Pentaquark Symmetries, Selection Rules and another potentially Narrow State

We identify essential differences between the pentaquark and chiral soliton models of \10bar$_5$ and {\bf 8}$_5$ pentaquarks and conventional {\bf 8}$_3$ states, which are experimentally measurable. We show how the decays of $\Xi_5$ states in particular can test models of the pentaquarks, recommend study of the relative branching ratios of e.g. $\Xi^{-}_5 \to \Xi^-\pi^0:\Xi^0\pi^-$, and predict that the decay amplitude $\Xi_5 \to \Xi^*\pi$ is zero at leading order in pentaquark models for any mixture of \10bar and the associated {\bf 8}$_5$. We also include a pedagogic discussion of wavefunctions in the pentaquark picture and show that pentaquark models have this {\bf 8}$_5$ with $F/D=1/3$, in leading order forbidding $\Xi_5 \to \Lambda K$. The role of Fermi-Dirac symmetry in the $qqqq$ wavefunction and its implications for the width of pentaquarks are briefly discussed. The relative couplings $g^2(\Theta_Q N K_Q^*)/g^2(\Theta_Q N K_Q) = 3$ for $Q \equiv s,c,b$. A further potentially narrow state $\Lambda$ in {\bf 8}$_5$ with $J^P = 3/2^+$ is predicted around 1650 MeV.


Introduction
The possible discovery of an exotic and metastable baryon with positive strangeness, the Θ + (1540), has led to an explosion of interest in chiral soliton models (a version of which is cited as having predicted this state) and their relation to quark models. In this letter we propose explicit experimental tests that are sensitive to the assumed dynamics and thereby can distinguish among models. Such a state was predicted in a version of the chiral soliton model [1,2] to be in a10 of flavour SU (3) and to have J P = 1/2 + . Subsequent to the original observation [3], several interpretations of the state have been suggested within quark models (for an early review see [4]). Their common feature is that its constitution be ududs with one unit of orbital angular momentum in the wavefunction; they differ in the ways that the interquark dynamics causes the 1/2 + state to be the lightest and to have an anomalously narrow width. If the Θ + is an isoscalar, then a common feature of all models is that it is a member of a10, which contains further exotic states, Ξ +,−− . There are three main differences among the implications of models that can distinguish among various dynamics.
(i) The magnitude of the mass gap spanning the10 from Θ to Ξ is significantly smaller in pentaquark models than in the original formulation of the chiral soliton model [2,[4][5][6] though the latter is somewhat flexible as has recently been noted [7]; (ii) The first excited state of Θ is predicted [8] in pentaquark models to be a J P = 3/2 + isoscalar, in a J P = 3/2 +1 0, whereas there is no place for such a state in the present formulation of chiral soliton models; (iii) The hadron decays of non-exotic members of the10, in particular those of Ξ 0,− are especially sensitive to the interquark dynamics in pentaquark models. A specific example has been discussed in [9] but we shall show here that there is a more extensive set of relations and selection rules that arise in pentaquark models and which can discriminate among various dynamic and mixing schemes. In particular the relative strengths of decays Ξ − → Ξ − π 0 : Ξ 0 π − test10 -8 mixing [9]; Ξ − → ΛK − : Σ 0 K − have selection rules that test the decay dynamics that have been hypothesised [4,10,11] to suppress the pentaquark widths; and Ξ → Ξ * π is predicted to vanish for both10 and 8 5 initial pentaquark states in such dynamics. The electromagnetic mass splittings of the Ξ states also contain important information.
(iv) In pentaquark models the spin-orbit forces imply the existence of a nearby 8 5 J P = 3/2 + multiplet containing a Λ that should be narrow and unmixed barring isospin violating effects. We make some brief comments on point (ii) and then develop our main thesis, which focuses on points (iii) and (iv).

A low-lying
An essential difference between the pentaquark and chiral soliton (Skyrme) models appears to be in their implications for the first excited state of the Θ. In qqqqq with positive parity 1/2 + there is necessarily angular momentum present, which implies a family of siblings but with J P = 3/2 + . The spin-orbit forces among the quarks and antiquark lead to a mass gap between any member of the J P = 1/2 + and its J P = 3/2 + counterpart, which was calculated in ref. [8] to be significantly less than m π and possibly only O(10 − 50)MeV in the models of [5,6]. Similar remarks hold for all the members of the10, such as Ξ +,−− , and their non-exotic analogues that can also occur in 8 5 , such as Ξ 0,− . Such a10 family of J P = 3/2 + states does not occur in the present formulation of chiral soliton models, nor can it if the Wess-Zumino constraint selects allowed multiplets [12]. In the Skyrme model there are exotic states with J P = 3/2 + or higher but these are in 27 and 35 multiplets of SU (3) F . Such states are also expected in pentaquark models (e.g. isotensor resonance with states ranging from uuuus with charge +3 to dddds with charge -1) [13]. The essential difference then is that in the chiral soliton Skyrme models any spin 3/2 partner of the Θ will exist in a variety of charge states with I = 1, 2 whereas the unique feature of the pentaquark models [5,6] is that the first excited state is an isoscalar analogue of the Θ. (There may be versions of pentaquark models where this state is higher in mass but that it is isoscalar is universal in any quark model description).

Pentaquark wavefunctions, mixing and decays:
In pentaquark models where the (qqqq) is in6 F , then6 ⊗3 =10 ⊕ 8 5 leading to an 8 5 that is degenerate with the10 5 before mixing; chiral soliton models can accommodate an 8 (as a radial excitation of the ground state nucleon octet) though degeneracy is accidental. A challenge will be to decode the mixings between10 5 , this 8 5 and possible contamination with excited 8 3 in experiment. This is our main focus.
The essential dynamics that underpins correlations among the flavours and spins of quarks in QCD derives from a considerable literature that recognises that ud in colour3 with net spin 0 feel a strong attraction [14]. This might even cause the S-wave combination to cluster as [udu][ds] which is the S-wave KN system, while the P -wave positive parity exhibits a metastability such as seen for the Θ. Two particular ways of realising this are due to Karliner and Lipkin [6] and Jaffe and Wilczek [5]. In such models the basic correlation among quarks is to form antisymmetric flavour pairs, in3 of SU (3) F . In order to study the decays and mixings of these states it is important to have a well defined convention for their wavefunctions [15]. We define the3 F = (3 F ⊗ 3 F ) basis states as The Θ + ≡ AAA ≡ (ud)(ud)s and all other members of the10 follow by operating on this state sequentially by U − and I − until all states have been achieved. For reference they are listed in table 1. We shall always understand the first two labels to refer to the diquarks and the rightmost to refer to the antiquark in JW [5], and for KL [6] the latter pair of labels is understood to be in the triquark. The flavour correlations in the two models are thus identical.
In addition to the three manifestly exotic combinations AAA, BBB, CCC the nonexotic states can also form an octet. In the specific dynamics advocated in [5], the quark pairs are strongly correlated into scalar pairs with colour3. These scalar "diquarks" are then forced to satisfy Bose symmetry, which leads naturally to the following correlations. Their colour degree of freedom is antisymmetric3 ⊗3 → 3; their relative L = 1 provides an antisymmetric spatial state; their spin coupling is trivially symmetric; and Bose symmetry is completed by their flavour pairings being symmetric. This leads naturally to the positive parity10. For the 8 5 it leads to the mixed symmetric 8 MS states of table 1; in this extreme dynamics there are no mixed antisymmetric 8 MA analogues (e.g. p ≡ (AC − CA)A/ √ 2). This 8 MS decays to 8⊗8 with F/D = 1/3 as will become apparent later. Similar occurs for the KL correlation where the assumption that the triquark is in a6 F implies that the pentaquark system form10 ⊕ 8 with the same symmetry type as in table 1.
Thus the selection rules that we obtain are common to all these pentaquark models and a consequence of the assumed decay dynamics. The proposal of refs. [4,10,11] is that such pentaquarks can naturally have narrow widths due to the mismatch between the colour-flavour-spin state in an initial pentaquark and the meson-baryon colour singlet states into which they decay. For a simple attractive square well potential of range 1fm the width of a P -wave resonance 100MeV above KN threshold is of order 200MeV [5,10]. However, this has not yet taken into account any price for recoupling colour and flavour-spin to overlap the (ud)(ud)s onto colour singlets uud and ds say for the KN . If decays are assumed to arise by "fall-apart" [4,10,11,16] without need for gluon exchange to trigger the decay (even though gluon exchange may be important in determining the eigenstates), then in amplitude, starting with the Jaffe-Wilczek configuration, the colour recoupling costs 1 √ 3 . It is further implicitly assumed that the fall-apart decay to a specific channel occurs only when the flavour-spin correlation in the initial wavefunction matches that of the said channel. In such a case the flavour-spin correlation to any particular channel (e.g. K + n) costs a further 1 2 √ 2 , hence a total suppression in rate of 1 24 . This was originally noted in [10]. The minimal assumption then is that a diquark must cleave such that one quark enters the baryon and the other enters the meson. While this is necessary, implicitly it is assumed also to be sufficient: any components in the wavefunction that are not kinematically allowed to decay are assumed to be absolutely forbidden. Selection rules that we obtain here assume this and therefore are implicitly a test of this decay dynamics. There is also a penalty for the spatial overlaps. If once organised into colour singlets, the constituents then simply fall apart in a P -wave with no momentum transfer, only the L z = 0 part of the wavefunction contributes. This implies a further suppression from the L = 1 ⊗ S = 1/2 → J = 1/2; 3/2 coupling. Thus a total suppression of 1/72 for the 1/2 + and 1/36 for 3/2 + may be expected [8].
The general conclusion is that if such dynamics govern the decays, then in such models a width of O(1 − 10)MeV for Θ → KN may be reasonable. The above dynamics also implies that g 2 (ΘN K * )/g 2 (ΘN K) = 3 (this is also implicit in [11]). Although the N K * decay mode is kinematically inaccessible this relation may eventually be tested in photoproduction experiments [17]. Analogously this implies that g 2 (Θ c N D * )/g 2 (Θ c N D) = 3. The Θ c is predicted in ref [5] to lie below strong decay threshold but spin-orbit effects [8] could elevate its mass such that it is even above D * N threshold (see e.g. [6]). Thus if m(Θ c ) > 2.95 GeV, an enhanced intrinsic coupling to D * N could be searched for. With the wavefunctions in table 1 we can immediately account for the relative strengths of final states by carefully exploiting the symmetries of the wavefunctions. For example, the Θ ≡ These amplitudes for decays into meson (M ) and baryon (B) also depend on the flavour-spin symmetry of the baryon. If we make this explicit (φ, χ referring to the flavour and spin wavefunctions respectively and M A , M S denoting their mixed symmetry properties under interchange [18]) we have The same colour-orbital configuration for tetraquarks (qqqq) in overall spin S = 0 can be realised with diquarks in 6 F , S = 1. The pattern of decays from this configuration mirror those above except that the baryon's flavour-spin symmetry is swapped Thus if one imposed overall antisymmetry on the tetraquark wavefunction one encounters for the flavour-spin part of the wavefunction Noting that there is an L = 1 within the (qqqq) system, the above wavefunctions imply that the (+) phase decays to M + B(56) in a P -wave and the (-) phase decays to M + B(70(L = 1)) in an S-wave. The latter would naively be kinematically forbidden and as such lead to a suppressed width if the Θ were in this representation (which is in the 105 dimensional mixed symmetry representation of flavour-spin [16]). However, one needs also to confront the kinematically allowed decays to M +B(56) from the (+) phase state (in the symmetric 126 representation and discussed in [11]). In practice decays shared by the (3, S = 0)(3, S = 0) and (6, S = 1)(6, S = 1) states lead to mixing. If this is stronger than the mass gap between these two states, then one would obtain the above two configurations, leading to the possibility of the Θ as a narrow state in 105 partnered by a (yet unobserved) broad partner in 126 (see also [19]). By contrast, if the mixing is small on the scale of the mass gap, the wavefunction of the light eigenstate in this spin-zero tetraquark sector will be dominated by the (3, S = 0)(3, S = 0) configuration, which is the Jaffe-Wilczek model [5]. As noted in ref. [10], mixing with the spin-one tetraquark sector as manifested in the KL correlation can lead to a lower eigenstate. The discussion in the rest of our present paper does not depend on this dynamical question.
Within the assumption that decays are driven by the fall-apart dynamics, the flavour patterns follow for all of these configurations. It is especially instructive to apply our study of the fall-apart to the Ξ 5 states. In what follows we assume that only the ground state baryon + 0 − meson channels are kinematically accessible.
If other channels such as 1 − mesons could be accessed these would cause the intrinsic suppression to be less dramatic.

Decays of Ξ 5 states
Starting with the wavefunction (see table 1) we can rewrite this in flavour space in the form (qqq)(qq).
which maps onto the following ground state hadrons These agree in relative magnitudes and phases with the standard de Swart results [20,21]; they agree in relative magnitudes with Oh et al. [22] but their phases differ from ours. Refs [9,22] do not discuss the 8 decays as these depend in general on an undetermined F/D. However with the pentaquark wavefunctions, as specified as in table 1, the octet from6 F ⊗3 F that is orthogonal to the10 is and for the assumed decay dynamics employed in [4,10,11], the particle decomposition is which corresponds to 8 → 8⊗ 8 with F/D=1/3 (or g 1 = √ 5g 2 in the de Swart convention [20,21]). With this one can therefore deduce the branching ratios for N, Σ, Λ states in 8 5 immediately from existing tables [20,21] and we do not discuss them further here.
For the Ξ 5 we see immediately distinctions between the two states. (i) Isospin (I = 3/2 versus I = 1/2) is responsible for the distinctive ratios and analogous for the ΣK modes.
(ii) There is a selection rule that ΛK − modes vanish. For the10 this is a trivial consequence of isospin; for the 8 5 it is a result of the pentaquark wavefunction, in particular that the qqqq flavour wavefunction of the pair of diquarks is symmetric in flavour, (i.e.6 F =3 F ⊗3 F ) leading to F/D=1/3. Rescattering from kinematically forbidden channels, such as Ξη can feed both KΣ and KΛ, though this is not expected to be a large effect if experience with light hadrons is relevant (such as the small width of the f 1 (1285) not being affected by rescattering from the kinematically closed KK * channel, and the predicted π 2 → b 1 π ∼ 0 [23] not being affected by rescattering from the allowed channels πf 2 ; πρ). Whether this carries over to pentaquarks may be tested qualitatively in models by comparing the relative suppression of Θ, Ξ −− and Ξ − states; if there is no rescattering and the Ξη channels are closed in the initial pentaquark wavefunction, its width will be further suppressed from 1/24 →∼ 3/115 and KΛ ∼ 0. In this case the width of Ξ − (after phase space effects have been removed) will be less than that of Ξ −− . A dominance of KΛ > KΣ can arise if there are pentaquark configurations having F = D. In this latter case the Σ 0 K − would be forbidden but ΛK − allowed. The ΛK : ΣK ratio in general can be used to constrain the F/D ratio and begin to discriminate between various dynamical schemes.
(iii) Decays to Ξ * π and Σ * K for Σ * , Ξ * in the 10 are forbidden (even if allowed by phase space). For the10 this is a result of10 = 8 ⊗ 10 as noted in ref [9] who also discuss SU (3) F breaking as a potential source of violation of this zero. However, this selection rule may be stronger in the pentaquark models of refs. [5,6] due to the diquarks having antisymmetric flavour (3) and spin zero, both of which prevent simple overlap of flavour-spin with the 10, S = 3/2 baryon decuplet resonances. Thus although SU (3) F allows 8 → 10 ⊗ 8 to occur, for the 8 5 states of table 1 it is again forbidden as a consequence of the antisymmetric flavour content of the wavefunction, at least within the models of suppressed decay widths considered here. While we discussed this for the Jaffe Wilczek wavefunction, Karliner and Lipkin have one of their quark pairs strongly correlated into a vector spin state within a triquark (e.g. uds) so the flavour antisymmetries and explicit scalar diquark in the residual wavefunctions suggest that this dynamics also would be challenged to accommodate a violation of this selection rule.
The I = 3/2 states will all be narrow. They are degenerate up to electromagnetic mass shifts. Across the I = 3/2 multiplet the mass split is Ξ −− −Ξ + = (d−u)+ e 2 /R [18] where the Coulomb contribution in known hadrons is ∼ 2 − 9MeV, hence a spread of 3-10MeV is expected. For the non exotic states m(Ξ 85 ) > m(Ξ1 0 ), with and hence degenerate to within better than 5 MeV. If the coupling to Ξ * π vanishes for the 8 5 as well as the10, then mixing by the common Ξπ decay channels will be destructive. If the widths are truly narrow the mass eigenstates become Ξ u ≡ (ds)(su)ū and Ξ d ≡ (ds)(ds)d separated by ∼ 10MeV. The heavier state b.r.(Ξ d → Ξ − π 0 ) = 2 × b.r.(Ξ d → Σ −K 0 ) (apart from phase effects) but it does not decay to either Ξ 0 π − nor Σ 0 K − . In contrast the lighter state Ξ u → Ξ 0 π − : Ξ − π 0 = 2, as for the pure 8 (but with opposite relative phase), while it does not decay to Σ −K 0 .
Violation of these relations would imply either mixing with excited 8 3 states, be due to pentaquark components in the wavefunction beyond those above, or because the width suppression is realised by some dynamics other than implicit in refs. [4,10,11]. In the former case one would expect the 8 3 components to decay without suppression and dominate the systematics of the widths. In this case there will be narrow Ξ −,0 with I=3/2 partnering the exotic Ξ +,−− and broad I=1/2 states that are akin to normal excited Ξ states. By contrast, were the Ξπ charge ratios to show mixing between the two Ξ 5 states with two narrow states such as Ξ d and Ξ u , then observation of any ΛK or Ξ * π would require components in the pentaquark wavefunction with different symmetries to those above.

Decays of p 5 and n 5 states
These follow immediately from SU (3) tables with F/D=1/3. In general there will be mixing between these as suggested by Jaffe and Wilczek. For the extreme p 5 (ss) and p 5 (dd) we have and while phase space only admits trivial p 5 (dd) decays to N π.
It is immediately apparent that the decays of P 11 (1440; 1710) do not fit well with this scheme. First, there is a dominance of nonstrange hadrons in the heavier P 11 (1710) with prominent ∆π in the decays of both P 11 (1440; 1710). This mode is not possible for the p 5 states in10 nor in 8 5 unless overwritten by rescattering or mixing with 8 3 . It is clear that P 11 (1440) is partnered by P 33 (1660) as in a traditional 56 multiplet of SU(6) qqq states. There is no obvious sign of pentaquarks here. A possibility is that the states are linear combinations of p 3 and p 5 ; the p 5 could even dominate the wavefunction but its O(1MeV) width is swamped by the O(100MeV) width of the unsuppressed p 3 component. The p 5 decays listed above would then show up as rare decays at the O(1%) level.
3. Λ 5 state with J P = 3/2 + There is one further potentially narrow state in pentaquark models, which has little opportunity for mixing with qqq states. This is the Λ 5 state that is the J P = 3/2 + spin-orbit partner of Λ 5 in 8 5 .
First note that10 contains Σ 5 but has no Λ 5 . The 8 5 contains a Λ 5 , and there will be no mixing with10 so long as isospin is good. If there were no mixing with Λ(qqq) excited states, this Λ 5 would be narrow, with width identical to that of the Θ apart from phase space factors.
The Λ 5 wavefunction shows that it has only one strange mass quark and hence is similar to the Θ in this regard. Ref. [5] estimate ∼ 1600 MeV for such a state (the excess ∼ 60 MeV relative to the Θ arising because the mass of a (us)d set is larger than (ud)s due to the relatively smaller downward mass shift in the (us) diquark). Scaling the spin-orbit splitting from [8] and allowing for the relative masses of thes/d and m(us)/m(ud) gives 40-70 MeV for the Λ 5 mass gap of 3/2 + − 1/2 + and hence 1600-1700 as a conservative estimate for the mass range for the partner Λ 5 (3/2 + ).
Perusal of the data [21] shows that, for the 1/2 + , mixing with qqq states is likely (given the existence of a candidate 56, 0 + multiplet containing P 11 (1440), Λ(1600), Σ(1660), Ξ(?)). However there is no 3/2 + multiplet with a Λ(1600 − 1700) seen, nor is one expected in standard qqq models. The first such is the set containing P 13 (1720), Λ(1890) · · · . Thus there is a significant gap between Λ(1890) and our predicted Λ 5 (3/2 + ). The branching ratios for either the spin 1/2 or 3/2 states can be determined from the breakdown Decays to Σ * π should be suppressed, even if they are kinematically accessible. The production rate of the spin 1/2 state in γp → K + Λ 5 should be similar to that of γn → K − Θ (perhaps a factor of four smaller if K exchange drives the production and g(KN Θ) = 2g(KN Λ 5 )). If the arguments about L ⊗ S coupling and fall-apart dynamics are correct, then we can expect the spin 3/2 state to be enhanced by a factor of two relative to the spin 1/2 counterpart. A search in γp → K + Λ 5 therefore seems appropriate.

Summary
We advocate study of decays of the Ξ states, especially the ratios of various charge modes, and searching for Ξ * π and ΛK as tests of the underlying dynamics that forms the states. We also stress the importance of isolating the J P = 3/2 + states that must occur in10 and 8 5 in pentaquark models but which have no analogue in the chiral soliton model. These states are predicted to be within a few tens of MeV of their 1/2 + counterparts in highly correlated models such as those of Jaffe-Wilczek or Karliner-Lipkin. Were the mass gap to be significantly larger, then it could point to the presence of other components in the pentaquark wavefunction. By contrast, the absence of such states together with the appearance of J P = 3/2 + in higher representations such as 27 or 35 would support the chiral soliton models. The Λ 5 (3/2 + ) state may be relatively light and