Measurement of $\phi$-meson production at forward rapidity in $p$$+$$p$ collisions at $\sqrt{s}$=510 GeV and energy dependence of $\sigma_\phi$ from $\sqrt{s}$=200 GeV to 7 TeV

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 partial cross section in the rapidity and $p_T$ ranges $1.2u003c|y|u003c2.2$ and $2u003cp_Tu003c7$ GeV/$c$ is $sigma_phi=[2.28 pm 0.09,{rm (stat)} pm 0.14,{rm (syst)} pm 0.27, {rm (norm)}] times 10^{-2}$~mb. The energy dependence of $sigma_phi$ ($1.2u003c|y|u003c2.2, ; 2u003cp_Tu003c5$ 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 ({sc pythia6, pythia8, phojet, ampt, epos3,} and {sc epos-lhc}).