Spontaneous breaking of chiral symmetry for confining potentials

Using the Bogoliubov-Valatin variational method, we show that the chiral-invariant vacuum is unstable for a color, fourth-component vector powerlike potential $r^{\alpha }(0<\alpha <3)$ independently of the strength of the coupling constant. The fermion self-energy is negative and dominates over the positive potential energy, destabilizing the vacuum by $\overline{\psi }\psi$ pair condensation. This self-energy is finite but infrared singular, reflecting the behavior of the potential at large distances. We give an analytical proof of the fact that the energy of the unbroken vacuum is not minimum. The proof extends to logarithmic potentials as $\alpha \to 0$, but breaks down for $\alpha \ge 3$ (number of spatial dimensions) due to severe infrared singularities. If the confining potential possesses a spin-spin piece, there are critical values of its strength, depending on the power $\alpha$, beyond which the stability of the chiral-invariant vacuum is restored. In the case of the harmonic oscillator $\alpha =2\mathrm{}$, the gap equation reduces to a non-linear second-order differential equation. We find (besides the usual chiral degeneracy) an infinite number of solutions breaking chiral symmetry, higher in energy as the number of their nodes increases. We compute the expectation value of $\overline{\psi }\psi$ and the mass gap for the new vacuum, the lowest solution in energy. The infrared singularity of the massless fermion self-energy is removed for the stable broken solution.