A model-independent parameterization of $B\to \pi \pi \mathrm{\ell }\nu$ decays

We introduce a novel parameterization of $B\to \pi \pi \mathrm{\ell }\nu$ 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 Omnès functions, thus allowing for a data-driven determination of the expansion parameters. We study the underlying composition of the di-pion system in $B\to \pi \pi \mathrm{\ell }\nu$ decays through fits to differential spectra of $B^{+}\to {\pi }^{+}{\pi }^{-}{\mathrm{\ell }}^{+}\nu$ 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 (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 $B^{+}\to {\pi }^0{\pi }^0{\mathrm{\ell }}^{+}\nu$ decay and obtain a sizeable branching fraction of $\mathcal{B}(B^{+}\to {\pi }^0{\pi }^0{\mathrm{\ell }}^{+}\nu )=2.9_{-0.7}^{+0.9}\times {10}^{-5}$.