The STAR Collaboration reports measurements of the transverse single-spin asymmetry (TSSA) of inclusive $pi^0$ at center-of-mass energies ($sqrt s$) of 200 GeV and 500 GeV in transversely polarized proton-proton collisions in the pseudo-rapidity region 2.7 to 4.0. The results at the two different energies show a continuous increase of the TSSA with Feynman-$x$, and, when compared to previous measurements, no dependence on $sqrt s$ from 19.4 GeV to 500 GeV is found. To investigate the underlying physics leading to this large TSSA, different topologies have been studied. $pi^0$ with no nearby particles tend to have a higher TSSA than inclusive $pi^0$. The TSSA for inclusive electromagnetic jets, sensitive to the Sivers effect in the initial state, is substantially smaller, but shows the same behavior as the inclusive $pi^0$ asymmetry as a function of Feynman-$x$. To investigate final-state effects, the Collins asymmetry of $pi^0$ inside electromagnetic jets has been measured. The Collins asymmetry is analyzed for its dependence on the $pi^0$ momentum transverse to the jet thrust axis and its dependence on the fraction of jet energy carried by the $pi^0$. The asymmetry was found to be small in each case for both center-of-mass energies. All the measurements are compared to QCD-based theoretical calculations for transverse-momentum-dependent parton distribution functions and fragmentation functions. Some discrepancies are found, which indicates new mechanisms might be involved.
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