Heavy pentaquark spectroscopy in the diquark model

$QQ^\prime qq\bar q$ pentaquarks are studied in a potential model, under the hypothesis that they are composite objects of two diquarks and one antiquark. The interaction between two coloured objects includes two contributions, one based on the $q\bar q$ potential in QCD, computed in the gauge/string duality approach, and another describing the spin-spin interaction. The model has been extended to investigate pentaquarks with different quark content, as $Qqqq\bar q$ and $Qqqq\bar Q$, the latter including the states observed by LHCb, $P_c(4380)^+$ and $P_c(4450)^+$.


Introduction
Quantum ChromoDynamics (QCD) forsees the existence of quark-antiquark states (mesons) and three-quark states (baryons), as well as multiquark states [1]- [2] such as tetraquarks, comprising two quarks and two antiquarks, and pentaquarks, comprising four quarks and one antiquark. The only requirement for these states is to be color singlets. Although most of ground state mesons and baryons are experimentally well known, many recently observed states are under discussion since their quark content and/or spin/parity are uncertain [3]- [4]; for a review on possible exotic states see [5]- [6]- [7]. One of the most intriguing cases is the X(3872), first observed by the Belle Collaboration [8]. The spin parity assignment 1 ++ is compatible with a meson state in the quark model [9], however its decay channels suggested a possible interpretation as a four-quark state [10]- [11]. Recently, LHCb observed two resonances in the J/ψ p channel in Λ 0 b decay, labeled P + c , with mass 4380±8±29 MeV and 4449.8±1.7±2.5 MeV, opposite parity and spin 3/2 and 5/2, compatible with heavy pentaquark ccuud states [12]. Previously, in 2003 researchers from the SPring-8 laboratory in Japan [13], ITEP in Russia [14], Jefferson Lab in Virginia [15] and from the ELSA accelerator in Germany [16] announced the observation of the Θ + pentaquark, consisting of four light quarks and a strange antiquark, but such evidence has not been confirmed by later experiments [17].
Considering that many experimental results will be allowed in the next few years by the increasing luminosity at experiments like LHCb at CERN and Belle-II at SuperKEKB, in this paper we compute the masses of heavy pentaquarks using a potential model. Our results can be compared with outcomes of different studies in the past few years in this sector. For example, the masses of QQqqq, Q being a heavy quark and q a light quark, have been computed in [18], while [19]- [20]- [21] focus on the hidden-charm ccqqq pentaquarks, having the same quark content as the states P c (4380) + and P c (4450) + observed by LHCb. In [22] a classification of all possible QQqqq states and quantum numbers has been presented. QQqqq states have been considered in, e.g., [23]- [24].
In the investigation of multiquark states, one of the most discussed issues is the existence of possible internal structures. In this respect, the main hypotheses for pentaquarks are that they could be compact states, relying on the interaction among two diquarks and an antiquark [25], or molecular states, relying on the interaction between a baryon and a meson [26]- [27]. Following the former approach, in Section 3.1 we compute the masses of QQ ′ qqq pentaquarks using the model introduced in Section 2. An attempt to study pentaquarks with a different quark content is put forward in Sections 3.2-3.3. Section 4 contains discussions and conclusions.

Model
We study pentaquarks in the potential model introduced in [28], in which meson masses are computed by solving the wave equation: where m 1 and m 2 are the masses of the consituent quark and antiquark, V (r) is the quarkantiquark potential, M and ψ are the mass and wavefunction of the meson. Eq. (1) arises from the Bethe-Salpeter equation in QCD by considering an instantaneous local potential of interaction. Differently from the Schrödinger equation, it has relativistic kinematics. Eq.
(1) can be also used to study pentaquarks if a pentaquark is considered as the bound state of two diquarks and an antiquark. The strategy consists in computing at first diquark masses and wavefunctions from interactions between single quarks, then the mass and wavefunction of the four-quark state formed by two diquarks, and finally the mass and wavefunction of the pentaquark resulting from the interaction between the four-quark state and one antiquark, as shown in Fig. 1. We call this model A. In this picture, each interaction is between two objects, as in Eq. (1). Model A is based on the diquark-diquark-antiquark description of pentaquarks [25], and on SU(3) color group arguments, according to which two quarks (in the 3 representation of the group) can attract each other forming a diquark (in the3 representation), and similarly two diquarks can attract each other forming a four-quark state (in the 3 representation of the group), and finally a four-quark state plus an antiquark form a color singlet (the pentaquark). We adopt the one-gluon-exchange approximation, in which the potential of interaction between two quarks (3 ⊗ 3) is equal to half the qq potential (3 ⊗3) [29].
The qq potential used in (1) for each two-body interaction has three terms: where V 0 is a constant term, V QCD (r) represents the color interaction and V spin (r) the spin-spin interaction.
For V QCD (r) we use the potential found in [30] in a phenomenological model inspired by the AdS/QCD correspondence by computing the expectation value of a rectangular Wilson loop. The potential is given in parametric form: where r is the distance between the quark and antiquark, c and g are parameters.
The term V spin (r) is given by [31]: where σ is a parameter defining the smeared delta function and S is the spin of the interacting particle. The parameter A is proportional to the strong coupling constant α s in the one-gluonexchange approximation.
A cutoff at small distance is introduced to cure the singularity of the wavefunction, fixing the potential (2) at the value V (r M ) for r r M , with r M = k M in case m 1 = m 2 , and r M = k ′ M in case m 1 = m 2 [32]- [33]. k and k ′ are two parameters and M is the mass of the final state.
Notice that both the one-gluon-exchange approximation and the use of an instantanous potential can be properly applied only to heavy states, in which at least one of the two interacting particles is heavy, i.e. contains a charm or bottom quark. Therefore, we compute masses of pentaquarks containing at least one heavy quark. Moreover, at each step we only consider states with orbital angular momentum ℓ=0.
We solve the Salpeter equation (1) through the Multhopp method [33], which allows one to transform an integral equation into a set of linear equations containing variables called Multhopp's angles. We fix the parameters of the model as in [28], where the masses of heavy mesons have been fitted to their experimental values: Two values for the parameter A in (4) have been introduced, in order to take into account the two scales, O(m c ) and O(m b ), at which α s must be computed: A b is used for states comprising at least a beauty quark and A c otherwise. The model, with this choice of parameters, has been able to predict with very good accuracy the mass of η b [34], observed soon after by the BaBar Collaboration [35].
As a first step, diquark masses are obtained by solvinq Eq. (1) with potential (2) divided by a factor 2 and a cutoff at r = r M , as done in [28]. In the second step, we use the Salpeter equation to study the interaction between two diquarks. The diquark-diquark potential is assumed to be the same as between two quarks in a diquark: this suggests to adopt again the potential (2) divided by a factor 2. However, diquarks are extended objects, so we take into account the structure of the diquarks by defining a smeared potential [28]: In this equation ψ d is the diquark wavefunction, N is a normalization factor. Since |ψ d (r)| 2 is strongly peaked at r ∼ 0, we cut the integral at the peak value of the function u d (r) = rψ d (r) [28]. In the last step, the potential producing a singlet state is obtained from a convolution of (2) with the diquark-diquark ψ dd wavefunction [36]: with N ′ a normalization factor.

QQ ′ qqq
Using the model A introduced in the previous Section, we compute the masses of pentaquarks comprising two heavy quarks. Each heavy quark forms a diquark with one light quark, therefore these states can be well described in this framework. The results are shown in Table  1, where q = {u, d}, q ′ = {u, d, s} and Q, Q ′ = {c, b}. Since we set ℓ = 0 in all the cases, the states have negative parity. We adopt the following notation: [Qq] diquark has spin 0, and {Qq} has spin 1. As for tetraquarks [28], a large number of states with different spin is found when combining two diquarks and an antiquark. There are five spin-1/2, four spin-3/2 and one spin-5/2 states, as expected when combining five spin-1/2 particles. For the sake of completeness we should mention that not all the states can be considered in the spectrum since one must take spin, flavour and colour representations such that the total wavefunction of identical fermions (bosons) is antisymmetric (symmetric). More details about this topic can be found in [22]- [24].

Qqqqq
Let us consider pentaquarks with one heavy quark and four light quarks. Model A introduced in Section 2 can not be used here, since only one heavy diquark can be constructed, while the  other diquark would contain two light quarks. Nevertheless, we try to determine the masses of these states by considering the singlet state resulting from subsequent interaction of the heavy quark with a light quark, as a sequence of the two-body interactions sketched in Fig. 2. This model, labeled B, is described hereinafter. Although such a configuration of quarks inside the pentaquark is not usually expected, it is interesting to investigate this possibility, which can also be applied to QqqqQ states (see Section 3.3), the candidates for the peaks observed by LHCb in the hidden charm sector. The strategy of the computation is described in the following scheme, in which each step consists in solving Eq. (1) for the indicated particles: The first three interactions are between states in the same representation of the colour group (3 or3), while only the last one produces a colour singlet. We treat the last interaction in the same way as the one between a quark and an antiquark, i.e. by solving Eq. has been used to study the first three interactions as well, with a potential equal to half the qq potential [29]. The correction (6) to the potential has been considered. The masses of pentaquarks with one charm or one beauty are shown in Table 2. Since ℓ = 0, all the states have negative parity. Different states correspond to different spin combinations, in which [ ] indicates the combination having the lowest spin, while {} the one with the highest.

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In order to compare models B and A, we have computed again the spectra of QQqqq, now assuming that the interaction among quarks works as in model B, with results shown in Table  3. By looking at Tables 1 and 3, we find that the mass difference between spin-1/2 states is at most 50 MeV in the charm sector and 30 MeV in the bottom sector, while it is at most 10 MeV for states with spin 3/2 and 5/2 in the charm and bottom sectors.

QqqqQ
Model B can be used to study QqqqQ states, in order to compare the outcomes with the masses of the states observed at LHCb. The values of the masses are shown in Table 4. The states with hidden charm and spin 3/2 have masses in the range 4.64 − 4.71 GeV, the spin-5/2 one has mass 4.76 GeV, while the experimental masses are equal to 4380 ± 8 ± 29 MeV and 4449.8 ± 1.7 ± 2.5 MeV, respectively. It has been experimentally argued that the observed states P c (4380) + and P c (4450) + have opposite parity, but, since we do not know which one has negative parity, we try comparing both with theoretical predictions. The mass differences between theoretical and experimental results are equal to 260 − 330 MeV and 310 MeV, and cannot be justified by the low experimental error. The theoretical error is expected to be small as well: we can safely estimate it being 80 MeV, as the one found when studying meson spectra with the same model and set of parameters [28], so the predicted masses are significantly higher than the experimental ones.

More on QQqqq
Within this framework, i.e. finding pentaquark masses by a sequence of two-body interactions involving at least one heavy particle, another configuration for QQqqq states is allowed. Indeed,

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one can consider the interaction between a Qq diquark and a Qqq state, as done, e.g., in [37]- [38]. The spirit of the computation is similar to what has been done in previous Sections, and is explained by Fig. 3. The results for the pentaquark masses are in Table 5. A comparison among Tables 1-3-5 shows that the masses found in this configuration are larger than the ones found in model A and B.
Q qq q Q Q qq Q q

Conclusions
We have computed pentaquark masses in a potential model. We have exploited a relativistic wave equation describing the interaction between two states, and tried to accomodate

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pentaquarks in this framework by considering them as emerging from three subsequent interactions, as shown in Fig. 1, in the diquark-diquark-antiquark picture. In particular, the scheme has been used for pentaquarks with two heavy quarks (QQ ′ qqq). Then, we have studied Qqqqq pentaquarks introducing a different scheme of interaction, depicted in Fig. 2. The model has also been applied to QQqqq pentaquarks, with the same quark content as the states P c (4380) + and P c (4450) + observed by LHCb in the hidden-charm sector. Comparing the predicted mass of spin 3/2 and 5/2 cqqqc states with the experimental ones, we have found values ∼ 300 MeV higher than those measured by LHCb. Futher investigations could help to shed light on this discrepancy, and clarify if it is due to the approximations involved in the model. As a future study, it would be interesting to improve this potential model, making it more suitable for the description of exotic states, for instance by considering known masses of tetraquarks or pentaquarks as inputs when fixing the parameters, or by improving the choice of the potential of interaction, going beyond the one-gluon-exchange approximation.