The differential cross section for $Z^{0}$ production, measured as a function of the boson's transverse momentum ($p_{\mathrm{T}}$), provides important constraints on the evolution of the transverse momentum dependent parton distribution functions (TMDs). The transverse single spin asymmetry (TSSA) of the $Z^{0}$ is sensitive to one of the polarized TMDs, the Sivers function, which is predicted to have the opposite sign in $p+p$$\rightarrow W/Z+ X$ from that which enters in semi-inclusive deep inelastic scattering. In this Letter, the STAR Collaboration reports the first measurement of the $Z^{0}/\gamma^{*}$ differential cross section as a function of its $p_{\mathrm{T}}$ in $p$$+$$p$ collisions at a center-of-mass energy of 510 GeV, together with the $Z^{0}/\gamma^{*}$ total cross section. We also report the measurement of $Z^{0}/\gamma^{*}$ TSSA in transversely polarized $p$$+$$p$ collisions at 510 GeV.
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