Resonance effects in pion and kaon decay constants

In this article we study impact of the lightest vector and scalar resonance multiplets in the pion and kaon decay constants up to next-to-leading order in the $1\mathrm{/}{N}_C$ expansion, i.e., up to the one-loop level. The $F_{\pi }$ and $F_K$ predictions obtained within the framework of Resonance Chiral Theory are confronted with lattice simulation data. The vector loops (and the $\rho -\pi \pi$ coupling $G_V$ in particular) are found to play a crucial role in the determination of the Chiral Perturbation Theory couplings $L_4$ and $L_5$ at next-to-leading order in $1\mathrm{/}{N}_C$. Puzzling, values of $G_V<40$ MeV seem to be necessary to agree with current phenomenological results for $L_4$ and $L_5$. Conversely, a value of $G_V>60$ MeV compatible with standard $\rho -\pi \pi$ determinations turns these chiral couplings negative. However, in spite of the strong anti-correlation with $L_4$, the $SU(3)$ chiral coupling $F_0$ remains stable all the time and stays within the range $78\sim 86$ MeV when $G_V$ is varied in a wide range, from $40$ up to $70$ MeV. Finally, we would like to remark that the leading order expressions used in this article for the $\eta -{\eta }^{\mathrm{\prime }}$ mixing, mass splitting of the vector multiplet masses and the quark mass dependence of the $\rho (770)$ mass are found in reasonable agreement with the lattice data.