Comments on compositeness in the SU(2) linear sigma model

First we summarize the quark-level linear $\sigma$ model compositeness conditions and verify that indeed $m_{\sigma }=2m_q$ when $m_{\pi }=0$ and $N_c=3$, rather than in the $N_c\to \mathrm{\infty }$ limit, as is sometimes suggested. Later we show that this compositeness picture also predicts a chiral symmetry restoration temperature $T_c=2f_{\pi }$, where $f_{\pi }$ is the pion decay constant. We contrast this self-consistent Z=0 compositeness analysis with prior studies of the compositeness problem.