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11.6 The quadrupole moment of the Omega

In fact the spin–spin potential (11.4) which we repeatedly used throughout this chapter is only a part of the chromomagnetic interaction

vc.m.= VSS σ1 ⋅ σ2 + VTS12 , S12 = 3σ1 ⋅ ˆrσ2 ⋅ ˆr − σ1 ⋅ σ2 ,
(11.26)

where S12 is the tensor operator. The tensor component V T plays a role for orbital momenta l > 0. In any potential model that includes only a vector-exchange piece V V and a scalar-exchange piece V S, the ratio of tensor to spin–spin is well defined. In the particular one-gluon model [103] for equal masses

C ∑ VSS = ---- σi ⋅ σjδ(3)(rij) , 2m2 i<j ∑ VT = -3--C-- Sij , 8π 2m2 i<j r3ij C = 8π-αs. 9
(11.27)

In Ref. [108] are listed the tests of chromomagnetism in baryon spectroscopy. Let us discuss here one example, the quadrupole deformation of Ω−.

The effect was first mentioned by Goldhaber and Sternheimer [109], who speculated on the fine and hyperfine structure of exotic atoms consisting of an atomic nucleus and an Ω−. The splitting pattern would provide a measurement of the quadrupole moment as well of the magnetic moment of the Ω−. In Ref. [109], a quadrupole moment Q = 1 fm2 was assumed for numerical illustration, on the grounds that the Ω− has a mass comparable to that of the deuteron and hence might also have Q of the same order of magnitude. In fact much smaller values of Q are obtained from current quark models [108, 110].

Let us adopt the same familiar notation 2S+1ℓ J as in quarkonium spectroscopy. The analogue of expansions (11.17) and (11.20) is [110]

(n) (p) (q) Ψ = ∑ u---(ξ) | 4S(n) ⟩ + ∑ w---(ξ)| 4D(p) ⟩ + ∑ u--(ξ)-| 2S(q) ⟩ , ξ5∕2 3∕2 ξ5∕2 3∕2 ξ5∕2 3∕2 n p q
(11.28)

where n, p and q denote the successive hyperspherical harmonics of given spin and orbital momentum content. For an Ω− with spin S z = 3∕2, one needs only

4 (0) −3∕2 3 3 | S 3∕2⟩ = π | 2,2⟩ ,
(11.29)

where the spin 3/2 wave function is given in Section 3.5, and

√ ---[∘ -- 4 (0) ---12- 2 2 2 3 1 | D 3∕2⟩ = π3∕2ξ2 5(ρ+ + λ+ ) | 2,− 2⟩ ∘ -- − 45(ρ+ρ3 + λ+ λ3) | 32, 12⟩ ∘ --- ] + 2(ρ23 + ρ+ ρ− + λ23 + λ+ λ− ) | 3, 3⟩ , 15 2 2
(11.30)

where ρ± = ∓(ρx ± ρy)∕√ -- 2 and ρ3 = ρz are the usual standard components of ρ. Indeed, the unperturbed wave function is dominated by its hyperscalar component and the quadrupole operator

∑ e Qzz = ei(3z2i − r2i) = -s(3ρ23 − ρ 2 + 3λ23 − λ 2) , i 2
(11.31)

connects ∣4S(0) 3∕2⟩ only with ∣4D(0) 3∕2⟩. We end with equations very similar to those involved for the neutron charge radius

e ∫∞ Q Ω = ⟨Qzz ⟩ = √-s-- u0 ξ2w0dξ , 10 0 √ -- ′′ 63-- C--96--2-u0(ξ)- w 0 − 4ξ2w0 + ms [E0 − V0,0]w0 = ms π2√5-- ξ3 .
(11.32)

Restricting the quadrupole deformation to a mixing of closest unperturbed states is exact for harmonic confinement, but leads us to underestimate QΩ by 20% for a smooth potential β = 0.1. The latter model, with the parameters (11.19), gives QΩ = 0.004 fm2.

Note that the dependence on the exponent β is rather simple, if one restricts oneself to power-law potentials of the type (11.18). The strength of confinement, B, is determined from any excitation energy, for instance the orbital splitting E ≡ E(2+) − E(0+), while the strength of hyperfine corrections is measured by the mass shift ΔM = ⟨3VSS ⟩ of the ground state. One can easily show [110] that QΩ scales as ΔM∕(E2m s) (and, in turn, ΔM is related by scaling to measurable mass shifts like Δ − N). In other words, choosing a confining potential fixes the reduced quadrupole moment qΩ = QΩE2m s∕ΔM, and qΩ varies by less than a factor of 2 when going from a very smooth to a very sharp confinement. The phenomenological uncertainties come mainly from the choice of strengths B and C, i.e., from different adjustments of excitation energies and hyperfine splittings. The same scaling laws also hold for the charge radius of the neutron.

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