Search for a doubly-charged $DDK$ bound state in $Upsilon(1S,2S)$ inclusive decays and via direct production in $e^+e^-$ collisions at $sqrt{s}$ = 10.520, 10.580, and 10.867 GeV

We report the results of a first search for a doubly-charged $DDK$ bound state, denoted the $R^{++}$, in $Upsilon(1S)$ and $Upsilon(2S)$ inclusive decays and via direct production in $e^+e^-$ collisions at $sqrt{s}$ = 10.520, 10.580, and 10.867 GeV. The search uses data accumulated with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider. No significant signals are observed in the $D^{+}D_{s}^{*+}$ invariant-mass spectra of all studied modes. The 90% credibility level upper limits on their product branching fractions in $Upsilon(1S)$ and $Upsilon(2S)$ inclusive decays (${cal B}(Upsilon(1S,2S) to R^{++} + anything) times {cal B}(R^{++} to D^{+} D_{s}^{*+})$), and the product values of Born cross section and branching fraction in $e^+e^-$ collisions ($sigma(e^+e^- to R^{++} + anything) times {cal B}(R^{++} to D^{+} D_{s}^{*+})$) at $sqrt{s}$ = 10.520, 10.580, and 10.867 GeV under different assumptions of $R^{++}$ masses varying from 4.13 to 4.17 GeV/$c^2$, and widths varying from 0 to 5 MeV are obtained.