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The heavy flavor studies at RHIC help improve the knowledge of the bottom/charm quark production and can test Quantum Chromodynamics (QCD). Compared to the LHC/Tevatron, the RHIC heavy flavor production originates from different partonic sub-processes and has a complementary kinematic coverage. The PHENIX forward rapidity silicon vertex detector (FVTX) provides precise determination of the event vertex, tracking and the Distance of Closest Approach (DCA) of charged tracks. This detector allows direct access to the $B$ meson production via measurements of non-prompt $J/\psi \to \mu^{+} + \mu^{-}$ within $1.2<|y|<2.2$ rapidity in $p$+$p$ collisions at $\sqrt{s} = $ 510 and 200 GeV. Comparison among PHENIX measurements of the $B \rightarrow J/\psi$ fraction with integrated $J/\psi$ $p_{T}$ up to 5 GeV$/c$ and higher energy results at the Tevatron and the LHC presents a smooth center of mass energy dependence from 0.2 to 13 TeV in $p$+$p$ ($p$+$\bar{p}$) collisions. The Next-To-Leading order Perturbative QCD (NLO pQCD) calculations are in reasonable agreement with the extracted total $b\bar{b}$ cross section based on the $B \rightarrow J/\psi$ fraction measurements at PHENIX.

Energy dependent forward $B \rightarrow J/\psi$ measurements in $p+p$ collisions at PHENIX