A model-independent parameterization of $B\rightarrowππ\ellν$ decays

We introduce a novel parameterization of $B\rightarrowππ\ellν$ 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 Omnès functions, thus allowing for a data-driven determination of the expansion parameters. We study the underlying composition of the di-pion system in $B\rightarrowππ\ellν$ decays through fits to differential spectra of $B^{+} \rightarrow π^{+}π^{-} \ell^+ ν$ 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 (π^+ π^-)_S \ell^+ ν) &= 2.2^{+1.4}_{-1.0}\times 10^{-5}\,,\\ \mathcal{B}(B^+\rightarrow (π^+ π^-)_P \ell^+ν) &= 19.6^{+2.8}_{-2.7}\times 10^{-5}\,,\\ \mathcal{B}(B^+\rightarrow (π^+ π^-)_D \ell^+ ν) &= 3.5^{+1.3}_{-1.1}\times 10^{-5}\,.\\ \end{align*} In addition, we derive predictions for the thus far unobserved $B^+\rightarrow π^0π^0\ell^+ν$ decay and obtain a sizeable branching fraction of $\mathcal{B}(B^+\rightarrow π^0π^0\ell^+ν) = 2.9^{+0.9}_{-0.7}\times 10^{-5}$.