QCD Inequalities for the Nucleon Mass and the Free Energy of Baryonic Matter

The positivity of the integrand of certain Euclidean space functional integrals for two flavor QCD with degenerate quark masses implies that the free energy per unit volume for QCD with a baryon chemical potential ${\mu }_B$ (and zero isospin chemical potential) is greater than the free energy with an isospin chemical potential ${\mu }_I=(2{\mu }_B/N_c)$ (and zero baryon chemical potential). The same result applies to QCD with any number of heavy flavors in addition to the two light flavors so long as the chemical potential is understood as applying to the light quark contributions to the baryon number. This relation implies a bound on the nucleon mass: there exists a particle $X$ in QCD (presumably the pion) such that $M_N\ge (N_c{m}_X/2I_X)$ where $m_X$ is the mass of the particle and $I_X$ is its isospin.