Temperature and volume dependence of pion-pion scattering lengths

The s-wave pion-pion scattering lengths $a_0$ and $a_2$ are studied at finite temperature and in finite spatial volume under the framework of the Nambu--Jona-Lasinio model. With the proper time regularization, the behavior beyond the pseudo transition temperature is presented. The scattering length $a_0$ shows singularity near the Mott temperature and $a_2$ is a continuous but non-monotonic function of temperature. We present the finite volume effect on the scattering length and have found that $a_0$ can be negative and its singularity disappears at small volume size which may hint the existence of chiral phase transition as volume decreases.