We present various properties of the production of the $X(3872)$ and $psi(2S)$ states based on 10.4 fb$^{-1}$ collected by D0 experiment in Tevatron $p bar p$ collisions at $sqrt{s}$ = 1.96 TeV. For both states, we measure the nonprompt fraction $f_{NP}$ of the inclusive production rate due to decays of $b$-flavored hadrons. We find the $f_{NP}$ values systematically below those obtained at the LHC. The $f_{NP}$ fraction for $psi(2S)$ increases with transverse momentum whereas for the X(3872) it is constant within large uncertainties, in agreement with the LHC results. The ratio of prompt to nonprompt $psi(2S)$ production, $(1 - f_{NP}) / f_{NP}$, decreases only slightly going from the Tevatron to the LHC, but for the $X(3872)$ this ratio decreases by a factor of about 3. We test the soft-pion signature of the $X(3872)$ modeled as a weakly-bound charm-meson pair by studying the production of the $X(3872)$ as a function of the kinetic energy of the $X(3872)$ and the pion in the $X(3872)pi$ center-of-mass frame. For a subsample consistent with prompt production, the results are incompatible with a strong enhancement in the production of the $X(3872)$ at small kinetic energy of the $X(3872)$ and the $pi$ in the $X(3872)pi$ center-of-mass frame expected for the $X$+soft-pion production mechanism. For events consistent with being due to decays of $b$ hadrons, there is no significant evidence for the soft-pion effect but its presence at the level expected for the binding energy of 0.17 MeV and the momentum scale $Lambda=M(pi)$ is not ruled out.
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