Probing QCD dynamics and the standard model with $D_{(s)} \to P^+ P^0 \gamma$ decays

We compute 10 radiative three-body decays of charged charmed mesons $D^+ \to P^+ P^0 \gamma$ and $D_s \to P^+ P^0 \gamma$, $P=\pi, K$, in leading order QCDF, HH$\chi$PT and the soft photon approximation. We work out decay distributions and asymmetries in the standard model and with new physics in the electromagnetic dipole operators. The forward-backward asymmetry is suitable to probe the QCD frameworks, in particular the $s$-channel dependent weak annihilation contributions in QCDF against the markedly different resonance structure in HH$\chi$PT. These studies can be performed with Cabibbo-favored modes $D_s \to \pi^+ \pi^0 \gamma$, $D^+ \to \pi^+ \overline{K}^0 \gamma$ and $D_s \to K^+ \overline{K}^0 \gamma$ with ${\cal{O}}(10^{-4}-10^{-3})$-level branching ratio, which are standard model-like and induced by different hadronic dynamics. Understanding of the latter can therefore be improved in a data-driven way and sharpens the interpretation of standard model tests. Singly Cabibbo-suppressed modes such as $D^+ \to \pi^+ \pi^0 \gamma$, $D_s \to \pi^+ K^0 \gamma$, $D_s \to K^+ \pi^0 \gamma$ with branching ratios within $\sim 10^{-5}-10^{-4}$ are sensitive to new physics that can be signalled in the forward-backward asymmetry and in the CP-asymmetry of the rate, ideally in the Dalitz region but also in single differential distributions. Results complement those with neutral $D^0 \to PP \gamma$ decays.