Measurements of the transverse-momentum-dependent cross sections of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>J</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>ψ</mml:mi></mml:math> production at mid-rapidity in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>proton</mml:mi><mml:mo>+</mml:mo><mml:mi>proton</mml:mi></mml:mrow></mml:math> collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msqrt><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msqrt><mml:mo>=</mml:mo><mml:mn>510</mml:mn></mml:mrow></mml:math> and 500 GeV with the STAR detector

We present measurements of the differential cross sections of inclusive $J/\ensuremath{\psi}$ meson production as a function of transverse momentum (${p}_{T}^{J/\ensuremath{\psi}}$) using the ${\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ and ${e}^{+}{e}^{\ensuremath{-}}$ decay channels in $\mathrm{proton}+\mathrm{proton}$ 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 ${\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ channel is for $0&lt;{p}_{T}^{J/\ensuremath{\psi}}&lt;9\text{ }\text{ }\mathrm{GeV}/c$ and rapidity range $|{y}^{J/\ensuremath{\psi}}|&lt;0.4$, and that from the ${e}^{+}{e}^{\ensuremath{-}}$ channel is for $4&lt;{p}_{T}^{J/\ensuremath{\psi}}&lt;20\text{ }\text{ }\mathrm{GeV}/c$ and $|{y}^{J/\ensuremath{\psi}}|&lt;1.0$. The $\ensuremath{\psi}(2S)$ to $J/\ensuremath{\psi}$ ratio is also measured for $4&lt;{p}_{T}^{\text{meson}}&lt;12\text{ }\text{ }\mathrm{GeV}/c$ through the ${e}^{+}{e}^{\ensuremath{-}}$ decay channel. Model calculations, which incorporate different approaches toward the $J/\ensuremath{\psi}$ production mechanism, are compared with experimental results and show reasonable agreement within uncertainties.