Model-independent parametrization of B → ππℓν decays

We introduce a novel parametrization of 𝐵 →𝜋⁢𝜋⁢ℓ⁢𝜈 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 →𝜋⁢𝜋⁢ℓ⁢𝜈 decays through fits to differential spectra of 𝐵$^+$ →𝜋$^+$⁢𝜋−⁢ℓ$^+$⁢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 ℬ⁡(𝐵$^+$ →(𝜋$^+$⁢𝜋$^−$)$_𝑆$ℓ$^+$⁢𝜈) =2.2$^{+1.4}_{−1.0}$ ×10$^{−5}$, ℬ⁡(𝐵$^+$ →(𝜋$^+$⁢𝜋$^−$)$_𝑃$⁢ℓ$^+$⁢𝜈) = 19.6$^{+2.8}_{−2.7}$ ×10$^{−5}$, and ℬ⁡(𝐵+ →(𝜋+⁢𝜋−)𝐷⁢ℓ+⁢𝜈) =3.5$^{+1.3}_{−1.1}$ ×10$^{−5}$. In addition, we derive predictions for the thus far unobserved 𝐵$^+$ →𝜋$^0$⁢𝜋$^0$⁢ℓ$^+$⁢𝜈 decay and obtain a sizable branching fraction of ℬ⁡(𝐵$^+$ →𝜋$^0$⁢𝜋$^0$⁢ℓ$^+$⁢𝜈) =2.9$^{+0.9}_{−0.7}$ ×10$^{−5}$.