Stability of pentaquarks with a two- plus three-body chromoelectric interaction

TL;DR

This paper investigates the stability of pentaquarks using a schematic SU(3) color symmetry model that incorporates two- and three-body chromoelectric interactions, extending previous work on baryons and multiquark states.

Contribution

It introduces a novel schematic model with algebraic arguments for chromoelectric interactions including two- and three-body parts to analyze pentaquark stability.

Findings

Chromoelectric interactions affect multiquark stability.

Model extends understanding of baryons, tetraquarks, and hexaquarks.

Provides insights into pentaquark spectrum stability.

Abstract

We study the stability of pentaquarks within a schematic model based on SU(3) color symmetry, by taking into account algebraic arguments leading to a chromoelectric interaction containing two- and three-body parts. It has already been proven that such an interaction can influence the spectrum of ordinary baryons and the stability of tetraquarks and hexaquarks.

Tables

Table 1. Table 1: Expectation values of the operator ( 9 ).

$state$	${\left[{\left(12\right)}^S\overline{5}\right]}_{\left[211\right]}$	${\left[{\left(12\right)}^A\overline{5}\right]}_{\left[211\right]}$	${\left[{\left(12\right)}^A\overline{5}\right]}_{\left[22\right]}$
${𝒞}_{ij\overline{k}}$	$\frac{5}{18}$	- $\frac{5}{9}$	$\frac{5}{18}$

Table 2. Table 2: The contributing terms to the matrix elements of operators defined by Eqs. ( 5 ) and ( 6 ) for the specific cases ⟨ 3 | d a ​ b ​ c ​ F 1 a ​ F 2 b ​ F 4 c | 3 ⟩ quantum-operator-product 3 superscript 𝑑 𝑎 𝑏 𝑐 subscript superscript 𝐹 𝑎 1 subscript superscript 𝐹 𝑏 2 subscript superscript 𝐹 𝑐 4 3 \langle 3|d^{abc}F^{a}_{1}F^{b}_{2}F^{c}_{4}|3\rangle and ⟨ 3 | d a ​ b ​ c ​ F 1 a ​ F 2 b ​ F ¯ 5 c | 3 ⟩ quantum-operator-product 3 superscript 𝑑 𝑎 𝑏 𝑐 subscript superscript 𝐹 𝑎 1 subscript superscript 𝐹 𝑏 2 subscript superscript ¯ 𝐹 𝑐 5 3 \langle 3|d^{abc}F^{a}_{1}F^{b}_{2}\overline{F}^{c}_{5}|3\rangle . The first column gives the color indices ( a ​ b ​ c ) 𝑎 𝑏 𝑐 (abc) , the second the corresponding constant d a ​ b ​ c superscript 𝑑 𝑎 𝑏 𝑐 d^{abc} , the third is the multiplicity of each term and the fourth and fifth columns the value of the matrix element for a given d a ​ b ​ c superscript 𝑑 𝑎 𝑏 𝑐 d^{abc} .

$\left(abc\right)$	$d^{abc}$	$m^{abc}$	$⟨3\left|d^{abc}{F}_1^a{F}_2^b{F}_4^c\right|3⟩$	$⟨3\left|d^{abc}{F}_1^a{F}_2^b{\overline{F}}_5^c\right|3⟩$
118	$\sqrt{3}/3$	3	-1/288	-1/144
146	1/2	6	-1/384	-1/192
157	1/2	6	-1/384	-1/192
228	$\sqrt{3}/3$	3	-1/288	-1/144
247	-1/2	6	-1/384	-1/192
256	1/2	6	-1/384	-1/192
338	$\sqrt{3}/3$	3	-1/288	-1/144
344	1/2	3	-1/384	-1/192
355	1/2	3	-1/384	-1/192
366	-1/2	3	-1/384	-1/192
377	-1/2	3	-1/384	-1/192
448	-$\sqrt{3}/6$	3	-1/1152	-1/576
558	-$\sqrt{3}/6$	3	-1/1152	-1/576
668	-$\sqrt{3}/6$	3	-1/1152	-1/576
778	-$\sqrt{3}/6$	3	-1/1152	-1/576
888	-$\sqrt{3}/3$	1	-1/288	-1/144

Equations

$$H=\sum _i\frac{{{\vec{p}}_i}^2}{2m_i}-\frac{{\vec{P}}^2}{2M}+V_{2b}+V_{3b}$$

$$V_{2b}=\sum _{i<j}{V}_{ij}(\frac{7}{3}+F_i^a\cdot F_j^a)$$

$$F_i^a=\frac{1}{2}{\lambda }_i^a,(a=1,...,8)$$

$$V_{3b}=V_{ijk}={\mathcal{V}}_{ijk}{\mathcal{C}}_{ijk}$$

$${\mathcal{C}}_{ijk}=d^{abc}{F}_i^a{F}_j^b{F}_k^c,$$

$${\mathcal{C}}_{ij\overline{k}}=-d^{abc}{F}_i^a{F}_j^b{\overline{F}}_k^c$$

$${\overline{F}}_i^a=-\frac{1}{2}{\lambda }_i^{a\ast },(a=1,...,8)$$

$${\mathcal{C}}_{ijk}=\frac{1}{6}[C_{i+j+k}^{(3)}-\frac{5}{2}{C}_{i+j+k}^{(2)}+\frac{20}{3}]$$

$${\mathcal{C}}_{ij\overline{k}}=-\frac{1}{6}[C_{i+j+\overline{k}}^{(3)}-\frac{5}{2}{C}_{i+j}^{(2)}+\frac{50}{9}]$$

$$C^{(2)}=\frac{1}{3}({\lambda }^2+{\mu }^2+\lambda \mu +3\lambda +3\mu )$$

$$C^{(3)}=\frac{1}{18}(\lambda -\mu )(2\lambda +\mu +3)(\lambda +2\mu +3).$$

$$⟨{\mathcal{V}}_{ijk}⟩=c(⟨V_{ij}⟩+⟨V_{jk}⟩+⟨V_{ki}⟩).$$

$${\chi }_1=\frac{5}{3}+\frac{10}{9}c$$

$$⟨{\mathcal{V}}_{ijk}⟩=\frac{c}{3}(⟨V_{ij}⟩+⟨V_{jk}⟩+⟨V_{ki}⟩),$$

$$|1⟩=|(123{)}_1(4\overline{5}{)}_1⟩$$

$$|2⟩=|(123{)}_8^{\rho }(4\overline{5}{)}_8⟩$$

$$|3⟩=|(123{)}_8^{\lambda }(4\overline{5}{)}_8⟩.$$

$$|[{(12)}^S\overline{5}{]}_{[211]}(34{)}_{[11]}⟩$$

$$|[{(12)}^A\overline{5}{]}_{[211]}(34{)}_{[11]}⟩$$

$$|[{(12)}^A\overline{5}{]}_{[22]}(34{)}_{[2]}⟩.$$

$$\begin{array}[]{c|ccc}&|{[{(12)}^{A}\,{\overline{5}}]}_{[211]}(34)_{[11]}\rangle&|{[{(12)}^{A}\,{\overline{5}}]}_{[22]}(34)_{[2]}\rangle&|{[{(12)}^{S}\,{\overline{5}}]}_{[211]}(34)_{[11]}\rangle\\ \hline\cr|({123})_{1}({4{\overline{5}}})_{1}\rangle&\sqrt{\frac{1}{3}}&\sqrt{\frac{2}{3}}&0\\ |({123})^{\rho}_{8}({4{\overline{5}}})_{8}\rangle&-\sqrt{\frac{2}{3}}&\sqrt{\frac{1}{3}}&0\\ |({123})^{\lambda}_{8}({4{\overline{5}}})_{8}\rangle&0&0&1\end{array}$$

$$⟨i|V_{2b}|i⟩=3⟨i|\frac{7}{3}+F_1\cdot F_2|i⟩$$

$$+3⟨i|\frac{7}{3}+F_3\cdot F_4|i⟩+4⟨i|\frac{7}{3}+F_4\cdot F_{\overline{5}}|i⟩.$$

$$⟨F_i\cdot F_j⟩=⟨F_i\cdot F_{\overline{j}}⟩=-\frac{1}{3}$$

$$⟨V_{2b}⟩=20\sum _{i<j}{V}_{ij}.$$

$$⟨i|V_{3b}|j⟩=4⟨i|{\mathcal{C}}_{123}|j⟩+6⟨i|{\mathcal{C}}_{12\overline{5}}|j⟩$$

$$\begin{array}[]{c|ccc}&\langle 1|1\rangle&\langle 2|2\rangle&\langle 3|3\rangle\\ \hline\cr\langle 1|1\rangle&20+\frac{40}{9}~{}c&\frac{10\sqrt{2}}{6}~{}c&0\\ \langle 2|2\rangle&\frac{10\sqrt{2}}{6}~{}c&20-\frac{20}{9}~{}c&0\\ \langle 3|3\rangle&0&0&20+\frac{10}{9}c\end{array}$$

$$\begin{gathered} e_{1,2}=20+\frac{10}{9}c\pm \frac{10}{\sqrt{6}}c, \\ {e}_3=20+\frac{10}{9}c. \end{gathered}$$

$$-\frac{3}{2}<c<\frac{2}{5}.$$

$$-1.0\le c<0.4.$$

$$\Delta E=E(q^4\overline{q})-E(q^3)-E(q\overline{q})$$

$$E_i=(N+6)\hslash {\omega }_i$$

$${\omega }_i=\sqrt{5/6}{\omega }_0.$$

Full text

I Introduction

First mentioned by Gell-Mann in his original paper on the quark model of baryons and mesons GellMann:1964nj , the existence of exotic hadrons as systems with more than three quarks (antiquarks) has become an important challenge for QCD-inspired quark models. In practice the problem has been initiated by Jaffe who noticed that the color-magnetic hyperfine interaction produced a stable $uuddss$ hexaquark, called the $H$-particle, Jaffe:1976yi and stable $q^{2}{\overline{q}}^{2}$ systems Jaffe:1976ig

Ten years later, still based on the one-gluon exchange model Gignoux, Silvestre-Brac and Richard Gignoux:1987cn and independently Lipkin Lipkin:1987sk found that the strange anticharm pentaquarks $uuds{\overline{c}}$ and $udds{\overline{c}}$ are also stable against strong decays. To be stable, these particles had to contain a heavy flavor. Generally, one expects an increase in the stability if a multiquark system contains heavy flavors $Q=c$ or $b$. The above pentaquarks have negative parity, i. e. the parity of the antiquark. Later on, it was shown Stancu:1998sm that the Goldstone-boson exchange model favors positive parity pentaquarks and stabilizes systems of type $uudd{\overline{Q}}$, without strange quarks. On the experimental side the H1 Collaboration has found evidence for a narrow resonance interpreted as a $uudd\bar{c}$ pentaquark Aktas:2004qf , so far unconfirmed by other experiments. An earlier experiment of the E791 Collaboration Aitala:1999ij failed to obtain statistically significant signals for a narrow strange anticharm pentaquark. An early short review on the stability of multiquark systems can be found, for example, in Ref. Stancu:1999xc . The physics of exotic hadrons has recently been reviewed in the spirit of quark models Richard:2016eis in a more advanced picture with an expansion in the Fock space to allow additional quark-antiquark pairs in the wave function of baryons.

The recent observation of the hidden-charm pentaquarks $P_{c}(4380)$ with $J^{P}=3/2^{-}$ and $P_{c}(4450)$ with $J^{P}=5/2^{+}$ by the LHCb collaboration Aaij:2015tga has triggered a new strong wave of interest in pentaquarks. Competing models have already been proposed, see Ref. Burns:2015dwa for a recent review on the phenomenology of pentaquarks.

The stability of a few quark systems, like the stability of a few-charge systems depends on the masses of the fermions which are involved Richard:2017vry . Most of the results discussing the mass dependence are related to heavy-light tetraquarks of type $QQ\bar{q}\bar{q}$ as a function of the quark mass ratio $M/m$ Zouzou:1986qh ; Brink:1998as ; Janc:2004qn . To reach stability one needs a rather large $M/m$. The results are model dependent and are also influenced by the performance of the variational method, the most elaborate being that of Ref. Janc:2004qn , where the stability of the $cc\bar{u}\bar{d}$ was achieved. The study of the $Q\bar{Q}q\bar{q}$ is more complicated because the lowest threshold is $Q\bar{Q}+q\bar{q}$ [13]. In constituent quark models the chromomagnetic interaction also plays an important role. But in the limit of very heavy quarks this interaction vanishes and the result depends on a pure chromoelectric interaction. The kinetic energy decreases by increasing the constituent mass, so that one gets more binding.

Here we discuss a mechanism which could influence the stability of pentaquarks against strong decays in quark models due to a pure chromoelectric interaction, which contains a two- plus a three-body confining potential as suggested in Ref. Dmitrasinovic:2001nu . Usually constituent quark models contain a two-body $F_{i}\cdot F_{j}$ confining interaction only. Based on the algebraic argument that SUC(3) is an exact local symmetry of QCD which implies that QCD-inspired Hamiltonian models are invariant under a global SUC(3) symmetry, in Ref. Dmitrasinovic:2001nu it was suggested that the quark model Hamiltonian should be expressed in terms of every invariant operator of SU(3). This implies that the Hamiltonian should contain both two- and three-body confining forces. In other words, the two-body confining force $F_{i}\cdot F_{j}$ can be expressed in terms of the quadratic (Casimir) operator and the three-body force (see next section) in terms of the cubic invariant. If added to the Hamiltonian, the three-body force has implications on the spectrum of ordinary baryons Dmitrasinovic:2001nu ; Papp:2002xh and on the stability of tetraquarks Dmitrasinovic:2001nu ; Pepin:2001is ; Dmitrasinovic:2003cb and six-quark systems (the nucleon-nucleon system) Pepin:2001is . Here we discuss its effect on the stability of pentaquarks containing both light and/or heavy quarks, inasmuch as the confining interaction is flavor independent.

In Refs. Dmitrasinovic:2001nu ; Papp:2002xh ; Pepin:2001is it was shown that the three-body confining interaction introduced in the next section can modify the gap between physical color singlet states and the unphysical color octet states which plague the spectrum of a quark model Hamiltonian containing two-body color confinement only. For negative values of the relative strength $c$ between the three- and two-body confinement interaction the color states are shifted high up in the spectrum. Then they can be neglected in calculations and the baryon can be described as a $q^{3}$ state. For $c\geq 0$ the color states are located in the observed region of the spectrum. Hence they have to be considered as giving rise to $(qqq)(q{\overline{q}})$ configurations to be admixed with $q^{3}$ configurations. It has been suggested that most of the low lying baryon resonances could have substantial $q^{4}{\overline{q}}$ admixtures, see, for example, Ref. Helminen:2000jb . The three-body force presented here can lower the energy of a $q^{4}{\overline{q}}$ state close to that of a $q^{3}$ state. Then one can expect significant admixtures of $q^{4}{\overline{q}}$ with $q^{3}$ configurations in the baryon wave functions, as mentioned in Ref. Richard:2016eis .

In Ref. Pepin:2001is it was shown that the three-body confining force increases the coupling between physical states and hidden-color (octet-octet) states in $q^{6}$ systems if $c$ is negative and has therefore some influence on the nucleon-nucleon interaction derived in a microscopic approach based on the quark structure of baryons.

In the present approach the arguments are purely algebraic but a three-body confining interaction could possibly mimic the little information we have from QCD lattice results Bali:2000gf ; Takahashi:2002bw ; Alexandrou:2001ip ; Suganuma:2011ci ; Bicudo:2015ofa . Also, it would be useful to establish some connection between flux-tube models for pentaquarks and the present work.

The paper is organized as follows. In the next section we briefly recall the three-body confining interaction of Refs. Dmitrasinovic:2001nu ; Papp:2002xh ; Pepin:2001is . In Sec. III we introduce the basis color states specific to the $q^{4}{\overline{q}}$ system and derive a useful unitary transformation between states obtained in two relevant coupling schemes. In Sec. IV we present results regarding the role of the three-body confinement interaction on the stability of pentaquarks and implications on the spectra of ordinary baryons. An estimate of the effect of this interaction is made in the case of a simple model of harmonic oscillator confinement. The last section is devoted to conclusions. Useful group theoretical details are described in Appendices A - D.

II Two- and three-body confining interactions

Let us consider the Hamiltonian

[TABLE]

where $\vec{p}_{i}$ and $m_{i}$ are the momentum and the mass of the quark (antiquark) $i$, $\vec{P}$ and $M$ are the total momentum and the mass of the $q^{4}{\overline{q}}$ system and $V_{2b}+V_{3b}$ is the confinement interaction. For the 2-body confinement interaction we choose the form Dmitrasinovic:2001nu

[TABLE]

where

[TABLE]

is the color charge operator of the quark $i$. In constituent quark models only a two-body color confinement interaction is usually considered. The common form of its color part is $-3/2~{}F^{a}_{i}\cdot F^{a}_{j}$, which differs from the expression above by an additional constant. As pointed out in Ref. Dmitrasinovic:2001nu the constant 7/3 ensures the stability of $q{\overline{q}}$ pairs. One can easily pass from one form to the other.

The three-body color confinement interaction proposed in Ref. Dmitrasinovic:2001nu has the following form

[TABLE]

where ${\mathcal{V}}_{ijk}$ is the radial part and ${\mathcal{C}}_{ijk}$ is the three-body color operator. For a $q^{3}$ system or a multiquark $q^{3}$ subsystem this has the form

[TABLE]

where $d^{abc}$ are some real constants, symmetric under any permutation of indices Stancu:1991rc . The three-body color operator acting in a $q^{2}{\overline{q}}$ subsystem is defined as

[TABLE]

where

[TABLE]

is the color charge operator of an antiquark. These operators can be expressed in terms of the quadratic invariant $C^{(2)}$ and the cubic invariant $C^{(3)}$ of SU(3) as Dmitrasinovic:2001nu ; Pepin:2001is

[TABLE]

and

[TABLE]

We recall that for a given irrep of SU(3) labelled by $(\lambda\mu)$, the eigenvalues of these invariants are

[TABLE]

and

[TABLE]

(Note that an extra factor of 1/2 is needed in Ref. Stancu:1991rc for the eigenvalue of the cubic invariant.) Then for a $q^{3}$ system the expectation value of (8) is $\frac{10}{9}$ for a singlet ($\lambda\mu$) = (00) and $-\frac{5}{36}$ for an octet ($\lambda\mu$) = (11). The expectation values of (9), necessary in the study of a $q^{4}{\overline{q}}$ system, will be given in the next section.

This is an exploratory study where a schematic form for the radial part of ${\mathcal{V}}_{ijk}$ of (4) is assumed. As in Refs. Dmitrasinovic:2001nu ; Papp:2002xh ; Pepin:2001is we introduce a parameter $c$ representing the strength of the three-body relative to the two-body interaction. For a $q^{3}$ system we assume that

[TABLE]

This is consistent with a triangular shape for a three-body interaction in baryons. With such a form the contribution of $V_{2b}$ and $V_{3b}$ add up together and the expectation value of the color part of this sum is

[TABLE]

for a color singlet $q^{3}$ system Dmitrasinovic:2001nu ; Papp:2002xh ; Pepin:2001is . To avoid multiple counting, in a $q^{4}{\overline{q}}$ system we take

[TABLE]

because each two-body interaction contributes three times. These assumptions will be used in Sec. IV. Next we discuss the basis states.

III The basis states

For describing $q^{4}{\overline{q}}$ systems one can introduce various coupling schemes, each being convenient for a particular form of the interaction operator. For our discussion suppose that the particles 1, 2, 3 and 4 are quarks and 5 is an antiquark. Then one can first couple three quarks together and then couple this subsystem to a $q{\overline{q}}$ pair. In this way one introduces the so called asymptotic channels having a physical color singlet - color singlet state $|(123)_{1}(4{\overline{5}})_{1}\rangle$ and two unphysical color octet - color octet states $|(123)_{8}(4{\overline{5}})_{8}\rangle$. This coupling scheme is useful to calculate matrix elements of the operator (5) and it gives the appropriate basis at finite distances. At very large distances only the singlet-singlet state survives, the octet-octet states being pushed up by the quark-quark interaction. They were first called hidden-color states in the context of the nucleon-nucleon problem described as a six quark system Harvey:1988nk .

Asymptotic channels are convenient to be used for all multiquark systems. For example the tetraquarks have one singlet-singlet and one octet-octet asymptotic channel Stancu:1991rc ; Brink:1994ic ; Stancu:2009ka . The hexaquarks have one octet-octet asymptotic channel when orbital excitations of permutation symmetry $[42]$ are included Pepin:2001is .

The pentaquark system has the particularity that it has two octet-octet channels, as shown in Appendix C. Thus the asymptotic channels are

[TABLE]

where the $\rho$ and $\lambda$ superscripts, traditionally used for baryons, correspond to the two linear independent basis vectors of the mixed irreducible representation $[21]$ of the permutation group S3. These are basis vectors where the $\rho$ index indicates that the pair 12 is in an antisymmetric state and the $\lambda$ index indicates that the pair 12 is in symmetric state.

On the other hand to estimate the contribution of the three-body interaction (6) it is convenient first to couple two quarks, say 1 and 2 to the antiquark ${\overline{5}}$ and then to the subsystem of the remaining pair of quarks, 3 and 4, to get again total color singlets. One can construct the following normalized independent states

[TABLE]

The upper index $S(A)$ indicates that the pair 12 is in a symmetric (antisymmetric) state. Then, the coupling of this pair to the antiquark ${\overline{5}}$ leads to three possible color states. The first two contain a triplet SU(3) $q^{2}{\overline{q}}$ state denoted by $[211]$ and the third contains an antisextet SU(3) state denoted by $[22]$. Their forms are explicitly given in Appendix C. Other coupling schemes giving rise to a complete set of independent $q^{4}\bar{q}$ color singlets are also possible Park:2017jbn .

Of course, the states between different coupling schemes are related to each other. In the present case, we found that the asymptotic channels (III) are related to the intermediate coupling channels (III) by the following unitary transformation

[TABLE]

derived in Appendix C. The first two rows give transformation coefficients identical to those found for tetraquark systems Stancu:1991rc . This means that from permutation symmetry point of view the structure of the corresponding asymptotic basis vectors is the same in both cases. However the state $|{[{(12)}^{S}\,{\overline{5}}]}_{[211]}(34)_{[11]}\rangle$ does not exist in tetraquark systems, being incompatible with the definition of an antiquark as an antisymmetric $qq$ pair. Thus there is only one octet-octet state in tetraquarks, as it was mentioned above.

The transformation (17) is used in the following to calculate matrix elements of the three-body confining interaction potential $V_{3b}$.

IV Results

Here we first calculate the matrix elements of the color part of $V_{2b}$ and $V_{3b}$ and then discuss their contribution to the total energy of a $q^{4}{\overline{q}}$ system. The diagonal matrix elements of the two-body interaction $(\ref{V2b})$ are calculated from

[TABLE]

where $i,j$ = 1, 2 and 3 are the asymptotic states (III). From Appendix B one has

[TABLE]

both for the singlet-singlet color and the octet-octet color channels. For 6 pairs of quarks plus 4 pairs quark-antiquark the expectation value of the color two-body operator becomes - 10/3 but adding the shift of 7/3 from the definition (2) or (IV) times 10 one obtains a total of 20 for every asymptotic channel. Therefore the integration in the color space of the two-body interaction (2) for any of the asymptotic states (III) gives

[TABLE]

The off-diagonal matrix elements of $V_{2b}$ are vanishing for all asymptotic states (III), as shown in Appendix B.

The matrix elements of the three-body interaction (4) in the basis (III) can be written as

[TABLE]

where ${\mathcal{C}}_{123}$ and ${\mathcal{C}}_{12{\overline{5}}}$ are defined by (8) and (9) respectively. The matrix elements of ${\mathcal{C}}_{123}$ are straightforwardly obtained from the expectation values indicated below Eq. (11). Using Table 1 one can see that the contribution of the operator ${\mathcal{C}}_{12{\overline{5}}}$ vanishes for the color singlet-color singlet state $|1\rangle$. The nonvanishing part is due to four times the contribution of ${\mathcal{C}}_{123}$ which is four times the contribution of a $q^{3}$ system.

To calculate all the other matrix elements of ${\mathcal{C}}_{12{\overline{5}}}$ we make use of the unitary transformation (17) which expresses the asymptotic channels as linear combination of the basis vectors (III) and then use the expectation values given in Table 1. The Ansatz (14) allows to introduce a common radial factor for $V_{2b}$ and $V_{3b}$, namely $\sum\limits_{i<j}V_{ij}$. In this way we have obtained the color part of $V_{2b}+V_{3b}$ as given by the matrix

[TABLE]

where $i$ = 1, 2, 3 are the asymptotic states (III). The eigenvalues of this matrix are

[TABLE]

To get the full contribution of the confinement one must multiply each eigenvalue by $\sum\limits_{i<j}V_{ij}$. The eigenvector associated with $e_{1}$ is dominantly color singlet - color singlet (the state $|1\rangle$ appears with 91 % probability) and the eigenvector associated with $e_{2}$ is dominantly color octet - color octet (the state $|2\rangle$ appears with 91 % probability) irrespective of the value of $c$. The eigenvector $e_{3}$ is a pure color octet-color octet state as it can be seen from Eq. (22). It is stable against strong decay into a baryon plus a meson.

In Ref. Dmitrasinovic:2001nu the range proposed for the relative strength $c$ was

[TABLE]

The upper limit ensures that the lowest color singlet appears below the lower color octet in a $q^{3}$ system and the lower limit is required by the stability condition of the nucleon, $\langle V_{2b}+V_{3b}\rangle>0.$ In Ref. Pepin:2001is some arbitrariness was noticed regarding the lower limit. For our discussion it is enough to restrict the interval to

[TABLE]

In the stability problem against strong decays of a $q^{4}{\overline{q}}$ system we are interested in the quantity

[TABLE]

where $E(q^{4}{\overline{q}})$ represents the lowest energy of the $q^{4}{\overline{q}}$ system and $E(q^{3})+E(q{\overline{q}})$ is the threshold energy for the decay of $q^{4}{\overline{q}}$ into a baryon described as a $q^{3}$ system and a meson $q{\overline{q}}$. The condition $\Delta E<0$ is interpreted as stability, otherwise the system is unstable against strong decays. Of course the Hamiltonian (1) must also be used in the calculation of $E(q^{3})$. In each case a hyperfine interaction should be added but this is beyond the scope of the present study. As mentioned in Sec. II the contribution of $V_{2b}+V_{3b}$ to a color singlet $q^{3}$ state is proportional to $\chi_{1}$ given by (13). In order to keep the ground state $E(q^{3})$ unchanged in the presence of the three-body force we have to rescale $E(q^{3})$ by dividing it by $\chi_{1}$ as in Ref. Papp:2002xh . For consistency one must also divide the eigenvalues (23) by the same quantity.

It is useful to make an estimate for a harmonic oscillator confinement. For five quarks (antiquarks) of equal masses the eigenvalues of (1) are Stancu:1997dq

[TABLE]

where $N$ is the number of quanta and we take $N=0$. The frequency $\omega_{i}$ of a five body system is related to the frequency $\omega_{0}$ of a three body system by Helminen:2000jb

[TABLE]

We normalize the eigenvalues $E_{i}$ of (1) such as in the absence of three-body forces ($c$=0) to obtain this relation. In addition, as we deal with 10 pairs of quarks we have to divide each eigenvalue of $H$ by 10 to be consistent with Ref. Papp:2002xh . Then, in terms of $e_{i}$ of Eqs. (23) we have

[TABLE]

where $\chi_{1}$ is defined by Eq. (13). For $\hbar\omega_{0}$ = 395 MeV Papp:2002xh the three resulting eigenvalues are displayed in Fig. 1.

Without a three-body force ($c$ = 0) the states are degenerate. The three-body confining force ($c\neq 0$) introduces a splitting. A negative $c$ increases all $E_{i}$ with respect to the values taken at $c$ = 0. For example one has $E_{1}(c=-1)-E_{1}(c=0)$= 1.06 GeV. This means that the pentaquarks become less and less stable with a decreasing negative $c$. The lowest eigenvalue is $E_{1}$ which is dominantly color singlet $q^{3}$ color singlet $q{\overline{q}}$. For $c>0$ the lowest state becomes $E_{2}$ which is dominantly color octet - color octet. One has $E_{2}(c=0)-E_{2}(c=0.35)$ = 267 MeV, which is a substantial lowering.

Thus the three-body force helps now in stabilizing the $q^{4}{\overline{q}}$ system against strong decays because $E_{2}$ is dominantly a color octet - color octet state thus it has a small amplitude to decay strongly. However the gap between $E_{2}$ and the other states is not so large. For example, $E_{2}(c=0.35)-E_{1}(c=0.35)$= - 138 MeV. This means that through the addition of a hyperfine interaction to the Hamiltonian (1) the proportion of singlet-singlet and octet-octet contributions might change.

The $c>0$ case also brings a new perspective to baryon spectroscopy. If the energy of the $q^{4}{\overline{q}}$ states are close to those of the $q^{3}$ states one should expect significant admixtures of $q^{4}{\overline{q}}$ states in the baryon resonance wave functions. The low lying $q^{4}{\overline{q}}$ states provide an approximation to the open baryon-meson channels that give rise to large strong decay widths of the empirical resonances Helminen:2000jb . Let us give an estimate for energies relevant for the Roper resonance. In constituent quark models this is usually described as a 2 $\hbar\omega_{0}$ excitation. In these models the presence of a negative additional constant $V_{0}$ in the Hamiltonian is also required for a realistic description of the spectrum. Accordingly, if we define the ”unperturbed” energy (no hyperfine interaction) of a $q^{3}$ system as $E(q^{3})=3m+3V_{0}+5\hbar\omega_{0}$ with $m=340$ MeV and $V_{0}$= - 269 MeV as in Ref. Helminen:2000jb and use, as above, $\hbar\omega_{0}$ = 395 MeV we obtain $E(q^{3})$ = 2188 MeV. If we estimate the energy of the lowest $q^{4}{\overline{q}}$ in a similar manner we have $E(q^{4}{\overline{q}})=5m+5V_{0}+E_{2}$. At $c=0.35$ one has $E_{2}$= 1897 Mev from which one obtains $E(q^{4}{\overline{q}})$ = 2252 MeV, very close to $E(q^{3})$. Thus the three-body confining interaction with a positive $c$ could provide a mechanism to lower the $q^{4}{\overline{q}}$ states, which is an alternative to the mechanism proposed in Ref. Helminen:2000jb based on a schematic flavor and spin dependent interaction where the strength of this interaction is adjusted to allow admixtures.

V Conclusions

Here we have shown that the three-body confining interaction (4)-(6) with a negative $c$ destabilizes the pentaquarks while a positive $c$ helps in stabilizing them. A similar conclusion has been drawn in Ref. Pepin:2001is in connection with tetraquarks. The conclusion also holds for tetraquarks or pentaquarks containing heavy flavors, inasmuch as the confinement interaction is flavor independent.

The unitary transformation (17) derived here relates two complete sets of color singlet bases needed to describe $q^{4}{\overline{q}}$ systems. Each contain three independent states. The set (III) is the most suitable to study the stability against strong decays. The set (III) may be useful in diquark-triquark type models Karliner:2003dt . Thus in a $q^{4}{\overline{q}}$ system there are three asymptotic channels in the color space as compared to two for $q^{2}{\overline{q}}^{2}$ systems.

As mentioned in the introduction the stability of multiquark systems depends on the constituent masses as well. In particular, for unequal masses the kinetic term deserves a special attention, by studying the role of both its symmetric and asymmetric parts, as it has been done for tetraquarks in Ref. Richard:2017vry . We are aware that the problem becomes complex if the pentaquark contains more than two flavors, as for example, those proposed in Refs. Gignoux:1987cn ; Lipkin:1987sk .

Similar to tetraquarks Brink:1994ic , one can construct full wave functions including the position, the spin and the flavor degrees of freedom to calculate the total energy of $q^{4}{\overline{q}}$ by adding a hyperfine interaction to see which is the role of octet-octet states in the stability. In this way one may perhaps avoid the fall apart decay mode of the ground state pentaquarks obtained in Ref. Park:2017jbn .

It would certainly be interesting to make contact with the many-quark confining forces from flux-tube models inspired by the strong coupling limit of QCD as proposed in Ref. Vijande:2007ix for tetraquarks and extended to pentaquarks in Ref. Richard:2009rp . For pentaquarks two distinct contributions should be considered, as depicted in Fig. 10 of Ref. Richard:2016eis . These are a disconnected and a connected flux-tube diagrams, the first corresponding to the baryon-meson configuration and the second to a Steiner tree diagram Richard:2009rp . In a color basis, the first should be a color singlet-color singlet state and the second should be related to two distinct color triplet - color antitriplet states. These must be associated to a cluster of four quarks and the antitriplet must be associated to the antiquark. One has to establish the relation of this basis to that of Eq. (III) each involving different intermediate couplings. Thus more algebraic work is needed to make a link between flux-tube confining potentials and the present study. If successful, a linear radial dependence, as supported by lattice studies for the three-body confining potential, could be considered in the future.

Appendix A The SU(3) antiquark generators

We find it useful to exhibit the action of the antiquark generators, Eq. (7),

[TABLE]

on antiquark states $\bar{q}_{i}$, needed in Appendix D. These are

[TABLE]

In the flavor space one can identify

[TABLE]

As a check, using the above relations and the symmetric constants $d^{abc}$ from Table 2 one can obtain the eigenvalue of the cubic invariant operator for antiquarks, in the Lorentz vector model Dmitrasinovic:2001nu

[TABLE]

by acting on $\bar{d}$, the highest weight state of the antitriplet $(\lambda\mu)$ = (01). The result is

[TABLE]

which means that the eigenvalue of the cubic invariant has an opposite sign to that of the triplet $u,d,s$, having $(\lambda,\mu)$ = (10), consistent with Eq. (11).

Appendix B Matrix elements for two-body operators

Here we prove Eq. (19) in a simplified way. This is a more explicit proof of the relation derived in Ref. Genovese:1997tm . We first need to evaluate the color operator matrix element of a subsystem of four quarks

[TABLE]

where $C^{(2)}$ is the Casimir operator eigenvalue (10) for a system of four quarks and $C^{(2)}_{q}=4/3$ is the eigenvalue of the Casimir operator for a quark, light or heavy. The color state of the four-quark subsystem is $[211]_{C}$ so that one has $C^{(2)}=4/3$. Hence $\langle O^{C}\rangle_{[211]}=-2$. As there are 6 pairs in (42), each pair of quarks will have

[TABLE]

For the antiquark the Casimir operator eigenvalue is $C^{(2)}_{\bar{q}}=4/3$, where $\bar{q}$ can be light or heavy. Then for a pentaquark, in a color-singlet state $[222]$, formed of a subsystem of four quarks and an antiquark, one can define the color operator eigenvalue

[TABLE]

where we have used the Casimir operator eigenvalue $C^{(2)}_{[222]}$ = 0 of the whole system in a color singlet, and $C^{(2)}_{q}$ = $C^{(2)}_{\bar{q}}$ = 4/3. As each quark shares 1/4 one can conclude that in average, for every asymptotic channel (III) we have

[TABLE]

which means that the color expectation value for any pair, quark-quark or a quark-antiquark is $-1/3$. There are in all 10 pairs so that for a two-body color confinement with a constant $V_{ij}$, the color operator gives

[TABLE]

There is another simplified, but slightly more elaborate, way to derive Eq. (19). We introduce the scalar product $F_{123}\cdot F_{4\bar{5}}$ defined as

[TABLE]

or alternatively

[TABLE]

In terms of the Casimir operators eigenvalues of the whole system $C^{(2)}_{[222]}$ and of the separate subsystems $C^{(2)}_{123}$ and $C^{(2)}_{4\bar{5}}$ one has

[TABLE]

For the asymptotic channel $|1\rangle$ we get

[TABLE]

On the other hand it is reasonable to assume that

[TABLE]

so that we have

[TABLE]

The nonvanishing terms give

[TABLE]

As we know

[TABLE]

we obtain

[TABLE]

as above. For the hidden-color (octet-octet) states $|2\rangle$ and $|3\rangle$ Eq. (49) gives

[TABLE]

from where the analog of the relation (52) becomes

[TABLE]

Then, for a color octet three quark states we take the average between symmetric and antisymmetric states of particles 1 and 2 to get

[TABLE]

Then it follows that for any of the hidden color states one has

[TABLE]

which is the same as for the singlet-singlet state.

This proof is convenient for our purpose where the confinement is a constant in the configuration space. But if a radial shape is introduced then one has to proceed as in the appendix of Ref. Park:2017jbn , which gives a consistent result with ours.

Using the explicit form of the color basis states $|1\rangle$, $|2\rangle$ and $|3\rangle$ of Appendix C one can calculate the required off-diagonal matrix elements of $V_{2b}$. For quark-quark pairs they are

[TABLE]

and for quark-antiquark pairs one has

[TABLE]

This implies that

[TABLE]

which proves that the off-diagonal matrix element of $V_{2b}$ between states $|1\rangle$ and $|2\rangle$ is zero. Similarly one can prove that the other off-diagonal matrix element of $V_{2b}$ are vanishing. For this purpose one needs to know that

[TABLE]

and

[TABLE]

Appendix C Two useful color singlet bases for pentaquarks

We have used the tensor method Stancu:1991rc to write down explicit expressions for the normalized basis states appearing in the unitary transformation (17) in terms of their quark content. By equalizing the coefficients of identical terms of both sides we have obtained a system of linear equations for the matrix elements of the unitary transformation.

Let us denote by $q_{i}$ and ${\overline{q}}_{i}$ ($i=$ 1,2 and 3) the three possible color states of a quark and an antiquark respectively. By using the tensor method Stancu:1991rc the asymptotic channel states can be easily constructed as scalarsRichard:2009rp from the product between a flavor singlet (flavor octet) $q^{3}$ state and a flavor singlet (flavor octet) $q{\overline{q}}$ state. The three possible normalized colorless $q^{4}{\overline{q}}$ states are

[TABLE]

[TABLE]

and

[TABLE]

Note that the normal order of particles is understood everywhere. The difference between (C) and (C) is that in the first the particles 1 and 2 are in an antisymmetric state while in the second they are in a symmetric state. That is why they carry the upperscripts ${\rho}$ and ${\lambda}$ respectively, like the mixed symmetry $q^{3}$ states used in baryon spectroscopy.

For the basis vectors in the intermediate coupling we use the tensor method first to construct $qq{\overline{q}}$ states.

1. The three components of $[(12)^{S}{\overline{5}}]_{[211]}^{i}$ state are obtained from by contraction with $\delta^{k}_{i}$

[TABLE]

from the tensor $T^{ij}_{k}$ which is symmetric in the upper indices

[TABLE]

The normalized components are

[TABLE]

2. The three components of the state $[(12)^{A}{\overline{5}}]_{[211]}^{i}$ form the antisymmetric tensor

[TABLE]

In this antisymmetric product the first factor represents a $qq$ subsystem described by

[TABLE]

and the second an antiquark seen as a $qq$ pair

[TABLE]

Then the first normalized components is

[TABLE]

The other two are obtained by circular permutations.

To form a $q^{4}{\overline{q}}$ singlet state together with particles 3 and 4 in the above two cases one must construct the scalar product $\frac{1}{\sqrt{3}}T^{i}T_{i}$ where $T^{i}$ is either of the form (C) or (C) plus circular permutations. The tensor $T_{i}$ made of the particles 3 and 4 is defined by (72).

3. The six components of $[(12)^{A}{\overline{5}}]_{[22]}^{i}$ are defined by the symmetric tensor

[TABLE]

where the first factor in the right hand side should be replaced by (72) and the second by (C). Then, for example the first component becomes

[TABLE]

To form a $q^{4}{\overline{q}}$ singlet state one needs the symmetric tensor

[TABLE]

associated to the particles 3 and 4. Then the $q^{4}{\overline{q}}$ singlet state is defined by the scalar product $\frac{1}{\sqrt{6}}T^{ij}T_{ij}$.

In this way we have obtained all states needed to determine the unitary transformation (17).

Appendix D Matrix elements of ${\mathcal{C}}_{ijk}$ and ${\mathcal{C}}_{ij\overline{k}}$

One can calculate the matrix elements of ${\mathcal{C}}_{ijk}$ either from Eq. (5) or from Eq. (8). The matrix elements of ${\mathcal{C}}_{ij\overline{k}}$ can be calculated either from Eq. (6) or from Eq. (9).

The first method is more tedious but it provides a check for the second. It implies to apply successively the generators $F^{a}_{i}$ or $\overline{F}^{a}$ on the states $|1\rangle$, $|2\rangle$ or $|3\rangle$. As an example, in Table 2 we show the contribution of all nonvanishing terms to Eqs. (5) and (6). Their sums gives the expectation value of ${\mathcal{C}}_{124}$ and of ${\mathcal{C}}_{12\overline{5}}$ respectively, for the color wave function $|3\rangle$ defined in Eq. (III). By taking into account the multiplicity $m^{abc}$ of each term one obtains - 5/36 consistent with Eq. (8) and 5/18 consistent with Eq. (9), respectively. The latter value is exhibited in column 2 of Table 1.

Thus, the third column gives the expectation value of the cubic invariant ${\mathcal{C}}_{124}$ for the state $|3\rangle$. It is very easy to calculate expectation value of ${\mathcal{C}}_{123}$ for the asymptotic state $|1\rangle$ using its definition (65). The result will coincide with the column 3 and for symmetry reasons all ${\mathcal{C}}_{ijk}$ are identical. As a matter of fact the result is also identical to column 3 of Table II of Ref. Pepin:2001is where a particular singlet color state of six quark systems has been taken as an example. This is a state which can be rewritten as a part of a $q^{4}\bar{q}$ wave function where the subsystems $q^{3}$ and $q\bar{q}$ are octets in the color space.

Acknowledgments

The author acknowledges support from the Fonds de la Recherche Scientifique - FNRS under the Grant No. 4.4501.05.