Cabibbo-favored (CF)
amplitudes contributing to this process. With this assumption, the
$C\\!P$-violating phase is independent of the final state,
$\phi_{f}\approx\phi\approx\arg(q/p)$ [6, 7].

The current world average of the mixing and $C\\!P$-violating parameters
yields $x=(3.7\pm 1.2)\times 10^{-3}$, $y=(6.8\,^{+\,0.6}_{-\,0.7})\times
10^{-3}$, $|q/p|=0.951\,^{+\,0.053}_{-\,0.042}$, and
$\phi=-0.092\,^{+\,0.085}_{-\,0.079}$ [8]. Measurements using decays such as
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ have resulted in precise
measurements of $y$ and have allowed for the observation of mixing [9, 10].
However, the data remain marginally compatible with $x=0$, and are consistent
with $C\\!P$ symmetry. Theoretical predictions for the mixing parameters are
of similar magnitude but less precise [11, 12], while predictions of the
$C\\!P$-violating phase are around 0.002 [13] and are well below the current
experimental precision.

Figure 1: Depiction of the interference of mixing and decay if a ${{D}^{0}}$
and a ${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ meson decay to a
common final state $f$.

Sensitivity to the mixing and $C\\!P$-violating parameters is offered by the
self-conjugate, multibody
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decay [14,
15, 16, 17, 18]. Inclusion of the charge-conjugate process is implied unless
stated otherwise. This final state is accessible in both ${{D}^{0}}$ and
${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ decays and leads to
interference between the mixing and decay amplitudes, as demonstrated
pictorially in Fig. 1. The dynamics of the decay are expressed as a function
of two invariant masses following the Dalitz-plot formalism, in which a three-
body decay is parametrized by a pair of two-body invariant masses [19, 20].
The squared invariant mass $m^{2}({{K}^{0}_{\mathrm{S}}}\pi^{\pm})$ is denoted
as $m_{\pm}^{2}$ for ${{D}^{0}}$ decays and $m_{\mp}^{2}$ for ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ decays. A mixture of DCS and CF
decay amplitudes results in large variations of the strong phase and, with
mixing, causes a decay-time evolution of the density of decays across the
phase space. A joint analysis of the Dalitz-plot and decay-time distributions
may be used to determine the mixing parameters. Splitting the sample by flavor
of the charm meson at production probes for $C\\!P$-violating effects. Usage
of multibody decay modes is typically challenging, as it requires knowledge of
the variation of the hadronic parameters and excellent control of
efficiencies, resolutions, and background effects.

This Letter reports on a measurement of the mixing and $C\\!P$ violation
parameters in
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays
using the “bin-flip” method [21], a model-independent approach which obviates
the need for detailed models of the efficiency, resolution, and contributing
amplitudes. Mixing and $C\\!P$ violation are parametrized by $z_{{C\\!P}}$ and
$\Delta z$, which are defined by $z_{{C\\!P}}\pm\Delta
z\equiv-\left(q/p\right)^{\pm 1}(y+ix)$. The results are expressed in terms of
the $C\\!P$-even mixing parameters
$x_{{C\\!P}}\equiv-{{\rm\,Im}\\!\left(z_{{C\\!P}}\right)}$ and
$y_{{C\\!P}}\equiv-{{\rm\,Re}\\!\left(z_{{C\\!P}}\right)}$, and of the
$C\\!P$-violating differences $\Delta x\equiv-{{\rm\,Im}\\!\left(\Delta
z\right)}$ and $\Delta y\equiv-{{\rm\,Re}\\!\left(\Delta z\right)}$.
Conservation of $C\\!P$ symmetry implies $x_{{C\\!P}}=x$, $y_{{C\\!P}}=y$, and
$\Delta x=\Delta y=0$. The method has already been employed by the LHCb
collaboration, yielding the single most precise measurement of $x_{{C\\!P}}$
and $\Delta x$ [18].

In the bin-flip method, data are partitioned into disjoint regions (bins) of
the Dalitz plot, which are defined to preserve nearly constant strong-phase
differences $\Delta\delta(m_{-}^{2},m_{+}^{2})$ between the ${D}^{0}$ and
${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ amplitudes within each
bin [22]. Two sets of eight bins are formed symmetrically about the
$m_{+}^{2}=m_{-}^{2}$ bisector, as illustrated in Fig. 2. The region
satisfying $m_{+}^{2}>m_{-}^{2}$, which includes regions dominated by the CF
${{D}^{0}}\\!\rightarrow{{K}^{*}}(892)^{-}\pi^{+}$ decay, is given a positive
index $+b$, while the opposite region, where the relative contribution from
decays following an oscillation is enhanced, is given a negative index $-b$.
The data are further split into 13 bins of decay time, chosen such that the
bins are approximately equally populated. The squared-mass and decay-time
resolutions are typically $0.006$$\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$
and $60\text{\,fs}$, respectively, which are smaller than the bin sizes used.
Thus, they are neglected and accounted for in the systematic uncertainties.

Figure 2: “Binning” of the
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ Dalitz
plot. Colors indicate the absolute value of the bin index $b$.

For each decay-time interval ($j$), the ratio of the number of decays in each
negative Dalitz-plot bin ($-b$) to its positive counterpart ($+b$) is
measured. The usage of ratios minimizes the need for precise modeling of the
efficiency variation across phase space and decay time. For small mixing
parameters and $C\\!P$-conserving decay amplitudes, the expected ratios for
initially produced ${D}^{0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$) mesons, $R_{bj}^{+}$
($R_{bj}^{-}$), are [21]

$R_{bj}^{\pm}\approx\frac{r_{b}+r_{b}\dfrac{\langle
t^{2}\rangle_{j}}{4}{{\rm\,Re}\\!\left(z_{{C\\!P}}^{2}-\Delta
z^{2}\right)}+\dfrac{\langle t^{2}\rangle_{j}}{4}\left|z_{{C\\!P}}\pm\Delta
z\right|^{2}+\sqrt{r_{b}}\langle
t\rangle_{j}{{\rm\,Re}\\!\left[X_{b}^{*}(z_{{C\\!P}}\pm\Delta
z)\right]}}{1+\dfrac{\langle
t^{2}\rangle_{j}}{4}{{\rm\,Re}\\!\left(z_{{C\\!P}}^{2}-\Delta
z^{2}\right)}+r_{b}\dfrac{\langle
t^{2}\rangle_{j}}{4}\left|z_{{C\\!P}}\pm\Delta
z\right|^{2}+\sqrt{r_{b}}\langle
t\rangle_{j}{{\rm\,Re}\\!\left[X_{b}(z_{{C\\!P}}\pm\Delta z)\right]}}.$ (1)

The parameter $r_{b}$ is the value of $R_{bj}$ at $t=0$, while $X_{b}$ is the
amplitude-weighted strong-phase difference between opposing bins. Finally,
$\langle t\rangle_{j}$ ($\langle t^{2}\rangle_{j}$) corresponds to the average
(squared) decay time in each positive Dalitz-plot region where the mixed
contribution is negligible, in units of the ${D}^{0}$ lifetime
$\tau=\hbar/\Gamma$ [5], calculated directly from background-subtracted data.
The other parameters are determined from a simultaneous fit of the observed
$R_{bj}^{\pm}$ ratios, in which external information on
$c_{b}\equiv{{\rm\,Re}\\!\left(X_{b}\right)}$ and
$s_{b}\equiv-{{\rm\,Im}\\!\left(X_{b}\right)}$ [22, 23] is used as a
constraint.

Samples of ${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$
decays are reconstructed from proton–proton ($pp$) collisions collected by the
LHCb experiment from 2016 to 2018, corresponding to an integrated luminosity
of 5.4$\text{\,fb}^{-1}$. The strong-interaction decay
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ is used to identify the flavor of
the neutral charm meson at production. Throughout this Letter, ${D}^{*+}$
indicates the ${{D}^{*}}(2010)^{+}$ meson and soft pion indicates the pion
from its decay. The LHCb detector [24, 25] is a single-arm forward
spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the
study of particles containing $b$ or $c$ quarks.

Decays of ${{K}^{0}_{\mathrm{S}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ are
reconstructed in two different categories: the first involving
${K}^{0}_{\mathrm{S}}$ mesons that decay early enough for the pions to be
reconstructed in all tracking detectors; and the second containing
${K}^{0}_{\mathrm{S}}$ mesons that decay later such that track segments of the
pions cannot be formed in the vertex detector, which surrounds the $pp$
interaction (primary vertex) region, resulting in a worse momentum resolution.
The latter category contains more candidates but has slightly worse mass and
decay-time resolution as well as larger efficiency variations.

The online event selection consists of a hardware stage, selecting events
based on calorimeter and muon detector information, followed by two software
stages. In the first software stage, the pion pair from the ${{D}^{0}}$ decay
is required to satisfy criteria on momenta and final-state charged-particle
displacements from any primary vertex for at least one pion (one-track) or
both together with a vertex quality requirement (two-track). The second
software stage fully reconstructs
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}},{{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$
candidates using further requirements on particle identification, momenta, and
track and vertex quality. Specific ranges of displacement and invariant mass
are imposed on the reconstructed ${{D}^{0}}$ and ${{K}^{0}_{\mathrm{S}}}$
candidates. Due to differing efficiencies, the sample is split into four
categories, depending on whether or not the ${K}^{0}_{\mathrm{S}}$ meson is
reconstructed in the VELO and whether or not they satisfy the one-track
requirement.

Offline, a kinematic fit constrains the tracks to form vertices according to
the decay topology, the ${K}^{0}_{\mathrm{S}}$ candidate mass to the known
value [5], and the ${D}^{*+}$ candidate to a primary vertex [26]. In the
reconstruction of the Dalitz-plot coordinates, an additional constraint on the
${D}^{0}$ candidate mass to the known value improves the resolution. Charm
mesons originating from the decays of $b$ hadrons are suppressed by requiring
that the ${D}^{0}$ and soft pion candidates originate from a primary vertex.
Candidates are rejected if two of the reconstructed tracks use the same hits
in the vertex detector. About 6% of the candidates are from collision events
in which multiple candidates are reconstructed, usually by pairing the same
${D}^{0}$ candidate with different soft pions. When this occurs, one candidate
is chosen randomly, and the rest are removed from the sample.

Figure 3:  Distribution of ${\Delta m}$ for the selected
${{D}^{*+}}\rightarrow{{D}^{0}}(\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{+}}$
candidates. The projection of the fit result is superimposed.

Signal yields are determined by fitting the distribution of the mass
difference between the ${D}^{*+}$ and ${D}^{0}$ candidates, denoted as $\Delta
m$. The signal probability density function is empirically described by a
combination of a Johnson $S_{U}$ distribution [27] and two Gaussian functions,
one of which shares a mean with the Johnson $S_{U}$. The background is
dominated by real ${D}^{0}$ decays incorrectly combined with a charged
particle not associated with a ${D}^{*+}$ decay, and is modeled with a smooth
phase-space-like model, $\theta({\Delta m}-m_{\pi})e^{-c({\Delta
m}-m_{\pi})}\left({\Delta m}-m_{\pi}\right)^{\alpha}$, where $\theta(x)$ is
the Heaviside step function, $m_{\pi}$ is the charged-pion mass [5], and
$\alpha$ and $c$ are free parameters. Figure 3 shows the $\Delta m$
distribution of the entire sample, from which the fit identifies $(30.585\pm
0.011)\times 10^{6}$ signal decays. This represents a factor of 15 larger
yield compared to the previous measurement.

To determine the yields used to form the ratios $R_{bj}^{\pm}$, separate fits
are performed for each set of Dalitz-plot and decay-time bins $bj$. The signal
model assumes the same parameters for each pair of positive and negative
Dalitz-plot bins, and fixes some parameters from a fit integrated over decay
time. Fits are performed independently for ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ candidates, as well as for each
of the four data subsamples. The measured signal yields are then corrected for
two effects that do not cancel in the ratio: experimentally induced
correlations between the phase space and decay time, and charge-dependent
efficiencies (detection asymmetries).

Online requirements on the displacement and momenta of the ${D}^{0}$ decay
products introduce efficiency variations that are correlated between the
phase-space coordinates and the ${D}^{0}$ decay time. The effect depends
predominantly on the invariant mass of two pions from the ${D}^{0}$ decay,
with the efficiency to reconstruct the candidates at low values decreasing
significantly at low ${D}^{0}$ decay times. This can bias the measured yield
ratios and produce mixing-like trends. To remove this bias, an approach that
estimates the relative efficiencies using data is developed. The Dalitz plot
is divided into small, rectangular-like regions formed symmetrically across
the bisector. Note that these include the portions above and below the
bisector, unlike the bins shown in Fig. 2. In the limit of $C\\!P$ symmetry,
the contribution of mixing to such symmetric regions depends only on
$y_{{C\\!P}}$ and the hadronic parameters of the ${{D}^{0}}$ decay [21]. As
oscillations result in a migration of decays from one side of the Dalitz plot
to the other, and the regions are symmetric with respect to the bisector,
there is no effect from $x_{{C\\!P}}$. Given a set of inputs for $y_{{C\\!P}}$
and the hadronic parameters, the contribution of mixing to the decay-time
distributions of these regions can be accounted for, such that the remaining
differences between regions come from the efficiency correlations. Relative
efficiency maps that align the decay-time distributions in all these symmetric
regions can then be calculated. Per-candidate weights assigned by the
efficiency maps are integrated over the data in each bin using the sPlot
method [28] with $\Delta m$ as the discriminating variable. This provides
correction factors for each of the fitted signal yields.

In calculating the efficiency maps, the strong phase variation within a
Dalitz-plot bin is approximated as constant, such that it can be described by
the external inputs ($s_{b}$). As $y_{{C\\!P}}$ and $s_{b}$ are parameters of
the fit, the correction maps and corresponding correction factors are
calculated for a range of values. The smallness of mixing results in smooth
variations of the correction factors for a given Dalitz-plot bin, which allows
for precise interpolation between the calculated points with polynomials.
These polynomials are then incorporated into the fit as a correction that
depends on $y_{{C\\!P}}$ and $s_{b}$. The correction is calculated for each
yield ratio, but is averaged over the initial flavor of the candidates. The
procedure has been validated with pseudoexperiments, and a systematic
uncertainty is assigned due to the approximation that $s_{b}$ is constant
within a bin.

Corrections are also applied in order to take into account detection
asymmetries. Due to utilizing ratios of yields, the analysis is insensitive to
detection asymmetries of the ${K}^{0}_{\mathrm{S}}$, as well as the soft pion
used to tag the flavor of the candidate. However, the kinematics of the pions
produced in the ${D}^{0}$ decay depend on the Dalitz-plot coordinate and
${D}^{0}$ flavor. This can result in asymmetric efficiency variations for
${D}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ candidates
that imitate $C\\!P$ violation. The two-track ${{\pi}^{+}}{{\pi}^{-}}$
asymmetry, ${A_{\mathrm{det}}({{\pi}^{+}}{{\pi}^{-}})}$, is determined by
measuring detection asymmetries in control samples of
${{D}^{+}_{s}}\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{D}^{+}_{s}}\rightarrow\phi{{\pi}^{+}}$ decays, in which the $\phi$ meson is
reconstructed through a ${K}^{+}$ ${K}^{-}$ pair. A randomly chosen
${{\pi}^{+}}$ in the
${{D}^{+}_{s}}\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay is paired
with the ${\pi}^{-}$ to form a proxy for the ${{\pi}^{+}}{{\pi}^{-}}$ pair of
interest. The ${{D}^{+}_{s}}\rightarrow\phi{{\pi}^{+}}$ sample is used to
cancel asymmetries induced from the remaining ${{\pi}^{+}}$,
$A_{\text{det}}({{\pi}^{+}})$, and other sources, such as the trigger
selection, $A_{\text{trigger}}({{D}^{+}_{s}})$, and the production of
${{D}^{+}_{s}}$ and ${{D}^{-}_{s}}$ mesons in $pp$ collisions,
$A_{\text{prod}}({{D}^{+}_{s}})$. For asymmetries of $\mathcal{O}(1\%)$, the
raw asymmetries $A_{\rm meas}$ can be approximated as

$\displaystyle A_{\rm
meas}({{D}^{+}_{s}}\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}})$
$\displaystyle\approx{A_{\mathrm{det}}({{\pi}^{+}}{{\pi}^{-}})}\ +$
$\displaystyle
A_{\text{det}}({{\pi}^{+}})+A_{\text{prod}}({{D}^{+}_{s}})+A_{\text{trigger}}({{D}^{+}_{s}}),$
(2) $\displaystyle A_{\rm meas}({{D}^{+}_{s}}\rightarrow\phi{{\pi}^{+}})$
$\displaystyle\approx$ $\displaystyle
A_{\text{det}}({{\pi}^{+}})+A_{\text{prod}}({{D}^{+}_{s}})+A_{\text{trigger}}({{D}^{+}_{s}}).$

The difference of the two measured asymmetries gives the detection asymmetry
of the ${\pi}^{+}$ ${\pi}^{-}$ pair. The control samples are weighted to match
the kinematics of the pions from the
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ sample.
This weighting is done separately for each Dalitz-plot bin. The detection
asymmetries are of the order of $10^{-3}$ and are used as corrections to the
measured yields. They are included as constraints in the fit along with the
associated covariance matrix $\Delta V_{\rm{asym}}$ describing uncertainties
coming from the limited size of the calibration samples.

The mixing parameters are determined by minimizing a least-squares function

$\displaystyle\chi^{2}\equiv$
$\displaystyle\sum_{+,-}\sum_{b,j}\frac{\left[N^{\pm}_{-bj}-R^{\pm}_{+bj}N^{\pm}_{+bj}/(C_{bj}(1\pm\Delta
A_{b}))\right]^{2}}{(\sigma^{\pm}_{-bj})^{2}+\left[R_{+bj}^{\pm}\sigma^{\pm}_{+bj}/(C_{bj}(1\pm\Delta
A_{b}))\right]^{2}}$
$\displaystyle+\sum_{b,b^{\prime}}\left(X^{\rm{EXT}}_{b}-X_{b}\right)(V_{\rm{EXT}}^{-1})_{bb^{\prime}}\left(X^{\rm{EXT}}_{b^{\prime}}-X_{b^{\prime}}\right)$
$\displaystyle+\sum_{b,b^{\prime}}\left(\Delta A^{\rm{asym}}_{b}-\Delta
A_{b}\right)(\Delta V_{\rm{asym}}^{-1})_{bb^{\prime}}\left(\Delta
A^{\rm{asym}}_{b^{\prime}}-\Delta A_{b^{\prime}}\right),$ (3)

where the yields $N$ and their measured uncertainties $\sigma$ are scaled by
factors for the correlation removal, $C_{bj}$, and detection asymmetry
correction, $\Delta
A^{b}\equiv{A_{\mathrm{det}}^{b}({{\pi}^{+}}{{\pi}^{-}})}-{A_{\mathrm{det}}^{-b}({{\pi}^{+}}{{\pi}^{-}})}$.
The different subsamples are fitted simultaneously, separated between
${D}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ flavors
denoted as $+$ and $-$, including all decay-time intervals $j$ and Dalitz-plot
bins $b$. The parameters $X_{b}$ are constrained with a Gaussian penalty term
using the values $X_{b}^{\rm{EXT}}$ and covariance matrix $V_{\rm{EXT}}$ from
a combination of CLEO and BESIII measurements[22, 23]. In the fit, the
parameters $r_{b}$ are determined independently for each subsample, as they
are affected by the sample-specific variation of the efficiency over the
Dalitz plot [21]. To avoid experimenter’s bias, the values of $x_{{C\\!P}}$,
$y_{{C\\!P}}$, $\Delta x$, and $\Delta y$ were not examined until the full
procedure had been finalized. Figure 4 shows the yield ratios with fit
projections overlaid for each of the eight Dalitz-plot bins. Deviations from
constant values are due to mixing. The fit projection when $x_{{C\\!P}}$ is
fixed to zero is also included and shows the inability of a nonzero
$y_{{C\\!P}}$ value to produce the deviations on its own. Also shown are the
differences of ratios between ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ decays, where a significant slope
would indicate $C\\!P$ violation.

Systematic uncertainties are assessed from ensembles of pseudoexperiments.
These use the
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ model of
Ref. [29] to describe the amplitude at $t=0$, and the decay-time dependence is
incorporated for a range of values of the mixing and $C\\!P$ violation
parameters. Different sources of systematic uncertainty are included, and the
effect on the measured parameters evaluated. The dominant systematic
uncertainty on the mixing parameters comes from reconstruction and selection
effects, and amounts to $0.20\times 10^{-3}$ ($0.76\times 10^{-3}$) for
$x_{{C\\!P}}$ ($y_{{C\\!P}}$). This includes neglecting the decay-time and
$m_{\pm}^{2}$ resolutions and efficiencies, as well as the correction to
remove the efficiency correlations. The most important effect for
$y_{{C\\!P}}$ is the approximation of the strong phase to be constant within
each bin in the procedure to remove correlations. Contamination from
$b$-hadron decays contributes $0.20\times 10^{-3}$ ($0.15\times 10^{-3}$) to
the $x_{{C\\!P}}$ ($y_{{C\\!P}}$) uncertainty. Potential mismodeling in the
signal yield fits contributes $0.36\times 10^{-3}$ to the $y_{{C\\!P}}$
uncertainty. Time-dependent detection asymmetries are present mainly in bins
that give the best sensitivity to $\Delta y$, resulting in a systematic
uncertainty of $0.12\times 10^{-3}$.

The consistency of the results is tested by repeating the analysis in subsets
of the data, divided according to magnet polarity, trigger and
${K}^{0}_{\mathrm{S}}$ category, data-taking period, ${D}^{*+}$ meson
kinematics, and other categories. The largest variation occurs for the value
of $x_{{C\\!P}}$ as a function of ${D}^{*+}$ meson pseudorapidity, where the
compatibility, considering statistical uncertainties only, amounts to a
p-value of 1.5%, depending on the details of the sample split, whereas the
overall p-value for all $x_{{C\\!P}}$ variations observed is above 8%. The
observed variations of the observables $x_{{C\\!P}}$, $y_{{C\\!P}}$, $\Delta
x$ and $\Delta y$ are all consistent with statistical fluctuations.

Figure 4: (Top) $C\\!P$-averaged yield ratios and (bottom) differences of
${D}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ yield
ratios as a function of $t/\tau$, shown for each Dalitz-plot bin with fit
projections overlaid.

The mixing and $C\\!P$ violation parameters are measured to be

$\displaystyle x_{{C\\!P}}$ $\displaystyle=(\phantom{-}3.97\pm 0.46\pm
0.29)\times 10^{-3},$ $\displaystyle y_{{C\\!P}}$
$\displaystyle=(\phantom{-}4.59\pm 1.20\pm 0.85)\times 10^{-3},$
$\displaystyle\Delta x$ $\displaystyle=(-0.27\pm 0.18\pm 0.01)\times 10^{-3},$
$\displaystyle\Delta y$ $\displaystyle=(\phantom{-}0.20\pm 0.36\pm 0.13)\times
10^{-3},$

where the first uncertainty is statistical and the second systematic. The
statistical uncertainty contains a subleading component due to the limited
precision of the external measurements of the strong phases and control
samples used for the detection asymmetry. This amounts to approximately
$(0.23$, $0.66$, $0.04$, and $0.08)\times 10^{-3}$ for $x_{{C\\!P}}$,
$y_{{C\\!P}}$, $\Delta x$, and $\Delta y$, respectively. The measurements are
statistically limited, though the systematic uncertainty on $y_{{C\\!P}}$ is
comparable to the statistical uncertainty. The results are used to form a
likelihood function of $x$, $y$, $|q/p|$, and $\phi$ using a likelihood-ratio
ordering that assumes the observed correlations to be independent of the true
parameter values [30]. The best fit point is

$\displaystyle x$ $\displaystyle=(3.98_{-\,0.54}^{+\,0.56})\times 10^{-3},$
$\displaystyle y$
$\displaystyle=(\phantom{0}4.6_{-\,1.4}^{+\,1.5\phantom{0}})\times 10^{-3},$
$\displaystyle|q/p|$ $\displaystyle=0.996\pm 0.052,$ $\displaystyle\phi$
$\displaystyle=0.056_{-\,0.051}^{+\,0.047}.$

In summary, a measurement of mixing and $C\\!P$ violation in
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays has
been performed with the bin-flip method, using $pp$ collision data collected
by the LHCb experiment and corresponding to an integrated luminosity of
5.4$\text{\,fb}^{-1}$. This resulted in the first observation of a nonzero
value of the mass difference $x$ of neutral charm meson mass eigenstates with
a significance of more than seven standard deviations, and significantly
improves limits on mixing-induced $C\\!P$ violation in the charm sector.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and NERSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from ARC and ARDC (Australia); AvH Foundation
(Germany); EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union);
A*MIDEX, ANR, IPhU and Labex P2IO, and Région Auvergne-Rhône-Alpes (France);
Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP,
Fundamental Research Funds for the Central Universities, and Sci. & Tech.
Program of Guangzhou (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal
and GENCAT (Spain); the Leverhulme Trust, the Royal Society and UKRI (United
Kingdom).

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##  1 Supplemental material

Table 1 summarizes the measured values along with their uncertainties and
correlations. Table 2 gives the derived values for $x$, $y$, $q/p$ and $\phi$
together with the 95.5% confidence interval. Table 3 shows a summary of the
uncertainties in this analysis. Fig. 5 shows the Dalitz plot of the
background-subtracted
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ candidates
used in the analysis. No efficiency corrections are applied. All samples are
combined.

Figure 5: Dalitz plot of background-subtracted ${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ candidates.  Table 1: Fit results of $x_{{C\\!P}}$, $y_{{C\\!P}}$, $\Delta x$, and $\Delta y$. The first contribution to the uncertainty is statistical, the second systematic. Parameter | Value | Stat. correlations | Syst. correlations
---|---|---|---
| [$10^{-3}$] | $y_{{C\\!P}}$ | $\Delta x$ | $\Delta y$ | $y_{{C\\!P}}$ | $\Delta x$ | $\Delta y$
$x_{{C\\!P}}$ | $\phantom{-}3.97\pm 0.46\pm 0.29$ | $0.11$ | $-0.02$ | $-0.01$ | $0.13$ | $0.01$ | $0.01$
$y_{{C\\!P}}$ | $\phantom{-}4.59\pm 1.20\pm 0.85$ |  | $-0.01$ | $-0.05$ |  | $-0.02$ | $0.01$
$\Delta x$ | $-0.27\pm 0.18\pm 0.01$ |  |  | $0.08$ |  |  | $0.31$
$\Delta y$ | $\phantom{-}0.20\pm 0.36\pm 0.13$ |  |  |  |  |  |
Table 2: Point estimates and 95.5% confidence-level (CL) intervals for $x$, $y$, $|q/p|$ and $\phi$. The uncertainties include statistical and systematic contributions. Parameter | Value | 95.5% CL interval
---|---|---
$x$ [$10^{-3}$] | $\phantom{-0}3.98\,_{-\,0.54}^{+\,0.56\phantom{0}}$ | $\phantom{-00}[2.9,5.0]\phantom{00}$
$y$ [$10^{-3}$] | $\phantom{-00}4.6\,_{-\,1.4}^{+\,1.5\phantom{00}}$ | $\phantom{-00}[2.0,7.5]\phantom{00}$
$|q/p|$ | $\phantom{-00}0.996\pm 0.052$ | $[\phantom{-}0.890,1.110]$
$\phi$ | $-0.056\,_{-\,0.051}^{+\,0.047}$ | $[-0.172,0.040]$
Table 3: Uncertainties in units of $10^{-3}$. The total systematic uncertainty is the sum in quadrature of the individual components. The uncertainties due to the external inputs and detection asymmetry calibration samples are included in the statistical uncertainty. These are also reported separately, along with the contributions due to the limited sample size, to ease comparison with other sources.  Source | $x_{{C\\!P}}$ | $y_{{C\\!P}}$ | $\Delta x$ | $\Delta y$
---|---|---|---|---
Reconstruction and selection | $0.199$ | $0.757$ | $0.009$ | $0.044$
Secondary charm decays | $0.208$ | $0.154$ | $0.001$ | $0.002$
Detection asymmetry | $0.000$ | $0.001$ | $0.004$ | $0.102$
Mass-fit model | $0.045$ | $0.361$ | $0.003$ | $0.009$
Total systematic uncertainty | $0.291$ | $0.852$ | $0.010$ | $0.110$
Strong phase inputs | 0.23 | 0.66 | 0.02 | 0.04
Detection asymmetry inputs | 0.00 | 0.00 | 0.04 | 0.08
Statistical (w/o inputs) | 0.40 | 1.00 | 0.18 | 0.35
Total statistical uncertainty | $0.46$ | $1.20$ | $0.18$ | $0.36$

LHCb collaboration

R. Aaij32, C. Abellán Beteta50, T. Ackernley60, B. Adeva46, M. Adinolfi54, H.
Afsharnia9, C.A. Aidala86, S. Aiola25, Z. Ajaltouni9, S. Akar65, J.
Albrecht15, F. Alessio48, M. Alexander59, A. Alfonso Albero45, Z. Aliouche62,
G. Alkhazov38, P. Alvarez Cartelle55, S. Amato2, Y. Amhis11, L. An48, L.
Anderlini22, A. Andreianov38, M. Andreotti21, F. Archilli17, A. Artamonov44,
M. Artuso68, K. Arzymatov42, E. Aslanides10, M. Atzeni50, B. Audurier12, S.
Bachmann17, M. Bachmayer49, J.J. Back56, P. Baladron Rodriguez46, V.
Balagura12, W. Baldini21, J. Baptista Leite1, R.J. Barlow62, S. Barsuk11, W.
Barter61, M. Bartolini24, F. Baryshnikov83, J.M. Basels14, G. Bassi29, B.
Batsukh68, A. Battig15, A. Bay49, M. Becker15, F. Bedeschi29, I. Bediaga1, A.
Beiter68, V. Belavin42, S. Belin27, V. Bellee49, K. Belous44, I. Belov40, I.
Belyaev41, G. Bencivenni23, E. Ben-Haim13, A. Berezhnoy40, R. Bernet50, D.
Berninghoff17, H.C. Bernstein68, C. Bertella48, A. Bertolin28, C.
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M.S. Bieker15, S. Bifani53, P. Billoir13, M. Birch61, F.C.R. Bishop55, A.
Bitadze62, A. Bizzeti22,k, M. Bjørn63, M.P. Blago48, T. Blake56, F. Blanc49,
S. Blusk68, D. Bobulska59, J.A. Boelhauve15, O. Boente Garcia46, T.
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Borisyak42, M. Borsato17, J.T. Borsuk35, S.A. Bouchiba49, T.J.V. Bowcock60, A.
Boyer48, C. Bozzi21, M.J. Bradley61, S. Braun66, A. Brea Rodriguez46, M.
Brodski48, J. Brodzicka35, A. Brossa Gonzalo56, D. Brundu27, A. Buonaura50, C.
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Byczynski48, S. Cadeddu27, H. Cai73, R. Calabrese21,f, L. Calefice15,13, L.
Calero Diaz23, S. Cali23, R. Calladine53, M. Calvi26,j, M. Calvo Gomez85, P.
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L. Capriotti20,d, A. Carbone20,d, G. Carboni31, R. Cardinale24, A. Cardini27,
I. Carli4, P. Carniti26,j, L. Carus14, K. Carvalho Akiba32, A. Casais Vidal46,
G. Casse60, M. Cattaneo48, G. Cavallero48, S. Celani49, J. Cerasoli10, A.J.
Chadwick60, M.G. Chapman54, M. Charles13, Ph. Charpentier48, G.
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M. Lucio Martinez32, V. Lukashenko32, Y. Luo3, A. Lupato62, E. Luppi21,f, O.
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M.W. Majewski34, J.J. Malczewski35, S. Malde63, B. Malecki48, A. Malinin81, T.
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L. Shekhtman43,v, Z. Shen5, V. Shevchenko81, E.B. Shields26,j, E. Shmanin83,
J.D. Shupperd68, B.G. Siddi21, R. Silva Coutinho50, G. Simi28, S. Simone19,c,
N. Skidmore62, T. Skwarnicki68, M.W. Slater53, I. Slazyk21,f, J.C.
Smallwood63, J.G. Smeaton55, A. Smetkina41, E. Smith50, M. Smith61, A.
Snoch32, M. Soares20, L. Soares Lavra9, M.D. Sokoloff65, F.J.P. Soler59, A.
Solovev38, I. Solovyev38, F.L. Souza De Almeida2, B. Souza De Paula2, B.
Spaan15, E. Spadaro Norella25,i, P. Spradlin59, F. Stagni48, M. Stahl65, S.
Stahl48, P. Stefko49, O. Steinkamp50,83, O. Stenyakin44, H. Stevens15, S.
Stone68, M.E. Stramaglia49, M. Straticiuc37, D. Strekalina83, F. Suljik63, J.
Sun27, L. Sun73, Y. Sun66, P. Svihra62, P.N. Swallow53, K. Swientek34, A.
Szabelski36, T. Szumlak34, M. Szymanski48, S. Taneja62, A.R. Tanner54, A.
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Institute Of High Energy Physics (IHEP), Beijing, China
5School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
6University of Chinese Academy of Sciences, Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut
Polytechnique de Paris, Palaiseau, France
13LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
14I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
15Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
16Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
17Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
18School of Physics, University College Dublin, Dublin, Ireland
19INFN Sezione di Bari, Bari, Italy
20INFN Sezione di Bologna, Bologna, Italy
21INFN Sezione di Ferrara, Ferrara, Italy
22INFN Sezione di Firenze, Firenze, Italy
23INFN Laboratori Nazionali di Frascati, Frascati, Italy
24INFN Sezione di Genova, Genova, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Milano-Bicocca, Milano, Italy
27INFN Sezione di Cagliari, Monserrato, Italy
28Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
29INFN Sezione di Pisa, Pisa, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31INFN Sezione di Roma Tor Vergata, Roma, Italy
32Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
33Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
36National Center for Nuclear Research (NCBJ), Warsaw, Poland
37Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
38Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
39Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
40Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
41Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia
42Yandex School of Data Analysis, Moscow, Russia
43Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
44Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
45ICCUB, Universitat de Barcelona, Barcelona, Spain
46Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
47Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
48European Organization for Nuclear Research (CERN), Geneva, Switzerland
49Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
50Physik-Institut, Universität Zürich, Zürich, Switzerland
51NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
52Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
53University of Birmingham, Birmingham, United Kingdom
54H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
55Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
56Department of Physics, University of Warwick, Coventry, United Kingdom
57STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
58School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
59School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
60Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
61Imperial College London, London, United Kingdom
62Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
63Department of Physics, University of Oxford, Oxford, United Kingdom
64Massachusetts Institute of Technology, Cambridge, MA, United States
65University of Cincinnati, Cincinnati, OH, United States
66University of Maryland, College Park, MD, United States
67Los Alamos National Laboratory (LANL), Los Alamos, United States
68Syracuse University, Syracuse, NY, United States
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 56
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Physics and Micro Electronic College, Hunan University, Changsha City,
China, associated to 7
72Guangdong Provincial Key Laboratory of Nuclear Science, Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Institute of Quantum Matter, South China
Normal University, Guangzhou, China, associated to 3
73School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
74Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 13
75Universität Bonn - Helmholtz-Institut für Strahlen und Kernphysik, Bonn,
Germany, associated to 17
76Institut für Physik, Universität Rostock, Rostock, Germany, associated to 17
77Eotvos Lorand University, Budapest, Hungary, associated to 48
78INFN Sezione di Perugia, Perugia, Italy, associated to 21
79Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 32
80Universiteit Maastricht, Maastricht, Netherlands, associated to 32
81National Research Centre Kurchatov Institute, Moscow, Russia, associated to
41
82National Research University Higher School of Economics, Moscow, Russia,
associated to 42
83National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 41
84National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
41
85DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 45
86University of Michigan, Ann Arbor, United States, associated to 68

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bHangzhou Institute for Advanced Study, UCAS, Hangzhou, China
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Genova, Genova, Italy
iUniversità degli Studi di Milano, Milano, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Modena e Reggio Emilia, Modena, Italy
lUniversità di Padova, Padova, Italy
mScuola Normale Superiore, Pisa, Italy
nUniversità di Pisa, Pisa, Italy
oUniversità della Basilicata, Potenza, Italy
pUniversità di Roma Tor Vergata, Roma, Italy
qUniversità di Siena, Siena, Italy
rUniversità di Urbino, Urbino, Italy
sMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
tAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
uP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
vNovosibirsk State University, Novosibirsk, Russia
wDepartment of Physics and Astronomy, Uppsala University, Uppsala, Sweden
xHanoi University of Science, Hanoi, Vietnam