Radiative transitions and the P wave levels in charmonium

A value of $\text{1}\text{2}$ or less for the ratio $\text{[E(2}{}^{\mathrm{++}}\text{) ? E(1}{}^{\mathrm{++}}\text{)]}\text{[E(1}{}^{\mathrm{++}}\text{) ? E(0}{}^{\mathrm{++}}\text{)]}$ of the P level splittings in approximate agreement with the assignment of the states at 3.41, 3.50 and 3.55 to the 0⁺⁺,and 2⁺⁺ P-wave levels, is obtained with a short-range Coulomb (Lorentz vector) potential together with a long-range linear (Lorentz scalar) confining potential. The radiative transition widths Γ(ψ′ → 3.41 + γ), Γ(ψ′ → 3.50 + γ), Γ(χ′ → 3.55 + γ) are significantly smaller than those obtained in previous (one-channel) charmonium calculations. The best results were obtained by allowing the Coulomb coupling constant α_s to have a momentum dependence suggested by asymptotic freedom formulae.