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The squared momentum transfer spectra of $\eta$ and $\eta ^0$, produced inhigh-energy photon-proton ($\gamma p$) $\rightarrow \eta(\eta^0)+p$ processesin electron-proton ($ep$) collisions performed at CEBAF, NINA, CEA, SLAC, DESY,and WLS are analyzed. The Monte Carlo calculations are used in the analysis ofthe squared momentum transfer spectra, where the transfer undergoes from theincident $\gamma$ to emitted $\eta(\eta^0)$ or equivalently from the targetproton to emitted proton. In the calculations, the Erlang distribution andTsallis-Levy function are used to describe the transverse momentum ($p_T$)spectra of emitted particles. Our results show that the average transversemomentum ($\langle p_T\rangle$), the initial-state temperature ($T_i$), and thefinal-state temperature ($T_0$) roughly decrease from the lower center-of-massenergy ($W$) to the higher one in the concerned energy range of a few GeV,which is different from the excitation function from heavy-ion collisions inthe similar energy range.

Excitation functions of related temperatures of {\eta} and {\eta}0 emission sources from squared momentum transfer spectra in high-energy collisions