defined by $${\bf A}_{\mu} \,=\, \left(\partial_{\mu}\sigma\right)
\mbox{\boldmath $\pi$}
- \left(\partial_{\mu} \mbox{\boldmath $\pi$} \right)\sigma \, ,$$ is conserved. It is believed that chiral symmetry is restored by a phase transition, which is perhaps of second order, at a critical temperature $T_c \approx 160$ MeV; we come back to this point later.

Any quantitative description of Nature at finite momentum and energy must include the vector mesons [@vector]. Since we are proposing here a qualitatively new phenomenon we shall neglect them, as well as strangeness.

The up and down quarks, while very light, are not massless; therefore, neither is the pion. Historically, there have been three ways to add a symmetry breaking term to the linear sigma model [@camp]:

[**1:**]{} $V_{SB} = -f_{\pi}m_{\pi}^2 \sigma$, which is linear in the $\sigma$ field. The PCAC relation $$\partial_{\mu}{\bf A}^{\mu} \,=\, f_{\pi}m_{\pi}^2
\mbox{\boldmath $\pi$}$$ is realized as an operator equation.

[**2:**]{} $V_{SB} = \frac{1}{2} m_\pi^2 \mbox{\boldmath $\pi$}^2$, which is quadratic in the pion field. The PCAC relation is realized only in the weak field limit.

[**3:**]{} $V_{SB} = {\rm constant}\times{\bar N}N$, which is quadratic in the nucleon field. The divergence of the axial vector current is proportional to the axial vector nucleon current.

In principle one 