# Confirming the molecule explain for the $\Xi(2030)$

Jing-wen Feng  School of Physics and Electronic Engineering, Sichuan Normal
University, Chengdu 610101, China  Cai Cheng<EMAIL_ADDRESS>School of
Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610101,
China  Yin Huang111corresponding author<EMAIL_ADDRESS>School of
Physical Science and Technology, Southwest Jiaotong University, Chengdu
610031,China

###### Abstract

Since its discovery in 1977, the spin-parity of $\Xi(2030)$ has not been fully
determined experimentally. The latest Particle Data Group (PDG) listing
suggests it may be a baryon with $J=5/2$. Therefore, studying the mass
spectrum and decay properties of $\Xi(2030)$ has become a current hot topic to
definitively establish its spin-parity. As the three-quark model fails to
explain $\Xi(2030)$, we previously proposed it may be a molecule primarily
composed of $\bar{K}\Sigma$ with $J^{P}=5/2^{+}$, based on its mass spectrum
study. To verify its molecular state interpretation, this work proposes
studying the strong decays of $\Xi(2030)$ assuming it is a $P$-wave
$J^{P}=5/2^{+}$ meson-baryon molecule predominantly composed of
$\bar{K}\Sigma$. We calculated all experimentally measured two-body and three-
body final state decay widths of $\Xi(2030)$, including
$\Xi(2030)\to\bar{K}\Lambda,\bar{K}\Sigma,\pi\Xi,\pi\Xi^{*}$, and
$\Xi(2030)\to\pi\pi\Xi,\pi\bar{K}\Sigma,\pi\bar{K}\Lambda$. The results
indicate that both the total decay width and partial decay widths agree well
with experimental values within the error margins. This supports that
$\Xi(2030)$ is a molecule with spin-parity $J^{P}=5/2^{+}$, predominantly
composed of $\bar{K}^{*}\Sigma$. Compared to the experimental central values,
our results are slightly smaller, which suggests that $\Xi(2030)$ may contain
additional components besides meson-baryon molecular components, such as three
quark structures.

##  I Introduction

Currently, the utilization of the hadronic molecular state concept is a
prominent subject in particle physics, aiming to elucidate and predict exotic
states that exhibit a more intricate internal structure than conventional
quark states Godfrey:1985xj ; Koniuk:1979vy ; Isgur:1978wd . This line of
research is primarily motivated by the fact that hadron molecular states
manifest as actual entities in nature, rather than solely existing within
theoretical models. An exemplar of such a molecular state is the deuteron, a
weakly bound state resulting from the binding of a neutron and a proton
through the nuclear force. And the nuclear force can bind more neutrons and
protons together to form heavier molecular states, such as atomic nuclei
${}^{208}Pb$. Another well-known candidate for a molecular state is the
$\Lambda(1405)$, which is widely believed to have a dominant $\bar{K}N$
component Oset:1997it ; Oller:2000fj ; Nemoto:2003ft ; Hall:2014uca ;
Miyahara:2015bya ; Kamiya:2016jqc . In particular, results from lattice-QCD
simulations Nemoto:2003ft ; Hall:2014uca  strongly support the interpretation
of $\Lambda(1405)$ as a molecular state of $\bar{K}N$ and its coupled channel.
Although the nature of the $X(3872)$ cannot be definitively determined as a
$\bar{D}D^{*}$ molecule, its discovery Belle:2003nnu  marks the beginning of a
new era in the exploration of molecular states. Furthermore, the LHCb
Collaboration has reported several hidden-charm pentaquark states LHCb:2019kea
; LHCb:2015yax ; LHCb:2016ztz ; LHCb:2016lve  that exhibit characteristics
aligning with $\Sigma_{c}\bar{D}^{(*)}$ molecules Chen:2019bip ; Guo:2019fdo ;
Xiao:2019aya ; He:2019ify ; Xiao:2019mvs ; Roca:2015dva ; Chen:2015moa ;
Chen:2015loa ; Yang:2015bmv ; Huang:2015uda . These groundbreaking finding has
provided significant impetus for the exploration of molecular states.

It is worth noting that the molecules discussed above are just a few of the
well-known candidate for molecular states. There are many molecular states
containing other components, especially those with a significant presence of
strange quarks. Indeed, there are predictions suggesting that the $f_{0}(980)$
and $a_{0}(980)$ are composed of a $\bar{K}K$ molecular component
Ahmed:2020kmp ; Dai:2014zta . This molecular structure involving $\bar{K}K$
can provide a natural explanation for the significant isospin violations
observed in the $\eta(1405/1475)\to{}\pi\pi\pi$ reaction Wu:2011yx . The
interaction between the $K$ meson and its excited state, $\bar{K}^{*}$, can
form a bound state corresponding to the $f_{1}(1285)$ resonance Roca:2005nm ;
Geng:2015yta ; Aceti:2015zva ; Zhou:2014ila ; Lutz:2003fm ; Lu:2016nlp , which
is supported by experimental evidence from the BESIII experiment
BESIII:2015you . There exist strong attraction between $K^{*}$ and
$\bar{K}^{*}$, and the $f_{0}(1710)$ can be interpreted as a
$K^{*}\bar{K}^{*}$ molecule Nagahiro:2008bn ; Geng:2010kma . Several analyses
of the $\phi$ and $K\Lambda(1520)$ photo-production data suggest the existence
of two possible hidden-strangeness molecular states He:2012ud ; Kim:2011rm ;
Xie:2010yk ; He:2014gga ; LEPS:2010ovn ; Qian:2010rr ; Xie:2013mua , which
could be associated with the nucleon resonances $N(2100)$ and $N(1875)$,
respectively. Indeed, the $\Sigma^{*}K$ and $\Sigma{}K^{*}$ interactions
provide a good understanding of these two statesHe:2017aps ; He:2015yva . The
spectroscopy and decay widths of the $\Xi(1620)$ and $\Xi(1690)$ can be well
explained by considering them as $\bar{K}\Lambda-\bar{K}\Sigma$ molecular
states Khemchandani:2016ftn ; Huang:2020taj . Additionally, the interaction
between the $\bar{K}^{*}$ and $\Sigma$ gives rise to a $P$-wave molecular
state associated with the $\Xi(2030)$ Hei:2023eqz . The presence of a
$\bar{K}\Xi(1530)$ hadronic molecular component is crucial in understanding
the observed $\Omega(2120)$ Huang:2018wth ; Valderrama:2018bmv ; Lin:2018nqd ;
Ikeno:2020vqv ; Ikeno:2022jpe . The molecular states with
$\bar{K}^{(*)}\Xi^{(*)}$ or $K^{(*)}\Omega$ components are also discussed in
Refs. Garcia-Recio:2006ztd ; Gamermann:2011mq .

In addition to the possible existence of molecules with strange quark in the
light flavor sector, it has been discovered that hadrons containing heavy
quarks, such as $c$ or $b$, have the ability to interact with
$\bar{K}^{(*)}/K^{(*)}$ mesons, leading to the formation of bound states
Guo:2017jvc . These bound states can be found in the Particle Data Group
ParticleDataGroup:2022pth . Notably, two well-known charmed-strange mesons,
namely $D_{s0}(2317)$ and $D_{s1}(2460)$, can be regarded as hadronic
molecules of $DK$ and $D^{*}K$, respectively. This interpretation arises from
the fact that their masses are approximately 160 and 70 MeV lower than the
predicted values based on the quark model Godfrey:1985xj , and they are about
40 MeV below the thresholds for $DK$ and $D^{*}K$, respectively. The molecular
assignments for $D_{s0}(2317)$ and $D_{s1}(2460)$ have been confirmed by Ref.
Kong:2021ohg . Furthermore, the authors in Ref. Kong:2021ohg  have identified
several additional bound states composed of $D\bar{K}^{*}$,$D^{*}\bar{K}^{*}$,
$\bar{K}^{(*)}B^{(*)}$, and $K^{(*)}B^{(*)}$ components, respectively. Among
them, the $D^{*}\bar{K}^{*}$ molecule corresponds to the experimentally
observed $X_{0}(2900)$.

We can find that the experimental information on the kaonic molecules is
abundant. However, the majority of these states are in the $S$-wave, except
for the $\Xi(2030)$, which can be identified as a $P$-wave molecular state
with mainly $\bar{K}^{*}\Sigma$ component Hei:2023eqz . In a similar vein,
when considering other components in the molecule, there are only several
candidates for $P$-wave molecule, such as $D_{s}\bar{D}_{s0}(2370)$
Zhu:2021exs  and $\bar{B}^{(*)}N$ Jian:2022rln , which correspond to the
experimentally observed states $Y(4274)$ and $\Lambda_{b}(6146,6152)$,
respectively. Of course, there are theoretical predictions for the existence
of $P$-wave meson-baryon molecule, which include $\bar{D}^{(*)}\Lambda_{c}$
and $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ components Yamaguchi:2016ote .

Figure 1: Feynman diagrams for the $\Xi(2030)$
$\to\bar{K}\Lambda$,$\bar{K}\Sigma$,$\pi\Xi^{(*)}$,$\pi\pi\Xi$,$\pi\bar{K}\Lambda$,
and $\pi\bar{K}\Sigma$ reactions. We also show the definition of the
kinematics ($p_{1},p_{2},p_{3},p_{4},p_{5}$) that we use in the present
calculation.

One of the main factors contributing to the scarcity of $P$-wave molecules is
the repulsive centrifugal potential present in the interaction between hadrons
at the $P$-wave, which greatly inhibits the formation of bound states.
Interestingly, the deuteron includes an undeniable $D$-wave $NN$ component,
prompting further exploration of possible candidates for high wave molecules,
especially in $P$-wave. A possible candidate for a $P$-wave molecular state is
the $\Xi(2030)$, discovered as early as 1977 Amsterdam-CERN-Nijmegen-
Oxford:1977bvi . Currently, its spin-parity is not fully determined, with only
the assumption that its most likely spin is $J=5/2$ ParticleDataGroup:2022pth
. However, theoretical calculations Xiao:2013xi  reveal that if it is a
conventional three-quark state with a spin-parity of $J^{P}=5/2^{+}$, its
strong decay width does not align with experimental observations. Therefore,
the authors revisit the idea in the literature Hei:2023eqz , proposing that it
might be a $P$-wave molecular state with dominate $\bar{K}^{*}\Sigma$
component (almost 100%). However, a basic analysis of its mass spectrum alone
cannot conclusively determine its molecular state structure, as it may
interact with quark states. If the molecular state is predominant, we can also
derive its mass spectrum. Another established method involves examining its
strong decay width, combined with mass spectrum results, to aid in determining
its molecular state structure.

In this study, we investigate the possible strong decay width of $\Xi(2030)$
by treating it as a molecular state with a nearly 100% $\bar{K}^{*}\Sigma$
component. A definitive understanding of the internal structure of $\Xi(2030)$
can be achieved by comparing the obtained decay width with experimental data.
This paper is organized as follows. In Sec. II, we will present the
theoretical formalism. In Sec. III, the numerical result will be given,
followed by discussions and conclusions in the last section.

##  II FORMALISM AND INGREDIENTS

In this work, we discuss the possibility that $\Xi(2030)$ is a $P$-wave
molecular state with a predominant component of $\bar{K}^{*}\Sigma$ by
studying its strong decay properties. The Feynman diagrams considered in this
study are depicted in Fig. (1). These diagrams encompass $t$-channel exchanges
involving $\pi$, $\rho$, $\omega$, and $K$ mesons for the transition from
$\Xi(2030)$ to $\bar{K}\Lambda$, $\bar{K}\Sigma$, and $\pi\Xi^{(*)}$ two-body
final state reactions. Additionally, $t$-channel $\bar{K}$ meson exchange is
considered for three-body final states, specifically in the reactions
$\Xi(2030)\to{}\pi\pi\Xi$, $\pi\bar{K}\Lambda$, and $\pi\bar{K}\Sigma$.
Furthermore, a tree diagram is included to represent the
$\Xi(2030)\to\pi\bar{K}\Sigma$ reaction.

To calculate the diagrams depicted in Fig. (1), it is necessary to determine
the effective Lagrangian densities corresponding to the relevant interaction
vertices. In the case of $\Xi(2030)\Sigma\bar{K}^{*}$ coupling, we adopt the
Lagrangian densities employed in Ref.Lu:2016nnt

$\displaystyle{\cal{L}}^{J^{P}=5/2^{+}}_{\Xi(2030)\Sigma\bar{K}^{*}}$
$\displaystyle=g_{\Xi(2030)\Sigma\bar{K}^{*}}\int{}d^{4}y\Phi(y^{2})\bar{\Sigma}(x+\omega_{\bar{K}^{*}}y)$
$\displaystyle\times\partial_{\mu}\bar{K}^{*}_{\nu}(x-\omega_{\Sigma}y)\Xi(2030)^{\mu\nu}+H.c.,$
(1)

where $\omega_{\bar{K}^{*}}=m_{\bar{K}^{*}}/(m_{\bar{K}^{*}}+m_{\Sigma})$ and
$\omega_{\Sigma}=m_{\Sigma}/(m_{\bar{K}^{*}}+m_{\Sigma})$. The $\Phi(y^{2})$
is effective correlation function. Its physical meaning is to describe the
spatial distribution of components within molecules. At the same time, it also
serves to prevent ultraviolet divergences in Feynman diagrams during loop
diagram calculations. Its form is not fixed, but one of main properties is
that it must vanishe fast in the ultraviolet region. We adopt the commonly
used forms in hadronic molecular decays, and the Fourier transform can be
written as

$\displaystyle\Phi(p_{E}^{2}/\Lambda^{2})\doteq{}\exp{(-p_{E}^{2}/\Lambda^{2})},$
(2)

where $p_{E}$ is the Euclidean Jacobi momentum, and $\Lambda$ is a free size
parameter, typically about 1 GeV, which varies in different systems.

The coupling constant $g_{\Xi(2030)\Sigma\bar{K}^{*}}$ in Eq. 1 can be
established through the compositeness condition Weinberg:1962hj ; Salam:1962ap
. This condition stipulates that the renormalization constant of the hadronic
molecular wave function must be zero,

$\displaystyle
Z_{\Xi(2030)}=1-\frac{\partial\Sigma^{T}_{\Xi(2030)}(p)}{\partial{}p^{2}}|_{p^{2}=m^{2}_{\Xi(2030)}}=0,$
(3)

Figure 2: The mass operator of $\Xi(2030)$.

where the transverse part, denoted as $\Sigma^{T}_{\Xi(2030)}(p)$, of the mass
operator $\Sigma^{\mu\nu\alpha\beta}_{\Xi(2030)}(p)$ can be obtained through
the following relationships,

$\displaystyle\Sigma^{\mu\nu\alpha\beta}_{\Xi(2030)}(p)=\frac{1}{2}(g^{\mu\alpha}_{\perp}g^{\nu\beta}_{\perp}+g^{\mu\beta}_{\perp}g^{\nu\alpha}_{\perp})\Sigma^{T}_{\Xi(2030)}(p)+...,$
(4)

with $g^{\mu\alpha}_{\perp}=g^{\mu\alpha}-p^{\mu}p^{\nu}/p^{2}$. Utilizing the
effective Lagrangian shown in Eq. 1, the required mass operator
$\Sigma^{\mu\nu\alpha\beta}_{\Xi(2030)}(p)$ can be calculated based on Fig. 2.
Its detailed expression is computed as follows

$\displaystyle\Sigma^{\mu\nu\alpha\beta}_{\Xi(2030)}(p)$
$\displaystyle=g^{2}_{\Xi(2030)\Sigma\bar{K}^{*}}\int\frac{d^{4}q}{(2\pi)^{4}i}\Phi^{2}(q-\omega_{\bar{K}^{*}}p)$
$\displaystyle\times\frac{1}{q\\!\\!\\!/-m_{\Sigma}}(p-q)^{\alpha}(p-q)^{\mu}$
$\displaystyle\times\frac{-g^{\beta\nu}+(p-q)^{\beta}(p-q)^{\nu}/m^{2}_{\bar{K}^{*}}}{(p-q)^{2}-m^{2}_{\bar{K}^{*}}},$
(5)

where the $p$ and $q$ are the four-momentum of the $\Xi(2030)$ and $\Sigma$,
respectively. $m_{\bar{K}^{*}}$ and $m_{\Sigma}$ are the masses of the meson
$\bar{K}^{*}$ and baryon $\Sigma$, respectively. Than, we can obtain the
coupling constant of the molecule $\Xi(2030)$ to its components

$\displaystyle\frac{1}{g^{2}_{\Xi(2030)\Sigma\bar{K}^{*}}}$
$\displaystyle=\int{}d\alpha\int{}d\beta-\sum_{i}\frac{\Lambda^{4}}{{\cal{C}}_{i}\pi^{2}{\cal{Z}}^{4}}[\frac{{\cal{F}}_{i}(m_{\bar{K}^{*}},\omega_{\bar{K}^{*}},m_{\Sigma})}{2{\cal{Z}}}$
$\displaystyle+{\cal{H}}_{i}(m_{\Sigma}){\cal{F}}_{i}^{{}^{\prime}}(m_{\bar{K}^{*}},\omega_{\bar{K}^{*}},m_{\Sigma})],$
(6)

with

$\displaystyle{\cal{F}}(x,y,z)=\frac{1}{x^{2}}$
$\displaystyle\times\exp\\{-\frac{1}{\Lambda^{2}}[\alpha{}y^{2}+\beta{}(x^{2}-m^{2})-2m^{2}z^{2}+\frac{m^{2}(\beta+2z)^{2}}{{\cal{Z}}}]\\},$
$\displaystyle{\cal{H}}(y)=y-\frac{m}{{\cal{Z}}},$ (7)

where $i=\bar{K}^{0}\Sigma^{0}$ and $i=\bar{K}^{-}\Sigma^{+}$, respectively,
and ${\cal{C}}_{\bar{K}^{*0}\Sigma^{0}}=256$ and
${\cal{C}}_{\bar{K}^{*-}\Sigma^{+}}=192$. ${\cal{Z}}=2+\alpha+\beta$ with
$\alpha$ and $\beta$ as parameters, which will be integrated from 0 to
infinity. $m$ is the mass of the baryon $\Xi(2030)$, and
${\cal{F}}_{i}^{{}^{\prime}}$ is the derivative with respect to $m$.

To calculate the diagrams depicted in Fig. (1), it is also necessary to
determine the effective Lagrangian densities for the interaction vertex
involving the coupling of meson to the baryon octet Khemchandani:2016ftn ;
Oset:1997it ; Borasoy:1998pe

$\displaystyle{\cal{L}}_{VBB}$
$\displaystyle=-g\\{\langle\bar{B}\gamma_{\mu}[V^{\mu},B]\rangle+\langle\bar{B}\gamma_{\mu}B\rangle\langle{}V^{\mu}\rangle$
$\displaystyle+\frac{1}{4M}(F\langle\bar{B}\sigma_{\mu\nu}[V^{\mu\nu},B]\rangle+D\langle\bar{B}\sigma_{\mu\nu}\\{V^{\mu\nu},B\\}\rangle)\\},$
(8) $\displaystyle{\cal{L}}_{PBB}$
$\displaystyle=\frac{F_{1}}{2}\langle\bar{B}\gamma_{\mu}\gamma_{5}[u^{\mu},B]\rangle+\frac{D_{1}}{2}\langle\bar{B}\gamma_{\mu}\gamma_{5}\\{u^{\mu},B\\}\rangle,$
(9) $\displaystyle{\cal{L}}_{PBPB}$
$\displaystyle=\frac{i}{4f^{2}}\langle\bar{B}\gamma^{\mu}[(P\partial_{\mu}P-\partial_{\mu}PP)B$
$\displaystyle-B(P\partial_{\mu}P-\partial_{\mu}PP)]\rangle,$ (10)

where the $M$ represent the masses of the baryon and $\langle...\rangle$
refers to an $SU(3)$ trace. $\\{A,B\\}=AB+BA$ and $[A,B]=AB-BA$. The constants
$D=2.4$ and $F=0.82$ were found to reproduce well the magnetic moments of the
baryons Jido:2002yz . $F_{1}=0.51$, $D_{1}=0.75$ Khemchandani:2016ftn ;
Borasoy:1998pe , and at the lowest order $u^{\mu}=-\sqrt{2}\partial^{\mu}P/f$
with $f$ = 93 MeV, The tensor field of the vector mesons
$V^{\mu\nu}=\partial^{\mu}V^{\nu}-\partial^{\nu}V^{\mu}$ and
$\sigma^{\mu\nu}=\frac{i}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})$.
$B$, $V^{\mu}$, and $P$ are the SU(3) baryon octet, vector meson, and
pseudoscalar matrices, respectively,

$\displaystyle B$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&\Sigma^{+}&p\\\
\Sigma^{-}&-\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&n\\\
\Xi^{-}&\Xi^{0}&-\frac{2}{\sqrt{6}}\Lambda\\\ \end{array}\right),$ (14)
$\displaystyle V_{\mu}$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}(\rho^{0}+\omega)&\rho^{+}&K^{*+}\\\
\rho^{-}&\frac{1}{\sqrt{2}}(-\rho^{0}+\omega)&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\\\ \end{array}\right)_{\mu},$ (18) $\displaystyle P$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{2}{\sqrt{6}}\eta\\\ \end{array}\right),$ (22)

The Lagrangians involving the interaction of the coupling of vector meson to
pseudoscalar meson vertex $VVP$ and $VPP$ are given by Bando:1984ej ;
Oset:2002sh ; Garzon:2012np

$\displaystyle{\cal{L}}_{VVP}$
$\displaystyle=\frac{G^{{}^{\prime}}}{\sqrt{2}}\epsilon^{\mu\nu\alpha\beta}\langle\partial_{\mu}V_{\nu}\partial_{\alpha}V_{\beta}P\rangle,$
(23) $\displaystyle{\cal{L}}_{VPP}$
$\displaystyle=-ig\langle[P,\partial_{\mu}P]V^{\mu}\rangle,$ (24)

where $\epsilon^{\mu\nu\alpha\beta}$ is the Levi-Cività tensor with
$\epsilon^{0123}=1$. The coupling constant
$G^{{}^{\prime}}=\frac{3g^{{}^{\prime}2}}{4\pi^{2}f}$ with
$g^{{}^{\prime}}=-\frac{G_{V}m_{\rho}}{\sqrt{2}f^{2}}$, $G_{V}=55$ MeV, and
$f=93$ MeV. The coupling constant $g$ can be fixed from the strong decay width
of $K^{*}\to{}K\pi$. With help of the Lagrangian shown in Eq. 24, the two body
decay width $\Gamma(K^{*+}\to{}K^{0}\pi^{+})$ is related to $g$ as

$\displaystyle\Gamma(K^{*+}\to{}K^{0}\pi^{+})=\frac{g^{2}}{6\pi{}m^{2}_{K^{*+}}}{\cal{P}}^{3}_{\pi{}K^{*}}=\frac{2}{3}\Gamma_{K^{*+}},$
(25)

where the ${\cal{P}}_{\pi{}K^{*}}$ is the three momentum of the $\pi$ in the
rest frame of the $K^{*}$. Using the experimental strong decay width
$\Gamma_{K^{*+}}=50.3\pm 0.8$ MeV and the masses of the particles
ParticleDataGroup:2022pth , we obtain $g=4.64$.

In order to evaluate the decay amplitudes, we also need the following
effective Lagrangians to calculate the relevant vertices Xie:2010zza

$\displaystyle{\cal{L}}_{\Xi^{*}\Sigma{}K}=\frac{f_{\Xi^{*}\Sigma{}K}}{m_{K}}\partial^{\nu}\bar{K}\bar{\Xi}^{*}_{\nu}\gamma_{5}\vec{\tau}\cdot\vec{\Sigma}+H.c.,$
(26)

where the coupling constant $f_{\Xi^{*}\Sigma{}K}=3.22$ and $m_{K}$ is the
mass of the $K$ meson.

Putting all the pieces together, we obtain the following strong decay
amplitudes,

$\displaystyle{\cal{M}}^{\\{\pi^{0},\pi^{+}\\}}_{a}$
$\displaystyle=(i)^{3}\frac{D_{1}gg_{\Xi(2030)\Sigma\bar{K}^{*}}}{3\sqrt{2}f}\\{1,\frac{2}{\sqrt{3}}\\}\int\frac{d^{4}q}{(2\pi)^{4}i}\Phi[$
$\displaystyle\times(p_{1}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]\bar{u}(p_{3})q\\!\\!\\!/\gamma_{5}\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\nu\sigma}(k)$
$\displaystyle\times{}p_{2}^{\nu}(p_{4}^{\alpha}+q^{\alpha})\frac{-g^{\sigma\alpha}+p_{2}^{\sigma}p_{2}^{\alpha}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}\frac{1}{q^{2}-m^{2}_{\pi}},$
(27) $\displaystyle{\cal{M}}^{\\{\rho^{0},\rho^{+}\\}}_{a}$
$\displaystyle=i(i)^{3}\frac{DG^{{}^{\prime}}g_{\Xi(2030)\Sigma\bar{K}^{*}}}{12\sqrt{2}m_{\Sigma}}\\{1,\frac{2}{\sqrt{3}}\\}\int\frac{d^{4}q}{(2\pi)^{4}i}$
$\displaystyle\times\Phi[(p_{1}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]\bar{u}(p_{3})(q\\!\\!\\!/\gamma_{\eta}-\gamma_{\eta}q\\!\\!\\!/)$
$\displaystyle\times{}\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\theta\lambda}(k)\epsilon^{\sigma\tau\alpha\beta}p_{2\sigma}q_{\alpha}p_{2}^{\theta}$
$\displaystyle\times\frac{-g^{\lambda\beta}+p_{2}^{\lambda}p_{2}^{\beta}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}\frac{-g^{\tau\eta}+q^{\tau}q^{\eta}/m^{2}_{\rho}}{q^{2}-m^{2}_{\rho}},$
(28) $\displaystyle{\cal{M}}^{\\{\pi^{0},\pi^{+}\\}}_{b}$
$\displaystyle=(i)^{3}\frac{F_{1}gg_{\Xi(2030)\Sigma\bar{K}^{*}}}{f}\\{0,-\sqrt{\frac{2}{3}}\\}\int\frac{d^{4}q}{(2\pi)^{4}i}\Phi[$
$\displaystyle\times(p_{1}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]\bar{u}(p_{3})q\\!\\!\\!/\gamma_{5}\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\nu\sigma}(k)$
$\displaystyle\times{}p_{2}^{\nu}(p_{4}^{\alpha}+q^{\alpha})\frac{-g^{\sigma\alpha}+p_{2}^{\sigma}p_{2}^{\alpha}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}\frac{1}{q^{2}-m^{2}_{\pi}},$
(29) $\displaystyle{\cal{M}}^{\eta}_{b}$
$\displaystyle=(i)^{3}\frac{D_{1}gg_{\Xi(2030)\Sigma\bar{K}^{*}}}{3\sqrt{6}f}\int\frac{d^{4}q}{(2\pi)^{4}i}\Phi[(p_{1}\omega_{\bar{K}^{*}}-p_{2}$
$\displaystyle\times\omega_{\Sigma})^{2}_{E}]\bar{u}(p_{3})q\\!\\!\\!/\gamma_{5}\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\nu\sigma}(k)p_{2}^{\nu}(p_{4}^{\alpha}+q^{\alpha})$
$\displaystyle\times{}\frac{-g^{\sigma\alpha}+p_{2}^{\sigma}p_{2}^{\alpha}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}\frac{1}{q^{2}-m^{2}_{\eta}},$
(30) $\displaystyle{\cal{M}}^{\\{\rho^{0},\rho^{+}\\}}_{b}$
$\displaystyle=-i(i)^{3}G^{{}^{\prime}}g_{\Xi(2030)\Sigma\bar{K}^{*}}\\{0,\sqrt{2}\\}\int\frac{d^{4}q}{(2\pi)^{4}i}$
$\displaystyle\times\Phi[(p_{1}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]\bar{u}(p_{3})\epsilon^{\alpha\beta\theta\lambda}q_{\alpha}p_{2\theta}$
$\displaystyle\times[g\gamma_{\eta}+\frac{F}{4}(q\\!\\!\\!/\gamma_{\eta}-\gamma_{\eta}q\\!\\!\\!/)]\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\sigma\tau}(k)p_{2}^{\sigma}$
$\displaystyle\times{}\frac{-g^{\tau\lambda}+p_{2}^{\tau}p_{2}^{\lambda}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}\frac{-g^{\eta\beta}+q^{\eta}q^{\beta}/m^{2}_{\rho^{-}}}{q^{2}-m^{2}_{\rho}},$
(31) $\displaystyle{\cal{M}}^{\\{\omega,\phi\\}}_{b}$
$\displaystyle=-i(i)^{3}\frac{G^{{}^{\prime}}g_{\Xi(2030)\Sigma\bar{K}^{*}}}{\sqrt{3}}\\{\sqrt{3},\sqrt{\frac{3}{2}}\\}\int\frac{d^{4}q}{(2\pi)^{4}i}$
$\displaystyle\times\Phi[(p_{1}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]\bar{u}(p_{3})[g\gamma_{\eta}+\\{\frac{D}{4}(q\\!\\!\\!/\gamma_{\eta}$
$\displaystyle-\gamma_{\eta}q\\!\\!\\!/),0\\}]\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\sigma\tau}(k)p_{2}^{\sigma}\epsilon^{\alpha\beta\theta\lambda}q_{\alpha}p_{2\theta}$
$\displaystyle\times\frac{-g^{\tau\lambda}+p_{2}^{\tau}p_{2}^{\lambda}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}\frac{-g^{\eta\beta}+q^{\eta}q^{\beta}/m^{2}_{\omega/\phi}}{q^{2}-m^{2}_{\omega/\phi}},$
(32) $\displaystyle{\cal{M}}^{\\{\bar{K}^{0},K^{-}\\}}_{c}$
$\displaystyle=-(i)^{3}\frac{(D_{1}+F_{1})gg_{\Xi(2030)\Sigma\bar{K}^{*}}}{2\sqrt{6}f}\\{1,-\sqrt{2}\\}\int\frac{d^{4}q}{(2\pi)^{4}i}$
$\displaystyle\times\Phi[(p_{1}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]\bar{u}(p_{3})q\\!\\!\\!/\gamma_{5}\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\nu\sigma}(k)$
$\displaystyle\times{}p_{2}^{\nu}(p_{4}^{\alpha}+q^{\alpha})\frac{-g^{\sigma\alpha}+p_{2}^{\sigma}p_{2}^{\alpha}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}\frac{1}{q^{2}-m^{2}_{\bar{K}}},$
(33) $\displaystyle{\cal{M}}^{\\{\bar{K}^{0},K^{-}\\}}_{d}$
$\displaystyle=-(i)^{3}\frac{f_{\Xi^{*}\Sigma{}K}gg_{\Xi(2030)\Sigma\bar{K}^{*}}}{\sqrt{6}m_{K}}\\{1,-\sqrt{2}\\}\int\frac{d^{4}q}{(2\pi)^{4}i}$
$\displaystyle\times\Phi[(p_{1}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]q\cdot\bar{u}(p_{3})\gamma_{5}\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\nu\sigma}(k)$
$\displaystyle\times{}p_{2}^{\nu}(p_{4}^{\alpha}+q^{\alpha})\frac{-g^{\sigma\alpha}+p_{2}^{\sigma}p_{2}^{\alpha}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}\frac{1}{q^{2}-m^{2}_{\bar{K}}},$
(34) $\displaystyle{\cal{M}}^{\\{\bar{K}^{0},K^{-}\\}}_{e}$
$\displaystyle=-i(i)^{3}\frac{gg_{\Xi(2030)\Sigma\bar{K}^{*}}}{8\sqrt{6}f^{2}}\int\frac{d^{4}q}{(2\pi)^{4}i}\Phi[(p_{1}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]$
$\displaystyle\times\bar{u}(p_{3})(p\\!\\!\\!/_{4}+q\\!\\!\\!/)\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\nu\sigma}(k)p_{2}^{\nu}\frac{-g^{\sigma\tau}+p_{2}^{\sigma}p_{2}^{\tau}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}$
$\displaystyle\times{}(p_{5}^{\tau}-q^{\tau})\frac{1}{q^{2}-m^{2}_{\bar{K}}}\\{1,-\sqrt{2}\\},$
(35) $\displaystyle{\cal{M}}^{\\{\bar{K}^{0},K^{-}\\}}_{f}$
$\displaystyle=-i(i)^{3}\frac{gg_{\Xi(2030)\Sigma\bar{K}^{*}}}{2\sqrt{6}}\Phi[(p_{3}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]\bar{u}(p_{3})$
$\displaystyle\times{}u_{\nu\alpha}(k)p_{2}^{\alpha}\frac{-g^{\mu\nu}+p_{2}^{\mu}p_{2}^{\nu}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}(p_{5}^{\mu}-p_{4}^{\mu})\\{1,-\sqrt{2}\\},$
(36) $\displaystyle{\cal{M}}^{\\{\bar{K}^{0},K^{-}\\}}_{g}$
$\displaystyle=i(i)^{3}\frac{gg_{\Xi(2030)\Sigma\bar{K}^{*}}}{2\sqrt{6}f^{2}}\int\frac{d^{4}q}{(2\pi)^{4}i}\Phi[(p_{1}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]$
$\displaystyle\times\bar{u}(p_{3})(p\\!\\!\\!/_{4}+q\\!\\!\\!/)\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\nu\sigma}(k)p_{2}^{\nu}\frac{-g^{\sigma\tau}+p_{2}^{\sigma}p_{2}^{\tau}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}$
$\displaystyle\times{}(p_{5}^{\tau}-q^{\tau})\frac{1}{q^{2}-m^{2}_{\bar{K}}}\\{0,1\\},$
(37) $\displaystyle{\cal{M}}^{\\{\bar{K}^{0},K^{-}\\}}_{h}$
$\displaystyle=i(i)^{3}\frac{gg_{\Xi(2030)\Sigma\bar{K}^{*}}}{\sqrt{6}f^{2}}\int\frac{d^{4}q}{(2\pi)^{4}i}\Phi[(p_{1}\omega_{\bar{K}^{*}}-p_{2}\omega_{\Sigma})^{2}_{E}]$
$\displaystyle\times\bar{u}(p_{3})(p\\!\\!\\!/_{5}+q\\!\\!\\!/)\frac{p\\!\\!\\!/_{1}+m_{\Sigma}}{p_{1}^{2}-m^{2}_{\Sigma}}u_{\nu\sigma}(k)p_{2}^{\nu}\frac{-g^{\sigma\tau}+p_{2}^{\sigma}p_{2}^{\tau}/m^{2}_{\bar{K}^{*}}}{p_{2}^{2}-m^{2}_{\bar{K}^{*}}}$
$\displaystyle\times{}(p_{4}^{\tau}-q^{\tau})\frac{1}{q^{2}-m^{2}_{\bar{K}}}\\{0,\sqrt{2}\\},$
(38)

where $\\{A,B\\}$ are for $\\{\pi^{0},\pi^{+}\\}$ exchange, respectively.

Once the amplitudes are calculated, we can obtain the partial decay widths.
The corresponding formulas for the two-body and three-body decay widths can be
found in Ref. ParticleDataGroup:2022pth ; we do not provide the detailed forms
here.

##  III RESULTS AND DISCUSSIONS

In this work, we study the strong decay of $\Xi(2030)$ as a $P$-wave molecule
with $J^{p}=5/2^{-}$ using the effective Lagrangian approach. If the estimated
strong decay width matches well with experimental observations, we can
validate the molecular interpretation of $\Xi(2030)$, primarily as a
$\bar{K}^{*}\Sigma$ component, as first proposed in Ref. Hei:2023eqz .
Currently, the accuracy of our calculations hinges primarily on the parameter
$\Lambda$, necessitating a preliminary discussion of its value. As mentioned
earlier, $\Lambda$ is usually set to 1 GeV, but we have observed variations in
$\Lambda$ ranging from 0.91 to 1.0 GeV through fitting experimental data
Dong:2017rmg . It’s important to note that this range is determined using the
same theoretical framework as applied in this study. Moreover, we have found
that many exotic states can be effectively described as molecules with
$\Lambda$ values ranging from 0.90 to 1.10 GeV. For further details, we refer
the reader to review Refs. Yang:2021pio ; Dong:2008gb ; Huang:2020taj  and
their references. Therefore, we choose to explore the range
$\Lambda=0.90-1.10$ GeV to investigate whether $\Xi(2030)$ can be interpreted
as a molecule primarily composed of $\bar{K}^{*}\Sigma$ molecular components.

Taking into account the $\Lambda$ values adopted in this work, we first
discuss the results of the calculated coupling constants. By substituting Eq.
7 into Eq. 6, we determine the dependence of the coupling constants on the
parameter $\Lambda$. Integrating parameters $\alpha$ and $\beta$ from 0 to
infinity, we present the numerical results of the coupling constants as
$\Lambda$ varies from 0.90 to 1.10 GeV in Fig. 3.

Figure 3: The coupling constants of the $\Xi(2030)$ that assuming it is a
$\bar{K}^{*}\Sigma$ molecule as a function of the parameter $\Lambda$.

From the results, it is evident that the coupling constants decrease as
$\Lambda$ increases, indicating a high sensitivity to changes in $\Lambda$,
with the range of variation decreasing from 58.26 to 49.92. Specifically, at
$\Lambda=1.0$ GeV, the coupling constant is
$g_{\Xi(2030)\Sigma\bar{K}^{*}}=53.73$.

After determining the coupling constants, we calculate the partial decay
widths for the two-body final state transitions
$\Xi(2030)\to\bar{K}\Lambda,\bar{K}\Sigma,\pi\Xi,\pi\Xi^{*}$ and the three-
body final state transition $\Xi(2030)\to\pi\pi\Xi,\pi\bar{K}\Sigma$. These
results are depicted in Fig.4 and vary with the parameter $\Lambda=0.9-1.1$
GeV. It is important to note that the partial decay widths in the channels
$\Xi(2030)\to\bar{K}^{0}\Lambda,\bar{K}^{0}\Sigma^{0},\pi^{0}\Xi^{0},\pi^{0}\Xi^{*0}$
and $\Xi(2030)\to\pi^{0}\pi^{0}\Xi^{0},\pi^{0}\bar{K}^{0}\Sigma^{0}$ are
estimated here; other channels can be determined using isospin symmetry. The
sum of these partial decay widths yields the total decay width of $\Xi(2030)$,
as also shown in Fig. 4.

Figure 4: The decay widths of $\Xi(2030)$ to the final states
$\Lambda\bar{K}$, $\Sigma\bar{K}$, $\Xi^{*}\pi$, $\Xi\pi\pi$ and
$\Sigma\bar{K}\pi$, along with the total decay width. The cyan error bands
correspond to the experimental values ParticleDataGroup:2022pth .

From the results shown in Fig. 4, it is evident that the total decay width,
depicted by the solid black line, decreases gradually as $\Lambda$ increases.
As $\Lambda$ increases from 0.9 GeV to 1.1 GeV, the predicted total decay
width decreases from 17.041 MeV to 13.161 MeV, closely matching the
experimental value of $20_{-5}^{+15}$ MeV ParticleDataGroup:2022pth ,
indicated by a cyan error bands. This suggests that if the spin-parity of the
$\Xi(2030)$ is $J^{P}=5/2^{+}$, the assignment as a $P$-wave molecule
predominantly composed of $\bar{K}^{*}\Sigma$ component for the $\Xi(2030)$ is
supported. The results also indicate that the main contribution comes from the
$\bar{K}\Sigma$ channel (dotted blue line), with the $\bar{K}\Lambda$ channel
(red dashed line) being the second most important, and the other decay
channels being much smaller. We list the detailed numerical results in Tab. 1.
For ease of comparison, we also provide the experimental results
ParticleDataGroup:2022pth  in the last column.

Table 1: Partial decay widths of $\Xi(2030)$ and the total decay width with $\Lambda=0.9-1.1$ GeV, compared with experimental values from Ref. ParticleDataGroup:2022pth . The values in parentheses in last line correspond to $\Lambda=1.0$ GeV. All widths are given in units of MeV.  Model |  Our work |  Exp. ParticleDataGroup:2022pth
---|---|---
$\Xi(2030)\to\bar{K}\Sigma$ |  $9.816-13.224$ |  $12-28$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}\to\bar{K}\Lambda$ |  $2.547-2.880$ |  $3-7$
$~{}~{}~{}~{}~{}~{}~{}~{}\to\pi\Xi$ |  $0.376-0.391$ |  small
$~{}~{}~{}~{}~{}~{}~{}~{}\to\pi\Xi^{*}$ |  $0.00579-0.00746$ |  small
$~{}~{}~{}~{}~{}~{}~{}~{}\to\pi\pi\Xi[not~{}\pi\Xi^{*}]$ |  $0.00367-0.0120$ |  small
$~{}~{}~{}~{}~{}~{}~{}~{}\to\pi\bar{K}\Sigma$ |  $0.409-0.528$ |  small
$~{}~{}~{}~{}~{}~{}~{}~{}\to\pi\bar{K}\Lambda$ |  small |  small
$~{}~{}~{}~{}~{}~{}~{}~{}Total$ |  $13.161-17.041(15.054)$ |  $20_{-5}^{+15}$

From Tab. 1, it can be seen that at $\Lambda=0.9-1.1$ GeV, the numerical
results for the two main decay channels, $\Xi(2030)\to\bar{K}\Sigma$ and
$\Xi(2030)\to\bar{K}\Lambda$, are 9.816-13.224 MeV and 2.547-2.880 MeV,
respectively, contributing 74.58%-77.60% and 16.9%-19.35% to the total decay
width. These results are very consistent with the experimentally measured
decay widths $\Gamma(\Xi(2030)\to\bar{K}\Sigma)=12-28$ MeV and
$\Gamma(\Xi(2030)\to\bar{K}\Lambda)=3-7$ MeV, which account for 80% and 20% of
the total decay width, respectively ParticleDataGroup:2022pth . This further
supports the reasonableness of considering $\Xi(2030)$ as a molecule primarily
composed of $\bar{K}^{*}\Sigma$ with a spin-parity of $J^{p}=5/2^{+}$ . Tab. 1
also shows that the estimated two-body decay widths of $\Xi(2030)$ into
$\Xi\pi$ and $\Xi^{*}\pi$ are approximately 0.376-0.391 MeV and
0.00579-0.00746 MeV, respectively. In experiments, specific values for these
two decay channels have not been provided and are only considered to be small.
Our numerical results are similarly small, accounting for 2.294%-2.857% and
0.0438%-0.0440% of the total decay width, respectively, which appears to be
consistent with the experimental observations. This means our results also
support that the transitions $\Xi(2030)\to\Xi\pi$ and $\Xi(2030)\to\Xi^{*}\pi$
contribute only a minor portion to the total decay width.

The same conclusion about the small contributions also applies to the
$\Xi(2030)\to\pi\pi\Xi$ (not via the intermediate state $\pi\Xi^{*}$),
$\Xi(2030)\to\pi\bar{K}\Lambda$, and $\Xi(2030)\to\pi\bar{K}\Sigma$ reactions
observed in experiments, which is also very consistent with the results we
have provided (see Table 1). For the $\Xi(2030)\to\pi\pi\Xi$ reaction, the
decay width is up to $0.00367-0.0120$ MeV, which accounts for 0.0279%-0.0704%
of total width. That means the transition $\Xi(2030)\to\pi\pi\Xi$ gives a
minor contribution. Our calculations also find that the decay width of
$\Xi(2030)\to\pi\bar{K}\Sigma$ is $0.409-0.528$ MeV, which also accounts for
only a small part of the total decay width, being 3.098%-3.107%.

Figure 5: The decay width of $\Xi(2030)\rightarrow\Sigma\bar{K}\pi$ with meson
exchange and tree-level contribution.

It is noteworthy that the decay $\Xi(2030)\to\pi\bar{K}\Sigma$ includes a
tree-level contribution (see Fig. 1 (f)), which typically should be dominant
and the decay width may even exceed that the main decay channel
$\Xi(2030)\to\bar{K}\Sigma$. However, we find that within the parameter range
$\Lambda=0.9-1.1$ GeV, the decay width due to the tree-level contribution is
only 0.918-1.185 MeV (see Fig. 5), which is much smaller than the decay width
of $\Xi(2030)\to\bar{K}\Sigma$, the main decay channel observed in
experiments. A possible explanation for this may be that the phase space for
the transition $\Xi(2030)\to\pi\bar{K}\Sigma$ is smaller than that of the
$\Xi(2030)\to\bar{K}\Sigma$ reaction. In Fig. 5, we also present the other
contributions to the $\Xi(2030)\to\bar{K}\Sigma$ decay. We can observe that
the contribution from $\pi$-meson exchange is smaller than that of the tree-
level diagram, and the contribution from $K$-meson exchange is the smallest.
Additionally, the interference effects between these contributions are
significant, resulting in the total decay width being smaller than the tree-
level decay width.

In Table 1, we do not provide the numerical result for the decay width of
$\Xi(2030)\to\pi\bar{K}\Lambda$ mainly because the decay of $\Xi(2030)$ to
$\pi\bar{K}\Lambda$ via the $\bar{K}^{*}\Sigma$ molecule is forbidden due to
the absence of the $\bar{K}\Sigma\to\bar{K}\Lambda$ and
$\pi\Sigma\to\pi\Lambda$ coupling vertices in Eq. 10. However, we find that
$\Xi(2030)$ also contains a very small $\bar{K}^{*}\Lambda$ component
Hei:2023eqz  that can decay to $\pi\bar{K}\Lambda$. Therefore, we can directly
deduce that the decay width of $\Xi(2030)$ to $\pi\bar{K}\Lambda$ via the
$\bar{K}^{*}\Lambda$ component is very small, which is also comparable to
experimental results.

Figure 6:  Partial decay widths of the $\Xi(2030)\rightarrow\Lambda\bar{K}$
and $\Xi(2030)\rightarrow\Sigma\bar{K}$. The cyan error bands correspond to
the experimental values ParticleDataGroup:2022pth .

It is noteworthy that while our results support that $\Xi(2030)$ is a molecule
primarily composed of the $\bar{K}^{*}\Sigma$ component, the calculated total
width, particularly with $\Lambda=1.0$, results in a width of
$\Gamma_{\Xi(2030)}=15.054$ MeV, which is smaller than the experimental
central value of $\Gamma_{\Xi(2030)}=20.0$ MeV. This suggests that $\Xi(2030)$
might not be solely a $P$-wave meson-baryon molecule Hei:2023eqz ; it could
also include additional three-quark components. A detailed comparison between
the theoretical calculations and experimental results for the two main decay
channels, $\Xi(2030)\to\Sigma\bar{K}$ and $\Xi(2030)\to\Lambda\bar{K}$, is
presented in Fig. 6. We can find that the obtained decay widths for these two
channels are smaller than the experimental central values, further supporting
the conclusion that a molecular assignment for $\Xi(2030)$ is insufficient.

Fig. 6 also indicates that $\pi$ meson exchange plays a predominant role in
the decays $\Xi(2030)\rightarrow\Lambda\bar{K}$ and
$\Xi(2030)\rightarrow\Sigma\bar{K}$, contributing significantly more than the
total width. However, contributions from other mesons, including $\rho$,
$\omega$, $\eta$, and $\phi$, are almost negligible. Yet, their interference
effects are very strong, resulting in a total decay width smaller than that
from $\pi$ meson exchange alone.

##  IV SUMMARY

Inspired by the failure of the quark model to explain $\Xi(2030)$ as a three-
quark state, we verify our proposal that $\Xi(2030)$ is a molecule with spin-
parity $J^{P}=5/2^{+}$ and predominantly composed of $\bar{K}^{*}\Sigma$ by
studying the allowed two-body and three-body strong decays of $\Xi(2030)$.
With the $\bar{K}^{*}\Sigma$ assignment, we calculated the partial decay
widths of $\Xi(2030)$ into the two-body final states
$\bar{K}\Lambda,\bar{K}\Sigma,\pi\Xi,\pi\Xi^{*}$, and the three-body final
states $\pi\pi\Xi,\pi\bar{K}\Sigma,\pi\bar{K}\Lambda$, some of which occur
through hadronic loops. The decay process is described by $t$-channel
exchanges of $\pi$, $K$, $\rho$, $\omega$, and $\phi$ mesons, as well as tree-
level contributions.

Our numerical results indicate that $\bar{K}\Lambda$ and $\bar{K}\Sigma$ are
the two main decay channels of $\Xi(2030)$, with decay widths of
$\Gamma(\Xi(2030)\to\bar{K}\Sigma)=9.816-13.224$ MeV and
$\Gamma(\Xi(2030)\to\bar{K}\Lambda)=2.547-2.880$ MeV, accounting for
74.58%-77.60% and 16.9%-19.35% of the total decay width, respectively.
However, other decay channels provide very small contributions and can be
almost neglected. These decay widths result in a total width of 13.161-17.041
MeV. When comparing with the experimental results, we find that both the total
decay width and the partial decay widths agree well with the experimental
values within the error margins. This supports the hypothesis that $\Xi(2030)$
is a molecule with spin-parity $J^{P}=5/2^{+}$, predominantly composed of
$\bar{K}^{*}\Sigma$. However, our results are slightly smaller than the
experimental central values, suggesting that $\Xi(2030)$ may also contain
additional components, such as three-quark structures, besides the meson-
baryon molecular components.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
under Grant No.12005177.

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# A Study on the Vulnerability of Test Questions against ChatGPT-based
Cheating

Shanker Ram  Lynbrook High School
San Jose, USA
<EMAIL_ADDRESS>Chen Qian  Department of Computer Science
University of California at Santa Cruz
Santa Cruz, California
<EMAIL_ADDRESS>
###### Abstract

ChatGPT is a chatbot that can answer text prompts fairly accurately, even
performing very well on postgraduate-level questions. Many educators have
found that their take-home or remote tests and exams are vulnerable to
ChatGPT-based cheating because students may directly use answers provided by
tools like ChatGPT. In this paper, we try to provide an answer to an important
question: how well ChatGPT can answer test questions and how we can detect
whether the questions of a test can be answered correctly by ChatGPT. We
generated ChatGPT’s responses to the MedMCQA dataset, which contains over
10,000 medical school entrance exam questions. We analyzed the responses and
uncovered certain types of questions ChatGPT answers more inaccurately than
others. In addition, we have created a basic natural language processing model
to single out the most vulnerable questions to ChatGPT in a collection of
questions or a sample exam. Our tool can be used by test-makers to avoid
ChatGPT-vulnerable test questions.

###### Index Terms:

machine learning, data analysis, ChatGPT, NLP

##  I Introduction

ChatGPT is an advanced chatbot developed by OpenAI, and it is proficient at
answering test questions from any educational level. Because of the recent
increase in online learning, using ChatGPT to cheat on tests is more
prevalent. According to Yahoo Finance [1], some teachers have caught over 26%
of students attempting to cheat on tests using ChatGPT. In order to combat AI
usage on tests, we conduct this study to find techniques test-makers can use
to detect questions that chatbots can answer and create those that are not
vulnerable to ChatGPT.

We first conduct a preliminary study on the accuracy of using ChatGPT to
answer existing test questions. After initially testing ChatGPT on over 10,000
questions, with the use of the Python ChatGPT API, we found that it answered
these medical school entrance exam questions correctly about 60% of the time.
ChatGPT’s above average performance on postgraduate-level questions is
alarming, given the frequency of cheating using AI on tests in college and
high school. [2]. We tried looking at basic features in the data, including
question lengths and the number of words per question. However, we soon
realized that ChatGPT was advanced enough to answer correctly irrespective of
these metrics. Our important discovery is that the true markers of a
question’s susceptibility against ChatGPT lies in more obscure features.

By scouring the data and analyzing the questions ChatGPT got right and wrong,
we discovered many helpful similarities in the questions it got right and the
ones it got wrong. By applying the techniques we present, test-makers can
improve the quality of the questions they create such that ChatGPT cannot
answer them accurately.

We also created an NLP model with a neural network so that given the text of a
question, the model would be able to immediately answer whether ChatGPT would
answer the question correctly or incorrectly, as well as the likelihood of
that outcome happening. We attempted this through the use of Google Colab’s
built-in GPU, making use of the PyTorch framework [3] in Python. The finished
model correctly answers whether a question is vulnerable to ChatGPT with a 60%
accuracy; in addition, when the model had a confidence score of over 85% in
its answer, which happened about half of the time, its accuracy rose to over
70%. Overall, this model is a satisfactory tool in at least marginally
filtering out GPT-susceptible questions.

##  II Dataset

Selecting the correct dataset for this task was important, since we need a
dataset with college-level multiple-choice questions, over 10,000 problems,
and answers to each of those problems. Additionally, we prefer problems
related to life science because processing questions with math notation was
tedious and inconsistent in Python. We ended up choosing MedMCQA [4], which
has 200,000 multiple-choice medical school entrance exam questions and
answers; as an additional bonus, each of the questions has labels with the
branch of medicine it tests, providing another feature to later analyze in the
results.

###  II-A Data Collection

In order to gather the responses from ChatGPT for each of the questions, we
used the ChatGPT Python API; we settled on testing about 13000 questions since
it took about 2 seconds for the API to process and respond to each prompt.
Additionally, it should be noted that ChatGPT is a nondeterministic model [5],
meaning the same prompt can be entered multiple times with different
responses. This means that not all of the data gathered is necessarily
representative of that done by using the ChatGPT model available on the OpenAI
website. However, although the Python API often tends to output different
responses than the website, the overall accuracies on questions are actually
similar for both. To test this, we manually entered a subset of 100 random
problems from the MedMCQA dataset into the ChatGPT model on the OpenAI website
to test its performance. The same 100 problems were also given to the ChatGPT
Python API. The results are shown in Table 1.

TABLE I: Accuracy for API vs. website Accuracy on 100 random questions from the dataset (website) |  Accuracy on 100 random questions from the dataset (Python API)
---|---
63% |  63%

Remarkably, both the API and the website got exactly 63 of the 100 questions
right, although not the same 63. Regardless, the existing sources of error are
extremely small, so the trends present in the data can still be extrapolated
and used to our advantage when making test questions.

##  III Apparent Trends in Data

ChatGPT performs very differently when faced with varying types of questions.
In some cases, the differences had no effect, while in others, GPT’s accuracy
is reduced by over 50%.

###  III-A Structuring and Complexity of Questions has no Effect on ChatGPT

When applying basic NLP techniques [6] toward the questions, including
calculating words per question, characters per question, sentences per
question, words per sentence, characters per sentence, characters per word,
special figures per question, and numerical figures per question, we found
that these metrics did not correspond to the problems ChatGPT answered
correctly or the problems it answered incorrectly, as shown in Fig. 1. This
implies that any basic features will not separate complex problems from simple
problems for ChatGPT.

After testing many simple NLP techniques, we moved on to advanced lexical
richness metrics. We tried 10 of these metrics [7], including the Flesch
Reading Ease score, the Flesch Kincaid Grade Level test, the SMOG index
readability score, the Coleman-Liau index, the Automated readability index,
the Dale-Chall readability score, Gunning Fog index, and the Gulpease index.
Like with the basic NLP techniques, all of the lexical richness metrics were
the same independent of whether ChatGPT got the problem right or wrong, as
shown in Fig. 2. We also tried comparing the cosine similarity [8] between
problems ChatGPT got right and wrong, and again, the results were very
similar.

In conclusion, it is not possible to fool ChatGPT by making problems sound
more complex or using complicated synonyms of words to try and confuse it.
Questions must be changed in different ways to actually have a chance of
stopping ChatGPT from correctly answering them.

Figure 1: Categorical scatterplot with slight random jitter to avoid
overplotting Figure 2: Categorical scatterplots with slight random jitter to
avoid overplotting

###  III-B Using Multi-Select Problems has Little to no Effect on ChatGPT’s
Performance

Figure 3: Example of multi-select problem, the correct answer is A and D

We assumed ChatGPT would perform worse on multi-select problems, an example of
which is shown in Fig. 3, and it did perform slightly worse, dropping 4-5% in
accuracy on these types of questions. Still, this does not justify using
multi-select problems over other types of problems to combat ChatGPT since

a) While ChatGPT is worse at answering these types of questions, so are test
takers.

b) Test-makers would need to spend much more time creating these questions
instead of normal MCQ.

c) The relatively small drop is not large enough to completely rely on these
types of questions.

###  III-C Adding Extra Option Choices has no Effect on ChatGPT’s Performance

Adding extra option choices would presumably create a much larger challenge
[9], but to our surprise, ChatGPT had almost the same accuracy on questions
regardless of the number of answer choices. When adding a fifth answer choice
to MCQ, ChatGPT’s accuracy was just slightly lower, decreasing by about 2%.

When analyzing the responses provided to these questions, it is easy to see
that the reason for this phenomenon is that ChatGPT treats all questions like
free-response questions, where it does not even consider the answer choices
until selecting its final answer. In fact, the only case where the number of
answer choices affects ChatGPT is when it has to guess, where the probability
of a guess being correct is higher with only four answer choices. This
explains the 2% drop seen earlier.

The good news for question-makers is that they do not have to worry about
trying to make free-response questions [10] to combat ChatGPT, since ChatGPT
answers both MCQ and FRQ with relatively equal accuracy.

###  III-D ChatGPT Struggles More with Questions with the Word “except” in
Them

To be more specific, questions in the dataset with the word “except” in them
typically had the phrase “All of the following are true except”. An example of
what such a problem could look like is shown in Fig. 4.

Interestingly, ChatGPT performed about 6% worse on questions worded similarly
to the example above. There are many possibilities for why this could be the
case but after analyzing ChatGPT’s responses to questions like this, one
theory is that for normal questions, ChatGPT has to verify that one statement
is true; in “except” questions, it has to verify that three statements are
true. This is a potential reason for ChatGPT’s worse performance.

However, it must be noted that 6% is not that large of a gap. On a 50-question
test, applying this technique could save about 3 questions from being answered
accurately by ChatGPT, but this is still not a huge impact.

Figure 4: Example of problem with ”except” in it

###  III-E A Major Indicator of ChatGPT’s Success at Answering a Question
Correctly is the Topic the Question is Based On

As expected, the topic of the question ChatGPT responds to is a huge factor in
whether it gets the question right or wrong. As it turns out, ChatGPT is very
good at answering questions with topics like biochemistry or psychiatry, with
accuracy rates over 70%, and it is much worse at answering questions with
topics like dentistry, with accuracy rates under 50% for these types of
questions.

A simple theory as to why ChatGPT is worse at answering certain questions
comes from the topics themselves. Since ChatGPT was trained off of data from
the Internet, it can be assumed it would be more proficient at answering
questions related to more common topics. As information related to
biochemistry and psychiatry is more readily available [11] on the Internet in
comparison to dentistry, there could have been less training data for ChatGPT
to train on dentistry. A full list of accuracies for each topic is shown in
Table 2.

TABLE II: Topic of Question vs. Accuracy Topic |  ChatGPT’s accuracy in answering questions with this topic (%)
---|---
Anatomy |  54
Biochemistry |  71
Surgery |  59
Ophthalmology |  55
Physiology |  63
Social and Preventive Medicine |  58
Gynecology and Obstetrics |  56
Anesthesia |  51
Psychiatry |  72
Microbiology |  64
Medicine |  60
Pharmacology |  70
Dental |  48
EMT |  55
Forensic Medicine |  50
Pediatrics |  58
Orthopedics |  53
Radiology |  52
Pathology |  64
Skin |  67

###  III-F ChatGPT Drastically Overpredicts the Options “All of the above” and
“None of the above”

When first running the code on questions with “None of the above” and “All of
the above” as possible answer choices, we found that ChatGPT is very good at
answering questions with “All of the above” as an answer choice and it is very
poor at answering “None of the above” as an answer choice. However, this data
did not tell the full story.

When looking at the data for questions with the answer choice “None of the
above” again, we split these questions up into two categories—questions where
the correct answer was “None of the above”, and questions where the correct
answer was not “None of the above”. For the questions where the correct answer
was “None of the above”, ChatGPT’s accuracy was a high 77%, and for the
questions where “None of the above” was incorrect, ChatGPT’s accuracy was a
shockingly low 48%. This discrepancy is shown in Table 3.

TABLE III: Accuracy vs. presence of ”None of the above” option ChatGPT’s accuracy on questions where “None of the above” is the right answer |  ChatGPT’s accuracy on questions where “None of the above” is not the right answer |  ChatGPT’s accuracy on questions without “None of the above” as an answer choice
---|---|---
78% |  47% |  59.5%

When repeating the same process for questions with the answer choice “All of
the above”, we found similar results, but exaggerated even further. For the
questions where the correct answer was “All of the above”, ChatGPT’s accuracy
was an incredibly high 98%, and for the questions where “All of the above” was
incorrect, ChatGPT’s accuracy was much lower at 16%. This discrepancy is shown
in Table 4.

TABLE IV: Accuracy vs. presence of ”All of the above” option ChatGPT’s accuracy on questions where “All of the above” is the right answer |  ChatGPT’s accuracy on questions where “All of the above” is not the right answer |  ChatGPT’s accuracy on questions without “All of the above” as an answer choice
---|---|---
98% |  16% |  59.5%

This huge discrepancy in accuracy can be explained by ChatGPT’s tendency to
select “None of the above” or ”All of the above” even when it is not the right
answer. It seems to overpredict these options as the answer in these cases.
Table 5 illustrates this tendency. For example, when option choice ”D” is
”None/All of the above” and it is wrong, ChatGPT still selects ”D” 81% of the
time.

TABLE V: Accuracy based on whether ”(D) None/All of the Above” was correct Phrase |  Percentage of ChatGPT’s responses which are “(D) None/All of the Above” when the phrase is the right answer |  Percentage of ChatGPT’s responses which are “(D) None/All of the Above” when the phrase is not the right answer
---|---|---
“None of the above” |  77 |  37
“All of the above” |  98 |  81

In conclusion, creating questions with the answer choices “All of the above”
or “None of the above”, but without the answer being those choices, is a very
strong way to deter ChatGPT from correctly answering such questions. As shown
before, ChatGPT’s accuracy in answering these types of questions is extremely
low, because of its tendency to overpredict “(D)” in these cases.

##  IV Natural Language Processing Model

We propose to build a tool that can answer whether a question will be answered
accurately by ChatGPT, in order to help test makers to quickly identify and
avoid test questions that are vulnerable to ChatGPT-based cheating. We made
use of the Pytorch neural network framework and the Google Colab GPU for
training a model for finding the vulnerability of a question.

###  IV-A Preprocessing of Data

Initially, we applied NLP preprocessing techniques to get each question into a
more numerical format, which was easier to process for the neural network.
This included tokenizing each question with the use of the BERT base model
(uncased) [12], which splits the text of the question up into words and word
pieces (cases where one word can consist of multiple tokens). These tokens
were then converted into token IDs, which are given by the BERT model. This
process is demonstrated in Table 6.

TABLE VI: Word to Token IDs conversion Word |  Token(s) |  Token ID(s)
---|---|---
Reorganize |  [”re”, ”##org”, ”##ani”, ”##ze”] |  [2128, 21759, 7088, 4371]
the |  [”the”] |  [1996]
hyper-parameters |  [”hyper”, ”##para”, ”##meter”, ”##s”] |  [23760, 28689, 22828, 2015]
in |  [”in”] |  [1999]
the |  [”the”] |  [1996]
model. |  [”model”, ”.”] |  [2944, 1012]

From here, the data was eventually converted to PyTorch tensors and was ready
for the neural network to train on.

We chose not to remove stop words, which are words that are so common that
they are ignored by typical tokenizers. Some examples are ”a”, ”the”, ”of”,
and ”in”. This technique was not used because it actually led to a decrease in
accuracy for the model.

###  IV-B Dataset Splits

Before training of the model started, we split the 11,657 questions and
responses into a training dataset (75%, used for the training model), a
validation dataset (20%, used for decreasing loss during training), and a
testing dataset (5%, used for determining accuracy of the final model).

###  IV-C Neural Network Architecture

The model was instantiated with ”bert-base-uncased”, a pretrained BERT model.
This model is pretrained on the English language and it uses a masked language
modeling objective, meaning it tries to predict missing words in a sentence.

The input into the model is first fed into twelve layers from the pretrained
BERT model, which tokenizes and preprocesses the words from the problem. The
hidden states (outputs) from these layers are fed into a BERT pooler layer,
which uses a tanh() activation function. Finally, the hidden states from that
layer are given to a dropout layer and a linear transformation is applied to
that output before the final hidden states are received. This model is
illustrated in Fig. 5.

###  IV-D Training

Figure 5: Model Architecture

We used the adam optimization algorithm [13] with a learning rate of $2e-5$
and an epsilon value of $1e-8$. The neural network was trained for 4 epochs.
The model’s highest validation accuracy was 0.6 and its lowest training loss
was 0.52, as shown in Fig. 6. We ended up with a model that was correct in its
answer of whether ChatGPT would correctly answer a question about 60% of the
time, which was satisfactory.

Figure 6: Training loss of model

###  IV-E Other Models

We have also tried other possible models, including Naïve Bayes Classifiers
(NBC), decision trees [14], random forests [15], k-nearest neighbors (KNN),
and logistic regression. For each of these models, we used the base model
without adjusting any hyperparameters. Compared to the other models tested,
neural networks provide the most proficient results in eliminating GPT-
susceptible questions. The accuracies on the testing datasets for each of
these models are shown in Table 7.

TABLE VII: Accuracy for different models Model |  Accuracy (%)
---|---
Neural Network |  61
NBC |  47.4
Decision Tree |  57.3
Random Forest |  61.3
KNN |  55.7
Logistic Regression |  56.4

###  IV-F Testing of Model

Although the model’s accuracy was 60%, when it was over 85% certain in its
answer, which happened about half of the time, its accuracy rose to over 70%,
which can filter out the majority of ChatGPT-vulnerable questions, given
questions from the MedMCQA dataset.

We then proceeded to make a tool with the NLP model where an instructor can
provide a certain set of questions and request to have a certain percentage of
them removed, where the most susceptible questions to ChatGPT are taken out.
Thus, we recorded the final accuracies based off of the testing dataset. The
results are shown in Table 8.

TABLE VIII: Performance by final model Percentage of the questions to be removed |  ChatGPT’s accuracy on the remaining questions after removal (%)
---|---
0 |  61
5 |  60.2
10 |  58.9
15 |  57.9
20 |  56
30 |  54.7
40 |  53
50 |  53.6

We find that ChatGPT’s accuracy barely goes down when the percentage of
questions removed goes up. However, we suspect when the model is trained more
specifically on the exact type of questions being tested, this accuracy drop
will increase.

##  V Alternate Dataset Testing

In order to confirm the trends in ChatGPT’s answers discussed earlier, we
gathered the same data on another smaller dataset. We compiled a dataset of
physics problems since we wanted to test if our observations would hold true
for a completely different type of problem. We decided on using a compilation
of physics problems from a user on HuggingFace, and ended up with almost 500
physics problems to run through the Python API. On this dataset, ChatGPT’s
accuracy was closer to 55%, so the questions from the data challenged it more
than the questions from the MedMCQA dataset. This can be attributed to the
fact that physics problems typically require much more computation than the
average medical school entrance exam question, and ChatGPT struggles more with
computations.

We started off by testing the observations we made with “None of the above”
questions, where ChatGPT is much more likely to select “(D) None of the above”
than any other options. In this case, when “D” was the right answer, the
accuracy was 75%, and when “D” was not the right answer, the accuracy was 50%.
This confirms our observations from before, proving ChatGPT’s tendency to
overpredict “D”. The accuracies are shown in Table 9.

TABLE IX: Accuracy vs. presence of ”None of the above” option ChatGPT’s accuracy on questions where “None of the above” is the right answer |  ChatGPT’s accuracy on questions where “None of the above” is not the right answer |  ChatGPT’s accuracy on questions without “None of the above” as an answer choice
---|---|---
75% |  50% |  55%

Next, we tested the “All of the above” observations, where ChatGPT is much
more likely to select “(D) All of the above” than any other option. In this
case, when “D” was the right answer, the accuracy was 100%, and when “D” was
not the right answer, the accuracy was 27%. Our observations were again
confirmed, and like with the previous dataset, ChatGPT’s tendency to
overpredict “D” is more exaggerated with “All of the above” questions. The
accuracies are shown in Table 10.

TABLE X: Accuracy vs. presence of ”All of the above” option ChatGPT’s accuracy on questions where “All of the above” is the right answer |  ChatGPT’s accuracy on questions where “All of the above” is not the right answer |  ChatGPT’s accuracy on questions without “All of the above” as an answer choice
---|---|---
100% |  27% |  55%

It must be noted that the data gathered from the second dataset comes from a
very small sample. However, even though the results come from such a small
sample, they can be trusted since they confirm observations coming from a much
larger sample.

##  VI Performance by Different Versions of GPT

GPT-4 performs alarmingly better than any of the other models OpenAI has
created. If someone were to use this model in an attempt to cheat, the
likelihood of success would be substantially higher. However, most people use
the gpt-3.5-turbo model, since that is the model being used on the main
website for free. Additionally, GPT-4 is not free and consequently, a much
lower percentage of people can afford to use it. The accuracies from different
OpenAI models are shown in Table 11.

TABLE XI: Performance by different models of ChatGPT Model |  Accuracy (%)
---|---
gpt-3.5-turbo |  60
gpt-3.5-turbo-16k |  49
text-davinci-003 |  47
gpt-4 |  85

##  VII Summary and conclusion

Based on our results, we find that test questions can be written in certain
ways to prevent ChatGPT from answering them correctly. This includes the usage
of certain phrases, such as ”except”, or the topic the questions test. There
are also methods that can mistakenly be thought to fool ChatGPT, which do not,
such as increasing question length, or using more complicated words. With the
use of the techniques described in this paper, teachers can decrease the
number of questions they put out that are susceptible to being answered
correctly by ChatGPT. Additionally, they could attempt to filter out the GPT-
susceptible questions in their tests by using an NLP model similar to the one
we showed earlier.

In the future, we hope to test ChatGPT in other fields, such as math, computer
science, and literature. We also hope to increase the accuracy of the current
NLP model, such that it can filter out ChatGPT-susceptible questions with an
even higher accuracy. Furthermore, we hope to validate or revise the trends
noticed for ChatGPT for other chatbots such as Bard AI, Claude, or Bing AI.

## Acknowledgment

C. Qian was partially supported by NSF Grants 1750704, 1932447, and 2114113.

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