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 $m_\sigma = 2 m_q$ when $m_\pi = 0$ and $N_c=3$, rather than in the $N_c\to\infty$ 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.