_White Paper for RF01 working group in Snowmass 2021_

Charm Physics in the High-Luminosity Super $\tau$-Charm Factory

Hai-Yang Cheng,a 111E-mail<EMAIL_ADDRESS>Xiao-Rui Lyu,b
222E-mail<EMAIL_ADDRESS>Zhi-Zhong Xingb,c 333E-mail<EMAIL_ADDRESS>
a Institute of Physics, Academia Sinica, Taiwan 11529, China

b University of Chinese Academy of Sciences, Beijing 100049, China

c Institute of High Energy Physics, Beijing 100049, China

(on behalf of the STCF working group)

###### Abstract

This is a white paper on the STCF charm physics contributing to the Snowmass
2021 proceedings in the RF01 working group.

##  1 Introduction

The discovery of the charm quark in 1974 was a great milestone in the
development of particle physics and the establishment of the standard model
(SM). A high-luminosity Super $\tau$-Charm Factory (STCF) [1, 2, 3], which is
capable of producing about $10^{9}\sim 10^{10}$ quantum-coherent
$D^{0}\bar{D}^{0}$ meson pairs and $D_{(s)}^{+}$ mesons, more than $10^{8}$
$\Lambda_{c}^{+}$ baryons as well as heavier charmed baryons, will be an
important low-background playground to test the standard model (SM) and probe
new physics, regarding to the experience at BESIII [4]. In particular, it will
serve as a unique tool to determine the Cabbibo-Kobayashi-Maskawa (CKM) matrix
elements $V_{cd}$ and $V_{cs}$, to measure $D^{0}$-$\bar{D}^{0}$ mixing
parameters, to probe CP violation in the charm sector, to search for rare and
forbidden charmed hadron decays, and to study other fundamental problems
associated with the charmed hadron. Many of the golden measurements at STCF
will be dominated by systematic uncertainties, which requires a state-of-art
detector with excellent performance, especially in identifying the types of
different charged particles, detecting low-momentum charged particles and
measuring photons [5].

##  2 Charmed meson

###  2.1 $D^{+}_{(s)}$ leptonic decays

Direct determination of the CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$ is
one of the most important targets in charm physics. These two quark flavor
mixing quantities not only govern the rates of leptonic $D^{+}$ and
$D^{+}_{s}$ decays, but also play a crucial role in testing the unitarity of
the CKM matrix. Precise measurement of $|V_{cd}|$ and $|V_{cs}|$ is a priority
of the STCF experiment.

Table 1: For the studies on $D^{+}_{(s)}\to\ell^{+}\nu_{\ell}$, the obtained precisions at BESIII and projected precisions at STCF and Belle II. Considering that the LQCD uncertainty of $f_{D_{(s)}^{+}}$ has been updated to be about 0.2% [11], the $|V_{cd}|$ measured at BESIII has been re-calculated, and is marked with ∗. Preliminary results are marked with †. For Belle II, we assume that the systematic uncertainties can be reduced by a factor of 2 compared to Belle’s results. | BESIII | STCF | Belle II
---|---|---|---
Luminosity | 2.93 fb-1 at 3.773 GeV | 1 ab-1 at 3.773 GeV | 50 ab-1 at $\Upsilon(nS)$
${\mathcal{B}}(D^{+}\to\mu^{+}\nu_{\mu})$ |  $5.1\%_{\rm stat}\,1.6\%_{\rm syst}$ [6] | $0.28\%_{\rm stat}$ | –
$f_{D^{+}}$ (MeV) |  $2.6\%_{\rm stat}\,0.9\%_{\rm syst}$ [6] | $0.15\%_{\rm stat}$ | –
$|V_{cd}|$ |  $2.6\%_{\rm stat}\,1.0\%_{\rm syst}^{*}$ [6] | $0.15\%_{\rm stat}$ | –
${\mathcal{B}}(D^{+}\to\tau^{+}\nu_{\tau})$ |  $20\%_{\rm stat}\,{10\%_{\rm syst}}$ [7] | $0.41\%_{\rm stat}$ | –
$\displaystyle\frac{{\mathcal{B}}(D^{+}\to\tau^{+}\nu_{\tau})}{{\mathcal{B}}(D^{+}\to\mu^{+}\nu_{\mu})}$ |  $21\%_{\rm stat}\,13\%_{\rm syst}$ [7] | $0.50\%_{\rm stat}$ | –
Luminosity | 3.2 fb-1 at 4.178 GeV | 1 ab-1 at 4.009 GeV | 50 ab-1 at $\Upsilon(nS)$
${\mathcal{B}}(D^{+}_{s}\to\mu^{+}\nu_{\mu})$ |  $2.8\%_{\rm stat}\,2.7\%_{\rm syst}$ [8] | $0.30\%_{\rm stat}$ | $0.8\%_{\rm stat}\,1.8\%_{\rm syst}$
$f_{D^{+}_{s}}$ (MeV) |  $1.5\%_{\rm stat}\,1.6\%_{\rm syst}$ [8] | $0.15\%_{\rm stat}$ | –
$|V_{cs}|$ |  $1.5\%_{\rm stat}\,1.6\%_{\rm syst}$ [8] | $0.15\%_{\rm stat}$ | –
$f_{D^{+}_{s}}/f_{D^{+}}$ |  $3.0\%_{\rm stat}\,1.5\%_{\rm syst}$ [8] | $0.21\%_{\rm stat}$ | –
${\mathcal{B}}(D^{+}_{s}\to\tau^{+}\nu_{\tau})$ | $2.2\%_{\rm stat}\,2.6\%_{\rm syst}^{\dagger}$ | $0.24\%_{\rm stat}$ | $0.6\%_{\rm stat}\,2.7\%_{\rm syst}$
$f_{D^{+}_{s}}$ (MeV) | $1.1\%_{\rm stat}\,1.5\%_{\rm syst}^{\dagger}$ | $0.11\%_{\rm stat}$ | –
$|V_{cs}|$ | $1.1\%_{\rm stat}\,1.5\%_{\rm syst}^{\dagger}$ | $0.11\%_{\rm stat}$ | –
$\overline{f}^{\mu\&\tau}_{D^{+}_{s}}$ (MeV) | $0.9\%_{\rm stat}\,1.0\%_{\rm syst}^{\dagger}$ | $0.09\%_{\rm stat}$ | $0.3\%_{\rm stat}\,1.0\%_{\rm syst}$
$|\overline{V}_{cs}^{\mu\&\tau}|$ | $0.9\%_{\rm stat}\,{1.0\%_{\rm syst}}^{\dagger}$ | $0.09\%_{\rm stat}$ | –
$\displaystyle\frac{{\mathcal{B}}(D^{+}_{s}\to\tau^{+}\nu_{\tau})}{{\mathcal{B}}(D^{+}_{s}\to\mu^{+}\nu_{\mu})}$ | $3.6\%_{\rm stat}\,3.0\%_{\rm syst}^{\dagger}$ | $0.38\%_{\rm stat}$ | $0.9\%_{\rm stat}\,3.2\%_{\rm syst}$

The most precise way to determine $|V_{cd}|$ and $|V_{cs}|$ at STCF is via
pure-leptonic decays $D_{(s)}^{+}\to\ell^{+}\nu_{\ell}$ (for
$\ell=e,\mu,\tau$), as the semi-leptonic decay suffers from large
uncertainties of LQCD calculations of form factors. The product of the decay
constant $f_{D_{(s)}^{+}}$, and $|V_{cd(s)}|$ is directly accessed by
measuring the widths of $D_{(s)}^{+}\to\ell^{+}\nu_{\ell}$. Then with the
input of $f_{D_{(s)}^{+}}$ from LQCD, the value of $|V_{cd(s)}|$ or
$f_{D_{(s)}^{+}}$ can be obtained. Listed in Table 1 are the most precise
determinations of $|V_{cs(d)}|$ and $f_{D^{+}_{(s)}}$ [6, 7, 8] at BESIII and
the projected precisions at STCF [9, 10]. Note that for
${\mathcal{B}}(D^{+}\to\tau^{+}\nu_{\tau})$, several $\tau^{+}$ decay
channels, such as $\tau^{+}\to\pi^{+}\overline{\nu}_{\tau}$,
$e^{+}\overline{\nu}_{\tau}\nu_{e}$, $\mu^{+}\overline{\nu}_{\tau}\nu_{\mu}$,
and $\rho^{+}\overline{\nu}_{\tau}$, are combined to improve statistical
sensitivities.

The systematic uncertainties at STCF are to be optimized to a subleading
level, as the statistical uncertainties are expected be less than 0.5%. To
reduce the systematic uncertainty due to background and fitting, it becomes
optimal for STCF to study $D_{s}^{+}\to\ell^{+}\nu_{\ell}$ using
$e^{+}e^{-}\to D_{s}^{+}D_{s}^{-}$ at 4.009 GeV. So far, $f_{D_{(s)}^{+}}$ are
calculated by LQCD with precisions of about 0.2% [11], which are given as
$f_{D}^{+}=212.7\pm 0.6$ MeV, $f_{D_{s}}^{+}=249.9\pm 0.4~{}\,\mathrm{MeV}$
and $f_{D_{s}}^{+}/f_{D}^{+}=1.1749\pm 0.0016$. At the time of STCF, their
precisions are expected to be below 0.1%. This means that the sizes of
systematic uncertainties at STCF are crucial and necessary to be improved to
the level of 0.1%. Among them, the efficiencies of muon and electron
identifications will be the critical issues, which is required to be optimized
in order to constrain the total uncertainty to the level of 0.1%.

On the other hand, the precise measurements of the semi-leptonic branching
fractions for $D_{(s)}\to h\ell^{+}\nu_{\ell}$, where $h$ is a charmless
hadron, will be used to calibrate LQCD calculations of the involved form
factors, by introducing the $|V_{cd(s)}|$ from global CKM fits (such as
CKMfitter [12, 13] and UTfit [14, 15]). For the case of $D_{(s)}\to
V(h_{1}h_{2})\ell^{+}\nu_{\ell}$ ($V$ denotes a vector meson, decaying into
hadrons $h_{1}$ and $h_{2}$), a time reversal (T) invariance can be tested in
high precision by constructing triple product T-odd observables [16]. This
will serve as a sensitive probe of CP violation mechanisms beyond standard
model and new physics [17], such as those with multi-Higgs doublets or
leptoquarks. Ref. [18] proposes combined measurements of $D\to
K_{1}(1270)\ell^{+}\nu_{\ell}$ and $B\to K_{1}(1270)\gamma$ to unambiguously
determine the photon polarization in $b\to s\gamma$ in a clean way to probe
right-handed couplings in new physics. A feasibility study shows that a
statistical sensitivity of $1.5\times 10^{-2}$ for the ratio of up-down
asymmetry can be reached based on about 60 thousands signals of $D^{0}\to
K_{1}(1270)^{-}e^{+}\nu$ with 1 ab-1 of data at 3.773 GeV at STCF [19].

Lepton flavor universality (LFU) can also be tested in charmed meson leptonic
decays. LFU violation may happen in $c\to s$ transitions due to an amplitude
that includes a charged Higgs boson, that arises in a two-Higgs-doublet model,
interfering with the SM amplitude involving a $W^{\pm}$ boson [20]. In the SM,
the ratio of the partial widths of $D^{+}_{(s)}\to\tau^{+}\nu_{\tau}$ and
$D^{+}_{(s)}\to\mu^{+}\nu_{\mu}$ is predicted to be

$\displaystyle
R_{D^{+}_{(s)}}=\frac{\Gamma(D^{+}_{(s)}\to\tau^{+}\nu_{\tau})}{\Gamma(D^{+}_{(s)}\to\mu^{+}\nu_{\mu})}=\frac{m^{2}_{\tau^{+}}\left(1-\frac{m^{2}_{\tau^{+}}}{m^{2}_{D^{+}_{(s)}}}\right)^{2}}{m^{2}_{\mu^{+}}\left(1-\frac{m^{2}_{\mu^{+}}}{m^{2}_{D^{+}_{(s)}}}\right)^{2}}.$
(1)

Using the world average values of the masses of leptons and $D^{+}_{(s)}$
[21], one obtains $R_{D^{+}}=2.67\pm 0.01$ and $R_{D^{+}_{s}}=9.75\pm 0.01$.
The preliminary measured value of $R_{D^{+}_{(s)}}$ reported by BESIII is
$3.21\pm 0.64$ $(10.2\pm 0.5)$, which agrees with the SM predicted values.
However, these measurements are currently statistically limited. At STCF, as
listed in Table 1, the statistical precision on $R_{D_{(s)}^{+}}$ will be
comparable to the uncertainties of the predictions in the SM. Hence, it will
provide meaningful test on LFU via these channels [9, 10].

Another LFU test would be via the semi-leptonic decay modes, where the semi-
tauonic decay is kinematically forbidden or suppressed. Measurements of the
ratios of the partial widths of $D^{0(+)}\to h\mu^{+}\nu_{\mu}$ over those of
$D^{0(+)}\to he^{+}\nu_{e}$ in different $q^{2}$ intervals constitute a
complementary test of LFU to those using tauonic decays. BESIII reported
precise measurements of the ratios
${\mathcal{B}}(D^{0}\to\pi^{-}\mu^{+}\nu_{\mu})/{\mathcal{B}}(D^{0}\to\pi^{-}e^{+}\nu_{e})=0.922\pm
0.030\pm 0.022$ and
${\mathcal{B}}(D^{+}\to\pi^{0}\mu^{+}\nu_{\mu})/{\mathcal{B}}(D^{+}\to\pi^{0}e^{+}\nu_{e})=0.964\pm
0.037\pm 0.026$ [22]. These results are consistent with the SM predictions,
within $1.7\sigma$ and $0.5\sigma$ [22], respectively. These measurements are
currently statistically limited [23, 22], and will be significantly improved
with 1 ab-1 of data taken at 3.773 GeV at STCF.

###  2.2 $D^{0}$-$\bar{D}^{0}$ mixing and CP violation

The phenomenon of meson-antimeson mixing has been of great interest in the
long history of particle physics. Contrary to $B$-meson and Kaon systems, CP-
violation in mixing of $D$-mesons has not been observed. STCF will be an ideal
place for the study of $D^{0}$-$\bar{D}^{0}$ mixing and CP-violation. By
convention the mass states of two neutral $D$ mesons are written as

$\displaystyle|D_{1}\rangle$ $\displaystyle=$ $\displaystyle
p|D^{0}\rangle+q|\bar{D}^{0}\rangle\;,$ $\displaystyle|D_{2}\rangle$
$\displaystyle=$ $\displaystyle p|D^{0}\rangle-q|\bar{D}^{0}\rangle\;,$ (2)

where $|p|^{2}+|q|^{2}=1$. The $D^{0}$-$\bar{D}^{0}$ mixing parameters are
defined by $x\equiv(M_{2}-M_{1})/\Gamma$ and
$y\equiv(\Gamma_{2}-\Gamma_{1})/(2\Gamma)$, where $M_{1,2}$ and $\Gamma_{1,2}$
are the masses and widths of $D_{1,2}$, respectively. Also
$\Gamma\equiv(\Gamma_{1}+\Gamma_{2})/2$ and $M\equiv(M_{1}+M_{2})/2$. This
system is unique because it is the only meson-antimeson system whose mixing
(or oscillation) takes place via the intermediate states with down-type
quarks. It is also the only meson-antimeson system whose mixing parameters $x$
and $y$ are notoriously hard to calculate in the SM, as those involve large
long-distance uncertainties in this nonperturbative regime. One expects $x\sim
y\sim\sin^{2}\theta_{\rm C}\times[{\rm SU(3)~{}breaking}]^{2}$ as the second-
order effect of the flavor SU(3) symmetry breaking. A more careful analysis
yields the order-of-magnitude estimates $x\lesssim y$ and
$10^{-3}<|x|<10^{-2}$ [24]. A global fit to the world measurements of $x$ and
$y$, carried out by the Heavy Flavor Averaging Group [25, 26], gives
$1.6\times 10^{-3}\lesssim x\lesssim 6.1\times 10^{-3}$ and $5.2\times
10^{-3}\lesssim y\lesssim 7.9\times 10^{-3}$ at the $95\%$ confidence-level
intervals [25, 26]. We see that the allowed region of $x$ and $y$ are
essentially consistent with the theoretical estimates (i.e., $x\lesssim y\sim
7\times 10^{-3}$). Much more precise measurements of these two
$D^{0}$-$\bar{D}^{0}$ mixing parameters can be achieved at STCF. While their
accurate values might not help much to clarify the long-distance effects in
$D^{0}$-$\bar{D}^{0}$ mixing, they will help a lot to probe the presumably
small effects of CP violation in neutral $D$-meson decays and mixing [28].

The charm sector is a precision laboratory to explore possible CP-violating
new physics, because the SM-induced CP-violating asymmetries in $D$-meson
decays are typically in the range from $10^{-4}$ to $10^{-3}$[27] and are very
challenging to be detected in experiment. The CP-violating asymmetries in the
singly Cabibbo-suppressed $D$-meson decays are now expected to be much larger
than those in the Cabibbo-favored and doubly Cabibbo-suppressed decays [28],
where such asymmetries vanish. There are in general three different types of
CP-violating effects in neutral and charged $D$-meson decays [29]: 1) CP
violation in $D^{0}$-$\bar{D}^{0}$ mixing; 2) CP violation in the direct
decay; 3) CP violation from the interplay of decay and mixing. Besides these
three types of CP-violating effects in $D$-meson decays, one may expect the
effect of CP violation induced by $K^{0}$-$\bar{K}^{0}$ mixing in some decay
modes with $K_{\rm S}$ or $K_{\rm L}$ in their final states. Its magnitude is
typically $2{\rm Re}(\epsilon_{K})\simeq 3.3\times 10^{-3}$, which may be
comparable with or even larger than the charmed CP-violating effects [30, 31].
So far a lot of effort has been put into searching for CP violation in
$D$-meson decays. The LHCb Collaboration has recently discovered CP violation
in combined $D^{0}\to\pi^{+}\pi^{-}$ and $D^{0}\to K^{+}K^{-}$ decays with the
significance of 5.3$\sigma$. The time-integrated CP-violating asymmetry is
given as

$\displaystyle\Delta a_{CP}$ $\displaystyle=$ $\displaystyle\frac{\Gamma(D\to
K^{+}K^{-})-\Gamma(\bar{D}\to K^{+}K^{-})}{\Gamma(D\to
K^{+}K^{-})+\Gamma(\bar{D}\to
K^{+}K^{-})}-\frac{\Gamma(D\to\pi^{+}\pi^{-})-\Gamma(\bar{D}\to\pi^{+}\pi^{-})}{\Gamma(D\to\pi^{+}\pi^{-})+\Gamma(\bar{D}\to\pi^{+}\pi^{-})}$
(3) $\displaystyle=$ $\displaystyle(-0.154\pm 0.029)\%,$

where $D$($\bar{D}$) is a $D^{0}$($\bar{D}{}^{0}$) at time $t$=0 [32], and it
mainly arises from direct CP violation in the charm-quark decay [33]. This
result is consistent with some theoretical estimates within the SM (see, e.g.,
Refs. [34, 35, 36, 37, 38, 39, 40, 41]), but the latter involve quite large
uncertainties. STCF will have a $10^{-4}$ level of sensitivity on
systematically searching for CP violation in different types of charm meson
decays. Especially, advantages of kinematical constraints to the initial four-
momenta of $e^{+}e^{-}$ collisions will make STCF competitive in studies of
CP-violating asymmetries in multi-body $D$-decays [42], such as 4-body
hadronic decays and therein CP asymmetries in local Dalitz region. As the CKM
mechanism of CP violation in the SM fails to explain the puzzle of the
observed matter-antimatter asymmetry in the Universe by more than 10 orders of
magnitude [43], it is well motivated to search for new (heretofore
undiscovered) sources of CP violation associated with both quark and lepton
flavors. In this connection the charm-quark sector is certainly a promising
playground.

Note that STCF will be a unique place for the study of $D^{0}$-$\bar{D}^{0}$
mixing and CP violation by means of quantum coherence of $D^{0}$ and
$\bar{D}^{0}$ mesons produced at the energy points near threshold. In fact, a
$D^{0}\bar{D}^{0}$ pair can be coherently produced through reactions
$e^{+}e^{-}\to(D^{0}\bar{D}^{0})_{\rm CP=-}$ at 3.773 GeV and $e^{+}e^{-}\to
D^{0}\bar{D}^{*0}\to\pi^{0}(D^{0}\bar{D}^{0})_{\rm CP=-}$ or
$\gamma(D^{0}\bar{D}^{0})_{\rm CP=+}$ at 4.009 GeV. One may therefore obtain
useful constraints on $D^{0}$-$\bar{D}^{0}$ mixing and CP-violating parameters
in the respective decays of correlated $D^{0}$ and $\bar{D}^{0}$ events [29].
For example, the $D^{0}$-$\bar{D}^{0}$ mixing rate $R_{M}=(x^{2}+y^{2})/2$ can
be accessed via the same charged final states
$(K^{\pm}\pi^{\mp})(K^{\pm}\pi^{\mp})$ or
$(K^{\pm}\ell^{\mp}\nu)(K^{\pm}\ell^{\mp}\nu)$ with a sensitivity of $10^{-5}$
with 1 ab-1 data at 3.773 GeV. Considering $e^{+}e^{-}\to\gamma
D^{0}\bar{D}^{0}$ at 4.009 GeV, $D^{0}\bar{D}^{0}$ pairs are in C-even states
and charm mixing contribution is doubled as compared with the time-dependent
(un-correlated) case. With 1 ab-1 data at 4.009 GeV, it is expected that the
measurement sensitivities of the mixing parameters ($x$, $y$) will reach a
level of 0.05%, and those of $|q/p|$ and $\arg(q/p)$ will be 1.5% and
$1.4^{\circ}$, respectively [44]. These sensitivities are complementary to the
future precision measurements foreseen at Belle II and the LHCb upgrades.
Another case is that the decay mode $\left(D^{0}\bar{D}^{0}\right)_{\rm
CP=\pm}\to\left(f_{1}f_{2}\right)_{\rm CP=\mp}$, where $f_{1}$ and $f_{2}$ are
proper CP eigenstates (e.g., $\pi^{+}\pi^{-}$, $K^{+}K^{-}$ and $K_{\rm
S}\pi^{0}$), is a CP-forbidden process and can only occur due to CP violation.
The rate of a pair of CP-even final states $f_{+}$ (such as
$f_{+}=\pi^{+}\pi^{-}$) can be expressed as

$\Gamma^{++}_{D^{0}\bar{D}^{0}}=\left[\left(x^{2}+y^{2}\right)\left(\cosh^{2}a_{m}-\cos^{2}\phi\right)\right]\Gamma^{2}(D\to
f_{+}),$ (4)

where $\phi=\arg(p/q)$, $R_{M}=|p/q|$, and $a_{m}=\log R_{M}$ [45].

CPT is conserved in all local Lorentz-invariant theories, which includes the
SM and its all commonly-discussed extensions. When CPT is conserved, CP
violation implies time reversal (T) symmetry violation. Yet, CPT violation
might arise in string theory or some extra-dimensional models with Lorentz-
symmetry violation in four dimensions. Hence, direct observation of T
violation without the presumption of CPT conservation is very important [46].
Experimental studies of the time evolution of CP-correlated
$D^{0}$-$\bar{D}^{0}$ states at STCF could be complementary to CPT-violation
studies at the super-$B$ factories and the LHCb experiments [47]. However,
this becomes very challenging with symmetric $e^{+}e^{-}$ collisions, as the
produced $D$ mesons have very low momentum in the laboratory frame, and hence
have too small flight distances to be detected. Only asymmetric $e^{+}e^{-}$
collision mode can be feasible for this topic.

The quantum correlation of the $D^{0}\bar{D}^{0}$ meson pair has a unique
feature to probe the amplitudes of the $D^{0}$ decays and determine the
strong-phase difference between their Cabibbo-favored and doubly Cabibbo-
suppressed amplitudes [48]. Measurements of the strong-phase difference are
well motivated in several aspects: understanding the non-perturbative QCD
effects in the charm sector; serving as essential inputs to extract the angle
$\gamma$ of the CKM unitarity triangle (UT), and relating the measured mixing
parameters in hadronic decay $(x^{\prime},y^{\prime})$ to the mass and width
difference parameters $(x,y)$ [25].

The measurements of the CKM unitary triangle (UT) angles $\alpha$, $\beta$,
and $\gamma$ in $B$ decays are important tests of the CKM unitarity and search
for possible CP violation beyond the SM. Any discrepancy in the measurements
of the UT involving tree- and loop-dominated processes would indicate the
existence of heavy new degrees of freedom contributing to the loops. Among the
three CKM angles, $\gamma$ is of particular importance because it is the only
CP-violating observable that can be determined using tree-level decays.
Currently the world-best single measurement of $\gamma$ is from LHCb:
$\gamma=(69\pm 5)^{\circ}$ [49]. The precision measurement of $\gamma$ will be
one of the top priorities for the LHCb upgrade(s) and Belle II experiments.

The most precise method to measure $\gamma$ is based upon the interference
between $B^{+}\to\bar{D}^{0}K^{+}$ and $B^{+}\to D^{0}K^{+}$ decays [50, 51,
52]. In the future, the statistical uncertainties of these measurements will
be greatly reduced by using the large $B$ meson samples recorded by LHCb and
Belle II. Hence, limited knowledge of the strong phases of the $D$ decays will
systematically restrict the overall sensitivity. A 20 fb-1 of data set at
3.773 GeV at BESIII would lead to a systematic uncertainty of $\sim$0.4∘ for
the $\gamma$ measurement [53]. Hence, to match the future statistical
uncertainty of less than $0.4^{\circ}$ in the future LHCb upgrade II, STCF
would provide important constraints to reduce the systematic uncertainty from
$D$ strong-phase to be less than 0.1∘ and allow detailed comparisons of the
$\gamma$ results from different decay modes.

###  2.3 Rare and forbidden decays

With high luminosity, clean collision environment and excellent detector
performance, STCF has great potential to perform searches for rare and
forbidden $D$-meson decays, which may serve as a useful tool for probing new
physics beyond the SM. They can be classified into three categories: (1)
decays via the flavor-changing neutral current (FCNC), such as
$D^{0(+)}\to\gamma V^{0(+)}$, $D^{0}\to\gamma\gamma$,
$D^{0}\to\ell^{+}\ell^{-}$, $D\to\ell^{+}\ell^{-}X$ channels (for
$\ell=e,\mu$), and $D\to\nu\overline{\nu}X$, which provide a SM-allowed
transition between $c$ and $u$ quarks; (2) decays with lepton flavor violation
(LFV), such as $D^{0}\to\ell^{+}\ell^{\prime-}$ and
$D\to\ell^{+}\ell^{\prime-}X$ channels (for $\ell\neq\ell^{\prime}$), which
are forbidden in the SM; (3) decays with lepton number violation (LNV), such
as $D^{+}\to\ell^{+}\ell^{\prime+}X^{-}$ and
$D^{+}_{s}\to\ell^{+}\ell^{\prime+}X^{-}$ channels (for either
$\ell=\ell^{\prime}$ or $\ell\neq\ell^{\prime}$), which are also forbidden in
the SM. The discoveries of neutrino oscillations have confirmed LFV in the
lepton sector, and LNV is possible if massive neutrinos are the Majorana
particles. It is therefore meaningful to search for the LFV and LNV phenomena
in the charm-quark sector.

Although the FCNC decays of $D$ mesons are allowed in the SM, they can only
occur via the loop diagrams and hence are strongly suppressed. The long-
distance dynamics is expected to dominate the SM contributions to such decays,
but their branching fractions are still tiny. For instance, ${\cal
B}(D^{0}\to\gamma\gamma)\sim 1\times 10^{-8}$ and ${\cal
B}(D^{0}\to\mu^{+}\mu^{-})\sim 3\times 10^{-13}$ in the SM [54], but they can
be significantly enhanced by new physics [55]. Current experimental bounds on
these two typical FCNC channels are ${\cal B}(D^{0}\to\gamma\gamma)<8.5\times
10^{-7}$ and ${\cal B}(D^{0}\to\mu^{+}\mu^{-})<6.2\times 10^{-9}$ [21].
However, the following semi-leptonic decays of
$D^{0}\to\pi^{+}\pi^{-}\mu^{+}\mu^{-}$, $K^{+}K^{-}\mu^{+}\mu^{-}$ and
$K^{-}\pi^{+}\mu^{+}\mu^{-}$ have been observed at LHCb with the BF level of
$10^{-7}$ [21]. Besides the removal of helicity suppression as dominates the
highly suppressed BF for $D^{0}\to\mu^{+}\mu^{-}$, the observed BFs for the
semi-leptonic decays indicate non-trivial contributions from complicated long-
distance effects. At STCF, it is more optimal to study the di-electron modes
$D\to e^{+}e^{-}X$ [56], which provide sensitivities of $10^{-8}\sim 10^{-9}$
for $m_{e^{+}e^{-}}$ in the range less polluted by the long-range resonance
contributions. Compared to Belle II and LHCb, STCF has competitive
sensitivities in the channels which contains neutral final states, such as
photon and $\pi^{0}$, because of clean backgrounds. Furthermore, STCF has
advantage to best constrain the upper limit of BF for $D$ rare decays with
neutrinos, such as $D^{0}\to\pi^{0}\nu\overline{\nu}$ and
$D^{0}\to\gamma\nu\overline{\nu}$.

No evidence has been found for the forbidden $D_{(s)}$-meson decays with
either LFV or LNV, or both of them. The present experimental bounds on the LFV
decays are generally set at the level of $10^{-6}$ to $10^{-5}$ (with an
exception of ${\cal B}(D^{0}\to\mu^{\pm}e^{\mp})<1.3\times 10^{-8}$) [21]. A
STCF will provide more stringent limits on such interesting LFV and LNV decay
modes, with a sensitivity of $10^{-8}$ to $10^{-9}$ or smaller, taking
advantage of its clean environment and accurate charge discrimination.

###  2.4 Charmed meson spectroscopy

STCF will also act as a good playground to study the production of charmed
mesons and explore the charmed meson spectroscopy. So far, all the 1$S$ and
1$P$ $D_{(s)}$ states have been found in experiment [57]. However, for other
quantum states, almost all other predicted excited states in QCD-derived
effective models are missing. Furthermore, there are many excited open-charm
states reported in experiment, which are still controversial in understanding
their natures. Some of them are candidates of exotic mesons. For instance, the
narrow $D^{*}_{sJ}(2632)$ state is observed by SELEX, but CLEO, BaBar and
FOCUS all reported negative search results. The unexpected low masses of the
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ bring in various exotic explanations,
such as $D^{(*)}K$ molecule state [58]. It has been claimed that the strong
$S$-wave $D^{(*)}K$ scattering contributes to the mass drop. More systematic
researches on the open-charm meson spectroscopy are highly desired.

At STCF, excited charmed meson states $D^{**}$ can be produced via direct
$e^{+}e^{-}$ production processes, such as $e^{+}e^{-}\to
D^{**}\bar{D}^{(*)}(\pi)$, in the energy rang from 4.1 to 7.0 GeV. Then, the
higher excited open-charm states can be studied through their hadronic or
radiative decays [59] to lower open-charm states. Systematical studies at STCF
on the open-charm meson spectra provide important data to explore the non-
perturbative QCD dynamics in the charm regime and test various theoretical
models.

##  3 Charmed baryon

Theoretical interest in hadronic weak decays of charmed baryons peaked around
the early 1990s and then faded away. Nevertheless, there are two major
breakthroughs in recent charmed-baryon experiments in regard to hadronic weak
decays of $\Lambda_{c}^{+}$. BESIII has played an essential role in these new
developments [4]. LHCb made significant progress on re-ordering the lifetime
hierarchy of charmed baryon lifetimes from
$\tau_{\Xi_{c}^{+}}>\tau_{\Lambda_{c}^{+}}>\tau_{\Xi_{c}^{0}}>\tau_{\Omega_{c}^{0}}$
to
$\tau_{\Xi_{c}^{+}}>\tau_{\Omega_{c}^{0}}>\tau_{\Lambda_{c}^{+}}>\tau_{\Xi_{c}^{0}}$
[62, 63, 64, 65]. Motivated by these experimental progresses, there exist
growing theoretical activities in the study of hadronic weak decays of singly
charm baryons.

Charm baryon spectroscopy provides an excellent ground for studying the
dynamics of light quarks in the environment of a heavy quark. In the past
decade, many new excited charmed baryon states have been discovered by BaBar,
Belle, CLEO and LHCb. $B$ decays and the $e^{+}e^{-}\to c\bar{c}$ continuum
are both very rich sources of charmed baryons. Many efforts have been made to
identify the quantum numbers of these new states and understand their
properties.

###  3.1 Hadronic weak decays

  * •

Nonleptonic decays of singly charmed baryons

$\Lambda_{c}$ decays The branching fractions of the Cabibbo-favored two-body
decays of $\Lambda_{c}^{+}$ are listed in Table 2. Many of them such as
$\Sigma^{+}\phi$, $\Xi^{(*)}K^{(*)+}$ and $\Delta^{++}K^{-}$ can proceed only
through $W$-exchange. Experimental measurement of them implies the importance
of $W$-exchange, which is not subject to color suppression in charmed baryon
decays. Both Belle [66] and BESIII [67] have measured the absolute branching
fraction of the decay $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$. A new average of
$(6.28\pm 0.32)\%$ for this benchmark mode is quoted by the Particle Data
Group (PDG) [21]. Doubly Cabibbo-suppressed decay $\Lambda_{c}^{+}\to
pK^{+}\pi^{-}$ has been observed by Belle [68] and LHCb [69]. It is thus
important to search for the two-body modes which are doubly Cabibbo-
suppressed: $pK^{0(*)}$ and $nK^{+(*)}$.

Table 2: The measured branching fractions of the Cabibbo-favored two-body decays of $\Lambda_{c}^{+}$ (in units of %) taken from 2020 Particle Data Group [21]. BESIII measurements of $\Lambda_{c}^{+}\to\Sigma^{+}\eta,\Sigma^{*+}\eta,\Sigma^{+}\eta^{\prime}$ are included. [60, 61]. Decay | $\mathcal{B}$ | Decay | $\mathcal{B}$ | Decay | $\mathcal{B}$
---|---|---|---|---|---
$\Lambda^{+}_{c}\to\Lambda\pi^{+}$ |  1.30$\pm$0.07 | $\Lambda^{+}_{c}\to\Lambda\rho^{+}$ | $<6$ | $\Lambda^{+}_{c}\to\Delta^{++}K^{-}$ | $1.08\pm 0.25$
$\Lambda^{+}_{c}\to\Sigma^{0}\pi^{+}$ |  1.29$\pm$0.07 | $\Lambda^{+}_{c}\to\Sigma^{0}\rho^{+}$ |  | $\Lambda^{+}_{c}\to\Sigma^{*0}\pi^{+}$ |
$\Lambda^{+}_{c}\to\Sigma^{+}\pi^{0}$ |  1.25$\pm$0.10 | $\Lambda^{+}_{c}\to\Sigma^{+}\rho^{0}$ | $<1.7$ | $\Lambda^{+}_{c}\to\Sigma^{*+}\pi^{0}$ |
$\Lambda^{+}_{c}\to\Sigma^{+}\eta$ |  0.53$\pm$0.15 | $\Lambda^{+}_{c}\to\Sigma^{+}\omega$ |  1.70$\pm$0.21 | $\Lambda^{+}_{c}\to\Sigma^{*+}\eta$ | $0.96\pm 0.17$
$\Lambda^{+}_{c}\to\Sigma^{+}\eta^{\prime}$ |  1.34$\pm$0.57 | $\Lambda^{+}_{c}\to\Sigma^{+}\phi$ |  0.38$\pm$0.06 | $\Lambda^{+}_{c}\to\Sigma^{*+}\eta^{\prime}$ |
$\Lambda^{+}_{c}\to\Xi^{0}K^{+}$ |  0.55$\pm$0.07 | $\Lambda^{+}_{c}\to\Xi^{0}K^{*+}$ |  | $\Lambda^{+}_{c}\to\Xi^{*0}K^{+}$ |  0.43$\pm$0.09
$\Lambda^{+}_{c}\to pK_{S}$ |  1.59$\pm$0.08 | $\Lambda^{+}_{c}\to p\bar{K}^{*0}$ |  1.96$\pm$0.27 | $\Lambda^{+}_{c}\to\Delta^{+}\bar{K}^{0}$ |

Various theoretical approaches to weak decays of heavy baryons have been
investigated, including the current algebra approach, factorization scheme,
pole model, relativistic quark model, quark diagram scheme and SU(3) flavor
symmetry. In general, the predicted decay rates by most of the models except
current algebra are below experimental measurements. Moreover, the decay
asymmetries of the two-body hadronic weak decays of charmed baryons can be
investigated, which are defined as $\alpha\equiv\frac{2{\rm
Re}(s^{*}p)}{|s|^{2}+|p|^{2}}$. Here $s$ and $p$ stand for the parity-
violating $s$-wave and parity-conserving $p$-wave amplitudes in the decay,
respectively. The pole model, the covariant quark model and its variant all
predict a positive decay asymmetry $\alpha$ for both
$\Lambda_{c}^{+}\to\Sigma^{+}\pi^{0}$ and $\Sigma^{0}\pi^{+}$, while it is
measured to be $-0.45\pm 0.31\pm 0.06$ for $\Sigma^{+}\pi^{0}$ by CLEO [70].
In contrast, current algebra always leads to a negative decay asymmetry for
aforementioned two modes: $-0.49$ in Ref. [71], $-0.31$ in Ref. [72], $-0.76$
in Ref. [73] and $-0.47$ in Ref. [74]. The issue with the sign of
$\alpha_{\Sigma^{+}\pi^{0}}$ was finally resolved by BESIII. The decay
asymmetry parameters of
$\Lambda_{c}^{+}\to\Lambda\pi^{+},\Sigma^{0}\pi^{+},\Sigma^{+}\pi^{0}$ and
$pK_{S}$ were recently measured by BESIII [75], for example,
$\alpha_{\Sigma^{+}\pi^{0}}=-0.57\pm 0.12$ was obtained. Hence, the negative
sign of $\alpha_{\Sigma^{+}\pi^{0}}$ measured by CLEO is confirmed by BESIII.
$\Xi_{c}$ and $\Omega_{c}$ decays The absolute branching fractions of
$\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$ and $\Xi_{c}^{+}\to\Xi^{-}\pi^{+}\pi^{+}$ were
recently measured by Belle [76, 77] to be

$\displaystyle\mathcal{B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})=(1.80\pm 0.50\pm
0.14)\%,\quad\mathcal{B}(\Xi_{c}^{+}\to\Xi^{-}\pi^{+}\pi^{+})=(2.86\pm 1.21\pm
0.38)\%.$ (5)

With these measurements, branching fractions of other $\Xi_{c}^{0}$ and
$\Xi_{c}^{+}$ decays can be inferred. No absolute branching fractions have
been measured for the $\Omega_{c}^{0}$. The hadronic weak decays of the
$\Omega_{c}^{0}$ were recently studied in great detail in Ref. [78], where
most of the decay channels in $\Omega_{c}^{0}$ decays were found to proceed
only through the $W$-exchange diagram.

It is conceivable that nonleptonic decay modes of $\Lambda_{c}^{+}$ and
$\Xi_{c}^{+,0}$ can be measured at STCF with significantly improved precision.
Priority will be ascribed to the decay asymmetries $\alpha$ in various charm
baryon decays and the absolute branching fractions of $\Omega_{c}^{0}$ decays.

  * •

Charm-flavor-conserving nonleptonic decays

There is a special class of weak decays of charmed baryons that can be studied
reliably, namely, heavy-flavor-conserving nonleptonic decays. Some examples
are the singly Cabibbo-suppressed decays $\Xi_{c}\to\Lambda_{c}\pi$ and
$\Omega_{c}\to\Xi^{\prime}_{c}\pi$. In these decays, only the light quarks
inside the heavy baryon will participate in weak interactions, while the heavy
quark behaves as a “spectator”. The synthesis of the heavy quark and chiral
symmetries provides a natural setting for investigating these reactions [79].
The predicted branching fractions for the charm-flavor-conserving decays
$\Xi_{c}^{0}\to\Lambda_{c}^{+}\pi^{-}$ and
$\Xi_{c}^{+}\to\Lambda_{c}^{+}\pi^{0}$ are of the order of $10^{-3}\sim
10^{-4}$ [79]. Very recently, the first measurement of the charm-flavor-
conserving decay $\Xi_{c}^{0}\to\Lambda_{c}^{+}\pi^{-}$ has been achieved by
the LHCb with the branching fraction $(0.55\pm 0.02\pm 0.18)\%$ [80], which is
in general larger than the theoretical predictions. STCF should be able to
cross-check this and search for another $c$-flavor-conserving weak decay,
namely, $\Xi_{c}^{+}\to\Lambda_{c}^{+}\pi^{0}$.

  * •

Semileptonic decays

Exclusive semileptonic decays of charmed baryons: $\Lambda_{c}^{+}\to\Lambda
e^{+}(\mu^{+})\nu_{e(\mu)}$, $\Xi_{c}^{+}\to\Xi^{0}e^{+}\nu_{e}$ and
$\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e}$ have been observed experimentally. Their
rates depend on the ${\cal B}_{c}\to{\cal B}$ form factors $f_{i}(q^{2})$ and
$g_{i}(q^{2})$ ($i=1,2,3$) defined as

$\displaystyle\langle{\cal B}_{f}(p_{f})|V_{\mu}|{\cal B}_{c}(p_{i})\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{f}(p_{f})[f_{1}(q^{2})\gamma_{\mu}+if_{2}(q^{2})\sigma_{\mu\nu}q^{\nu}+f_{3}(q^{2})q_{\mu}]u_{i}(p_{i}),$
$\displaystyle\langle{\cal B}_{f}(p_{f})|A_{\mu}|{\cal B}_{c}(p_{i})\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{f}(p_{f})[g_{1}(q^{2})\gamma_{\mu}+ig_{2}(q^{2})\sigma_{\mu\nu}q^{\nu}+g_{3}(q^{2})q_{\mu}]\gamma_{5}u_{i}(p_{i}).$
(6)

These form factors have been evaluated using the non-relativistic quark model
[81, 82, 83, 84], MIT bag model [81], relativistic quark model [85, 86, 87],
light-front quark model [88], QCD sum rules [89, 90, 91] and lattice QCD [92,
93]. Many of the early predictions of $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e})$ are smaller than the first measurement of the absolute
branching fraction of $(3.6\pm 0.4)\%$ by BESIII [94]. Lattice QCD
calculations in [92] yield good agreement with experiment for both
$\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$ and
$\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu}$. Needless to say, the semileptonic
decays of $\Lambda_{c}^{+}$ (including the yet-to-be-observed
$\Lambda_{c}^{+}\to ne^{+}\nu_{e}$), $\Xi_{c}^{+,0}$ and $\Omega_{c}^{0}$ will
be thoroughly studied at STCF, which can be used to discriminate between
different form-factor models.

###  3.2 Electromagnetic and weak radiative decays

The electromagnetic decays of interest in the charmed baryon sector are: (i)
$\Sigma_{c}\rightarrow\Lambda_{c}+\gamma,\Xi^{\prime}_{c}\rightarrow\Xi_{c}+\gamma$,
(ii)
$\Sigma^{\ast}_{c}\rightarrow\Lambda_{c}+\gamma,\Xi^{\ast}_{c}\rightarrow\Xi_{c}+\gamma$,
(iii)
$\Sigma^{\ast}_{c}\rightarrow\Sigma_{c}+\gamma,\Xi^{\ast}_{c}\rightarrow\Xi^{\prime}_{c}+\gamma,\Omega^{\ast}_{c}\rightarrow\Omega_{c}+\gamma$,
and (iv) $\Lambda_{c}(2595,2625)\to\Lambda_{c}+\gamma$,
$\Xi_{c}(2790,2815)\to\Xi_{c}+\gamma$. Among them, the decay modes
$\Xi^{\prime 0}_{c}\to\Xi_{c}^{0}\gamma$,
$\Xi^{\prime+}_{c}\to\Xi^{+}_{c}\gamma$ and
$\Omega_{c}^{*0}\to\Omega_{c}^{0}\gamma$ have been seen experimentally.

Table 3: Electromagnetic decay rates (in units of keV) of $s$-wave charmed baryons in heavy hadron chiral perturbation theory to (i) LO [95, 96], (ii) NLO [97] and (iii) NNLO [98]. | $\Sigma^{+}_{c}\to\Lambda_{c}^{+}\gamma$ | $\Sigma_{c}^{*+}\to\Lambda_{c}^{+}\gamma$ | $\Sigma_{c}^{*++}\to\Lambda_{c}^{++}\gamma$ | $\Sigma_{c}^{*0}\to\Sigma_{c}^{0}\gamma$ | $\Xi^{\prime+}_{c}\to\Xi_{c}^{+}\gamma$
---|---|---|---|---|---
(i) | 91.5 | 150.3 | 1.3 | 1.2 | 19.7
(ii) | 164.2 | 893.0 | 11.6 | 2.9 | 54.3
(iii) | 65.6 | 161.8 | 1.2 | 0.49 | 5.4
| $\Xi^{*+}_{c}\to\Xi_{c}^{+}\gamma$ | $\Xi^{*0}_{c}\to\Xi_{c}^{0}\gamma$ | $\Xi^{\prime 0}_{c}\to\Xi_{c}^{0}\gamma$ | $\Omega_{c}^{*0}\to\Omega_{c}^{0}\gamma$ |
(i) | 63.5 | 0.4 | 1.0 | 0.9 |
(ii) | 502.1 | 0.02 | 3.8 | 4.8 |
(iii) | 21.6 | 0.46 | 0.42 | 0.32 |

The calculated results of [95, 96], [97] and [98] denoted by (i), (ii) and
(iii), respectively, in Table 3 can be regarded as the predictions of heavy
hadron chiral perturbation theory (HHChPT) to the leading order (LO), next-to-
leading order (NLO) and next-to-next-to-leading order (NNLO), respectively. It
is not clear why the predictions of HHChPT to NLO are quite different from
that to LO and NNLO for the following three modes:
$\Sigma_{c}^{*+}\to\Lambda_{c}^{+}\gamma$,
$\Sigma_{c}^{*++}\to\Sigma_{c}^{++}\gamma$ and
$\Xi^{*+}_{c}\to\Xi_{c}^{+}\gamma$. It is naively expected that all HHChPT
approaches should agree with each other to the lowest order of chiral
expansion provided that the coefficients are inferred from the nonrelativistic
quark model. This issue can be clarified by STCF through the measurement of
these decay rates.

Very recently, Belle has observed the electromagnetic decays of orbitally
excited charmed baryons $\Xi_{c}(2790)$ and $\Xi_{c}(2815)$ for the first time
[99]. The partial widths of $\Xi_{c}(2815)^{0}\to\Xi_{c}^{0}\gamma$ and
$\Xi_{c}(2790)^{0}\to\Xi_{c}^{0}\gamma$ are measured to be $320\pm
45^{+45}_{-80}$ keV and $\sim 800$ keV, respectively. However, no signal was
found for the analogous decays of $\Xi_{c}(2815)^{+}$ and $\Xi_{c}(2790)^{+}$.

Weak radiative decays such as $\Lambda_{c}^{+}\to\Sigma^{+}\gamma$ and
$\Lambda_{c}^{+}\to p\gamma$ can proceed through the bremsstrahlung processes
$cd\to us\gamma$ (Cabibbo-favored) and $cd\to ud\gamma$ (Cabibbo-suppressed),
respectively. The branching fraction of the former was estimated to be of
order $10^{-4}$ [100].

###  3.3 CP violation

The CKM matrix contains a phase which implies the existence of CP violation.
This means CP-violation can be studied with baryons as well. The predicted CP-
violating asymmetries are, however, small for charmed baryons. The search for
CP violation in charmed baryon decays has taken on new momentum with the large
samples of $\Lambda_{c}$ obtained by BESIII and LHCb. For two-body decays of
the $\Lambda_{c}^{+}$, CP violation can be explored through the measurement of
CP-violating asymmetry, ${\cal
A}=(\alpha+\bar{\alpha})/(\alpha-\bar{\alpha})$, which corresponds to the
asymmetry of $\alpha$ for the $\Lambda_{c}^{+}$ decays and $\bar{\alpha}$ for
$\bar{\Lambda}_{c}^{-}$ decays. For example, ${\cal A}$ in
$\Lambda_{c}^{+}\to\Lambda\pi^{+}$ and
$\bar{\Lambda}_{c}^{-}\to\bar{\Lambda}\pi^{-}$ was measured by FOCUS to be
$-0.07\pm 0.19\pm 0.24$ [101]. In STCF, much more sensitive searches for CP
violation will be carried out by combining the single tag $\Lambda_{c}^{+}$
data [102] and double tag $\Lambda_{c}^{+}\bar{\Lambda}{}_{c}^{-}$ data, where
the pairs of $\Lambda_{c}^{+}\bar{\Lambda}{}_{c}^{-}$ are quantum-correlated
regarding to their spins aligned to the initial spins of the virtual photons.
Especially, with polarized beams [103], unique advantage of enhanced
sensitivities on the decay asymmetries and the CP violations can be achieved
with prior knowledge of the spin direction of the produced $\Lambda_{c}^{+}$.
As for three-body decays, LHCb has measured $\Delta A_{CP}$ as the difference
between CP asymmetries in $\Lambda_{c}^{+}\to pK^{+}K^{-}$ and
$\Lambda_{c}^{+}\to p\pi^{+}\pi^{-}$ decay channels. The result is $\Delta
A_{CP}=(0.30\pm 0.91\pm 0.61)\%$ [104], to be compared with a generic SM
prediction of a fraction of 0.1% [105]. In order to probe the SM contribution
to such asymmetries, one has to increase the available statistics by at least
a factor of 100.

For multi-hadrons in the final state of $\Lambda_{c}^{+}$ decays such as
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}\pi^{0}$,
$\Lambda_{c}^{+}\to\Lambda\pi^{+}\pi^{+}\pi^{-}$ and $\Lambda_{c}^{+}\to
pK_{S}\pi^{+}\pi^{-}$, CP violation can be exploited through several T-odd
observables. Owing to its characters of high luminosity, broad center-of-mass
energy acceptance, abundant production and clean environment, STCF will
provide a great platform for this kind of study. A fast Monte Carlo simulation
[106] of 1 ab-1 $e^{+}e^{-}$ annihilation data at $\sqrt{s}=4.64$ GeV, which
are expected to be available at the future STCF, indicates that a sensitivity
at the level of (0.25-0.5)% is accessible for the above-mentioned three decay
modes. This will be enough to measure non-zero CP-violating asymmetries as
large as 1%.

###  3.4 Spectroscopy

Table 4: Antitriplet and sextet states of charmed baryons. Mass differences $\Delta m_{\Xi_{c}\Lambda_{c}}\equiv m_{\Xi_{c}}-m_{\Lambda_{c}}$, $\Delta m_{\Xi^{\prime}_{c}\Sigma_{c}}\equiv m_{\Xi^{\prime}_{c}}-m_{\Sigma_{c}}$, $\Delta m_{\Omega_{c}\Xi^{\prime}_{c}}\equiv m_{\Omega_{c}}-m_{\Xi^{\prime}_{c}}$ are all in units of MeV.  | $J^{P}(nL)$ | States | Mass difference
---|---|---|---
${\bf\bar{3}}$ |  ${1\over 2}^{+}(1S)$ |  $\Lambda_{c}(2287)^{+}$, $\Xi_{c}(2470)^{+},\Xi_{c}(2470)^{0}$ |  $\Delta m_{\Xi_{c}\Lambda_{c}}=183$
|  ${1\over 2}^{-}(1P)$ |  $\Lambda_{c}(2595)^{+}$, $\Xi_{c}(2790)^{+},\Xi_{c}(2790)^{0}$ | $\Delta m_{\Xi_{c}\Lambda_{c}}=198$
|  ${3\over 2}^{-}(1P)$ |  $\Lambda_{c}(2625)^{+}$, $\Xi_{c}(2815)^{+},\Xi_{c}(2815)^{0}$ | $\Delta m_{\Xi_{c}\Lambda_{c}}=190$
|  ${3\over 2}^{+}(1D)$ |  $\Lambda_{c}(2860)^{+}$, $\Xi_{c}(3055)^{+},\Xi_{c}(3055)^{0}$ | $\Delta m_{\Xi_{c}\Lambda_{c}}=201$
|  ${5\over 2}^{+}(1D)$ |  $\Lambda_{c}(2880)^{+}$, $\Xi_{c}(3080)^{+},\Xi_{c}(3080)^{0}$ | $\Delta m_{\Xi_{c}\Lambda_{c}}=196$
${\bf 6}$ |  ${1\over 2}^{+}(1S)$ |  $\Omega_{c}(2695)^{0}$, $\Xi^{\prime}_{c}(2575)^{+,0},\Sigma_{c}(2455)^{++,+,0}$ |  $\Delta m_{\Omega_{c}\Xi^{\prime}_{c}}=119$, $\Delta m_{\Xi^{\prime}_{c}\Sigma_{c}}=124$
|  ${3\over 2}^{+}(1S)$ |  $\Omega_{c}(2770)^{0}$, $\Xi^{\prime}_{c}(2645)^{+,0},\Sigma_{c}(2520)^{++,+,0}$ |  $\Delta m_{\Omega_{c}\Xi^{\prime}_{c}}=120$, $\Delta m_{\Xi^{\prime}_{c}\Sigma_{c}}=128$

The observed antitriplet and sextet states of charmed baryons are listed in
Table 4. By now, the $J^{P}={1\over
2}^{+},\frac{1}{2}^{-},\frac{3}{2}^{+},\frac{3}{2}^{-}$ and $\frac{5}{2}^{+}$
antitriplet states $\Lambda_{c},\Xi_{c}$ and $J^{P}={1\over 2}^{+},{3\over
2}^{+}$ sextet states $\Omega_{c},\Xi^{\prime}_{c},\Sigma_{c}$ are
established. The highest state $\Lambda_{c}(2940)^{+}$ in the $\Lambda_{c}$
family was first discovered by BaBar in the $D^{0}p$ decay mode [107] but its
spin-parity assignment is quite diverse (see [108] for a review). The
constraints on its spin and parity were recently found to be
$J^{P}=\frac{3}{2}^{-}$ by LHCb [109]. It was suggested in [110] that the
quantum numbers of $\Lambda_{c}(2940)^{+}$ are most likely
$\frac{1}{2}^{-}(2P)$ based on the Regge analysis. However, it was argued in
[111] that $\Lambda_{c}(2940)^{+}$ is a $\frac{3}{2}^{-}(2P)$ state and there
is a state $\frac{1}{2}^{-}(2P)$ higher than the $\Lambda_{c}(2P,3/2^{-})$.
This issue can be clarified by STCF.

In 2017 LHCb has explored the charmed baryon sector of the $\Omega_{c}$ and
observed five narrow excited $\Omega_{c}$ states decaying into
$\Xi_{c}^{+}K^{-}$: $\Omega_{c}(3000)$, $\Omega_{c}(3050)$,
$\Omega_{c}(3066)$, $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$ [112]. Except
$\Omega_{c}(3119)$, the first four states were also confirmed by Belle later
[113]. This has triggered a lot of interest in possible identification of
their spin-parity quantum numbers.

Within the energy region of STCF up to $7$ GeV, it is suitable to study the
spectroscopy of singly charmed baryon states $\Lambda_{c}$, $\Sigma_{c}$,
$\Xi_{c}^{(\prime)}$, $\Omega_{c}$ and their excited states in the energy
range of $5\sim 7$ GeV. It is important for SCTF to explore their possible
structure and spin-parity quantum number assignments, especially for the five
new and narrow $\Omega_{c}$ resonances. If the energy region is extended to
above 7.4 GeV, the production of the double charmed baryon $\Xi^{++}_{cc}$ is
allowed. This will also enable the study of some more detailed of the recently
discovered double charmed baryons.

##  4 Summary

To summarize, STCF will be an ideal facility to study charm physics using the
unique pair production of the charmed hadrons near threshold. In particular,
the charm mixing and CP-violating parameters, as well as the strong phase of
the neutral $D$ mesons, can be precisely determined utilizing the quantum
correlation of the $D^{0}\bar{D}^{0}$ pair system. With 1 ab-1 data at 4.009
GeV, it is expected that the measurement sensitivities of the mixing
parameters ($x$, $y$) will reach a level of 0.05%, and those of $|q/p|$ and
$\arg(q/p)$ will be 1.7% and $1.3^{\circ}$, respectively. $D$ strong-phase
measurement at STCF will be necessary to constrain the uncertainty of the CP-
violating phase $\gamma$ being less than 0.1∘ and allow detailed comparisons
of the $\gamma$ results from different analysis channels in the future LHCb
upgrade II. These measurements are complementary to those studies at LHCb and
Belle II, and provide crucial information to understand the mechanism of CP
violation. In addition, the leptonic decays of the charmed mesons and baryons
will be systematically explored in world-best precisions, which serves for
overconstraint of the CKM matrix, clean probe of the dynamics of strong force,
stringent test of LFU and search for new physics.

Furthermore, in the energy region above the charmed baryon threshold, studies
of production and decay properties of the (excited) charmed baryons can be
extensively carried out, which are still lacking of experimental data. The
absolute measurements of the semi-leptonic and nonleptonic decays of the
$\Lambda_{c}^{+}$, $\Xi_{c}^{+,0}$ and $\Omega_{c}^{0}$ baryons will be
significantly improved at STCF. Priority will be ascribed to the decay
asymmetries $\alpha$ in various charm baryon decays, which can be further used
to search for CP violation effects, and the absolute branching fractions of
$\Omega_{c}^{0}$ decays. Moreover, measurements of and searches for rare- and
forbidden decays of charmed hadrons with up to two orders-of-magnitude
improvements in sensitivity could be realized as part of a search for new
physics.

Many of these measurements at STCF will benefit from the clean reaction
environment, strictly constrained kinematics and well-controlled systematic
uncertainties, which necessitates a detector with excellent performance,
especially in identifying the types of different charged particles, detecting
low-momentum charged particles and measuring photons. The unprecedented
precisions obtained with high statistics charm data will enable us to have a
much more in-depth understanding of the challenges facing the SM and hopefully
will provide some clues/solutions to them. Hence, it will play a crucial role
in leading the high intensity frontier of elementary particle physics
worldwide.

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# Traffic Sign Recognition Using Local Vision Transformer

Ali Farzipour, Omid Nejati Manzari, Shahriar B. Shokouhi
School of Electrical Engineering, Iran University of Science and Technology,
Tehran, Iran

###### Abstract

Recognition of traffic signs is a crucial aspect of self-driving cars and
driver assistance systems, and machine vision tasks such as traffic sign
recognition have gained significant attention. CNNs have been frequently used
in machine vision, but introducing vision transformers has provided an
alternative approach to global feature learning. This paper proposes a new
novel model that blends the advantages of both convolutional and transformer-
based networks for traffic sign recognition. The proposed model includes
convolutional blocks for capturing local correlations and transformer-based
blocks for learning global dependencies. Additionally, a locality module is
incorporated to enhance local perception. The performance of the suggested
model is evaluated on the Persian Traffic Sign Dataset and German Traffic Sign
Recognition Benchmark and compared with SOTA convolutional and transformer-
based models. The experimental evaluations demonstrate that the hybrid network
with the locality module outperforms pure transformer-based models and some of
the best convolutional networks in accuracy. Specifically, our proposed final
model reached 99.66% accuracy in the German traffic sign recognition benchmark
and 99.8% in the Persian traffic sign dataset, higher than the best
convolutional models. Moreover, it outperforms existing CNNs and ViTs while
maintaining fast inference speed. Consequently, the proposed model proves to
be significantly faster and more suitable for real-world applications.

###### Keywords:

vision transformer, Deep Learning, Traffic Sign Recognition, self-driving
vehicles

##  I Introduction

The importance of safety systems in self-driving cars and driver assistance
systems is obvious. These safety systems are an essential part of vehicles in
terms of value. Due to the recent popularity and trade of self-driving cars
and driver assistance systems, traffic sign recognition (see Figure 1) gained
considerable attention among machine vision tasks [1]. Conversely, deep
learning techniques are now the dominant solution for machine learning and
especially machine learning issues. Naturally, machine vision became the
epitome of deep learning. Convolutional Neural Network (CNN) is one of the
most well-known architectures with a remarkable ability to maintain local
dependencies in deep learning realm. Their filters perceive a local region of
the input image. In other words, CNNs are very efficient in learning local
features. These Networks performed well in various machine vision applications
and achieved excellent results.

After the success of CNNs, transformers were introduced. Before machine vision
transformers became the state-of-the-art(SOTA) solution in Natural Language
Processing (NLP) studies. After that, researchers began to modify transformers
and use them in machine vision tasks. This approach led to the introduction of
vision transformers (ViT) [2]. Unlike CNNs, these transformers do not detect
correlations in nearby pixels very well. But they are very efficient in
learning global dependencies by maintaining the relationship between every
pixel through the image. First, basic Vits achieved encouraging results in
machine vision tasks, but they required a high amount of data for training and
still needed help to reach SOTA CNN models. So researchers tried to modify the
primary vision transformer’s architecture and introduced more complicated
models [3] which performed much better than basic ones and even outperformed
some of the best convolutional networks.

Figure 1:  The provided images display various traffic signs sourced from the
GTSRB dataset [4].

As local and global features play a vital role in classification tasks,
researchers have begun using convolutions and transformers in their newly
designed hybrid models. These models outperformed pure transformer-based
models, and some of them outperformed state-of-the-art convolutional models as
well. So, we introduced a new hybrid model to answer shortcuts of
convolutional and pure transformer-based networks and use both advantages. Our
proposed model consists of convolutional and transformer-based blocks.
Convolutional blocks are used for perceiving and learning the correlation
between near pixels. Transformer-based blocks are used for sensing global
relationships. We also added a locality module in one of the transformer-based
blocks to add more local perception to the model.

So far, our contributions in this work are: 1- We proposed a new hybrid model
and added a locality module. 2- We analyzed our model’s performance on German
Traffic Sign Recognition Benchmark (GTSRB) [4] and Persian Traffic Sign
Dataset (PTSD) [5]. We compared it with other SOTA convolutional and
transformer-based models. 3- We used AugMix [6] data augmentation to add more
robustness to the model, which raised accuracy. This paper has been arranged
in the following order: Section 2 investigates previous studies on TSR and
classification tasks. Section 3 explains applied modification both in terms of
architecture and computation. In section 4, we will demonstrate and analyze
our results. The results contain all previous models and the newly modified
model’s performance in both datasets. Lastly, in the final part, we summarize
the findings of our work and make suggestions for further research.

Figure 2: Network architecture of the vanilla ViT.

##  II Related Works

Various methods have been explored for developing traffic sign recognition
systems. However, since 2013, using convolutional neural networks has become
standard in object recognition tasks due to their capability of learning a
hierarchy of features by producing low-level features from high-level features
[7]. In [8], the author has proposed a committee of convolutional layers for
feature extraction. This committee enhances the model’s ability to identify
patterns within the image. This model achieved an accuracy of 98.41% with only
2.61M parameters. In [9], another convolutional model was introduced by
designing an inception module that can attend to the entire image. This model
is highly accurate and has a relatively small number of parameters. In recent
work[10], ensemble learning was used to recognize traffic signs. Ensemble
learning involves using multiple models simultaneously and in combination for
decision-making. In this study, pre-trained DensNet [11], VGG [12] and ResNet
[13] models were utilized. Additionally, the number of data samples was
significantly increased, with the amount of images in the GTSRB dataset rising
from 51,000 to 96,000. The proposed model was found to be more accurate than
other models in the comparison made in this study. However, it did not achieve
99% accuracy.

After the success of Transformers in the NLP area, researchers tried using
them in vision tasks. The first severe vision transformer introduced was VIT
[2]. After VIT, some other models were introduced, mostly VIT’s improvements.
The work [14] presents a comparison between convolutional models and vision
transformers in the field of TSR. The study evaluated seven convolutional
models, such as DensNet, VGG, ResNet and MobileNet and five vision
transformers like VIT and PVT [15]. The results showed that advanced
convolutional models outperformed primary vision transformers. Particularly,
the test phase accuracy on the GTSRB dataset for the VGG model was the
greatest. In incorporation of convolution inductive biases into the early
blocks of transformer led to outperforming SOTA models. In [16], newer vision
transformer models, such as CCT [17], which used convolution to build image
segments, were compared with DensNet for traffic sign recognition. This study
showed that not all vision transformer models are suitable for this task. The
comparison results demonstrated that pre-trained DensNet was superior.
However, the CCT model was able to outperform the untrained DensNet model.

##  III METHODOLOGY

In this section, we will introduce our proposed model. Before explaining its
architecture, we will briefly overview the vision transformer’s architecture
and equations to better understand the proposed model. After that, we will
describe our model and its mechanism.

###  III-A Overview of Vision Transformer

The primary transformer architecture contains of a decoder and an encoder,
which have similar structures. Since we are focused on the classification of
traffic signs, only the encoder portion of the transformer is required. The
encoder typically comprises two parts.As illustrated in Figure 2 The first
part includes a self-attention mechanism that evaluates the relationship
between each token and all the other tokens. The second part is a simple
fully-connected neural network with two layers that extracts more and richer
features from each token. From a mathematical perspective, since transformers
operate on one-dimensional data, a two-dimensional input image with dimensions
of C × H × W must be converted into a sequence of tokens. less mathematical
calculations are required, so we first divide the input image into P × P
patches, resulting in a set of images with dimensions P × P × C. The number of
this set equals:

$N=(H\times W)/P^{2}$ (1)

Also, the size of each token is:

$d=C\times P^{2}$ (2)

However, this number can be very high. If we use this sequence directly in the
transformer, we would need to multiply a matrix with dimensions N×d by a
matrix with dimensions d × N, which increases the computational complexity and
memory consumption of the model and thus decrease the speed. To address this
issue, linear mapping reduces the tokens’ dimensionality to the model’s the
desired number. For instance, if the desired number is D, the input matrix
will be multiplied by a matrix with dimensions N × D. A convolutional layer or
a simple MLP achieves linear mapping in practice.

The input tokens are fed into the self-attention mechanism. First, Key (K),
Query (Q), and Value (V) matrices are created through mapping from the input
using fully connected networks (FCs). The dimensions of these matrices are
identical to the input dimensions (N × D). Next, the self-attention mechanism
applies relation 3 to these matrices.

$\mathrm{Z}=\operatorname{softmax}\left(\frac{\mathbf{Q}K^{T}}{\sqrt{D}}\right)\mathbf{V}$
(3)

The output spatial dimensions of the self-attention mechanism remain the same
as the input dimensions. The resulting output is passed to the feed-forward
section after the addition of the normalized sum. The FFN section mainly
comprises of two fully connected (FC) layers that extract richer and more
complex features. This is obtained by growing the amount of neurons in the
first layer and multiplying the dimension of features. On the other hand, the
second layer has the same number of neurons as the input, ensuring that the
dimensions of the input and output of the feed-forward section remain the
same. This approach is similar to the residual block in ResNet [13].

###  III-B Next Generation of Local Vision Transformer

The objective of our model is to extract both local features of an image. To
achieve this, we suggest a novel hybrid network that combines convolutional
and transformer layers. The convolutional are particularly effective for
capturing high-frequency features such as edges and corners, while
transformers are better suited for capturing low-frequency features such as
global patterns. The Next-ViT [18] model inspires our model and includes NCB
and NTB blocks. We will describe the architecture and equations of these
blocks and then discuss the strategy for arranging them in the model called
NHS.

Figure 3: The architecture of proposed Next Generation of Local Vision
Transformer.

###  III-C Next Convolution Block

Our goal with the NCB block is to leverage convolutional layers while
maintaining the high performance of transformers. A token mixer is utilized to
achieve this, emphasizing the importance of Multi-Head Convolutional Attention
(MHCA). We shall go into great depth about MHCA in the section that follows.
The NCB block consists of one MHCA and one MLP, which together can be
described by the following equations:

$\displaystyle\hat{z}^{l}$ $\displaystyle=MHCA(z^{l-1})+z^{l-1}$
$\displaystyle z^{l}$ $\displaystyle=MLP(\hat{z}^{l})+\hat{z}^{l}$

Here $z^{l-1}$ represents the input from the $l-1$ component, while
$\hat{z}^{l}$ and $z^{l}$ represent the output of the MHCA and the $lth$ NCB.

####  III-C1 MHCA

To improve the speed of the attention-based token mixer, we introduce an
attention mechanism with the convolutional operation named Convolutional
Attention (CA). Furthermore, inspired by basic transformers, we utilize a
multi-head architecture to attend to features from various representation
subspaces, which allows for efficient local data learning. The general
functioning of MHCA can be summarized as bellow:

$\operatorname{MHCA}(x)=\operatorname{Concat}\left(\mathrm{CA}_{1}\left(x_{1}\right),\mathrm{CA}_{2}\left(x_{2}\right),\ldots,\mathrm{CA}_{h}\left(x_{h}\right)\right)W^{P}$
(4)

In this equation, the Multi-Head Channel Attention (MHCA) mechanism is
employed to extract information from the divided feature z, which is split
into h heads. To further enhance the interaction of information from multiple
heads, a projection layer (WP) is utilized. Single-head convolutional
attention, or CA, is referred to in this context and is defined as:

$\mathrm{CA}(X)=O\left(\mathrm{W}\cdot\mathrm{T}_{\\{i,j\\}}\right),\text{
where }\mathrm{T}_{\\{i,j\\}}\in X$ (5)

In this equation, $T_{i}$ and $T_{j}$ represent neighboring tokens within the
input feature $z$. $O$ denotes an inner product operation with $W$ as a
learnable parameter and input patches. Through iterative optimization of
parameter $W$, $O$ can effectively learn the connectivity between various
tokens within the local receptive field.

###  III-D Next Transformer Block

Accurate recognition requires the capture of both local and global
information. While NCB has demonstrated proficiency in learning local
features, we have introduced an additional block to capture global features.
Transformer blocks are known for their effectiveness in capturing low-
frequency signals, which provide global information, but they are not as
proficient in capturing high-frequency information. To address this issue, we
have produced the Next Transformer component, which operates in a multi-
frequency manner to capture both high-frequency and low-frequency signals. The
first step in our approach involves the use of a novel mechanism called
Efficient Multi-Head Self Attention (E-MHSA), that could be formulated as
follows:

$\operatorname{ESA}(x)=\operatorname{Concat}\left(\operatorname{SA}_{1}\left(x_{1}\right),\mathrm{SA}_{2}\left(x_{2}\right),\ldots,\mathrm{SA}_{h}\left(x_{h}\right)\right)W^{O}$
(6)

Where z represents the input feature like before, and SA means spatial
reduction self-attention operator, which performs as:

$\operatorname{SA}(X)=\operatorname{Attention}\left(X\cdot
W^{Q},\mathrm{P}_{s}\left(X\cdot W^{K}\right),\mathrm{P}_{s}\left(X\cdot
W^{V}\right)\right)$ (7)

Here, the term Attention refers to routine calculation as Eq(3). $W^{Q}$,
$W^{K}$, and $W^{V}$ are linear layers, and $P_{s}$ is an avg-pool operator
with a stride of $S$ to reduce computational cost. It is clear that the time
required for the E-MHSA is directly proportional to the number of channels it
has. NTB utilizes a point-wise convolution to expedite the process by reducing
the channel dimension. A shrinking ratio, denoted as r, is introduced to
achieve this. In addition, NTB incorporates an MHCA module that works the
E-MHSA to extract multi-frequency features. Following MHCA, both MHCA and
E-MHSA outputs are concatenated to blend high- and low-frequency data
information. An MLP is often used to extract key characteristics near the
ending of a neural network. However, following the approach presented in [19],
we utilize a Local Feed-Forward (LFF) module exclusively in the first NTB
block, while MLP is still utilized in the rest of the NTB blocks. The locality
module is an inverted residual block [20] that employs convolutions, thereby
enhancing the local vision of the network as compared to a simple MLP. The
locality module in the first NTB block is used because the initial blocks
focus on capturing local features, such as high-frequency features. The
complete implementation of NTB (excluding the first block for the last
equation) can be calculated as bellow:

$\displaystyle\bar{z}^{l}$
$\displaystyle=\operatorname{Proj}\left(z^{l-1}\right)$ (8)
$\displaystyle\ddot{z}^{l}$
$\displaystyle=\mathrm{E-MHSA}\left(\bar{z}^{l}\right)+\bar{z}^{l}$
$\displaystyle\dot{z}^{l}$
$\displaystyle=\operatorname{Proj}\left(\ddot{z}^{l}\right)$
$\displaystyle\tilde{z}^{l}$
$\displaystyle=\operatorname{MHCA}\left(\dot{z}^{l}\right)+\dot{z}^{l}$
$\displaystyle\hat{z}^{l}$
$\displaystyle=\operatorname{Concat}\left(\ddot{z}^{l},\tilde{z}^{l}\right)$
$\displaystyle z^{l}$
$\displaystyle=\operatorname{LFF}\left(\hat{z}^{l}\right)+\hat{z}^{l}$

TABLE I: Achievement of SOTA models for Traffic Sign Recognition on the GTSRB.

Method Accuracy (%) Parameters (M) GFLOPS (G) EfficientNetV2 99.33 20.23 8.48
InceptionResnetV2 99.11 54.37 6.54 ResNet34 98.94 21.31 3.68 DenseNet121 98.9
7.00 2.86 Next-Vit 99.35 30.78 5.8 PIT 99.18 22.9 2.87 VITAE-S 99.15 23.66 6.2
PVT-V2 98.89 24.87 4.04 VITAE-6m 98.64 6.37 2.17 VIT-b16 98.57 86.57 17.56
R50+VIT-b-16 98.84 97.9 22.36 Next-LVT (ours) 99.66 33.14 7.66

####  III-D1 Next Hybrid Strategy

Traditionally, convolutions are frequently used in shallow phases of
conventional hybrid models, and transformer blocks are stacked in the final
stage or stages. However, this approach can fail to capture global information
in the early stages, resulting in suboptimal performance. We propose a new
method called Next Hybrid Strategy (NHS) to address this issue. The NHS
strategy follows an (NCB×N+NTB×1) pattern, where each stage has one $NTB$ and
$NNCB$. To allow the model to learn global features in shallow layers, the
transformer block is explicitly included at the end of each step. Finally, we
repeat each stage by the factor of $L$. So the final pattern of the model will
be $(NCB\texttimes N+NTB\texttimes 1)\texttimes L$, as shown in Figure 3.

##  IV Experiments

In this part, we initiate the discussion by examining the datasets utilized to
train our enhanced Transformer model, and provide a summary of the
experimental configurations. Following that, we conduct a comprehensive
analysis of the experimental outcomes, wherein We evaluate the effectiveness
of our Transformer model with recent breakthroughs in the field of traffic
sign recognition.

###  IV-A Datasets and Implementation details

After discussing the model’s architecture, we will now shift focus to the
experiments. In total, we conducted 23 experiments on two datasets: GTSRB and
PTSD. We trained and tested various models on these datasets. We ran our
experiments using Google Colab, which provided us with a Tesla T4 GPU and 15
gigabytes of RAM. We implemented our experiments using the PyTorch framework.
Now our training and testing setup is as follows. With a linear learning
scheduler that split the learning by $10$ after three training epochs, we set
the starting learning rate at $0.007$. For the GTSRB dataset, we trained and
tested for 20 epochs, and For the PTSD dataset, we trained and tested for 15
epochs. Our test batch size was set at 256 and our training batch size to 64.
We used the Cross-Entropy loss function and the SGD optimizer to determine the
loss during training. As for our dataset’s setup, the respective dataset
sources provided the train and test sets. For the GTSRB dataset, 39,209
training photos and 12,630 test images totaling 43 distinct classes of traffic
signs were used. For the PTSD dataset, we had a total of 14,405 training
images and 2,421 test images, also containing 43 different types of traffic
signs. It is worth noting that the amount of images in the train and test sets
varied between the two datasets. We also employed AugMix [6] as our primary
data augmentation technique to enhance the robustness of our model.

TABLE II: Performance evaluation of suggested Transformer and SOTA models for
PTSD.

Method Accuracy (%) Parameters (M) GFLOPS (G) EfficientNetV2 99.67 20.23 8.48
InceptionResnetV2 99.46 54.37 6.54 DenseNet121 99.67 7.00 2.86 Next-Vit 99.7
30.78 5.8 PIT 99.67 22.9 2.87 VITAE-S 99.3 23.66 6.2 PVT-V2 99.55 24.87 4.04
VITAE-6m 99.5 6.37 2.17 Next-LVT (ours) 99.8 33.14 7.66

###  IV-B RESULTS

In Table.1, we present the results of our experiments on the GTSRB dataset,
including the accuracy, whole amount of parameters, and total amount of GFLOPs
for each of the 14 models. Table.2 presents the same metrics for our
experiments on the PTSD dataset. These metrics provide valuable information
about our model’s computational complexity and efficiency and allow easy
comparison between different models. Table 1 shows that our model, called
Next-generation Local Vision Transformer (Next-LVT), achieved an accuracy of
99.66%, surpassing other convolutional, transformer, and hybrid models. While
our model has a slightly higher number of total parameters and GFLOPs, it
remains suitable for real-time applications. Similarly, Table 2 demonstrates
that our model achieved the highest accuracy of 99.8%. The differences in the
PTSD dataset were smaller than those in the GTSRB dataset, indicating that
PTSD is a less complex and diverse dataset.

##  V Conclusion

This paper addresses the importance of safety systems in self-driving cars and
driver assistance systems, specifically on traffic sign recognition. We
suggest a new novel model that combines the strong points of both CNNs and
transformers. Our model consists of convolutional and transformer-based
blocks, leveraging the advantages of each approach. A locality module is also
introduced to excel the model’s local perception. The suggested model is
evaluated on the GTSRB and PTSD and compared against SOTA convolutional and
transformer architectures. The results demonstrate that our hybrid network
outperforms pure transformer-based models and surpasses some of the best
convolutional networks. The proposed model achieves improved accuracy in
traffic sign recognition tasks by incorporating local and global features.
Furthermore, using data augmentation techniques, such as AugMix, enhances the
model’s robustness. Future research directions may involve further
improvements to the hybrid model and exploring its application in other
computer vision tasks.

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