Falconer’s $(K, d)$ distance set conjecture can fail for strictly convex sets $K$ in $\mathbb R^d$

Bishop, Christopher; Drillick, Hindy; Ntalampekos, Dimitrios

doi:10.4171/rmi/1254

Abstract We present measurements of the branching fractions of eight$$ {\overline{B}}^0 $$

{\bar{B}}^{0}

→D^(*)+K⁻$$ {K}_{(S)}^{\left(\ast \right)0} $$

K_{(S)}^{(*) 0}

,B⁻→D^(*)0K⁻$$ {K}_{(S)}^{\left(\ast \right)0} $$

K_{(S)}^{(*) 0}

decay channels. The results are based on data from SuperKEKB electron-positron collisions at the Υ(4S) resonance collected with the Belle II detector, corresponding to an integrated luminosity of 362 fb⁻¹. The event yields are extracted from fits to the distributions of the difference between expected and observedBmeson energy, and are efficiency-corrected as a function ofm(K⁻$$ {K}_{(S)}^{\left(\ast \right)0} $$

K_{(S)}^{(*) 0}

) andm(D^(*)$$ {K}_{(S)}^{\left(\ast \right)0} $$

K_{(S)}^{(*) 0}

) in order to avoid dependence on the decay model. These results include the first observation of$$ {\overline{B}}^0 $$

{\bar{B}}^{0}

→D⁺K⁻$$ {K}_S^0 $$

K_{S}^{0}

,B⁻→D*⁰K⁻$$ {K}_S^0 $$

K_{S}^{0}

, and$$ {\overline{B}}^0 $$

{\bar{B}}^{0}

→D*⁺K⁻$$ {K}_S^0 $$

K_{S}^{0}

decays and a significant improvement in the precision of the other channels compared to previous measurements. The helicity-angle distributions and the invariant mass distributions of theK⁻$$ {K}_{(S)}^{\left(\ast \right)0} $$

K_{(S)}^{(*) 0}

systems are compatible with quasi-two-body decays via a resonant transition with spin-parityJ^P= 1⁻for theK⁻$$ {K}_S^0 $$

K_{S}^{0}

systems andJ^P= 1⁺for theK⁻K*⁰systems. We also present measurements of the branching fractions of four$$ {\overline{B}}^0 $$

{\bar{B}}^{0}

→D^(*)+$$ {D}_s^{-} $$

D_{s}^{-}

,B⁻→D^(*)0$$ {D}_s^{-} $$

D_{s}^{-}

decay channels with a precision compatible to the current world averages.

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