The pentaquark spectrum from Fermi statistics

We study compact hidden charm pentaquarks in the Born–Oppenheimer approximation, previously introduced for tetraquarks, assuming the heavy pair to be in a color octet. We show that Fermi statistics applied to the complex of the three light quarks, also in color octet, requires S-wave pentaquark ground states to consist of three octets of flavour-SU(3)f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_f$$\end{document}, two with spin 1/2 and one with spin 3/2, in line with the observed, strangeness S=0,-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=0,-1$$\end{document}, spectrum. Additional lines corresponding to decays into J/ψ+Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi +\Sigma $$\end{document} and J/ψ+Ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi +\Xi $$\end{document} are predicted. In the language of non-relativistic SU(6), ground state pentaquarks form either a 56\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{56}}$$\end{document} or a 20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{20}}$$\end{document} representation, distinguished by presence or absence of pentaquarks decaying in the spin 3/2 decuplet, e.g. in J/ψ+Δ++\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi +\Delta ^{++}$$\end{document}. Observation of a strangeness S=-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=-2$$\end{document} or isospin I=3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I=3/2$$\end{document} pentaquarks would be a clear signature of compact, QCD bound pentaquarks.

To describe compact, hidden charm or beauty pentaquarks, the Born-Oppenheimer (BO) approximation is particularly appropriate since it improves when the mass of the heavy constituents increases with respect to the mass of the light ones, see [9][10][11].In this framework, one considers the heavy constituents (c and c) as color sources at rest, with fixed relative distance R, and computes the lowest energy E(R) of the three light quarks, either analytically, in perturbation theory with techniques borrowed from molecular physics [9], or non-perturbatively, in lattice QCD [12].
Besides distance and spin, one has to specify the color quantum numbers of the sources.For colored quarks, the natural choice is to assume the heavy quark-antiquark pair to be in a color octet, coupled to the light quarks to make an overall color singlet (sum over A = 1, . . ., 8 understood).B A describes the complex of the three light quarks, which live in a color octet as well.Specializing to a proton-like pentaquark, we specify B A = B A [u(x 1 ), u(x 2 ), d(x)] in terms of the light quark coordinates.
In this paper, we address the restrictions posed by Fermi statistics to the complex of the three light quarks in (1), considering exchange of color, coordinates, flavor and spin.We summarize flavor and spin in representations of non-relativistic SU(6) ⊃ SU(3) f ⊗ SU(2) spin [13].
We show that the simplest hypothesis of complete symmetry under coordinates, which would imply pentaquarks in the mixed-symmetry 70 representation of SU (6) [14], is excluded.
The BO approximation allows mixed symmetry in the coordinates also for ground state, S-wave, pentaquarks by distributing quarks in orbitals around the fixed sources.We show that, in this case, one obtains a consistent solution for the SU(6) representations 56 and 20 but not for the 70.
The elimination of the 70 greatly reduces spin and flavor multiplicity of ground state pentaquarks.Including the spin of the c − c pair, we obtain, for either 56 or 20, three octets of flavour-SU(3) f , two with spin 1/2 and one with spin 3/2, in line with the S = 0, −1 spectrum observed in [1].
The plan of the paper is as follows.We discuss the properties of three light quarks operators in Sect II.Sect.III illustrates the way to deal with the representations of the group of three objects, S 3 , for color, coordinates, flavor and spin.In Sect.IV we present the essentials of the BO approximation for pentaquarks and derive a consistency condition on QCD couplings of light to heavy quarks, for quarks distributed in different BO orbitals.In Sect.V we illustrate the conditions required by Fermi statistics on the (flavour ⊗ spin) wave function of light quarks and in Sect.VI compute the resulting SU(6) wave functions.Finally, in Sect.VII we presents our results and illustrate the prospects of future calculations of pentaquark mass spectra in the BO approximation.More technical details are contained in four Appendices.

II. BASIC LIGHT QUARK OPERATORS
We describe pentaquarks with operators of the generic form given in Eq. ( 1) B A is obtained from two basic octets, constructed in turn from antisymmetric or symmetric diquark operators.
Given the form of B A in terms of P A and Φ A , relations (4) to (6) allow us to find the QCD couplings qQ and q Q.We define Exchanging quarks.Since P A and Φ A are the only octet operators with the given three quarks, we must be able to express P A (u 1 , d|u 2 ) in terms of octets made by u 1 u 2 and d, i.e. find a and b such that Relation ( 9) is what we need to move quarks around and bring a given pair together.However, operators P, Φ are not equally normalised.Starting from a and b, we compute the normalisation factors to obtain a relation between normalised kets with α 2 + β 2 = 1.A simple calculation leads to the transformation table

III. THE GROUP S3 AND ITS REPRESENTATIONS
The group S 3 of the six permutations of three elements can be seen as the group of symmetries of an equilateral triangle by thinking of these as permuting the three vertices.This group consists of the identiy transformation corresponding to three cycles of length one, two rotations by 120 • and 240 • , which are permutations corresponding to cycles of length three (e.g. the vertex a → b → c → a) and three reflections in the three altitudes of the triangle, each consisting of a cycle of length two combined with a cycle length one leaving one vertex unchanged.
The two dimensional representation is given by [15] and Therefore Two different mixed representations of S 3 acting on different variables, M 1 and M 2 , may combine to an A, S or M representation, according to the scheme [17] For color we define where q 1 , q 2 , q 3 correspond to the vertices in the triangle a, b, c.Eqs. ( 15) and ( 16) reproduce the results in Table I.The left side columns of Table I correspond to D(τ 3 ), or b ⇄ c reflection, whereas the right side columns corresponds to D(τ 2 ), or a ⇄ c reflection.
Following [17], we list explicitly in App.B the building blocks of mixed and symmetric representations with respect to color, flavour, spin and coordinates, restricting to proton-like states (flavour octet or decuplet).

IV. BORN-OPPENHEIMER APPROXIMATION: A QCD CONSISTENCY CONDITION
We follow the perturbative scheme illustrated in [11] for hydrogen molecules and ions in QED and in [9] for hidden charm hadrons in QCD.The starting point is the interaction of each light particle with the fixed sources, following the instructions given in Sect.II. 1.In the simplest one orbital scheme, one solves, analitytically or numerically [11], the Scrödinger equation of q in the presence of the static sources.The orbital is the corresponding ground state with wave function, ψ 0 (x q ) and energy ǫ 0 .We represent the orbital in Fig. 1 with an ellipse around the heavy sources.Similarly to what done in atomic physics, we put the other light quarks in the same orbital, corresponding to the ground state wave function Ψ 0 is symmetric under the exchange of quark coordinates.In the atomic physics language, we attribute occupation number 3 to the orbital.Denoting by V res the sum of the light-to-light interactions, that did not intervene in the construction of the orbital, the BO potential, to first order in V res , is where V cc (R) is the QCD interaction between c and c.V BO is obviously a function of R and is the potential of the Schrödinger equation of the cc system.2. One can also consider a two orbitals scheme with two separate Schrödinger equations: orbital q − c (denoted by φ) and orbital c (denoted by ψ).In Fig. 2 (a) two light quarks sit in φ and one in ψ, the opposite in Fig. 2 (b).
We have two distinct possibilities for the ground state A consistency condition.QED charges of protons and electrons are fixed constants.This is not the case for two-body QCD charges, which depend on the superpositions of the relative color representations in which the pair occurs, as indicated in Sect.II.This leads to a consistency condition, namely that: quarks of different flavor in the same orbital (as in Fig. 1) must share the same QCD coupling to the heavy quarks at the center of the orbital with possibly λ cq = λ cq .We shall see that this condition is not trivially satisfied in the pentaquark.

V. THREE LIGHT QUARK OPERATORS AND FERMI STATISTICS
Color singlet baryons.S-wave, color singlet baryons are fully antisymmetric under color exchange and fully symmetric under coordinate exchange.Therefore quark (flavour ⊗ spin) must be symmetric.
If we summarise spin and flavour quantum numbers with representations of SU(6) ⊃ SU(3) f ⊗ SU(2) spin , this is the 56 representation of non-relativistic SU(6) [13], with content 56 : (8, 1/2) ⊕ (10, 3/2) ( A different case is that of excited, negative parity baryons: color is fully antisymmetric, but coordinates are in a mixed state (two quarks in S-wave and one quark in P -wave), see Ref. [16,17].The three quarks must form a mixed-symmetry representation in flavour and spin, to obtain full symmetry when combined with coordinates.In SU(6) language, negative parity baryons are in the 70 representation, which decomposes as The third three-quark SU(6) representation, the fully antisymmetric is forbidden by Fermi statistics, for both ground state and P wave baryons.
Pentaquarks.S-wave pentaquarks have the three light quarks in color octet, distributed in one or more orbitals.Total antisymmetry under quark exchange required by Fermi statistics may be reached in different ways, summarized in Tab.II.
The simplest possibility is to assume complete symmetry of the coordinates.In this case, full antisymmetry under quark exchange requires the light quark complex to have mixed symmetry under spin and flavour exchange, to be combined with colour to a totally antisymmetric state, first row of Tab.II: the three light quarks must form a 70 representation [14], with the flavour-spin content reported in Eq. ( 25).
However, one needs to take into account the consistency condition stated in Sect.IV.In the last row we require that the mixed symmetries for color and coordinate are composed in a mixed symmetry which, combined with spin-flavor SU (6), makes a totally anti-symmetric state.
1.In Born-Oppenheimer parlance, symmetry under coordinate exchanges means that light quarks populate a single orbital, Fig. 1.Accordingly, they must share the same QCD coupling to c and to c We analyse the three cases in Sect.VI, and show that only 56 and 20 obey the consistency condition stated in Sect.IV.

VI. OCTETS AND DECUPLETS WITH LIGHT QUARKS IN TWO ORBITALS
We give in this Section the explicit form of the light quarks operators corresponding to the SU (6) representations 56, 20 and 70, with quarks distributed in two orbitals denoted by φ and ψ and mixed-symmetry in the coordinates (see (B7) and rows 2 to 4 in Tab.II).
We give a complete discussion of the spin 1/2 octet, (8, 1/2) of 56 and report the results for the other cases, referring to App.C for details.
In the following, we use the following abbreviated notations [q 1 , q 2 ]q 3 ⇔ ((q 1 , q 2 )3 c q 3 ) 8c (q 1 , q 2 )q 3 ⇔ ((q 1 , q 2 ) 6c q 3 ) 8c [q 1 , q 2 ] s ∨ (q 1 , q 2 ) s ⇔ (q 1 , q 2 )3 c,S=s ∨ (q 1 , q 2 ) 6c,S=s {[q 1 , q 2 ]q 3 } s ∨ {(q 1 , q 2 )q 3 } s ⇔ ([q 1 , q 2 ]q 3 ) S=s ∨ ((q 1 , q 2 )q 3 ) S=s Whenever we write (cc) we mean color octet (cc) 8 A. The spin 1/2 octet of 56 Following Eqs. ( 17), the fully antisymmetric wave function in color and coordinate is For color 6 or 3 , when q 1 and q 3 are in φ and q 2 in ψ we introduce the abbreviation The situation is analogous to the molecular structure of the ion H − 2 = 2p 3e − We obtain Antisymmetry of Eq. (30) under the exchange (1, 2) is explicit.To check the case (1, 3) one has to use Tab.I.A tedious but direct calculation shows that indeed (1, 3)P = −P (31) Using Tab.I, we bring all q φ inside the parentheses and obtain the more symmetric form (dropping an irrelevant overall sign) We have to combine Eq. ( 32) with the flavour-spin fully symmetric expression obtained from the mixed-symmetry templates, Eqs.(B3) and (B5) in App.B to obtain The d 3 = d condition.We may use (32), (33) and Tab.I to bring the d i in the same position.One obtains the superposition of three replicas which are cyclical permutations of the assignments of u and d flavours to quarks 1, 2, 3.
In practical calculations, we may decide that the d flavour is in quark q 3 , keep the term containing d 3 = d and discard the rest.To obtain a normalised state we have to multiply by √ 3.In a way, the condition is analogous to fixing the gauge in a gauge invariant theory and restrict to one of the (infinitely many) orbits generated by gauge transformations.In our case, the group is the group of quark permutations and there are only three orbits.Applying the condition d = d 3 to (33) only the second term in (33) is non vanishing and we obtain 3 or more explicitely Combining ( 35) and (30), the operator that creates P 56 octet takes the proton-like form (the extension to the other members of the octet is given in App.D) It is convenient to have an expression which puts together quarks that belong to the same orbital.This is obtained by using (35) with (32) and further expressing the wave function in terms of the total spin of the pair inside orbital φ 3 Reassuringly, one verifies that the choice: Color couplings of d φ and d ψ can be easily calculated from the above formulas This operator describes a state with the orbitals, φ and ψ, populated according to φ(qq), ψ(q) We have a second possibility where ψ has occupation number two.In total The first possibility has lower energy, −7/18 < −2/18, in the one gluon exchange approximation.
We register the case with Color couplings u φ,ψ 3 − c/c are the same as in (39).
C. The spin 1/2 octet of 20 We find or, equivalently Before closing, we comment on the possibility to compute the mass spectrum of pentaquarks.In our two-orbitals scheme there are three ground state candidates, namely 56 (a) Fig. 2  We start from the first case, 56 a .A glance at Eq. (5) shows that orbitals at large R are in triality zero, color configurations 8 c − 8 c or 1 c − 1 c .As argued in [18], soft gluons may screen colors in the 8 c − 8 c configuration and the BO potential vanishes at infinity [9].For tetraquarks this behaviour is supported by the recent lattice QCD calculation of [12].The pentaquark goes asymptotically into a superposition of charmed baryon-anticharmed meson states.The pentaquark at intermediate distances, is a kind of molecular state bound by QCD interactions.In this case, the mass spectrum would be fully computable, similarly to the doubly charm tetraquark mass [19][20][21].
Orbitals in the second case, 56 b , 20 b , are shown in Fig. 2(b).The configuration reminds closely the compact pentaquark: [cu](c[ud]) proposed in [9].Orbitals are in color confined configurations 3c − 3 c or 6 c − 6c , see (4).We must add to the BO potential a string potential rising to infinity for R → ∞ [9].As it happens for charmonia, pentaquark masses contain one undetermined constant and one can compute only mass differences with respect to ground state, i.e. those due to the hyperfine interactions.

C 2 (
R) is the quadratic Casimir operator 1 of the color representation R of the pair cq.If the pair is in a superposition of two or more SU(3) c representations with amplitudes a, b, . . .we use[9] D(σ 1 ) is a anti-clockwise rotation by 120 • of the position vectors of the vertices of the triangle (taken from the center), and D(σ 2 ) is a rotation by −120 • .D(τ 1 ) represents a reflection (through the y axis).A rotation D(σ 1 ) changes a → b → c → a.If after this rotation a D(τ 1 ) reflection is done, which amounts to c ⇄ a, we get D(τ 1 )D(σ 1 ) = D(τ 3 ), which corresponds to a b ⇄ c reflection on the original triangle.Similarly D(τ 1 )D(σ 2 ) = D(τ 2 ), corresponding to a a ⇄ c reflection.Let us name the eigenvectors of the reflection D(τ 1 ) as

FIG. 1 :
FIG. 1:For the three light quarks to be in the same orbital, it is necessary that they carry the same color charge with respect to both c and c.

FIG. 2 :
FIG. 2: Two different possibilities for quark occupation distributed in two orbitals.In (a) two quarks sit in the c orbital (denoted by φ); in (b) two quarks sit in the c orbital (denoted by ψ).Full lines indicate interactions that make the orbitals, dotted lines indicate additional interactions taken into account to first order perturbation theory.The dot-dashed line represents the Born-Oppenheimer potential.For consistency, as discussed in Sect.IV, light quarks in the same orbital must have the same interaction with c and c.

2 .
27) with possibly λ cq = λ cq .An explicit calculation for the case of the (8, 3/2) ⊂ 70, see App.A, shows however that the couplings of d and u quarks are different: light quarks in a color octet cannot populate a single orbital.The next simplest possibility, in analogy with L = 1 baryons, is to arrange light quarks in two orbitals 2 , one around c and the other around c, Fig. 2, (a) or (b).Combining color mixed-symmetry with coordinate mixed-symmetry, one may obtain color-coordinate symmetry of type A, S, or M .To obey Fermi statistics, these possibilities must be associated to flavor-spin symmetries of type S, A or M , respectively, namely to the SU(6) representations 56, 20 or 70 (second to fourth rows of Tab.II).

TABLE I :
Transformation table for quark rearrangements inside P A and Φ A color octets.

TABLE II :
To get a totally anti-symmetric (A) state, we need M × M → A or M × M → S depending on the SU(6) or coordinate entries.