Measurements of $B \rightarrow J/\psi$ at forward rapidity in $p$+$p$ collisions at $\sqrt{s}=510$ GeV

We report the first measurement of the fraction of J/ψ mesons coming from B-meson decay (FB→J/ψ) in p+p collisions at s=510  GeV. The measurement is performed using the forward silicon vertex detector and central vertex detector at PHENIX, which provide precise tracking and distance-of-closest-approach determinations, enabling the statistical separation of J/ψ due to B-meson decays from prompt J/ψ. The measured value of FB→J/ψ is 8.1%±2.3%(stat)±1.9%(syst) for J/ψ with transverse momenta 0<pT<5  GeV/c and rapidity 1.2<|y|<2.2. The measured fraction FB→J/ψ at PHENIX is compared to values measured by other experiments at higher center of mass energies and to fixed-order-next-to-leading-logarithm and color-evaporation-model predictions. The bb¯ cross section per unit rapidity [dσ/dy(pp→bb¯)] extracted from the obtained FB→J/ψ and the PHENIX inclusive J/ψ cross section measured at 200 GeV scaled with color-evaporation-model calculations, at the mean B hadron rapidity y=±1.7 in 510 GeV p+p collisions, is 3.63-1.70+1.92  μb. It is consistent with the fixed-order-next-to-leading-logarithm calculations.