Measurement of branching fractions for $D$ meson decaying into $phi$ meson and a pseudoscalar meson

The four decay modes $D^{0}tophipi^{0}$, $D^{0}tophieta$, $D^{+}tophipi^{+}$, and $D^{+}tophi K^{+}$ are studied by using a data sample taken at the centre-of-mass energy $sqrt{s} = 3.773$~GeV~with the BESIII detector, corresponding to an integrated luminosity of 2.93 fb$^{-1}$. The branching fractions of the first three decay modes are measured to be ${cal B}(D^{0}tophipi^{0})=(1.168pm0.028pm0.028)times10^{-3}$, ${cal B}(D^{0}tophieta)=(1.81pm0.46pm0.06)times10^{-4}$, and ${cal B}(D^{+}tophipi^{+})=(5.70pm0.05pm0.13)times10^{-3}$, respectively, where the first uncertainties are statistical and the second are systematic. No significant signal is observed of the decay $D^+tophi{K^+}$ and the upper limit of its branching fraction is set for the first time, to be $2.1times10^{-5}$ at the $90%$ confidence level. The ratio of ${cal B}(D^0tophipi^0)$ to $cal B(D^+tophipi^+)$ is calculated to be $(20.49pm0.50pm0.45)%$, which is consistent with the theoretical prediction based on isospin symmetry between these two decay modes.