$\pi$-$K$ scattering lengths at finite temperature in the Nambu--Jona-Lasinio model

The transition amplitude for $\pi K$ scattering is evaluated within the SU(3) Nambu–Jona-Lasinio model. Ordering terms according to the expansion in $1\mathrm{/}{N}_c$ leads to a box-like diagram, $t$ channel diagrams that admit scalar isoscalar $(\sigma ,{\sigma }^{\mathrm{\prime }})$ exchanges, and a $u$ channel exchange of a scalar isodoublet ${\sigma }_K$ that has quantum numbers corresponding to the $K_0^{\ast }(1430)$. Both the Pauli-Villars and O(3) regularization procedures are used to evaluate the T=0 values of the $l=0$ scattering lengths $a_0^{3\mathrm{/}2}$ and $a_0^{1\mathrm{/}2}$. The finite temperature dependence is studied. We find that the variation in the $t$ channel in the calculation of $a_0^{3\mathrm{/}2}$ leads to a change in $a_0^{3\mathrm{/}2}$ of a factor of about two over the temperature range of T=150 MeV.