$B\to \rho$ semileptonic decays and $|V_{ub}|$

We reevaluate the $B\to\rho\,l^+\nu_l$ decay width as a full $B\to\pi\pi\,l^+\nu_l$ four-particle decay, in which the two final pions are produced via an intermediate $\rho$ meson. The decay width can be written as a convolution of the $B\to\rho\,l^+\nu_l$ decay width, for an off-shell $\rho$, with the $\rho\to\pi\pi$ line shape. This allows to fully incorporate the effects of the finite $\rho$ meson width. As shown, consideration of the $\rho$ meson width effects increase the $|V_{ub}|$ value by some 8%, rendering it in better agreement with the determination based in the $B\to\pi$ decay. We take the $q^2$ dependence of the $B\to \rho$ semileptonic form factors from a dispersive Omn\`es representation. The Omn\`es subtraction constants and the overall normalization parameter $|V_{ub}|$ are fitted to light cone sum rules and lattice QCD theoretical form-factor calculations, in the low and high $q^2$ regions respectively, together to the CLEO, BaBar and Belle experimental partial branching fraction distributions. The extracted value from this global fit is $|V_{ub}|=(3.12\pm 0.13)\times 10^{-3}$, in agreement with the average $B\to \pi$ exclusive value $|V_{ub}|=(3.23\pm 0.31)\times 10^{-3}$ quoted by the Particle Data Group. The extracted value increases to $|V_{ub}|=(3.51\pm 0.16)\times 10^{-3}$ if only the most recent Belle Collaboration data is used. This latter value is in agreement with different theoretical determinations based in the $B\to\pi$ semileptonic decay and the values obtained by the CKMfitter and UTfit groups. In any case a clear tension with the $|V_{ub}|$ value extracted from inclusive semileptonic $b\to u$ decays still persists.