11institutetext: Guangxi Key Laboratory of Machine Vision and Intelligent
Control, Wuzhou University, 82 Fumin Third Road, Wanxiu District, Wuzhou,
China 22institutetext: Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Shijingshan District, Beijing, China
33institutetext: College of Physics, Jilin University, 2699 Qianjin Street,
Changchun, China 44institutetext: University of Chinese Academy of Sciences,
19A Yuquan Road, Shijingshan District, Beijing, China

# Performance studies of jet flavor tagging and measurement of $R_{b}(R_{c})$
using ParticleNet at CEPC

Libo Liao<EMAIL_ADDRESS>11 Shudong Wang 2244 Weimin Song 33 Zhaoling
Zhang 33 Gang Li<EMAIL_ADDRESS>(corresponding author)22

(Received: date / Revised version: date)

###### Abstract

Jet flavor tagging plays a crucial role in the measurement of relative partial
decay widths of $Z$ boson, denoted as $R_{b}$($R_{c}$), which is considered as
a fundamental test of the Standard Model and sensitive probe to new physics.
In this study, a Deep Learning algorithm, ParticleNet, is employed to enhance
the performance of jet flavor tagging. The combined efficiency and purity of
$c$-tagging is improved by more than 50% compared to the Circular Electron
Positron Collider (CEPC) baseline software. In order to measure
$R_{b}$($R_{c}$) with this new flavor tagging approach, we have adopted the
double-tagging method. The precision of $R_{b}$($R_{c}$) is improved
significantly, in particular to $R_{c}$, which has seen a reduction in
statistical uncertainty by 40%.

###### pacs:

07.05.KfData analysis: algorithms and implementation; data management and
12.15.−yElectroweak interactions and 14.70.HpZ bosons

##  1 Introduction

The measurement of the relative partial decay widths of $Z$ boson,
$R_{q}=\Gamma_{q\bar{q}}/\Gamma_{h}$, where $\Gamma_{q\bar{q}}$ and
$\Gamma_{h}$ are the partial decay width of $Z\to q\bar{q}$ and the total
hadronic decay width respectively, plays a crucial role in testing the
Standard Model (SM) Glashow ; Weinberg  and searching for new physics.
Particularly, $R_{b}$ is sensitive to the loop corrections to the $Zb\bar{b}$
vertex, potentially sensitive to new physics contributions SUSY . The decay
width to a quark-antiquark final state can be expressed as Vysotsky:1996he

$\displaystyle\Gamma(Z\to q\bar{q})$ $\displaystyle=$
$\displaystyle\frac{G_{F}M_{Z}^{3}}{2\sqrt{2}\pi}(g^{2}_{Aq}R_{Aq}+g^{2}_{Vq}R_{Vq})~{},$
(1)

where $g_{Aq}$ and $g_{Vq}$ are the axial and vector coupling constants,
respectively, and $R_{Aq}$ and $R_{Vq}$ are radiation factors to account for
the final state Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD)
corrections. The electroweak radiative corrections to the propagator and the
$Zq\bar{q}$ vertex are effectively accounted for in the $g_{A}$ and $g_{V}$
couplings. The QED and QCD corrections at first order are flavour blind and
can be represented as

$R_{Aq}\approx R_{Vq}\approx 1+\frac{\alpha_{s}(M_{Z})}{\pi}~{},$ (2)

so that the counterparts of the denominator and numerator cancel each other
out in the ratio $\Gamma_{q\bar{q}}/\Gamma_{h}$.

The latest world averages of $R_{b}$ and $R_{c}$, which are dominated by the
measurements of experiments on the LEP and the SLC L3:1999aer ; OPAL:1998kxc ;
DELPHI:1998cnd ; ALEPH:1997xqy ; SLD:2005zyw ; ALEPH:2005ab , and the
combination results of Gfitter Group Gfitter2018  for $R_{b}$ and $R_{c}$ are
shown in Table 1. It is apparent that the theoretical uncertainties given by
the Gfitter Group are smaller than the experimental results by about two
orders of magnitude. Therefore, It is a promising way to search new physics
beyond the Standard Model by reducing the uncertainties in both experimental
and theoretical domains.

Table 1: $R_{b}$ and $R_{c}$ values in experiment and Gfitter. | Experiment | Gfitter results
---|---|---
$R_{b}$ | $0.21629\pm 0.00066$ | $0.21582\pm 0.00011$
$R_{c}$ | $0.1721\pm 0.0030$ | $0.17224\pm 0.00008$

Various approaches have been used to measure the $R_{b}$($R_{c}$), such as
double tagging, multi-tagging, etc. However, the precision was limited by the
statistics and detector performance. Recently, a few electron-positron
colliders, such as the CEPC CDR-D  and the FCC-ee FCC:2018evy , were proposed
to perform precision Higgs and electroweak studies. These facilities are going
to deliver huge statistics of data at $Z$ pole, $W$ threshold, and about 240
GeV to maximize the production cross section of Higgs-struhlung process, and
so on. It is natural that these experiments will adopt both new detector and
software technologies to achieve the best performance in the detection and
reconstruction of physics objects, especially for jets.

To measure $R_{b}$($R_{c}$), jets are essential physics objects. Therefore,
good jet reconstruction algorithms are key ingredients, in particular, jet
flavor tagging. Jets from different quarks have different characteristics. For
instance, the final states of $b$-jets usually have a wider energy
distribution, and the vertex displacement of tracks in a $b$-jet are larger
than those of other jets because of the long lifetimes of $b$-flavored
hadrons, and so on.

The LCFIPlus LCFIPlus  based on the TMVA package TMVA2007 , is used for the
International Linear Collider (ILC) ilc1 ; ilc2 , the CEPC, and the FCC-ee
physics performance study and detector optimization. The CEPC delivers great
$b/c$-tagging performance thanks to its high precision vertex detector. The
$b$-jets can be tagged with an efficiency of 80% at a purity of 90%. Compared
with $b$-tagging, $c$-jet tagging is particularly challenging as charm hadrons
have relatively shorter lifetimes than bottom ones and suffer more
backgrounds. Therefore, an efficiency of 60% and a purity of 60% can be
achieved for the $c$-jet tagging. The FCC-ee also investigated jet flavor
tagging by developing its own deep learning flavor tagging tool,
ParticleNetIDEA Bedeschi:2022rnj ; Gautam:2022szi . It is suggested that this
methodology yields performance that is commensurate with those reported in the
present study. In this article, the performance of jet flavor tagging of the
CEPC baseline detector is improved using a new deep learning (DL) algorithm,
ParticleNet Qu:2019gqs . In addition, another novel DL algorithm, Particle
Flow Network (PFN) Komiske:2018cqr , is used for comparison and cross-
checking. The article is organized as follows. The simulation, reconstruction
software, and Monte Carlo (MC) samples are introduced in Section 2; the DL
algorithms and the results of jet flavor tagging are presented in Section 3;
then the measurement of $R_{b}$($R_{c}$) is discussed in Section 4; and a
summary is given in Section 5.

##  2 Detector, software, and samples

The study is based on the CEPC baseline detector, which is advanced design
from the International Large Detector ild  on the ILC and optimized to meet
the physics requirement of the CEPC, as shown in Fig. 1. The baseline detector
is designed according to the Particle Flow Algorithm Arbor , which could reap
a better precision and efficiency of reconstructed objects by using the most
suitable sub-detectors. From the inside out, the detector includes a silicon
pixel vertex detector, a silicon tracker, a time projection chamber (TPC), a
calorimetry system which includes an electromagnetic calorimeter (ECAL) and a
hadronic calorimeter (HCAL) of very high granularity, and a muon detector
embedded inside the return yoke of a solenoid magnet system which provides a
magnetic field of 3 Tesla.

Figure 1:  The CEPC baseline detector. The left is the $r-\phi$ view of the
detector. In the barrel from inside to outside, the detector is composed of a
silicon pixel vertex detector, a silicon inner tracker, a TPC, a silicon
external tracker, an ECAL, an HCAL, a solenoid of 3 Tesla, and a muon
detector. The right is the silicon pixel vertex detector, which consists of 3
concentric cylindrical double-layers of high spatial resolution.

The vertex detector consists of six layers of silicon pixel sensors at radii
between 1.6 and 6.0 cm with excessive spatial resolution of $\sim
5~{}\mu\text{m}$. The resolution in $r\phi$ plane can be parameterized by

$\sigma_{r\phi}=a\oplus\frac{b}{p\text{(GeV)}\sin^{3/2}\theta},$ (3)

where $\sigma_{r\phi}$ denotes the impact parameters resolution, $p$ is the
track momentum, and $\theta$ is the polar track angle, $a=5~{}\mu\text{m}$ and
$b=10~{}\mu\text{m}\cdot\text{GeV}$. The silicon tracker is made of 4
components, which are the Silicon Inner Tracker, the Silicon External Tracker,
the Forward Tracking Detector, and the End-cap Tracking Detector. The Time
Projection Chamber is designed within the framework of the LCTPC collaboration
LCTPC  and provided a large number of hits to enhance track finding
efficiency. The ECAL and HCAL are each composed of 1 barrel and 2 end-cap
sections. The detailed description of the CEPC baseline detector is in Ref.
CDR-D .

The MC samples for this study are produced with the CEPC full simulation,
reconstruction, and analysis framework cepcsoft . The physics processes are
generated with WHIZARD 1.9.5 whizard . PYTHIA 6 pythia6  is then used for
hadronization. MokkaPlus MokkaPlus , a GEANT4-based geant4  detector
simulation tool, is used to model the detector response. Arbor Arbor  is used
to reconstruct physics objects including tracks, photons, and neutral hadrons,
and LCFIPlus LCFIPlus  is used to reconstruct (secondary) vertices and jets.

There are 3 hadronic decay modes of $Z$ boson used for jet flavor tagging in
this study, which are $e^{+}e^{-}\to Z\to b\bar{b}$, $c\bar{c}$, and
$q\bar{q}(u\bar{u}/d\bar{d}/s\bar{s})$. For each process, 450,000 events are
produced, which has 900,000 jets in total. The jets are reconstructed using
$e^{+}e^{-}$-$k_{t}$ algorithm in LCFIPlus LCFIPlus , where all particles,
including the reconstructed primary and second vertices, are clustered into
two jets.

##  3 Jet flavor tagging with ParticleNet

In this study, ParticleNet is utilized as the nominal algorithm, while PFN is
employed as a comparison and cross-check method.

Based on the particle cloud representation, which treats a jet as an unordered
group of particles, an effective algorithm, ParticleNet, has been developed.
It is a customized neural network model using Dynamic Graph Convolutional
Neural Network (DGCNN) DGCNN  for jet tagging.

ParticleNet has several advantages. First, it can deal with the varying number
of particles in an event, which is common in experimental high energy physics.
Second, the algorithm is designed to respect the particle permutation
invariance, which refers to the fact that the algorithm does not assume any
special order of the particles in a jet. Third, ParticleNet makes extensive
use of EdgeConv DGCNN  operations to update the graph representation
dynamically. The study in Ref. DGCNN  shows that it is beneficial to recompute
the graph using nearest neighbors in the feature space produced by each layer.
With dynamic graph updates, the jet (sub-)structure can be probed
hierarchically, which leads to better performance than keeping the graph
static. Last but not least, ParticleNet could exploit local neighborhood
information explicitly while most of the other DL algorithms could only use
global symmetric features.

###  3.1 Visualizing the data sets

The jet flavor tagging algorithm is based on features of the data sets. In
this study, these features could be categorized into three types. The first
type is related to jet kinematics, such as multiplicity, momentum
distribution, etc. The second is the impact parameters of the charged tracks,
which are very informative for $b$-tagging. The last one is the types of
particles in a jet, i.e., particle identification (PID). Those could be
expected that the multiplicity of $b$-jet should be larger than the others
because of higher masses of $b$-flavored hadrons and that the tracks should
have larger vertex displacement because of their longer lifetimes, etc.
Considering three types of jets to be studied, some distributions are shown in
Fig. 2. Figure 2 is the multiplicity versus momenta of tracks, where it can be
seen that the number of tracks in $b$ jets is slightly larger than those of
$c$ jets and light ($q$) jets, which is consistent with the decay properties
of heavier $B$ hadrons. The distribution of impact parameters versus the
momenta is shown in Fig. 2. Clear patterns can be observed: $b$ jets have the
significant contribution of larger impact parameters and of energetic tracks
compared with $c$ and $q$ jets. Figure 2 shows the momentum weighted fractions
of different particle types in the three physics processes. It is clear that
$b$ quarks produce more energetic leptons, $c$ quarks produce slightly more
energetic kaons. All the above are consistent with our expectations.

In this data-set, kinematic information, i.e., the four momentum
($p_{x}$,$p_{y}$,$p_{z}$,$E$), and vertex displacements of each particle when
available, are listed in Tab. 2. The ($\cos\theta$, $\phi\sin\theta$) are used
as coordinates Li:2020vav  to compute the distances between particles in the
first EdgeConv block. They are also used together with some other variables,
such as $\Delta R$, PID, $E$, $Q$, $\log E$, $\log P$, $D_{0}$, $Z_{0}$,
$D_{0}/\sigma_{D_{0}}$, $Z_{0}/\sigma_{Z_{0}}$, and the prob which is defined
as

$\text{prob}=\int_{\chi^{2}}^{\infty}p(x,N)dx,$ (4)

where $\chi^{2}=(D_{0}/\sigma_{D_{0}})^{2}+(Z_{0}/\sigma_{Z_{0}})^{2}$,
$p(x,N)$ is the probability density function of the chi-square distribution,
and $N(=2)$ is the number of degrees of freedom.

Figure 2: The feature plots of $b,c,$ and $q$ in jet level. The 2-dimensional diagram of charged multiplicity versus momentum distribution is shown in the top panel; the 2-dimensional diagrams of momentum versus D0 are shown in the middle panel; the fractions of all particle types in $b\bar{b}$, $c\bar{c}$, and $q\bar{q}$ weighted by momentum in the bottom panel, where the PID is based on the MC truth. Table 2: Variables used in the DL algorithms.  Variable | Definition
---|---
$\cos\theta$ | cosine of polar angle of particle
$\phi\sin\theta$ | azimuth angle times sine of polar theta of particle
$\Delta R$ |  $\sqrt{\delta\theta^{2}+\delta\phi^{2}}$, angular separation between the particle and the jet axis
PID | particle ID
$E$ | energy of a particle
$Q$ | electric charge of a particle
$\log E$ | logarithm of the particle’s energy
$\log P$ | logarithm of the particle’s momentum
$D_{0}$ | impact parameter of a track in the r-$\phi$ plane
$Z_{0}$ | impact parameter of a track along the $z$ axis
$D_{0}/\sigma_{D_{0}}$ | significance of the impact parameter in the r-$\phi$ plane
$Z_{0}/\sigma_{Z_{0}}$ | significance of the impact parameter along the $z$ axis
prob | the probability for a certain Chi-squared and number of degrees of freedom

###  3.2 Deep learning algorithms and configuration

The ParticleNet used in this paper consists of three EdgeConv blocks, a global
average pooling layer, and two fully connected layers. The number of channels
$C$ for each EdgeConv block is (64, 64,64), (128, 128, 128), and (256, 256,
256), respectively. After the EdgeConv blocks, a channel-wise channelwise
global average pooling operation is applied to aggregate the learned features
over all particles in the cloud. This is followed by a fully connected layer
with 256 neurons and the ReLU activation relu . A dropout layer dropout  with
a drop probability of 0.1 is included to prevent overfitting. A fully
connected layer with $N$ neurons, followed by a softmax function, is used to
generate the output, where the $N$ is the number of categories in a
classification task. For the number of nearest neighbors $k$ for all three
blocks, some optimization is performed, which shows that 12 for jet tagging is
optimal. The configuration of PFN is directly taken from the Ref.
Komiske:2018cqr , since it is only used for cross-checking.

###  3.3 Training and evaluation

Both ParticleNet and PFN are implemented and running with 8 Intel® Xeon® Gold
6240 CPU cores and 4 NVIDIA® Tesla® V100-SXM2-32GB GPU cards at the IHEP GPU
farm. During training, the common properties of the neural networks include a
categorical cross-entropy loss function, the Adam optimization algorithm adam
, a batch size of 1,024, and a starting learning rate of 0.005. 900,000 jets
are used for each process, therefore, the total number of jets is 2,700,000.
The full data-set is split into training, validation, and test samples
according to the fraction of 7:1.5:1.5. The monitoring of loss and accuracy on
training and validation shows that the algorithm converges well and there is
no obvious over-training.

The computation consumption of ParticleNet and PFN algorithms could be
estimated. Only the total consumption of GPU and CPU is used for estimation
since all the computing resources could only be accessed indirectly via a
workload manager server. ParticleNet takes about 190 minutes for training (30
epochs) and 3 minutes for inference. PFN takes about 30 minutes for training
(80 epochs) and less than a minute for inference. Both two methods could be
finished on a reasonable time scale. However, it could be a problem for the
study of $10^{12}~{}Z$ bosons and solved by hardware development in the next
decades.

###  3.4 Performance

Both ParticleNet and PFN outperform the LCFIPlus in terms of jet flavor
tagging. The accuracy of two novel algorithms, which is defined as the
fraction of correctly classified jets, are summarized in Tab. 3, together with
those in Ref. yangfan . PartcileNet could achieve an accuracy of about 87.6%,
which is at least 9% better than those in Ref. yangfan .

Table 3: The accuracy of different algorithms for jet flavor tagging. In this study, ParticleNet is trained 9 times using randomly initialized weights, and the results from the median-accuracy are shown, while PFN is trained only once and the uncertainty from randomly initialized weights is negligible. Algorithm | ParticleNet | PFN | DNN | BDT | GBDT | gcforest | XGBoost
---|---|---|---|---|---|---|---
Accuracy | 0.876 | 0.850 | 0.788 | 0.776 | 0.794 | 0.785 | 0.801

The numerical results of efficiencies and Area Under Curve (AUC) of both
algorithms in different jet flavor tagging are listed in Tab. 4. The
efficiencies, also called recalls, are determined by taking the largest score
of a jet predicted by the classifiers, which are the same as the corresponding
diagonal terms of the confusion matrix which is shown in Fig. 3. It can be
seen that the performance of $b$-tagging is always better than $c$\- and
$q$-tagging. The observation is consistent with the results in Ref. yangfan .

Figure 3: The confusion matrix with ParticleNet. The training is repeated 9
times using randomly initialized weights, and the results of the training with
median-accuracy are adopted.

Figure 4: Efficiencies for selecting jets with the wrong flavor when tagging
$b$ jets (the left panel) and $c$ jets (the right panel). The points and the
rectangles are the efficiencies of the CEPC baseline and this study,
respectively. The training is repeated 9 times using randomly initialized
weights, and the results of the training with median-accuracy are shown.

The performance of ParticleNet and PFN are generally better than those in Ref.
yangfan . This could be explained from two sides, one is that much richer
information about a jet is used including four momenta, impact parameters, and
PIDs, and the other is that ParticleNet and PFN have a strong inductive bias
for representing high energy events. ParticleNet outperforms PFN, which is
consistent with the study in Ref. Qu:2019gqs , and the authors explained that
"the Deep Sets (PFN) approach does not explicitly exploit the local spatial
structure of particle clouds, but only processes the particle clouds in a
global way."

Table 4: The performance of ParticleNet and PFN in jet tagging. ParticleNet is trained 9 times using randomly initialized weights and the one with median-accuracy is taken.  tag | ParticleNet | PFN
---|---|---
Efficiency | AUC | Efficiency | AUC
$b$ | 0.908 | 0.986 | 0.870 | 0.979
$c$ | 0.798 | 0.951 | 0.765 | 0.930
$q$ | 0.923 | 0.974 | 0.911 | 0.966

An alternative way to show the flavor tagging performance is the tagging
efficiencies versus the corresponding wrong flavour efficiencies, as the plots
in Fig. 4. For $b$-tagging, the main background is from $c$ jets. In the case
of $c$-tagging, the situation is different. The main background is from light
flavour jets at efficiency above 80%, while it is dominated by misidentified
$b$ jets at lower efficiency.

To demonstrate the physics impacts of jet flavor tagging, a detailed
comparison in terms of the product of efficiency and purity, $\epsilon\rho$,
is performed. Taking the measurement of $R_{b}$($R_{c}$) as an example, the
Eq. 5 gives the connection between its statistical uncertainty and
$\epsilon\rho$. It is known for decades that maximizing $\epsilon\rho$ is
identical to minimizing the statistical uncertainties.

$(\Delta R_{i})^{2}\propto\frac{1}{\epsilon_{i}\rho_{i}}.$ (5)

To compare the performance of various jet flavor tagging methods, some working
points are chosen. Table 5 summarizes the numerical results, where LCFIPlus
and XGBoost are taken as references. The table shows that the performance of
ParticleNet is much better than the others, especially in $c$-tagging.
ParticleNet is more than 50% better compared with LCFIPlus when the efficiency
of $c$-taggging is 60%. A specific example to illustrate the impact is that
ParticleNet could improve the statistical uncertainty in counting $c$ jets by
30% compared with the XGBoost. PFN also achieves comparable improvement and
confirms the correctness of ParticleNet.

Table 5: The performance of the specific method in different working points, where the results of LCFIPlus are reported in Ref. CDR-D , and the results of XGBoost are reported in Ref. yangfan  tag | $\epsilon_{S}$(%) | $\epsilon\times\rho$
---|---|---
LCFIPlus | XGBoost | ParticleNet | PFN
$b$ | 80 | - | 0.747 | 0.786 | 0.763
90 | 0.72 | 0.713 | 0.821 | 0.752
$c$ | 60 | 0.36 | - | 0.554 | 0.485
70 | - | - | 0.605 | 0.497
80 | - | 0.345 | 0.597 | 0.467
90 | - | 0.292 | 0.532 | 0.402

##  4 Measurement of relative decay width

In the LEP, $R_{b}$ is measured with various methods, which are based on
counting the events with either one or both hemisphere tagged. In this study,
jet is akin to hemisphere in LEP and it would be used in the rest of this
paper. The observed number of jets of flavor $i$ (single tag),
$N_{s}^{i,\mathrm{obs}}$, and the observed number of jet pairs (double tag),
$N_{d}^{i,\mathrm{obs}}$, are given by:

$\begin{split}N_{s}^{i,\mathrm{obs}}=&2N^{h,\mathrm{pro}}\cdot(R_{b}\varepsilon_{ib}+R_{c}\varepsilon_{ic}+R_{q}\varepsilon_{iq})~{},\\\
N_{d}^{i,\mathrm{obs}}=&N^{h,\mathrm{pro}}\cdot[R_{b}\varepsilon_{ib}^{2}(1+C_{ib})+R_{c}\varepsilon_{ic}^{2}(1+C_{ic})\\\
&+R_{q}\varepsilon_{iq}^{2}(1+C_{iq})]~{},\end{split}$ (6)

where $i(j)=b,c,q$ are flavors of jets, $C_{ij}$ is the correlation between a
jet pair of flavor $j$ when both are tagged as $i$, $\varepsilon_{ij}$ is the
efficiency of a $j$ jet being tagged as a $i$ jet, $N^{h,\mathrm{pro}}$ is
total number of $Z$ hadronic events produced in collisions, $R_{i}$ is the
relative decay widths of $Z$ to jet pair of $i$.

$R_{c}$ measurement is more challenging than $R_{b}$, since the $c$-tagging
has less efficiency and less purity than $b$-tagging. Therefore, several
methods are employed, such as double tag measurement, charm counting, etc. In
fact, the key ingredient of a relative partial width measurement is
classifying the signal and background correctly, i.e., jet flavor tagging.

To measure $R_{b}$($R_{c}$), the double tag method is deployed, which solves
the Eq. (6) to get $R_{b}$($R_{c}$) ($R_{q}=1-R_{b}-R_{c}$ by definition) when
a working point is determined. All the $\varepsilon_{ij}$ could be determined
by MC simulation and the correlation between jets could be neglected
temporarily. Signal regions of $b$, $c$, and $q$ candidates are defined as the
red lines, i.e, working point, in Fig. 5. There are 2 equations for each
region, and 6 in total. As over-determined equations, they could be solved by
the least square method. Using the same integrated luminosity assumed in Ref.
Li:2021zlv , a toy MC approach is used to calculate the statistical
uncertainty of $R_{b}$($R_{c}$). A total number of $10^{11}$ $Z$ hadronic
decay events is sampled according to Poisson distribution, and then this
number is sampled into three categories, $b\bar{b}$, $c\bar{c}$, and
$q\bar{q}$, according to multinomial distribution. The detection and selection
procedures are also simulated according to multinomial distribution. Finally,
three observed numbers are obtained by adding sampling results. Now
$R_{b}$($R_{c}$) could be calculated with the least square method, as well as
its statistical uncertainty.

Figure 5: The two-dimensional distribution of $b$-likeliness versus
$c$-likeliness, where the red lines indicate one of the jet classifications.
The upper left triangle is the candidate area of $b$, the lower left rectangle
is the candidate area of $q$, and the lower right triangle is the candidate
area of $c$.

The results are summarized in Table 6. The measurements of the LEP/SLC
ALEPH:2005ab , FCC-ee Bernardi:2022hny , and Ref. Li:2021zlv  are also listed
for comparison purposes. The uncertainties of relative decay width in LEP/SLC
ALEPH:2005ab  are primarily limited by statistics. The template fit Li:2021zlv
got excellent precision by using a much larger sample size and more
information. The double tag also achieves comparable precision as the template
fit on $R_{b}$, but for $R_{c}$, the precision is improved by nearly 40%,
thanks to the superior $c$-tagging performance of the DL algorithm compared to
LCFIPlus. The statistical uncertainty in FCC-ee is $0.3\times 10^{-6}$.
However, it should be noticed that the statistics used at FCC-ee are 50 times
larger than those used in Ref. Li:2021zlv  and this study. If the same
integrated luminosity is assumed, it would be $2.1\times 10^{-6}$. So the
results of Ref. Li:2021zlv  and this study are much better because the FCC-ee
simply extrapolates the results from LEP, while the other two studies employ
innovative analysis methods and enhanced detector designs.

Table 6: Statistical uncertainties ($10^{-6}$) of relative decay widths. The results of LEP/SLC ALEPH:2005ab , FCC-ee Bernardi:2022hny , and template fit Li:2021zlv  are reported. The flavor tagging methods employed in Template fit and Double tag are also listed. | $\sigma_{R_{b}}$ | $\sigma_{R_{c}}$ | $\sigma_{R_{q}}$ | flavor tagging method
---|---|---|---|---
LEP+SLC | 659 | 3015 | - | -
FCC-ee | 2.1(0.3) | - | - | -
Template fit | 1.2 | 2.3 | 2.1 | LCFIPlus
Double tag | 1.3 | 1.4 | - | ParticleNet

##  5 Summary and discussion

This study utilizes two DL algorithms to enhance the performance of jet flavor
tagging. ParticleNet, in particular, shows significant improvement in jet
flavor tagging, especially with regard to $c$-tagging. In terms of the product
of purity and efficiency, the $c$-tagging is improved by over 50% compared to
the CEPC baseline software when the efficiency of $c$-tagging is 60%. It’s
understandable why ParticleNet achieves significantly better performance.
Compared to the traditional methods, ParticleNet can maximize the usage of
information in a jet, as it uses lower-level information, such as momenta,
energies, and impact parameters, as input. On the other hand, the point-cloud
(set) representation, which preserves some important symmetries, has better
expressive power for jets Lorentz-symmetry .

$R_{b}$($R_{c}$) is used as a test bed to demonstrate the physics impacts of
the new DL algorithm. The results indicate that the precision of $R_{c}$ can
be improved by a factor of 1.6 compared to those in Ref. Li:2021zlv . In a
high-precision study as this, the systematic uncertainties pose a significant
challenge and require careful investigation in future studies.

#### Acknowledgements

This work is partially supported by the National Natural Science Foundation of
China (NSFC) (12075271, 12047569) and Guangxi University Young and Middle-aged
Teachers Research Basic Research Ability Improvement Project (2023KY0707).

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