††thanks: Corresponding author

# Investigation of the strange pentaquark candidate $P_{\psi
s}^{\Lambda}(4338)^{0}$ recently observed by LHCb

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  School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box
19395-5531, Tehran, Iran  Y. Sarac<EMAIL_ADDRESS>Electrical and
Electronics Engineering Department, Atilim University, 06836 Ankara, Turkey
H. Sundu<EMAIL_ADDRESS>Department of Physics
Engineering, Istanbul Medeniyet University, 34700 Istanbul, Turkey

###### Abstract

The recently observed strange pentaquark candidate, $P_{\psi
s}^{\Lambda}(4338){}^{0}$, is investigated to provide information about its
nature and substructure. To this end, its mass and width through the decay
channels $P_{\psi s}^{\Lambda}(4338){}^{0}\rightarrow J/\psi\Lambda$ and
$P_{\psi s}^{\Lambda}(4338){}^{0}\rightarrow\eta_{c}\Lambda$ are calculated by
applying two- and three-point QCD sum rules, respectively. The state is
considered as a $\Xi_{c}\bar{D}$ meson-baryon molecular structure with spin-
parity quantum numbers $J^{P}=\frac{1}{2}^{-}$. The obtained mass, $m_{P_{\psi
s}^{\Lambda}(4338){}^{0}}=4338\pm 130~{}\mathrm{MeV}$, and width,
$\Gamma_{P_{\psi s}^{\Lambda}(4338){}^{0}}=10.40\pm 1.93~{}\mathrm{MeV}$, are
consistent with the experimental data within the presented uncertainties. This
allows us to assign a $\Xi_{c}\bar{D}$ molecular structure of
$J^{P}=\frac{1}{2}^{-}$ for the $P_{\psi s}^{\Lambda}(4338){}^{0}$ state.

##  I Introduction

Exotic states such as pentaquarks and tetraquarks have become one of the focus
of investigations in particle physics since the proposal of the quark model
Gell-Mann . Because their existence was not prohibited either by the quark
model or QCD, they attracted attention from the beginning and were
investigated extensively for a long time. Finally, expectations eventuated and
the announcement of the first observation of such states was made in 2003 for
a tetraquark state, $X(3872)$, by the Belle Collaboration Choi:2003ue . Later,
the confirmation of this state came from various collaborations CDF:2003cab ;
BaBar:2004oro ; D0:2004zmu ; CDF:2009nxk ; LHCb:2011zzp ; CMS:2013fpt . In
2015 a different member of the exotic states, namely the pentaquark state,
containing five valance quarks was announced to be observed by the LHCb
Collaboration Aaij:2015tga . The two states, $P_{c}(4380)$ and $P_{c}(4450)$,
were observed in the $J/\psi+p$ decay channel Aaij:2015tga  and later, in
2019, the analyses with a larger data sample revealed that the previously
announced $P_{c}(4450)$ state split into two states, $P_{c}(4440)$ and
$P_{c}(4454)$, and another pick, $P_{c}(4312)^{+}$, also came into sight
Aaij:2019vzc . The reported resonance parameters for these states were as
follows Aaij:2015tga ; Aaij:2019vzc : $m_{P_{c}(4380)^{+}}=4380\pm 8\pm
29~{}\mbox{MeV}$, $\Gamma_{P_{c}(4380)^{+}}=205\pm 18\pm 86~{}\mbox{MeV}$,
$m_{P_{c}(4440)^{+}}=4440.3\pm 1.3^{+4.1}_{-4.7}~{}\mbox{MeV}$,
$\Gamma_{P_{c}(4440)^{+}}=20.6\pm 4.9^{+8.7}_{-10.1}~{}\mbox{MeV}$,
$m_{P_{c}(4457)^{+}}=4457.3\pm 0.6^{+4.1}_{-1.7}~{}\mbox{MeV}$,
$\Gamma_{P_{c}(4457)^{+}}=6.4\pm 2.0^{+5.7}_{-1.9}~{}\mbox{MeV}$,
$m_{P_{c}(4312)^{+}}=4311.9\pm 0.7^{+6.8}_{-0.6}~{}\mbox{MeV}$ and
$\Gamma_{P_{c}(4312)^{+}}=9.8\pm 2.7^{+3.7}_{-4.5}~{}\mbox{MeV}$. In 2021 and
2022 there occurred two more pentaquark states’ reports which possess a
strange quark. These two states $P_{cs}(4459)$ LHCb:2020jpq  and $P_{c}(4337)$
LHCb:2021chn  were reported to have the following masses and widths:
$m_{P_{cs}(4459)^{0}}=4458.8\pm 2.9^{+4.7}_{-1.1}~{}\mathrm{MeV}$,
$\Gamma_{P_{c}(4459)^{0}}=17.3\pm 6.5^{+8.0}_{-5.7}~{}\mathrm{MeV}$ and
$m_{P_{cs}(4337)^{+}}=4337^{+7}_{-4}{}^{+2}_{-2}~{}\mathrm{MeV}$,
$\Gamma_{P_{c}(4337)^{+}}=29^{+26}_{-12}{}^{+14}_{-14}~{}\mathrm{MeV}$.

The experimental observations of these non-conventional states have increased
the theoretical interest in these states and triggered extensive theoretical
investigations over their identifications and various properties. Their
substructures were still obscure, which has motivated many affords to explain
this point by assigning them either being molecules or compact states. In
Refs. Wang:2015epa ; Maiani:2015vwa ; Anisovich:2015zqa ; Li:2015gta ;
Lebed:2015tna ; Anisovich:2015cia ; Wang:2015ava ; Wang:2015ixb ;
Ghosh:2015ksa ; Wang:2015wsa ; Wang:2016dzu ; Zhang:2017mmw ;
Giannuzzi:2019esi ; Wang:2019got ; Wang:2020rdh ; Ali:2020vee ; Azizi:2021utt
; Wang:2020eep ; Azizi:2021pbh ; Ozdem:2021ugy ; Zhu:2015bba ; Gao:2021hmv
the pentaquark states were investigated by taking their substructure as
diquark-diquark-antiquark or diquark-triquark forms. Owing to their proximity
to the relevant meson baryon threshold and small widths, the molecular
structure has been another commonly considered structure for the pentaquark
states. With molecular structure assumption, the properties of these states,
such as their mass spectrum and various interactions, were investigated with
the application of different approaches including the contact-range effective
field theory Liu:2019tjn ; Liu:2020hcv ; Peng:2020gwk ;
PavonValderrama:2019nbk , the effective Lagrangian approach Cheng:2021gca ;
Ling:2021lmq ; Xiao:2019mvs ; Lu:2016nnt ; Yang:2021pio , the QCD sum rule
method Chen:2016otp ; Azizi:2016dhy ; Azizi:2018bdv ; Azizi:2020ogm ;
Wang:2022neq ; Wang:2022gfb ; Wang:2022ltr ; Wang:2021itn ; Wang:2019hyc ;
Xu:2019zme , one-boson exchange potential model Wang:2021hql ; Chen:2022onm ;
Yalikun:2021dpk ; Yalikun:2021bfm ; Chen:2021kad ; Wang:2021hql ; Pan:2020xek
; Liu:2019zvb ; Wang:2019nwt ; Chen:2019asm ; Wang:2019aoc ; Chen:2016ryt ;
Chen:2016heh  and quasipotential Bethe-Salpeter He:2019ify ; He:2019rva ;
Zhu:2021lhd ; Wang:2022mxy . Besides, one can find other works in
Refs.Chen:2015loa ; He:2015cea ; Chen:2015moa ; Roca:2015dva ;
Meissner:2015mza ; Xiao:2019gjd ; Wang:2019nvm ; Xiao:2019gjd ; Chen:2020opr ;
Wu:2021caw ; Yan:2021nio ; Chen:2020uif ; Du:2021fmf ; Phumphan:2021tta ;
Lu:2021irg ; Chen:2021obo ; Peng:2022iez ; Yang:2022ezl ; Zhu:2022wpi ;
Wang:2022ztm ; Giachino:2022pws ; Ortega:2022uyu ; Li:2023zag ; Dong:2020nwk ;
Voloshin:2019aut ; Peng:2020hql ; Chen:2020kco ; Chen:2021tip ; Xiao:2021rgp ;
Du:2021bgb ; Hu:2021nvs ; Feijoo:2022rxf ; Yalikun:2023waw ; Nakamura:2022jpd
and the references therein adopting the molecular interpretation for
pentaquark states. They were also investigated with the possibility that they
were arising from kinematical effects Guo1 ; Mikhasenko:2015vca ; Liu1 ;
Bayar:2016ftu ; Guo2 ; Nakamura:2021qvy . Though there exist so many works
over them, they were in need of many more to clarify or support their still
uncertain properties. On the other hand, the possible pentaquark states other
than the observed ones and possessing strange, bottom or charm quarks were
also quested for with their expectation to be observed in the future
Liu:2022qns ; Stancu:2022xax ; Azizi:2022qll ; Wang:2022ugk ; Valera:2022elt ;
Giachino:2022pws ; Hu:2022qlr ; Wang:2022tib ; Yan:2022wuz ; Wang:2021hql ;
Wang:2015wsa ; Azizi:2021pbh ; Wang:2022neq ; Shi:2021wyt ; Deng:2022vkv ;
Wang:2020bjt ; Azizi:2018dva ; Meng:2019fan ; Ozdem:2022iqk ; Liu:2021ixf ;
Feijoo:2015kts ; Chen:2015sxa ; Cheng:2015cca ; Xing:2021yid ; Azizi:2017bgs ;
Huang:2021ave ; Zhu:2020vto ; Ferretti:2018ojb ; Shimizu:2016rrd ; Liu:2017xzo
; Liu:2021efc ; Liu:2021tpq ; Yang:2022bfu ; Xie:2020ckr ; Zhang:2020dwp ;
Paryev:2020jkp ; Xie:2020niw ; Cao:2019gqo ; Wang:2019zaw ; Wang:2019ato ;
Huang:2018wed ; Yang:2018oqd ; Yamaguchi:2017zmn ; Gutsche:2019mkg ;
Zhang:2020cdi ; Ferretti:2020ewe ; Wang:2021xao ; An:2020jix ; Yan:2021glh ;
Zhang:2020vpz .

Among these pentaquark states, the present work focuses on the one which was
observed very recently by the LHCb collaboration LHCb:2022jad  in the
amplitude analyses of $B^{-}\rightarrow J/\psi\bar{p}$ decay. The measured
mass and width for the state, which was labeled as $P_{\psi
s}^{\Lambda}(4338){}^{0}$, were reported as $m_{P_{\psi
s}^{\Lambda}}=4338.2\pm 0.7\pm 0.4~{}\mathrm{MeV}$ and $\Gamma_{P_{\psi
s}^{\Lambda}}=7.0\pm 1.2\pm 1.3~{}\mathrm{MeV}$, respectively with the
preferred spin-parity quantum numbers $J^{P}=\frac{1}{2}^{-}$. Having a mass
and narrow width in consistency with meson-baryon molecular interpretation
this structural form is adopted in Refs. Ortega:2022uyu ; Zhu:2022wpi . In
Ref. Zhu:2022wpi , a coupled-channel calculation was applied considering
molecular states and the results obtained for $\Xi_{c}\bar{D}$ interaction
indicated a wider peak than the observed one in the experiment. With the
constituent quark model formalism $P_{\psi s}^{\Lambda}(4338){}^{0}$ was
suggested to be a baryon-meson molecule state with
$(I)J^{P}=(0)\frac{1}{2}^{-}$ and mass and width $m_{P_{\psi
s}^{\Lambda}(4338){}^{0}}=4318.1~{}\mathrm{MeV}$ and $\Gamma_{P_{\psi
s}^{\Lambda}(4338){}^{0}}=0.07~{}\mathrm{MeV}$, respectively Ortega:2022uyu .
In the Ref. Ozdem:2022kei , the light cone QCD sum rules method is implemented
to calculate the magnetic moments of the $P_{\psi s}^{\Lambda}(4338){}^{0}$
and $P_{\psi s}^{\Lambda}(4459){}^{0}$ states. The $P_{\psi
s}^{\Lambda}(4338){}^{0}$ state and the other candidate pentaquark states were
investigated in Ref. Wang:2022neq  using QCD sum rules method and adopting the
molecular structure, and the analyses were in favor of the $P_{\psi
s}^{\Lambda}(4338){}^{0}$ state having a $\Xi_{c}\bar{D}$ molecular structure
with spin-parity and isospin quantum numbers $J^{P}=\frac{1}{2}^{-}$ and
$(I,I_{3})=(0,0)$, respectively. The references Karliner:2022erb ; Yan:2022wuz
; Meng:2022wgl ; Chen:2022wkh  also investigated the $P_{\psi
s}^{\Lambda}(4338){}^{0}$ state in association to the molecular form.

As already mentioned, there exist many studies devoted to describing the
nature of the pentaquark states. These studies, performed with various
approaches covering different structures for the pentaquark states, gave
results consistent with the experimentally observed parameters. This fact
makes the subject more intriguing and open to new investigations. Therefore it
is necessary to provide further information to support or check the proposed
alternative structures for a better identification of their obscure sub-
structure. Moreover, the works over these exotic states both test our
knowledge and provide support for the improvement of our understanding of the
QCD in its non-perturbative regime. With these motivations, in the present
work, we investigate two dominant strong decays of the recently observed
pentaquark state $P_{\psi s}^{\Lambda}(4338){}^{0}$ into $J/\psi\Lambda$ and
$\eta_{c}\Lambda$ states, sticking to a meson-baryon molecular interpretation.
To this end, we apply the QCD sum rule method, which put forward its success
with plenty of predictions consistent with the experimental observations
Shifman:1978bx ; Shifman:1978by ; Ioffe81 . The interpolating field for the
state is chosen in the $\Xi_{c}\bar{D}$ molecular form. For completeness,
firstly, we obtain the mass of the state and current coupling constant using
the considered interpolating current, which are subsequently to be used as
inputs in strong coupling constant analyses.

The rest of the paper has the following organization. In Sec. II the QCD sum
rule for the mass of the considered state is presented with the numerical
calculation of the corresponding results for the mass and current coupling
constant. Sec. III contains the details of the QCD sum rules to calculate the
strong coupling constants for the $P_{\psi s}^{\Lambda}(4338){}^{0}\rightarrow
J/\psi\Lambda$ decay and their numerical analyses as well. Sec. IV presents
the similar QCD sum rule calculation and the analyses for the strong coupling
constants and the width corresponding to the $P_{\psi
s}^{\Lambda}(4338){}^{0}\rightarrow\eta_{c}\Lambda$ channel. Last section
gives a short discussion and conclusion.

##  II QCD sum rule for the mass of $P_{\psi s}^{\Lambda}(4338){}^{0}$ state

To better understand the substructure of the pentaquark states, one way is the
comparison of the observed properties of these particles with the related
theoretical findings. One of the important observables is the mass of these
states. Beside the mass, the current coupling constant is also a very
important input that is needed to calculate the observables related to the
decays of the particles like their width. The present section gives the
details of QCD sum rules calculations for the mass and current coupling of the
strange pentaquark candidate $P_{\psi s}^{\Lambda}(4338){}^{0}$. The
calculations start with the following two-point correlation function:

$\Pi(q)=i\int d^{4}xe^{iq\cdot x}\langle
0|\mathcal{T}\\{J_{P_{cs}}(x)\bar{J}_{P_{cs}}(0)\\}|0\rangle.$ (1)

In this equation, $\mathcal{T}$ is the time ordering operator and $J_{P_{cs}}$
represents the interpolating current for the $P_{\psi
s}^{\Lambda}(4338){}^{0}$ pentaquark state, which is denoted as $P_{cs}$ in
what follows. The current to interpolate this state is the $\Xi_{c}\bar{D}$
molecular type with spin-parity $J^{P}=\frac{1}{2}^{-}$ :

$\displaystyle J_{P_{cs}}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}d^{T}_{a}C\gamma_{5}s_{b}c_{c}\bar{c}_{d}i\gamma_{5}u_{d},$
(2)

where $C$ represents the charge conjugation operator, subindices
$a,~{}b,~{}c,~{}d$ are used to represent the color indices, and
$u,~{}d,~{}s,~{}c$ are the quark fields. To proceed in the calculations, one
follows two separate paths resulting in two corresponding expressions
containing the hadronic parameters on one side and QCD fundamental parameters
on the other side. They are therefore called as the hadronic and QCD sides,
respectively. The physical parameter under quest is obtained via a match of
these two sides by means of a dispersion relation. Both sides contain various
Lorentz structures and the matching is carried out considering the same
structures obtained in these representations. The Borel transformation and
continuum subtraction are the final operations applied on both sides to
suppress the contributions of higher states and continuum.

For the computation of the hadronic side, a complete set of the intermediate
states with same quark content and carrying the same quantum numbers of the
considered state is inserted inside the correlator. Treating the interpolating
currents as annihilation or creation operators, and performing the integration
over four-$x$ the correlator becomes

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

where the contributions coming from higher states and continuum are
represented by $\cdots$, and one particle pentaquark state with momentum $q$
and spin $s$ is represented by $|P_{cs}(q,s)\rangle$. To proceed, we need the
following matrix element:

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

given in terms of the Dirac spinor $u_{P_{cs}}(q,s)$ and current coupling
constant $\lambda_{P_{cs}}$. Substituting Eq. (4) into Eq. (3) and applying
the summation over spin

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

the result for the hadronic side is achieved as

$\displaystyle\Pi^{\mathrm{Had}}(q)=\frac{\lambda_{P_{cs}}^{2}(\not\\!q+m_{P_{cs}})}{m_{P_{cs}}^{2}-q^{2}}+\cdots,$
(6)

which turns into following final form after the Borel transformation:

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

where $\tilde{\Pi}^{\mathrm{Had}}(q)$ denotes the Borel transformed form of
the correlator and $M^{2}$ is the Borel mass parameter.

The QCD side of the calculations requires the usage of the interpolating field
explicitly in the correlator, Eq. (1). This is followed by the possible
contractions of the quark fields via Wick’s theorem, which turns the result
into the one containing quark propagators as

$\displaystyle\Pi^{\mathrm{QCD}}(q)$ $\displaystyle=$ $\displaystyle-i\int
d^{4}xe^{iqx}\epsilon_{abc}\epsilon_{a^{\prime}b^{\prime}c^{\prime}}\Big{\\{}\mathrm{Tr}[S_{s}^{bb^{\prime}}(x)\gamma_{5}CS_{d}^{T}{}^{aa^{\prime}}(x)C\gamma_{5}]\mathrm{Tr}[S_{u}^{dd^{\prime}}(x)\gamma_{5}S_{c}^{d^{\prime}d}(-x)\gamma_{5}]\Big{\\}}S_{c}^{cc^{\prime}}(x).$
(8)

The light and heavy quark propagators necessary for further calculations have
the following explicit forms Yang:1993bp ; Reinders:1984sr :

$\displaystyle S_{q,}{}_{ab}(x)$ $\displaystyle=$ $\displaystyle
i\delta_{ab}\frac{\not{x}}{2\pi^{2}x^{4}}-\delta_{ab}\frac{m_{q}}{4\pi^{2}x^{2}}-\delta_{ab}\frac{\langle\overline{q}q\rangle}{12}+i\delta_{ab}\frac{\not{x}m_{q}\langle\overline{q}q\rangle}{48}-\delta_{ab}\frac{x^{2}}{192}\langle\overline{q}g_{\mathrm{s}}\sigma
Gq\rangle+i\delta_{ab}\frac{x^{2}\not{x}m_{q}}{1152}\langle\overline{q}g_{\mathrm{s}}\sigma
Gq\rangle$ (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

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

with $G^{\alpha\beta}_{ab}=G^{\alpha\beta}_{A}t_{ab}^{A}$,
$GG=G_{A}^{\alpha\beta}G_{A}^{\alpha\beta}$; $a,~{}b=1,~{}2,~{}3$;
$A=1,~{}2,\cdots,8$ and $t^{A}=\frac{\lambda^{A}}{2}$ where $\lambda^{A}$ are
the Gell-Mann matrices. The propagator for $u$, $d$ or $s$ quark is
represented by the sub-index $q$. The final results for this side are obtained
after the Fourier and Borel transformations as

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

where $s_{0}$ is the threshold parameter entering the calculations after the
continuum subtraction application using the quark hadron duality assumption.
$\rho_{i}(s)$ represents the spectral densities which are the imaginary parts
of the results obtained as $\frac{1}{\pi}\mathrm{Im}\Pi_{i}^{\mathrm{QCD}}$
with $i$ corresponding to either the result obtained from the coefficient of
the Lorentz structure $\not\\!q$ or $I$. The results of such calculations
contain long expressions and, to avoid giving overwhelming expressions in the
text, the explicit results of spectral densities will not be presented here.
The quantities that we seek in this section, namely mass and the current
coupling constant of the pentaquark state, are obtained by the match of the
coefficients of the same Lorentz structures obtained in both the hadronic and
QCD sides. These matches are represented as

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

and

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

The next step is the analysis of the obtained results, for which one may apply
any of the present structures. To this end, we choose the $I$ structure. The
input parameters needed in the calculation of the mass and current coupling
constant are given in Table 1, which are also used for the coupling constant
calculations to be given in the next section.

Parameters | Values
---|---
$m_{c}$ |  $1.27\pm 0.02~{}\mathrm{GeV}$ 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
$m_{\eta_{c}}$ |  $(2983.9\pm 0.4)~{}\mathrm{MeV}$ Zyla:2020zbs
$m_{\Lambda}$ |  $(1115.683\pm 0.006)~{}\mathrm{MeV}$ Zyla:2020zbs
$\lambda_{\Lambda}$ |  $(0.013\pm 0.02)~{}\mathrm{GeV}^{3}$ Aliev:2002ra
$f_{J/\psi}$ |  $(481\pm 36)~{}\mathrm{MeV}$ Veliev:2011kq
$f_{\eta_{c}}$ |  $(320\pm 40)~{}\mathrm{MeV}$ Colangelo:1992cx
Table 1: Some input parameters used in the analyses of mass, current coupling
constants and coupling constant of the $P_{cs}\rightarrow J/\psi\Lambda$
decay.

In addition to the given input parameters, there are two auxiliary parameters
needed in the analyses: the Borel parameter $M^{2}$ and the continuum
threshold $s_{0}$. Following the standard criteria of the QCD sum rule method,
their suitable intervals are fixed. These criteria include a relatively slight
variation of the results with the change of these auxiliary parameters, the
dominant contribution of the focused state compared to the higher states and
continuum, and the convergence of the operator product expansion (OPE) used in
the QCD side’s calculation. Sticking to these criteria, we establish working
regions of these parameters from the analyses. Seeking a region, for which the
higher-order terms on OPE side contribute less compared to the lowest ones,
and the ground state dominates over the higher ones, the working interval of
the Borel parameter is determined as:

$\displaystyle 3.0~{}\mbox{GeV}^{2}\leq M^{2}\leq 4.0~{}\mbox{GeV}^{2}.$ (14)

The determination of the continuum threshold interval has a connection to the
energy of the possible excited states of the considered pentaquark state. With
this issue in mind, we fix its interval as

$\displaystyle 23~{}\mbox{GeV}^{2}\leq s_{0}\leq 25~{}\mbox{GeV}^{2}.$ (15)

By using all the inputs as well as the working windows of the auxiliary
parameters, we depict the variation of the mass with respect to the auxiliary
parameters for the considered structure in Fig. 1.

Figure 1: Left: Variation of the the mass as function of $M^{2}$ at different
values of threshold parameter $s_{0}$. Right: Variation of the the mass as
function of $s_{0}$ at different values of threshold parameter $M^{2}$.

This figure shows the mild dependence of the mass on the variations of the
auxiliary parameters in their working windows. The residual dependence appear
as the uncertainties in the results.

The resultant values for the mass and the current coupling constant are:

$\displaystyle m_{P_{cs}}=4338\pm
130~{}\mathrm{MeV},~{}~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}~{}\lambda_{P_{cs}}=(7.24\pm
0.21)\times 10^{-4}~{}\mathrm{GeV}^{6}.$ (16)

The result obtained for the mass has a good consistency with the observed mass
of $P_{\psi s}^{\Lambda}(4338){}^{0}$ state announced as $m_{P_{\psi
s}^{\Lambda}}=4338.2\pm 0.7\pm 0.4~{}\mathrm{MeV}$ LHCb:2022jad .

As is mentioned, the results obtained in this section are necessary inputs for
the next sections which are devoted to the strong decays of the considered
pentaquark state, namely $P_{\psi s}^{\Lambda}(4338){}^{0}\rightarrow
J/\psi\Lambda$ and $P_{\psi
s}^{\Lambda}(4338){}^{0}\rightarrow\eta_{c}\Lambda$.

##  III QCD sum rule to analyze the $P_{\psi
s}^{\Lambda}(4338){}^{0}\rightarrow J/\psi\Lambda$ decay

The bare mass investigations of pentaquark states present in the literature,
performed to explain the properties of the newly observed states, indicated
that different assumptions for the substructures of these states might give
consistent mass predictions with the observed ones. These necessitate deeper
investigations which serve as support for previous findings. With this
motivation, to clarify more the substructure and the quantum numbers of the
observed $P_{\psi s}^{\Lambda}(4338){}^{0}$ state, in this section, we
investigate the $P_{\psi s}^{\Lambda}(4338){}^{0}\rightarrow J/\psi\Lambda$
decay and calculate its width. To this end, the main ingredients are the
strong coupling constants entering the low energy amplitude of the decay. To
calculate these coupling constants via the QCD sum rule method, we use the
following three-point correlation function:

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

with the interpolating currents given in Eq. (2) and

$\displaystyle J^{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\epsilon^{lmn}\sum_{i=1}^{2}\Big{[}2(u^{T}_{l}CA_{1}^{i}d_{m})A_{2}^{i}s_{n}+(u^{T}_{l}CA_{1}^{i}s_{m})A_{2}^{i}d_{n}+(d^{T}_{n}CA_{1}^{i}s_{m})A_{2}^{i}u_{l}\Big{]},$
$\displaystyle J_{\mu}^{J/\psi}$ $\displaystyle=$
$\displaystyle\bar{c}_{l}\gamma_{\mu}c_{l}.$ (18)

In Eq. (18) sub-indices, $l,~{}m,~{}n$, are used to represent the color
indices, and $u,~{}s,~{}c$ stand for quark fields, $A_{1}^{1}=I$,
$A_{1}^{2}=A_{2}^{1}=\gamma_{5}$, $A_{2}^{2}=\beta$ which is a mixing
parameter, and $C$ represents the charge conjugation operator. Similar steps
of the calculation followed in the previous section also apply here.
Calculation of hadronic and QCD sides are followed by their proper matches
considering the coefficients of the same Lorentz structures from both sides.

For the hadronic side, we insert complete sets of hadronic states that have
the same quantum numbers with the interpolating fields. Taking the four
integral results in

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

with $\cdots$ denoting the contribution of the higher states and continuum;
and $p$, $p^{\prime}$ and $q$ being the respective momenta of the $P_{cs}$,
$\Lambda$ and $J/\psi$ states. The required matrix elements for calculations
have the following forms:

$\displaystyle\langle 0|J^{P_{cs}}|P_{cs}(p,s)\rangle$ $\displaystyle=$
$\displaystyle\lambda_{P_{cs}}u_{P_{cs}}(p,s),$ $\displaystyle\langle
0|J^{\Lambda}|\Lambda(p^{\prime},s^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\lambda_{\Lambda}u_{\Lambda}(p^{\prime},s^{\prime}),$
$\displaystyle\langle 0|J_{\mu}^{J/\psi}|J/\psi(q)\rangle$ $\displaystyle=$
$\displaystyle f_{J/\psi}m_{J/\psi}\varepsilon_{\mu},$ (20)

where $\varepsilon_{\mu}$ and $f_{J/\psi}$ represent the polarization vector
and the decay constant of the $J/\psi$ state ; and $\lambda_{P_{cs}}$ and
$\lambda_{\Lambda}$ are the current coupling constants of the $P_{cs}$ and
$\Lambda$ states, respectively. $|P_{cs}(p,s)\rangle$ corresponds to the one-
particle pentaquark state with its spinor $u_{P_{cs}}$ and $u_{\Lambda}$ is
the spinor of $\Lambda$ state. The matrix element,$\langle
J/\psi(q)\Lambda(p^{\prime},s^{\prime})|P_{cs}(p,s)\rangle$ is given in terms
of the considered strong coupling constants, $g_{1}$ and $g_{2}$, as

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

Substituting the matrix elements, Eq. (20) and Eq. (21), into the Eq. (19)
using the summation over the spins of the spinors and polarization vector
given as

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

the hadronic side is achieved as

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

Among the present Lorentz structures the ones used in the analyses are given
explicitly, and the remaining ones are represented by
$\mathrm{other~{}structures}$. Here $Q^{2}=-q^{2}$. The Borel parameters
$M^{2}$ and $M^{\prime 2}$, present in the last result, are determined from
the analyses following similar criteria given in the previous section.

As for the QCD side, the insertion of the interpolating currents given in Eqs.
(2) and (18) inside the correlator in Eq. (17), and after the possible
contractions of quark fields using the Wick’s theorem, the result takes the
following form in terms of the quark propagators:

$\displaystyle\Pi_{\mu}^{\mathrm{QCD}}(p,q)$ $\displaystyle=$ $\displaystyle
i^{2}\int d^{4}xe^{-ip\cdot x}\int d^{4}ye^{ip^{\prime}\cdot
y}\frac{1}{\sqrt{6}}\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}\sum_{i=1}^{2}\Big{[}\mathrm{Tr}[S_{s}^{bb^{\prime}}(y-x)\gamma_{5}CS_{d}^{T}{}^{ca^{\prime}}(y-x)CA_{1}^{i}]A_{2}^{i}S_{u}^{ad^{\prime}}(y-x)$
(24) $\displaystyle\times$
$\displaystyle\gamma_{5}S_{c}^{d^{\prime}l}(x)\gamma_{\mu}S_{c}^{lc^{\prime}}(-x)-2A_{2}^{i}S_{s}^{cb^{\prime}}(y-x)\gamma_{5}CS_{d}^{T}{}^{ba^{\prime}}(y-x)CA_{1}^{i}S_{u}^{ad^{\prime}}(y-x)\gamma_{5}S_{c}^{d^{\prime}l}(x)\gamma_{\mu}S_{c}^{lc^{\prime}}(-x)$
$\displaystyle-$ $\displaystyle
A_{2}^{i}S_{d}^{ca^{\prime}}(y-x)\gamma_{5}CS_{s}^{T}{}^{bb^{\prime}}(y-x)CA_{1}^{i}S_{u}^{ad^{\prime}}(y-x)\gamma_{5}S_{c}^{d^{\prime}l}(x)\gamma_{\mu}S_{c}^{lc^{\prime}}(-x)\Big{]}.$

Considering the same Lorentz structures given explicitly in Eq. (23), we
obtain the QCD side and represent the lengthy results shortly as in the
following form:

$\displaystyle\Pi_{\mu}^{\mathrm{QCD}}(p,q)$ $\displaystyle=$
$\displaystyle\Pi_{1}\,\not\\!pp_{\mu}\gamma_{5}+\Pi_{2}\,\not\\!p\gamma_{\mu}\gamma_{5}+\mathrm{other\,\,\,structures}.$
(25)

To proceed in the calculations, the light and heavy quark propagators given in
Eqs. (9) and (10) are used explicitly and the four-dimensional Fourier
integrals are performed. The imaginary parts of the obtained results
constitute the spectral densities to be used in the following 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)

where $\rho_{i}(s,s^{\prime},q^{2})=\frac{1}{\pi}\mathrm{Im}[\Pi_{i}]$; and
$\rho_{i}^{\mathrm{pert}}(s,s^{\prime},q^{2})$ and $\rho_{i}^{\mathrm{non-
pert}}(s,s^{\prime},q^{2})$ represent the results of the perturbative and non-
perturbative parts, respectively, with $i=1,~{}2,..,12$ corresponding to all
the Lorentz structures existing in the results. The analyses in the present
work are performed via resulting matches of the hadronic and QCD sides
obtained from the structures $i=1,~{}2$, which results in two coupled sum
rules equations including both $g_{1}$ and $g_{2}$:

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

where the Borel transformations on the variables $-p^{\prime 2}$ and $-p^{2}$
have been performed, and $\tilde{\Pi}_{i}$ in the results represent the Borel
transformed $\Pi_{i}$ expressions obtained in the QCD side. Solution of these
equations for $g_{1}$ and $g_{2}$ give

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

The numerical analyses of the $g_{1}$ and $g_{2}$ given in the Eq. (29)
require the input parameters given in Table 1 and some additional auxiliary
parameters such as Borel parameters $M^{2}$, $M^{\prime 2}$ and threshold
parameters $s_{0}$ and $s^{\prime}_{0}$ and the mixing parameter $\beta$
present in the interpolating current of the $\Lambda$ state. The similar
standard criteria of the method used for the mass calculation in the previous
section, namely weak dependence on the auxiliary parameters, pole dominance,
and the convergence of the operator product expansion (OPE) used on the QCD
side, are applied in the determination of the auxiliary parameters of this
section as well. Taking into account their relations and considering the
possible excited resonances of the considered states, the threshold parameters
are fixed as:

$\displaystyle 23.0\,\,\mathrm{GeV}^{2}$ $\displaystyle\leq$ $\displaystyle
s_{0}\leq 25.0\,\,\mathrm{GeV}^{2},$ $\displaystyle 1.7\,\,\mathrm{GeV}^{2}$
$\displaystyle\leq$ $\displaystyle s^{\prime}_{0}\leq
2.3\,\,\mathrm{GeV}^{2},$ (30)

in which the interval of $s_{0}$ is the same as the one used in the previous
section. At this point, we shall note that the threshold parameter may be
expected to be the same, for instance, as in Ref. Azizi:2021utt , given the
close masses of the particles involved. However, the analyses performed on the
considered state with the given requirements in the previous section are
effective in the final determination of these parameters. In each distinct
study, it becomes necessary to reassess these requirements. Therefore, in the
analyses of the results, one needs to check these requirements in every
different work from scratch and obtain the a proper interval satisfying the
given requirements for identifying these parameters. Besides, as is seen in
Ref. Azizi:2021utt , the structures assigned for these two particles are
different. While the structure in Ref. Azizi:2021utt  was a diquark-diquark-
antiquark, the present work adopts a molecular one. As a result, the
discrepancy in threshold values arising from the analyses can also be
attributed to these inner quark structures assigned to the considered states.
For the Borel parameters, considering the pole dominance and convergence of
the OPE lead us to the following intervals:

$\displaystyle 3.0\,\,\mathrm{GeV}^{2}$ $\displaystyle\leq$ $\displaystyle
M^{2}\leq 4.0\,\,\mathrm{GeV}^{2},$ $\displaystyle 1.5\,\,\mathrm{GeV}^{2}$
$\displaystyle\leq$ $\displaystyle M^{\prime 2}\leq 2.5\,\,\mathrm{GeV}^{2}.$
(31)

$M^{2}$ again spans the same interval given in the previous section. The
working interval of the last auxiliary parameter, $\beta$, is determined from
a parametric plot of the results given as a function of $\cos\theta$ with
$\beta=\tan\theta$ in which the relatively stable regions are considered to
fix $\beta$ intervals. These analyses give the following intervals:

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

In all of these intervals for the auxiliary parameters, we expect weak
dependence of the results on these parameters. To depict this, we provide the
graphs of the strong coupling constant $g_{1}$ as functions of these auxiliary
parameters in Fig. 2 as examples: The criteria are satisfied, dependencies are
mild and the uncertainties remain inside the limits allowed by the method.

Figure 2: Left: Variation of the the coupling constant $g_{1}$ as function of
$M^{2}$ and $M^{\prime 2}$ at central values of threshold parameters $s_{0}$
and $s^{\prime}_{0}$ and at $Q^{2}=2.5~{}\mathrm{GeV}^{2}$. Right: Variation
of the the coupling constant $g_{1}$ as function of $s_{0}$ and
$s^{\prime}_{0}$ at central values of Borel parameters $M^{2}$ and $M^{\prime
2}$ and at $Q^{2}=2.5~{}\mathrm{GeV}^{2}$.

The analyses give the results reliable only for some regions of the $Q^{2}$,
and therefore to get the coupling constants’ values at
$Q^{2}=-m_{J/\psi}^{2}$, we need to expand the analyses to the region of
interest using a proper fit function given as

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

The fit parameters providing a good overlap with the results in the reliable
region of the QCD sum rule results and the values of the coupling constants
obtained from the fit functions at $Q^{2}=-m_{J/\psi}^{2}$ are presented in
Table 2.

Coupling constant | $g_{0}$ | $c_{1}$ | $c_{2}$ | $g_{i}(-m_{J/\psi}^{2})$
---|---|---|---|---
$g_{1}$ | $-1.10\pm 0.13$ | $6.43$ | $-26.13$ | $(-4.71\pm 0.52)\times 10^{-5}$
$g_{2}$ | $15.57\pm 1.86$ | $4.43$ | $-3.81$ | $0.61\pm 0.07$
Coupling constant | $g_{0}$ | $c_{1}$ | $c_{2}$ | $g(-m_{\eta_{c}}^{2})$
$g$ | $7.71\pm 0.85$ | $5.98$ | $-6.23$ | $0.11\pm 0.02$
Table 2:  Values of the fit parameters for the fit functions of coupling
constants, $g_{1}$, $g_{2}$ and $g$ and the coupling constant values at
$Q^{2}=-m_{J/\psi}^{2}$ and $Q^{2}=-m_{\eta_{c}}^{2}$.

The results contain the errors arising from the uncertainties inherited from
both the input parameters and the determinations of the intervals of the
auxiliary parameters.

The strong coupling constants determined from the QCD sum rules analyses are
applied for the width calculation of the decay $P_{cs}\rightarrow
J/\psi\Lambda$, which is performed via the relation

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

The function $f(x,y,z)$ in the width formula is given as

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

The result obtained for the width is

$\displaystyle\Gamma(P_{cs}\rightarrow J/\psi\Lambda)$ $\displaystyle=$
$\displaystyle\left(7.22\pm 1.78\right)~{}\mathrm{MeV},$ (36)

in a nice agreement with the experiment.

##  IV QCD sum rule to analyze the $P_{\psi
s}^{\Lambda}(4338){}^{0}\rightarrow\eta_{c}\Lambda$ decay

This section provides the investigation of another possible decay channel of
the $P_{\psi s}^{\Lambda}(4338)$ state, namely $P_{\psi
s}^{\Lambda}(4338){}^{0}\rightarrow\eta_{c}\Lambda$ decay, and the
corresponding width for this channel. For the calculation of the corresponding
strong coupling constant, the three-point correlation function is

$\Pi(p,q)=i^{2}\int d^{4}xe^{-ip\cdot x}\int d^{4}ye^{ip^{\prime}\cdot
y}\langle
0|\mathcal{T}\\{J^{\Lambda}(y)J^{\eta_{c}}(0)\bar{J}^{P_{cs}}(x)\\}|0\rangle.$
(37)

In addition to the interpolating currents, $J^{P_{cs}}$ and $J^{\Lambda}$
previously given, the interpolating current of $\eta_{c}$ is needed:

$\displaystyle J^{\eta_{c}}$ $\displaystyle=$
$\displaystyle\bar{c}_{l}\gamma_{5}c_{l}.$ (38)

We follow similar steps with the previous section to calculate the width of
the strong decay under study in this section. The hadronic side for this decay
is obtained via insertion of complete sets of hadronic states carrying the
same quantum numbers of the applied interpolating currents as

$\displaystyle\Pi^{\mathrm{Had}}(p,q)=\frac{\langle
0|J^{\Lambda}|\Lambda(p^{\prime},s^{\prime})\rangle\langle
0|J^{\eta_{c}}|\eta_{c}(q)\rangle\langle\eta_{c}(q)\Lambda(p^{\prime},s^{\prime})|P_{cs}(p,s)\rangle\langle
P_{cs}(p,s)|\bar{J}^{P_{cs}}|0\rangle}{(m_{\Lambda}^{2}-p^{\prime
2})(m_{\eta_{c}}^{2}-q^{2})(m_{P_{css}}^{2}-p^{2})}+\cdots,$ (39)

where $\cdots$ represent the contribution of the higher states and continuum.
$p$, $p^{\prime}$ and $q$ are the momenta of the $P_{cs}$, $\Lambda$ and
$\eta_{c}$ states, respectively. Besides the matrix elements given in Eq.
(20), we need the matrix element relevant to the $\eta_{c}$ state in terms of
the related decay constant $f_{\eta_{c}}$ which is given as

$\displaystyle\langle 0|J^{\eta_{c}}|\eta_{c}(q)\rangle$ $\displaystyle=$
$\displaystyle\frac{f_{\eta_{c}}m_{\eta_{c}}^{2}}{2m_{c}}.$ (40)

The matrix element,
$\langle\eta_{c}(q)\Lambda(p^{\prime},s^{\prime})|P_{cs}(p,s)\rangle$,
defining this transition is

$\displaystyle\langle\eta_{c}(q)\Lambda(p^{\prime},s^{\prime})|P_{cs}(p,s)\rangle=g\bar{u}_{\Lambda}(p^{\prime},s^{\prime})u_{P_{cs}}(p,s),$
(41)

where $g$ is the corresponding strong coupling constant. Using the summation
over the spins of the $P_{cs}$ and $\Lambda$ states given in the Eq (22), we
get the hadronic side as

$\displaystyle\tilde{\Pi}^{\mathrm{Had}}(p,q)$ $\displaystyle=$ $\displaystyle
e^{-\frac{m_{P_{cs}^{2}}}{M^{2}}}e^{-\frac{m_{\Lambda}^{2}}{M^{\prime
2}}}\frac{g\lambda_{\Lambda}\lambda_{P_{cs}}f_{\eta_{c}}m_{\eta_{c}}^{2}}{2m_{c}(m_{\eta_{c}}^{2}+Q^{2})}\not{p}\not{p^{\prime}}+\mathrm{other~{}structures}+\cdots,$
(42)

where $Q^{2}=-q^{2}$ and only the Lorentz structure used in the analyses is
presented explicitly, and $\mathrm{other~{}structures}+\cdots$ denote the
contributions coming from excited states and the other present structures.

The QCD side is obtained after plugging in the interpolating currents present
in the Eq. (37) and applying contractions of quark fields using Wick’s
theorem. The result for this side is obtained as

$\displaystyle\Pi^{\mathrm{QCD}}(p,q)$ $\displaystyle=$ $\displaystyle
i^{2}\int d^{4}xe^{-ip\cdot x}\int d^{4}ye^{ip^{\prime}\cdot
y}\frac{1}{\sqrt{6}}\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}\sum_{i=1}^{2}\Big{[}\mathrm{Tr}[S_{s}^{bb^{\prime}}(y-x)\gamma_{5}CS_{d}^{T}{}^{ca^{\prime}}(y-x)CA_{1}^{i}]A_{2}^{i}S_{u}^{ad^{\prime}}(y-x)$
(43) $\displaystyle\times$
$\displaystyle\gamma_{5}S_{c}^{d^{\prime}l}(x)\gamma_{5}S_{c}^{lc^{\prime}}(-x)-2A_{2}^{i}S_{s}^{cb^{\prime}}(y-x)\gamma_{5}CS_{d}^{T}{}^{ba^{\prime}}(y-x)CA_{1}^{i}S_{u}^{ad^{\prime}}(y-x)\gamma_{5}S_{c}^{d^{\prime}l}(x)\gamma_{5}S_{c}^{lc^{\prime}}(-x)$
$\displaystyle-$ $\displaystyle
A_{2}^{i}S_{d}^{ca^{\prime}}(y-x)\gamma_{5}CS_{s}^{T}{}^{bb^{\prime}}(y-x)CA_{1}^{i}S_{u}^{ad^{\prime}}(y-x)\gamma_{5}S_{c}^{d^{\prime}l}(x)\gamma_{5}S_{c}^{lc^{\prime}}(-x)\Big{]}.$

Following the same steps in the previous section, the result obtained for this
side for the Lorentz structure $\not{p}\not{p^{\prime}}$ is matched with that
of the hadronic side for the same Lorentz structure to get the coupling
constant $g$ as

$\displaystyle
g=e^{\frac{m_{P_{cs}^{2}}}{M^{2}}}e^{\frac{m_{\Lambda}^{2}}{M^{\prime
2}}}\frac{2m_{c}(m_{\eta_{c}}^{2}+Q^{2})}{\lambda_{\Lambda}\lambda_{P_{cs}}f_{\eta_{c}}m_{\eta_{c}}}\tilde{\Pi},$
(44)

where we again perform the Borel transformations on the variables $-p^{\prime
2}$ and $-p^{2}$, and $\tilde{\Pi}$ represents the Borel transformed $\Pi$ for
the QCD side of the calculation corresponding to the mentioned Lorentz
structure, $\not{p}\not{p^{\prime}}$.

In the analyses of this decay channel, we adopt the input parameters given in
Table 1, and the same auxiliary parameters $M^{2}$, $M^{\prime 2}$, $s_{0}$,
$s^{\prime}_{0}$ and $\beta$ given in the previous section, which works also
well for this channel.

To get the value of coupling constant $g$ at $Q^{2}=-m_{\eta_{c}}^{2}$, we
again need a proper fit function due to the fact that, as in the Sec. III, the
results are only reliable in some regions of the $Q^{2}$. For this purpose, we
use the same form of the fit function given in Sec. III with the fit
parameters presented in Table 2, providing a good overlap for the obtained
results of the QCD sum rule in the reliable region. Note that Table 2 also
contains the numerical value of the coupling constant $g$ at
$Q^{2}=-m_{\eta_{c}}^{2}$.

With the obtained coupling constant, the width of the
$P_{cs}\rightarrow\eta_{c}\Lambda$ decay is attained using the following width
formula:

$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle
g^{2}\frac{f(m_{P_{cs}},m_{\eta_{c}},m_{\Lambda})}{8\pi
m_{P_{cs}}^{2}}[(m_{\Lambda}+m_{P_{cs}})^{2}-m_{\eta_{c}}^{2}],$ (45)

with the function $f(x,y,z)$ given in Sec. III. Finally the width is obtained
for this decay channel as

$\displaystyle\Gamma(P_{cs}\rightarrow\eta_{c}\Lambda)$ $\displaystyle=$
$\displaystyle\left(3.18\pm 0.74\right)~{}\mathrm{MeV}.$ (46)

##  V Discussion and conclusion

In a recent report, the LHCb collaboration announced the observation of a new
candidate pentaquark state with strangeness in $J/\psi\Lambda$ channel. The
observed mass and the width of the state were reported as $m=4338.2\pm 0.7\pm
0.4~{}\mathrm{MeV}$ and $\Gamma=7.0\pm 1.2\pm 1.34~{}\mathrm{MeV}$,
respectively LHCb:2022jad  with preferred spin and parity quantum numbers
being $J^{P}=\frac{1}{2}^{-}$. To elucidate its inner structure and certify
its quantum numbers, further theoretical investigations are necessary. With
this purpose, in the present work, the $P_{\psi s}^{\Lambda}(4338){}^{0}$
state was assigned a molecular $\Xi_{c}\bar{D}$ structure with spin parity
$J^{P}=\frac{1}{2}^{-}$ and its decays to $J/\psi\Lambda$ and
$\eta_{c}\Lambda$ states were investigated using the three-point QCD sum rule
approach. For completeness, firstly, the chosen interpolating current was
applied to calculate the mass and the current coupling constant of the
considered state using the two-point QCD sum rule method. These quantities are
main inputs in the decay calculations. The obtained mass, $m_{P_{cs}}=4338\pm
130~{}\mathrm{MeV}$, is in good consistency with the observed one. Our
prediction for the mass is also consistent with the mass predictions based on
the molecular assumption present in the literature, such as $m=4327.4$ MeV
Yang:2022ezl , $m=4341.0$ MeV Ortega:2022uyu , $m=4336.34$ MeV and
$m=4329.11.34$ MeV Giachino:2022pws , and $m=4.34^{+0.07}_{-0.07}$ GeV
Wang:2022neq .

As stated, the predicted mass and the current coupling constant comprise the
main input parameters for the width calculations of the $P_{\psi
s}^{\Lambda}(4338){}^{0}\rightarrow J/\psi\Lambda$ and $P_{\psi
s}^{\Lambda}(4338){}^{0}\rightarrow\eta_{c}\Lambda$ channels which are taken
into account as dominant decay channels of the considered state. To compute
the widths of these channels, we first calculated the relevant strong coupling
constants and subsequently used them to get the corresponding widths. The
resultant widths are obtained as $\Gamma(P_{cs}\rightarrow
J/\psi\Lambda)=\left(7.22\pm 1.78\right)~{}\mathrm{MeV}$ and
$\Gamma(P_{cs}\rightarrow\eta_{c}\Lambda)=\left(3.18\pm
0.74\right)~{}\mathrm{MeV}$ whose total, $\Gamma=\Gamma(P_{cs}\rightarrow
J/\psi\Lambda)+\Gamma(P_{cs}\rightarrow\eta_{c}\Lambda)=(10.40\pm
1.93)~{}\mathrm{MeV}$, also agrees with the experimentally observed width
within the presented uncertainties.

Within the existing literature, numerous works have investigated the decays of
observed pentaquark states. In order to elucidate the internal structures of
the observed pentaquark states, the width calculations of various transition
channels are necessary for either comparing the result with experimental
findings or potentially uncovering new channels for further investigations.
$J/\psi\Lambda$ is the channel that the observation of $P_{\psi
s}^{\Lambda}(4338){}^{0}$ was reported. However, the width of this particular
pentaquark state also receives contributions from other channel,
$\eta_{c}\Lambda$, that was considered in the present study. This is the case
for different pentaquark states. If we consider theoretical studies present in
the literature on the observed pentaquark states, for instance, with spin-
parity quantum numbers $J^{P}=\frac{1}{2}^{-}$, as in our case, the widths
were obtained for their decays using different form factor sets in Ref.
Lin:2019qiv  as $P_{c}(4312)\rightarrow J/\psi p=0.001~{}\mathrm{MeV}$,
$P_{c}(4312)\rightarrow\eta_{c}p=0.01~{}\mathrm{MeV}$, $P_{c}(4312)\rightarrow
J/\psi p=0.1~{}\mathrm{MeV}$,
$P_{c}(4312)\rightarrow\eta_{c}p=0.4~{}\mathrm{MeV}$ for $P_{c}(4312)$ state,
$P_{c}(4440)\rightarrow J/\psi p=0.03~{}\mathrm{MeV}$,
$P_{c}(4440)\rightarrow\eta_{c}p=3^{-4}~{}\mathrm{MeV}$,
$P_{c}(4440)\rightarrow J/\psi p=0.6~{}\mathrm{MeV}$,
$P_{c}(4440)\rightarrow\eta_{c}p=0.07~{}\mathrm{MeV}$ for $P_{c}(4440)$, and
$P_{c}(4457)\rightarrow J/\psi p=0.02~{}\mathrm{MeV}$,
$P_{c}(4457)\rightarrow\eta_{c}p=2^{-4}~{}\mathrm{MeV}$,
$P_{c}(4457)\rightarrow J/\psi p=0.2~{}\mathrm{MeV}$,
$P_{c}(4457)\rightarrow\eta_{c}p=0.02~{}\mathrm{MeV}$ for $P_{c}(4457)$. In
Ref. Li:2023zag  the following predictions were obtained:
$P_{c}(4312)\rightarrow J/\psi p=0.17^{-0.04}_{+0.04}~{}\mathrm{MeV}$,
$P_{c}(4312)\rightarrow\eta_{c}p=0.085^{-0.016}_{+0.018}~{}\mathrm{MeV}$. In
Ref. Wang:2019spc  the widths were attained as $P_{c}(4312)\rightarrow J/\psi
p=0.32\pm 0.08~{}\mathrm{MeV}$, $P_{c}(4312)\rightarrow\eta_{c}p=0.89\pm
0.25~{}\mathrm{MeV}$, $P_{c}(4440)\rightarrow J/\psi p=2.92\pm
0.55~{}\mathrm{MeV}$, $P_{c}(4440)\rightarrow\eta_{c}p=0.15\pm
0.03~{}\mathrm{MeV}$, $P_{c}(4457)\rightarrow J/\psi p=0.45\pm
0.13~{}\mathrm{MeV}$, $P_{c}(4457)\rightarrow\eta_{c}p=0.02\pm
0.01~{}\mathrm{MeV}$. In Ref. Xu:2019zme  the width predictions were given as
$P_{c}(4312)\rightarrow J/\psi p=1.67^{+0.92}_{-0.56}~{}\mathrm{MeV}$,
$P_{c}(4312)\rightarrow\eta_{c}p=5.54^{+0.75}_{-0.5}~{}\mathrm{MeV}$. The
widths of same transitions were obtained in Ref. Dong:2020nwk  as
$P_{c}(4312)\rightarrow J/\psi
p=0.0448^{+0.0197(+0.0309)}_{-0.0161(-0.0287)}~{}\mathrm{MeV}$,
$P_{c}(4312)\rightarrow\eta_{c}p=0.0892^{+0.0392(+0.0615)}_{-0.0321(-0.0571)}~{}\mathrm{MeV}$.
And also the transitions to these final states, for possible spin-parity
$J^{P}=\frac{3}{2}^{-}$ case, were also considered in Ref. Lin:2019qiv  with
the following findings: $P_{c}(4440)\rightarrow J/\psi p=0.02~{}\mathrm{MeV}$,
$P_{c}(4440)\rightarrow\eta_{c}p=8^{-5}~{}\mathrm{MeV}$,
$P_{c}(4440)\rightarrow J/\psi p=1.8~{}\mathrm{MeV}$,
$P_{c}(4440)\rightarrow\eta_{c}p=0.008~{}\mathrm{MeV}$ for $P_{c}(4440)$,
$P_{c}(4457)\rightarrow J/\psi p=0.01~{}\mathrm{MeV}$,
$P_{c}(4457)\rightarrow\eta_{c}p=6^{-5}~{}\mathrm{MeV}$,
$P_{c}(4457)\rightarrow J/\psi p=0.6~{}\mathrm{MeV}$,
$P_{c}(4457)\rightarrow\eta_{c}p=0.003~{}\mathrm{MeV}$ for $P_{c}(4457)$. As
is seen, in some cases, the widths for the channels involving $J/\psi$ in the
final state have dominant contributions, while in others, the widths of
channels involving $\eta_{c}$ state are large. The width result obtained for
the $P_{\psi s}^{\Lambda}(4338){}^{0}\rightarrow\eta_{c}\Lambda$ decay channel
in this work indicated that this channel has a significant contribution to the
total width, though it is smaller than that of $P_{\psi
s}^{\Lambda}(4338){}^{0}\rightarrow J/\psi\Lambda$ decay channel that the
state $P_{\psi s}^{\Lambda}(4338){}^{0}$ was observed.

The results obtained for the mass and the total width of $P_{\psi
s}^{\Lambda}(4338){}^{0}$ are consistent with the experimental findings within
the presented uncertainties and favor the $\Xi_{c}\bar{D}$ molecular nature of
the $P_{\psi s}^{\Lambda}(4338){}^{0}$ state with quantum numbers
$J^{P}=\frac{1}{2}^{-}$.

## ACKNOWLEDGEMENTS

K. Azizi is thankful to Iran Science Elites Federation (Saramadan) for the
partial financial support provided under the grant number ISEF/M/401385.

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# On the experimental feasibility of quantum state reconstruction via machine
learning

Sanjaya Lohani  Tulane University, New Orleans, LA 70118, USA
<EMAIL_ADDRESS>Thomas A. Searles  Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Howard University, Washington, DC 20059, USA  Brian
T. Kirby  Tulane University, New Orleans, LA 70118, USA  United States Army
Research Laboratory, Adelphi, MD 20783, USA<EMAIL_ADDRESS>Ryan
T. Glasser  Tulane University, New Orleans, LA 70118, USA<EMAIL_ADDRESS>
###### Abstract

We determine the resource scaling of machine learning-based quantum state
reconstruction methods, in terms of both inference and training, for systems
of up to four qubits. Further, we examine system performance in the low-count
regime, likely to be encountered in the tomography of high-dimensional
systems. Finally, we implement our quantum state reconstruction method on a
IBM Q quantum computer and confirm our results.

## Main

Quantum state tomography is a standard procedure for reconstructing the state
of an unknown quantum system through a series of repeated measurements on an
ensemble of identically prepared systems. The number of measurement settings
required for full state tomography of a multi-qubit state scales exponentially
with the qubit number[1]. Further, the determination of a physically valid
density matrix that corresponds to the measured data is itself resource-
intensive[2, 3]. Many methods for reconstructing valid density matrices from
measured data have been developed, such as Bayesian [4], maximum likelihood
estimation (MLE)[5, 6, 3], projected gradient descent[7], and linear
regression[8]. Recently, several methods for state reconstruction based on
machine learning have been proposed and demonstrated [9, 10, 11, 12, 13, 14,
15]. These machine learning-based techniques offer several potential
advantages compared to their conventional counterparts, such as the ability to
front-load expensive computations and robustness to missing measurements and
experimental noise[16]. Despite these advantages, it is not yet clear how the
machine learning models’ training scales with system size, nor how they
perform in situations where limited copies of the unknown state are available.

In this brief communication, we implement quantum state tomography for systems
of up to four qubits on the IBM Q quantum processor and reconstruct the
measured states via a neural network. In all cases, we either match or exceed
the fidelities achieved compared to MLE. Using our results, we can estimate
the resource scaling of both training and inference for our system. Further,
we investigate the impact of statistical noise on our neural network system by
limiting how many copies of an unknown state are available for tomographic
measurements. We find that when performing state reconstruction on data
obtained from only a few copies of a state, meaning significant statistical
noise is present in the measurement results, we obtain enhanced reconstruction
fidelity by using training data that includes comparable statistical noise. In
other words, the variations in measurement results due to limited counting
statistics are themselves learnable, to some degree, by our network. These
results inform the applicability of neural network quantum state
reconstruction for high-dimensional systems where resource limitations in
computation and the repeatability of measurements are important factors.

Figure 1: Network architecture and reconstructing quantum states for various
sizes of qubits. (a) Example of an end-to-end tomography setup with a qubit
size of two. Abbreviations H, D, and R, respectively represents horizontal,
diagonal and right-circular polarization basis. The density matrices (DM) are
predicted at the DM prediction layer. (b) Fidelity for the predicted density
matrices versus the number of qubits. The error bars show one standard
deviation from the mean. The inset shows the cross-validation fidelity for the
ideal test set. (c) Network inferences versus the number of qubits. The
whiskers on the box-plot represent fidelity from 1 to 99 percentiles.

An end-to-end tomography setup with a neural network is shown in Fig. 1 (a).
First, we make predictions for unknown test quantum states with a network
solely trained with ideal, simulated tomography data (see “Methods”), the
results of which are shown in Fig. 1 (b, c). With the trained network, the
fidelity between the reconstructed and corresponding target density matrices
systems of one, two, three, and four qubits are shown in Fig. 1 (b). The
reconstruction fidelity per epoch for the random quantum states from the
validation set in the cases of one and four qubits are shown in the inset. We
implement the network using tensorflow[17] with an AMD Ryzen 9 3900x 12-core
processor and GeForce RTX 2060 GPU. We find that the network takes only 14
seconds for a single qubit, and 240 seconds for four qubits in order to reach
saturation. The red, blue, and green lines respectively represent the fidelity
for the test set from the ideal scenario, the IBM Q simulator with noise, and
the real IBM Q device as described in “Methods”. For all the test cases and
systems with qubits of size of one and two, we find the trained networks
reconstruct the quantum states with near-unity fidelity. Additionally, we find
a significant increase in size of the quantum circuit as well as number of
gates for higher values of the qubits (see “Supplementary Fig. S1’).
Consequently, noise in the tomography data from the real IBM Q machine also
increases. This results in a decrease in the average fidelity for the three
and four qubit cases. Even with such noisy tomography values from the real IBM
device, we show that the network trained with ideal tomography sets
reconstructs random quantum states of size of three qubits with an average
fidelity of 0.89. Examples of the target and predicted density matrices for a
qubit size of three are shown in “Supplementary Fig. S2”. Similarly, for a
qubit size of four, random quantum states are reconstructed from the
tomography data obtained from the IBM Q simulator containing noise with an
average fidelity of 0.83. Note that the IBM Q backend transpiles a huge
circuit for a qubit size of four as illustrated in ‘Supplementary Fig. S1’. As
a result, we end up with extremely noisy tomography data with the real IBM Q
for a qubit size of four.

Next, we evaluate the computational cost on training and testing of the
network. As the number of qubits, ‘d’, increases, the number of quantum
circuits and over-complete tomography values are, respectively, increased on
the order of $3^{d}$ and $6^{d}$. Because of this we increase the network
training parameters accordingly in order to achieve similar fidelities for
systems of all qubit sizes. The average training time per epoch versus number
of qubits is shown by the green dotted line in Fig. 1 (c). The box plots
represent the statistics of the time per epoch from 1 to 99 percentiles. As
expected we find that the training time increases with the number of qubits.
The sudden rise in the training time for qubit size of four is due to drastic
increase in the trainable parameters as illustrated in the inset. However,
once the network is trained, we find that the network makes the prediction of
a density matrix for the qubit size of one, and four, respectively, within 3.4
milliseconds and 0.08 seconds. The inference on a density matrix
reconstruction for data obtained from the ideal, IBM Q simulator with noise,
and real IBM Q hardware cases are, respectively, shown by a red, blue and
green solid line in Fig. 1 (c).

Figure 2: Network performance with varying shots. (a) Reconstruction fidelity
for ranges of shots. For larger shots, the fidelity with the networks, and
maximum likelihood estimation (MLE) are shown in the inset. (b) Reconstruction
statistics at shots 15 (left) , and 8192 shots (right). The whiskers on the
box-plots represent fidelities from $Q_{1}\,-\,1.5\,(Q_{3}\,-\,Q_{1})$ to
$Q_{3}\,+\,1.5\,(Q_{3}\,-\,Q_{1})$, where $Q_{1}$ and $Q_{3}$ represent the
first quartile and third quartile of the fidelity values. (c) Inference on
maximum likelihood estimation method, and the trained network. The error bar
shows one standard deviation from the mean. (d) Squared differences between
the ideal tomography and tomography measured from the real IBM Q computer at
various shots.

We now turn to investigating the reconstruction fidelity at various shots,
which represents the number of times the same quantum circuit is executed. In
order to illustrate the proof of concept, we fix the qubit size at two and
vary the number of shots when using different reconstruction methods. Here we
implement three methods: a network trained with ideal tomography values “NN
(Ideal)”, a network trained with the tomography values simulated at 15 shots
“NN (Shots: 15)”, and finally the maximum likelihood estimation (MLE). For
each method, experimental shots ranging from 5 to 8192 were executed in order
to make predictions for a test set containing tomography data for 20 random
quantum states from the real IBM Q computer. The average reconstructed
fidelity at various shots is shown in Fig. 2 (a). Moreover, the fidelity at
higher shots is shown in the inset. As expected, we find that the network
trained with ideal tomography (green line) works better and reaches a fidelity
of 0.99 at 8192 shots. Additionally, we show the fidelity at 5 and 15 shots
are enhanced with the network that is trained with the simulated data at 15
shots (a blue line). Furthermore, in order to illustrate the performance
efficacy of the networks, we also reconstruct random quantum states from the
same test-set with MLE as shown by a red line. The statistics of the fidelity
for the predicted density matrices at shots 15 (left) and 8192 (right) are
shown as the box plots in Fig. 2 (b). We find that the networks trained with
ideal tomography and simulated tomography simply outperform MLE. And notably,
the network trained for simulated tomography at 15 shots produces higher
average fidelity than MLE in lower shots regime.

Similarly, we also make an inference on per density matrix prediction with the
MLE and the neural network. We find a significant improvement in computational
cost with the network, which is 6.2 milliseconds, compared to MLE, which takes
3.89 seconds, as shown in Fig. 2 (c). The error-bar represents a standard
deviation from the mean. Finally, we show the extent of statistical noise as a
squared difference between the tomography values from the real IBM Q device
and the ideal measurement scenarios at various shots in Fig. 2 (d). As
expected, we find noise on the tomography from real IBM Q computer decreases
with increasing shots.

In conclusion, we have experimentally investigated the resource and
computation time scaling of quantum state tomography using machine learning
techniques. We implement quantum state tomography on random quantum states
using the IBM Q quantum computer for systems of up to four qubits. We also
implement MLE for state estimation and find our neural network system
outperforms it in several scenarios. We show that the neural network approach
may be useful in larger dimensional systems since the number of repeated
circuits necessary for accurate tomographic reconstruction is small. Lastly,
we reiterate that while a neural network’s training time increases for quantum
systems of increasing dimension, the network need only be trained a single
time. Once trained, the inference time when using the neural network in
practice is negligible. As such, the results provide evidence for machine
learning as a tool for full quantum state tomography in near-term
intermediate-scale quantum hardware.

## Methods

### Generating random quantum states and measuring the tomography

The random pure quantum states, corresponding density matrices ($\rho$), and
$\tau$ vectors, arrays of elements of the lower triangular matrices obtained
after applying the Cholesky decomposition to the density matrices[18], are
generated as described in [13]. In order to have the ideal tomography set, we
apply the relation given by

$\textrm{Measurements}\,(m)\,=\,\textrm{Tr}(\rho\mathcal{\hat{P}}),$ (1)

where ‘Tr’ represents the trace, and $\mathcal{\hat{P}}$ is the set of
projectors for the given size of the qubits. Similarly, for the tomography
from the IBM Q-qasm simulator with noise and the real IBM Q computer, we first
initialize the qubits with the given random quantum states and allow the
system to self-optimize the quantum circuits[19]. Note that we perform the
light order of optimization on the circuits with ‘qiskit.execute’ from the
Qiskit Terra API[20].

### Training the network

We custom build a convolutional neural network with a convolutional unit
followed by a max-pooling layer and a second convolutional unit in a row.
After that we connect two dense layers each followed by a dropout layer, which
is finally attached to an output layer where density matrices are predicted as
shown in Fig. 1 (a). First, In order to train the network, we simulate 35,500
random quantum states using the Haar measure and their tomography measurements
for systems with number of qubits ranging from one to four. The simulated data
are then split into a training set containing 35,000 tomography measurements,
and a validation set with 500 tomography measurements to cross-validate the
network performance. Additionally, so as to reduce the bias-variance trade-off
and increase the fidelity at the output, we use separate networks with
different hyper-parameters for different sizes of qubits. Note that we
implement the batch size of four in the training of all the networks. Once the
networks are trained with simulated data, we then generate test sets which are
completely unknown to the trained network. For the test-sets we consider three
scenarios, a simulated test-set without any noise (“Ideal”), a test-set
generated from the ibmq-qasm-simulator at shots of 2192 with a noise model
provided by IBM Q-backend ‘$ibmq\\_rome$’ (“Simulator with IBMQ noise”), and
finally, a test set containing the tomography data directly measured from the
real IBM Q computer (5 qubits machine: $ibmq\\_rome$). Note that each test set
consists of tomography measurements for 20 random quantum states, again for
each of the systems from one to four qubits.

Moreover, our custom built network makes the predictions for the $\tau$ arrays
at the error layer as shown in Fig. 1 (a), which are finally compared with the
target $\tau$ arrays and the mean squared loss (MSE) is evaluated and finally
optimized with the Adagrad optimizer. Also, note that there are no training
parameters between the error layer and the density matrix (DM) prediction
layer. We build the DM prediction layer in the same graph of the network in
order to efficiently evaluate the fidelity metric per epoch to cross-validate
the training as well as avoid the post-processing.

### Maximum likelihood estimation

For the two qubit case, the over-complete tomography measurement consists of
36 data points $m_{i}$, where $i$ runs from 1 to 36. The expected value is
assumed to be $\bar{m_{i}}$. For simplicity we assume the noise on the
tomography is Gaussian, which leads the probability ($P$) of obtaining a
tomography set of $m_{i}$ to,

$P(m_{1},m_{2},...,m_{36})\,=\,\prod_{i=1}^{36}\exp{\frac{-(m_{i}-\bar{m_{i}})^{2}}{2\sigma_{i}^{2}}}.$
(2)

Here $\sigma_{i}$ is approximated as $\sqrt{\bar{m_{i}}}$. As in eqn (1), the
expected tomography for the density matrix
$\rho(\tau_{1},\tau_{2},...\tau_{16})$ is

$\bar{m_{i}}\,=\,\textrm{Tr}\big{(}\rho(\tau_{1},\tau_{2},...\tau_{16})\mathcal{\hat{P}}_{i}\big{)}.$
(3)

Thus, the negative log-likelihood of the given density matrix
$\rho(\tau_{1},\tau_{2},...\tau_{16})$ that results $m_{i}$ measurements is

$-\mathcal{L}=\sum_{i=1}^{36}\frac{\Big{[}\textrm{Tr}\big{[}\rho(\tau_{1},\tau_{2},...\tau_{16})\mathcal{\hat{P}}_{i}\big{]}-m_{i}\Big{]}^{2}}{2\textrm{Tr}\big{[}\rho(\tau_{1},\tau_{2},...\tau_{16})\mathcal{\hat{P}}_{i}\big{]}}.$
(4)

In order to maximize the likelihood, we minimize the right-hand side of eqn
(4) using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimizer of the
tensorflow-probability library. Note that we always randomly initialize the
values for $\tau_{1},\tau_{2},...\tau_{16}$ using the Glorot-Uniform
initializer of the tensorflow-keras library, and implement the gradient
tolerance of $10^{-8}$.

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## Acknowledgements

This material is based upon work supported by, or in part by, the Army
Research Laboratory and the Army Research Office under contract/grant numbers
W911NF-19-2-0087 and W911NF-20-2-0168. The views and conclusions contained in
this document are those of the authors and should not be interpreted as
representing the official policies, either expressed or implied, of the Army
Research Laboratory or the U.S. Government. The U.S. Government is authorized
to reproduce and distribute reprints for Government purposes notwithstanding
any copyright notation herein. T.A.S. acknowledges support from the IBM-HBCU
Quantum Center and the Martin Luther King Visiting Scholars Program at MIT.
Additionally, we acknowledge use of the IBM Q for this work. The views
expressed are those of the authors and do not reflect the official policy or
position of IBM Quantum. We would like to thank Joseph Lukens at Oak Ridge
National Laboratory and Hanhee Paik at IBM for helpful discussions.

## Author contributions statement

S.L. developed the setups, ran all simulations and measurements on IBM Q
machine. S.L., B.T.K. and R.T.G. conceived of and led the project. All authors
contributed to discussions and interpretations of the results, and wrote the
manuscript.

## Data availability

The data that support the findings of this study are available from the
corresponding authors on reasonable request.

## Competing interests

The authors declare no competing interests.

## Supplementary

### Supp.: Quantum circuits and tomography measurements

Figure S1: Examples of transpiled circuits for (a) three qubits, (f) four
qubits, and corresponding probability measurements. (a, f) In the light-green
box is the tomography portion of the circuit, while the preceding elements
(light-red box), making up the bulk of the overall circuit, generate a random
quantum state transpiled by the IBM Q backend ‘$ibmq\\_rome$’.

As mentioned in the manuscript, in order to perform the over-complete
tomography for three qubits requires 27 quantum circuits, which increases to
81 for four qubits. Shown in Fig. S1 [a, f] is one example circuit for each of
the three and four qubit cases, respectively. The light-green boxes in Fig. S1
[a, f] show the circuits’ tomography portion, while the remainder of the
elements in the light-red boxes are the transpiled circuit used to generate
random states. The corresponding projectors used to generate the probability
amplitudes shown in Fig. S1 (b-e) and (g-j) are the result of measuring in the
‘H (first qubit)’, ‘D (second qubit)’, ‘R (third qubit)’ bases for three
qubits and in the ‘D (first qubit)’, ‘D (second qubit)’, ‘D (third qubit)’, ‘D
(forth qubit)’ for four qubits. For each measurement scenario we show the
results for the ideal projection, ibmq-qasm simulator, ibmq-qasm simulator
with IBMQ noise, and the real IBMQ computer.

### Supp.: Reconstructing quantum states

The density matrices reconstructed directly from the network and the
corresponding target density matrices are shown in Fig. S2. The network used
is solely trained with the data obtained from the ideal tomography scenarios,
whereas the test-set contains the tomography data obtained from the ideal
mode, IBMQ qasm simulator with noise, and the real IBMQ computer.

Figure S2: Examples of the target and predicted density matrices for
tomography obtained from the ideal mode, IBMQ qasm simulator with noise, and
the real IBMQ computer.