Transverse single-spin asymmetries of midrapidity $\pi^0$ and $\eta$ mesons in polarized $p$$+$$p$ collisions at $\sqrt{s}=200$ GeV

We present a measurement of the transverse single-spin asymmetry for $\pi^0$ and $\eta$ mesons in $p^\uparrow$$+$$p$ collisions in the pseudorapidity range $|\eta|<0.35$ and at a center-of-mass energy of 200 GeV with the PHENIX detector at the Relativistic Heavy Ion Collider. In comparison with previous measurements in this kinematic region, these results have a factor of 3 smaller uncertainties. As hadrons, $\pi^0$ and $\eta$ mesons are sensitive to both initial- and final-state nonperturbative effects for a mix of parton flavors. Comparisons of the differences in their transverse single-spin asymmetries have the potential to disentangle the possible effects of strangeness, isospin, or mass. These results can constrain the twist-3 trigluon collinear correlation function as well as the gluon Sivers function.