Hadron Liquid with a Small Baryon Chemical Potential at Finite Temperature

First, within one diagram of $\Phi$ we discuss general properties of a system of heavy fermions of one kind (including antiparticles) interacting with rather light bosons of one kind. Fermion chemical potential is assumed to be small, $\mu_f \lsim T$. Already for the low temperature, $T\ll {min} (T_{\rm bl.f}, m_{b})$, the fermion mass shell proves to be partially blurred due to multiple fermion rescatterings on virtual bosons, $m_{b}$ is the boson mass, $T_{\rm bl.f}$ $(\ll m_f)$ is the typical temperature corresponding to a complete blurring of the gap between fermion-antifermion continua, $m_f$ is the fermion mass. As the result, the ratio of the number of fermion-antifermion pairs to the number provided by the ordinary Boltzmann distribution becomes larger than unit ($R_N >1$). For $T\gsim m_{b}^* (T)$ (hot hadron liquid, blurred boson continuum), $m_{b}^* (T)$ is the effective boson mass, the abundance of all particles dramatically increases. The effective fermion mass $m_f^* (T)$ decreases with the temperature increase. For $T\gsim T_{\rm bl.f}$ fermions are essentially relativistic particles. Due to the interaction of the boson with fermion-antifermion pairs, $m_{b}^* (T)$ decreases leading to the possibility of the ``hot Bose condensation'' for $T>T_{cb}$. The phase transition might be of the second order or of the first order depending on the species under consideration. We estimate $R_N \sim 1.5$ for $T\sim m_{\pi}/2$; $T_{\rm bl.f}$ proves to be near $T_{cb}$; both values are in the vicinity of the pion mass $m_{\pi}$.