The PHENIX experiment at the Relativistic Heavy Ion Collider has measured the differential cross section of $\phi$(1020) meson production at forward rapidity in $p$$+$$p$ collisions at $\sqrt{s}=$510 GeV via the dimuon decay channel. The integrated cross section in the rapidity and $p_T$ ranges $1.2<|y|<2.2$ and $2<p_T<7$ GeV/$c$ is $\sigma_\phi=2.79 \pm 0.20\,{\rm (stat)} \pm 0.17\,{\rm (syst)} \pm 0.34\, {\rm (norm)} \times 10^{-2}$~mb. The energy dependence of $\sigma_\phi$ ($1.2<|y|<2.2$; $2<p_T<5$ GeV/$c$) is studied using the PHENIX measurements at $\sqrt{s}=$200 and 510 GeV and the Large-Hadron-Collider measurements at $\sqrt{s}=$2.76 and 7 TeV. The experimental results are compared to various event generator predictions (pythia6, pythia8, phojet, ampt, epos3, and epos-lhc).
The $\phi$-meson-production cross section d$\sigma_{\phi}$/dy in $p$ + $p$ collisions at $\sqrt{s}$ = 510 GeV integrated in the transverse-momentum range 2 < $p_T$ < 7 GeV/$c$.
The $\phi$-meson-production cross section d$\sigma_{\phi}$/dy in $p$ + $p$ collisions at $\sqrt{s}$ = 510 GeV integrated in the transverse-momentum range 2 < $p_T$ < 7 GeV/$c$.
The $\phi$-meson-differential-production cross section d${}^{2}$$\sigma_{\phi}/dp_T dy$ for 1.2 < |y| < 2.2 in $p$ + $p$ collisions at $\sqrt{s}$ = 510 GeV.
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