A model-independent parameterization of $B\rightarrow\pi\pi\ell\nu$ decays

We introduce a novel parameterization of $B\rightarrow\pi\pi\ell\nu$ form factors relying on partial-wave decompositions and series expansions in suitable variables. We bound the expansion coefficients through unitarity and include left-hand cut contributions using established dispersive methods. The two-hadron lineshapes are treated in a model-independent manner using Omn\`es functions, thus allowing for a data-driven determination of the expansion parameters. We study the underlying composition of the di-pion system in $B\rightarrow\pi\pi\ell\nu$ decays through fits to differential spectra of $B^{+} \rightarrow \pi^{+}\pi^{-} \ell^+ \nu$ measured by the Belle experiment. In contrast to previous works, we are able to study the full phase-space and are not limited to certain kinematic regions. As a consequence, we extract branching fractions for the different partial waves of the di-pion system. We find: \begin{align*} \mathcal{B}(B^+\rightarrow (\pi^+ \pi^-)_S \ell^+ \nu) &= 2.2^{+1.4}_{-1.0}\times 10^{-5}\,,\\ \mathcal{B}(B^+\rightarrow (\pi^+ \pi^-)_P \ell^+\nu) &= 19.6^{+2.8}_{-2.7}\times 10^{-5}\,,\\ \mathcal{B}(B^+\rightarrow (\pi^+ \pi^-)_D \ell^+ \nu) &= 3.5^{+1.3}_{-1.1}\times 10^{-5}\,.\\ \end{align*} In addition, we derive predictions for the thus far unobserved $B^+\rightarrow \pi^0\pi^0\ell^+\nu$ decay and obtain a sizeable branching fraction of $\mathcal{B}(B^+\rightarrow \pi^0\pi^0\ell^+\nu) = 2.9^{+0.9}_{-0.7}\times 10^{-5}$.