Mass splittings in a linear sigma model for multiflavor gauge theories

We calculate the tree-level mass spectrum for a linear sigma model describing the scalar and pseudoscalar mesons of a $SU(3)$ local gauge theory with Dirac fermions in the fundamental representation. $N_1$ fermions have a mass $m_1$ and $N_2$ a mass $m_2$. Using recent lattice data with $m_1=m_2$ and $N_1+N_2$= 8 or 12, we predict the mass splittings for $m_2=m_1+\delta m$. At first order in $\delta m$, an interesting inverted pattern appears in the $0^{++}$ sector, where mesons with lighter fermions are heavier. This feature could be tested in ongoing calculations provided that $m_1$ and $\delta m$ are sufficiently small. We discuss possible improvements of the approach.