The dependence of the ratio between the B$_\mathrm{s}^0$ and B$^+$ hadron production fractions, $f_\mathrm{s} / f_\mathrm{u}$, on the transverse momentum ($p_\mathrm{T}$) and rapidity of the B mesons is studied using the decay channels B$_\mathrm{s}^0$$\to$ J$/\psi\,\phi$ and B$^+$$\to$ J$/\psi$ K$^+$. The analysis uses a data sample of proton-proton collisions at a center-of-mass energy of 13 TeV, collected by the CMS experiment in 2018 and corresponding to an integrated luminosity of 61.6 fb$^{-1}$. The $f_\mathrm{s} / f_\mathrm{u}$ ratio is observed to depend on the B $p_\mathrm{T}$ and to be consistent with becoming asymptotically constant at large $p_\mathrm{T}$. No rapidity dependence is observed. The ratio of the B$^0$ to B$^+$ hadron production fractions, $f_\mathrm{d} / f_\mathrm{u}$, measured using the B$^0$$\to$ J$/\psi$ K$^{*0}$ decay channel, is found to be consistent with unity and independent of $p_\mathrm{T}$ and rapidity, as expected from isospin invariance.
A study of the particle multiplicity between jets with large rapidity separation has been performed using the D\O\ detector at the Fermilab Tevatron $p\bar{p}$ Collider operating at $\sqrt{s}=1.8$\,TeV. A significant excess of low-multiplicity events is observed above the expectation for color-exchange processes. The measured fractional excess is $1.07 \pm 0.10({\rm stat})~{ + 0.25}_{- 0.13}({\rm syst})\%$, which is consistent with a strongly-interacting color-singlet (colorless) exchange process and cannot be explained by electroweak exchange alone. A lower limit of $0.80\%$ (95\% C.L.) is obtained on the fraction of dijet events with color-singlet exchange, independent of the rapidity gap survival probability.
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