We consider the polarization function $P_Z^{\exp }\left(t\right)$ measured by the muon-spin relaxation ($\mu \mathrm{SR}$) technique for the ${\mathrm{SrCr}}_8{\mathrm{Ga}}_4{\mathrm{O}}_{19}$ and ${\mathrm{ZnCu}}_3{\left(\mathrm{OH}\right)}_6{\mathrm{Cl}}_2$ spin-liquid systems. We show the functional form of $P_Z^{\exp }\left(t\right)$ to imply that, in the temperature range of order 0.1 K, the spectral-weight function $F\left(\omega \right)$ of the magnetic correlations scales with $1/{\left|\omega \right|}^{1-x}$ ($0<x<1$) in the energy range of $\hslash \omega \approx 1\mu \mathrm{eV}$. We derive the parameters involved in $F\left(\omega \right)$ from fits to available experimental data. Inelastic neutron scattering data probing $F\left(\omega \right)$ in the millielectronvolt energy range are consistent with a more conventional behavior. These differences could be due to a variety of spin-dynamics mechanisms, i.e., intrinsic to the kagome layer or related to the magnetic defects that have been evidenced in these compounds, acting at different energies. New $\mu \mathrm{SR}$ measurements are proposed and theoretical developments are suggested to pinpoint the mechanisms at play.

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• Frustrated magnetism
• Magnetism
• Quantum spin liquid
• Spin dynamics
• Spin liquid