Phenomenology of the Pentaquark Antidecuplet

We consider the mass splittings and strong decays of members of the lowest-lying pentaquark multiplet, which we take to be a parity-odd antidecuplet. We derive useful decompositions of the quark model wave functions that allow for easy computation of color-flavor-spin matrix elements. We compute mass splittings within the antidecuplet including spin-color and spin-isospin interactions between constituents and point out the importance of hidden strangeness in rendering the nucleon-like states heavier than the S=1 state. Using recent experimental data on a possible S=1 pentaquark state, we make decay predictions for other members of the antidecuplet.


I. INTRODUCTION
Recently, a number of laboratories have announced observation of a strangeness +1 baryon [1,2,3] with a mass of 1540 MeV and a narrow decay width. Such a state cannot be a 3-quark baryon made from known quarks, and it is natural to interpret it as a pentaquark state, that is, as a state made from four quarks and one antiquark, q 4q . The current example of the strangeness S = +1 baryon is positively charged and is called Z + in the particle data tables and θ + in some recent works [3]. The Z + of necessity has an s and four non-strange quarks. The parity, spin, and isospin of the experimental state are currently unmeasured.
In this paper, we study consequences of describing the Z + within the context of conventional constituent quarks models, in more focused detail than was done in earlier work [4,5,6] and with new results. In these models, all quarks are in the same spatial wave function, and spin dependent mass splittings come from either color-spin or flavor-spin exchange. The Z + made this way has negative parity. We treat it as a flavor antidecuplet, with spin-1/2 because this state has, at least by elementary estimates, the lowest mass by a few hundred MeV among the Z + 's that can be made with all quarks in the ground spatial state.
The pentaquark by now has some history of theoretical study. In the context of constituent quark models, it was analyzed relatively early on [4,5,6], but the subject was not pursued, probably for lack of experimental motivation. (The first of [4] gives a simple estimate of the Z + mass of 1615 MeV and then states "There definitely is no Z * (I = 0) state at such a low mass.") Much of the effort shifted to studying pentaquarks involving charmed as well as strange quarks [7,8], before the recent flurry of theoretical attention [9].
Pentaquarks have also been studied in the context of the Skyrme model [10,11]. Ref. [11] in particular makes a striking prediction, based on the assumption that the Z + is a member of a flavor antidecuplet and that the nucleon-like members of this decuplet are the observed N * (1710) states, that the Z + would have a mass of about 1530 MeV and a width less than 15 MeV. Note that in this case the Z + is a positive parity state.
We may elaborate on the Z + states and masses in quark models briefly before proceeding.
In outline, there are several ways to make a Z + , and one can obtain Z + 's which are isospin 0, 1, or 2. The mass splittings between the states can be estimated using, say, the color-spin interactions described in more detail in the next section. Techniques and useful information may be found in [4,8,12]. The lightest Z + state is the isosinglet (in the 10) with spin-1/2.
The isosinglet spin-3/2 is a few hundred MeV heavier. The heaviest states are the isotensor spin-1/2 and (somewhat lighter) spin-3/2 states. The mass gap between the lightest and heaviest of the Z + 's is triple the mass gap between the nucleon and the ∆(1232), if one does not account for changes in the quarks's spatial wave functions (e.g., due to changes in the Bag radius), or the better part of a GeV. The isovector masses lie in between the two limits.
These statements are considered in quantitative detail in Ref. [13] In the next section, we will discuss the color-flavor-spin wave functions of the antidecuplet that contains the Z + . This is a necessary prelude to a discussion of the mass splittings and decays of the full decuplet, which follows in Section III. One intriguing result is the roughly equal mass spacing of the antidecuplet, with the Z + lightest. Normally one expects the strange state to be heavier that the non-strange one. The explanation of this counterintuitive behavior is hidden strangeness, that is, there is a fairly high probability of finding an ss pair in the non-strange state. We also show that there is a markedly different pattern of kinematically allowed decays, depending of whether spin-isospin or spin-color exchange interactions are relevant in determining the mass spectrum. We close in Section IV with some discussion.

II. WAVE FUNCTION
There are two useful ways to compose the pentaquark state. One is to build the q 4 state from two pairs of quarks and then combine with theq. The other is to combine a q 3 state with a qq to form the pentaquark. We first represent the pentaquark state in terms of states labelled by the quantum numbers of the first and second quark pairs. Since the antiquark is always in a (3,3,1/2) (color,flavor,spin) state, we know immediately that the remaining four-quark (q 4 ) state must be a color 3. The flavor of a generic q 4 state can be either a 3,6, 15 M , or 15 S (where S and M refer to symmetry and mixed symmetry under quark interchange, respectively). However, only the6 can combine with the3 antiquark to yield an antidecuplet. Finally, the spin of the q 4 state can be either 0 or 1 if the total spin of the state is 1/2. However, it is not difficult to show that any state constructed with the correct quantum numbers using the spin-zero q 4 wave function will be antisymmetric under the combined interchange of the two quarks in the first pair with the two quarks in second pair; this is inconsistent with the requirement that the four-quark state be totally antisymmetric.
(2.1) Figure The requirement of total antisymmetry of the q 4 wave function, determines the relative coefficients. We find that the properly normalized state is given by where we have suppressed the quantum numbers of the antiquark, (3,3,1/2), which are the same in each term. Also tacit on the right-hand side is that each q 4 state is combined to (3,6,1). The signs shown in Eq. (2.2) depend on sign conventions for the states on the right-hand side. For the Z + component,spin ↑, we find Here we have written the color wave function in tensor notation for compactness, with It is often convenient for calculational purposes to have a decomposition of the pentaquark wave function in terms of the quantum numbers of the first three quarks, and of the remaining quark-antiquark pair. The quark-antiquark pair can be either in a 1 or 8 of color, which implies that we must have the same representations for the three-quark (q 3 ) system, in order that a singlet may be formed. As for flavor, the q 3 and qq systems must both be in 8's: the qq pair cannot be in a flavor singlet, since there is no way to construct a 10 from the remaining three quarks, and the q 3 state must be an 8 since the remaining possibilities (1 and 10) do not yield an antidecuplet when combined with the qq flavor octet. Finally, the qq spin can be either 0 or 1, which implies that the q 3 spin can be either 1/2 or 3/2. The states consistent with q 3 antisymmetry are then Again, we may find the coefficients by requiring that the total wave function is antisymmetric under interchange of the four quarks. Alternatively, we may take the overlap of any of these states with the wave function that we have already derived in Eqs. (2.2)-(2.5). The details and explicit results will be presented in a longer publication [13]. We find where N 5 , Σ 5 and Ξ 5 represent the strangeness 0,−1 and −2 members of the 10, respectively.
The nonstrange member of the 10 is heavier than the Z + because it has, on average, m s /3 more mass from its constituent strange and antistrange quarks.
We also note that our decomposition in Eq. (2.6) allows us to easily compute overlaps with states composed of physical octet baryons and mesons. For example, the first term in Eq. (2.6) may be decomposed for the Z + The sizes of the coefficients of these terms affect the rate of the "break-apart" decay modes, such as Z + → NK + . We therefore find that the smallness of the observed Z + decay width In the bag model, the mass of a hadronic state is given by where Ω i /R is the relativistic energy of the i th constituent in a bag of radius R, and x is a root of The parameter Z 0 is a zero-point energy correction, and B is the bag energy per unit volume.
In the conventional bag model, Z 0 = 1.84 and B 1/4 = 0.145 GeV. The term α s C I represents the possible interactions among the constituents. We first take into account the color-spin interaction originating from single gluon exchange, so that where α s = 2.2 is the value of the strong coupling appropriate to the bag model, and µ(m i , m j ) is a numerical coefficient that depends on the masses of the of the i th and j th quarks. For the case of two massless quarks, µ(0, 0) ≈ 0.177; the analytic expression for arbitrary masses can be found in Ref. [15].
We take into account the effect of SU(3) breaking (i.e., the strange quark mass) in both Ω i and in the coefficients µ(m i , m j ). To simplify the analysis, we break the sum in Eq. (3.4) into two parts, quark-quark and quark-antiquark terms, and adopt an averaged value for the parameter µ in each, µ qq and µ qq . Using the wave function in Eqs. (2.2)-(2.5) we find that the relevant spin-flavor-color matrix elements are given by where j = 5 corresponds to the antiquark. This evaluation was done by group theoretic techniques [13], as well as brute-force symbolic manipulation [19]. To understand how we evaluate the coefficients µ qq and µ qq let us consider a nucleon-like state in the antidecuplet, the p 5 . The probability of finding an ss pair in the p 5 state is 2/3. In this case, 1/2 of the possible qq pairs will involve a strange quark. On the other hand, the probability that the p 5 will contain five non-strange constituents is 1/3. Thus, we take We also use the averaged kinetic energy terms The bag mass prediction is then obtained by numerically minimizing the mass formula with respect to the bag radius R. Applying this procedure to the p 5 and Z + states, we find the antidecuplet mass splitting We use the observed Z + mass, 1542 MeV, and the splitting ∆M 10 to estimate the masses of the p 5 , Σ 5 , and Ξ 5 states; we find 1594, 1646, and 1698 MeV, respectively. Decay predictions from SU(3) symmetry are summarized in Table I.
While we used the bag model as a framework for evaluating the mass spectra above, we believe our results are typical of any constituent quark model.
We adopt a simpler approach in evaluating the effect of spin-isospin constituent interactions, The Skyrme model also has predictions [11] for the masses and decays of the antidecuplet.
The mass splittings there were about 180 MeV between each level of the decuplet (with the Z + still the lightest), considerably larger splittings than we find in a constituent quark model where the mass splittings come from strange quark masses and from color-spin interactions.
Mass splittings using isospin-spin interactions were, on the other hand, more comparable to the Skyrme model results.
Decays of the antidecuplet into a ground state octet baryon and an octet meson involve a decay matrix element and phase space. Ratios of decay matrix elements for pure antidecuplets, such as we show in Table I, are fixed by SU(3) F symmetry. They are the same in any model, as may be confirmed by comparing Table I to results in [11]. We have neglected mixing; Ref. [11] does consider mixing but does not find large consequences for the decays.
The differences between relative decay predictions are then due to differences in phase space, and the differences are due to masses and due to parity. Negative parity states decaying to ground state baryon and pseudoscalar meson have S-wave phase space, while positive parity  are present, decays of the p 5 and Σ 5 to final states in which both decay products have nonzero strangeness are kinematically forbidden. In addition, the Ξ 5 states are narrower than those in Ref. [11], so that experimental detection might be possible and dramatic. If instead, spin-isospin interactions dominate, all the decays in Table I become kinematically accessible.
The work summarized here sets the groundwork for further investigation. Of particular interest to us is the relation between bag model predictions for the absolute pentaquark mass (rather than the mass splittings considered here) and the mass of other multiquark exotic states. The conventional MIT bag predicts a Z + mass that is too large relative to the experimental value (we find that a prediction of about 1700 MeV is typical); however, these numbers can be easily reconciled by allowing bag model parameters to float [16,17]. An appropriate analysis requires a simultaneous fit to pentaquark and low-lying non-exotic hadron masses, and consideration of center-of-mass corrections. Whether such fits simultaneously allow for sufficiently heavy six-quark states, given a choice of constituent interactions, is an open question. Our analysis also gives insight into other pentaquark states. For example, there are nucleon-like states in the pentaquark octet (states in the same spin-color representation as the Z + ) which are potentially light. However, we find that these states also have hidden strangeness, placing them within one-third of the strange quark mass below the Z + , if no other effects are considered, and at or above the Z + mass if spin-isospin interactions are taken into account. This is one example of the value of extending our present analysis to other pentaquark multiplets. A more detailed discussion of these topics, as well as of the group theoretical issues described here will be presented in a longer publication [13].