a mass greater than about 1 GeV. For definiteness, we choose $m_{\sigma} = 1$ GeV. We shall vary $\epsilon_1$ between 0 and $f_{\pi} m_{\pi}^2$. There is no further freedom given the above constraints.

Finite temperature
------------------

To estimate what may happen at finite temperature we will calculate thermal fluctuations to one loop order and furthermore take the high temperature limit. Quantitatively this cannot be very accurate. The relevant coupling constant is large: $\lambda \approx (m_{\sigma}
/f_{\pi})^2/2 \approx 50$. Qualitatively the result should be alright; however, see [@rw; @rob].

To proceed, we expand the fields about an arbitrary point, as in eqs. (9-10), and determine the masses of the fluctuations. If $\bf v$ points in the third direction in isospin space then the eigenvalues of the mass-squared matrix are $$\begin{aligned}
m_1^2 &=& \lambda v^2 - c^2 \, , \nonumber \\
m_2^2 &=& \lambda v^2 - c^2 \, , \nonumber \\
m_3^2 &=& 2 \lambda v^2 - c^2 - \frac{\epsilon_2}{2}
- \sqrt{\lambda^2v^4 - \epsilon_2 \lambda v^2 \cos(2\theta)
+\frac{\epsilon_2^2}{4}} \, , \nonumber \\
m_0^2 &=& 2 \lambda v^2 - c^2 - \frac{\epsilon_2}{2}
+ \sqrt{\lambda^2v^4 - \epsilon_2 \lambda v^2 \cos(2\theta)
+\frac{\epsilon_2^2}{4}} \, .\end{aligned}$$ In the high temperature limit of the one loop approximation one 