Measurements of $B \rightarrow J/ψ$ 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 ($F_{B{\rightarrow}J/ψ}$) in $p$+$p$ collisions at $\sqrt{s}=$ 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 $F_{B{\rightarrow}J/ψ}$ is 8.1\%$\pm$2.3\% (stat)$\pm$1.9\% (syst) for $J/ψ$ with transverse momenta $0<p_T<5$ GeV/$c$ and rapidity $1.2<|y|<2.2$. The measured fraction $F_{B{\rightarrow}J/ψ}$ 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 $b\bar{b}$ cross section per unit rapidity ($dσ/dy(pp{\rightarrow}b\bar{b})$) extracted from the obtained $F_{B{\rightarrow}J/ψ}$ and the PHENIX inclusive $J/ψ$ cross section measured at 200 GeV scaled with color-evaporation-model calculations, at the mean $B$ hadron rapidity $y={\pm}1.7$ in 510 GeV $p$$+$$p$ collisions, is $3.63^{+1.92}_{-1.70}μ$b, and it is consistent with the fixed-order-next-to-leading-logarithm calculations.