Temperature and volume dependence of pion-pion scattering lengths

The s-wave pion-pion scattering lengths and are studied at finite temperature and in finite spatial a(0) a(2) volume under the framework of the Nambu-Jona-Lasinio model. The behavior beyond the pseudo transition temperature is investigated using proper time regularization. The scattering length exhibits singularity near the Mott a(0) temperature, and is a continuous but non-monotonic function of temperature. We present the effect of finite a(2) volume on the scattering length and find that can be negative and its singularity disappears at small volumes, a(0) which may hint at the existence of a chiral phase transition with decreasing volume.