A linear potential in a light cone QCD inspired model

The general equation from previous work is specialized to a linear potential V (r) = −a + F r acting in the space of spherically symmetric S wave functions. The fine and hyperfine interaction creates then a 1 r -dependence in the effective potential energy equation and a position dependent mass m̃(r) in the effective kinetic energy of the associated Schrödinger equation. The results are compared with the available experimental and theoretical spectral data on the π and ρ. Solving the eigenvalue problem within the analytically tractable Airy-function approach induces a certain amount of arbitrariness (fudge factors). Despite of this, the agreement with experimental data is good and partially better than other calculations, including Godfrey and Isgur [9] and Baldicchi and Prosperi [10]. The short comings of the present model can be removed easily in more elaborate work. PACS. 11.10.Ef – 12.38.Aw – 12.38.Lg – 12.39.-x 1 The S-state Hamiltonian For spherically symmetric S states the previously derived Hamiltonian reduces in Fourier approximation to [1,2,3] H = p 2 2mr + V + Vhf + VK + VD , Vhf = σ1σ2 6m1m2 ∇V , VK = V m1m2 p , VD = − [ V 16m1m2 p + ∇ V 4m1m2 ]( m1 m2 + m2 m1 ) . (1) There are no more interactions than the central potential, the hyperfine, the kinetic, and the Darwin interaction, but also no less. For s-states the total spin squared is a good quantum number S = [(σ1 + σ2)/2] 2 = S(S + 1), thus σ1σ2 = 2S(S + 1)− 3 = { +1, for S = 1, triplet, −3, for S = 0, singlet. (2) Because it is shorter, σ1σ2 is kept explicit in the equations as an abbreviation for Eq.(2). Choosing a linear potential, V (r) = −a+ F r , (3) with the force parameter F , often called string tension σ, the Hamiltonian (1) becomes a non-local Schrödinger equation with a 1 r -potential H = [ 1 2mr + V (r) m1m2 − V (r) 16m1m2 ( m1 m2 + m2 m1 ) ] p + [ Fr − a ] + [ F 3m1m2 σ1σ2 − F 2m1m2 ( m1 m2 + m2 m1 ) ] 1 r ,