The cross section of bottom quark-antiquark ($bbar{b}$) production in $p$+$p$ collisions at $sqrt{s}=510$ GeV is measured with the PHENIX detector at the Relativistic Heavy Ion Collider. The results are based on the yield of high mass, like-sign muon pairs measured within the PHENIX muon arm acceptance ($1.2<|y|<2.2$). The $bbar{b}$ signal is extracted from like-sign dimuons by utilizing the unique properties of neutral $B$ meson oscillation. We report a differential cross section of $dsigma_{bbar{b}rightarrow mu^pmmu^pm}/dy = 0.16 pm 0.01~(mbox{stat}) pm 0.02~(mbox{syst}) pm 0.02~(mbox{global})$ nb for like-sign muons in the rapidity and $p_T$ ranges $1.2<|y|<2.2$ and $p_T>1$ GeV/$c$, and dimuon mass of 5--10 GeV/$c^2$. The extrapolated total cross section at this energy for $bbar{b}$ production is $13.1 pm 0.6~(mbox{stat}) pm 1.5~(mbox{syst}) pm 2.7~(mbox{global})~mu$b. The total cross section is compared to a perturbative quantum chromodynamics calculation and is consistent within uncertainties. The azimuthal opening angle between muon pairs from $bbar{b}$ decays and their $p_T$ distributions are compared to distributions generated using {sc ps pythia 6}, which includes next-to-leading order processes. The azimuthal correlations and pair $p_T$ distribution are not very well described by {sc pythia} calculations, but are still consistent within uncertainties. Flavor creation and flavor excitation subprocesses are favored over gluon splitting.
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