Measurements of the branching fractions for the semi-leptonic decays $D^+_s\toϕe^{+}ν_{e}$, $ϕμ^{+}ν_μ$, $ημ^{+}ν_μ$ and $η'μ^{+}ν_μ$
By analyzing 482 pb$^{-1}$ of $e^+e^-$ collision data collected at the center-of-mass energy $\sqrt s=4.009$ GeV with the BESIII detector, we measure the %absolute branching fractions for the semi-leptonic decays $D_{s}^{+}\to ϕe^{+}ν_{e}$, $ϕμ^{+}ν_μ$, $ημ^{+}ν_μ$ and $η'μ^{+}ν_μ$ to be ${\mathcal B}(D_{s}^{+}\toϕe^{+}ν_{e})=(2.26\pm0.45\pm0.09)$\%, ${\mathcal B}(D_{s}^{+}\toϕμ^{+}ν_μ)=(1.94\pm0.53\pm0.09)$\%, ${\mathcal B}(D_{s}^{+}\toημ^{+}ν_μ)=(2.42\pm0.46\pm0.11)$\% and ${\mathcal B}(D_{s}^{+}\toη'μ^{+} ν_μ) = (1.06\pm0.54\pm0.07)$\%, where the first and second uncertainties are statistical and systematic, respectively. The branching fractions for the three semi-muonic decays $D_s^+\toϕμ^+ν_μ, ημ^+ν_μ$ and $η' μ^+ν_μ$ are determined for the first time and that of $D^+_s\to ϕe^+ν_e$ is consistent with the world average value within uncertainties.