Pi-Pi scattering lengths at finite temperature

The $s$-wave $\pi-\pi$ scattering lengths $a^I(T)$ at finite temperature $T$ and isospin $I=0,2$ are calculated within the SU(2) Nambu--Jona-Lasinio model. $a^2(T)$ displays a singularity at the Mott temperature $T_M$, defined as $m_\pi(T_M) = 2m_q(T_M)$, while $a^0(T)$ is singular in addition at the lower temperature $T_d$, where $m_\sigma(T_d) = 2m_\pi(T_d)$, $m_\sigma$ and $m_\pi$ being the masses of the $\sigma$ and $\pi$ mesons, respectively. Numerically we find $T_d = 198$MeV and $T_M=215$MeV. We speculate on possible experimental consequences.