9.5 Comparison of the wave functions

Within the above Gaussian approximation, the r.m.s. separation between quarks, d_ij ≡^1∕2, experiences the same value in the baryon as in the

pseudomeson with constituent masses 3 4m. This approximation corresponds to choose the best harmonic approximation ar² + b to the potential V . When one treats the anharmonicity λv(r) in perturbation, the pseudomesons, in comparison to baryons, receive large contributions from higher states of the harmonic oscillator. This results in larger shifts, hence in the inequality (9.39), and also in larger radii. An illustration is given in Table 9.2, where are compared the r.m.s. separation in baryons and pseudomesons, for (again !) simple potentials ϵ(β)r^β. Except for β = 2, the quarks are more tightly connected in baryons. The rigorous comparison of the d_ij’s is studied in a recent paper [96].

The situation is slightly more complicated when one compares the wave functions at a given point. For instance, in Refs. [97, 98], the zero-range correlation in a baryon, δ₁₂ = is compared with its value in the pseudomeson. For a power–law confinement ϵ(β)r^β, δ ₁₂ is smaller than in the pseudomeson for β < 2, larger for β > 2 and equal for the harmonic case β = 2. This result is rigorous for β ≤ 2, and in fact for any potential such that d∕dr(V ′∕r) < 0 [98]. For β > 2, it is very plausible and has been checked by accurate numerical calculations.