Accurately determining the smallest CKM matrix element, $|V_{ub}|$, remains a critical challenge of the Standard model due to the discrepancies between its exclusive and inclusive determinations. The decay process $B \to \rho \ell \bar{\nu}$ is perfect to address this issue, particularly through lattice QCD calculations. In this talk, I will discuss our ongoing efforts in computing the vector and axial form factors for $B \to \rho \ell \bar{\nu}$, focusing specifically on the unstable nature of the $\rho$, which appears as an enhancement in P-wave $\pi\pi$ scattering. I will highlight the analysis approach, outline our methodological strategies, and present our recent results on a gauge ensemble with 2+1 flavors of dynamical clover fermions, where the light quark mass corresponds to a pion mass of 320 MeV.
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