Ratios of the light quark masses: Cubic curve versus ellipse

The bounds on the light quark masses are obtained by fitting the squares of pseudoscalar meson masses $m_{\pi }^2$ and $m_K^2$ to second order in $1/N_c$ expansion. The result is an algebraic cubic curve whose coefficients are the known Weinberg values for the quark mass ratios $m_u/m_d$ and $m_s/m_d$. Additional restrictions arise when using the ratio $m_s/m_{ud}=27.23\left(10\right)$ quoted by FLAG for lattice simulations with four quark flavors. This provides a tight constraint on the ratio $m_u/m_d=0.455\left(8\right)$.