In this article we address the problem of getting the temperature dependence of the $\pi \mathrm{\text{−}}\pi$ scattering lengths in the frame of the linear sigma model. Using the real time formalism, we calculate all the relevant one loop diagrams. The temperature corrections are only considered in the pion sector, due to the Boltzmann suppression for heavier fields like the sigma meson. From this analysis we obtain the thermal behavior of the s waves scattering lengths $a_0^0$ and $a_0^2$ associated to isospin $I=0$ and $I=2$, respectively. If we normalize with the zero temperature value it turns out that $\frac{a_0^0(T)}{a_0^0}$ grows with temperature, whereas the opposite occurs with $\frac{a_0^2(T)}{a_0^2}$. Finally we compare our results with other determinations of the scattering lengths based on the Nambu-Jona-Lasinio model and chiral perturbation theory.

• Received 23 January 2008

DOI:https://doi.org/10.1103/PhysRevD.77.105006