Measurements of the transverse-momentum-dependent cross sections of $J\mathrm{/}\psi$ production at mid-rapidity in proton+proton collisions at $\sqrt{s}=$ 510 and 500 GeV with the STAR detector

We present measurements of the differential cross sections of inclusive $J\mathrm{/}\psi$ meson production as a function of transverse momentum ($p_T^{J\mathrm{/}\psi }$) using the ${\mu }^{+}{\mu }^{-}$ and $e^{+}{e}^{-}$ decay channels in proton+proton collisions at center-of-mass energies of 510 and 500 GeV, respectively, recorded by the STAR detector at the Relativistic Heavy Ion Collider. The measurement from the ${\mu }^{+}{\mu }^{-}$ channel is for 0 $< p_{T}^{J/\psi} <$9GeV\mathrm{/}$c$andrapidityrange$|y^{J/\psi}| < $ 0.4, and that from the $e^{+}{e}^{-}$ channel is for 4 $<p_T^{J\mathrm{/}\psi }<$ 20 GeV/$c$ and $\mathrm{\mid }{y}^{J\mathrm{/}\psi }\mathrm{\mid }<$ 1.0. The $\psi (2S)$ to $J\mathrm{/}\psi$ ratio is also measured for 4 $<p_T^{\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{n}}<$ 12 GeV/$c$ through the $e^{+}{e}^{-}$ decay channel. Model calculations, which incorporate different approaches toward the $J\mathrm{/}\psi$ production mechanism, are compared with experimental results and show reasonable agreement within uncertainties.