Search for a doubly charged DDK bound state in ϒ(1S, 2S) inclusive decays and via direct production in e+e− collisions at 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 $\mathrm{\ensuremath{\Upsilon}}(1S)$ and $\mathrm{\ensuremath{\Upsilon}}(2S)$ inclusive decays and via direct production in ${e}^{+}{e}^{\ensuremath{-}}$ 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 ${e}^{+}{e}^{\ensuremath{-}}$ 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 $\mathrm{\ensuremath{\Upsilon}}(1S)$ and $\mathrm{\ensuremath{\Upsilon}}(2S)$ inclusive decays ($\mathcal{B}(\mathrm{\ensuremath{\Upsilon}}(1S,\text{ }2S)\ensuremath{\rightarrow}{R}^{++}+\text{anything})\ifmmode\times\else\texttimes\fi{}\mathcal{B}({\mathrm{R}}^{++}\ensuremath{\rightarrow}{\mathrm{D}}^{+}{\mathrm{D}}_{\mathrm{s}}^{*+})$), the product values of Born cross section and branching fraction in ${e}^{+}{e}^{\ensuremath{-}}$ collisions ($\ensuremath{\sigma}({e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\phantom{\rule{0ex}{0ex}}{R}^{++}+\text{anything})\ifmmode\times\else\texttimes\fi{}\mathcal{B}({\mathrm{R}}^{++}\ensuremath{\rightarrow}{\mathrm{D}}^{+}{\mathrm{D}}_{\mathrm{s}}^{*+})$) at $\sqrt{s}=10.520$, 10.580, and 10.867 GeV under different assumptions of ${R}^{++}$ masses varying from 4.13 to $4.17\text{ }\text{ }\mathrm{GeV}/{c}^{2}$ and widths varying from 0 to 5 MeV are obtained.