We present measurements of the differential production cross sections of the inclusive $J/\psi$ meson as a function of transverse momentum ($p_{T}^{J/\psi}$) 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 $< p_{T}^{J/\psi} <$ 9 GeV/$c$ and rapidity range $|y^{J/\psi}| < $ 0.4, and that from the $e^{+}e^{-}$ channel is for 4 $< p_{T}^{J/\psi} <$ 20 GeV/$c$ and $|y^{J/\psi}| < $ 1.0. The $\psi(2S)$ to $J/\psi$ ratio is also measured for 4 $< p_{T}^{\rm meson} <$ 12 GeV/$c$ through the $e^{+}e^{-}$ decay channel. Model calculations, which incorporate different approaches toward the $J/\psi$ production mechanism, are compared with experimental results and show reasonable agreement within uncertainties. A more discriminating comparison to theoretical models at low $p_T$ can be performed in the future, if the calculations are carried out within our fiducial volume, eliminating the uncertainty due to the $J/\psi$ polarization.
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