Large $N_c$ scaling of meson masses and decay constants

We perform an $\textit{ab initio}$ calculation of the $N_c$ scaling of the low-energy couplings of the chiral Lagrangian of low-energy strong interactions, extracted from the mass dependence of meson masses and decay constants. We compute these observables on the lattice with four degenerate fermions, $N_f=4$, and varying number of colours, $N_c=3-6$, at a lattice spacing of $a\simeq 0.075$ fm. We find good agreement with the expected $N_c$ scaling and measure the coefficients of the leading and subleading terms in the large $N_c$ expansion. From the subleading $N_c$ corrections, we can also infer the $N_f$ dependence, that we use to extract the value of the low-energy couplings for different values of $N_f$. We find agreement with previous determinations at $N_c=3$ and $N_f=2, 3$ and also, our results support a strong paramagnetic suppression of the chiral condensate in moving from $N_f=2$ to $N_f=3$.