Di-jet production constraining $\Delta g(x)$ at low $\mathbf{x}$ in polarized $\mathbf \vec{p}+\vec{p}$ collisions at RHIC

Polarized $\vec{p}+\vec{p}$ collisions at $\sqrt{s}=200\,$GeV and at $\sqrt{s}=500\,$GeV at RHIC provide a 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 $\Delta g(x)$. A recent global analysis provides for the first time evidence of a non-zero value of the gluon polarization $\int_{\tiny 0.05}^{\tiny 1}\Delta g (x)\, dx \,(Q^{2}=10\,{\rm GeV}^{2}) = 0.20^{+0.06}_{-0.07}$. 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. Forward di-jet production at STAR beyond the current acceptance of $-1<\eta<+2$, in particular at $\sqrt{s}=500\,$GeV, provides access to low $x$ values at the level of $10^{-3}$ where current uncertainties of $\Delta g(x)$ remain very large. Recent STAR di-jet results constraining $\Delta g(x)$ will be briefly summarized followed by a detailed presentation of the physics case of forward di-jet production at $\sqrt{s}=500\,$GeV for $2.5<\eta<4$ requiring an upgrade of the STAR forward detection system, including a discussion of the kinematic coverage and projected uncertainties.