QCD analysis of B → π π form factors and the | V u b | extraction from B l 4 decays

We present a comprehensive QCD analysis of the <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:math> form factors within the framework of light-cone sum rules, employing two-pion light-cone distribution amplitudes ( <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mn>2</d:mn> <d:mi>π</d:mi> <d:mi>DA</d:mi> </d:math> s). For the first time, we derive the twist-three <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mn>2</f:mn> <f:mi>π</f:mi> <f:mi>DA</f:mi> </f:math> s explicitly by exploiting the conformal symmetry of massless QCD. This breakthrough enables the first systematic calculation of higher-power corrections to the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>B</h:mi> <h:mo stretchy="false">→</h:mo> <h:mi>π</h:mi> <h:mi>π</h:mi> </h:math> -type transitions. Leveraging the recently measured partial decay fractions of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mi>B</k:mi> <k:mo>+</k:mo> </k:msup> <k:mo stretchy="false">→</k:mo> <k:msup> <k:mi>π</k:mi> <k:mo>+</k:mo> </k:msup> <k:msup> <k:mi>π</k:mi> <k:mo>−</k:mo> </k:msup> <k:msup> <k:mi>l</k:mi> <k:mo>+</k:mo> </k:msup> <k:msub> <k:mi>ν</k:mi> <k:mi>l</k:mi> </k:msub> </k:math> by the Belle detector, we obtain <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mo stretchy="false">|</n:mo> <n:msub> <n:mi>V</n:mi> <n:mrow> <n:mi>u</n:mi> <n:mi>b</n:mi> </n:mrow> </n:msub> <n:mo stretchy="false">|</n:mo> <n:mo>=</n:mo> <n:mrow> <n:mo stretchy="false">(</n:mo> <n:mn>4.27</n:mn> <n:mo>±</n:mo> <n:mn>0.49</n:mn> <n:msub> <n:mo stretchy="false">|</n:mo> <n:mrow> <n:mi>data</n:mi> </n:mrow> </n:msub> <n:mo>±</n:mo> <n:mn>0.55</n:mn> <n:msub> <n:mo stretchy="false">|</n:mo> <n:mtext>LCSRs</n:mtext> </n:msub> <n:mo stretchy="false">)</n:mo> </n:mrow> <n:mo>×</n:mo> <n:msup> <n:mn>10</n:mn> <n:mrow> <n:mo>−</n:mo> <n:mn>3</n:mn> </n:mrow> </n:msup> </n:math> within the dominant region of the <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi>ρ</v:mi> </v:math> resonance. We clarify that the relatively small value of <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mo stretchy="false">|</x:mo> <x:msub> <x:mi>V</x:mi> <x:mrow> <x:mi>u</x:mi> <x:mi>b</x:mi> </x:mrow> </x:msub> <x:mo stretchy="false">|</x:mo> </x:math> extracted from <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:mi>B</bb:mi> <bb:mo stretchy="false">→</bb:mo> <bb:mi>ρ</bb:mi> <bb:mi>l</bb:mi> <bb:mi>ν</bb:mi> </bb:math> , when compared to the golden channel <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>B</eb:mi> <eb:mo stretchy="false">→</eb:mo> <eb:mi>π</eb:mi> <eb:mi>l</eb:mi> <eb:mi>ν</eb:mi> </eb:math> , arises due to the finite width and nonresonant effects. We anticipate future measurements of purely isovector and isoscalar <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:msub> <hb:mi>B</hb:mi> <hb:mrow> <hb:mi>l</hb:mi> <hb:mn>4</hb:mn> </hb:mrow> </hb:msub> </hb:math> decays, which will aid in reducing uncertainties by enhancing our understanding of the <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mi>π</jb:mi> <jb:mi>π</jb:mi> </jb:math> phase shifts in the two-pion distribution amplitudes through heavy flavor decays.