Evidence for $CP$ violation in $B^{+} \to K^{*}(892)^{+} \pi^{0}$ from a Dalitz plot analysis of $B^{+} \to K^{0}_{\rm S} \pi^{+} \pi^{0}$ decays

We report a Dalitz plot analysis of charmless hadronic decays of charged $B$ mesons to the final state $K_{\rm S}^{0}\pi^{+}\pi^{0}$ using the full BaBar dataset of $470.9\pm2.8$ million $B\bar{B}$ events collected at the $\Upsilon(4S)$ resonance. We measure the overall branching fraction and $CP$ asymmetry to be ${\cal B}\left(B^{+}\to K^{0}\pi^{+}\pi^{0}\right) = \left(45.9 \pm 2.6 \pm 3.0^{+8.6}_{-0.0}\right)\times10^{-6}$ and $A_{CP}\left(B^{+}\to K^{0}\pi^{+}\pi^{0}\right) = 0.07 \pm 0.05 \pm 0.03^{+0.02}_{-0.03}$, where the uncertainties are statistical, systematic, and due to the signal model, respectively. This is the first measurement of the branching fraction for $B^{+} \to K^{0}\pi^{+}\pi^{0}$. We find first evidence of a $CP$ asymmetry in $B^{+}\to K^{*}(892)^{+}\pi^{0}$ decays: $A_{CP}\left(B^{+}\to K^{*}(892)^{+}\pi^{0}\right)=-0.52 \pm 0.14 \pm 0.04^{+0.04}_{-0.02}$. The significance of this asymmetry, including systematic and model uncertainties, is $3.4$ standard deviations. We also measure the branching fractions and $CP$ asymmetries for three other intermediate decay modes.