First measurement of the CKM angle $\phi_3$ with $B^{\pm}\to D(K_{\rm S}^0\pi^+\pi^-\pi^0)K^{\pm}$ decays

We present the first model-independent measurement of the CKM unitarity triangle angle ϕ$_{3}$ using B$^{±}$→ D( $ {K}_{\mathrm{S}}^0 $ π$^{+}$π$^{−}$π$^{0}$) K$^{±}$ decays, where D indicates either a D$^{0}$ or $ \overline{D}^{0}$ meson. Measurements of the strong-phase difference of the D → $ {K}_{\mathrm{S}}^0 $ π$^{+}$π$^{−}$π$^{0}$ amplitude obtained from CLEO-c data are used as input. This analysis is based on the full Belle data set of 772 × 10$^{6}$B $ \overline{B} $ events collected at the Υ(4S) resonance. We obtain ϕ$_{3}$ = ( $ {5.7}_{-8.8}^{+10.2} $ ±3.5±5.7)$^{°}$ and the suppressed amplitude ratio r$_{B}$ = 0.323±0.147±0.023±0.051. Here the first uncertainty is statistical, the second is the experimental systematic, and the third is due to the precision of the strong-phase parameters measured from CLEO-c data. The 95% confidence interval on ϕ$_{3}$ is (−29.7, 109.5)$^{°}$, which is consistent with the current world average.