Critical coupling constants in chiral-symmetry breaking

Finger, Horn, and Mandula showed in a Tamm-Dancoff approximation in QCD that Coulomb exchange induces instability of the chiral-invariant vacuum for ${\alpha }_s>{\alpha }_s^{\mathrm{crit}}$ in the sense that for ${\alpha }_s>{\alpha }_s^{\mathrm{crit}}$ there are negative eigenvalues of an operator $H$ derived by them. We show here that another critical coupling constant ${\alpha }_s^{\prime \mathrm{crit}}>{\alpha }_s^{\mathrm{crit}}$ is more meaningful. For ${\alpha }_s>{\alpha }_s^{\prime \mathrm{crit}}$ there is no positive self-adjoint extension of $H$; for ${\alpha }_s<{\alpha }_s^{\prime \mathrm{crit}}$ there is at least one; when ${\alpha }_s^{\mathrm{crit}}<{\alpha }_s<{\alpha }_s^{\prime \mathrm{crit}}$, among several self-adjoint extensions, a positive one is singled out by the physical requirement of a finite kinetic-energy expectation value. Considering, respectively, Coulomb exchange with the Tamm-Dancoff approximation, Coulomb exchange with the Bogoliubov-Valatin approximation, and Coulomb exchange plus transverse-gluon exchange, the values are $\frac{4}{3}{\alpha }_s^{\prime \mathrm{crit}}=\frac{8\pi }{(4+{\pi }^2)}, \frac{4}{\pi }, \frac{12}{5\pi }$, to be compared with $\frac{4}{3}{\alpha }_s^{\mathrm{crit}}=\frac{3}{2}, 1, \frac{\pi }{(3\mathrm{}\pi -4)}$.