††thanks: Corresponding author

# Investigation of hidden-charm double strange pentaquark candidate $P_{css}$
via its mass and strong decay to $\Xi^{0}J/\psi$

K. Azizi<EMAIL_ADDRESS>Department of Physics, University of Tehran,
North Karegar Avenue, Tehran 14395-547, Iran  Department of Physics, Doǧuş
University, Dudullu-Ümraniye, 34775 Istanbul, Turkey  Y. Sarac
<EMAIL_ADDRESS>Electrical and Electronics Engineering Department,
Atilim University, 06836 Ankara, Turkey  H. Sundu
<EMAIL_ADDRESS>Department of Physics, Kocaeli University, 41380
Izmit, Turkey

###### Abstract

This work presents an analysis of a hidden charmed pentaquark candidate state
with double strange quark in its quark content. The investigation is performed
in two parts, which provide the mass prediction of the considered state and
its partial decay width for the $P_{css}\rightarrow\Xi^{0}J/\psi$ channel. For
the analyses, two-point and three-point QCD sum rule methods are applied to
get the mass and the width, respectively. The mass for this candidate state is
obtained as $m_{P_{css}}=4600\pm 155$ MeV with corresponding current coupling
constant $\lambda_{P_{css}}=(0.81\pm 0.21)\times 10^{-3}$ GeV6. These results
are used in the analysis of the partial width of this state for the decay
$P_{css}\rightarrow\Xi^{0}J/\psi$. For this decay, the width is obtained as
$\Gamma=9.52\pm 2.85$ MeV. These results may shed light on the future
experimental searches in which such types of states are probed and may provide
information to discriminate between such possible observations.

††preprint:

##  I Introduction

In the last few decades, the significant progress in experiments has brought
new particles to light. With their quark substructure, some of these newly
observed particles differ from conventional hadrons, which are composed of
quark-antiquark (mesons) or three quarks (baryons). Quarks are combined in
color singlet states to form these hadrons, and nature does not rule out the
non-conventional hadrons, which consist of other multiquark combinations in
color singlet forms such as tetraquarks and pentaquarks. This fact has always
attracted interests toward these non-conventional states. They were firstly
proposed by Gell-Mann Gell-Mann , and their existences and properties have
been investigated thoroughly since then from different points of view using
different approaches in both experiments and theory. Their first observation
was announced in 2003 by Belle Collaboration for a tetraquark state, $X(3872)$
Choi:2003ue . This observation was later confirmed by other collaborations
such as the BABAR Collaboration BaBar:2004oro ,CDF II Collaboration
CDF:2003cab ; CDF:2009nxk , D0 Collaboration D0:2004zmu , LHCb Collaboration
LHCb:2011zzp , and CMS Collaboration CMS:2013fpt . Observation of other
tetraquark states followed this observation.

On the other hand, hidden charm pentaquark states, which are other members of
these exotic states, were firstly observed in 2015 by the LHCb Collaboration
Aaij:2015tga  in the $J/\psi p$ invariant mass spectrum of the
$\Lambda_{b}^{0}\rightarrow J/\psi pK^{-}$ process. The resonance properties
for these pentaquark states were reported as $m_{P_{c}(4380)^{+}}=4380\pm 8\pm
29~{}\mathrm{MeV}$, $\Gamma_{P_{c}(4380)^{+}}=205\pm 18\pm 86~{}\mathrm{MeV}$
and $m_{P_{c}(4450)^{+}}=4449.8\pm 1.7\pm 2.5~{}\mathrm{MeV}$,
$\Gamma_{P_{c}(4450)^{+}}=39\pm 5\pm 19~{}\mathrm{MeV}$ Aaij:2015tga . In
2016, full amplitude analysis for $\Lambda_{b}^{0}\rightarrow J/\psi p\pi^{-}$
decays Aaij:2016ymb  supported this observation. And later, in 2019, the LHCb
collaboration updated the analysis with a combined data set and confirmed the
previously observed states, and also observed additional structures which
became more significant with additional data set compared to the previous
analysis Aaij:2019vzc . The resonance parameters for the additionally observed
state, $P_{c}(4312)^{+}$, were presented as $m_{P_{c}(4312)^{+}}=4311.9\pm
0.7^{+6.8}_{-0.6}~{}\mathrm{MeV}$ and $\Gamma_{P_{c}(4312)^{+}}=9.8\pm
2.7^{+3.7}_{-4.5}~{}\mathrm{MeV}$ Aaij:2019vzc . In this analysis, for the
previously observed peak of $P_{c}(4450)^{-}$, there occurred a split into two
peaks with the following mass and width: $m_{P_{c}(4440)^{+}}=4440.3\pm
1.3^{+4.1}_{-4.7}~{}\mathrm{MeV}$, $\Gamma_{P_{c}(4440)^{+}}=20.6\pm
4.9^{+8.7}_{-10.1}~{}\mathrm{MeV}$ and $m_{P_{c}(4457)^{+}}=4457.3\pm
0.6^{+4.1}_{-1.7}~{}\mathrm{MeV}$, $\Gamma_{P_{c}(4457)^{+}}=6.4\pm
2.0^{+5.7}_{-1.9}~{}\mathrm{MeV}$ Aaij:2019vzc . The next observation
consistent with possible pentaquark state came with a strange quark in its
quark content LHCb:2020jpq , which was observed in the $J/\psi\Lambda$
invariant mass distribution analysis of the $\Xi_{b}^{-}\rightarrow
J/\psi\Lambda K^{-}$ decays and denoted as $P_{cs}(4459)$, with mass
$m_{P_{cs}(4459)}=4458.8\pm 2.9^{+4.7}_{-1.1}$ MeV and width $17.3\pm
6.5^{+8.0}_{-5.7}$ MeV. Very recently the LHCb Collaboration announced an
evidence for a pentaquark candidate state in the $B_{s}^{0}\rightarrow J/\psi
p\bar{p}$ decays with mass and width
$m_{P_{c}(4337)}=4337^{+7}_{-4}{}^{+2}_{-2}$ MeV and
$\Gamma_{P_{c}(4337)}=29^{+26}_{-12}{}^{+14}_{-14}$ MeV, respectively
LHCb:2021chn .

Though the possible pentaquark states have been investigated before all these
observations, with these experimental findings, they have recaptured the
theoretical attentions over themselves again, and they were extensively
investigated for their different properties using various theoretical
approaches. Their spectroscopy and interactions were examined to explain their
substructures and quantum numbers. Among the considered possible substructures
is the molecular interpretation suggested based on their closeness to the
meson baryon thresholds. The molecular structure was taken into account in the
one-pion-exchange model Chen:2016heh , in the QCD sum rules method
Chen:2016otp ; Azizi:2016dhy ; Azizi:2018bdv ; Azizi:2020ogm , in the
quasipotential Bethe-Salpeter equation approach He:2019ify ; Zhu:2021lhd , in
the contact-range effective field theory approach Liu:2019tjn , in the
effective Lagrangian approach Xiao:2019mvs ; Lu:2016nnt , in the constituent
chiral quark model Hu:2021nvs  and in the chiral perturbation theory
Meng:2019dba . The meson baryon molecular interpretation was also adopted in
Refs. Chen:2015loa ; Chen:2015moa ; He:2015cea ; Meissner:2015mza ;
Roca:2015dva ; Chen:2020opr ; Yan:2021nio ; Wu:2021caw ; Chen:2020uif ;
Phumphan:2021tta ; Du:2021fmf ; Lu:2021irg ; Gao:2021hmv ; Yalikun:2021dpk .
Ref. Chen:2021obo  used the dynamically constrained phase space coalescence
model and the parton and hadron cascade model and studied three kinds of
possible structures for the $P_{c}$ states, which are the pentaquark state,
the nucleus-like state, and the molecular state. These states were also
considered to be compact pentaquark states. In Refs. Wang:2015epa ;
Maiani:2015vwa ; Giannuzzi:2019esi ; Li:2015gta ; Lebed:2015tna ;
Anisovich:2015cia ; Wang:2015ava ; Wang:2015ixb ; Ghosh:2015ksa ; Wang:2015wsa
; Zhang:2017mmw ; Wang:2019got ; Wang:2020rdh ; Ali:2020vee ; Wang:2016dzu ;
Azizi:2021utt ; Anisovich:2015zqa ; Wang:2020eep ; Gao:2021hmv , the
substructures of the pentaquark states were treated to be diquark-diquark-
antiquark forms. Another possible substructure, diquark-triquark, was also
investigated in Refs. Wang:2016dzu ; Zhu:2015bba ; Gao:2021hmv . The
possibility of their being kinematical effect was regarded in Refs. Guo1 ;
Guo2 ; Mikhasenko:2015vca ; Liu1 ; Bayar:2016ftu . In Ref. Nakamura:2021dix ,
the newly observed $P_{c}(4337)$ was considered by developing a model to
analyze the data on $B_{s}^{0}\rightarrow J/\psi p\bar{p}$. Despite all these
works, both the substructure and quantum numbers of these observed states are
still not certain and they need further investigations both theoretically and
experimentally for their clarifications. While many works were devoted to
explain the properties of the observed pentaquark states, others were carrying
the aim of offering new possible pentaquark states with different quark
contents to be observed in future experimental investigations. Among these
works are the ones given in Refs. Liu:2020cmw ; Chen:2015sxa ; Feijoo:2015kts
; Irie:2017qai ; Chen:2016ryt ; Zhang:2020cdi ; Paryev:2020jkp ;
Gutsche:2019mkg ; Azizi:2017bgs ; Azizi:2018dva ; Cao:2019gqo ; Zhang:2020vpz
; Xie:2020ckr ; Meng:2019fan ; Liu:2021tpq ; Wang:2020bjt ; Wang:2019nvm ;
Liu:2021ixf ; Giron:2021fnl ; Santopinto:2016pkp ; Xiao:2019gjd .

There are many works devoted to the investigation of these states. However,
their properties are still not certain, and there are ambiguities in their
substructures and the quantum numbers as well. Different works suggested
different interpretations for the properties of the observed pentaquark
states. To clarify all these ambiguities, we need more investigations on the
properties of both the observed states and the new possible ones to be
observed. In these investigations, either complementary reactions or some
other decay modes can be considered for the presently observed pentaquark
states, or the new particles that are possible to be observed can be probed,
to provide inputs to the experiments, by their spectroscopic properties or
possible decay modes. The analyses for such processes can be found in Refs.
Feijoo:2015kts ; Liu:2021ixf ; Chen:2015sxa ; Wang:2015jsa ;
Kubarovsky:2015aaa ; Karliner:2015voa ; Cheng:2015cca ; Liu:2020ajv ;
Yang:2021pio ; Stancu:2021rro ; Dong:2020nwk ; Voloshin:2019aut ; Ling:2021lmq
; Xing:2021yid ; Cheng:2021gca ; Wang:2019hyc ; Xu:2019zme ; Liu:2021ojf .

It is natural to expect observations of new pentaquark states, with different
quark contents than the observed ones, in the near future. So far, with this
expectation, some of such states were investigated. For instance, the
pentaquark states with bottom quark were probed in Refs. Chen:2016heh ;
Liu:2017xzo ; Azizi:2017bgs ; Huang:2021ave ; Zhu:2020vto ; Ferretti:2018ojb ;
Shimizu:2016rrd . The pentaquark candidates with single or triple charm quark
were investigated in Ref. Azizi:2018dva  with a molecular interpretation. With
the expectation to observe the fully-heavy pentaquark states, these types of
pentaquark structures were considered in Refs. Zhang:2020vpz ; Wang:2021xao ;
An:2020jix ; Yan:2021glh . Another possible candidate for such states is the
ones including strange quarks. The pentaquark with double strange quark was
examined in Refs. Wang:2020bjt  with molecular interpretation using the one-
boson-exchange model. In Ref. Meng:2019fan , the pentaquark with triple
strange quarks was considered. In Refs. Ferretti:2020ewe ; Ferretti:2021zis
the hadro-quarkonium model was used to search for the double strange
pentaquarks. The pentaquarks with triple strange quark were analyzed in Ref.
Wang:2021hql  in the meson-baryon molecular picture via the one-boson exchange
model. In this respect, theoretical works involving the predictions for
possible new pentaquark states may shed light on experimental investigations.
With this motivation, in this work, we investigate a possible hidden charm
pentaquark state involving double strange quarks. To this end, we apply the
QCD sum rules method Shifman:1978bx ; Shifman:1978by ; Ioffe81  which is among
the powerful methods, and provided consistent predictions so far with the
observed hadrons. In the application of the method, the main input is the
proper interpolating field for the considered state. For the present study, we
work with an interpolating field in the diquark-diquark-antiquark form and
with $J^{P}=\frac{1}{2}^{-}$ spin-parity quantum numbers. Using the two-point
QCD sum rule analysis, we first calculate the mass and current coupling
constant of the candidate pentaquark state. Then we use these findings in the
calculation of the width of its strong decay to $\Xi^{0}J/\psi$.

The outline of the article is as follows: In Sec. II we present the two-point
QCD sum rule analysis to calculate the mass and current coupling constant of
the considered state (in what follows it is labeled as $P_{css})$ and present
the analyses of the obtained results. Sec. III is devoted to the calculation
of the strong decay width of the $P_{css}$ state to the states $\Xi^{0}J/\psi$
via the application of three-point QCD sum rules. The last section presents
the summary and conclusion.

##  II The mass of $P_{css}$ state

In this section, the mass of the candidate double strange pentaquark state is
obtained. The starting point is to consider the following two-point
correlation function:

$\Pi(q)=i\int d^{4}xe^{iq\cdot x}\langle
0|\mathcal{T}\\{\eta_{P_{css}}(x)\bar{\eta}_{P_{css}}(0)\\}|0\rangle,$ (1)

where $\mathcal{T}$ is the time ordering operator and $\eta^{P_{css}}$ is the
interpolating current for the candidate pentaquark state $P_{css}$. In this
work, we use an interpolating current in the diquark-diquark-antiquark form
with spin-$\frac{1}{2}$ and negative parity. The interpolating current has the
following explicit form:

$\displaystyle\eta_{P_{css}}$ $\displaystyle=$
$\displaystyle\epsilon^{ila}\epsilon^{ijk}\epsilon^{lmn}u^{T}_{j}C\gamma_{5}s_{k}s^{T}_{m}C\gamma_{5}c_{n}C\bar{c}^{T}_{a}.$
(2)

The interpolating current is constructed considering the valance quark content
and the assumed quantum numbers of the state. $u$, $s$, and $c$ in the
interpolating current correspond to the respective quark fields, and $C$ is
the charge conjugation matrix.

The two-point correlation function is calculated following two paths. The
first path gives its result in terms of hadronic degrees of freedom, such as
the mass and current coupling constant of the considered state. Due to that,
this side is named as the hadronic or physical side of the calculation. The
calculation following the second path results in terms containing QCD degrees
of freedom such as masses of the quarks, quark-gluon condensates, and QCD
coupling constant, hence called as QCD side of the calculation. These two
sides are matched via dispersion relation and give us the QCD sum rules
through which we get the physical quantities under question. On both sides,
there appear various Lorentz structures. In the match of the results, one
considers one of the appropriate Lorentz structures present on each side, and
to further improve the results, Borel transformation is applied to both sides
to suppress the contributions of higher states and continuum.

In the calculation of the physical side, a complete set of intermediate
states, with the same quantum numbers of the considered one, is substituted
into the correlation function treating the interpolating current as
annihilation and creation operator for the hadrons which have the same quantum
numbers and quark contents of the interpolating current. Performing the
integration over four-$x$, we obtain the correlator in this part as

$\displaystyle\Pi^{\mathrm{Had}}(q)=\frac{\langle
0|\eta_{P_{css}}|P_{css}(q,s)\rangle\langle
P_{css}(q,s)|\bar{\eta}_{P_{css}}|0\rangle}{m_{P_{css}}^{2}-q^{2}}+\cdots,$
(3)

where $\cdots$ is used to represent the higher states and continuum.
$|P_{css}(q,s)\rangle$ represents the one-particle pentaquark state with
momentum $q$ and spin $s$. The matrix elements in the Eq. (3) have the
following form given in terms of the current coupling constant $\lambda$ and
Dirac spinor, $u_{P_{css}}(q,s)$, with spin $s$:

$\displaystyle\langle 0|\eta_{P_{css}}|P_{css}(q,s)\rangle$ $\displaystyle=$
$\displaystyle\lambda u_{P_{css}}(q,s).$ (4)

When the last equation is substituted inside Eq. (3), with the summation over
the spin

$\displaystyle\sum_{s}u_{P_{css}}(q,s)\bar{u}_{P_{css}}(q,s)=\not\\!q+m,$ (5)

the result of the hadronic side turns into

$\displaystyle\Pi^{\mathrm{Had}}(q)=\frac{\lambda^{2}(\not\\!q+m_{P_{css}})}{m_{P_{css}}^{2}-q^{2}}+\cdots.$
(6)

The hadronic side finally takes the following form after the Borel
transformation:

$\displaystyle\tilde{\Pi}^{\mathrm{Had}}(q)=\lambda^{2}e^{-\frac{m_{P_{css}}^{2}}{M^{2}}}(\not\\!q+m_{P_{css}})+\cdots,$
(7)

where we represent the Borel transformed form of the correlator by
$\tilde{\Pi}^{\mathrm{Had}}(q)$ and the Borel mass by $M^{2}$. In the final
result we have two Lorentz structures, $\not\\!q$ and $I$ that can be
considered in the analyses. We take into account both of these structures and
give the average value of the results obtained from each.

What comes next is the calculation of the correlator for the QCD side. For the
calculation of this side, the interpolating current is used explicitly inside
the Eq. (1), and Wick’s theorem is applied to make the possible contractions
between the quark fields. This operation turns the correlator into the one
written in terms of the light and heavy quark propagators, which reads as

$\displaystyle\Pi^{\mathrm{QCD}}(q)$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{iqx}\epsilon_{ila}\epsilon_{ijk}\epsilon_{lmn}\epsilon_{i^{\prime}l^{\prime}a^{\prime}}\epsilon_{i^{\prime}j^{\prime}k^{\prime}}\epsilon_{l^{\prime}m^{\prime}n^{\prime}}\Big{\\{}-\mathrm{Tr}[S_{s}(x)^{kk^{\prime}}\gamma_{5}CS_{u}^{T}{}^{jj^{\prime}}(x)C\gamma_{5}]\mathrm{Tr}[S_{c}(x)^{nn^{\prime}}\gamma_{5}CS_{s}^{T}{}^{mm^{\prime}}(x)C\gamma_{5}]$
(8) $\displaystyle+$
$\displaystyle\mathrm{Tr}[S_{s}(x)^{mk^{\prime}}\gamma_{5}CS_{u}^{T}{}^{jj^{\prime}}(x)C\gamma_{5}S_{s}(x)^{km^{\prime}}\gamma_{5}CS_{c}^{T}{}^{nn^{\prime}}(x)C\gamma_{5}]\Big{\\}}CS_{c}^{T}{}^{aa^{\prime}}(-x)C.$

For the calculation of this side, we use the light and heavy quark propagators
explicitly in Eq. (8) as follows:

$\displaystyle S_{q,}{}_{ab}(x)$ $\displaystyle=$ $\displaystyle
i\delta_{ab}\frac{\not{x}}{2\pi^{2}x^{4}}-\delta_{ab}\frac{m_{q}}{4\pi^{2}x^{2}}-\delta_{ab}\frac{\langle\overline{q}q\rangle}{12}+i\delta_{ab}\frac{\not{x}m_{q}\langle\overline{q}q\rangle}{48}-\delta_{ab}\frac{x^{2}}{192}\langle\overline{q}g_{\mathrm{s}}\sigma
Gq\rangle+i\delta_{ab}\frac{x^{2}\not{x}m_{q}}{1152}\langle\overline{q}g_{\mathrm{s}}\sigma
Gq\rangle$ (9)
$\displaystyle-i\frac{g_{\mathrm{s}}G_{ab}^{\alpha\beta}}{32\pi^{2}x^{2}}\left[\not{x}{\sigma_{\alpha\beta}+\sigma_{\alpha\beta}}\not{x}\right]-i\delta_{ab}\frac{x^{2}\not{x}g_{\mathrm{s}}^{2}\langle\overline{q}q\rangle^{2}}{7776},$

and

$\displaystyle S_{c,{ab}}(x)$ $\displaystyle=$
$\displaystyle\frac{i}{(2\pi)^{4}}\int d^{4}ke^{-ik\cdot
x}\left\\{\frac{\delta_{ab}}{\\!\not\\!{k}-m_{Q}}-\frac{g_{s}G^{\alpha\beta}_{ab}}{4}\frac{\sigma_{\alpha\beta}(\\!\not\\!{k}+m_{Q})+(\\!\not\\!{k}+m_{Q})\sigma_{\alpha\beta}}{(k^{2}-m_{Q}^{2})^{2}}\right.$
(10)
$\displaystyle\left.+\frac{\pi^{2}}{3}\langle\frac{\alpha_{s}GG}{\pi}\rangle\delta_{ij}m_{Q}\frac{k^{2}+m_{Q}\\!\not\\!{k}}{(k^{2}-m_{Q}^{2})^{4}}+\cdots\right\\},$

where $G^{\alpha\beta}_{ab}=G^{\alpha\beta}_{A}t_{ab}^{A}$ and
$GG=G_{A}^{\alpha\beta}G_{A}^{\alpha\beta}$ with $a,~{}b=1,2,3$ and
$A=1,2,\cdots,8$ and $t^{A}=\frac{\lambda^{A}}{2}$ where $\lambda^{A}$ is the
Gell-Mann matrices.The sub-index $q$ is used to represent $u$ ($s$) quark.
After the application of the Fourier and Borel transformations we get

$\displaystyle\tilde{\Pi}_{i}^{\mathrm{QCD}}(s_{0},M^{2})=\int_{(2m_{c}+2m_{s})^{2}}^{s_{0}}dse^{-\frac{s}{M^{2}}}\rho_{i}(s)+\Gamma_{i}(M^{2}),$
(11)

where $\rho_{i}(s)$ are the spectral densities. We obtain the spectral
densities from the imaginary parts of the results as
$\frac{1}{\pi}\mathrm{Im}\Pi_{i}^{\mathrm{QCD}}$ with $i$ representing Lorentz
structures, $\not\\!q$ or $I$. In order not to overwhelm the article with long
expressions we will not present the explicit forms of $\rho_{i}(s)$ and
$\Gamma_{i}(M^{2})$ here and focus on just the results obtained from them. In
Eq. (11), $s_{0}$ represents the threshold parameter that arises after the
application of the continuum subtraction using the quark-hadron duality
assumption, and $M^{2}$ represents the Borel parameter. To obtain the physical
quantities, mass and the current coupling constant, we match the results
obtained from both the hadronic and QCD sides considering the coefficient of
the same Lorentz structure. The results for both structures are:

$\displaystyle\lambda^{2}e^{\frac{m_{P_{css}}^{2}}{M^{2}}}$ $\displaystyle=$
$\displaystyle\tilde{\Pi}_{\not\\!q}^{\mathrm{QCD}}(s_{0},M^{2}),$ (12)

and

$\displaystyle\lambda^{2}m_{P_{css}}e^{-\frac{m_{P_{css}}^{2}}{M^{2}}}$
$\displaystyle=$ $\displaystyle\tilde{\Pi}_{I}^{\mathrm{QCD}}(s_{0},M^{2}).$
(13)

In principle, for the analyses, any of the Lorentz structure can be used. In
our calculation we extract the results from both of them and obtain the final
quantities by considering their averages. To obtain the results we need to use
some input parameters which are presented in Table 1.

Parameters | Values
---|---
$m_{c}$ |  $1.27\pm 0.02~{}\mathrm{GeV}$ Zyla:2020zbs
$m_{b}$ |  $4.18^{+0.03}_{-0.02}~{}\mathrm{GeV}$ Zyla:2020zbs
$m_{s}$ |  $93^{+11}_{-5}~{}\mathrm{MeV}$ Zyla:2020zbs
$\langle\bar{q}q\rangle(1\mbox{GeV})$ |  $(-0.24\pm 0.01)^{3}$ $\mathrm{GeV}^{3}$ Belyaev:1982sa
$\langle\bar{s}s\rangle$ |  $0.8\langle\bar{q}q\rangle$ Belyaev:1982sa
$m_{0}^{2}$ |  $(0.8\pm 0.1)$ $\mathrm{GeV}^{2}$ Belyaev:1982sa
$\langle\overline{q}g_{s}\sigma Gq\rangle$ | $m_{0}^{2}\langle\bar{q}q\rangle$
$\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle$ |  $(0.012\pm 0.004)$ $~{}\mathrm{GeV}^{4}$Belyaev:1982cd
$m_{J/\psi}$ |  $(3096.900\pm 0.006)~{}\mathrm{MeV}$ Zyla:2020zbs
$m_{\Xi^{0}}$ |  $(1314.86\pm 0.20)~{}\mathrm{MeV}$ Zyla:2020zbs
$\lambda_{\Xi^{0}}$ |  $(3.8\pm 0.2)\times 10^{-2}~{}\mathrm{GeV}^{3}$ Wang:2007yt
$f_{J/\psi}$ |  $(481\pm 36)~{}\mathrm{MeV}$ Veliev:2011kq
Table 1: Some input parameters used in the analyses of mass and coupling
constants.

This table also contains the parameters necessary for the coupling constant
calculations of the next section. Besides the input parameters given in Table
1, the mass calculations contain two more auxiliary parameters, which are the
threshold parameter $s_{0}$ and the Borel parameter $M^{2}$. The proper
intervals of these parameters are determined from the analyses based on the
standard criteria of the sum rule method. One of the criteria is a mild
dependence of the results on these parameters. The second one is the dominance
of the contribution of the considered state over the higher ones and the
continuum. And the last one is the convergence of the operator product
expansion (OPE). To obtain the proper intervals, we consider these criteria
and get the regions for which these criteria are satisfied. The working
interval of the Borel parameter $M^{2}$ can be deduced by imposing the higher-
order terms in the OPE side to have small contributions, and the dominant
contribution comes from the lowest state. We analyze the results accordingly
and deduce the interval of the Borel parameters as:

$\displaystyle 5.0~{}\mbox{GeV}^{2}\leq M^{2}\leq 7.0~{}\mbox{GeV}^{2},$ (14)

and we extract the interval for the threshold parameter, which also has a
connection to the energy of the possible excited state, as:

$\displaystyle 24.5~{}\mbox{GeV}^{2}\leq s_{0}\leq 26.5~{}\mbox{GeV}^{2}.$
(15)

With the application of these parameters and the input parameters given in
Table 1, finally, we get the mass and current coupling constant as

$\displaystyle m_{P_{css}}=4600\pm
175~{}\mathrm{MeV}~{}~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}~{}\lambda_{P_{css}}=(0.81\pm
0.21)\times 10^{-3}~{}\mathrm{GeV}^{6}.$ (16)

The results obtained here serve as input parameters for the next section in
which we discuss the strong decay of the considered state to $\Xi^{0}J/\psi$
final state.

##  III The strong decay $P_{css}\rightarrow\Xi^{0}J/\psi$

This section is devoted to the calculation of the width for the decay
$P_{css}\rightarrow\Xi^{0}J/\psi$. The calculation is performed via the three-
point correlation function which has the following form:

$\Pi_{\mu}(p,q)=i^{2}\int d^{4}xe^{-ip\cdot x}\int d^{4}ye^{ip^{\prime}\cdot
y}\langle
0|\mathcal{T}\\{\eta^{\Xi^{0}}(y)\eta_{\mu}^{J/\psi}(0)\bar{\eta}^{P_{css}}(x)\\}|0\rangle.$
(17)

In this three-point correlation function, $\eta^{P_{css}}$ still represents
the interpolating current for the candidate $P_{css}$ state, which is given in
Eq. (2). The interpolating currents of final states, $\Xi^{0}$ and $J/\psi$
states, are represented by $\eta^{\Xi^{0}}$ and $\eta_{\mu}^{J/\psi}$,
respectively, and these currents have the following forms

$\displaystyle\eta^{\Xi^{0}}$ $\displaystyle=$
$\displaystyle-2\epsilon^{lmn}\sum_{i=1}^{2}\Big{[}(s^{T}_{l}CA_{1}^{i}u_{m})A_{2}^{i}s_{n}\Big{]},$
$\displaystyle\eta_{\mu}^{J/\psi}$ $\displaystyle=$
$\displaystyle\bar{c}_{l}\gamma_{\mu}c_{l},$ (18)

with the sub-indices, $l,~{}m,~{}n$, we represent the color indices, and
$u,~{}s,~{}c$ are the quark fields, $A_{1}^{1}=I$,
$A_{1}^{2}=A_{2}^{1}=\gamma_{5}$, and $A_{2}^{2}=\beta$ is an arbitrary
parameter, and $C$ is the charge conjugation operator. The parameter $\beta$
is fixed from the analyses of the results, and its working interval is to be
presented. The standard steps of the QCD sum rule are followed again, and the
physical quantities, which are the coupling constants $g_{1}$ and $g_{2}$ for
the strong vertex, are obtained via the match of the coefficients of the same
Lorentz structures attained from the hadronic and QCD sides.

According to the standard QCD sum rule procedure, we first calculate the
hadronic side via insertion of the complete sets of hadronic states with the
same quantum numbers of the interpolating fields, and the four integrals are
performed to obtain the following representation

$\displaystyle\Pi_{\mu}^{\mathrm{Had}}(p,q)=\frac{\langle
0|\eta^{\Xi^{0}}|\Xi^{0}(p^{\prime},s^{\prime})\rangle\langle
0|\eta_{\mu}^{J/\psi}|J/\psi(q)\rangle\langle
J/\psi(q)\Xi^{0}(p^{\prime},s^{\prime})|P_{css}(p,s)\rangle\langle
P_{css}(p,s)|\bar{\eta}^{P_{css}}|0\rangle}{(m_{\Xi^{0}}^{2}-p^{\prime
2})(m_{J/\psi}^{2}-q^{2})(m_{P_{css}}^{2}-p^{2})}+\cdots,$ (19)

where $\cdots$ represents the contributions coming from the higher states and
continuum, the $p$, $p^{\prime}$, and $q$ are the momenta of the $P_{css}$ and
$\Xi^{0}$ and $J/\psi$ states, respectively. To proceed, the matrix elements
in the last result are required, and they are represented in terms of the
masses and current coupling constants as:

$\displaystyle\langle 0|\eta^{P_{css}}|P_{css}(p,s)\rangle$ $\displaystyle=$
$\displaystyle\lambda_{P_{css}}u_{P_{css}}(p,s).$ $\displaystyle\langle
0|\eta^{\Xi^{0}}|\Xi(p^{\prime},s^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\lambda_{\Xi^{0}}u_{\Xi^{0}}(p^{\prime},s^{\prime}),$
$\displaystyle\langle 0|\eta_{\mu}^{J/\psi}|J/\psi(q)\rangle$ $\displaystyle=$
$\displaystyle f_{J/\psi}m_{J/\psi}\varepsilon_{\mu}.$ (20)

The $\varepsilon_{\mu}$, $f_{J/\psi}$, $\lambda_{P_{css}}$,
$\lambda_{\Xi^{0}}$ in the matrix elements correspond to the polarization
vector and the decay constant of the $J/\psi$ state and the current coupling
constants of the $P_{css}$ and $\Xi^{0}$ states, respectively.
$|P_{css}(p,s)\rangle$ represents the one-particle pentaquark state whose
spinor is given by $u_{P_{css}}$ and the spinor for the $\Xi^{0}$ state is
$u_{\Xi^{0}}$. The matrix element defining the coupling constants, $g_{1}$ and
$g_{2}$, is

$\displaystyle\langle
J/\psi(q)\Xi^{0}(p^{\prime},s^{\prime})|P_{css}(p,s)\rangle=\epsilon^{*\nu}\bar{u}_{\Xi^{0}}(p^{\prime},s^{\prime})\big{[}g_{1}\gamma_{\nu}-\frac{i\sigma_{\nu\alpha}}{m_{\Xi^{0}}+m_{P_{css}}}q^{\alpha}g_{2}\big{]}\gamma_{5}u_{P_{css}}(p,s).$
(21)

When we substitute the matrix elements given in Eq. (20) and Eq. (21) into the
Eq. (19) together with the following summation over the spins of the spinors
and polarization vector

$\displaystyle\sum_{s}u_{P_{css}}(p,s)\bar{u}_{P_{css}}(p,s)$ $\displaystyle=$
$\displaystyle({\not{p}}+m_{P_{css}}),$
$\displaystyle\sum_{s^{\prime}}u_{\Xi^{0}}(p^{\prime},s^{\prime})\bar{u}_{\Xi^{0}}(p^{\prime},s^{\prime})$
$\displaystyle=$ $\displaystyle({\not{p}^{\prime}}+m_{\Xi^{0}}),$
$\displaystyle\varepsilon_{\alpha}\varepsilon^{*}_{\beta}$ $\displaystyle=$
$\displaystyle-g_{\alpha\beta}+\frac{q_{\alpha}q_{\beta}}{m_{J/\psi}^{2}},$
(22)

we get the physical side as

$\displaystyle\tilde{\Pi}_{\mu}^{\mathrm{Had}}(p,q)$ $\displaystyle=$
$\displaystyle
e^{-\frac{m_{P_{css}^{2}}}{M^{2}}}e^{-\frac{m_{\Xi^{0}}^{2}}{M^{\prime
2}}}\frac{f_{J/\psi}\lambda_{\Xi^{0}}\lambda_{P_{css}}m_{\Xi}^{0}}{m_{J/\psi}(m_{\Xi^{0}}+m_{P_{css}})(m_{J/\psi}^{2}+Q^{2})}\big{[}g_{1}(m_{\Xi^{0}}+m_{P_{css}})^{2}+g_{2}m_{J/\psi}^{2}\big{]}\not\\!pp^{\prime}_{\mu}\gamma_{5}$
(23) $\displaystyle+$ $\displaystyle
e^{-\frac{m_{P_{css}^{2}}}{M^{2}}}e^{-\frac{m_{\Xi^{0}}^{2}}{M^{\prime
2}}}\frac{f_{J/\psi}\lambda_{\Xi^{0}}\lambda_{P_{css}}}{m_{J/\psi}(m_{\Xi^{0}}+m_{P_{css}})(m_{J/\psi}^{2}+Q^{2})}\big{[}-g_{1}(m_{\Xi^{0}}+m_{P_{css}})\big{(}m_{P_{css}}^{3}+m_{P_{css}}Q^{2}$
$\displaystyle+$ $\displaystyle
m_{\Xi^{0}}(2m_{J/\psi}^{2}+m_{\Xi^{0}}^{2}+Q^{2})\big{)}+g_{2}m_{J/\psi}^{2}(m_{P_{css}}^{2}+m_{P_{css}}m_{\Xi^{0}}-m_{\Xi^{0}}^{2}+Q^{2})\big{]}p_{\mu}\gamma_{5}$
$\displaystyle+$ $\displaystyle\mathrm{other~{}structures}+\cdots.$

In Eq. (23), $Q^{2}=-q^{2}$, $M^{2}$ and $M^{\prime 2}$ are the Borel
parameters which are determined from the analysis of the results following
standard criteria of the QCD sum rule method. In this part, only the terms
containing the Lorentz structures $\not\\!pp^{\prime}_{\mu}\gamma_{5}$,
$p_{\mu}\gamma_{5}$ are given explicitly, and the result contains more Lorentz
structures than the given ones. We present only these because we apply them
directly in our further analyses. The contributions coming from the other
structures and excited states and continuum are represented as
$\mathrm{other~{}structures}+\cdots$.

After the physical side, the QCD side is calculated using the interpolating
fields given in Eqs. (2) and (18) explicitly in the correlation function, Eq.
(17). Performing the possible contractions between the quark fields via Wick’s
theorem, the result is achieved in terms of the light and heavy quark
propagators. This part results in

$\displaystyle\Pi_{\mu}^{\mathrm{OPE}}(p,p^{\prime},q)$ $\displaystyle=$
$\displaystyle i^{2}\int d^{4}xe^{-ip\cdot x}\int d^{4}ye^{ip^{\prime}\cdot
y}\epsilon^{klm}\epsilon^{i^{\prime}l^{\prime}a^{\prime}}\epsilon^{i^{\prime}j^{\prime}k^{\prime}}\epsilon^{l^{\prime}m^{\prime}n^{\prime}}2\sum_{i=1}^{2}\bigg{\\{}Tr[\gamma_{5}CS_{u}^{Tkj^{\prime}}(y-x)(CA_{1}^{i})^{T}S_{s}^{lk^{\prime}}(y-x)]$
(24) $\displaystyle\times$ $\displaystyle
A_{2}^{i}S_{s}^{mm^{\prime}}(y-x)\gamma_{5}CS_{c}^{Tnn^{\prime}}(-x)C\gamma_{\mu}CS_{c}^{Ta^{\prime}n}(x)C-A_{2}^{i}S_{s}^{mk^{\prime}}(y-x)\gamma^{5}CS_{u}^{Tkj^{\prime}}(y-x)(CA_{1}^{i})^{T}S_{s}^{lm^{\prime}}(y-x)$
$\displaystyle\times$
$\displaystyle\gamma_{5}CS_{c}^{Tnn^{\prime}}(-x)C\gamma_{\mu}CS_{c}^{Ta^{\prime}n}(x)C\bigg{\\}}.$

In the last equation $S_{q}^{ab}(x)=S_{u(s)}^{ab}(x)$ and $S_{c}^{ab}(x)$ are
the light and heavy quark propagators given in the Eqs. (9) and (10),
respectively. This side emerges with the same Lorentz structures as in the
hadronic side. We again isolate the coefficients of the structures used in the
analyses, which are $\not\\!pp^{\prime}_{\mu}\gamma_{5}$, $p_{\mu}\gamma_{5}$,
and represent the QCD side as

$\displaystyle\Pi_{\mu}^{OPE}(p,q)$ $\displaystyle=$
$\displaystyle\Pi_{1}\,\not\\!pp^{\prime}_{\mu}\gamma_{5}+\Pi_{2}\,p_{\mu}\gamma_{5}+\mathrm{other\,\,\,structures},$
(25)

where the other structures term represents the contributions coming from the
other Lorentz structures. The calculations of the coefficients $\Pi_{i}$ are
performed using the light and the heavy quark propagators explicitly in the
Eq. (24). Then the results are transformed to momentum space, and the four
integrals are computed. Finally, after the application of the Borel
transformation, the spectral densities are obtained from the imaginary parts
of the results as $\rho_{i}(s,s^{\prime},q^{2})=\frac{1}{\pi}Im[\Pi_{i}]$ and
used in the following dispersion relation:

$\displaystyle\Pi_{i}=\int ds\int
ds^{\prime}\frac{\rho_{i}^{\mathrm{pert}}(s,s^{\prime},q^{2})+\rho_{i}^{\mathrm{non-
pert}}(s,s^{\prime},q^{2})}{(s-p^{2})(s^{\prime}-p^{\prime 2})}.$ (26)

The last equation represents the final form of the QCD side of the calculation
with $\rho_{i}^{\mathrm{pert}}(s,s^{\prime},q^{2})$ and
$\rho_{i}^{\mathrm{non-pert}}(s,s^{\prime},q^{2})$ being the perturbative and
non-perturbative parts of the spectral densities, respectively, where
$i=1,2,..,12$. Again we will not give the lengthy mathematical expressions of
the results here and focus on their analyses.

After the computation of each side, the matches of the results coming from the
coefficients of the same Lorentz structures are used to attain the physical
quantities, $g_{1}$ and $g_{2}$. From the considered structures, we obtained
the following two coupled relations:

$\displaystyle
e^{-\frac{m_{P_{css}^{2}}}{M^{2}}}e^{-\frac{m_{\Xi^{0}}^{2}}{M^{\prime
2}}}\frac{f_{J/\psi}\lambda_{\Xi^{0}}\lambda_{P_{css}}m_{\Xi^{0}}}{m_{J/\psi}(m_{\Xi^{0}}+m_{P_{css}})(m_{J/\psi}^{2}+Q^{2})}\big{[}g_{1}(m_{\Xi^{0}}+m_{P_{css}})^{2}+g_{2}m_{J/\psi}^{2}\big{]}=\tilde{\Pi}_{1}$
$\displaystyle
e^{-\frac{m_{P_{css}^{2}}}{M^{2}}}e^{-\frac{m_{\Xi^{0}}^{2}}{M^{\prime
2}}}\frac{f_{J/\psi}\lambda_{\Xi^{0}}\lambda_{P_{css}}}{m_{J/\psi}(m_{\Xi^{0}}+m_{P_{css}})(m_{J/\psi}^{2}+Q^{2})}\big{[}-g_{1}(m_{\Xi^{0}}+m_{P_{css}})\big{(}m_{P_{css}}^{3}+m_{P_{css}}Q^{2}m_{\Xi^{0}}(2m_{J/\psi}^{2}+m_{\Xi^{0}}^{2}+Q^{2})\big{)}$
$\displaystyle+$ $\displaystyle
g_{2}m_{J/\psi}^{2}(m_{P_{css}}^{2}+m_{P_{css}}m_{\Xi^{0}}-m_{\Xi^{0}}^{2}+Q^{2})\big{]}=\tilde{\Pi}_{2},$
(28)

where $\tilde{\Pi}_{i}$ is used to represent the $\Pi_{i}$ after the Borel
transformations. The solutions of these two relations for the $g_{1}$ and
$g_{2}$ have the following forms:

$\displaystyle g_{1}$ $\displaystyle=$ $\displaystyle
e^{\frac{m_{P_{css}}^{2}}{M^{2}}}e^{\frac{m_{\Xi^{0}}^{2}}{M^{\prime
2}}}\frac{m_{J/\psi}(m_{J/\psi}^{2}+Q^{2})\left[(m_{P_{css}}^{2}+m_{P_{css}}m_{\Xi^{0}}-m_{\Xi^{0}}^{2}+Q^{2})\tilde{\Pi}_{1}-m_{\Xi^{0}}\tilde{\Pi}_{2}\right]}{2f_{J/\psi}\lambda_{\Xi^{0}}\lambda_{P_{css}}m_{\Xi^{0}}[m_{P_{css}}^{3}+m_{P_{css}}^{2}m_{\Xi^{0}}+m_{P_{css}}Q^{2}+m_{\Xi^{0}}(m_{J/\psi}^{2}+Q^{2})]}$
$\displaystyle g_{2}$ $\displaystyle=$ $\displaystyle
e^{\frac{m_{P_{css}}^{2}}{M^{2}}}e^{\frac{m_{\Xi^{0}}^{2}}{M^{\prime
2}}}(m_{P_{css}}+m_{\Xi^{0}})(m_{J/\psi}^{2}+Q^{2})$ (29)
$\displaystyle\times$
$\displaystyle\frac{\left[[m_{P_{css}}^{3}+m_{P_{css}}Q^{2}+m_{\Xi^{0}}(2m_{J/\psi}^{2}+m_{\Xi^{0}}^{2}+Q^{2})]\tilde{\Pi}_{1}+m_{\Xi^{0}}(m_{P_{css}}+m_{\Xi^{0}})\tilde{\Pi}_{2}\right]}{2f_{J/\psi}\lambda_{\Xi^{0}}\lambda_{P_{css}}m_{\Xi^{0}}m_{J/\psi}[m_{P_{css}}^{3}+m_{P_{css}}^{2}m_{\Xi^{0}}+m_{P_{css}}Q^{2}+m_{\Xi^{0}}(m_{J/\psi}^{2}+Q^{2})]}.$

These results also require some input parameters, which are given in Table 1
and there enters five more auxiliary parameters into the coupling constant
calculations, which are Borel parameters $M^{2}$ and $M^{\prime 2}$, threshold
parameters $s_{0}$ and $s^{\prime}_{0}$ and the arbitrary parameter $\beta$
present in the interpolating field of the $\Xi^{0}$ baryon. To determine their
working intervals, we seek their proper regions satisfying the standard QCD
sum rule criteria whose details are also stated in the mass calculation part
of this work. We extract the regions for the auxiliary parameters from our
analyses as

$\displaystyle 5.0\ \mathrm{GeV}^{2}\leq M^{2}$ $\displaystyle\leq$
$\displaystyle 7.0\ \mathrm{GeV}^{2},$ $\displaystyle 1.6\
\mathrm{GeV}^{2}\leq M^{\prime 2}$ $\displaystyle\leq$ $\displaystyle 3.0\
\mathrm{GeV}^{2},$ (30)

and

$\displaystyle 24.5\,\,\mathrm{GeV}^{2}$ $\displaystyle\leq$ $\displaystyle
s_{0}\leq 26.5\,\,\mathrm{GeV}^{2},$ $\displaystyle 2.0\,\,\mathrm{GeV}^{2}$
$\displaystyle\leq$ $\displaystyle s^{\prime}_{0}\leq
2.5\,\,\mathrm{GeV}^{2}.$ (31)

As for the parameter $\beta$, its interval is fixed by a parametric plot
analysis of the results. In this analysis, the results are plotted as a
function of $\cos\theta$, defining $\beta=\tan\theta$, and the regions with
least sensitivity to the variation of $\cos\theta$ are selected, which are
obtained as

$\displaystyle-1.0\leq\cos\theta\leq-0.5~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}0.5\leq\cos\theta\leq
1.0$ (32)

We use all these auxiliary parameters and input parameters given in Table 1 to
analyze our results and attain the coupling constants for the decay channel
under question. The results obtained from these sum rule analyses are reliable
in a range of $Q^{2}$ values. Therefore, to expand it to the region of
interest, we apply a proper fit function which has the following form

$\displaystyle g_{i}(Q^{2})$ $\displaystyle=$ $\displaystyle
g_{0}e^{c_{1}\frac{Q^{2}}{m_{P_{css}}^{2}}+c_{2}(\frac{Q^{2}}{m_{P_{css}}^{2}})^{2}}.$
(33)

The values obtained for fit parameters $g_{0}$, $c_{1}$ and $c_{2}$ are given
in the Table 2.

Coupling Constant | $g_{0}$ | $c_{1}$ | $c_{2}$ | $g_{i}(-m_{J/\psi}^{2})$
---|---|---|---|---
$g_{1}$ | $12.80$ | $1.57$ | $-0.91$ | $5.21\pm 0.78$
$g_{2}$ | $7.80$ | $1.40$ | $4.02$ | $9.45\pm 1.43$
Table 2:  Values of the parameters for the fit functions of coupling
constants, $g_{1}$ and $g_{2}$ and their values obtained from the fit
functions at $Q^{2}=-m_{J/\psi}^{2}$.

The last column of the Taple 2 presents also the values corresponding to the
coupling constants $g_{1}$ and $g_{2}$ extracted with these fit functions at
$Q^{2}=-m_{J/\psi}^{2}$. The errors in these $g_{1}$ and $g_{2}$ results arise
due to both the uncertainties of the input parameters and the uncertainties
carried by the determination of the working intervals of the auxiliary
parameters.

As a final task, the coupling constants are applied to get the width of the
considered decay channel, $P_{css}\rightarrow\Xi^{0}J/\psi$ with the following
width formula

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{f(m_{P_{css}},m_{J/\psi},m_{\Xi^{0}})}{16\pi
m_{P_{css}}^{2}}\Bigg{[}-\frac{2(m_{J/\psi}^{2}-(m_{\Xi^{0}}+m_{P_{css}})^{2})}{m_{J/\psi}^{2}(m_{\Xi^{0}}+m_{P_{css}})^{2}}\Big{(}g_{2}^{2}m_{J/\psi}^{2}(m_{J/\psi}^{2}+2(m_{\Xi^{0}}-m_{P_{css}})^{2})$
(34) $\displaystyle+$ $\displaystyle
6g_{1}g_{2}m_{J/\psi}^{2}(m_{\Xi^{0}}-m_{P_{css}})(m_{\Xi^{0}}+m_{P_{css}})+g_{1}^{2}(2m_{J/\psi}^{2}+(m_{\Xi^{0}}-m_{P_{css}})^{2})(m_{\Xi^{0}}+m_{P_{css}})^{2}\Big{)}\Bigg{]},$

where

$\displaystyle f(x,y,z)$ $\displaystyle=$
$\displaystyle\frac{1}{2x}\sqrt{x^{4}+y^{4}+z^{4}-2x^{2}y^{2}-2x^{2}z^{2}-2y^{2}z^{2}}.$
(35)

The corresponding width is obtained as

$\displaystyle\Gamma(P_{css}\rightarrow J/\psi\Xi^{0})$ $\displaystyle=$
$\displaystyle 9.52\pm 2.85~{}\mathrm{MeV}.$ (36)

##  IV Summary and conclusion

The number of pentaquark candidate states is growing due to the improved
analyses and the experimental techniques. Looking at the particle history, it
is natural to expect such new pentaquark states with different quark
substructures. Besides the state including a single strange quark
$P_{cs}(4459)^{0}$ observed by the LHCb Collaboration LHCb:2020jpq  in the
$J/\psi\Lambda$ invariant mass distribution of the $\Xi_{b}^{-}\rightarrow
J/\psi\Lambda K^{-}$ decays, the pentaquark states with double or triple
strange quark content may exist and to be observed. By this expectation, in
this work, we considered a possible pentaquark state with double strange
quark. To analyze various properties of such a possible state, we applied the
QCD sum rule method and used an interpolating current in the form of diquark-
diquark-antiquark with spin-parity quantum numbers $J^{P}=\frac{1}{2}^{-}$.
The QCD sum rule method was widely used in various analyses in literature and
gave reliable predictions consistent with the experimental observations and
the predictions of the other approaches.

In the first part, we calculated the mass for this state via two-point QCD sum
rule. In the analyses we considered both of the present Lorentz structures and
extracted average values for both the mass and current coupling constant of
the state, which are $m_{P_{css}}=4600\pm 175~{}\mathrm{MeV}$ and
$\lambda_{P_{css}}=(0.81\pm 0.21)\times 10^{-3}~{}\mathrm{GeV}^{6}$,
respectively.

The investigations in the literature for the presently observed pentaquark
states indicate that the spectral analyses may not be enough to fix the
properties of such states. Therefore, we also expanded our analyses with the
inclusion of the decay with computation for the strong decay of the considered
state, $P_{css}\rightarrow\Xi^{0}J/\psi$. To obtain the corresponding width,
we need the coupling constants, $g_{1}$ and $g_{2}$, defining this transition,
and they were calculated using three-point QCD sum rule approach at
$Q^{2}=-m_{J/\psi}^{2}$. Then these results were applied to get the
corresponding width as $\Gamma(P_{css}\rightarrow J/\psi\Xi^{0})=9.52\pm
2.85~{}\mathrm{MeV}$

Both the results of mass and width may shed light on the future observations
of such states in $J/\psi\Xi$ invariant mass. In the future, the
experimentally observed such states may be compared with the obtained results
to provide insights into the explanation of their sub-structures and possible
quantum numbers.

## ACKNOWLEDGEMENTS

K. Azizi is thankful to Iran Science Elites Federation (Saramadan) for the
partial financial support provided under the grant number ISEF/M/400150.

## References

  * (1) M. Gell-Mann,  Phys. Lett. 8, 214 (1964).
  * (2) S. K. Choi et al. [Belle Collaboration], Phys. Rev. Lett. 91, 262001 (2003) [hep-ex/0309032].
  * (3) B. Aubert et al. [BaBar], Phys. Rev. D 71, 071103 (2005) [arXiv:hep-ex/0406022 [hep-ex]].
  * (4) D. Acosta et al. [CDF], Phys. Rev. Lett. 93, 072001 (2004) [arXiv:hep-ex/0312021 [hep-ex]].
  * (5) T. Aaltonen et al. [CDF], Phys. Rev. Lett. 103, 152001 (2009) [arXiv:0906.5218 [hep-ex]].
  * (6) V. M. Abazov et al. [D0], Phys. Rev. Lett. 93, 162002 (2004) [arXiv:hep-ex/0405004 [hep-ex]].
  * (7) R. Aaij et al. [LHCb], Eur. Phys. J. C 72, 1972 (2012) [arXiv:1112.5310 [hep-ex]].
  * (8) S. Chatrchyan et al. [CMS], JHEP 04, 154 (2013) [arXiv:1302.3968 [hep-ex]].
  * (9) R. Aaij et al. [LHCb Collaboration],  Phys. Rev. Lett. 115, 072001 (2015) [arXiv:1507.03414 [hep-ex]].
  * (10) R. Aaij et al. [LHCb], Phys. Rev. Lett. 117, no.8, 082003 (2016) [arXiv:1606.06999 [hep-ex]].
  * (11) R. Aaij et al. [LHCb], Phys. Rev. Lett. 122, no.22, 222001 (2019) [arXiv:1904.03947 [hep-ex]].
  * (12) R. Aaij et al. [LHCb], Sci. Bull. 66, 1278-1287 (2021) [arXiv:2012.10380 [hep-ex]].
  * (13) R. Aaij et al. [LHCb], [arXiv:2108.04720 [hep-ex]].
  * (14) R. Chen, X. Liu and S. L. Zhu, Nucl. Phys. A 954, 406-421 (2016) [arXiv:1601.03233 [hep-ph]].
  * (15) H. X. Chen, E. L. Cui, W. Chen, X. Liu, T. G. Steele and S. L. Zhu, Eur. Phys. J. C 76, no.10, 572 (2016) [arXiv:1602.02433 [hep-ph]].
  * (16) K. Azizi, Y. Sarac and H. Sundu,  Phys. Rev. D 95, no. 9, 094016 (2017) [arXiv:1612.07479 [hep-ph]].
  * (17) K. Azizi, Y. Sarac and H. Sundu, Phys. Lett. B 782, 694-701 (2018) [arXiv:1802.01384 [hep-ph]].
  * (18) K. Azizi, Y. Sarac and H. Sundu, Chin. Phys. C 45, no.5, 053103 (2021) [arXiv:2011.05828 [hep-ph]].
  * (19) J. He, Eur. Phys. J. C 79, no.5, 393 (2019) [arXiv:1903.11872 [hep-ph]].
  * (20) J. T. Zhu, L. Q. Song and J. He, Phys. Rev. D 103, no.7, 074007 (2021) [arXiv:2101.12441 [hep-ph]].
  * (21) M. Z. Liu, Y. W. Pan, F. Z. Peng, M. Sánchez Sánchez, L. S. Geng, A. Hosaka and M. Pavon Valderrama, Phys. Rev. Lett. 122, no.24, 242001 (2019) [arXiv:1903.11560 [hep-ph]].
  * (22) C. J. Xiao, Y. Huang, Y. B. Dong, L. S. Geng and D. Y. Chen, Phys. Rev. D 100, no.1, 014022 (2019) [arXiv:1904.00872 [hep-ph]].
  * (23) Q. F. Lü and Y. B. Dong, Phys. Rev. D 93, no.7, 074020 (2016) [arXiv:1603.00559 [hep-ph]].
  * (24) X. Hu and J. Ping, [arXiv:2109.09972 [hep-ph]].
  * (25) L. Meng, B. Wang, G. J. Wang and S. L. Zhu, [arXiv:1911.09250 [hep-ph]].
  * (26) R. Chen, X. Liu, X. Q. Li and S. L. Zhu,  Phys. Rev. Lett. 115, no. 13, 132002 (2015) [arXiv:1507.03704 [hep-ph]].
  * (27) H. X. Chen, W. Chen, X. Liu, T. G. Steele and S. L. Zhu,  Phys. Rev. Lett. 115, no. 17, 172001 (2015) [arXiv:1507.03717 [hep-ph]].
  * (28) J. He,  Phys. Lett. B 753, 547 (2016) [arXiv:1507.05200 [hep-ph]].
  * (29) U. G. Meißner and J. A. Oller,  Phys. Lett. B 751, 59 (2015) [arXiv:1507.07478 [hep-ph]].
  * (30) L. Roca, J. Nieves and E. Oset,  Phys. Rev. D 92, no. 9, 094003 (2015) [arXiv:1507.04249 [hep-ph]].
  * (31) H. X. Chen, [arXiv:2011.07187 [hep-ph]].
  * (32) M. J. Yan, F. Z. Peng, M. S. Sánchez and M. P. Valderrama, [arXiv:2108.05306 [hep-ph]].
  * (33) Q. Wu, D. Y. Chen and R. Ji, Chin. Phys. Lett. 38, no.7, 071301 (2021) [arXiv:2103.05257 [hep-ph]].
  * (34) H. X. Chen, W. Chen, X. Liu and X. H. Liu, Eur. Phys. J. C 81, no.5, 409 (2021) [arXiv:2011.01079 [hep-ph]].
  * (35) K. Phumphan, W. Ruangyoo, C. C. Chen, A. Limphirat and Y. Yan, [arXiv:2105.03150 [hep-ph]].
  * (36) M. L. Du, V. Baru, F. K. Guo, C. Hanhart, U. G. Meißner, J. A. Oller and Q. Wang, JHEP 08, 157 (2021) [arXiv:2102.07159 [hep-ph]].
  * (37) J. X. Lu, M. Z. Liu, R. X. Shi and L. S. Geng, Phys. Rev. D 104, no.3, 034022 (2021) [arXiv:2104.10303 [hep-ph]].
  * (38) F. Gao and H. S. Li, [arXiv:2112.01823 [hep-ph]].
  * (39) N. Yalikun and B. S. Zou, [arXiv:2112.06426 [hep-ph]].
  * (40) C. h. Chen, Y. L. Xie, H. g. Xu, Z. Zhang, D. M. Zhou, Z. L. She and G. Chen, [arXiv:2111.03241 [hep-ph]].
  * (41) Z. G. Wang, Eur. Phys. J. C 76, no.2, 70 (2016) [arXiv:1508.01468 [hep-ph]].
  * (42) L. Maiani, A. D. Polosa and V. Riquer,  Phys. Lett. B 749, 289 (2015) [arXiv:1507.04980 [hep-ph]].
  * (43) F. Giannuzzi, Phys. Rev. D 99, no.9, 094006 (2019) [arXiv:1903.04430 [hep-ph]].
  * (44) G. N. Li, X. G. He and M. He,  JHEP 12, 128 (2015) [arXiv:1507.08252 [hep-ph]].
  * (45) R. F. Lebed,  Phys. Lett. B 749, 454 (2015) [arXiv:1507.05867 [hep-ph]].
  * (46) V. V. Anisovich, M. A. Matveev, J. Nyiri, A. V. Sarantsev and A. N. Semenova,  arXiv:1507.07652 [hep-ph].
  * (47) Z. G. Wang and T. Huang,  Eur. Phys. J. C 76, no. 1, 43 (2016) [arXiv:1508.04189 [hep-ph]].
  * (48) Z. G. Wang, Nucl. Phys. B 913, 163-208 (2016) [arXiv:1512.04763 [hep-ph]].
  * (49) R. Ghosh, A. Bhattacharya and B. Chakrabarti, Phys. Part. Nucl. Lett. 14, no.4, 550-552 (2017) [arXiv:1508.00356 [hep-ph]].
  * (50) Z. G. Wang, Eur. Phys. J. C 76, no.3, 142 (2016) [arXiv:1509.06436 [hep-ph]].
  * (51) J. X. Zhang, Z. G. Wang and Z. Y. Di, Acta Phys. Polon. B 48, 2013 (2017) [arXiv:1711.10728 [hep-ph]].
  * (52) Z. G. Wang, Int. J. Mod. Phys. A 35, no.01, 2050003 (2020) [arXiv:1905.02892 [hep-ph]].
  * (53) Z. G. Wang, H. J. Wang and Q. Xin, Chin. Phys. C 45, 063104 (2021) [arXiv:2005.00535 [hep-ph]].
  * (54) A. Ali, I. Ahmed, M. J. Aslam, A. Parkhomenko and A. Rehman, PoS ICHEP2020, 527 (2021) [arXiv:2012.07760 [hep-ph]].
  * (55) G. J. Wang, R. Chen, L. Ma, X. Liu and S. L. Zhu,  Phys. Rev. D 94, no. 9, 094018 (2016) [arXiv:1605.01337 [hep-ph]].
  * (56) K. Azizi, Y. Sarac and H. Sundu, Phys. Rev. D 103, no.9, 094033 (2021) [arXiv:2101.07850 [hep-ph]].
  * (57) V. V. Anisovich, M. A. Matveev, J. Nyiri, A. V. Sarantsev and A. N. Semenova, Int. J. Mod. Phys. A 30, no.32, 1550190 (2015) [arXiv:1509.04898 [hep-ph]].
  * (58) Z. G. Wang, Int. J. Mod. Phys. A 36, no.10, 2150071 (2021) [arXiv:2011.05102 [hep-ph]].
  * (59) R. Zhu and C. F. Qiao,  Phys. Lett. B 756, 259 (2016) [arXiv:1510.08693 [hep-ph]].
  * (60) F. K. Guo, U. G. Meißner, W. Wang and Z. Yang, Phys. Rev. D 92, no.7, 071502 (2015) [arXiv:1507.04950 [hep-ph]].
  * (61) F. K. Guo, U. G. Meißner, J. Nieves and Z. Yang, Eur. Phys. J. A 52, no.10, 318 (2016) [arXiv:1605.05113 [hep-ph]].
  * (62) M. Mikhasenko,  arXiv:1507.06552 [hep-ph].
  * (63) X. H. Liu, Q. Wang and Q. Zhao, Phys. Lett. B 757, 231-236 (2016) [arXiv:1507.05359 [hep-ph]].
  * (64) M. Bayar, F. Aceti, F. K. Guo and E. Oset, Phys. Rev. D 94, no.7, 074039 (2016) [arXiv:1609.04133 [hep-ph]].
  * (65) S. X. Nakamura, A. Hosaka and Y. Yamaguchi, Phys. Rev. D 104, no.9, L091503 (2021) [arXiv:2109.15235 [hep-ph]].
  * (66) X. Liu, H. Huang and J. Ping, Phys. Rev. C 102 (2020) no.2, 025204 [arXiv:2005.09646 [hep-ph]].
  * (67) H. X. Chen, L. S. Geng, W. H. Liang, E. Oset, E. Wang and J. J. Xie, Phys. Rev. C 93, no.6, 065203 (2016) [arXiv:1510.01803 [hep-ph]].
  * (68) A. Feijoo, V. K. Magas, A. Ramos and E. Oset, Eur. Phys. J. C 76, no.8, 446 (2016) [arXiv:1512.08152 [hep-ph]].
  * (69) Y. Irie, M. Oka and S. Yasui, Phys. Rev. D 97, no.3, 034006 (2018) [arXiv:1707.04544 [hep-ph]].
  * (70) R. Chen, J. He and X. Liu, Chin. Phys. C 41, no.10, 103105 (2017) [arXiv:1609.03235 [hep-ph]].
  * (71) Q. Zhang, B. R. He and J. L. Ping, [arXiv:2006.01042 [hep-ph]].
  * (72) E. Y. Paryev, [arXiv:2007.01172 [nucl-th]].
  * (73) T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 100, no.9, 094031 (2019) [arXiv:1910.03984 [hep-ph]].
  * (74) K. Azizi, Y. Sarac and H. Sundu, Phys. Rev. D 96, no.9, 094030 (2017) [arXiv:1707.01248 [hep-ph]].
  * (75) K. Azizi, Y. Sarac and H. Sundu, Phys. Rev. D 98, no.5, 054002 (2018) [arXiv:1805.06734 [hep-ph]].
  * (76) X. Cao, F. K. Guo, Y. T. Liang, J. J. Wu, J. J. Xie, Y. P. Xie, Z. Yang and B. S. Zou, Phys. Rev. D 101, no.7, 074010 (2020) [arXiv:1912.12054 [hep-ph]].
  * (77) J. R. Zhang, Phys. Rev. D 103, no.7, 074016 (2021) [arXiv:2011.04594 [hep-ph]].
  * (78) Y. P. Xie and V. P. Goncalves, Phys. Lett. B 814, 136121 (2021) [arXiv:2012.07501 [hep-ph]].
  * (79) F. L. Wang, R. Chen and X. Liu, Phys. Rev. D 103, no.3, 034014 (2021) [arXiv:2011.14296 [hep-ph]].
  * (80) Q. Meng, E. Hiyama, K. U. Can, P. Gubler, M. Oka, A. Hosaka and H. Zong, Phys. Lett. B 798, 135028 (2019) [arXiv:1907.00144 [nucl-th]].
  * (81) Y. Liu, M. A. Nowak and I. Zahed, [arXiv:2108.04334 [hep-ph]].
  * (82) B. Wang, L. Meng and S. L. Zhu, Phys. Rev. D 101, no.3, 034018 (2020) [arXiv:1912.12592 [hep-ph]].
  * (83) Y. Liu, M. A. Nowak and I. Zahed, Phys. Rev. D 104, no.11, 114022 (2021) [arXiv:2108.07074 [hep-ph]].
  * (84) J. F. Giron and R. F. Lebed, Phys. Rev. D 104, no.11, 114028 (2021) [arXiv:2110.05557 [hep-ph]].
  * (85) E. Santopinto and A. Giachino, Phys. Rev. D 96, no.1, 014014 (2017) [arXiv:1604.03769 [hep-ph]].
  * (86) C. W. Xiao, J. Nieves and E. Oset, Phys. Lett. B 799, 135051 (2019) [arXiv:1906.09010 [hep-ph]].
  * (87) Q. Wang, X. H. Liu and Q. Zhao, Phys. Rev. D 92, 034022 (2015) [arXiv:1508.00339 [hep-ph]].
  * (88) V. Kubarovsky and M. B. Voloshin, Phys. Rev. D 92, no.3, 031502 (2015) [arXiv:1508.00888 [hep-ph]].
  * (89) M. Karliner and J. L. Rosner, Phys. Lett. B 752, 329-332 (2016) [arXiv:1508.01496 [hep-ph]].
  * (90) H. Y. Cheng and C. K. Chua, Phys. Rev. D 92, no.9, 096009 (2015) [arXiv:1509.03708 [hep-ph]].
  * (91) W. Y. Liu, W. Hao, G. Y. Wang, Y. Y. Wang, E. Wang and D. M. Li, Phys. Rev. D 103, no.3, 034019 (2021) [arXiv:2012.01804 [hep-ph]].
  * (92) F. Yang, Y. Huang and H. Q. Zhu, Sci. China Phys. Mech. Astron. 64, no.12, 121011 (2021) [arXiv:2107.13267 [hep-ph]].
  * (93) F. Stancu, Phys. Rev. D 104, no.5, 054050 (2021) [arXiv:2108.05841 [hep-ph]].
  * (94) Y. Dong, P. Shen, F. Huang and Z. Zhang, Eur. Phys. J. C 80, no.4, 341 (2020) [arXiv:2002.08051 [hep-ph]].
  * (95) M. B. Voloshin, Phys. Rev. D 100, no.3, 034020 (2019) [arXiv:1907.01476 [hep-ph]].
  * (96) X. Z. Ling, J. X. Lu, M. Z. Liu and L. S. Geng, Phys. Rev. D 104, no.7, 074022 (2021) [arXiv:2106.12250 [hep-ph]].
  * (97) Y. Xing and Y. Niu, Eur. Phys. J. C 81, no.11, 978 (2021) [arXiv:2106.09939 [hep-ph]].
  * (98) C. Cheng, F. Yang and Y. Huang, [arXiv:2110.04746 [hep-ph]].
  * (99) Z. G. Wang and X. Wang, Chin. Phys. C 44, 103102 (2020) [arXiv:1907.04582 [hep-ph]].
  * (100) Y. J. Xu, C. Y. Cui, Y. L. Liu and M. Q. Huang, Phys. Rev. D 102, no.3, 034028 (2020) [arXiv:1907.05097 [hep-ph]].
  * (101) J. Liu, D. Y. Chen and J. He, Eur. Phys. J. C 81, no.11, 965 (2021) [arXiv:2108.00148 [hep-ph]].
  * (102) Y. Huang and H. Q. Zhu, Phys. Rev. D 104, no.5, 056027 (2021) [arXiv:2107.03773 [hep-ph]].
  * (103) J. T. Zhu, S. Y. Kong, Y. Liu and J. He, Eur. Phys. J. C 80, no.11, 1016 (2020) [arXiv:2007.07596 [hep-ph]].
  * (104) J. Ferretti, E. Santopinto, M. Naeem Anwar and M. A. Bedolla, Phys. Lett. B 789, 562-567 (2019) [arXiv:1807.01207 [hep-ph]].
  * (105) Y. Shimizu, D. Suenaga and M. Harada, Phys. Rev. D 93, no.11, 114003 (2016) [arXiv:1603.02376 [hep-ph]].
  * (106) Y. Liu and I. Zahed, Phys. Rev. D 95, no.11, 116012 (2017) [arXiv:1704.03412 [hep-ph]].
  * (107) Z. G. Wang, Nucl. Phys. B 973, 115579 (2021) [arXiv:2104.12090 [hep-ph]].
  * (108) H. T. An, K. Chen, Z. W. Liu and X. Liu, Phys. Rev. D 103, no.7, 074006 (2021) [arXiv:2012.12459 [hep-ph]].
  * (109) Y. Yan, Y. Wu, X. Hu, H. Huang and J. Ping, [arXiv:2110.10853 [hep-ph]].
  * (110) J. Ferretti and E. Santopinto, JHEP 04, 119 (2020) [arXiv:2001.01067 [hep-ph]].
  * (111) J. Ferretti and E. Santopinto, [arXiv:2111.08650 [hep-ph]].
  * (112) F. L. Wang, X. D. Yang, R. Chen and X. Liu, Phys. Rev. D 103, no.5, 054025 (2021) [arXiv:2101.11200 [hep-ph]].
  * (113) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov,  Nucl. Phys. B 147, 385 (1979).
  * (114) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov,  Nucl. Phys. B 147, 448 (1979).
  * (115) B. L. Ioffe,  Nucl. Phys. B 188, 317 (1981) Erratum: [Nucl. Phys. B 191, 591 (1981)].
  * (116) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020).
  * (117) V. M. Belyaev and B. L. Ioffe, Sov. Phys. JETP 56, 493-501 (1982) [Zh. Eksp. Teor. Fiz. 83, 876 (1982)].
  * (118) V. M. Belyaev and B. L. Ioffe, Sov. Phys. JETP 57, 716 (1983) [Zh. Eksp. Teor. Fiz. 84, 1236 (1983)].
  * (119) Z. G. Wang, Phys. Rev. D 75, 054020 (2007) [arXiv:hep-ph/0701176 [hep-ph]].
  * (120) E. V. Veliev, K. Azizi, H. Sundu, G. Kaya and A. Turkan,  Eur. Phys. J. A 47, 110 (2011) [arXiv:1103.4330 [hep-ph]].


# Toward A Reinforcement-Learning-Based System for Adjusting Medication to
Minimize Speech Disfluency

Pavlos Constas4, Vikram Rawal1, Matthew Honorio Oliveira1, Andreas Constas1,
Aditya Khan1, Kaison Cheung1, Najma Sultani1, Carrie Chen3, Micol Altomare1,
Michael Akzam1, Jiacheng Chen1, Vhea He1, Lauren Altomare1, Heraa Murqi1, Asad
Khan1, Nimit Amikumar Bhanshali1, Youssef Rachad1, Michael Guerzhoy12

###### Abstract

We propose a Reinforcement-Learning-based system that would automatically
prescribe a hypothetical patient medications that may help the patient with
their mental-health-related speech disfluency, and adjust the medication and
the dosages in response to data from the patient. We demonstrate the
components of the system: a module that detects and evaluates speech
disfluency on a large dataset we built, and a Reinforcement Learning algorithm
that automatically finds good combinations of medications. To support the two
modules, we collect data on the effect of psychiatric medications for speech
disfluency from the literature, and build a plausible patient simulation
system. We demonstrate that the Reinforcement Learning system is, under some
circumstances, able to converge to a good medication regime. We collect and
label a dataset of people with possible speech disfluency and demonstrate our
methods using that dataset. Our work is a proof of concept: we show that there
is promise in the idea of using automatic data collection to address
disfluency.

## Introduction

Speech disfluency is a common medical issue. It can be caused by, among other
factors, conditions such as depression, anxiety, and insomnia (see Section
Medications Literature Review). Speech disfluency includes stuttering as well
as issues like pauses that are too long, repetitions, “false starts,” and
“repairs” of previous utterances Shriberg, Bates, and Stolcke (1997). In this
paper, we propose a hypothetical Reinforcement Learning system for helping
physicians adjust the medications for people with speech disfluency. In the
paper, we focus on a proof-of-concept system. Of course, regulatory and
clinical issues would need to be addressed before actually implementing such a
system. However, we show that such a system can in principle be implemented,
and automatic disfluency detection can work well enough to support such a
system. On a high level, the system works as follows: the person with a speech
disfluency interacts with a system that measures their speech fluency; the
system tries to assign different medications in different doses to the person,
to minimize their speech disfluency. Using Reinforcement Learning (RL), the
system trades off exploring to find the best medication combination for the
person and exploring effective medication combinations that have been found.
We demonstrate two components of the system: a subsystem for detecting how
disfluent the person’s speech is, and a subsystem to minimize the speech
disfluency by finding a combination of medications that works using RL. We
train our disfluency detection system to predict the labels we assigned to
clips in the dataset we collected, and then we demonstrate the effectiveness
of the disfluency detection subsystem on a dataset we collect. To demonstrate
the feasibility of the RL subsystem, we attempt to construct a plausible
dataset we measure the effectiveness of our speech disfluency detection
subsystem, as well as search the literature for the plausible timespans and
onset times for the effects of medications. We then run a patient simulation
with plausible inputs and demonstrate that our RL subsystem can find
strategies to minimize the speech disfluencies of our plausibly-simulated
patients. To be able to evaluate our subsystems, we collect a dataset of
public videos of people with possible disfluencies and label the dataset using
a scalable strategy that allows us to obtain precise and standardized ratings
by having each video be rated by multiple raters. The rest of the paper is
organized as follows: we explain our data collection and labelling process. We
then describe our disfluency rating process, and report results on that
subsystem. We then describe our patient simulation process and report results
of the RL system’s performance on the simulated patients. For the patient
simulation to be plausible, we connect the patient simulation to how
accurately speech can be rated for fluency by our system and to the plausible
effects of medications (note that if the measurement of speech fluency is too
noisy and/or the medications’ effect is too subtle or the onset is too long,
learning would likely not be possible). Finally, we summarize our literature
search results for medications that could plausibly affect speech fluency.

## Data Collection

### Methods

The objective of our data collection process was to obtain a series of audio
samples from individuals with possible mental-health-related speech
disfluency, across a period. To this end, we collected 19 channels from
searching on YouTube for mental health-related vlog channels by YouTubers, as
well as the D-vlog dataset Yoon et al. (2022) (channels of YouTubers with
depression). For each YouTuber represented in the videos, we scraped their
channel for other videos which contained significant stretches of unedited
spoken audio. Terms used to query for videos from each channel were a subset
of the following keywords $\\{$“depression”, “story”, “vlog”, “depression
vlog”, “anxiety”, “tested”, “figure”, “rambling”, “issues”, “anxiety vlog”,
“webcam”$\\}$. For each video, only the audio was extracted. In total, we
obtained $195$ audio clips. Particularly, there are $9$ to $11$ audio clips
for each channel, with an average of $10$ audio clips per channel.

### Rating System

A rating system was devised to assess the severity of the disfluency in the
video data. A group of 17 undergraduate students from the [REDACTED]
volunteered as raters for the videos. The 19 YouTuber channels in the dataset
were examined for their disfluencies in them. Raters were tasked with
assessing the disfluency severity in each video on a scale of 1 to 7, which
was adapted from the Stuttering Severity Instrument-Third Edition (SSI-3)
Glyndon (1994) (but note the “disfluency” is a more general term than
“stuttering”.) The evaluation consisted of two rating stages, each stage
lasting approximately one week. In each stage, raters randomly received
several channels and were asked to rate their audio samples in the dataset.
This was arranged so that each audio sample in the dataset would have up to 3
raters. Each rater received different channels between stages. To ensure
independent evaluation, raters were advised against sharing their assessments
between themselves. Data from the initial stage was not used in our
experiments. The round was used for acquainting raters with the variation of
disfluency observed in the dataset. At the end of this phase, each rater was
privately given summary statistics regarding their performance in the round,
including the mean and standard deviation of their ratings across audio
samples, as well as a spreadsheet containing a measure of their bias for each
audio sample (where bias is the distance of their rating from the mean rating
across all raters for that audio sample). This process was aimed at allowing
raters to recognize possible misconceptions or errors in previous judgments,
ensuring better consistency in the final rating cycle. The ratings from the
second stage were the finalized ratings that would be used for fine-tuning our
automatic labelling language model. The ratings were standardized, as
described below.

## Rater Performance Analysis

In this Subsection, we analyze the rater data, and show that raters are
somewhat consistent in their ratings of the same clips. This indicates that we
can use the standardized ratings (see below) as targets when estimating the
fluency of speakers in audio clips.

### Data Model

To assess the performance of the raters, we conducted a regression analysis.
The model we utilized was

$r_{ij}=\alpha_{i}+\beta_{j}+\varepsilon_{ij}$

where $r_{ij}$ is the rating given to audio clip $j$ by rater $i$,
$\alpha_{i}$ is the rater bias, $\beta_{j}$ is the true average disfluency of
the channel, and $\varepsilon_{ij}$ is the random error (see Raymond &
Viswesvaran (1991) for a similar model). This model is estimated using least-
squares regression. Using this model, the performance of the raters was
assessed by randomly splitting the dataset into a 70% training set, and a 30%
validation set.

### Analysis

Below, we perform an exploratory analysis of the non-standardized ratings. We
compute the Root-Mean-Square Error (RMSE) on the training set and the
validation set when predicting the disfluency scores using the data model. The
RMSE on the training set was 0.776/6.000 and the RMSE on the validation set
was 0.877/6.000. The validation RMSE we would obtain if the data model
predicted the average rating every time would be 1.394/6.000. the $R^{2}$
value of the model on the training set was 0.440, indicating that the rater
coefficients and the clip coefficients have explanatory power. For each clip
on the validation set, we compute the standard deviation of the ratings
assigned by different raters to the same clip. The median standard deviation
is 0.623/6.000. This suggests that the median disagreement between raters was
just over half a rating point on a given clip The standard deviations are
given on a scale of 6.0 since the scores range from 1 to 7. On the validation
set, we also found that the median standard deviation of ratings across all
clips was 0.623. This suggests that the median disagreement between raters was
just over half a rating point on a given clip.

### Standardized ratings

Different raters use different standards for fluency. We therefore obtained
standardized ratings. We accomplish this by subtracting the rater bias
$\alpha_{i}$ (see the Data Model section) for rater $i$ for each rating
$r_{ij}$ by rater $i$. Then, when we compute the average standardized rating
for every clip, we average ratings that are actually on the same scale.

## Disfluency Pipeline

In this section, we described our subsystem for assessing the disfluency of a
person in the input clip. We use Whisper
111https://github.com/openai/whisperRadford et al. (2023) to transcribe the
audio. We then use an Auto-Correlational Neural Network-based tagger Lou,
Anderson, and Johnson (2020) to tag the Whisper transcript. Finally, we fine-
tune GPT-2 Radford et al. (2019) on the tagged transcript as input in order to
predict the disfluency scores we assigned.

### Transcribing audio with Whisper

The YouTube videos are transcribed using an Automated Speech Recognition (ASR)
model; the larger version of a Whisper model was used to yield more accurate
results. This was further enhanced by the inclusion of additional prompts to
the model which represent common disfluency utterances (e.g., ”uh uhh um umm
so like ah ahh well”) as the model suppresses them by default.

### Tagging

The parsed text transcripts were subsequently fed into a Disfluency Tagging
Auto-Correlational Neural Network (DT-ACNN) Lou, Anderson, and Johnson (2020)
– a system designed to categorize each word within the text transcript as
either “fluent” or “disfluent”. In Lou, Anderson, and Johnson (2020), the
Switchboard corpus of conversational speech Godfrey, Holliman, and McDaniel
(1992) dataset was used. For the task of predicting a per-word “fluent” or
“disfluent” label, the authors report a Recall score of 90.0%, Precision score
of 82.8%, and F1 score of 86.2% on the dataset. The reported results indicate
the effectiveness of the ACNN model in disfluency detection.

### Fine-tuning GPT-2

We fine-tune GPT-2 to predict the average standardized disfluency score in our
dataset from the tagged Whispter transcripts, as well as from the words-per-
minute (WPM) measure. We train a binary classifier and a regression. The
binary classifier predicts whether the disfluency rating is less than or equal
to 3 or if it is greater than 3. The regression predicts the disfluency rating
directly. For the regression task, we train with the embedding size 768, using
the Mean Squared Loss (MSE), and the AdamW optimizer with parameters
$\beta_{1}=0.9$, $\beta_{2}=0.999$, $\epsilon=10^{-9}$. The token limit of
GPT2 is 1024. Inputs that exceed this limit were truncated. The following
hyperparameters were used during training: a learning rate of $4.5\times
10^{-4}$, a batch size of 4 (dictated by computational limitations), and a
total of 50 epochs, even though 27 epochs were identified as optimal.
Additionally, a weight decay of 0.01 was applied. P-tuning Liu et al. (2023)
was implemented to lessen the demands on computational resources. In this
approach, a soft prompt with a set of 100 tokens is introduced at the
beginning of the input. These tokens aid in guiding the model during
classification. The model uses a Prompt Encoder to optimize the prompt, with
an encoding layer comprised of 128 units. The model performance was evaluated
by randomly splitting the dataset into an 80% training set, and a 20%
validation set.

### Results

Preliminary results for the non-standardized ratings in the regression setting
as shown in Fig. 1. We observe that our system is currently able to predict
the validation rating to about within 0.15/6 of the actual rating on average
(the standard error is obtained by taking the square root of the MSE) for
YouTubers not in the training set.

Figure 1: Regression Fine-tuning Loss Curve

### Analysis

Preliminary results indicate that we are able to predict the non-standardized
rating well. Further improvements are possible by fine-tuning GPT-2 on
standardized ratings, since noise would be reduced, and by fine-tuning a
larger language model.

## Patient Simulation and Reinforcement Learning

### Overview

In this Section, we explore the plausibility of using Reinforcement Learning
together with signals from our speech disfluency detector to find an effective
medication regimen for people with speech disfluency. We first describe how we
simulate people with speech disfluency in a medically plausible way. We then
demonstrate that our reinforcement learning algorithm could find an effective
medication regimen in a plausible scenario.

### Person with Disfluency Simulation

We model a medication administration environment, aiming to determine the most
effective medication regimen for people experiencing depression, anxiety,
insomnia, and resulting speech fluency issues. The person’s health state
evolves based on Hidden Markov Models (HMMs). Each health issue (depression,
anxiety, insomnia) has its unique HMM, governing how the patient’s state
progresses. The patient’s speech fluency is also influenced by these health
states. The Medication object represents different types of medications, each
with varying effects on the aforementioned health issues. These effects
include beneficial impacts on the conditions and potential side effects.
Medications have properties like dosage, half_life, and time_to_effect, which
dictate how they function over time. We simulate people with disfluency by
evolving the HMM state. The patient model has the following attributes:

  * •

_Depression, Anxiety, Insomnia Scores_ : These attributes represent the
initial underlying conditions of the patient. Represented as an integer
between 1 and 5. A higher number denotes a more severe state.

  * •

_Depression, Anxiety, Insomnia Hidden Markov Models_ : Models that represent
the behaviour of how the severity of the patient’s depression, anxiety, and
insomnia change over time based on their initial states and also through
interaction with medicine. These directly impact observed speech fluency
context.

  * •

_Regular Speech Fluency Score_ : Indicates the patient’s natural ability to
speak fluently, modelled as a continuous value between 0 and 1.

  * •

_Medication Accumulation_ : A list that keeps track of all medications that
are currently in the patient’s system.

Alongside this, we also model an individual medication with the following
attributes:

  * •

_Name_ : The name of the medication.

  * •

_Depression, Anxiety, Insomnia Effects_ : Captures the medication’s average
effect and variability on each condition given the standard dosage.

  * •

_Dosage_ : The amount of medication administered relative to the standard
dose. (e.g. Dosage = 1.5 means 1.5x the standard dose). This attribute scales
the effects of the medication on the patient.

  * •

_Time to Effect_ : The amount of days it takes for the medication to start
showing effects.

  * •

_Half-Life_ : The amount of days it takes for the medication dosage in the
patient’s system to reduce by half.

### Hidden Markov Model (HMM)

A Hidden Markov Model (HMM) is a statistical model that represents
probabilistic sequences of observable data. These sequences are generated
based on hidden states which cannot be directly observed. The model assumes
the Markov property as the future state depends only on the current state, and
not on previous states. In the context of this program, the “observable data”
is the patient’s speech fluency score while the “hidden states” are the
underlying depression, anxiety, and insomnia conditions that affect the
severity. Each health condition — depression, anxiety, and insomnia — has its
associated HMM. The key components of these HMMs are:

  * •

The initial probability distribution over the initial state of the condition.
Initialized as a uniform distribution, indicating that any severity level is
equally likely at the start.

  * •

The transition matrix, which defines the probability of transitioning from one
state (severity level) to another in consecutive time steps. For instance, if
a patient is currently at a severity level of 3 for depression, the transition
matrix, the transition matrix will dictate the probability of them improving
to level 2, worsening to level 4, or remaining at level 3 in the next step.

  * •

We use a Gaussian Hidden Markov Model as the observable context is assumed to
be generated from a Gaussian distribution. The means and covariances define
these distributions for each hidden state.

  * •

Each state has a mean context emission which are the same across the
depression, anxiety, and insomnia states, and are represented in Figure 2.

123450.80.70.60.50.4outputoutputoutputoutputoutput

Figure 2: A diagram of a Hidden Markov Model representing context emissions
generated from states.

### RL Environment

At each step, an agent can choose to administer a specific medication from the
available list. The environment then evolves based on the medication’s effects
and the underlying HMMs. The agent receives a reward based on the patient’s
measured fluency. The program employs the LinUCB Nelson et al. (2022)
algorithm to learn the optimal medication strategy. The goal is to maximize
the patient’s speech fluency while minimizing the side effects and the
severity of the health conditions. Over numerous iterations, the agent learns
to navigate this complex environment, aiming to discover the best treatment
regimen for the patient. The effects of the medication on the patient model is
implemented by applying the effects of the medication on each condition on
each condition’s transition matrix. Medications that have positive effects on
the conditions will skew their transition matrix to tend more towards lower
scores as time goes on and vice versa.

### Medication Selection Algorithm

The LinUCB algorithm is an Upper Confidence Bound method tailored for
contextual bandits with linear payoff functions Nelson et al. (2022). It’s
designed to strike a balance between exploration and exploitation in the
context of online learning, especially when the reward of an action can be
approximated as a linear function of its features. When deciding which action
to take, LinUCB chooses the action with the highest upper bound of this
confidence interval. This strategy balances exploration of uncertain actions
and exploitation of actions with known rewards. We use increase in speech
fluency as our reward, and in our case, speech fluency is modelled as a linear
function of:

$S=0.1\cdot D+0.2\cdot A+0.3\cdot I+0.4\cdot R$

where $S$ is current speech fluency, $D$, $A$, $I$ are the patient’s current
depression, anxiety, and insomnia scores respectively, labelled on a 5 point
scale, where 1 represents a patient who does not suffer from the given effect,
and 5 represents a patient who suffers from it in it’s most malignant form.
This data can be collected through regular patient check-ups and be formed as
a scale for each individual. $R$ is the patient’s regular speech fluency
score, a value that may be different for every person, and was computed as a
weighted sum utilizing factors such as age. Our implementation of the LinUCB
algorithm observes the current state of the patient, then for each medication
that is part of the environment, it estimates the reward using a linear
approximation. An upper confidence bound is calculated for the estimated
reward in which the medication with the highest upper confidence bound is
chosen according to the equation: $a_{t}=argmax_{a\epsilon
A}(x_{t}(\alpha)^{T}\theta_{a}+\alpha\sqrt{x_{t}(a)^{T}A_{a}^{-1}x_{t}(a)})$
where $x_{t}$(a) is the feature vector for action $a$ at time $t$,
$\theta_{a}$ is the parameter vector for action $a$ which we want to estimate,
and $\alpha$ is the hyperparameter controlling how exploitative/exploratory
the agent wants to be in making decisions. The higher the value of $\alpha$,
the more the agent will venture out and choose new medications, the lower the
value of $\alpha$ the more patient will choose medications it has current
knowledge of. In our implementation of the LinUCB algorithm, we chose the
value of $\alpha=10.0$ to get our results, as it demonstrated an acceptable
bound for the algorithms exploratory tendency and provided a representative
time frame for the onset of speech disfluency to decrease. For training our
algorithm, we set up a training loop simulating a single day where the agent
uses a dynamic step driver responsible for executing the agent’s policy in the
environment for a specified number of steps. A step in our environment
consists of administering the selected medication to the patient and
subsequently updating the states of the patient based on the effects of the
medication. After each step it collects data for the agent which is stored in
a replay buffer which allows for efficient learning by sampling from these
experiences.

### Results

In Figure 3, we see the 3 most common outcomes of the medication selection
algorithm. In the top graph, we see a steady increase in the speech fluency of
a patient. The model converges, and we see that the model has success in
selecting accurate medications. The graph in the middle demonstrates an
initial increase and then a plateau in the medication selection. This is
indicative of the model struggling to find the most suitable medication for
the patient, resulting in a prolonged period of uncertainty and potentially
ineffective treatment. Finally, in the bottom graph, we observe a sharp
decline in speech fluency, suggesting that the medication selection algorithm
failed to provide effective treatment for the patient, leading to worsening
symptoms. These three distinct outcomes illustrate the variability in the
medication selection process and emphasize the importance of continuous
monitoring and adjustment of the algorithm to improve patient outcomes.

Figure 3: Speech Fluency over time

We ran a limited set of simulations where the medications had plausible
properties, and the precision of the measurement of the speech disfluency was
to within 10% (note that we obtained results indicating that this is
possible). Our preliminary results indicate that if speech disfluency can be
measured to within 10% of the true score and the medications have plausible
properties (similar to the ones seen in Medication Literature Review),
reinforcement learning is possible to pessimize speech disfluency.

## Medication Literature Review

A systematic literature review was conducted to determine common medications
used to treat major depressive disorder (“depression”) and other mental
illnesses that affect speech fluency. Table LABEL:t1 indicates 23 medications
whose onset time and response rates were used to inform the reinforcement
learning simulation.

## Ethical Considerations

In this paper, we outline and evaluate a proposal to adjust medications
automatically in order to improve the speech fluency of a simulated patient.
Although administering medications automatically is sometimes done (e.g., with
Insulin pumps), this can only be done after thorough clinical trials and with
informed consent from the patient. Patients must be thoroughly informed about
the nature of the automated system, its potential risks and benefits, and
their rights in the decision-making progress. This extends beyond initial
consent, and includes ongoing consent as the system adapts medication
regiments. We believe that if safe dosage ranges can be determined,
automatically adjusting medications can in principle be done. Potential
upsides include better-optimized medication regimes that are found faster (or
found at all). The system’s reliance on sensitive patient data raises
significant privacy and security concerns. Strict adherence to data protection
laws and ethical guidelines is mandatory. Finally, the long-term implications
of integrating such a system into healthcare practices must be considered.
This includes ongoing monitoring of the system’s impact on patient health
outcomes, adapting to new medical insights, and updating the system
accordingly.

## Limitations

This paper studies the feasibility of using automated measures of disfluency
together with a reinforcement learning system to adjust medications. Our model
is informed by the literature, but it is still a “toy” model. However, our
results indicate that it is possible to measure disfluency to a certain
precision (for some definition of “disfluency”), and that this precision can
be sufficient to serve as input to an RL system.

## Conclusions and Future Work

We propose and evaluate a new idea: people with speech disfluency can have
their disfluency measured automatically, and a Reinforcement Learning
algorithm can find an optimal medication regime for them through
exploration/exploitation. We have shown that the components of this system are
possible to build. In our preliminary experiments, we have shown that with an
accurate enough disfluency detection system and with medications with
plausible properties, a good medication regime can be found by a Reinforcement
Learning system. Future work includes improving our subsystems for
performance, ensuring that our simulations preserve patient safety (e.g. by
only trying safe dosages), and running more extensive simulations to show
under what conditions RL for optimizing medications is feasible.

## References

  * Abdul-Baki et al. (2009) Abdul-Baki, H.; El Hajj, I. I.; ElZahabi, L.; Azar, C.; Aoun, E.; Skoury, A.; Chaar, H.; and Sharara, A. I. 2009.  A randomized controlled trial of imipramine in patients with irritable bowel syndrome.  _World journal of gastroenterology: WJG_ , 15(29): 3636.
  * Allen, Sabir, and Sharma (2018) Allen, M. J.; Sabir, S.; and Sharma, S. 2018.  GABA receptor.
  * Chokhawala, Lee, and Saadabadi (2022) Chokhawala, K.; Lee, S.; and Saadabadi, A. 2022.  Lithium. National Library of Medicine.  _National Center for Biotechnology Information. PubMed Central_.
  * Cincotta and Rodefer (2010) Cincotta, S. L.; and Rodefer, J. S. 2010.  Emerging role of sertindole in the management of schizophrenia.  _Neuropsychiatric disease and treatment_ , 429–441.
  * Cookson et al. (2007) Cookson, J.; Keck Jr, P. E.; Ketter, T. A.; and Macfadden, W. 2007.  Number needed to treat and time to response/remission for quetiapine monotherapy efficacy in acute bipolar depression: evidence from a large, randomized, placebo-controlled study.  _International clinical psychopharmacology_ , 22(2): 93–100.
  * Dean (2020) Dean, L. 2020.  Venlafaxine therapy and CYP2D6 genotype.
  * Fariba and Saadabadi (2020) Fariba, K. A.; and Saadabadi, A. 2020.  Topiramate.
  * Flament et al. (1999) Flament, M. F.; Lane, R.; Zhu, R.; and Ying, Z. 1999.  Predictors of an acute antidepressant response to fluoxetine and sertraline.  _International clinical psychopharmacology_ , 14(5): 259–276.
  * Gibbons et al. (2012) Gibbons, R. D.; Hur, K.; Brown, C. H.; Davis, J. M.; and Mann, J. J. 2012.  Benefits from antidepressants: synthesis of 6-week patient-level outcomes from double-blind placebo-controlled randomized trials of fluoxetine and venlafaxine.  _Archives of general psychiatry_ , 69(6): 572–579.
  * Glyndon (1994) Glyndon, R. 1994.  _Stuttering severity instrument for children and adults. 3rd ed._ Austin, Texas: Pro-Ed.
  * Godfrey, Holliman, and McDaniel (1992) Godfrey, J. J.; Holliman, E. C.; and McDaniel, J. 1992.  SWITCHBOARD: Telephone speech corpus for research and development.  In _Acoustics, speech, and signal processing, ieee international conference on_ , volume 1, 517–520. IEEE Computer Society.
  * Heijnen et al. (2015) Heijnen, W. T.; De Fruyt, J.; Wierdsma, A. I.; Sienaert, P.; and Birkenhäger, T. K. 2015.  Efficacy of tranylcypromine in bipolar depression: a systematic review.  _Journal of clinical psychopharmacology_ , 35(6): 700–705.
  * Hui et al. (2019) Hui, T.; Kandola, A.; Shen, L.; Lewis, G.; Osborn, D.; Geddes, J.; and Hayes, J. 2019.  A systematic review and meta-analysis of clinical predictors of lithium response in bipolar disorder.  _Acta Psychiatrica Scandinavica_ , 140(2): 94–115.
  * Ionescu et al. (2016) Ionescu, D. F.; Shelton, R. C.; Baer, L.; Meade, K. H.; Swee, M. B.; Fava, M.; and Papakostas, G. I. 2016.  Ziprasidone augmentation for anxious depression.  _International clinical psychopharmacology_ , 31(6): 341.
  * Jia et al. (2022) Jia, Z.; Yu, J.; Zhao, C.; Ren, H.; and Luo, F. 2022.  Outcomes and predictors of response of duloxetine for the treatment of persistent idiopathic dentoalveolar pain: A retrospective multicenter observational study.  _Journal of Pain Research_ , 3031–3041.
  * Jiang et al. (2017) Jiang, K.; Li, L.; Wang, X.; Fang, M.; Shi, J.; Cao, Q.; He, J.; Wang, J.; Tan, W.; and Hu, C. 2017.  Efficacy and tolerability of escitalopram in treatment of major depressive disorder with anxiety symptoms: a 24-week, open-label, prospective study in Chinese population.  _Neuropsychiatric Disease and Treatment_ , 515–526.
  * Kumar and Balan (2007) Kumar, A.; and Balan, S. 2007.  Fluoxetine for persistent developmental stuttering.  _Clinical neuropharmacology_ , 30(1): 58–59.
  * Lavergne et al. (2005) Lavergne, F.; Berlin, I.; Gamma, A.; Stassen, H.; and Angst, J. 2005.  Onset of improvement and response to mirtazapine in depression: a multicenter naturalistic study of 4771 patients.  _Neuropsychiatric disease and treatment_ , 1(1): 59–68.
  * Lee et al. (2011) Lee, J.-S.; Ahn, J.-H.; Lee, J.-I.; Kim, J.-H.; Jung, I.; Lee, C.-U.; Lee, J.-Y.; Lee, S.-I.; and Kim, C.-Y. 2011.  Dose pattern and effectiveness of paliperidone extended-release tablets in patients with schizophrenia.  _Clinical Neuropharmacology_ , 34(5): 186–190.
  * Lewis et al. (2019) Lewis, G.; Duffy, L.; Ades, A.; Amos, R.; Araya, R.; Brabyn, S.; Button, K. S.; Churchill, R.; Derrick, C.; Dowrick, C.; et al. 2019.  The clinical effectiveness of sertraline in primary care and the role of depression severity and duration (PANDA): a pragmatic, double-blind, placebo-controlled randomised trial.  _The Lancet Psychiatry_ , 6(11): 903–914.
  * Liu et al. (2023) Liu, X.; Zheng, Y.; Du, Z.; Ding, M.; Qian, Y.; Yang, Z.; and Tang, J. 2023.  GPT understands, too.  _AI Open_.
  * Liu et al. (2020) Liu, Y.-T.; Chen, G.-T.; Huang, Y.-C.; Ho, J.-T.; Lee, C.-C.; Tsai, C.-C.; and Chang, C.-N. 2020.  Effectiveness of dose-escalated topiramate monotherapy and add-on therapy in neurosurgery-related epilepsy: A prospective study.  _Medicine_ , 99(52).
  * Lou, Anderson, and Johnson (2020) Lou, P. J.; Anderson, P.; and Johnson, M. 2020.  Disfluency Detection using Auto-Correlational Neural Networks.  arXiv:1808.09092.
  * Maan, Duong, and Saadabadi (2018) Maan, J. S.; Duong, T.; and Saadabadi, A. 2018.  Carbamazepine.
  * Moore and Saadabadi (2022) Moore, J. J.; and Saadabadi, A. 2022.  Selegiline.  In _StatPearls [Internet]_. StatPearls Publishing.
  * Nelson et al. (2022) Nelson, E.; Bhattacharjya, D.; Gao, T.; Liu, M.; Bouneffouf, D.; and Poupart, P. 2022.  Linearizing contextual bandits with latent state dynamics.  In _Uncertainty in Artificial Intelligence_ , 1477–1487. PMLR.
  * Pajonk (2004) Pajonk, F.-G. 2004.  Risperidone in acute and long-term therapy of schizophrenia—a clinical profile.  _Progress in Neuro-Psychopharmacology and Biological Psychiatry_ , 28(1): 15–23.
  * Pande et al. (2003) Pande, A. C.; Crockatt, J. G.; Feltner, D. E.; Janney, C. A.; Smith, W. T.; Weisler, R.; Londborg, P. D.; Bielski, R. J.; Zimbroff, D. L.; Davidson, J. R.; et al. 2003.  Pregabalin in generalized anxiety disorder: a placebo-controlled trial.  _American Journal of Psychiatry_ , 160(3): 533–540.
  * Panebianco et al. (2021) Panebianco, M.; Al-Bachari, S.; Hutton, J. L.; and Marson, A. G. 2021.  Gabapentin add-on treatment for drug-resistant focal epilepsy.  _Cochrane Database of Systematic Reviews_ , (1).
  * Parikh, Yilanli, and Saadabadi (2017) Parikh, N.; Yilanli, M.; and Saadabadi, A. 2017.  Tranylcypromine.
  * Peuskens (1995) Peuskens, J. 1995.  Risperidone in the treatment of patients with chronic schizophrenia: a multi-national, multi-centre, double-blind, parallel-group study versus haloperidol.  _The British Journal of Psychiatry_ , 166(6): 712–726.
  * Radford et al. (2023) Radford, A.; Kim, J. W.; Xu, T.; Brockman, G.; McLeavey, C.; and Sutskever, I. 2023\.  Robust speech recognition via large-scale weak supervision.  In _International Conference on Machine Learning_ , 28492–28518. PMLR.
  * Radford et al. (2019) Radford, A.; Wu, J.; Child, R.; Luan, D.; Amodei, D.; Sutskever, I.; et al. 2019\.  Language models are unsupervised multitask learners.  _OpenAI blog_ , 1(8): 9.
  * Shriberg, Bates, and Stolcke (1997) Shriberg, E.; Bates, R.; and Stolcke, A. 1997.  A prosody only decision-tree model for disfluency detection.  In _Fifth European Conference on Speech Communication and Technology_.
  * Singh and Saadabadi (2019) Singh, H. K.; and Saadabadi, A. 2019.  Sertraline.
  * Song et al. (2015) Song, H. R.; Bahk, W.-M.; Woo, Y. S.; Jeong, J.-H.; Kwon, Y.-J.; Seo, J. S.; Kim, W.; Kim, M.-D.; Shin, Y.-C.; Lee, S.-Y.; et al. 2015.  Efficacy and tolerability of generic mirtazapine (Mirtax) for major depressive disorder: multicenter, open-label, uncontrolled, prospective study.  _Clinical Psychopharmacology and Neuroscience_ , 13(2): 144.
  * Strawn et al. (2018) Strawn, J. R.; Geracioti, L.; Rajdev, N.; Clemenza, K.; and Levine, A. 2018.  Pharmacotherapy for generalized anxiety disorder in adult and pediatric patients: an evidence-based treatment review.  _Expert opinion on pharmacotherapy_ , 19(10): 1057–1070.
  * Tenjin et al. (2013) Tenjin, T.; Miyamoto, S.; Ninomiya, Y.; Kitajima, R.; Ogino, S.; Miyake, N.; and Yamaguchi, N. 2013.  Profile of blonanserin for the treatment of schizophrenia.  _Neuropsychiatric disease and treatment_ , 587–594.
  * Tolou-Ghamari et al. (2013) Tolou-Ghamari, Z.; Zare, M.; Habibabadi, J. M.; and Najafi, M. R. 2013.  A quick review of carbamazepine pharmacokinetics in epilepsy from 1953 to 2012.  _Journal of research in medical sciences: the official journal of Isfahan University of Medical Sciences_ , 18(Suppl 1): S81.
  * Townsend (2008) Townsend, C. A. 2008.  Selegiline transdermal patch (Emsam) for major depressive disorder.  _American Family Physician_ , 77(4): 505.
  * Trivedi et al. (2006) Trivedi, M. H.; Rush, A. J.; Wisniewski, S. R.; Nierenberg, A. A.; Warden, D.; Ritz, L.; Norquist, G.; Howland, R. H.; Lebowitz, B.; McGrath, P. J.; et al. 2006\.  Evaluation of outcomes with citalopram for depression using measurement-based care in STAR* D: implications for clinical practice.  _American journal of Psychiatry_ , 163(1): 28–40.
  * Waugh and Goa (2003) Waugh, J.; and Goa, K. L. 2003.  Escitalopram: a review of its use in the management of major depressive and anxiety disorders.  _CNS drugs_ , 17: 343–362.
  * Wilson and Tripp (2019) Wilson, M.; and Tripp, J. 2019.  Clomipramine.
  * Yoon et al. (2022) Yoon, J.; Kang, C.; Kim, S.; and Han, J. 2022.  D-VLOG: Multimodal Vlog dataset for Depression Detection.  _Proceedings of the AAAI Conference on Artificial Intelligence_ , 36(11): 12226–12234.
  * Zhao et al. (2012) Zhao, T.; Park, T.-W.; Yang, J.-C.; Huang, G.-B.; Kim, M.-G.; Lee, K.-H.; and Chung, Y.-C. 2012.  Efficacy and safety of ziprasidone in the treatment of first-episode psychosis: an 8-week, open-label, multicenter trial.  _International clinical psychopharmacology_ , 27(4): 184–190.

Drug Name |  Medication Onset Time |  Variance in Patient Response |  Treated Condition
---|---|---|---
Clomipramine |  6-12 weeks Wilson and Tripp (2019) |  Low remission rate (20%); Wilson and Tripp (2019) |  Obsessive compulsive disorder Wilson and Tripp (2019)
Imipramine |  4-5 weeks Abdul-Baki et al. (2009) |  80.6% patients after 12-week treatment, compared to 48.0% in the placebo Abdul-Baki et al. (2009) |  Depression, anxiety Abdul-Baki et al. (2009)
Selegiline |  1-2 weeks Townsend (2008) |  33-40% Townsend (2008) |  Parkinson’s disease, depression Moore and Saadabadi (2022)
Blonanserin |  4-6 weeks Tenjin et al. (2013) |  52-87% Tenjin et al. (2013) |  Schizophrenia, mania Tenjin et al. (2013)
Venlafaxine |  4-6 weeks Dean (2020) |  67.2% Gibbons et al. (2012) |  Depression, anxiety, panic disorders Dean (2020)
Sertindole |  4-6 weeks Cincotta and Rodefer (2010) |  85% Cincotta and Rodefer (2010) |  Schizophrenia Cincotta and Rodefer (2010)
Carbamazepine |  First few days Maan, Duong, and Saadabadi (2018) |  75-85% Tolou-Ghamari et al. (2013) |  Bipolar disorder, seizures Maan, Duong, and Saadabadi (2018)
Pregabalin |  3 days Pande et al. (2003) |  40% Strawn et al. (2018) |  Anxiety Strawn et al. (2018)
Lithium |  1-3 weeks Chokhawala, Lee, and Saadabadi (2022) |  33% Hui et al. (2019) |  Bipolar disorder Chokhawala, Lee, and Saadabadi (2022)
Ziprasidone |  1-2 weeks Ionescu et al. (2016) |  51.85% Zhao et al. (2012) |  Bipolar disorder and associated depression Ionescu et al. (2016)
Risperidone |  4 weeks Pajonk (2004) |  63.4% (4 mg), 65.8% (8 mg) Peuskens (1995) |  Treats irritability associated with autism and schizophrenia Peuskens (1995)
Gabapentin |  1-4 weeks Allen, Sabir, and Sharma (2018) |  17.2% - 37.6% (600 mg-1800mg dosage) Panebianco et al. (2021) |  Epilepsy Allen, Sabir, and Sharma (2018)
Mirtazapine |  1 week Lavergne et al. (2005) |  67.1% Song et al. (2015) |  Depression, anxiety, obsessive compulsive disorder Lavergne et al. (2005)
Topiramate |  2-4 weeks (epilepsy), 3 months (migraines) Fariba and Saadabadi (2020) |  88% Liu et al. (2020) |  Epilepsy, migraines Fariba and Saadabadi (2020)
Sertraline |  Within 6 weeks Lewis et al. (2019) |  59% Flament et al. (1999) |  Depression, PTSD, OCD, panic disorder, social anxiety disorder Singh and Saadabadi (2019)
Citalopram |  1-2 weeks to start working, 4-6 weeks for full benefit Trivedi et al. (2006) |  47% Trivedi et al. (2006) |  Depression, social anxiety disorder, PTSD Trivedi et al. (2006)
Duloxetine |  2-4 weeks Jia et al. (2022) |  77% Jia et al. (2022) |  Depression, anxiety Jia et al. (2022)
Tranylcypromine |  1-2 weeks to start working, 6-8 weeks for full improvement Parikh, Yilanli, and Saadabadi (2017) |  60%-80.7% Heijnen et al. (2015) |  Major depressive episodes Parikh, Yilanli, and Saadabadi (2017)
Escitalopram |  1-2 weeks to start working, 6-8 weeks for full improvement Waugh and Goa (2003) |  30.2% after 2 weeks of treatment and 75.8% after 8 weeks of treatment Jiang et al. (2017) |  Depression, generalized anxiety disorder (GAD), obsessive compulsive disorder (OCD) and panic attacks Waugh and Goa (2003)
Quetiapine |  1-2 weeks to start working, 2-3 months for full improvement Cookson et al. (2007) |  58.2% for quetiapine 600 mg/day and 57.6% for quetiapine 300 mg/day after 8 weeks of treatment Cookson et al. (2007) |  Schizophrenia, manic, psychotic and depressive episodes Cookson et al. (2007)
Paliperidone |  2-8 weeks for full improvement Lee et al. (2011) |  33.6% after 2 weeks Lee et al. (2011) |  Psychotic disorders including schizophrenia Lee et al. (2011)
Fluoxetine |  4-5 weeks Kumar and Balan (2007) |  41% Kumar and Balan (2007) |  OCD, certain eating disorders, panic attacks Kumar and Balan (2007)
Table 1: Summary of medication and variation in patient response for
depression and related mental health conditions.