Measurement of $CP$ asymmetries in $D^+_{(s)}\rightarrow \eta \pi^+$ and $D^+_{(s)}\rightarrow \eta^{\prime} \pi^+$ decays

Searches for $CP$ violation in the decays $D^+_{(s)}\rightarrow \eta \pi^+$ and $D^+_{(s)}\rightarrow \eta^{\prime} \pi^+$ are performed using $pp$ collision data corresponding to 6 fb$^{-1}$ of integrated luminosity collected by the LHCb experiment. The calibration channels $D^+_{(s)}\rightarrow \phi \pi^+$ are used to remove production and detection asymmetries. The resulting $CP$-violating asymmetries are $A^{CP}(D^+ \rightarrow \eta \pi^+) = (0.34 \pm 0.66 \pm 0.16 \pm 0.05)\%$, $A^{CP}(D^+_s \rightarrow \eta \pi^+) = (0.32 \pm 0.51 \pm 0.12)\%$, $A^{CP}(D^+ \rightarrow \eta^{\prime} \pi^+) = (0.49 \pm 0.18 \pm 0.06 \pm 0.05)\%$, $A^{CP}(D^+_s \rightarrow \eta^{\prime} \pi^+) = (0.01 \pm 0.12 \pm 0.08)\%$, where the first uncertainty is statistical, the second is systematic and the third, relevant for the $D^+$ channels, is due to the uncertainty on $A^{CP}(D^+ \to \phi \pi^+)$. These measurements, currently the most precise for three of the four channels considered, are consistent with the absence of $CP$ violation. A combination of these results with previous LHCb measurements is presented.