We calculate the physical mass and the width of the sigma meson by considering that it couples in vacuum to two virtual pions. The mass is calculated by using the spectral function, and we find that it is about 600 MeV. In addition, we obtained 220 MeV as the value for the width of its spectral function. The value obtained for the mass is in good agreement with that reported in the Particle Data Book for the σ-meson, which is also named $f_{0}$(600). This result also shows that 𝜎-meson can be considered as a two-pion resonance.

The propagation of 𝜎-meson in nuclear matter is studied in the Walecka model, by assuming that the sigma couples to a pair of nucleon–antinucleon states and to particle–hole states. The in-medium effect of $\sigma -\omega$ mixing is also studied. For completeness, the coupling of sigma to two virtual pions was also considered. It is found that the 𝜎-meson mass decreases with respect to its value in vacuum and that the contribution of the $\sigma -\omega$ mixing effect on the mass shift is relatively small.