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We report the first measurement of the fraction of $J/\psi$ mesons comingfrom $B$-meson decay ($F_{B{\rightarrow}J/\psi}$) in $p$+$p$ collisions at$\sqrt{s}=$ 510 GeV. The measurement is performed using the forward siliconvertex detector and central vertex detector at PHENIX, which provide precisetracking and distance-of-closest-approach determinations, enabling thestatistical separation of $J/\psi$ due to $B$-meson decays from prompt$J/\psi$. The measured value of $F_{B{\rightarrow}J/\psi}$ is 8.1\%$\pm$2.3\%(stat)$\pm$1.9\% (syst) for $J/\psi$ with transverse momenta $0<p_T<5$ GeV/$c$and rapidity $1.2<|y|<2.2$. The measured fraction $F_{B{\rightarrow}J/\psi}$ atPHENIX is compared to values measured by other experiments at higher center ofmass energies and to fixed-order-next-to-leading-logarithm andcolor-evaporation-model predictions. The $b\bar{b}$ cross section per unitrapidity ($d\sigma/dy(pp{\rightarrow}b\bar{b})$) extracted from the obtained$F_{B{\rightarrow}J/\psi}$ and the PHENIX inclusive $J/\psi$ cross sectionmeasured at 200 GeV scaled with color-evaporation-model calculations, at themean $B$ hadron rapidity $y={\pm}1.7$ in 510 GeV $p$$+$$p$ collisions, is$3.63^{+1.92}_{-1.70}\mu$b, and it is consistent with thefixed-order-next-to-leading-logarithm calculations.

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