Interpretation of the Theta+ as an isotensor pentaquark with weakly decaying partners

The Theta+(1540), recently observed at LEPS, DIANA and CLAS, is hypothesized to be an isotensor resonance. This implies the existence of a multiplet where the Theta++, Theta+ and Theta0 have isospin-violating strong decays, and the Theta+++ and Theta- have weak decays and so are long-lived. Production mechanisms for these states are discussed. The J^P assignment of the Theta is most likely 1/2^- or 3/2^-.

1 are above threshold for such fall-apart decays can be expected to be immeasurably broad, as no such states have been observed to date. The WA102 collaboration [6] sees roughly 15,000 events in f 2 (1270)ππ that are well described by a Breit-Wigner at 1950 MeV with a width of 450 MeV, associated with the state f 2 (1950). This is likely the broadest well established resonance. This would suggest that multi-quark states that fall-apart should have decay widths of at least 500 MeV.
The largest phase space for the fall apart decay of an isoscalar or isovector Θ + → nK + will be when the Θ + has J P = 1/2 − , and so decays in an S-wave. This is the lowest angular momentum possible for the Θ if its four quarks and anti-quark are in a spatial ground state.
If the width of such a state is 500 MeV, then other spin-parity assignments for the Θ + will allow decay widths suppressed by phase-space factors (p/β) 2L /(2J + 1), where β ≃ 400 MeV is a typical energy scale associated with the size of hadrons, and p = 270 MeV is the center of mass decay momentum.
If the Θ + is isoscalar or isovector with spin-parity 3/2 − or 5/2 − , also possible in a spatial ground state, the decay is D-wave, with a width significantly larger than the observed limit 1 . If the Θ + has spin-parity 1/2 + or 3/2 + , which require one unit of orbital angular momentum, the decay is P -wave with larger widths. If the Θ + has J P = 5/2 + , 7/2 − ,... then it decays in F -wave or higher, and it is possible that is an isoscalar or isovector state and phase space suppression accounts for the small observed width. However, it is unlikely that such a state would be a ground state, as the addition of orbital angular momentum will significantly increase its energy.
In what follows, an estimate is made of the width of the isotensor Θ + → nK + decay.
Isospin-violating strong decays in mesons have widths which are typically fractions of MeV.
The P -wave decay ω → π + π − is a G-parity and so isospin violating strong decay, with a partial width of 0.14 MeV, which is roughly 0.1% of the isospin conserving ρ → ππ width.
An upper bound on the width of an isotensor Θ + can be estimated as follows. The m d − m u mass difference and electromagnetic interactions cause the wave functions of the Θ + and 1 The state Λ(1520) decays toKN in D-wave, has a mass similar to the Θ + , but has a small width of 15.6 ± 1.0 MeV. However, its decay requires the creation of a light qq pair through a non-trivial decay operator. It is, therefore, necessarily narrower than a similar state with a fall-apart decay, where the final state configuration exists in the initial state and so the decay operator is trivial (unity).
State Quarks I z Decay modes Assuming a strong decay operator O which conserves isospin, the isospin violating Θ + → nK + decay width will be Θ + |O|nK + ≃ γ 2|O|3/2, 1/2 + α 1|O|1/2, 1/2 + β 0|O|1/2, 1/2 , ignoring terms second order in the small coefficients and suppressing I z values. Isospinviolating effects in the nucleon wave function are known to be very small [7,8], so γ is negligible. A larger mixing occurs in the Λ 0 -Σ 0 system, where the physical Λ is a mixture Λ 0 8 +ǫ Σ 0 8 of states of definite isospin with a mixing coefficient [9] of ǫ = −0.015, and similarly for the physical Σ 0 . Assuming that α and β are both of this size, and that the isospinconserving decays widths (of our state well above its fall-apart decay threshold) are all of the order of 500 MeV, a rough upper bound for the Θ + width is 4 (0.015) 2 (500 MeV) = 0.45 MeV. This is of the same order of magnitude as that of the isospin-violating ω → π + π − decay width, and is much smaller than the experimental upper bounds on the Θ + width.
2 Prediction of strongly decaying Θ ++ and Θ 0 and weakly decaying Θ +++ and Θ − If the Θ + is an I = 2 pentaquark, it is the isospin projection I z = 0 member of the multiplet depicted in Table 1. If discovered, the Θ +++ would be the first triply charged hadron.

State Decay mode
Pairs State Decay mode Pairs Table 2: Semi-leptonic and non-leptonic weak decay modes of the Θ +++ and Θ − in order of increasing phase space. Here l = e, µ. All decay modes listed are Cabibbo suppressed. Also indicated is whether the decay proceeds by single W annihilation (A), exchange (E), or production (P), as well as the number of qq pairs that are created by the strong interaction.
pπ + π + , are distinguished from those that proceed through an intermediate resonance, e.g.
∆ ++ π + → pπ + π + , even though they lead to the same final state. Because the ∆ is broad, a final state ∆Xl + ν l would be experimentally indistinguishable from NπXl + ν l , if ν l is undetected.
Typical splittings in I = 1 2 , 1 or 3 2 multiplets are n − p = 1.3 MeV, π ± − π 0 = 4.6 MeV and ∆ 0 − ∆ ++ = 2.3-2.7 MeV respectively [10]. From model expectations and data [11], the ∆ − − ∆ ++ splitting is likely close to 4.5 MeV. Based on this, the largest mass splitting in the Θ multiplet is expected to be less than 10 MeV. The relevant threshold for strong decay of the Θ is NK. Since the Θ + is below NKπ threshold by about 30 MeV, its isospin partners are also 2 . This precludes the strong decays Θ → NKπ, specifically Θ − → nπ − K 0 and Θ +++ → pπ + K + , and so these states must 3 decay weakly as indicated in Table 2. A multiplet where the central members (Θ 0 , Θ + and Θ ++ ) decay strongly, while the outlying members (Θ +++ and Θ − ) decay weakly, has no analogue for known mesons and baryons. 2 The lowest threshold is pπ 0 K ± at 1567 MeV. Only the pπ 0 K + decay can come from a pentaquark with the structure of the Θ + . 3 The electromagnetic decay Θ →Xγ is kinematically forbidden if there is no pentaquark X with the charge of the Θ, but at a lower mass.

Explanation of CLAS and DIANA data
The isotensor Θ + is produced via isospin conserving reactions of the form γN → Θ + K or γd → Θ + K − p, through the isovector component of the photon, at LEPS and CLAS 4 .
In the CLAS collaboration's photo-production experiment with a deuteron target, the Θ + is reconstructed from the invariant mass spectrum of K + n in the reaction γd → pK − K + (n).
The four momenta of all reaction products are specified, as the three charged particles in the final state are detected, and the neutron is then identified by missing mass. Note that a detailed study of this reaction was unable to explain the events interpreted as the Θ + by non-resonant rescattering processes [12]. Similarly, in the photo-production experiment with a proton target, the neutron in γp → π + K − K + (n) is identified by missing mass after detection of the charged particles.
In the deuteron target experiment, it should be possible to produce the isospin partner Θ ++ of the Θ + by the reaction γd → Θ ++ K − n → pK + K − (n), and detect it by examining the invariant mass of the K + p system [13]. It may be the case that production of the Θ ++ is suppressed relative to that of the Θ + . If, as is common in kaon photo-production experiments at these photon energies, the reactions γn → K − Θ + and γp → K − Θ ++ result in forward peaked K − distributions, these negatively charged particles will be bent into the beam direction by the magnetic field and will go largely undetected, unless they scatter off the spectator nucleon. Scattering cross sections for K − n are considerably smaller than those of K − p where they are measured, at kaon beam energies of 600 MeV or higher [10].
So if the K − particles are forward peaked and the K − n cross section remains small down to low energies, it is likely that the Θ ++ will remain undetected without a larger data sample.
Under ideal circumstances a formation reaction of the form K + n → Θ + → K 0 p, with a K + momentum of 440 MeV incident on a Xe target at rest and with the Xe ′ at rest in its ground state (quasi-free production), could account for the ITEP data. However, such a formation process is isospin violating for an isotensor Θ + . It is hence predicted that ITEP should not see the Θ + through this formation process. In the ITEP experiment the kinematics are not completely reconstructed for the reaction K + Xe → K 0 p Xe ′ , as the energy and momentum of the Xe ′ is unknown, and so it is not possible to know whether this quasi-free production process happens. If ITEP is indeed seeing the isotensor Θ + , this is expected to be through 4 Mechanisms for the production of Θ +++ and Θ − Here we focus on likely production mechanisms for the weakly decaying pentaquark states Θ, as opposed to anti-pentaquark statesΘ, because of the availability of nuclear targets.
The figure of merit is whether the production cross-section for Θ times the branching ratio of Θ into its final state is substantial.
On face value the most promising process for the production of the Θ +++ is pp → Θ +++ Σ − , which involves one ss pair creation. The search may be feasible at ITEP, JHF or COSY. The reaction pp → Θ + Σ + was previously suggested [14] in order to search for the Θ + . Another promising process is K + p → Θ +++ π − , involving the creation of one light quark pair. A search at JHF and ITEP is feasible. In these experiments the weakly-decaying Θ +++ could be either long-lived and produce a track in the detector, or relatively short-lived, and so should be observed in pπ + π + (see Table 2).
Production of the Θ − in pp or K + p collision would require more simultaneous quark pair creations, and so is strongly suppressed. The most promising production mechanism for Θ − appears to be nn → Θ − Σ + , although this would require a neutron beam more energetic than the maximum 800 MeV beam at LANSCE at LANL.
There is a secondary production mechanism for the Θ +++ and Θ − in photo-or electroproduction at JLab, SPring-8, Mainz, SAPHIR or HERA, following production of a K + or K 0 . The production of the Θ +++ would proceed as follows. In the reaction γd → Σ − π − Θ +++ a K + can be produced off the neutron (γn → Σ − K + ), which then interacts with the proton, K + p → Θ +++ π − . The final state Σ − π − Θ +++ consists entirely of weakly decaying particles. The process γd → Σ − π − Θ +++ involves in total two qq pair creations, and scattering of the K + with the proton, qualitatively similar to the possible production mechanisms at CLAS discussed in the previous section. The production mechanism for the Θ − is closely related, and occurs at a similar level in γd → Σ + π + Θ − . Here the effect is to produce a K 0 off the proton (γp → Σ + K 0 ), which then interacts with the neutron, It is conceivable to interpret the isotensor Θ + as a ∆K molecule below threshold. However, this interpretation faces two problems: • The ∆ is short-lived (115 − 125 MeV wide [10]), making a simple molecular picture unlikely.
• The Θ + is ∼ 190 MeV below the ∆K threshold, which is an atypically large binding energy compared to molecular candidates like the f 0 (980) and a 0 (980), which are ∼ 10 MeV below threshold. Furthermore, relating the binding energy of a molecule 1/E b ∼ 2µ r 2 r.m.s. to the root mean square separation between the ∆ and K, yields r r.m.s. ∼ 0.5 fm. This distance is smaller or similar to the sizes of the constituents, so that the picture of a molecule built from undeformed hadrons is not reasonable.
The Θ + is hence best modeled as a pentaquark, as opposed to a loosely bound molecular state. An isotensor pentaquark can only be constructed when each of the quark-pairs are isovector, which means that each of the quark-pairs ij and kl must be symmetric under exchange i → j, k → l, so that the flavor wave function is also totally symmetric. Assuming that the Θ + is the ground-state pentaquark, and that the ground-state has all the quarks and antiquark in relative S-waves (which would give the lowest energy if there is no strong repulsive core), the spatial wave function is totally symmetric. The Pauli principle and isospin symmetry require the overall fermion wave function to be totally antisymmetric under exchange of the four light quarks, so that the color-quark-spin wave function of the four quarks must be totally antisymmetric.
This implies that the four quarks must be in an antisymmetric representation of the colorspin symmetry group SU(6). This is [15] the 15 representation of SU (6), which is made up of a color 6 with spin zero, and a color triplet of spin 1. When combined with the color 3 antiquark, a color singlet pentaquark can only have the four quarks combined to a color-triplet with spin 1. This restricts the isotensor pentaquark to J P = 1/2 − or 3/2 − . These J P would only allow S-or D-wave decays Θ + → nK + , which, when compared with experiment, indicates that Θ + is anomalously narrow. Even though the observation of Θ + in nK + allows for any J P assignment, and an angular analysis must yet be performed to determine the parity of the Θ + , it is expected that J P = 1/2 − or 3/2 − . In Ref. [16] the Θ + is interpreted as an isoscalar or isovector pentaquark in a constituent quark model with a flavor-spin interaction that arises from Goldstone boson exchange between the quarks. This flavor-spin interaction provides an attractive interaction between the quarks, which can lower the energy of the state with the four light quarks in an P -wave state below that of the corresponding S-wave state, so that the ground state pentaquark has positive parity [17].

Discussion
The chiral-soliton model prediction [5] of a narrow isoscalar state Z + at 1530 MeV, situated at the top of an SU(3) f anti-decuplet of J P = 1/2 + states, partially motivated the searches for the experimental Θ + state described above. In Ref. [5], the width of the Z + is predicted to be 15 MeV. The mass of the Z + is found by "anchoring" the anti-decuplet to a nonexotic member of the anti-decuplet, which has nucleon flavor. This is identified with the established state N(1710) seen in Nπ elastic scattering, as well as in other channels. If the anti-decuplet P 11 is chosen to be N(1440), the Z + state would be stable against strong decays, and if it is chosen to be N(2100) the Z + would be very broad. The N(1710) state is described in the constituent quark model as a radial excitation of the nucleon, and its strong decays, photocouplings, and mass [18] are well understood in that model.
The total width of the N 10 (1710) state is predicted in the chiral soliton model to be 43 MeV, which is below the 50 to 250 MeV range quoted by the Particle Data Group (PDG) [10], and significantly below the PDG's "estimate" of 100 MeV. Similarly, the width of the Σ 10 , associated with the two-star P 11 state Σ(1880), is predicted to be about 70 MeV, which is again smaller than the 80-260 MeV range of widths quoted by the PDG. In the chiral soliton model, the widths of all members of the anti-decuplet are proportional to a calculated constant G 2 10 . If this was adjusted upward to accommodate the PDG estimate of the width of the N(1710), the predicted width of the Z + would increase, and would exceed the bounds set by the experimental data.
The essential mechanism for the narrow width of the Θ + , which is well above its strong nK + and pK 0 thresholds, is that its decay to nK + or pK 0 is isospin violating if it is isotensor. It is as though the strong decay threshold has been "raised". It is intriguing to note that such raised thresholds will occur for all isotensor qqqqQ states, with q an up or down quark, and Q a heavy quark, since their decay to N(qQ) also violates isospin conservation. A related phenomenon happens for the dddQū and uuuQd pentaquarks. The decays dddQū → ∆ − (Qū) or Σ Q π − are allowed, while dddQū → N(Qū) or Λ Q π are not. Also, the decays uuuQd → ∆ ++ (Qd) or Σ Q π + are allowed, while uuuQd → N(Qd) or Λ Q π are not.