Gluon Sivers function and transverse single spin asymmetries in $e + p^\uparrow \rightarrow \gamma + X$

We present estimates of transverse single-spin asymmetry in prompt photon production in the scattering of low virtuality photons off a polarized proton target and discuss the possibility of using this as a probe to get information about the gluon Sivers function (GSF). Using a generalized parton model (GPM) framework, we estimate the asymmetries at electron-ion collider energy ($\sqrt{s}$ =140 GeV) taking into account both direct and resolved photon processes and find that the dominant contribution, up to $10\%$, comes from quark Sivers function (QSF) while the contribution from GSF is found to be up to $2\%$. However, upon taking into account the effects of the process-dependent initial and final state interactions through the color-gauge invariant generalized parton model approach we find that the situation is significantly changed, with near zero contributions from the QSFs and up to a $1\%$ level contribution from the \textit{f}-type GSF. Our results indicate that this process may be useful for distinguishing between GPM and color-gauge invariant generalized parton models and can be used as a good probe of \textit{f}-type GSF.