A linear sigma model for multiflavor gauge theories

We consider a linear sigma model describing $2N_f^2$ bosons ($\sigma$, ${\bf a_0}$, $\eta '$ and ${\bf \pi}$) as an approximate effective theory for a $SU(3)$ local gauge theory with $N_f$ Dirac fermions in the fundamental representation. The model has a renormalizable $U(N_f)_L\bigotimes U(N_f)_R$ invariant part, which has an approximate $O(2N_f^2)$ symmetry, and two additional terms, one describing the effects of a $SU(N_f)_V$ invariant mass term and the other the effects of the axial anomaly. We calculate the spectrum for arbitrary $N_f$. Using preliminary and published lattice results from the LatKMI collaboration, we found combinations of the masses that vary slowly with the explicit chiral symmetry breaking and $N_f$. This suggests that the anomaly term plays a leading role in the mass spectrum and that simple formulas such as $M_\sigma^2\simeq (2/N_f-C_\sigma)M_{\eta '}^2$ should apply in the chiral limit. Lattice measurements of $M_{\eta '}^2$ and of approximate constants such as $C_\sigma$ could help locating the boundary of the conformal window. We show that our calculation can be adapted for arbitrary representations of the gauge group and in particular to the minimal model with two sextets, where similar patterns are likely to apply.