A narrow "peanut" pentaquark

We analyse the decay Θs(1/2)→ NK in a non-relativistic Fock space description using three and five constituent quarks for the nucleon and the pentaquark, respectively. Following Jaffe and Wilczek [1], we assume that quark-quark correlations in spin-zero state play an important role for the pentaquark internal structure. Within this scenario, a strong dynamical suppression of the decay width is shown to be possible only if the pentaquark has an asymmetric "peanut" structure with the strange antiquark in the center and the two extended composite diquarks rotating around. In this case a decay width of ≃ 1 MeV may be a natural possibility. The existence of pentaquarks is not yet undoubtedly established. But if these particles exist, the exotic members of the pentaquark multiplet must have a very small decay width of order 1 MeV or even lower. For the possible origin of the small pentaquark width many qualitative suggestions have been put forward. In a scenarios proposed by Jaffe and Wilczek [1] the positive-parity spin-1/2 pentaquark consists of an antiquark and two scalar diquarks in a relative P-wave state. In this talk I present the results of a fully dynamical quark-model calculation of the pentaquark width done together with B.Stech and S.Simula [2] using a non-relativistic Fock space representation for the JP = 1 2 + pentaquark in the Jaffe-Wilczek scenario. The decay amplitude T (Θ → KN) is related to the matrix element 〈N(p)|s̄γμγ5d|Θ(p)〉 = gA(q)ūN(p)γμγ5uΘ(p)+gP(q)qμ ūN(p)γ5uΘ(p) +gT (q )ūN(p )σμνqγ5uΘ(p), q = p− p′. Here the form factors gi contain poles at q2 > 0 due to strange meson resonances with the appropriate quantum numbers. The residue of the pole in gP at q2 = M2 K is related to the amplitude of interest T (Θ → NK): for q2 → M2 K (M2 K −q)gP(q)ūN(p)γ5uΘ(p) → fKT (Θ → NK), where fK = 160 MeV is the kaon decay constant. The form factor gA contains the pole at q2 = (K∗ A) 2, but at q2 = M2 K it is a regular function. Making use of the relationship between the form factors gA and gP emerging in the limit of spontaneously broken chiral symmetry [2] gives T (Θ → NK) = MΘ +MN fK gA(M 2 K) · ūN(p)iγ5uΘ(p) and Γ(Θ) = Γ(Θ → Kn)+Γ(Θ → K0 p) ≃ 1 π |~q|3 f 2 K gA(M 2 K). For MΘ = 1540 MeV one finds |~q|= 270 MeV and Γ(Θ) = 240 gA MeV. For transitions between hadrons of the same quark structure gA ≃ 1 (e.g. for the nucleon gA ≃ 1.23). So for a normal resonance one would expect Γ(Θ) ≃ 200 MeV. To obtain a width of ≤ 10 MeV one needs a strongly suppressed value gA ≤ 0.2. In [2] we calculated the amplitude 〈N|s̄γμ γ5d|Θ〉 and the form factor gA(q) using a non-relativistic equal-time Fock space representation. The nucleon in this framework is described by its coordinate wave function depending on the relative coordinates ~ρN =~r2 −~r3 and~λN = 2(~r2 +~r3)−~r1, for which we take the Gaussian function ΨN(r1|r2,r3) ∼ exp ( − 1 2α2 ρN ~ρ2 N − 2 3α2 λN ~λ 2 N )