# Chirality-Induced Spin Filtering in Pseudo Jahn-Teller Molecules

Akihito Kato  Division of Natural and Environmental Sciences, The Open
University of Japan, Chiba 261-8586, Japan  Hiroshi M. Yamamoto  Research
Center of Integrative Molecular Systems, Institute for Molecular Science,
Okazaki, Aichi 444-8585, Japan  Jun-ichiro Kishine  Division of Natural and
Environmental Sciences, The Open University of Japan, Chiba 261-8586, Japan
Research Center of Integrative Molecular Systems, Institute for Molecular
Science, Okazaki, Aichi 444-8585, Japan

###### Abstract

Chirality-induced spin selectivity (CISS) refers to an ability to induce a
spin polarization of an electron transmitted through chiral materials. An
important experimental observation is that incredibly large spin polarization
is realized at room temperature even for the organic molecules that have weak
spin-orbit coupling (SOC), although SOC is an only interaction that can
manipulate the electrons’ spins in the setups. Therefore, the mechanism of the
CISS needs to be constructed in a way insensitive to or enhancing the
magnitude of the SOC strength. In this Letter, we describe a theoretical study
for CISS with a model chiral molecule that belongs to the point group
$\mathrm{C}_{3}$. In this molecule, electronic translational and rotational
degrees of freedom for an injected electron are coupled one another via the
nuclear vibrational mode with a pseudo Jahn-Teller effect. By properly taking
the molecular symmetry as well as the time reversal symmetry into account and
classifying the molecular ground states by their angular and spin momentum
quantum numbers, we show that the chiral molecule can act as an efficient
spin-filter. The efficiency of this spin-filtering can be nearly independent
of the SOC strength in this model, while it well exceeds the spin-polarization
relying solely on the SOC. The nuclear vibrations turned out to have the role
of not only mediating the translation-rotation coupling, but also enhancing
the spin-filtering efficiency.

## Introduction.

An object that cannot be overlapped with its mirror images, namely, that lacks
the reflection symmetries, is called chiral. The chiral symmetry breaking
gives rise to abundant functionality barron2004Molecular  such as chiral
magnetism muhlbauer09Skyrmion ; togawa2016Symmetry , chiral phononics
zhang2015Chiral ; zhu18Observation ; kishine2020ChiralityInduced , chiral
photonics kuwata-gonokami2005Giant , and nonreciprocal conductivity
rikken2002Observation ; tokura2018Nonreciprocal . Recent experimental
observations have confirmed that the chiral materials exhibit spin-selective
phenomena, including a large spin-polarization of photoelectrons transmitted
through helical molecules gohler2011Spin ; mishra2013Spindependent ;
kettner2015Spin  and a large spin dependence of the current-voltage
characteristics for the tunneling electrons through helical molecules
xie2011Spin ; kettner2018ChiralityDependent ; suda2019Lightdriven . Even
enantio-separation and asymmetric electrochemical reactions by magnetic
electrodes are reported banerjee-ghosh2018Separation ; metzger2020Electron .
Furthermore, generation of large spin current was established for the
inorganic chiral crystals inui2020ChiralityInduced ; nabei2020Currentinduced ;
shiota2021ChiralityInduced . These phenomena originated from the structural
chirality is collectively termed as the chirality-induced spin selectivity
(CISS) naaman2012ChiralInduced ; naaman2019Chiral ; naaman2020Chiral , where
the spin is oriented parallel or antiparallel to the velocity of an injected
electron depending on the materials’ handedness.

CISS has been actively studied and become an interdisciplinary research fields
spanning physics, chemistry, and biology. The reported spin-polarization up to
$60\text{\,}\%$ amounts to the effective magnetic field of an order of
$100\text{\,}\mathrm{T}$ naaman2012ChiralInduced , which is unrealistically
strong. So far, several attempts have been made on the electron motion in a
chiral molecule in the presence of spin-orbit coupling (SOC)
guo2012SpinSelective ; guo2014Spindependent ; matityahu2016Spindependent . The
present status of theoretical studies is well summarized in Ref. evers21Theory
. In order to find an alternative explanation, we would like to understand the
fact that CISS effect is observed even in the organic molecules with weak SOC
strength in a way that do not rely explicitly on the strength of the SOC. Our
belief here is based on the importance of interplay between nuclear and
electronic motions which are governed by the same symmetry restrictions in
chiral materials. Moreover, our objective also lies on a quest for CISS theory
that do not violate the time reversal symmetry. This is exactly what we
address in this Letter.

To this end, we consider a molecule under the pseudo Jahn-Teller effect
bersuker2006JahnTeller ; bersuker20Jahn , where the coupling between an
electronic translation and rotation are mediated by the nuclear vibrational
degrees of freedom. The Hamiltonian that includes the electron-nuclear
coupling and the SOC satisfies the time-reversal symmetry. This symmetry
confirms that the up-spin and the down-spin states are degenerate forming a
Kramers pair. This degenerate pair can be separated from each other by its
translational direction, which is made possible by the pseudo Jahn-Teller
coupling. Consequently, the chiral molecule acts as the spin-filter by
selecting the direction of electron translation. In this Letter, we will
demonstrate that the performance of this chiral spin-filter is determined
critically by the spin-selective transmission probability. In this model,
angular momentum (AM) quantum numbers play crucial role in obtaining quite
large efficiency insensitive to the SOC strength.

## Molecular Basis.

Molecular symmetry shapes a potential field that exerts on an electron
injected into the molecule to create a rotational motion from the
translational one. As a result, molecular orbital occupied by the injected
electron is spanned by both translational and rotational basis. This study
focuses on the point group $\mathrm{C}_{3}$ as the minimal point group that
allows chiral structure and has rotational basis with the one rotating in a
direction opposite to the other. Let $z$-axis be the three-fold rotation axis.
We write $\ket{\phi_{z}}$ for the translational state that are transformed as
$z$, and $\ket*{\bar{\phi}_{z}}\coloneqq\Theta\ket{\phi_{z}}$ for its time-
reversal with $\Theta$ being the time-reversal operator. Inclusion of
$\ket*{\bar{\phi}_{z}}$ allows us to construct the model Hamiltonian that
satisfy the time-reversal symmetry in an unambiguous way. Obviously,
$\expectationvalue{\hat{p}_{z}}{\phi_{z}}=-\expectationvalue*{\hat{p}_{z}}{\bar{\phi}_{z}}$
holds, where $\hat{p}_{z}$ is the $z$-component of the electron momentum
operator, and $\expectationvalue{\hat{p}_{z}}{\phi_{z}}>0$ is assumed. Because
these basis belong to the $\mathrm{A}$ irreducible representation of the point
group $\mathrm{C}_{3}$, they are invariant under the three-fold rotation
$C_{3}$. The basis for the rotation on the $xy$-plane, which belong to the
$\mathrm{E}_{1}$ and $\mathrm{E}_{2}$ irreducible representations, are denoted
by $\ket{\phi_{\pm}}$. These rotational states are transformed under $C_{3}$
and $\Theta$ as

$C_{3}\ket{\phi_{\pm}}=e^{\mp i2\pi/3}\ket{\phi_{\pm}},$ (1)

and

$\Theta\ket{\phi_{\pm}}=\ket{\phi_{\mp}},$ (2)

respectively.

In chiral materials, translational and rotational motions are coupled with
each other barron2004Molecular . However, in our model, the coupling between
the translational and rotational states of an electron is mediated by the
nuclear vibrational modes belonging to the $\mathrm{e}_{1}$ and
$\mathrm{e}_{2}$ representation. Indeed, the product representation
$\mathrm{e}_{i}\times\mathrm{E}_{i}$ for $i=1,2$ includes $\mathrm{A}$
representation. Consequently, there arises a finite matrix element
$\matrixelement*{\mathrm{A}}{\mathrm{e}_{i}}{\mathrm{E}_{i}}$ in a symbolic
form. The nuclei in the $\mathrm{e}_{1,2}$ representation also induce the
coupling between the rotational states. This coupling between degenerate
rotational states causes the spontaneous distortion, which is known as the
Jahn-Teller effect bersuker2006JahnTeller ; bersuker20Jahn . The coupling
between non-degenerate ($\mathrm{E}_{i}$ and $\mathrm{A}$) states, can also
cause a similar symmetry-breaking called the pseudo Jahn-Teller effect
bersuker2006JahnTeller ; bersuker20Jahn . The nuclear coordinates in the
$\mathrm{e}_{1,2}$ representation are written as $Q_{\pm}=\rho e^{\pm
i\varphi}$ with the radius $\rho$ and the angle $\varphi$. The symmetry
operations transform $Q_{\pm}$ in the following way:

$C_{3}Q_{\pm}=e^{\pm i2\pi/3}Q_{\pm},$ (3)

and

$\Theta Q_{\pm}=Q_{\mp}.$ (4)

To describe the spin-dependent process, the electronic spin must be added to
the basis. Let $\ket{\uparrow}$ and $\ket{\downarrow}$ denote the up-spin and
down-spin states, respectively. They are transformed under the symmetry
operations as

$C_{3}\ket{\uparrow\\!\\!/\downarrow}=e^{\mp
i\pi/3}\ket{\uparrow\\!\\!/\downarrow},$ (5)

and

$\Theta\ket{\uparrow\\!\\!/\downarrow}=\pm\ket{\downarrow\\!\\!/\uparrow}.$
(6)

## Model Hamiltonian.

The Hamiltonian, $H$, describing an electron propagating through the chiral
molecule consists of the electronic $H_{\mathrm{e}}$, the nuclear
$H_{\mathrm{n}}$, the electron-nuclear coupling $H_{\mathrm{en}}$
Hamiltonians, which are all spin-independent, and the SOC Hamiltonian
$H_{\mathrm{soc}}$,
$H=H_{\mathrm{e}}+H_{\mathrm{n}}+H_{\mathrm{en}}+H_{\mathrm{soc}}$. The
electronic Hamiltonian is given by

$H_{\mathrm{e}}=\epsilon_{\mathrm{tr}}(\outerproduct{\phi_{z}}{\phi_{z}}+\outerproduct*{\bar{\phi}_{z}}{\bar{\phi}_{z}})+\epsilon_{\mathrm{rot}}(\outerproduct{\phi_{+}}{\phi_{+}}+\outerproduct{\phi_{-}}{\phi_{-}}),$
(7)

where we set $\epsilon_{\mathrm{rot}}\equiv 0$ throughout the paper, and the
nuclear Hamiltonian is described as the two-dimensional harmonic oscillator,

$H_{\mathrm{n}}=-\frac{\hbar^{2}}{2M}\left(\frac{1}{\rho}\partialderivative{\rho}\rho\partialderivative{\rho}+\frac{1}{\rho^{2}}\partialderivative[2]{\varphi}\right)+\frac{M\omega^{2}}{2}\rho^{2},$
(8)

where $M$ and $\omega$ are the nuclear mass and frequency, and $\hbar$ is the
Planck’s constant. For derivation of $H_{\mathrm{en}}$, we use the symmetry
conditions $[H_{\mathrm{en}},C_{3}]=[H_{\mathrm{en}},\Theta]=0$, and take only
the first order in $Q_{\pm}$, which becomes (see Refs. zeng17General  and
Supplemental Material for details)

$\displaystyle H_{\mathrm{en}}=$ $\displaystyle
V_{+}Q_{-}\outerproduct{\phi_{z}}{\phi_{+}}+V_{-}Q_{+}\outerproduct{\phi_{z}}{\phi_{-}}$
$\displaystyle+V_{-}^{\ast}Q_{-}\outerproduct*{\bar{\phi}_{z}}{\phi_{+}}+V_{+}^{\ast}Q_{+}\outerproduct*{\bar{\phi}_{z}}{\phi_{-}}$
$\displaystyle+V_{0}Q_{-}\outerproduct{\phi_{+}}{\phi_{-}}+\mathrm{H.c.},$ (9)

where $\mathrm{H.c.}$ stands for the Hermite conjugate of all the preceding
terms. The first and second lines of the right-hand-side of Eq. (9) represent
the translation-rotation coupling with its strength $V_{\pm}$ and the third
line represents the rotation-rotation coupling with the coupling strength
$V_{0}$. Without loss of generality, these coupling constants can be always
set to be real.

Here, we stress that the chirality imposes

$V_{+}\neq V_{-}.$ (10)

Actually, the reflection symmetry leads to $V_{+}\equiv V_{-}$. We see this by
adding a vertical mirror, i.e., consider tha case of the point group
$\mathrm{C_{3v}}$. In this case, the reflection operator $\sigma_{\mathrm{v}}$
acts on the molecular basis as
$\sigma_{\mathrm{v}}\ket{\phi_{\pm}}=\ket{\phi_{\mp}}$,
$\sigma_{\mathrm{v}}Q_{\pm}=Q_{\mp}$, and
$\sigma_{\mathrm{v}}\ket{\uparrow\\!\\!/\downarrow}=\pm\ket{\downarrow\\!\\!/\uparrow}$.

The only ingredient for the spin-dependent process in our model is the SOC.
The derivation of $H_{\mathrm{soc}}$ is similar to that of $H_{\mathrm{en}}$
(see Ref. wang19Hamiltonian  and Supplemental Material for details): We use
the symmetry conditions $[H_{\mathrm{soc}},K]=0$ for
$K=C_{3},\sigma_{\mathrm{v}},\Theta$. Note that in this Letter, the only
source of the chirality is the inequality of Eq. (10) and we impose the
reflection symmetry on the SOC. For simplicity, we retain only the zeroth-
order terms in $Q_{\pm}$,

$H_{\mathrm{soc}}=\lambda(\outerproduct{\phi_{+}}{\phi_{+}}-\outerproduct{\phi_{-}}{\phi_{-}})\otimes\hat{\sigma}_{z},$
(11)

with
$\hat{\sigma}_{z}=\outerproduct{\uparrow}{\uparrow}-\outerproduct{\downarrow}{\downarrow}$
and the SOC strength $\lambda$. Eq. (11) indicates that spin-momentum locking
works in the chiral molecule: In the cases of $V_{+}>V_{-}$ and $V_{+}<V_{-}$,
the electron propagated with the positive momentum stabilizes the
counterclockwise ($\ket{\phi_{+}}$) and clockwise ($\ket{\phi_{-}}$)
rotations, respectively. For $\lambda>0$, this counterclockwise (clockwise)
rotation stabilizes the down-spin (up-spin) state.

## Observables.

To numerically solve the Schrödinger equation for $H$, we need the nuclear
eigenstates $\\{\ket{n,m}\\}$ to evaluate the matrix elements of $H$
koizumi94Geometric ; requist16Molecular . Then, the product state comprised of
the chiral molecule and injected electron is written as
$\ket{\Psi}=\sum_{a=z,\bar{z},+,-}\sum_{s=\uparrow,\downarrow}\sum_{n,m}C_{n,m}^{(a,s)}\ket{\phi_{a},s}\ket{n,m}$
with $C_{n,m}^{(a,s)}$ being the coupling coefficient
($\phi_{\bar{z}}\equiv\bar{\phi}_{z}$ is used for convenience). The electron
linear momentum and nuclear AM of $\ket{\Psi}$, which is analyzed later, are
calculated from
$P=\expectationvalue*{\hat{p}_{z}}{\Psi}=g\expectationvalue*{\hat{p}_{z}}{\phi_{z}}$,
where the momentum factor
$g=\sum_{s,n,m}[\absolutevalue*{C_{n,m}^{(z,s)}}^{2}-\absolutevalue*{C_{n,m}^{(\bar{z},s)}}^{2}]$,
and
$L_{\mathrm{n}}=\expectationvalue{(-i\hbar\partial_{\varphi})}{\Psi}=\sum_{a,s,n,m}m\hbar\absolutevalue*{C_{n,m}^{(a,s)}}^{2}$,
respectively footnote1 . Hereafter, the parameter values for the energy and
coupling constants are given in units of $\hbar\omega$ and
$(M\omega/\hbar)^{1/2}$, respectively.

## Spin Selective Filtering Effect.

The absence of the spin-flipping components in $H_{\mathrm{soc}}$ indicates
that the injected electron pass through the molecule without changing its
spin. This enables us to propose that the chiral molecule can act as the spin-
filter: Suppose that a ground state $\ket{\Psi}$ has the positive momentum,
$P=\expectationvalue*{\hat{p}_{z}}{\Psi}>0$, and the down-spin. Due to the
time-reversal invariance of $H$, $\Theta\ket{\Psi}$ is degenerate with
$\ket{\Psi}$, but has the negative momentum and the up-spin. Thus, these
states form Kramers pair. In injecting a down-spin electron with $P>0$ into
the molecule, the electron propagates in accordance with $\ket{\Psi}$, the
lowest state among eigenstates with the down-spin, to succeed in passing
through the molecule because of the positivity of the momentum. In contrast,
the injected up-spin electron with positive $P$ propagates in accordance with
$\Theta\ket{\Psi}$ and not with $\ket{\Psi}$ due to the mismatch of the spin.
However, because the momentum of $\Theta\ket{\Psi}$ is negative, the up-spin
electron cannot pass through the molecule (see Fig. 1(a) for the schematic
picture). Therefore, the spin is selectively filtered depending on the
translational direction of the lowest eigenstate among with the same spin as
the injected one. It is of particular importance that this mechanism of the
selective spin filtering requires the molecule to be chiral. Achiral molecule
with $V_{+}=V_{-}$ has the Hamiltonian invariant under the exchange of
$\ket{\phi_{z}}$ and $\ket*{\bar{\phi}_{z}}$, physically corresponding to the
$z$-inversion invariance of the system. This invariance causes the
disappearance of the momentum, $\expectationvalue*{\hat{p}_{z}}{\Psi}\equiv
0$, which means that achiral molecules cannot utilize the translational
direction of each eigenstate to filter the electronic spins. Therefore, the
spin-filtering is the chirality-induced effect. Note that this filtering
effect efficiently works only when the lowest state for the down-spin
propagating in the positive translational direction is energetically well-
separated from that for the up-spin. We call their energy difference as the
spin barrier difference (SBD) denoted by $\delta$, and explore the condition
that the SBD becomes large enough to explain CISS not relying on the SOC
strength $\lambda$.

In what follows, we consider the chiral case only, and without loss of
generality, $V_{+}>V_{-}>0$ is assumed.

Figure 1: (a) Schematic picture for the spin-filer of chiral molecules. (b),
(c) Energy-Momentum diagrams of spinless, spinful with $\lambda=0$, and
spinful with $\lambda>0$ systems for (b) $\delta^{\prime}=\epsilon_{\pm
1}-\epsilon_{0}>0$ of case(i) and (c) $\delta^{\prime}<0$ of case(ii). Up-
arrows and down-arrows represent the up-spin and down-spin, respectively.

## Classification of Eigenstates by AM Quantum Number.

The fact that $C_{3}$ commutes with $H$ allows us to classify all the energy
eigenstates by their AM quantum number $J$. We write $\ket{\Psi_{J}}$ for the
energy eigenstate with the AM quantum number $J$, which satisfies the
eigenvalue equations $H\ket{\Psi_{J}}=E_{J}\ket{\Psi_{J}}$ and
$C_{3}\ket{\Psi_{J}}=e^{-i2\pi J/3}\ket{\Psi_{J}}$ with $J=0,\pm 1$ for the
spinless systems and $J=\pm 1/2,3/2$ for the spinful systems. Note that $J$ is
determined up to the integer multiple of $3$, and thus, $J=-3/2$ is equivalent
with $J=3/2$. From the time-reversal invariance, $\ket{\Psi_{J}}$ and
$\ket{\Psi_{-J}}$ are degenerate, but with the opposite momentum and spin from
each other, immediately leading to
$\expectationvalue*{\hat{p}_{z}}{\Psi_{0}}=0$.

Now, we consider the effect of small but finite $\lambda$ on the eigenstates.
For distinction, we use a capital $J$ and small letters $j$ for the quantum
number in the spinful and spinless systems, respectively, and corresponding
state, energy, and momentum are denoted by $\ket*{\Psi_{J}}$, $E_{J}$, and
$P_{J}$ for the spinful systems and $\ket*{\psi_{j}}$, $\epsilon_{j}$, and
$p_{j}$ for the spinless systems, respectively. Two quantum numbers are
related by $J=j+1/2$ for a up-spin and $J=j-1/2$ for a down-spin. In the cases
of $\lambda=0$, the spinful states are given by
$\ket*{\Psi_{j+1/2}}\equiv\ket*{\psi_{j}}\ket{\uparrow}$ or
$\ket*{\Psi_{j-1/2}}\equiv\ket*{\psi_{j}}\ket{\downarrow}$, where the spatial
part of the states is exactly the spinless state $\ket*{\psi_{j}}$, indicating
that $E_{j\pm 1/2}\equiv\epsilon_{j}$ and $P_{j\pm 1/2}\equiv p_{j}$. In
contrast, in the cases of $\lambda>0$, where the spatial part of
$\ket*{\Psi_{j\pm 1/2}}$ is modified from $\ket*{\psi_{j}}$, these state have
the different energy and momentum; For $j=0$, the states with $J=\pm 1/2$ are
degenerate (interchanged by the time-reversal operation) and $P_{\pm 1/2}\neq
0$ due to finite $\lambda$, as seen in Fig. 2(a), where the momentum factor
$g$ scales linearly with $\lambda$. The spin direction of the state is
governed by the spin-momentum locking. For $\lambda>0$ and $V_{+}>V_{-}$, the
momentum of the down-spin, namely, the state with $J=-1/2$, is positive. For
$j=1$ with $p_{1}=-p_{-1}>0$ in the cases of $V_{+}>V_{-}$, the spin-momentum
locking stabilizes the down-spin state, namely, $E_{3/2}>E_{1/2}$. The origin
of the energy splitting in these states is the SOC.

## SBD.

Based on the above argument, we examine the SBD in the cases of (i)
$\delta^{\prime}\coloneqq\epsilon_{1}-\epsilon_{0}>0$ and (ii)
$\delta^{\prime}<0$. In the case (i) (see Fig. 1(b) for the schematic
picture), the chiral molecule filters the spin with the ground states
$\ket*{\Psi_{0\pm 1/2}}$. The SBD is, then, given by $\delta\equiv
E_{3/2}-E_{-1/2}=\delta^{\prime}+O(\lambda)$. Hence, under the condition that
$\delta^{\prime}$ is much larger than the contribution from the SOC, the SBD
is approximated as $\delta\approx\delta^{\prime}$, and can be nearly
independent of $\lambda$, as numerically shown in Fig. 2(b). This condition
will be discussed later. We propose that the efficient spin-filtering effect
exactly corresponds to this situation, where the effect is insensitve to the
magnitude of $\lambda$. On the other hand, in the case (ii) (see Fig. 1(c) for
the schematic picture), the spin-filtering occurs in the ground states
$\ket*{\Psi_{1/2=1-1/2}}$ and $\ket*{\Psi_{-1/2=-1+1/2}}$. The SBD is then
identical to $\delta\equiv E_{3/2}-E_{1/2}=O(\lambda)$, which is the energy
difference caused by the SOC and thus scales linearly with $\lambda$ as
presented in Fig. 2(b). Therefore, the resultant spin filter efficiency is
expected to be small.

Figure 2: (a) Momentum factor $g$ of the ground state for the case (i) and
(ii) are plotted as a function of the SOC strength $\lambda$. Momentum scales
linearly for the case (i), whereas it is nearly independent of $\lambda$ for
the case (ii). (b) The SBD $\delta\equiv E_{3/2}-E_{-1/2}$ for the case (i)
and $\delta\equiv E_{3/2}-E_{1/2}$ for the case (ii) are plotted as a function
of $\lambda$.

## Ground-State of Spinless Systems.

We justify the conditions, under which the case (i) (Fig. 1(b)) holds, namely,
$\ket*{\psi_{0}}$ becomes the ground state for the spinless systems. We start
this by presenting two limits that we can analytically identify the AM quantum
number of the ground state. The first is the limit of $V_{0}\equiv 0$, where
the coupling between the rotational states disappears and only the
translation-rotation coupling exist. In this limit,
$\hat{l}_{1}=-i\hbar\partial_{\varphi}-\hbar(\outerproduct{\phi_{+}}{\phi_{+}}-\outerproduct{\phi_{-}}{\phi_{-}})$
is conserved with the integer eigenvalue $l_{1}/\hbar\in\mathbb{Z}$. Among
them, the ground state is the state with $l_{1}=0$ that is simultaneously the
eigenstate of $C_{3}$ with $j=0$ footnote2 . Hence, the ground state is given
by $\ket*{\psi_{0}}$ in this limit. The second is the limit of $V_{\pm}\to 0$,
where the translation and rotation are decoupled, which means that the
injected electron directly pass through or is simply circulating in the
molecule. In this limit,
$\hat{l}_{2}=-i\hbar\partial_{\varphi}+(\hbar/2)(\outerproduct{\phi_{+}}{\phi_{+}}-\outerproduct{\phi_{-}}{\phi_{-}})$
is conserved with the half odd integer eigenvalue $l_{2}/\hbar=\pm(2n-1)/2$
with $n\in\mathbb{N}$. The ground states are given by the state with
$l_{2}/\hbar=\pm 1/2$ that are simultaneously the eigenstates of $C_{3}$ with
$j=\pm 1$. Therefore, in this limit, $\ket*{\psi_{\pm 1}}$ is the ground
state.

General situations are located in between these two limits and the ground
states are interchanged depending on the parameter values of
$\epsilon_{\mathrm{tr}}$, $V_{\pm}$, and $V_{0}$. Figure 3(a) presents the
energy difference $\delta^{\prime}$ as a function of the rotational coupling
constant $V_{0}$. As $V_{0}$ is decreased below $0.5$, in which the situation
approaches the first limit, $\delta^{\prime}$ increases, thus strongly
stabilizing the ground state with $j=0$. Decreasing $\epsilon_{\mathrm{tr}}$
and/or increasing $V_{\pm}$ also make the system approach the first limit, in
which the translation and rotation is strongly coupled. Therefore, the ground
state is again given by $\ket{\psi_{0}}$, which is consistent with the
numerical result presented in Fig. 3(b).

Figure 3: (a) The energy differences of the spinless system
$\epsilon_{1}-\epsilon_{0}$ are plotted as a function of $V_{0}$ for $V_{+}=1$
and $V_{+}=1.5$ with fixed $V_{-}=0.5$ and $\Delta=0.1$. (b) The energy
differences of the spinless system $\epsilon_{1}-\epsilon_{0}$ are plotted as
a function of $\epsilon_{\mathrm{tr}}$ for various values of $V_{\pm}$ with
fixed $V_{0}=0.5$.

Finally, we mention that the energy difference $\delta^{\prime}$ is correlated
with that of the nuclear AM. Figure 4(a) and (b) present the energy difference
$\delta^{\prime}=\epsilon_{1}-\epsilon_{0}$ and the nuclear AM,
$L_{\mathrm{n}}=\expectationvalue*{(-i\hbar\partial_{\varphi})}{\psi_{1}}$,
respectively, as a function of $\Delta_{V}\coloneqq V_{+}-V_{-}$ with fixed
$V_{+}$. The behavior in Figs. 4(a) and (b) can be accounted for by an
interplay between the nuclear rotational energy and the pseudo Jahn-Teller
effect. In the $\Delta_{V}<0.5$ region in Fig. 4(a), as $\Delta_{V}$
increases, this imbalance increases the nuclear AM and more differentiates
$\ket*{\psi_{\pm 1}}$ from $\ket*{\psi_{0}}$, thus increasing
$\delta^{\prime}$. However, increasing $\Delta_{V}$ by decreasing $V_{-}$ also
diminishes the pseudo Jahn-Teller effect, making the nuclear stable
configuration close to the high-symmetry point, $\rho\equiv 0$, and hence
reducing the nuclear average radius (see Fig. 4(c)-(e)), which leads to the
decrease of the nuclear AM and thus, the nuclear rotational energy. The
resultant energy difference decreases as presented in the large $\Delta_{V}$
region ($\Delta_{V}>1.2$) in Fig. 4(a). Therefore, the nuclear vibrations have
the role of not only mediating the translation-rotation coupling, but also
increasing the energy difference $\delta^{\prime}$, and consequently enhancing
the spin-filtering efficiency.

Figure 4: (a) Energy difference $\delta^{\prime}=\epsilon_{1}-\epsilon_{0}$
and (b) nuclear AM $L_{\mathrm{n}}$ of $\ket{\psi_{1}}$ are plotted as a
function of $\Delta_{V}=V_{+}-V_{-}$ with fixed $V_{+}$. Other parameters are
set to $\epsilon_{\mathrm{tr}}=0.1$ and $V_{+}=1.5$. (c)-(e) Ground state
Born-Oppenheimer potential surfaces are depicted for $V_{-}=0.1$ (c), $0.5$
(d), and $1$ (e). Crossed points display the location of the minimal points in
each potential surface, which are farther from the origin as $V_{-}$
increased.

## Concluding Remarks.

In this Letter, we have theoretically explored the mechanism of the spin-
filtering effect of chiral molecules. For that purpose, we have constructed a
model of the point-group $\mathrm{C}_{3}$ that consists of the electronic
translational and rotational states, which are coupled with each other by the
pseudo Jahn-Teller effect, or being mediated by the nuclear vibrational
degrees of freedom. The time-reversal symmetry of the Hamiltonian makes it
possible to possess, in combination with the SOC, the spin-momentum locking to
filter the injected spin depending on the translational direction. Only when
the injected spin state matches the lowest eigenstate of the same
translational direction, the electron can selectively pass through the
molecule with the ground state energy. This also enables us to classify the
molecular eigenstates by their AM quantum number. When the spinless system has
the ground state with $j=0$, the SBD that determines the spin-filter
efficiency is nearly independent of the SOC strength, and thus, makes it
possible for the chiral molecule to act as the efficient spin-filter protected
by the energy for AM quantization. Here we stress that $\delta^{\prime}$ can
be estimated at around $50\text{\,}\mathrm{meV}$ which is about twice as large
as room temperature energy and is therefore enough to explain $60\text{\,}\%$
spin-polarization in the photoelectron spectroscopy. In the above estimate, as
an example, we employed $\hbar\omega=$0.34\text{\,}\mathrm{eV}$$ (ca.
$2740\text{\,}\mathrm{c}\mathrm{m}^{-1}$) in Ref. allen05Quantum  and maximum
reduced $\delta^{\prime}$ value of $0.16$ in Fig. 3 (a). The ground state with
$j=0$ for the spinless systems is obtained when the translation-rotation
coupling constant $V_{\pm}$ is sufficiently large compared to the rotational
one $V_{0}$. In our future work, the relation between the model parameters and
the molecular structure should be clarified together with the direct
estimation of the spin-filtering efficiency for actual chiral molecules.
Additionally, the mechanism of the spin-filtering proposed in this Letter is
not the specific property of the point-group $\mathrm{C}_{3}$, and will be
straightforwardly generalized to other point or space groups to which many
chiral and/or gyrotropic materials belong.

In this Letter, we have eliminated the spin-flipping components from the SOC,
thus focusing only the spin-filtering effect and neglecting the spin-
polarizing effect of chiral molecules. However, a chirality-induced bulk
magnetization of CrNb3S6 seems to require the mechanisms that include the
spin-polarizing effect nabei2020Currentinduced . Therefore, CISS seems to
include both spin-filtering and spin-polarizing effects depending on the
situations. In our model, as given explicitly in the Supplemental Material, we
are also able to allow the nuclear-dependent SOC come into play, yielding the
spin-flipping terms such as
$(\outerproduct{\phi_{+}}{\phi_{+}}-\outerproduct{\phi_{-}}{\phi_{-}})\otimes(Q_{+}\hat{\sigma}_{+}+Q_{-}\hat{\sigma}_{-})$
and
$i(\outerproduct{\phi_{z}}{\phi_{z}}-\outerproduct*{\bar{\phi}_{z}}{\bar{\phi}_{z}})\otimes(Q_{+}\hat{\sigma}_{+}-Q_{-}\hat{\sigma}_{-})$
with $\hat{\sigma}_{+}=\outerproduct{\uparrow}{\downarrow}$ and
$\hat{\sigma}_{-}=\outerproduct{\downarrow}{\uparrow}$. These terms allow AM
transfer from nuclear phonon to electron spin, thus polarizing the spin in the
chiral molecule while respecting the conservation of AM. This AM conversion
may be further enhanced when the molecule involves conical intersections on
the adiabatic potential energy surfaces wu2021Electronic ; bian2021Modeling .
We here note that the current-induced magnetization phenomena, called the
Edelstein effect, is apparently similar to the CISS effect from a global
symmetry viewpoint. This effect is, however, interpreted as being mainly
dominated by orbital effects furukawa2017Observation ; yoda2018orbital  and
the resultant magnetization is much weaker than huge polarization caused by
the CISS effect. The relevance of the spin-flipping components of the SOC to
the spin-polarizing effect will be focused on in future studies.

###### Acknowledgements.

We thank A. S. Ovchinnikov and I. G. Bostrem for discussions in the early
stage of this work. This work was supported by the JSPS KAKENHI Grant No.
19H00891, No. 20J01875, and No. 21H01032.

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[1]Zheng-Yun You (尤郑昀)

1]School of Physics, Sun Yat-Sen University, Guangzhou, 510275, China

2]Institute of Modern Physics, Fudan University, Shanghai, 200433, China

3]Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, 1015, Switzerland

4]School of Physics, Peking University, Beijing, 100871, China

5]Institute of High Energy Physicss, Chinese Academy of Sciences, Beijing,
100049, China

6]University of Chinese Academy of Sciences, Beijing, 100049, China

# Visualization for physics analysis improvement and applications in BESIII

Zhi-Jun Li (李志军)  Ming-Kuan Yuan (袁铭宽)  Yun-Xuan Song (宋昀轩)  Yan-Gu Li (李彦谷)
Jing-Shu Li (李静舒)  Sheng-Sen Sun (孙胜森)  Xiao-Long Wang (王小龙)
<EMAIL_ADDRESS>Ya-Jun Mao (冒亚军)  [  [  [  [  [  [

###### Abstract

Modern particle physics experiments usually rely on highly complex and large-
scale spectrometer devices. In high energy physics experiments, visualization
helps detector design, data quality monitoring, offline data processing, and
has great potential for improving physics analysis. In addition to the
traditional physics data analysis based on statistical methods, visualization
provides unique intuitive advantages in searching for rare signal events and
reducing background noises. By applying the event display tool to several
physics analyses in the BESIII experiment, we demonstrate that visualization
can benefit potential physics discovery and improve the signal significance.
With the development of modern visualization techniques, it is expected to
play a more important role in future data processing and physics analysis of
particle physics experiments.

###### keywords:

particle physics experiments, visualization, physics analysis, BESIII

##  1 Introduction

Visualization has always been an essential technique since the birth of
nuclear and particle physics. In the early days of the 20th century, when the
field of particle physics emerged, nuclear emulsion, cloud chambers, and
bubble chambers were used to detect particles. The specific particles and
event processes are observed directly with a high level of intuitiveness
during the data analysis. With the advancement of particle physics
experimental techniques, this visual observation method can no longer meet the
increasing demands for higher precision and larger data volume for modern
particle physics experiments. Consequently, high-speed electronic data
processing and statistical-based analysis methods become popular for massive
data processing.

Nowadays, large-scale general-purpose spectrometers such as ATLAS [1], CMS
[2], LHCb [3, 4], BESIII [5], Belle [6] and BelleII [7], as well as other
collider based experiments, are playing significant roles in the field of
particle physics. High-precision large-scale spectrometers exhibit
considerable complexity and often translate particle information into a set of
non-intuitive electronic signals. Visualization of large-scale spectrometers
can recreate the behavior of particles in the detector more intuitively,
allowing the spectrometers to maintain high speed and high-precision data
processing while still being visually accessible [8, 9]. However,
visualization in particle physics applications often remains at the level of
visual display, and the potential value of its application in specific physics
analysis is usually overlooked.

In this study, we present the visualization-based method to assist physics
analysis. Building on traditional analysis methods, the visualization-based
analysis method can further find new discrimination variables, suppress the
background, and quickly assess the quality of observed data in modern particle
physics experiments. BESIII, an electron-positron collider experiment with low
background level, is suitable to demonstrate the advantages of the
visualization method for physics analysis. Taking BESIII as an example, the
BESIII Visualization software (BesVis) will be introduced and its application
in several specific physics analyses will be discussed.

The remainder of this paper is organized as follows. In Section 2, the
analysis method based on visualization is presented. In Section 3, we
introduce the BESIII experiment and its visualization software. The
application of visualization to assist BESIII analyses is introduced in
Section 4. Finally, the potential for further development and applications is
discussed in Section 5.

##  2 Methodologies

Modern particle physics data analyses usually rely on statistical methods
based on selection criteria to screen physically interesting signal events.
The raw hits in the detector from real experiments or simulations are the
original information, which will be reconstructed as an event or multi-events
based on the detector geometry. The position and time of hits, cluster of
multiple hits, particle vertex, momentum and energy of track, which are
considered as low-level input information, can be obtained from the raw hits
with the reconstruction to construct the physics event data. Based on the
reconstruction data, more physics variables are necessary for further
analysis, such as the invariant mass of particles, missing transverse energy
(MET), missing momentum ($P_{\rm{miss}}$), kinematic fit updated four-momentum
[10], which are treated as high-level event information. Events from different
processes have different features on these physics variables, thus providing a
portal to distinguish the signal and background in an analysis, which is
usually called “cut” in particle physics analysis. The statistical cut-based
method can apply cuts to reduce background and extract signals, processing a
massive number of events and quantifying the statistical features of these
events with great success in modern particle physics research. However, this
method typically requires a certain number of event statistics to reveal the
characteristics of the physics variables and usually depends on empirical
choices for constructing suitable physics variables, which generally lack an
intuitive understanding of the global features of events in the detector.
Before applying the cut-based event selection, construction of the high-level
physics variables also involves triggering, simulation, and reconstruction, in
which the low-level event information, such as raw hits, tracks, and their
association with detector geometry, may have been lost in the stages of data
processing.

Another method, the visualization method has been widely used in the early
development of particle physics, such as the observation of the strange
particle by W. B. Fowler [11] and the first observation of ternary and
quaternary fission of Uranium nuclei by Sanqiang Qian and Zehui He [12]. In
contrast to the multi-event, empirical, and software-dependent nature of the
cut-based method, the visualization method allows researchers to grasp the
entire response of an event in the detector as a whole picture. It offers
strong visual interpretability, with less reliance on empirical knowledge and
data processing software, and enables a rapid examination of the physical
characteristics of individual events. In modern particle physics, the cut-
based method is the basic data analysis method, while the visualization method
can help overcome the limitations of the cut-based method and enable both
approaches to complement each other, as shown in Table 1. The visualization
method relies on three key inputs from the cut-based method: low-level input
from raw hits and detector, input from reconstruction (RECO) data, and events
type input from the signal-background distinction. Based on these inputs,
there are four important outputs for the cut-based method that can help to
improve the physics analysis:

  * •

Find reconstruction problem. By displaying the raw hits, detector and
reconstruction information at the same time, it is easy to check whether the
reconstruction is consistent with the raw hits, which is widely applied in
data quality monitoring system and data reconstruction. Sec. 4.1 is an example
of its application in physics analysis.

  * •

Discover more useful cuts. “cut” variables in the cut-based method are usually
based on the experience, but the event display based on the visualization
method can show the overall information of an event, helping reveal additional
useful cut variables or new input for the cut variables, particularly those
related to the detector structure or event reconstruction. Sec. 4.3 (Sec. 4.4)
is an example of discovering a new cut (input) based on the visualization
method.

  * •

Suppress background. The remaining background after cut-based method can be
further checked with the visualization method at the event level, which can
illustrate the origin and behavior of the background, leading to the discovery
of more approaches to further suppress the background. Sec. 4.2 is an example
of studying the remaining background based on the visualization method.

  * •

Eliminate fake signal. The remaining signals after the cut-based method also
need to be further checked with the visualization method at the event level,
especially in discovery of any rare process or new physics signals. Any
performance that exceeds expectations in the visualization may indicate that
there are no real signals but unexpected backgrounds with the cut-based
method. Sec. 4.1 is an example of eliminating fake signals based on the
visualization method.

Figure 1 shows the steps in general statistical cut-based physics analysis and
the supplement of visualization in each step. Up to now, the visualization
method has been effectively applied in data quality monitoring systems,
physics simulations, and data reconstruction. Additionally, due to the
intuitiveness of event display software, visualized images are increasingly
used in public news, journal covers, and outreach articles to present the
physics processes visually. However, direct application of visualization in
specific physics analysis is still limited.

Figure 1: The steps in general statistical cut-based physics analysis (left part) and the supplement of visualization in each step (right part). The two methods can be combined to improve the physics analysis. Table 1: The advantages and disadvantages of statistical cut-based and visualization analysis. Characteristic | Cut-based analysis | Visualization
---|---|---
Processing a large number of events | ✓ | ✗
Quantifying the statistical features of multiple events | ✓ | ✗
Relying on other software and experience | ✓ | ✗
Highly intuitive | ✗ | ✓
Comprehensive detailed information for a single event | ✗ | ✓

##  3 BESIII and visualization

Figure 2: Visualization of BESIII detector, where the outermost red part
interleaved with white layers is MUC, the blue part is EMC, the yellow part is
TOF, the light grey part is MDC, and the central dark grey part is the beam
pipe.

Beijing Spectrometer III (BESIII) [5] is a general-purpose spectrometer for
$\tau$-charm physics study in the center-of-mass energy range from 2.0 to 4.7
GeV. BESIII records symmetric $e^{+}e^{-}$ collisions provided by the Beijing
Electron Positron Collider II (BEPCII) storage ring [13] and has collected
large data samples in this energy region [14]. The cylindrical core of the
BESIII detector covers 93% of the full solid angle and consists of a helium-
based multilayer drift chamber (MDC), a plastic scintillator time-of-flight
system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all
enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic
field. The solenoid is supported by an octagonal flux-return yoke with
resistive plate counter muon identification modules interleaved with steel
(MUC). Visualization of the BESIII detector is shown in Figure 2.

BesVis [15, 16] is a visualization software in the BESIII experiment, which is
developed with ROOT [17] in the BESIII Offline Software (BOSS). As shown in
Figure 3 (a), its graphical user interface (GUI) is based on ROOT GUI package
and the geometric description is provided in the format of Geometry
Description Markup Language (GDML) [15]. By reading GDML files, BesVis
generates detector geometry in ROOT format, ensuring complete consistency with
the detector description used in the simulation and reconstruction of offline
software.

BesVis supports both two-dimensional and three-dimensional visualization of
detectors and physics events. As shown in Fig. 3, an example event of
$\psi(2S)\to\pi^{+}\pi^{-}J/\psi,~{}J/\psi\to\gamma\eta_{c},~{}\eta_{c}\to\gamma\gamma$
from Monte Carlo (MC) simulation is displayed. Figure 3 (a) shows the two-
dimensional display, where the left sud-pad is XY view and the right sub-pad
is ZR view. From outer to inner, the pink ring represents the MUC, the blue
ring represents the EMC, the yellow ring represents the TOF, and the innermost
gray area represents the MDC. The detector units highlighted in red indicate
hits fired by the particles, while the black curves and crosses represent
reconstructed track information. For example, in Figure 3 (a), the two
circular tracks in the MDC represent the two oppositely charged pions, and the
three hit clusters in EMC without corresponding MDC tracks indicate that they
are three photons.

Figure 3 (b) is the three-dimensional display. To emphasize visualization of
the event, some parts of the sub-detectors are set to invisible. The hits on
the two barrel tracks indicate $\pi^{+}\pi^{-}$ interactions with MDC wires.
The blue EMC barrel wall shows three red hits that are not $\pi^{+}\pi^{-}$
interactions, corresponding to three photons, where the two large clusters
correspond to high-energy decay photons of $\eta_{c}$, and the small cluster
corresponds to low-energy radiated photons of $J/\psi$. The 2D display is
based on a 2D plane projection, while the 3D display is developed using OpenGL
in the ROOT framework. This allows users to examine the detectors and events
from different perspectives and gain a comprehensive understanding of the
spatial distribution and characteristics of the particles and their
interactions with the detector.

Figure 3: GUI of BesVis and display of an example event with 2D (a) and 3D (b)
visualization, where the event is
$\psi(2S)\to\pi^{+}\pi^{-}J/\psi,J/\psi\to\gamma\eta_{c},\eta_{c}\to\gamma\gamma$
from simulation.

Since the development of BesVis, it has played an important role in detector
design optimization, detector simulation, reconstruction algorithm tuning,
data quality monitoring, the paper publication with journal cover [18, 19],
outreach, and education for particle physics. With the promotion of the
visualization method in BESIII, BesVis has also gradually enhanced its
application in physics analysis in recent years.

##  4 Application in analysis

In this section, we will take the application of BesVis in the physics
analysis of BESIII as an example to introduce the role of visualization in
physics analysis, including signal validation, background rejection, and
assisting new analysis technique development. The following scenarios will be
discussed:

  * •

Visualization aids in identifying the authenticity of new physics signals in
search for invisible decay of $\Lambda$ baryons.

  * •

Visualization helps in studying background peaks in search for charmonium rare
semi-muonic decay $J/\psi\to D^{-}\mu^{+}\nu_{\mu}$.

  * •

Visualization assists in further reducing background and improving data
sensitivity to new physics in search for lepton flavor violation decay
$\psi(2S)\to e^{-}\mu^{+}$.

  * •

Visualization demonstrates the potential to implement machine learning
techniques for signal discrimination in observation of
$\Lambda^{+}_{c}\to{n}e^{+}{\nu}_{e}$ decays.

###  4.1 Validating potential discovery in invisible decay of $\Lambda$ baryon

In collider experiments, it is possible to generate dark matter, which
interacts very weakly with ordinary matter, making it invisible in detectors.
BESIII operates at the threshold energy point of charm production, its clean
background and complete reconstruction of particles enable the search of
invisible particles. Searching for the invisible decay of lambda baryons in
BESIII is one of the ways to search for dark matter [20].

In BESIII, $\Lambda$ baryons are often produced in pairs. For instance, they
can be generated through the decay of $J/\psi$ particles with a 3.097 GeV
threshold, resulting in $\Lambda\bar{\Lambda}$ pairs. By tagging
$\bar{\Lambda}\to\bar{p}\pi^{+}$, a clean sample of $\Lambda$ on the opposite
side can be obtained, allowing researchers to study the invisible decay of
$\Lambda$ baryons. The energy deposition $E_{EMC}$ of showers in EMC on the
opposite side of $\bar{\Lambda}$ can serve as a physical variable to
characterize the invisible signal. Since the invisible dark matter does not
interact with the detector, the signal of interest will distribute near zero
on the $E_{EMC}$ spectrum. Hence, an excess near zero on the $E_{EMC}$
spectrum implies a possible signal from invisible dark matter. After applying
traditional cut-based selection criteria, during the early stage of data
analysis, a prominent target “signal” peak was discovered. Using visualization
software to display such events beyond expectation with more information can
help differentiate the authenticity of the “dark matter signal”. If the dark
matter signal exists, there should be no hits in EMC on the opposite side of
$\bar{p}\pi^{+}$, as shown in Figure 4(a). However, as shown in Figure 4(b), a
significant amount of hits are observed, which indicates that these events are
not genuine dark matter signals.

Figure 4: (a) An event display simulated for
$J/\psi\to\Lambda\bar{\Lambda},\bar{\Lambda}\to\bar{p}\pi^{+},\Lambda\to\rm{invisible}$.
(b) A fake “dark matter signal” in observed data from cut-based analysis. The
blue region represents EMC, and the yellow region represents TOF. The inner
red hits indicate $\bar{p}\pi^{+}$ tracks, while the red squares within the
yellow region represent TOF hits. The outermost purple-red bars represent EMC
clusters. In (b), the additional EMC hits on the opposite of $\bar{p}\pi^{+}$
indicate that this event is a false signal, and the lack of TOF hits implies
the reason for the fake signal.

Visualization helps to eliminate the fake signals in the observed data through
comprehensive detailed information of the event, and further study in the
visualization found that there are no TOF hits of $\bar{p}\pi^{+}$ in the
“dark matter signal” and the hit time of the additional strange showers in EMC
is larger than the expectation, which implies the source of the false signals.
In fact, the previously discovered “dark matter signal” was due to a hidden
issue with time triggers in the BESIII offline software. In BESIII, photons
are usually selected within the time window of 0-700 ns, and the starting time
of this window (T0) [21], is usually determined by the triggering of charged
particles in MDC and TOF. When no charged particles hit the TOF detector, the
uncertainty in T0 can be relatively large, leading to the omission of some
photons that should have been detected. This results in the appearance of a
fake signal peak near zero in $E_{EMC}$ spectrum. The “dark matter signal”
disappeared once the requirement of at least one charged track hitting TOF was
imposed [20]. Visualization can help researchers quickly determine whether the
events beyond expectation are genuine signals, thus avoiding false
discoveries. In comparison to traditional cut-based physical analysis, the
visualization approach displays more low-level hit information of events and
is less affected by the possible issues during reconstruction and analysis
data processing, allowing for direct, intuitive, and quick assessment of the
authenticity of observed signals. It holds significant practical value in the
search for rare processes and new physics.

###  4.2 Studying peaking background in search for semi-muonic charmonium
decay

The charmonium $J/\psi$ can decay into a charmed meson through weak decay,
although its branching ratio is extremely low in the standard model (SM) [22,
23, 24, 25, 26]. Nevertheless, some new physics models beyond SM can
significantly enhance this process by several orders of magnitude [27]. Hence,
the search for weak decays of $J/\psi$ is a sensitive probe to test SM and
explore new physics possibilities [14, 28, 29, 30, 31], such as searching for
semi-muonic decay $J/\psi\to D^{-}\mu^{+}\nu_{\mu}$ [32].

In $J/\psi\to D^{-}\mu^{+}\nu_{\mu}$ analysis, the $D^{-}$ meson is
reconstructed using the $D^{-}\to K^{+}\pi^{-}\pi^{-}$ decay channel, so the
final state requires identifying four charged tracks corresponding to
$K^{+}\pi^{-}\pi^{-}\mu^{+}$, along with an undetectable neutrino carrying
missing information. To characterize the signal of this process, a physical
variable $U_{\rm{miss}}$ is defined as
$U_{\rm{miss}}=E_{\rm{miss}}-|\vec{P}_{\rm{miss}}|$, where $E_{\rm{miss}}$
represents missing energy, and $|\vec{P}_{\rm{miss}}|$ represents missing
momentum, both contributed by the neutrino in the final state. For the signal
process $J/\psi\to D^{-}\mu^{+}\nu_{\mu}$, $U_{\rm{miss}}$ follows a Gaussian-
like distribution centered around zero. If the semi-muonic weak decay is
present in the data, a peak around zero should be observed in the
$U_{\rm{miss}}$ spectrum.

Figure 5: (a) $J/\psi\to
K^{+}K^{-}\pi^{+}\pi^{-},K^{-}\to\mu^{-}\bar{\nu}_{\mu}$ background events,
where the four charged tracks cannot intersect at a single point. (b)
$J/\psi\to D^{-}\mu^{+}\nu_{\mu},D^{-}\to K^{+}\pi^{-}\pi^{-}$ signal events,
where the four charged tracks can intersect at a single point, indicating they
are from the same primary vertex. The display mode is called fisheye view,
which amplifies the central region for better observation of the particles’
vertices. The green and blue circles represent the stereo and straight wires
in MDC, respectively.

Based on the traditional cut-based analysis, a clear peak was found in the
$U_{\rm{miss}}$ spectrum near 0. Similarly, visualization can help quickly
inspect the physical behavior of this peak event. Using BesVis for event
display, it was observed that among the four charged track paths in the final
state, only three intersect at the same point near the vertex, while the
fourth track deviates, as shown in Figure 5 (a). In contrast, normal
$J/\psi\to D^{-}\mu^{+}\nu_{\mu},D^{-}\to K^{+}\pi^{-}\pi^{-}$ events, where
all four charged track are produced near the collision vertex, exhibit all
four tracks intersecting at the same point in the visualization, as shown in
Figure 5 (b).

Visualization of the event shows that one of the charged tracks in the peaking
background is not produced in the collision vertex, indicating that the source
of the background may come from a secondary decay of one particle in flight.
Actually, these peaking events come from the background $J/\psi\to
K^{+}K^{-}\pi^{+}\pi^{-},K^{-}\to\mu^{-}\bar{\nu}_{\mu}$. Although the
lifetime of $K^{\pm}$ is relatively long compared to the time scale of an
event in BESIII, there are still a very small fraction of $K^{\pm}$ that decay
within the selection range of good charged tracks and contribute to the
peaking background. In the event display, the charged track that does not
intersect with the other three tracks at a single point is precisely the muon
produced from the decay of $K^{\pm}$ after flying a short distance. The
visualization method can easily identify the physical characteristics of such
peaking background events. Furthermore, it can play a crucial role in
suppressing these backgrounds and improving the sensitivity to signals in
experimental data [32].

###  4.3 Reducing detector associated background in search for charged lepton
flavor violating decay

$e\nu_{e}$, $\mu\nu_{\mu}$, and $\tau\nu_{\tau}$ are three generations of
leptons in the SM, each generation has its own flavor. With the discovery of
neutrino oscillation, the flavor violation of neutral leptons has been
observed [33, 34]. However, charged lepton flavor violation (CLFV) is heavily
suppressed in the SM. Therefore, the discovery of any CLFV process would be a
clear signal of new physics beyond the SM [35, 36]. Searching for CLFV process
is an important goal in modern particle physics experiments, and one direction
is to study it in heavy quark system [37, 38, 39, 40, 41, 42, 43], such as
search for $\psi(2S)\to e^{\pm}\mu^{\mp}$.

In cut-based physical analysis, it is necessary to select $\psi(2S)\to
e^{\pm}\mu^{\mp}$ signal events based on the distinct characteristics of
electrons and muons. For instance, electrons tend to deposit a majority of
their energy in EMC, while muons only deposit a small fraction of their energy
in EMC. High-momentum muons with strong penetrating power can reach deep into
the outermost layer of MUC, whereas electrons have limited penetrating power
and can hardly reach MUC. By exploiting these characteristics, it is possible
to effectively suppress the background noise and investigate whether $e\mu$
events are present in the data. Although the background noise level has been
significantly reduced through traditional event selection, further reducing
the background is still crucial to enhance the sensitivity for discovering
rare new physics processes. The visualization method can be used to further
reduce the remaining background.

After applying visualization to scan the remaining background events (mostly
$e^{+}e^{-}\to e^{+}e^{-}$), it is found that these backgrounds are related to
the structure of BESIII detector. As shown in Figure 6, the direction of the
charged track is near $\rm{cos}\theta\approx 0.85$ (a) or
$\rm{cos}\theta\approx 0$ (b). Here, $\theta$ represents the polar angle
between the direction of the particle and the beam axis. In these two
directions, there are gaps between EMC crystals, the electrons will traverse
the gap and deposit only a small fraction of its energy in EMC. Meanwhile, if
a cosmic ray muon happens to pass through the MUC and coincide with the
electron track in time, the electron will be misidentified as a muon track
since it satisfies the muon selection criteria, leading to the
misidentification of an $e^{+}e^{-}$ background event as an $e^{+}\mu^{-}$
signal event. Although the probability of such coincidence is very low, it
becomes a major background while analyzing billions of events, which is a
common characteristic in searching for rare processes.

By performing the visualization-based analysis method, the origins of these
backgrounds can be discovered, identified, and understood, enabling further
suppression of these backgrounds. Undoubtedly, it can increase the sensitivity
of existing data in search for rare processes, allowing for a better search
for new physics or an improved constraint on new physics models.

Figure 6: Two background events in searching for $\psi(2S)\to
e^{\pm}\mu^{\mp}$. The black lines represent the reconstructed charged tracks.
(a) The particle has a $\rm{cos}\theta\approx 0.85$ and passes through the gap
between the EMC barrel and endcap. (b) The particle has a
$\rm{cos}\theta\approx 0$ and passes through a small gap between EMC crystals
pointing to the collision vertex.

###  4.4 Studying discrimination of neutron and $\Lambda$ baryon in
$\bar{\Lambda}^{-}_{c}$ decay

$\bar{\Lambda}^{-}_{c}$ is the lightest charmed baryon and studying its semi-
leptonic decays can provide a valuable test for non-perturbative QCD theory
[44]. The decay width of $\bar{\Lambda}^{-}_{c}$ semi-leptonic decays is
related to the form factor and the CKM matrix element, where the form factor
is a non-perturbative QCD parameter, and CKM matrix element describes quark
flavor transition [45, 46]. If it is the process $c\to s$,
$\bar{\Lambda}^{-}_{c}$ can decay into $\bar{\Lambda}l^{-}\bar{\nu}_{l}$
($l=e,\mu$), while for the process $c\to d$, $\bar{\Lambda}^{-}_{c}$ can decay
into $\bar{n}l^{-}\bar{\nu}_{l}$. Due to the Cabibbo suppression mechanism,
the branching ratio of $\bar{\Lambda}^{-}_{c}$ anti-neutron semi-leptonic
decay will be much lower than that of the anti-Lambda baryon.

Studying $\bar{\Lambda}^{-}_{c}\to\bar{n}l^{-}\bar{\nu}_{l}$ in BESIII
presents significant challenges as BESIII lacks a dedicated hadron
calorimeter, and detection of neutrons primarily relies on EMC. One key
difficulty lies in distinguishing the main background from
$\bar{\Lambda}^{-}_{c}\to\bar{\Lambda}l^{-}\bar{\nu}_{l}$, where
$\bar{\Lambda}\to\bar{n}\pi^{0}$, $\pi^{0}\to\gamma\gamma$. The ability of EMC
to identify the additional $\pi^{0}$ from the anti-neutron background will
determine the feasibility of this analysis in BESIII.

Before performing the complex statistical cut-based method in this analysis,
the visualization method can be used to intuitively show the differences of
EMC cluster shape between anti-neutron and $\bar{\Lambda}\to\bar{n}\pi^{0}$.
As shown in Figure 7 (a), the reaction of anti-neutron in the EMC often
exhibits a larger and more complex hit shape, whereas the two small hits at
the top in Figure 7 (b) represent the typical cluster shape of two photons in
the $\pi^{0}$ final state. The difference between the two pictures is quite
evident, which means distinguishing anti-neutron and anti-Lambda baryon with
the EMC cluster shape is feasible. However, achieving this task is still
highly complex, and practical implementation might require the use of machine
learning tools [47, 48, 49, 50] such as GNN (Graph Neural Network). The
visualization method can showcase the potential of such advanced techniques
before their implementation in complex cut-based analysis.

Figure 7: Comparison of reaction signals for $\bar{n}$ (a) and
$\bar{\Lambda}\to\bar{n}\pi^{0}$ (b) in the EMC. The blue part represents the
EMC, and the red parts indicate the hits of neutral particles in the EMC. In
(a) and (b), the large red hits in the lower half of the EMC are associated
with anti-neutron, while the two smaller red hits in the upper half of the EMC
in (b) come from $\pi^{0}\to\gamma\gamma$ decay.

##  5 Future development

In the previous section, we demonstrate the advantages of using visualization-
based analysis techniques in various physics analyses in BESIII experiment.
While there are additional physics applications [42, 51, 52, 53, 54], they
have not been extensively detailed in this paper. In addition to the currently
operational particle physics experiments, there are several experiments
currently under construction or in the design stage, such as JUNO [55], STCF
[56], and CEPC [57]. It is necessary to develop visualization software
tailored for these future experiments, and the visualization method can also
be applied to the upcoming physics analysis work. For example, if the search
for CLFV process like $e^{+}e^{-}\to e^{-}\mu^{+}$ is going to be conducted at
STCF or CEPC, the background mentioned in Sec. 4.3 may potentially become a
spurious fake new physics signal. In that case, it will be necessary to employ
careful event display to validate the authenticity of the new physics
discovery. JUNO is a large-scale neutrino experiment aiming at the
determination of neutrino mass hierarchy, in which the background rate is more
than six orders of magnitude higher than the reactor neutrino signal rate. The
visualization tools [58, 58, 59] have been developed in JUNO and are expected
to play an essential role in screening the rare signal events for neutrino
analyses.

In future developments, it is important that visualization software can keep
up with the advancements of software and computing technology from industry to
serve physics analysis better. Having a unified platform that can provide
consistent visualization services for different experiments would be a
favorable choice, including a universal detector description method [60], a
comprehensive detector geometry transformation interface [61], and a common
event data model [62]. Furthermore, if visualization could be utilized through
web browsers, its applications would become even more convenient. With such an
implementation, visualization will become more versatile and accessible, thus
offering better assistance for particle physics analysis.

##  6 Summarry

Visualization can play a unique role in modern particle physics data analysis.
In this manuscript, we introduce the potential value of visualization method
for improvement of physics analysis. The applications of event display in
several BESIII physics analyses have demonstrated its versatility and
complementarity to the traditional statistical analysis method. It is
recommended that the visualization method be generally taken in physics
analysis, especially in search for rare physics signals in the current
operational experiments and design of the next generation particle physics
experiments.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Grant Nos. 12175321, 12150005, 11975021, 11675275, and U1932101), National
Key Research and Development Program of China (Nos. 2023YFA1606000,
2020YFA0406300 and 2020YFA0406400), State Key Laboratory of Nuclear Physics
and Technology, Peking University (Nos. NPT2020KFY04 and NPT2020KFY05),
Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program;
National College Students Innovation and Entrepreneurship Training Program,
Sun Yat-sen University.

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