Measurements of absolute branching fractions of $D^{0(+)}\to KKK\pi$ decays

Using an $e^+e^-$ sample of $20.3\,\rm fb^{-1}$ collected at the center-of-mass energy $\sqrt{s}=$ 3.773 GeV with the BESIII detector, we report measurements of several four-body hadronic decays of the $D$ mesons. The absolute branching fractions are determined to be ${\mathcal B}(D^0\to K^0_S K^+K^-\pi^0 )=( 18.4^{+2.6}_{-2.5}\pm 2.4)\times 10^{-5}$, ${\mathcal B}(D^0\to K^0_S K^0_S K^-\pi^+ )=( 12.9^{+1.7}_{-1.6}\pm 2.5)\times 10^{-5}$, ${\mathcal B}(D^0\to K^0_S K^0_S K^+\pi^-)=(5.7^{+1.2}_{-1.1}\pm 1.3)\times 10^{-5}$, ${\mathcal B}(D^0\to K^+K^-K^-\pi^+ )=(17.4^{+1.8}_{-1.7}\pm { 2.2})\times 10^{-5}$, and ${\mathcal B}(D^+\to K^0_S K^+K^-\pi^+)=(13.8^{+2.4}_{-2.2}\pm 2.5)\times 10^{-5}$. Furthermore, significant $\phi$ signals are found in the decay channels involving $K^+K^-$ pair, and the corresponding branching fractions are measured as ${\mathcal B}(D^0\to \phi K^0_S\pi^0 )=( 22.7^{+5.4}_{-5.1}\pm 3.7)\times 10^{-5}$, ${\mathcal B}(D^0\to \phi K^-\pi^+ )=(25.2^{+3.5}_{-3.3}\pm 4.6)\times 10^{-5}$, ${\mathcal B}(D^+\to \phi K^0_S\pi^+)=(16.5 ^{+6.0}_{-5.3}\pm 2.6 )\times 10^{-5}$. The branching fractions of $D^0\to K^0_S K^+K^-\pi^0$, $D^0\to \phi K^0_S\pi^0$, and $D^+\to \phi K^0_S \pi^+$ are measured for the first time, and those of $D^0\to K^0_S K^0_SK^-\pi^+$, $D^0\to K^0_S K^0_SK^+\pi^-$, $D^0\to K^+K^-K^-\pi^+$, $D^0\to \phi K^-\pi^+$, and $D^+\to K^0_S K^+K^-\pi^+$ are measured with improved precision. The first uncertainties are statistical and the second are systematic.