Cross sections of $\eta$ mesons in p0p collisions at forward rapidity at $\sqrt{s}=500$ GeV and central rapidity at $\sqrt{s}=510$ GeV

We present the first measurements of the forward and midrapidity $\eta$-meson cross sections from $p$$+$$p$ collisions at $\sqrt{s}=500$ and $510$~GeV, respectively. We also report the midrapidity $\eta \mathrm{/}{\pi }^0$ ratio at 510 GeV. The forward cross section is measured differentially in $\eta$-meson transverse momentum ($p_T$) from 1.0 to 6.5~GeV/$c$ for pseudorapidity $3.0<\mathrm{\mid }\eta \mathrm{\mid }<3.8$. The midrapidity cross section is measured from 3.5 to 44 GeV/$c$ for pseudorapidity $\mathrm{\mid }\eta \mathrm{\mid }<0.35$. Both cross sections serve as critical inputs to an updated global analysis of the $\eta$-meson fragmentation functions.