Model-independent parametrization of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>π</mml:mi><mml:mi>π</mml:mi><mml:mo>ℓ</mml:mo><mml:mi>ν</mml:mi></mml:math> de

We introduce a novel parametrization of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>B</a:mi><a:mo stretchy="false">→</a:mo><a:mi>π</a:mi><a:mi>π</a:mi><a:mo>ℓ</a:mo><a:mi>ν</a:mi></a:math> 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>B</d:mi><d:mo stretchy="false">→</d:mo><d:mi>π</d:mi><d:mi>π</d:mi><d:mo>ℓ</d:mo><d:mi>ν</d:mi></d:math> decays through fits to differential spectra of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>B</g:mi><g:mo>+</g:mo></g:msup><g:mo stretchy="false">→</g:mo><g:msup><g:mi>π</g:mi><g:mo>+</g:mo></g:msup><g:msup><g:mi>π</g:mi><g:mo>−</g:mo></g:msup><g:msup><g:mo>ℓ</g:mo><g:mo>+</g:mo></g:msup><g:mi>ν</g:mi></g:math> 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 <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi mathvariant="script">B</j:mi><j:mo stretchy="false">(</j:mo><j:msup><j:mi>B</j:mi><j:mo>+</j:mo></j:msup><j:mo stretchy="false">→</j:mo><j:mo stretchy="false">(</j:mo><j:msup><j:mi>π</j:mi><j:mo>+</j:mo></j:msup><j:msup><j:mi>π</j:mi><j:mo>−</j:mo></j:msup><j:msub><j:mo stretchy="false">)</j:mo><j:mi>S</j:mi></j:msub><j:msup><j:mo>ℓ</j:mo><j:mo>+</j:mo></j:msup><j:mi>ν</j:mi><j:mo stretchy="false">)</j:mo><j:mo>=</j:mo><j:msubsup><j:mn>2.2</j:mn><j:mrow><j:mo>−</j:mo><j:mn>1.0</j:mn></j:mrow><j:mrow><j:mo>+</j:mo><j:mn>1.4</j:mn></j:mrow></j:msubsup><j:mo>×</j:mo><j:msup><j:mn>10</j:mn><j:mrow><j:mo>−</j:mo><j:mn>5</j:mn></j:mrow></j:msup></j:math>, <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi mathvariant="script">B</r:mi><r:mo stretchy="false">(</r:mo><r:msup><r:mi>B</r:mi><r:mo>+</r:mo></r:msup><r:mo stretchy="false">→</r:mo><r:mo stretchy="false">(</r:mo><r:msup><r:mi>π</r:mi><r:mo>+</r:mo></r:msup><r:msup><r:mi>π</r:mi><r:mo>−</r:mo></r:msup><r:msub><r:mo stretchy="false">)</r:mo><r:mi>P</r:mi></r:msub><r:msup><r:mo>ℓ</r:mo><r:mo>+</r:mo></r:msup><r:mi>ν</r:mi><r:mo stretchy="false">)</r:mo><r:mo>=</r:mo><r:mspace linebreak="goodbreak"/><r:msubsup><r:mn>19.6</r:mn><r:mrow><r:mo>−</r:mo><r:mn>2.7</r:mn></r:mrow><r:mrow><r:mo>+</r:mo><r:mn>2.8</r:mn></r:mrow></r:msubsup><r:mo>×</r:mo><r:msup><r:mn>10</r:mn><r:mrow><r:mo>−</r:mo><r:mn>5</r:mn></r:mrow></r:msup></r:math>, and <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi mathvariant="script">B</ab:mi><ab:mo stretchy="false">(</ab:mo><ab:msup><ab:mi>B</ab:mi><ab:mo>+</ab:mo></ab:msup><ab:mo stretchy="false">→</ab:mo><ab:mo stretchy="false">(</ab:mo><ab:msup><ab:mi>π</ab:mi><ab:mo>+</ab:mo></ab:msup><ab:msup><ab:mi>π</ab:mi><ab:mo>−</ab:mo></ab:msup><ab:msub><ab:mo stretchy="false">)</ab:mo><ab:mi>D</ab:mi></ab:msub><ab:msup><ab:mo>ℓ</ab:mo><ab:mo>+</ab:mo></ab:msup><ab:mi>ν</ab:mi><ab:mo stretchy="false">)</ab:mo><ab:mo>=</ab:mo><ab:msubsup><ab:mn>3.5</ab:mn><ab:mrow><ab:mo>−</ab:mo><ab:mn>1.1</ab:mn></ab:mrow><ab:mrow><ab:mo>+</ab:mo><ab:mn>1.3</ab:mn></ab:mrow></ab:msubsup><ab:mo>×</ab:mo><ab:msup><ab:mn>10</ab:mn><ab:mrow><ab:mo>−</ab:mo><ab:mn>5</ab:mn></ab:mrow></ab:msup></ab:math>. In addition, we derive predictions for the thus far unobserved <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msup><ib:mi>B</ib:mi><ib:mo>+</ib:mo></ib:msup><ib:mo stretchy="false">→</ib:mo><ib:msup><ib:mi>π</ib:mi><ib:mn>0</ib:mn></ib:msup><ib:msup><ib:mi>π</ib:mi><ib:mn>0</ib:mn></ib:msup><ib:msup><ib:mo>ℓ</ib:mo><ib:mo>+</ib:mo></ib:msup><ib:mi>ν</ib:mi></ib:math> decay and obtain a sizable branching fraction of <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:mi mathvariant="script">B</lb:mi><lb:mo stretchy="false">(</lb:mo><lb:msup><lb:mi>B</lb:mi><lb:mo>+</lb:mo></lb:msup><lb:mo stretchy="false">→</lb:mo><lb:msup><lb:mi>π</lb:mi><lb:mn>0</lb:mn></lb:msup><lb:msup><lb:mi>π</lb:mi><lb:mn>0</lb:mn></lb:msup><lb:msup><lb:mo>ℓ</lb:mo><lb:mo>+</lb:mo></lb:msup><lb:mi>ν</lb:mi><lb:mo stretchy="false">)</lb:mo><lb:mo>=</lb:mo><lb:msubsup><lb:mn>2.9</lb:mn><lb:mrow><lb:mo>−</lb:mo><lb:mn>0.7</lb:mn></lb:mrow><lb:mrow><lb:mo>+</lb:mo><lb:mn>0.9</lb:mn></lb:mrow></lb:msubsup><lb:mo>×</lb:mo><lb:msup><lb:mn>10</lb:mn><lb:mrow><lb:mo>−</lb:mo><lb:mn>5</lb:mn></lb:mrow></lb:msup></lb:math>.