  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.
