5.6 Extension to unequal masses

Let us consider now a baryon made of different quarks. We use the Jacobi coordinates of Eq. (3.52) and the corresponding reduced mass μ = 2m₁m₂∕(m₁ + m₂). If two quarks are identical, as in Λ, Σ or Ξ type of baryons, or almost identical, as s and u in Ξ_c⁺, they are assigned to labels 1 or 2. This helps to enforce the exact or approximate symmetry constraints. The hyperspherical treatment of

(5.25)

is then performed as for identical quarks. There are, however, more harmonics into the expansion, since some (if m₁ = m₂≠m₃) or all (if m₁≠m₂≠m₃) symmetry constrains are relaxed. This makes the use of potential harmonics more desirable.

In Ref. [58] is studied the case of harmonic forces, for which an exact solution is available. The results of the linear model V = 1 2 ∑ r_ij^β with β = 1, m ₁ = m₂ = 1, and m₃ = m′ are shown in Table 5.3. Another example, more suited to double-charm baryons [7] is given in Table 5.3. The hyperscalar approximation is obviously rather poor for the wave function. However, the convergence remains satisfactory as L_max increases.