Heavy-light scalar-quark system in the leading-log effective-action model: Exact solution

An exact solution is obtained in a renormalization-group-improved effective-action model of QCD, for a system containing an infinitely heavy antiquark and a scalar quark of arbitrary mass. The solution corresponds to a solitonlike $S$-wave state of finite radius $R$. The only input parameters are the QCD scale ${\Lambda }_{\overline{\mathrm{MS}}}$ (where $\overline{\mathrm{MS}}$ denotes the modified minimal subtraction scheme), the number of light quark flavors $N_f$, and the scalar-quark mass $m$. Color confinement arises from the nonlinear properties of the effective action. An analysis of the radial excitations ($m=0\mathrm{}$) of the $S$-wave soliton state clearly shows that this confinement is linear. We find for high radial excitations that the leading behavior of the total energy of the system ($m=0\mathrm{}$) is $U_N=Q\kappa R_N+\mathrm{const}$, where $Q=\sqrt{\frac{4}{3}}$, ${\kappa }^{\frac{1}{2}}=1.043\mathrm{}{\Lambda }_{\overline{\mathrm{MS}}} (N_f=3)$, and $R_N$ is the radius of the soliton corresponding to the $N\mathrm{th}$ radial excitation. A detailed description is given of the properties of the ground-state soliton as a function of $N_f$ and $m$. An analytical expression is obtained for the short-distance behavior of the scalar-quark wave function.