First measurement of the CKM angle ϕ$_{3}$ withS B$^{\pm}$ → D($ {K}_{\mathrm{S}}^0 $π$^{+}$+π$^{-}$π$^{0}$) K$^{\pm}$ decays

We present the first model-independent measurement of the CKM unitarity triangle angle ϕ3 using B±→ D($$ {K}_{\mathrm{S}}^0 $$π+π−π0) K± decays, where D indicates either a D0 or $$ \overline{D} $$0 meson. Measurements of the strong-phase difference of the D →$$ {K}_{\mathrm{S}}^0 $$π+π−π0 amplitude obtained from CLEO-c data are used as input. This analysis is based on the full Belle data set of 772 × 106B$$ \overline{B} $$ events collected at the Υ(4S) resonance. We obtain ϕ3 = ($$ {5.7}_{-8.8}^{+10.2} $$±3.5±5.7)° and the suppressed amplitude ratio rB = 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 ϕ3 is (−29.7, 109.5)°, which is consistent with the current world average.