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Using data collected with the BESIII detector in $e^+e^-$ collisions atcenter-of-mass energies between 4.178 and 4.226 GeV and corresponding to6.32~fb$^{-1}$ of integrated luminosity, we report the amplitude analysis andbranching-fraction measurement of the $D^+_s \to \pi^+ \pi^0 \eta^{\prime}$decay. We find that the dominant intermediate process is $D^+_s \to\rho^+\eta^{\prime}$ and the significances of other resonant and nonresonantprocesses are all less than $3\sigma$. The upper limits on the branchingfractions of $S$-wave and $P$-wave nonresonant components are set to $0.10\%$and $0.74\%$ at the $90\%$ confidence level, respectively. In addition, thebranching fraction of the $D^+_s \to \pi^+ \pi^0 \eta^{\prime}$ decay ismeasured to be $(6.15\pm0.25(\rm stat.)\pm0.18(\rm syst.))\%$, which receivessignificant contribution only from $D_s^+\to \rho^+\eta^{\prime}$ according tothe amplitude analysis.

Amplitude analysis and branching-fraction measurement of $D_{s}^{+} \to \pi^{+}\pi^{0}\eta^{\prime}$