Pc pentaquarks in a hybrid model of chiral and quark dynamics
Yasuhiro Yamaguchi1 , Atsushi Hosaka2,1 , Sachiko Takeuchi3,4 , and Makoto Takizawa5,6,7

Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, JapanIn Physics
practice (RCNP),
resonancesOsaka
decaying
strongly intoJ/ p Osaka
must havea minimal
quark content
Research Center for Nuclear
University,
Ibaraki,
567-0047,
Japan
+
of
ccuud,
andthus
arecharmonium-pentaquarks;
we
label
such
statesP,
irrespectiveof3cJapan College of Social Work, Kiyose, Tokyo 204-8555, Japan4
theinternalbindingmechanism.
In
order
to
ascertainifthestructures
seen
in
Fig.
2(b)
Theoretical Research Division, Nishina Center, RIKEN, Hirosawa, Wako, Saitama 351-0198,
Japan
5
are resonant in University,
nature and not
due to reflections
generatedJapanby the ⇤⇤ states, it is necessary
Showa Pharmaceutical
Machida,Tokyo 194-8543,
6
to Instituteperform afor
fullParticle
amplitudeinterference
bothJapan
decayand
J-PARC Branch, KEK Theory Center,andanalysis,
Nuclearallowing
Studies,forKEK,
Tokai, e↵ects
Ibarakibetween
319-1106,7
sequences.
Meson Science Laboratory,
Cluster for Pioneering Research, RIKEN, Saitama 351-0198 JapanThe fit uses five decay angles and the K p invariant mass mKp as independent variables.
First
we recenttried toachievement
fit the data with
an amplitudemodel
14 ⇤⇤Pstates
listed by
In this article we report ourin thecollaboration
forthatthecontains
pentaquarkc ’s.
Particle
Data Group[12].where
As thisnotdistancegive a satisfactory
A unique feature of our the
study
is a hybrid
picturethedidlong
structure descriptionof hadronicof the data,
we added
one Pc+ state,
and whenthat was
we included
a second
molecule driven by the onepion exchange
potential(OPEP)and not
the sufficient
short range
structureof five state. The
Pc+force
statesmediatedare foundby
to thehavegluons.
masses The
of 4380± 8 ± 29due
MeVtoand± 1.7
quarks dominated by thetwo
color
dynamics
the4449.8
pion isthe± 2.5 MeV,
necessity of the spontaneous
breakingchiralrole units arewith symmetry
corresponding
widths(SSB)of 205 of± 18± 86symmetry.
MeV and 39We
± 5emphasize± 19 MeV.the(Naturalof the tensor component usedof thethroughout
OPEP boththeir massorderingand decay widths.
The theformerthisinLetter.
Whenevertwo uncertainties
are quoted
first isis statistical
crucial for the determinationof spinandsystematic.)
parity of theand thesecondTheobserved
fractionsPof
c . the total sample due to the lower mass andhigher mass states are (8.4 ± 0.7 ± 4.2)% and (4.1 ± 0.5 ± 1.1)%, respectively. The best fit
solution has spin-parity J P values of (3/2 , 5/2+ ). Acceptable solutions are also found+
INTRODUCTION
evident,
while
the(3/2
lowerone) at
4380
MeVbestcamefor additional cases with opposite
parity,either, 5/2
or around(5/2+ , 3/2). Thetheirdetailed
analysis.
fit projections are shown infrom
Fig. 3.
BothmKp andamplitude
the peaking
structureBeing
in mJ/observedarepreproducedby thefit. The significancesof the thelower
mass andishigher
mass statesare 9 toin J/ψp events,
structure
naturally
associatedIn the past decades the Large Hadron
Collider(LHC)
12 standard
respectively. containing five quarks of cc̄uud, thus named
resonances
has been producing a large amountandof new
hadrondeviations,
data
1

2

Events/(15 MeV)

data
total fitbackgroundPc(4450)
Pc(4380)Λ(1405)
Λ(1520)Λ(1600)Λ(1670)Λ(1690)Λ(1800)Λ(1810)Λ(1820)Λ(1830)Λ(1890)Λ(2100)Λ(2110)

Data
2015

2.2

2.4

Events/(15 MeV)

Pc .

indicating the existence of hadrons that have the structure beyond the minimal contents of the standard quark
2200
model: q̄q for mesons and qqq for baryons,
where q de2000
LHCb ofnotes one of any kinds of quarks. The 1800
colliding(a)energy
order 10 TeV enables to produce many
1600 bottom quarks
1400 charm quarks.
which subsequently decay weakly into
Among many of them, in this report we1200
discuss the candi1000
date of the hidden charm pentaquarks P
whose minimal
800c
content is considered to be c̄cqqq, where
600 c is the charm
400 report is based
quark and q is one of u, d quarks. This200
on our collaboration that has been supported
in part by
0
1.4
1.6
1.8
2
the RCNP COREnet project.

800

LHCb

(b)

700
600
500
400
300
200
100
0

2.6

4

4.2

4.4

4.6

4.8

mK p [GeV]5mJ / ψp [GeV]−
3: Fit projections for (a) mKp and (b) mJ/ p for the reduced ⇤ model with two Pc statess KFigurethepoints show the
0 solid (red)u (see Table 1). The data are shown as solid (black) squares, while
Σ+DΣ + D* 0c fit.
results of the
J/ψThe solid (red) histogram shows the backgroundc distribution.c The (blue) open
c the shaded histogram represent
Data the Pc (4450)+ state, and the shaded histogramsquares with

Λb

b
u
d

W

Pc

Weighted candidates/(2 MeV)

⇤

+

1200

+ state. Each ⇤⇤ component
data
LHCb is also
topped with (purple) filled squares represents
2019 the Pc (4380)
utotal fitshown. The error bars on the points showing the fit
1000results are due to simulation statistics.

u pdFIG. 1.
A postulated quark diagram of the weak decay
Λ0b → J/ψK − p.
Q
q

2

800
600
400

+

+

P c(4312)

5q(1)

and performed the invariant mass analysis for J/ψp. A
postulated quark diagram is shown in Fig. 1. There, theyobserved nontrivial structures at masses around 4380
MeV and 4450 MeV. The data is shown in the upper panelof Fig. 2, where the higher peak at around 4450 MeV is+P c(4440)

P c(4457)200

π
The first reportfrom LHCb+ group came in 2015 [1]qwith evidence
of
two
P
’s.
They
studied the three-bodycqdecayQΛ0b → J/ψK − pbackground04200 4250 4300 4350 4400 4450 4500 4550 4600
m J / ψp [MeV]
Figure 6: Fit to the cos ✓P c -weighted mJ/ p distribution with three BW amplitudes and asixth-order polynomial background. This fit is used to determine the central values of the masses
and widths of the Pc+ states. The mass thresholds for the ⌃c+ D0 and ⌃c+ D⇤0 final states aresuperimposed.

2

FIG. 2. Invariant mass spectrum of Jψp in the weak decay
approximately 5 MeV and 2 MeV below the ⌃ D and ⌃ D thresholds, respectively, asof Λ0b → J/ψK −p. Dataarethem
takenfrom forRefs.
[1,of these2] systems.
illustrated
in Fig. 6, making
excellent candidates
bound states
+
c

0

+
c

⇤0

The Pc (4440)+ could be the second ⌃c D⇤ state, with about 20 MeV of binding energy, since
two states with J P = 1/2 and 3/2 are possible. In fact, several papers on hidden-charmstates created dynamically by charmed meson-baryon interactions [31–33] were publishedwell before the first observation of the Pc+ structures [1] and some of these predictionsfor ⌃c+ D0 and ⌃c+ D⇤0 states [28–30] are consistent with the observed narrow Pc+ states.
Such an interpretation of the Pc (4312)+ state (implies J P = 1/2 ) would point to theimportance of ⇢-meson exchange, since a pion cannot be exchanged in this system [10].
In summary, the nine-fold increase in the number of ⇤0b ! J/ pK decays recon-

In 2019, they have reported higher statistics data refining their analysis with three narrow peaks as shown in
the lower panel of Fig. 2 [2]. Their locations and widths
8

2
are

Q
q

Pc+ (4312) : M = 4311.9 ± 0.7+6.8−0.6 MeV ,qqQΓ = 9.8 ± 2.7+3.7−4.5 MeV ;

5q

π

+

Pc+ (4440) : M = 4440.3 ± 1.3+4.1−4.7 MeV ,Γ = 20.6 ± 4.9+8.7−10.1 MeV ;FIG. 3. Interaction between meson and baryon. The firstterm is due to the pion exchange between light quarks, andthe second the coupling of the meson-baryon (M B) states tofive quark (5q) states.

Pc+ (4457) : M = 4457.3 ± 0.6+4.1−1.7 MeV ,+5.7
Γ = 6.4 ± 2.0−1.9 MeV .
The previous higher peak at 4450 MeV splits into two
peaks at 4440 MeV and 4457 MeV. The previous lower
signal at 4380 MeV is still kept, but now a new single
peak at 4312 MeV has been unveiled.
These results are very suggestive as they are all locatedvery close to two-hadron thresholds,
Σc D̄ ∼ 4320 MeV
Σc D̄∗ ∼ 4463 MeV(2)having lead to the idea that these states are molecularlike states of these hadrons. The five quarks are clusteredas cc̄uud → (cud)-(c̄u), and so
Pc+ (4312) ∼ Σc D̄
Pc+ (4440), Pc+ (4457) ∼ Σc D̄∗(3)

By knowing the spin and parity of Σc , D̄, D̄∗ and assuming that the molecules are in S-wave, their spin and parity
are expected to be 1/2− for Pc+ (4312), and 1/2− or 3/2−
for Pc+ (4440), Pc+ (4457).
Motivated by these observations, we have started our
project in October 2016, when one of the authors Hosaka
stayed in Genova and discussed the problem with Santopinto and Yamaguchi who was a postdoc there. Prior
to this project we had some experiences of molecular
structure of hadrons, with the emphasis on the specialrole of the pion exchange force at long distances [3, 4].
As shown in the first term of Fig. 3, the pion is exchanged between light quarks q = u, d which are theingredients of the possible molecular components suchas Σc ∼ cud, D̄∗ ∼ c̄u. At the same time we consider theimportant role of short range interaction where quark dynamics is of direct relevance as shown in the second termof Fig. 3. We have then proposed a model of hadronicmolecules with the one pion exchange force and quark
dynamics.
This article reports our previous work from 2017 to2020 which have been performed in collaboration withthe group of Genova and several domestic institutions [5,6]. By this work Yasuhiro Yamaguchi was awarded the
JPS prize for young scientists, “Wakate Shorei Sho” in
2021.

THEORETICAL METHOD

Our model is a hybrid model of hadronic molecules ofmesons and baryons (M B) coupled by five quark states(5q). In principle, both of these components are described by five quarks with different cluster correlations.
Our assumption is that they are well developed and regarded as good basis states. Thus, our model Hamiltonian is expanded by the open-charm M B and 5q channels
as

 

HM B VKM B + Vπ V
H==
(4)
V † H5q
V†
H5q
and the corresponding Schrödinger equation is given by





HM B V
ψM B
ψM B
=E
(5)
ψ5q
ψ5q
V † H5q
In the second equation of the Hamiltonian (4) we haveshown the OPEP Vπ explicitly. Eliminating ψ5q in the
coupled equation (5) we obtain the equation for themeson-baryon channel ψM B which is the direct relevanceto the experimental observations,(KM B + Vπ + Uef f ) ψM B = EψM B .

(6)

where the effective interaction is given by
Uef f = Vπ + V

1V†
E − H5q(7)

with the corresponding diagram shown in Fig. 3.
Details of the computational method is found in
Refs. [5, 6]. Here we list some important ideas.
• The OPEP takes on the form in momentum space,q = p0 − p,
 πM M πBB ggA
Vπ (q) = − A
4fπ2(S 1 · q )(S 2 · q )×T̂1 · T̂2 ,(8)
q 2 + m2π
πM M
πBB
where gA
, gA
are the axial coupling constantsfor mesons and baryons, respectively, and S1,2 are
the spin operators for the relevant hadrons. The axial coupling constants are then determined from the2

3pion-quark coupling by constructing the wave func4370 MeV below the Σ∗c D̄ threshold, and three scattered
tions of mesons and baryons by the quark model.
around 4470 – 4530 MeV below the Σ∗c D̄∗ . As expected
We note that the resulting coupling constants satthese states have dominant component of the correspondisfy the heavy quark symmetry for D̄(∗) mesons anding threshold particles, forming the heavy-quark multi(∗)
plets;…the doublet of PHYS.
Σc D̄∗ andtripletof 091502
Σ∗c D̄∗ . (2020)
PC Σ̄
PENTAQUARKSWITHCHIRALTENSOR
AND
QUARK
REV.
D 101,
baryons.
The
form
factorisalso
constructedcby the quark model wave functions. In this way we! M B "for V .
can fix all parameters
g g ðŜ1 · πqÞðŜ2 · qÞπV ðqÞ ¼ − A 2A
ð6ÞT̂ 1 · T̂ 2 ;2
q2 þ• For the coupling4f
ofπ M B-5q,
wememploythe idea ofπ
the spectroscopic factor where we assume that thewhereŜ is theofspin
T̂ is theisospin operator.
strength
the operator
M B-5q andcoupling
is proportionaltogBA isthetheprobability
axial vectorofcouplingthedistancecorresponda M B constantstate at ofzero
toing meetbaryons.
a 5q1 state. The five quark states are formed byThe
couplingof the various
MB channels,
i and j, to [7].
thetakinginto accountcolor configurations
five-quark(5q)channels,
α, gives
rise toMan
effective
Thus therelativestrengthof various
B-5q
couplings areinteraction,
V 5qfixed.
,• Another dynamicalof theX issue is the
1 treatment
hijV 5qtermjji ¼of the
hijVjαihαjV † jji;the secð7Þ
non-local
effective interaction,
5q
E
−
E
α
α
ond term of Uef f . Observing that the energies of
5q states H5q are estimated around ∆E ∼ 400 MeV
where
V represents
therelevanttransitions
between energy
the MB regionand 5qhigherthan5q thethresholdis the eigenenergy
a 5q channel.
WechannelsandweEαapproximateof M B,
it as a ofcontactterm withfurther
introduced
theGaussian
following typeassumption:
smearedby thelocal potential ofrange 1 fm.
hijVjαi ¼ fhijαi;ð8Þ• Finally, one unknown parameter is the overall coupling
M B-5q which
couplings,
f [5, 6].
where
f isstrength
the onlyfor
freetheparameter
determines
the
This
is
the
only
one
parameterofourhybridmodel
overall strength of the matrix elements. In order to calculate
determinedin comparisonand
with
data.
the thathijαi,canwe beconstructthe meson-baryon
five-quark
functionsin using
the standard
nonrelativistic
Inwavethis way,once fexplicitly
is fixed by
one experimental
inquarkwith astrengths
harmonicare
oscillator
confiningpotential.
put
the modelinteraction
completely
determined.
The derived potential hijV 5q jji turned out to give similar
results to those derived from the quark cluster model [7].
RESULTS
ANDofDISCUSSIONSThe energies
and widths
the bound and resonant states
were obtained by solving the coupled-channel Schrödinger
πequation
theby
OPEP,
ðrÞ, and 5qmethod.
potentialWe
V 5q
ðrÞ,
Eq. (5) iswith
solvedusingVvariational
used
the Gaussian basis functions as trial functions [8]. To ob5q
þ VemployedðrÞÞΨðrÞthe
¼ complexEΨðrÞ; scalingð9ÞðK þV π ðrÞwe
tain resonancestates,method [9]. In Fig. 4, the results are shown for two f ’s
where
K whereis the kineticof Boththe meson-baryon
f /ff0 ∼ 6 energy
MeV [6].
experimentalsystemdata
0 = 50andtheoryΨðrÞ isresults
the wavemeson-baryon
systems
andarefunction
shown ofbythe
rectangular
boxes
with
withmiddle
r being
the relative
distancetobetween
the center
their
pointscorresponding
the masses
as indi-of
massbyofthe
the numbers
meson andthat ofand
thetheirbaryon.
The coupledcatednearby,verticallengthð%Þ
decay
widths.
The boxed
are forstateschannels
included
are all numbers
possible onesof thoseΣc D̄ð%Þ
and
ð%Þ by the experiment and the
P corresponding ones
observedΛc D̄ which can form a given J and isospin I ¼ 1=2.
of themodel. (9) is solved by using variational method. We
Equation
For smaller
value basis
f /f0functions= 50, observed
states Pc (4312)
used
the Gaussian
as trial functions[56]. InandPc (4440)
areresonancewell explained
only for theirmassesorder
to obtain
states,not
we employed
the complex
butalso fordecay[57].
widths. For Pc (4457), f /f0 = 50 does
scaling
methodnot hold
it may survive
as a virtual
In Fig.a 1resonance,
and Table but
I, experimental
data [1,34]
and ourstate.
For thef /f0The= centers
80, all ofstatesare inboundpredictions
are larger
compared.
the barsFig. 1
withtendencybut
the levels
are loweredduetoare similar
located at
the central
valuesof pentaquark
masseswhilestronger
attraction.
Thesetoobserved
stateswidths
have awith
domitheir lengths
correspondthe pentaquarkthenantcomponents
as indicated
Eq. is(3).
exceptionof Pc ð4380Þ
width, bywhich
too large and does
In addition
to shown
the observed
not
fit into theenergy states,
region. our
Themodelboxed predicts
numbersotherstatesnearofthe
thresholds;
one ataround–are themassesthe recently
observed
states[34],4350
and the1
In our previous publication [20], there were a few errors in thematrix elements, which are corrected in this paper. After thecorrections, however, important results of our discussions remainunchanged.

FIG. 1. Experimental data (EXP) [1,34] and our results of
FIG. 4. Experimental data (EXP) [1, 2] and our results ofmasses and widths for various P states. The horizontal dashedmasses and widths for various cPc states [6]. The horizontal
lines show the thresholds for corresponding channels and values
dashed lines show the thresholds for corresponding channels
in therightinaxis
isospinones
in units onesof MeV.
The
andvaluesthearerightaxisaveraged
are isospin
averagedin units
centers
of
the
bars
are
locatedatthecentralvaluesofpentaquarkof MeV. The centers of the bars are located at the central
massesofwhile
their lengths
correspondto lengths
the pentaquark
widthsvaluespentaquark
masseswhile theircorrespondtowithpentaquark
the exceptionof Pcwithð4380Þ
thewidthsthewidth.
exception of Pc (4380) width.

corresponding predictions in our model. The dashed lines
are for threshold values. Our predicted masses and the
decay widths are shown for the parameters f=f0 ¼ 50 and
f=f0 ¼ 80. Here, f 0 is the strength of the one-pionexchange diagonal term for the Σc D̄% meson-baryon channel, f 0 ¼ jCπΣ D̄% ðr ¼ 0Þj ∼ 6 MeV (see Ref. [45]). Settingcthe free parameter f=f0 at f=f0 ¼ 50, we observe that bothþ
masses and widths of Pþc ð4312Þ and Pc ð4440Þ are reproduced within the experimental errors. However, the statecorresponding to Pþc ð4457Þ is absent in our results, where
the attraction is not enough. Increasing the value of f=f0 to
70, the state with JP ¼ 1=2− appears below the Σc D̄%threshold, and at f=f0 ¼ 80 the mass and width of this state
are in reasonable agreement with Pþc ð4457Þ. However, as
shown in Fig. 1, the attraction at f=f0 ¼ 80 is stronger thanFIG.
5. f=fComparing
results
andofwithout
tenthat athence
the with
massesthe otherthestates
0 ¼ 50 andthe
sor
force
of the OPEP for the states around the Σc D̄∗ andshift
downward.
∗Σ∗c D̄
[6]. Thelabel
stands for theWe thresholds
find as expected
that
the ’without
dominantT’componentsofresult without the OPEP tensor force, while the label ’withthese states are nearby threshold channels and with theT’ stands for that with the OPEP tensor force. PThe same−
quantum isnumbers
convention
adopted as
as infollows:
Fig. 4. Σc D̄ with J ¼ 1=2
þ
%
P
−
þ
[Pc ð4312Þ], Σc D̄ with J ¼ 3=2 [Pc ð4440Þ] and with
JPDetailed¼ 1=2− discussions[Pþ
meson-baryon
molecularc ð4457Þ] are
discussed inRef. [6]; states.
here weLet
uscompareour
results
with
the
ones
reported bybriefly summarize important features.
other works. In Ref. [36], the assignments of the quantum
numbersPc ð4440Þ
andattraction
Pc ð4457Þ throughare different
from• The for
OPEP
providesthe tensor%
ours. force
Since inthese
twostatesarelocatednear
ΣthresholdD̄c is the samethe second order process. Thisand bothstates havethe thenarrowwidths, where
it is naturaltomechanismas fordeuteron,
only theconsider them to form the J ¼ 1=2 and 3=2 states in
S-wave. It is emphasized that in our model the spin 3=2
state (4440) is lighter than the spin 1=2 state (4457). In
Ref. [37], they studied seven heavy quark multiplets ofΣc D̄, Σc D̄% , Σ%c D̄, and Σ%c D̄% , and considered two options of

4
OPEP is sufficient to generate the bound states ofthe binding energy 2 MeV by setting the cutoff parameter in the form factor around 800 MeV which
is very consistent √with the core size of the nucleon
rc ∼ 0.6f m; Λ ∼ 6rc ∼ 800 MeV. In the present
case, the OPEP alone does not produce sufficientattraction. The main reason is that the couplingstrength is smaller than for the nucleon due to thesmaller number of light valence quarks that couple
to the pion.
• Another source of attraction is provided by the coupling to the 5q states. When their masses are higher
than the relevant threshold energies E, the negative
denominator E −M5q < 0 guarantees the attractive
contribution. The strength, however, is not determined theoretically in the present study which wasthen fixed in comparison with data.
• The level ordering of heavy quark multiplets isdominated by the tensor force of the OPEP. For
instance, the larger attraction due to the second
order tensor force lowers the 3/2− state of Σc D̄∗than the 1/2− . The same mechanism works for the
triplet of 5/2− , 3/2− , 1/2− of Σ∗c D̄∗ . This observation contrasts to other model prediction [10].
• The tensor force due to the OPEP is also important to reproduce the observed widths As shownin Fig. 5 where the results are compared with andwithout the OPEP, clearly indicating the importance of the OPEP.
Having observed the above features, important conclusion of our present study is the following prediction forthe spin and parity identification,
Pc (4312) : 1/2− , Pc (4440) : 3/2− , Pc (4457) : 3/2−

where Pc (4312) is a heavy quark spin singlet, andPc (4445), Pc (4457) form a heavy quark spin doublet. Furthermore, we expect the existence of other states formedby Σ∗c D̄∗ . At present, their spin and parity are not yet
determined experimentally. It is crucial to know the spin
and parity of these states as well as the further search forthe other states.

Acknowledgments
We thank all other collaborators, Elena Santopinto,
Hugo García-Tecocoatzi, Alessandro Giachino. These
works are supported in part by the Grant-in-Aid for Science Research (C) 26400273, No. JP17K05441 (C) and
Scientific Research on Innovative Areas No. 18H05407.

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