We introduce a novel parametrization of $B\to \pi \pi \ell \nu$ 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 line shapes 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 $B\to \pi \pi \ell \nu$ decays through fits to differential spectra of $B^{+}\to {\pi }^{+}{\pi }^{-}{\ell }^{+}\nu$ 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 $\mathcal{B}(B^{+}\to ({\pi }^{+}{\pi }^{-}{)}_S{\ell }^{+}\nu )={2.2}_{-1.0}^{+1.4}\times {10}^{-5}$, $\mathcal{B}(B^{+}\to ({\pi }^{+}{\pi }^{-}{)}_P{\ell }^{+}\nu )={19.6}_{-2.7}^{+2.8}\times {10}^{-5}$, and $\mathcal{B}(B^{+}\to ({\pi }^{+}{\pi }^{-}{)}_D{\ell }^{+}\nu )={3.5}_{-1.1}^{+1.3}\times {10}^{-5}$. In addition, we derive predictions for the thus far unobserved $B^{+}\to {\pi }^0{\pi }^0{\ell }^{+}\nu$ decay and obtain a sizable branching fraction of $\mathcal{B}(B^{+}\to {\pi }^0{\pi }^0{\ell }^{+}\nu )={2.9}_{-0.7}^{+0.9}\times {10}^{-5}$.

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• Branching fraction
• Cabibbo–Kobayashi–Maskawa matrix
• Form factors
• Hadron-hadron interactions
• Leptonic, semileptonic & radiative decays
• Scattering amplitudes
• Bottom mesons
• Bottom quark
• Leptons
• Pions
• Angular momentum
• Integral equations
• Real & complex analysis