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Future prospects of di-jet production constraining $Δg(x)$ at low $\mathbf{x}$ at STAR at RHIC

One of the main objectives of the high-energy spin physics program at RHIC at BNL is the precise determination of the polarized gluon distribution function, $Δg(x)$. Polarized $\vec{p}+\vec{p}$ collisions at $\sqrt{s}=200\,$GeV and at $\sqrt{s}=500\,$GeV at RHIC provide an unique way to probe the proton spin structure. Inclusive measurements, such as inclusive jet and hadron production, have so far been the prime focus of various results at $\sqrt{s}=200\,$GeV constraining $Δg(x)$ for $0.05<x<0.2$. A recent global analysis provides for the first time evidence of a non-zero value of the gluon polarization $\int_{\tiny 0.05}^{\tiny 1}Δg (x)\, dx \,(Q^{2}=10\,{\rm GeV}^{2}) = 0.20^{+0.06}_{-0.07}$. First results of di-jet production at $\sqrt{s}=200\,$GeV by the STAR collaboration will allow a better constraint of the underlying event kinematics. Extending the current program to smaller values of $x$ is a key goal for the future high-energy spin physics program at RHIC. Forward di-jet production measurements at STAR beyond the current acceptance from $-1<η<+2$ to $+2.5<η<+4$, in particular those carried out at $\sqrt{s}=500\,$GeV, provides access to low $x$ values at the level of $10^{-3}$ where current uncertainties of $Δg(x)$ remain very large. Those measurements will eventually be complemented by a future Electron-Ion Collider facility probing $Δg(x)$ in polarized $\vec{e}+\vec{p}$ collisions.