Compact pentaquark structures
Abstract
We study the possibility that at least one of the two pentaquark structures recently reported by Aaij:2015tga (); Aaij:2016phn (); Aaij:2016ymb () could be described as a compact pentaquark state, and we give predictions for new channels that can be studied by the experimentalists if this hypothesis is correct. We use very general arguments dictated by symmetry considerations, in order to describe the pentaquark states within a group theory approach. A complete classification of all possible states and quantum numbers, which can be useful both to the experimentalists in their search for new findings and to theoretical model builders, is given, without the introduction of any particular dynamical model. Some predictions are finally given by means of a GürseyRadicati (GR) inspired mass formula. We reproduce the mass and the quantum numbers of the lightest pentaquark state reported by () with a parameterfree mass formula, fixed on the wellestablished baryons. We predict other pentaquark resonances (giving their masses, and suggesting possible decay channels) which belong to the same multiplet as the lightest one. Finally, we compute the partial decay widths for all the predicted pentaquark resonances.
pacs:
Multiquark particles, 14.20.c Baryons;14.65.Dw Charmed quarks;12.39.x Phenomenological quark models;02.20.a Group theoryI Introduction
The collaboration has recently reported the observation of exotic structures in
decay Aaij:2015tga (), further supported by another two articles of the
collaboration Aaij:2016phn (); Aaij:2016ymb ().
The decay can proceed according to the diagram
in fig. 1, which involves conventional hadrons:
(1) 
or it can be characterised by exotic contributions, which are referred to as charmoniumpentaquark states (fig. 2):
(2) 
The collaboration found two resonant structures: a lower mass state at ,
with a width of , and a higher mass one at , with
a width of , in the invariant mass spectrum.
Moreover, according to the collaboration Aaij:2015tga (), the preferred assignments
are and , respectively.
Since the observation of the two resonant structures, many explanations have been proposed for
the pentaquark states.
Mesonbaryon molecules were suggested in
Karliner:2015ina (); Chen:2015loa (); Roca:2015dva (); He:2015cea (); Meissner:2015mza (); Chen:2015moa ().
Pentaquark states of diquarkdiquarkantiquark nature were suggested in Maiani:2015vwa (); Wang:2015epa (), and soliton states in Scoccola:2015nia ().
The molecular interpretation works well for the heaviest resonant state (see, for example
Karliner:2015ina ()).
Therefore, in this study, we focus on the lightest pentaquark structure (with ),
by means of a multiquark approach.
From the quantum numbers of the lightest resonant state, we show that
it can be described as
a pentaquark state with spin .
We show that the ground state multiplet of the charmonium pentaquark states
is a octet, and we studied all the charmonium pentaquark states which
belong to the octet, predicted their masses, and
suggested possible decay channels in which the experimentalists can observe them.
By using an effective Lagrangian Kim:2011rm () for
the coupling,
in combination with the branching ratio upper
limit extracted by Wang Wang:2015jsa (), and with our predicted masses,
we compute the partial
decay widths for the predicted pentaquark resonances.
Ii Classification of the multiplets as based on symmetry properties
In order to classify the pentaquark multiplets, we made use, as much as possible,
of symmetry principles,
without introducing any explicit dynamical models.
We made use of the Young tableaux technique, adopting for each representation the
notation
, where denotes the number of boxes in the th row
of the Young tableau,
and is the dimension of the representation.
In agreement with the hypothesis Aaij:2015tga (),
we think the charmonium pentaquark wave function as
where is a light quark and is the heavy charm quark.
Let us first discuss the possible configurations of quarks
in the system.
The can be in a colour octet or singlet with spin 0 or 1.
The colour wave function of the system must be
an singlet, so
the remaining three light quarks are also in a colorsinglet or in a coloroctet.
The orbital symmetry of the quark wave function depends on the
quantum numbers of the resonant state
.
Indeed, the parity of the pentaquark system is:
(3) 
and so must be even.
The total angular momentum is , and so or . In this paper, we hypothesise that the lightest charmonium pentaquark state
reported by the collaboration is a ground state pentaquark with ,
and so each quark is in wave.
The three light quarks must satisfy the Pauli principle.
As a consequence, since the orbital part is completely symmetric,
the spinflavour and the colour part
are conjugate: spinflavour symmetric state if they are in a colour singlet;
or spinflavour mixed symmetry state if they are in a colour octet.
Therefore, the allowed spinflavour pentaquark configurations are
a 56plet () and a 70plet () which correspond respectively
to the three light quarks in a colour singlet and in a colour octet.
In Tab. 1 the analysis of the flavour and spin content of the spinflavour
56plet and of the 70plet , i.e. their decomposition into
the representations of , is reported.
The 70plet contains an flavour octet and a decuplet , while the 56plet contains an flavour singlet , two octets and a decuplet . The allowed flavour representations to which the charmonium pentaquark states can belong are therefore:
(4) 
Since the charmonium pentaquark state, as reported by , has a quark content , it does not have strange quarks, and so the strangeness . In the case of flavours (), the hypercharge is defined as:
(5) 
where is the barionic number and is the strangeness.
Since the charmonium pentaquark state, as reported
by LHCb, has a quark content , it does not have
strange quarks, and so the strangeness ,
the charm is 0, the barionic number , and then must be equal to 1.
Therefore the pentaquark state must be found in a submultiplet of the
allowed flavour states of Eq. 4.
Following this reasoning, we must exclude the singlet , because it
does not have any submultiplets;
therefore, the remaining possible multiplets for the charmonium
pentaquark states are:
(6) 
Iii The extension of the GürseyRadicati mass formula
In order to determine the mass splitting between the multiplets of Eq. 6, we made use of a GürseyRadicati (GR)inspired formula F.Gursey (). As yet, there is experimental evidence of only two charmonium pentaquark states. This is not sufficient to determine all parameters in the GR mass formula, and then to predict the masses of the other pentaquarks. For this reason, we use the values of the parameters determined from the threequark spectrum (see Tab. 2), assuming that the coefficients in the GR formula are the same for different quark systems. The simplest GR formula extension which permits us to distinguish the different multiplets of is
(7) 
where is a scale parameter: this means that, for example, in baryons each quark gives a contribution of
roughly to the whole mass.
and are the isospin and hypercharge, respectively, while is
the eigenvalue of the
Casimir operator.
Finally, is a counter of quarks or antiquarks.
This term takes into account the mass difference
between a quark (or a antiquark) in relation to the light quarks ().
The coefficients and the scale parameter have been fixed by using
the wellestablished baryons spectrum.
Tab. 2 reports the baryons used to fix the parameters in Eq. 7 , the
multiplet which they were assigned to, the corresponding eigenvalues of
the Casimir operator , their quantum numbers,
and the values of .
In Tab. 3 all the parameters, with their corresponding values,
are reported.
baryon  spin  

multiplet  
0  0  
0  1  
1  1  
1  
1  
0  1  
0  1 
value ()  940,8  23,6  157,3  32,0  1365,7  52,5 

In order to show the reliability of the values obtained with the GR mass formula extension, we calculated the predicted mass of the two charmed baryons and reported by the PDG PDG () . The quantum number assignments and the predicted masses are reported in Tabs. 4 and 5, respectively.
baryon  spin  

multiplet  
1  1  
1 
baryon  isospin  exp. masses  predicted masses 

configurations  ()  ()  
2526  
2526  
2526  
2646  
2646 
Iv Application of the GR formula to the pentaquark states
In Eq. 6 we reported the possible multiplets for the charmonium pentaquark states. We hypothesise that the charmonium pentaquark state , reported by the collaboration, belongs to the lowest mass multiplet. According to the GR formula 7, the mass splitting between the different multiplets of Eq. 6 is due to the different eigenvalues of the Casimir operator , and so it is proportional to the coefficient (reported in Tab. 3). Since is positive (), the lowest mass multiplet is the one with the minimum Casimir operator eigenvalue, and so it is the octet (see Tab. 6). In Tab. 6, each multiplet, with the corresponding eigenvalues of the Casimir operator , is reported.
multiplet  

6  
3 
From Tab. 6, we can see that the lowest mass charmonium pentaquark state is the octet. Therefore, in this octet, we expect to find the charmonium pentaquark state reported by the collaboration. In the following, we focus on the octet charmonium pentaquark states, and we apply the GR mass formula 7, with the values of the parameters reported in Tab. 3, to each state of the octet, in order to predict the corresponding mass. As regards the notation, we indicate a charmonium pentaquark state (, with ) by , where is the number of strange quarks of a given pentaquark state, is the pentaquark’s electric charge, and the predicted mass. The state identified with the one reported by the collaboration (), and the other predicted charmonium pentaquark states of the octet, are reported in Fig. 3 . We observe that the charge state has just the same quantum numbers as the lightest resonance (charge, spin, parity) reported by the collaboration.
Its theoretical mass, predicted by means of our GR formula extension, is .
Despite the simplicity of the approach that we used, this result is in agreement with the mass reported by the
collaboration: .
Our compact pentaquark approach predicts that it is a member of an isospin doublet, with hypercharge .
If the compact pentaquark description is correct, also the other octet states should be found by the collaboration.
On the contrary, if the pentaquark is mainly a molecular state, it is not necessary that all the states of that multiplet exist.
V Decay channels
We will now explore the possible decay channels in which the other predicted states of the octet can be observed. These channels will be described in detail. The state is a part of an isospin doublet. In order to observe its isospin partner (), a possible decay channel could be:
(8) 
The corresponding Feynman diagram is reported in Fig. 4 .
With respect to the other charmonium pentaquark states of the octet, with strangeness, we have to focus on the decays of bottom baryons with strange quarks. Let us consider the following decay:
(9) 
This decay is present in nature and was discovered by the collaboration (V.M.Abazov_Xi_b ()). In analogy with the exotic decay of Fig. 2 , we can expect that also in the case of baryon there is another possible exotic decay channel:
(10) 
where and have the same quark content (), and belong to the isospin triplet, and to the isosinglet, respectively (see Fig. 3). Since they have the same quark content and both are neutral, they can both come from the decay. The charmonium pentaquark state can be observed in the following decay process:
(11) 
The difference between the two suggested decays for the baryon (Eq. 10, and Eq. 11)
is in the final state: in the case of the final state of Eq. 10 ,
a couple of quarks comes from the vacuum, while, in the decay of Eq. 11 ,
the couple of quarks is replaced with a couple .
The baryon is a member of an isodoublet.
The decay of its isospin partner
(12) 
is probably the most important one from the experimental point of view, since all the final state particles are charged and, therefore, easier to detect.
In order to have a final pentaquark state with two strange quarks , we need a double strange baryon in the initial state. The known decay channel of the baryon is:
(13) 
This decay was discovered by the detector at the Fermilab Tevatron collider V.M.Abazov_Omega_b (). Another possible decay channel may be, in analogy with the exotic decay channel of Fig. 2 :
(14) 
The state of Eq. 14 is a part of an isospin doublet (see Fig. 3). In order to observe its isospin partner (), a possible decay channel could be:
(15) 
The difference between the decays of Eq. 14 and that of Eq. 15 is, respectively, the creation of a couple and from the vacuum.
Vi Partial decay widths
We adopt the effective Lagrangian for the couplings from Ref. Kim:2011rm () as follows:
(16) 
where is the pentaquark field with spinparity , and are the nucleon and the fields, respectively. The matrices are defined as follows:
(17) 
As noticed by Wang Wang:2015jsa () in the pentaquark state decays into , the momentum of the final states are fairly small compared with the nucleon mass. Thus, the higher partial wave terms proportional to and can be neglected, so we only consider the first term in Eq. (16). This approximation leads to the following expression for the partial decay width in the channel Oh:2011 ():
(18) 
with
(19) 
The kinematic variables and in Eq. (18) are
defined as
and .
Unfortunately, the branching ratio is
not known at present, so the coupling constant of Eq. 19 is unknown.
However, by using our pentaquark mass predictions,
we can provide an expression of the partial decay widths for the
pentaquark states with open strangeness.
For example, the partial decay width in the channel
is given by:
(20) 
and the coupling constant is:
(21) 
The expressions for the partial decay widths of the , , and channels are listed in Table 7.
initial state  channel  partial width () 

Since the pentaquark states were observed in channel, it is natural to expect that they can be produced in photoproduction via the s and uchannel process. Wang et al. Wang:2015jsa () calculated the pentaquark states cross section in photoproduction and compared it with the present experimental data (Camerini (), Anderson (), Gittelman ()). The coupling between and the two pentaquark states are extracted by assuming it accounts for their total width and , respectively. As a result, they found that if one assumes that the channel saturates the total width of the two pentaquark states (that is ) one significantly overestimates the experimental data. In conclusion they found that to be consistent also with the present photoproduction data, it is necessary that the branching ratio for both the pentaquark states is . Thus, if we use the upper branching ratio limit extracted by Wang Wang:2015jsa (), that is , we obtain that the partial decay width for the channel is
(22) 
where as reported by the collaboration, is . The numerical results for the other channels are listed in Table 8.
initial state  channel  partial width () 

Vii Conclusions
The collaboration has recently reported the observation of two exotic structures in
channel Aaij:2015tga (), which they referred to as charmonium pentaquark states
( with a quark content )
further supported by another two articles by the collaboration Aaij:2016phn (); Aaij:2016ymb ().
The significance of each of these states is more than 9 standard deviations. The lightest one has a
mass of and a width of , while the heaviest has a mass
of and a width of .
The preferred assignments, according to the collaboration Aaij:2015tga (), are and ,
respectively.
The earliest prediction for the charmonium pentaquark with was given by J. J. Wu et al. Wu ().
The heaviest pentaquark state has been apparently well explained by means of a molecular approach Roca:2015dva (); Karliner:2015ina (),
and it was also predicted in a molecular approach before the discovery by Xiao:2013yca (),
in a coupledchannel
unitary approach.
As regards the lightest one, molecular models have also been proposed,
but the predictions are not so good as
for the heaviest state Roca:2015dva (); Karliner:2015ina ().
Some predictions of its mass and quantum numbers were given by Yuan:2012wz ()
in 2012, by means of a
potential quark model approach, but these predictions depend strongly on the
particular interaction used:
colourmagnetic interaction (CM) based on onegluon exchange, chiral interaction
(FS ) based on meson exchange,
and instantoninduced interaction (Inst.) based on the nonperturbative QCD vacuum
structure.
In this present study, we focused on describing the lightest resonant state
(),
by means of a multiquark approach.
An extension of the original GR mass formula F.Gursey () which correctly
describes the charmed baryon
sector was performed (Tab. 5), and also proved able
to give an unexpected prediction
for the mass of the lightest pentaquark state , which is in
agreement with the experimental
value within one standard deviation.
We found that the lightest pentaquark state belonged to the
octet .
The theoretical mass of the lightest pentaquark state
predicted by means of the GR formula extension (Eq. 7) is ,
in agreement with
the experimental mass . We also predicted other
pentaquark states, which belong to
the same multiplet as the lightest resonance ,
giving their mass, and suggesting
possible decay channels in which they can be observed.
We have finally computed the
partial decay widths for all the suggested octetpentaquark decay channels.
As the decay
is expected to be dominated by resonances
Aaij:2015tga (), we observe that
the poor knowledge about the excited states can
affect the estimation of the parameters of the two pentaquark
resonances.
Moreover, as was noticed by Wang Wang:2015jsa (),
if the two pentaquark candidates are genuine states,
their production in photoproduction should be a natural expectation.
For these reasons, on the one hand it is important to increase our knowledge about
the missing excited states with new experiments
(proceedings ()), in order
to improve the analysis and to extract with more precision the two pentaquark
masses and widths.
On the other hand, a refined measurement of the photoproduction
cross section would provide more information about the
nature of the pentaquark states.
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Conference: C16â 02â 01.1, p.127â 142 Proceedings
eâ Print: arXiv:1604.02141 [hepâ ph]