Renormalization-Group Evolution for the Three-Particle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi></mml:math>-Meson Soft Function

We determine for the first time the renormalization-group equation for the three-particle B-meson soft function dictating the nonperturbative strong interaction dynamics of the long-distance penguin contributions to the double radiative B-meson decays. The distinctive feature of the ultraviolet renormalization of this fundamental soft function consists in the pattern of mixing positive into negative support for an arbitrary initial condition. The exact solution to this integrodifferential evolution equation is then derived with the Laplace transform technique, allowing for the model-independent extraction of the asymptotic behavior at large and small partonic momenta.