1]School of electronic Information, Shanghai DianJi University, Shanghai,
201306, China

# Research on Image Processing and Vectorization Storage Based on Garage
Electronic Maps

Nan Dou<EMAIL_ADDRESS>Qi Shi<EMAIL_ADDRESS>Zhigang Lian
<EMAIL_ADDRESS>[

###### Abstract

For the purpose of achieving a more precise definition and data analysis of
images, this study conducted a research on vectorization and rasterization
storage of electronic maps, focusing on a large underground parking garage
map. During the research, image processing, vectorization and rasterization
storage were performed. The paper proposed a method for the vectorization
classification storage of indoor two-dimensional map raster data. This method
involves converting raster data into vector data and classifying elements such
as parking spaces, pathways, and obstacles based on their coordinate positions
with the grid indexing method, thereby facilitating efficient storage and
rapid querying of indoor maps. Additionally, interpolation algorithms were
employed to extract vector data and convert it into raster data. Navigation
testing was conducted to validate the accuracy and reliability of the map
model under this method, providing effective technical support for the digital
storage and navigation of garage maps.

###### keywords:

electronic maps, vectorization storage, rasterization storage, classification
storage, vehicle navigation

##  1 Background of map digital storage

This paper conducts research on the vectorization of electronic maps. When
maps are directly scanned into the computer, data is stored in raster format.
However, for more accurate location, length, and size definitions, as well as
for Geographic Information Systems (GIS) data analysis, spatial data topology
analysis, and joint analysis of spatial and attribute data, it is necessary to
convert raster data into vector data. This process is known as vectorization
[1], which transforms image data into graphic data while preserving the
corresponding topological structure. Graphic data includes lines, circular
arcs, curves, character strokes, etc. Vector data has higher accuracy,
allowing for more precise definition of location, length, and size. Therefore,
in GIS data analysis, transformation, spatial data topology analysis, and the
joint analysis of spatial and attribute data, vector data is indispensable.
Thus, converting raster data from scanned images into vector data becomes one
of the important tasks in GIS technology [2].

Raster to vector data conversion is a data transformation technique used to
convert between raster and vector representations. Each representation has its
own advantages, disadvantages, and applicable scenarios, so the decision to
perform raster to vector data conversion depends on the intended use [3].

##  2 Overview of Digital Storage Issues for Garage Maps

###  2.1 Problem Statement

In a large underground parking garage with multiple entrances and parking
spaces, the system needs to process its structural map through image
processing and convert it into vector data for storage, facilitating the
system’s map modeling [4].

###  2.2 Research Methodology

This article first uses image feature retrieval to match feature points
between the corrected raster image and the latitude and longitude template
image. Then, based on the matching results, it obtains the latitude and
longitude information and the horizontal and vertical coordinate information
on the raster image. Next, it solves for the parameters of the affine
transformation model [5]. Finally, it vectorizes the raster image and performs
coordinate transformation. The technical approach is illustrated in Figure 1
[6].

Figure 1: Technical road map

The digital storage methods for indoor maps primarily include rasterization,
vectorization, hybridization and so on. Among them, rasterization [7] is a
commonly used method, converts indoor maps into pixel grid data and achieves
digital storage through array-based representation. Vectorization transforms
indoor maps into vector data, enabling high-precision processing and visual
representation. Hybridization combines raster and vector data, comprehensively
leveraging their advantages to achieve better results. Indoor maps refer to
maps that describe and depict indoor spaces. With the continuous development
of artificial intelligence technology and indoor navigation applications, the
demand for indoor maps is growing. This paper proposes a method for
vectorization and categorical storage of indoor two-dimensional raster data.
By converting raster data into vector data and adopting a categorical storage
approach, this method achieves efficient storage and rapid querying of indoor
maps.

##  3 Image Feature Matching and Rasterization Processing

###  3.1 Image Feature Matching

For image feature matching [8], this paper sequentially matches the template
image with the raster image, obtaining the coordinates of the matched blocks.
The paper detects contours in the images and calculates the coordinates of the
four vertices of the second largest area contour. In longitude matching, the
paper calculates the image region from 0 to Xmin, while in latitude matching,
it calculates the image interval from top to bottom, ranging from 0 to Ymin .
Xmin is the minimum value of the abscissa, and Ymin is the minimum value of
the ordinate among the coordinates of the four vertices.

When calculating adjacent positions, it may lead to some inaccurate matches.
To avoid such situations, the paper detects the minimum parent rectangle of
the cropped raster image and calculates the center point of the rectangle.
Using this method, the feature point information for each parent rectangle can
be calculated. The calculation formula is as follows:

$\displaystyle\left\\{\begin{aligned} &f(A,B)=(Bx-Ax)(py-Ay)-(px-Ax)(By-Ay)\\\
&f(p_{2},p_{3})\times f(p_{4},p_{3})\geq 0\\\ &f(p_{2},p_{3})\times
f(p_{4},p_{1})\geq 0\end{aligned}\right.$ (1)

In the formula, A and B represent two adjacent vertices of the rectangle,
while x and y represent the abscissa and ordinate of any point inside the
rectangle. p1 to p4 are the four vertices of the rectangle, and p is a feature
point. If equation (1) is satisfied, it indicates that the feature point is
located inside the rectangle.

###  3.2 Image Rasterization

Image rasterization is the process of dividing an image into several pixel
units and representing each pixel unit as a fixed-size square. This process
usually involves aspects such as image segmentation, feature extraction,
pattern recognition, etc., and is an important research direction in the field
of digital image processing. In Python, the Pillow library can be used to
implement image rasterization operations, including opening the image,
resizing, converting the image, obtaining pixel color values, and converting
them to binary (0/1). Image rasterization is widely used in computer vision,
machine learning, and other fields, such as object detection, image
classification, image recognition, etc.

The rasterization of indoor maps involves transforming a two-dimensional map
into pixel grid data. The basic steps of rasterization methods include
binarization, grayscale conversion, feature extraction, and image
segmentation. Binarization converts the image into black and white, with pixel
values of only 0 and 1. Grayscale conversion transforms the image into a
grayscale image with pixel values ranging from 0 to 255. Feature extraction
extracts the feature information of the indoor map, such as walls, doors,
etc., based on the grayscale image. Image segmentation divides the indoor map
into multiple connected regions for subsequent processing and storage.

The specific steps of image rasterization are as follows:

  1. (1)

Open the map image,

  2. (2)

Resize the map image to the desired size,

  3. (3)

Convert the map image to grayscale,

  4. (4)

Obtain the color value of each pixel and convert it to binary,

  5. (5)

Save or use the binary pixel matrix.

The example of map image rasterization is shown in Figures 2 and 3:

Figure 2: Map grayscale processing Figure 3: The map images were reconstructed
using a binary matrix

##  4 Vectorization and Categorical Storage of Garage Map Raster Data

###  4.1 Raster Data Vectorization

Raster data vectorization is the process of converting raster data into vector
data. In GIS, raster data consists of grid-like data composed of pixels, while
vector data consists of spatial data composed of basic elements such as
points, lines, and polygons. Converting raster data to vector data allows for
more convenient spatial analysis and modeling, as vector data offers better
precision and visualization effects.

The main steps of raster data vectorization include:

  1. (1)

Pixel Classification: Each pixel in raster data is classified based on its
attribute values, such as classifying it as a parking space, obstacle, path,
etc.

  2. (2)

Edge Detection [9]: For each classified pixel, compare it with neighboring
pixels to detect boundary lines.

  3. (3)

Boundary Refinement: Refine and repair the detected boundary lines to ensure
they are continuous and smooth.

  4. (4)

Vectorization: Convert the boundary lines into vector basic elements such as
points, lines, and polygons to generate vector data.

  5. (5)

Data Editing: Edit and repair the generated vector data to ensure its accuracy
and completeness.

###  4.2 Map Vector Data Processing

Image processing techniques are used to process the garage map. The edge
detection algorithm is applied to extract the edges of the garage map,
followed by using morphological transformation algorithms to binarize the map.
Subsequently, the connected component algorithm is employed to separately
extract elements such as parking spaces, pathways, and obstacles, transforming
them into labeled vector data.

To extract pathways in the map, the connected component algorithm [10] can be
used. The basic idea of the algorithm is to traverse the two-dimensional
matrix, find all connected 0s, and form independent regions. Then, contour
extraction is performed on each region, obtaining the vector representation of
the region, which is then converted into vector data and stored in a database.

To extract parking spaces in the map, the polygon fitting algorithm [11] can
be used. Detecting rectangular parking spaces using the polygon fitting
algorithm involves the following steps:

  1. (1)

For each connected region (pathway), use contour detection algorithm to obtain
its set of contour points.

  2. (2)

For the contour point set of each connected region, use the polygon fitting
algorithm to fit a convex hull polygon.

  3. (3)

For the fitted convex hull polygon, calculate its perimeter and area.

  4. (4)

For convex hull polygons that meet certain area and perimeter restrictions,
determine whether they are rectangular, i.e., whether the four corner points
are approximately right angles.

  5. (5)

For convex hull polygons meeting the rectangular conditions, extract their
four corner coordinates to obtain rectangular parking spaces, and use the grid
index method to extract the coordinates of parking spaces and obstacles.

The map’s vector data processing is shown in Figure 4. After obtaining the
parking space matrix, the system extracts the corresponding coordinates for
each parking space. First, based on the position information of the parking
space matrix, the system determines the geometric position of each parking
space on the map. Then, through scanning and analyzing the map, it extracts
the specific coordinate information for each parking space. This coordinate
information can include the latitude and longitude of the parking space or the
pixel coordinates relative to the map. The extracted parking space coordinates
are stored in the system’s database for subsequent parking navigation and path
planning. By extracting parking space coordinates, the system can accurately
locate each parking space, providing precise parking navigation and path
planning services for vehicle owners.

Figure 4: The location coordinates of the parking space were extracted by
shape detection

###  4.3 Data Classification and Storage

This paper uses the grid index method to classify and store elements such as
parking spaces, pathways, and obstacles based on their coordinate positions.
Firstly, the garage map is divided into several grids, with each grid size
equal to that of a parking space. Then, parking spaces, pathways, and
obstacles are placed into their respective grids based on their coordinate
positions, forming a two-dimensional array. This greatly improves the
efficiency of querying parking spaces, pathways, and obstacles since only the
relevant grid needs to be queried.

A SQL database model is designed, including tables for parking spaces,
pathways, and obstacles. The parking space table includes information such as
parking space number, coordinate position, and parking space type. The pathway
table includes information such as pathway number, starting point coordinates,
and ending point coordinates. The obstacle table includes information such as
obstacle number, coordinate position, and obstacle type. Parking space types
and obstacle types can be determined based on features such as color or shape,
for example, red parking spaces are no-parking zones, rectangular parking
spaces are for small vehicles, and circular parking spaces are for large
vehicles. Model construction examples are shown in Tables 1, 2 and 3.

Table 1: Parking table Attribute Name | Data Type | Primary Key | Description
---|---|---|---
id | int | YES | Parking Space ID
x_coordinate | double | - | Parking Space X Coordinate
y_coordinate | double | - | Parking Space Y Coordinate
space_type | char | - | Parking Space Type
|  |  |
Table 2: Routing table Attribute Name | Data Type | Primary Key | Description
---|---|---|---
id | int | YES | Parking ID
start_x | double | - | Path Start X Coordinate
start_y | double | - | Path Start Y Coordinate
end_x | double | - | Path Start x Coordinate
end_y | double | - | Path Start y Coordinate
|  |  |
Table 3: Obstruct table Attribute Name | Data Type | Primary Key | Description
---|---|---|---
id | int | YES | Parking ID
x_coordinate | double | - | Obstacle X Coordinate
y_coordinate | double | - | Obstacle Y Coordinate
obstacle_type | char | - | Obstacle Type
|  |  |

##  5 Extraction of Vector Data and its Conversion to Raster Data

Vector data extraction and conversion to raster data can be achieved using
interpolation algorithms. Interpolation is a method of estimating unknown
position data by leveraging the values of known data points. It involves
inferring spatial relationships between known data points to compute the
values of unknown positions. When transforming vector data into raster data,
interpolation algorithms can be employed to calculate the values for each
raster grid.

One common interpolation algorithm is Inverse Distance Weighting (IDW), which
calculates the value of unknown positions based on the weighted distances.
When converting vector data to raster data, each vector data point can be
treated as a known data point. The values for each raster grid are then
computed based on the coordinates and attribute values of these known data
points. Specifically, for a raster grid position, the distances to all known
data points are calculated. The weights are determined based on these
distances and the attribute values of the known data points, and the raster
grid value is subsequently computed using these weights.

###  5.1 Vector Point Rasterization

Vector point rasterization: Essentially involves converting the coordinates
(x,y) of points to row and column numbers in a raster. The coordinate origin
of the raster is located at the top-left corner, where the row number I
increases downward, and the column number J increases to the left. Vector
point rasterization is illustrated in Figure 5.

Figure 5: The vector data points were converted to raster data

In the figure, (x0,y0)represents the origin of the raster, while dx and dy
respectively represent the length and width of the raster, i.e., its size.
During the conversion of vector data to raster data, raster data can be
expressed using a two-dimensional array.

###  5.2 Vector Line Rasterization

Eight-Direction Rasterization:It is a straightforward method based on the
inclinations of lines.It ensures that only one pixel is marked as black in
each row or column. This method decomposes a line into eight fixed directions
(horizontal, vertical, and diagonal) and determines the pixel positions on
each direction based on the inclination of the line. Figure 6 illustrates an
example of eight-direction rasterization. Using this algorithm, lines can be
effectively converted into discrete pixel representations, facilitating
subsequent processing and analysis.

Figure 6: Eight direction grid

Full Path Rasterization: This is essentially a "striping" method, where the
raster is divided into rows or columns, and all elements intersecting with the
vector line on each strip are filled. As shown in Figure 7.

Figure 7: Full path raster

###  5.3 Two-Dimensional Polygon Sampling

Grid sampling is a common method for two-dimensional polygon sampling,
enabling the transformation of raster data into regularly sampled grid points.
The basic steps for creating grid sampling for maps are as follows:

Create Grid Sampling: Based on the sampling interval and the bounding box of
polygon features, create regularly spaced grid points within each polygon’s
range. The following steps can be used for grid sampling:

Determine the sampling interval: Based on the needs and precision of the map,
determine the spacing between grid points, such as the horizontal and vertical
intervals within each polygon.

Iterate through polygon features: For each polygon feature, create grid points
within the bounding box of the feature at the specified sampling interval. Use
two nested loops to iterate through the coordinates within the bounding box
and determine the position of grid points based on the sampling interval.

Determine grid point positions: During the iteration, use the current
coordinates as the position of the grid points, and add each point to the
sampling point collection. An example of creating grid coordinate sampling is
shown in Figure 8.

Figure 8: Example plots of the grid sampling

##  6 Establishment of Map Models and Navigation Testing

Navigational testing of the map stored in the database through vector
rasterization can be conducted following these steps:

Extract map information: Extract map information from the database stored in
vector rasterized format and convert it into the required format for
navigation. For example, road information can be transformed into a graph of
nodes and edges, establishing a road network.

Set start and end points: Set start and end points on the road network graph,
and determine the navigation algorithm.

Navigation algorithm: Utilize the navigation algorithm based on the set start
and end points to calculate the shortest or optimal path.

Visualize navigation results: Visualize the navigation results to provide
users with a clear view of the navigation path, including navigation
instructions. For instance, display the navigation path, turning directions,
distance, time, etc., on the map.

Navigation testing: Conduct navigation testing to examine the accuracy and
reliability of the navigation results.

The navigation test is illustrated in Figures 9 and 10:

Figure 9: Vehicle navigation example diagram1 Figure 10: Vehicle navigation
example diagram2

##  7 Conclusion

In this study, a method for the structural mapping of a large underground
parking garage was proposed, involving image processing and vector data
conversion to achieve systematic map modeling. Here is the summary of the
method:

Image feature retrieval: Correction of raster images and latitude-longitude
template images is performed first. Subsequently, image feature retrieval is
employed to calculate feature points of the parent matrix, and matching is
conducted at last. Through the matched feature point correspondences, GIS
features on the raster image and their Cartesian coordinates are determined.

Affine transformation model solution: Utilizing the matching results, the data
for the affine transformation model is computed. By establishing the affine
transformation matrix, pixel coordinates in the raster image can be mapped to
actual geographic coordinates.

Raster image vectorization: Based on matching and affine transformation
results, the raster image is converted into vector data. Pixel points in the
raster image are connected to form line segments or polygonal elements,
creating vector polygon features. Each feature represents structures such as
entrances, exits, or parking spaces in the garage.

Vector data classification and storage: Grid indexing is employed to classify
and store elements like parking spaces, pathways, and obstacles based on their
coordinate positions.

Coordinate transformation: Conversion of coordinate information from the
raster image to coordinate values compatible with the map coordinate system.
This allows for map projection and spatial analysis of the vectorized data.

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# Heavy-quark opportunities and challenges at FCC-ee

Stéphane Monteil  Guy Wilkinson  Université Clermont Auvergne, CNRS/IN2P3,
LPC, Clermont-Ferrand, France  Department of Physics, University of Oxford,
Oxford, United Kingdom

(Received: / Revised version: )

###### Abstract

The abundant production of beauty and charm hadrons in the $5\times 10^{12}$
$Z^{0}$ decays expected at FCC-ee offers outstanding opportunities in flavour
physics that in general exceed those available at Belle II, and are
complementary to the heavy-flavour programme of the LHC. A wide range of
measurements will be possible in heavy-flavour spectroscopy, rare decays of
heavy-flavoured particles and $C\\!P$-violation studies, which will benefit
from the low-background experimental environment, the high Lorentz boost, and
the availability of the full spectrum of hadron species. This essay first
surveys the important questions in heavy-flavour physics, and assesses the
likely theoretical and experimental landscape at the turn-on of FCC-ee. From
this certain measurements are identified where the impact of FCC-ee will be
particularly important. A full exploitation of the heavy-flavour potential of
FCC-ee places specific constraints and challenges on detector design, which in
some cases are in tension with those imposed by the other physics goals of the
facility. These requirements and conflicts are discussed.

††journal: EPJ Plus

##  1 Introduction: FCC-ee as a flavour factory

Measurements in flavour physics have played a central role in the construction
of the Standard Model (SM). Studies in the kaon sector led to the GIM
mechanism and the prediction of the charm quark [1]; the observation of
$C\\!P$ violation suggested the existence of a third generation [2], and the
first measurements of $B^{0}-\overline{B}{}^{0}$ oscillations immediately
indicated the top quark to be very massive [3]. Now complete, the SM poses
many questions that are intimately connected to flavour. Why are there three
generations? What explains the hierarchy in quark masses and the distinctive
structure of the Cabibbo-Kobayashi-Maskawa matrix? What is the nature of the
non-SM contributions to $C\\!P$ violation that are required to explain the
matter-antimatter asymmetry of the universe? Furthermore, flavour physics is
known to be a powerful tool of discovery, as the contribution of loop
amplitudes to many of the processes of interest provide natural entry points
for new, massive particles. Indeed, the broad consistency of flavour
observables with SM predictions was the first warning sign that TeV-scale non-
SM physics (hence referred to a ‘new physics’) would not be observed at the
LHC. For these reasons flavour physics, in particular studies of beauty,
charm, and of $\tau$ leptons, is a vibrant field of study, with the current
flagship experiments being LHCb at the Large Hadron Collider, and Belle II
operating in the $e^{+}e^{-}$ environment at the $\Upsilon(4S)$.

Table 1: Advantageous attributes for flavour-physics studies at Belle II ($\Upsilon(4S)$), the LHC ($pp$) and FCC-ee ($Z^{0}$). Attribute | $\Upsilon(4S)$ | $pp$ | $Z^{0}$
---|---|---|---
All hadron species |  | ✓ | ✓
High boost |  | ✓ | ✓
Enormous production cross-section |  | ✓ |
Negligible trigger losses | ✓ |  | ✓
Low backgrounds | ✓ |  | ✓
Initial energy constraint | ✓ |  | (✓)

Table 1 compares the advantages for flavour-physics studies at an
$e^{+}e^{-}\to\Upsilon(4S)\to b\bar{b}$ experiment, such as Belle II, a $pp\to
b\bar{b}X$ experiment, such as LHCb, and an experiment that relies on
$e^{+}e^{-}\to Z^{0}\to b\bar{b}$ production, such as would be the case at
FCC-ee. It can be seen that the $Z^{0}$ environment combines most of the
advantages of Belle II and LHCb. For the former these are the high signal-to-
noise and fully efficient trigger, as well as a very high geometrical
acceptance; for the latter they are the production of the full spectrum of
hadrons, and the high boost. The momenta of $b$ and $c$ hadrons produced at
the $Z^{0}$ are not known a priori, in contrast to the $\Upsilon(4S)$,
although their distribution is very well understood. The momentum of the
produced tau leptons is of course perfectly known in both $e^{+}e^{-}$
environments.

The one disadvantage that the $Z^{0}$ has in comparison with the LHC is the
production cross section, but this is partially mitigated at FCC-ee by the
enormous luminosity that is foreseen. Table 2 gives the yields for each
$b$-hadron species that will be produced in $5\times 10^{12}$ $Z^{0}$ decays
111Note that about a factor of two more $Z^{0}$ decays can be recorded if the
design of the FCC-ee evolves towards a four interaction-points layout.. The
number of $b\bar{b}$ pairs from which these yields arise is around fifteen
times larger than that expected at Belle II. As will be explained below, the
particular advantages of the $Z^{0}$ environment will allow for many studies
that are complementary or more sensitive to those foreseen at LHCb and its
upgrades. There will also be great opportunities in charm and tau physics, for
which yields are also listed in Table 2. In tau physics, in particular, the
FCC-ee will have unsurpassed physics reach in almost all measurements, as is
discussed in companion essays in this volume.

Table 2: Yields of heavy-flavoured particles produced at FCC-ee for $5\times 10^{12}$ $Z^{0}$ decays. The charge conjugate states have the same production yields. These yields are computed using the Z branching fractions and hadronisation rate reported in Refs. [4, 5]. The $B_{c}$ hadronisation fraction is assumed to be $f_{B_{c}}=2\times 10^{-3}$ [6]. Particle species | $B^{0}$ | $B^{+}$ | $B^{0}_{s}$ | $\Lambda_{b}$ | $B_{c}^{+}$ | $c\overline{c}$ | $\tau^{-}\tau^{+}$
---|---|---|---|---|---|---|---
Yield ($\times 10^{9}$) | $310$ | $310$ | $75$ | $65$ | $1.5$ | $600$ | $170$

We also note that the proposed running strategy of FCC-ee, with the intention
to collect data at several collision energies, will open up flavour
possibilities that are not restricted to the $Z$ pole. The decays of on-shell
$W$ bosons will provide a particularly rich laboratory for studies of the CKM
matrix, as is described in more detail below.

In this brief report we first survey the likely landscape in quark-flavour
physics in the mid to late 2030s, for both theory and experiment. We then
outline the opportunities that will exist in this domain at FCC-ee,
highlighting certain key measurements. We conclude by presenting the key
detector characteristics that an experiment should possess in order to realise
these opportunities.

##  2 Quark-flavour physics at the dawn of FCC-ee: the theoretical and
experimental landscape

The sensitivity to virtual particles and phases that is afforded by flavour
observables give them excellent sensitivity to new physics at mass scales far
in excess of those achievable in direct searches. In this sense the goals of
flavour-physics studies at FCC-ee are similar and complementary to those
directed at electroweak precision observables. Already some intriguing hints
of new physics effects have been seen in possible violations of lepton
universality in both penguin-mediated processes and in tree decays. Rather
than speculate about the evolution of these anomalies, we here choose to make
more general remarks on the status of the flavour-measurement programme, with
a focus on where further experimental progress will be valuable during the
FCC-ee era.

According to current planning, the 2030s will see operation of the second
upgrade of LHCb, which will accumulate around 300 fb-1 in integrated
luminosity, and collect sample sizes in excess of 50 times larger than those
so far analysed [7]. ATLAS and CMS will continue to contribute strongly to the
LHC flavour programme in selected areas, for example the measurement of the
weak phase $\phi_{s}$ and studies of $B^{0}\to\mu^{+}\mu^{-}$ and
$B^{0}_{s}\to\mu^{+}\mu^{-}$. Belle II will integrate around 50 ab-1, with the
prospect of further running now under active discussion. In parallel to the
great steps forward in experimental knowledge these projects will bring, there
are expected to be corresponding advances in lattice QCD calculations, and the
development of other theoretical tools. Figure 1 shows the possible status of
the Unitarity Triangle after LHCb Upgrade II operation, based on results from
that experiment alone, and assuming plausible lattice QCD improvements. It
makes the pessimistic assumption that all results will be in agreement.
Similarly high precision will be achieved in studies of ‘rare decays’, i.e.
suppressed flavour-changing neutral-current (FCNC) processes.

Figure 1: Possible status of the Unitarity Triangle in the late 2030s,
assuming LHCb measurements alone and improvements in lattice QCD [7].

Many of the observables in $C\\!P$-violation studies, and others contributing
to Unitarity Triangle tests, are theoretically pristine, and so warrant
continued experimental attention throughout the coming decades. The angle
$\gamma$, for example, may be measured in $B^{-}\to DK^{-}$ decays with a
relative theoretical uncertainty of $10^{-6}$ or better [8]. Other
measurements, that may become limited by systematic biases at the LHC, will
benefit from the cleaner and very different environment of the FCC-ee.
Examples include studies of semileptonic $C\\!P$-violating asymmetries, and
determinations of $|V_{ub}/V_{cb}|$ performed with $B^{0}_{s}$ mesons and
$\Lambda_{b}$ baryons that are not accessible at Belle II. Here the hadronic
systematics may be different to those that apply for measurements with
$B^{0,\pm}$ mesons, and the spin of the baryon brings potentially useful new
information. Very importantly, there will remain much to be learned from
$b$-hadron decays involving $\tau$ leptons, which are very challenging at
LHCb, and where the $B$-factory studies will be limited by sample size, as
well as being restricted to $B^{0}$ and $B^{-}$ mesons.

##  3 Flavour-physics opportunities at FCC-ee: some key measurements

FCC-ee will perform sensitive studies of $C\\!P$-violating phenomena that will
allow for improved knowledge of the Unitarity Triangle angles $\alpha$,
$\beta$ and $\gamma$, and the phase $\phi_{s}$ between mixing and decay in the
$B^{0}_{s}-\overline{B}{}^{0}_{s}$ system. Consideration of the sample sizes
suggests that it will be possible to measure the relevant observables with
similar or better precision to previous experiments, as reported in Ref. [9].
A particular strength of the FCC-ee flavour-programme will be the ability to
make very sensitive studies of modes containing neutrals, with much larger
sample sizes than will be available at Belle II. This possibility will enable
FCC-ee to harness a very wide range of charm-meson decay modes in measurements
of $C\\!P$ asymmetries in $B^{-}\to DK^{-}$ (where $D$ indicates a
superposition of $D^{0}$ and $\bar{D}^{0}$ and $B_{s}\to
D_{s}^{(*)\pm}K^{\mp}$, which are sensitive to the angle $\gamma$. It is
expected that the flavour-tagging efficiency will be significantly higher than
at the LHC, bringing corresponding gains for time-dependent measurements.
Other exciting possibilities include modes relevant for the angle $\alpha$;
for example, precise measurements of the time-dependent $C\\!P$ asymmetries in
$B^{0}\to\pi^{0}\pi^{0}$ can be performed making use of Dalitz decays. Around
2.5 thousand $B^{0}\to\pi^{0}(\to\gamma\gamma)\pi^{0}(\to e^{+}e^{-}\gamma)$
events can be expected, assuming electron and $\pi^{0}\to\gamma\gamma$
reconstruction efficiencies similar to those achieved with the LEP detectors.

It is quite possible, however, that the most significant impact of FCC-ee in
this domain will come from the measurement of other observables. In making
this argument, it is useful to recall the results of a model-independent study
of possible BSM contributions to the $B^{0}-\overline{B}{}^{0}$ and
$B^{0}_{s}-\overline{B}{}^{0}_{s}$ mixing amplitudes, reported in Ref. [10].
The box diagrams that drive the oscillations and carry $C\\!P$-violating
phases are natural entry points for any heavy degrees-of-freedoms from BSM
physics. The mixing amplitudes can be modelled with complex numbers of modulus
$h_{q}$ ($q=d,s$), each multiplying the SM $B^{0}_{q}$ mixing Hamiltonian
matrix element. Figure 2 shows the constraints on these BSM parameters
(assuming no BSM signal) for three scenarios: current measurements, the
foreseen LHCb upgrade and Belle II experiments measurements, and at FCC-ee.
The projected performance at Belle II and LHCb are based on the numbers
reported in Refs. [7, 11], and reasonable assumptions concerning progress in
lattice QCD [12]. The FCC-ee inputs include extrapolations of the current
$\beta$ and $\gamma$ angle measurements to the expected statistics at the $Z$
pole [9]. As no dedicated studies have yet been performed, we conservatively
assume that the measurements of $|V_{ub}|$ and the mixing frequencies, as well
as knowledge of hadronic parameters, will remain unchanged from the end of the
HL-LHC era [12, 11]. The FCC-ee constraints benefit in addition from precise
determinations of three key parameters: $|V_{cb}|$, and the semileptonic
asymmetries $a^{d}_{\rm sl}$ and $a^{s}_{\rm sl}$.

Figure 2:  Sensitivity to $h_{d}-h_{s}$ parameters in $B_{d}$ and $B_{s}$
mixing with current sensitivity (top left), the anticipated constraints after
LHCb Upgrade I and Belle II operation (top right), and the FCC-ee expectation
(bottom). The dotted curves show the 99.7% C.L. ($3\sigma$) contours. All
plots are made with the SM inputs $h_{d}=h_{s}=0.$ Taken from Ref. [10].

  * 1.

The sides of the Unitarity Triangle are normalised by the CKM element
$V_{cb}$. In Fig. 2 the sensitivity to the BSM parameters, even with the very
large sample sizes of LHCb Upgrade I and Belle II accumulated by around 2030,
start to become limited by the knowledge of this parameter, which comes from
measurements of semileptonic $B$ decays. These measurements can be made both
with inclusive and exclusive decays, but in both cases require hadronic input,
e.g. from lattice QCD, for $|V_{cb}|$ to be extracted. At FCC-ee another
approach will open up that has no such systematic limitation. Several $10^{8}$
$W$ boson decays will be collected when operating FCC-ee at the $W^{+}W^{-}$
threshold and above. By benefitting from the excellent vertexing capabilities
that will be available at FCC-ee, and selecting events with $b$-tagged and
$c$-tagged jets [13, 14], it may be possible to improve the knowledge of
$|V_{cb}|$ up to an order of magnitude with respect to current knowledge. It
will also be possible to make very precise measurements of the magnitude of
other CKM elements, e.g. $|V_{cs}|$, using on-shell $W$ decays. Similar
studies, albeit with lower statistical precision, will be possible with the
large $t\bar{t}$ sample that will be accumulated (see Ref. [15] for a proposal
first made in the context of the LHC).

  * 2.

The semileptonic asymmetries

$a^{q}_{\rm
sl}\equiv\frac{\Gamma(\overline{B}{}^{0}_{q}\to\bar{f})-\Gamma({B^{0}_{q}}\to
f)}{\Gamma(\overline{B}{}^{0}_{q}\to\bar{f})+\Gamma({B^{0}_{q}}\to f)},$ (1)

which are so-called as semileptonic decays are typically chosen for the
flavour-specific final-state $f$, probe the $C\\!P$-violating phases in
$B^{0}_{q}-\bar{B}{}^{0}_{q}$ oscillations ($q=d,s$). In the SM these
asymmetries are very small, but precisely predicted: $a^{d}_{sl}=(-4.73\pm
0.42)\times 10^{-4}$ and $a^{s}_{sl}=(2.06\pm 0.18)\times 10^{-5}$ [16]. Hence
measurements of $a^{d}_{\rm sl}$ and $a^{s}_{\rm sl}$ are very valuable in
providing sensitivity to $h_{d}$ and $h_{s}$. LHCb Upgrade II will attain a
statistical sensitivity of a few $10^{-4}$ for both observables, but
controlling systematic effects from production and detection asymmetries at
the same level of precision will be very challenging [7]. A solenoidal
detector at FCC-ee will have clear advantages in this regard, and can reach an
overall sensitivity of a few $10^{-5}$ [17].

In addition to the measurement of CKM-related observables, FCC-ee will perform
studies of a wide range of suppressed flavour-changing-neutral-current (FCNC)
processes, such as dileptonic and semileptonic modes that are driven by
electroweak penguins and box diagrams in the SM, for example
$B^{0}\to\mu^{+}\mu^{-}$, $B^{0}_{s}\to\mu^{+}\mu^{-}$,
$B^{0}_{s}\to\tau^{+}\tau^{-}$, $b\to s(d)\ell^{+}\ell^{-}$, where $\ell$
represents either an electron or a muon, $b\to s(d)\tau^{+}\tau^{-}$ and $b\to
s(d)\nu\overline{\nu}$. This programme will be extended to the analysis of
favoured, but experimentally challenging, modes, where the SM predictions are
reliable and BSM effects could be pronounced, for example the decays
$B_{c}^{+}\to\mu^{+}\nu$ and $B_{c}^{+}\to\tau^{+}\nu$. Promising exploratory
studies have been performed for the latter mode at a $Z$ factory [18, 19]. The
analysis of these channels, together with that of radiative FCNCs in both the
beauty and charm sectors, will allow for stringent tests of the SM and have
high discovery potential in probing for new-physics extensions. FCC-ee will be
able to complement many of the studies performed at Belle II and LHCb, and
significantly extend the measurement programme in several critical areas.

To illustrate these possibilities more clearly, we here discuss the ‘flagship’
ultra-rare channels $B^{0}_{s}\to\mu^{+}\mu^{-}$ and $B^{0}\to\mu^{+}\mu^{-}$.
In Runs 1 and 2 at the LHC the focus has been on measuring the branching
fraction of $B^{0}_{s}\to\mu^{+}\mu^{-}$ [20, 21, 22] and comparing with the
precise predictions of the SM [23]. In coming years, and at FCC-ee, emphasis
will shift to the search for the even rarer sister mode,
$B^{0}\to\mu^{+}\mu^{-}$, and the measurement of the ratio of the
$B^{0}\to\mu^{+}\mu^{-}$ to $B^{0}_{s}\to\mu^{+}\mu^{-}$ branching fractions,
which is a powerful test of minimal flavour violation. Further important
observables will include the $B^{0}_{s}\to\mu^{+}\mu^{-}$ effective lifetime
[24], where proof-of-principle measurements already exist [20], and
measurements of $C\\!P$ asymmetries, which currently are not feasible. In all
these studies FCC-ee will excel, with particular benefits coming from the
excellent mass resolution, which will clearly separate the $B^{0}$ and
$B^{0}_{s}$ signals, high-performance flavour tagging (necessary for the
$C\\!P$-asymmetry measurements), and low backgrounds. Figure 3 shows the
signal peaks that are expected with $5\times 10^{12}$ $Z^{0}$ decays and
reasonable assumptions on the detector performance. These peaks contain around
540 and 70 $B^{0}_{s}\to\mu^{+}\mu^{-}$ and $B^{0}\to\mu^{+}\mu^{-}$ decays,
respectively [25]. Also included is the background arising from
$B^{0}\to\pi^{+}\pi^{-}$ decays, assuming a double $\pi\to\mu$
misidentification probability of $2\times 10^{-5}$, which lies under the
$B^{0}\to\mu^{+}\mu^{-}$ signal. Controlling this source of contamination will
be an important consideration in the detector optimisation.

Figure 3: Reconstructed invariant mass of $B^{0}\to\mu^{+}\mu^{-}$ and
$B^{0}_{s}\to\mu^{+}\mu^{-}$ signals for $5\times 10^{12}$ $Z^{0}$ decays.
Also shown is the background contribution from misidentified
$B^{0}\to\pi^{+}\pi^{-}$ events [25].

Final states involving $\tau$ leptons also warrant further discussion on
account of the unique opportunities that will exist at FCC-ee [26, 27, 28].
Studies reported in Refs. [29, 30] have established a proof-of-principle of
the kinematical fits that determine the missing neutrino missing momentum for
three-prongs $\tau$ decays, as described in a companion essay in this volume.
This approach requires the detector to be highly hermetic and have excellent
resolution for reconstructing the production and decay vertices. Figure 4
illustrates these capabilities for the mode $B^{0}\to
K^{*}(892)\tau^{+}\tau^{-}$, showing the reconstructed mass peak in a fast
simulation with a parametric detector response, and including the contribution
of a subset of background decay modes sharing the very same signal final
state. This signal channel is of great interest, particularly in light of the
hints of the violation of lepton universality seen in the companion decays
involving the first two generations, but cannot be well studied at Belle II or
LHCb Upgrade II unless its branching fraction is greatly enhanced with respect
to the SM expectations. Another example is the decay channel
$B_{c}^{+}\to\tau^{+}\nu$, where a recent analysis, already mentioned, has
demonstrated good prospects at FCC-ee [4]. These promising initial studies
must be refined, with all background sources comprehensively accounted for,
and the detector design optimised to maximise performance for this very
important category of decay channel.

Figure 4: Reconstructed invariant-mass of $B^{0}\to
K^{*}(892)\tau^{+}\tau^{-}$ candidates determined by using constraints on the
production and decay vertices. The assumed resolutions on the primary and
secondary vertices cartesian coordinates are 3 and 7 ${\rm\mu}$m,
respectively. Two background sources are considered: (red) $B^{0}\to
D_{s}^{-}\overline{K}{}^{*0}(892)\tau^{+}\nu_{\tau}$ and (pink)
$\overline{B}_{s}\to D_{s}^{-}D_{s}^{+}\overline{K}{}^{*0}(892)$, with the
relative branching fractions appropriately scaled to their estimated SM
values.

The brief survey of interesting observables does not exhaust the list of
$b$-physics opportunities at FCC-ee, but highlights the unique capabilities of
the facility. It is clear from the examples given that many important
measurements can be made that will complement well the strengths of the LHC
flavour programme. The ability to exploit the full range of $b$-hadrons, and
with a high Lorentz boost, gives FCC-ee an advantage over Belle II that is
even more important than its statistical superiority. Furthermore, there will
also be similarly important and exciting studies to perform in the domains of
charm physics and heavy-flavour spectroscopy.

##  4 Detector requirements: towards a dedicated heavy-flavour experiment

Although there are commonalities between the experimental requirements of all
physics goals at FCC-ee, for example the need to cope with collimated jets of
particles, a detector optimised for beauty, charm and, indeed, tau physics
will have capabilities that are rather different to those that are necessary
for, say, Higgs studies. The experimental layouts considered until now at FCC-
ee have, understandably, been designed with the higher-energy requirements in
mind. There are scenarios, however, under which an experiment largely or
wholly optimised for heavy-flavour physics is conceivable. For example, if the
machine design evolves to a four-interaction point collider then there will be
a strong argument for diversity in the attributes of the experiments that are
deployed. Here we sketch out the essential features of a heavy-flavour
experiment, paying particular attention to the demands on vertexing,
calorimetry and hadron particle identification (PID).

Precise vertexing is essential for all heavy-flavour measurements, for example
playing a particularly important role in time-dependent $B^{0}_{s}$ studies
where the proper-time uncertainty must be small compared to the $\sim$350 fs
oscillation period of the $B^{0}_{s}$ meson in order not to dilute the $C\\!P$
asymmetries. The performances of the vertex detectors proposed for the IDEA
and CLD detectors are probably adequate in this regard, with resolutions of a
few microns for multi-track vertices [31]. However, for other studies there
will be significant gains if even better performance can be attained. A case
in point is the study of $b$-hadron decays involving tau leptons, discussed
above. Here the partial reconstruction of the decay relies on a kinematic fit
that takes as input the tau decay vertex and, if available, that of the $b$
hadron. Current detector designs aim for a resolution of around 5 $\mu$m for a
three-track vertex, which will limit the effective invariant-mass resolution
for this class of decays. Starting with the ALPIDE-style sensor design [32,
33] already proposed for the IDEA experiment, various improvements can be
envisaged that would lead to reduced vertex uncertainties. These include a
smaller beam-pipe radius, the deployment of curved sensors, improved hit
resolution and the implementation of air cooling.

The relatively low-multiplicity environment of FCC-ee presents an outstanding
opportunity to make high precision studies of heavy-flavour decays involving
photons, $\pi^{0}$ and $\eta$ mesons. (Although important results in this
domain have emerged from LHCb, the situation there is a priori very
challenging, due to the high level of combinatorics). Achieving this goal
requires excellent energy resolution in order to obtain an invariant-mass
resolution that is comparable to that obtainable for decays involving charged
tracks only. This attribute will be even more critical at FCC-ee than at the
$B$-factories, as it will be necessary to separate $B^{0}$ and $B^{0}_{s}$
decays to the same final state. To pick but one example, without excellent
mass resolution, $b\to d\gamma$ penguin transitions in $B^{0}_{s}$ decays will
be overwhelmed by $B^{0}$ decays involving the much more prevalent $b\to
s\gamma$ process. It will be necessary to maintain this level of performance
down to low momenta. For example, an important class of $C\\!P$-violation
measurements requires separation of the decays $D^{\ast 0}\to D^{0}\pi^{0}$
and $D^{\ast 0}\to D^{0}\gamma$ [34]. The isolation criteria that must be
imposed in selecting decays such as $B_{c}^{+}\to\tau^{+}\nu_{\tau}$,
discussed above, point to similar requirements. Meeting these physics goals
mandates a crystal-based electromagnetic calorimeter, as appreciated by CLEO
and the $B$-factories. To set the scale, the BaBar detector was equipped with
a thallium doped caesium-iodide calorimeter that achieved an energy resolution
with a stochastic term better than 3% [35]. In contrast, hadron calorimetry is
not an essential capability for flavour measurements, but consideration should
be given to $K^{0}_{L}$ detection, which at BaBar was performed by combining
information from the ECAL and the instrumented flux return [36].

An extensive flavour-physics programme requires high-performance PID over a
wide momentum range. A powerful component of the flavour tagging, necessary
for time-dependent $C\\!P$-violation measurements, comes from noting the
charge of kaons in the event not associated with the signal decay. These kaons
are typically below 10 GeV/$c$. In contrast, kaons from the signal decay are
harder, with some modes giving rise to spectra extending beyond 30 GeV/$c$.
This wide span of momentum necessitates different solutions than those that
were adequate at the $B$-factories, with tighter space constraints than exist
for LHCb. The cluster-counting of ionisation deposits that is intended for the
IDEA drift chamber would largely meet these requirements, if it performs to
specification [31]. Other technologies, such as time-of-flight measurements
using a TORCH detector [37, 38], would provide useful performance, but not at
high momentum. A RICH system with more than one radiator could encompass the
necessary momentum range, but would unavoidably take up ${\cal{O}}(1\,{\rm
m})$ of radial space in the detector, which would then have consequences for
the tracking volume and also degrade the energy resolution, if placed in front
of the ECAL. Solving this problem is an interesting challenge for the overall
detector optimisation. More extensive discussion on this topic may be found in
a companion essay in this volume.

##  5 Conclusions

In this essay we have discussed the features of the $Z^{0}$ factory that make
it a very promising environment for heavy-flavour physics, combining most of
he advantages of Belle II and LHCb experiments, and have attempted to foresee
the experimental and theoretical landscape of flavour physics at the dawn of
the FCC-ee. We then reviewed selected key observables, where the impact of
FCC-ee will be most pronounced, and which are promising in probing for BSM
effects. These measurements place specific demands on the detector design,
particularly in the areas of vertexing, calorimetery and particle
identification. These requirements are not necessarily the same as those
required for electroweak and Higgs physics, and motivate a machine with four
interaction points. The outstanding physics opportunities and stimulating
detector challenges make the heavy-flavour programme one of the most exciting
and attractive prospects at FCC-ee.

## Acknowledgements

We thank Donal Hill for helpful discussions and contributing material to this
essay.

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