$B \to \rho \ell \bar{\nu}$ resonance form factors from $B \to \pi\pi \ell \bar{\nu}$ in lattice QCD

The decay $B \to \rho \ell \bar{\nu}$ is an attractive process for determining the magnitude of the smallest CKM matrix element, $|V_{ub}|$, and can provide new insights into the origin of the long-standing exclusive-inclusive discrepancy in determinations of this Standard-Model parameter. This requires a nonperturbative QCD calculation of the $B \to \rho$ form factors $V$, $A_0$, $A_1$, and $A_{12}$. The unstable nature of the $\rho$ resonance has prevented precise lattice QCD calculations of these form factors to date. Here, we present the first lattice QCD calculation of the $B \to \rho$ form factors in which the $\rho$ is treated properly as a resonance in $P$-wave $\pi \pi$ scattering. To this end, we use the Lellouch-L\"uscher finite-volume formalism to compute the $B \to \pi \pi$ form factors as a function of both momentum transfer and $\pi \pi$ invariant mass, and then analytically continue to the $\rho$ resonance pole. This calculation is performed with $2+1$ dynamical quark flavors at a pion mass of approximately 320 MeV, and demonstrates a clear path toward results at the physical point.