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We study $B\to\pi\pi$ form factors using QCD light-cone sum rules with$B$-meson distribution amplitudes. These form factors describe the semileptonicdecay $B\to \pi\pi \ell\bar{\nu}_{\ell}$, and constitute an essential input in$B\to \pi\pi \ell^+\ell^-$ and $B\to \pi\pi\pi$ decays. We employ thecorrelation functions where a dipion isospin-one state is interpolated by thevector light-quark current. We obtain sum rules where convolutions of the$P$-wave $\bar{B}^0\to \pi^+\pi^0$ form factors with the time-like pion vectorform factor are related to universal $B$-meson distribution amplitudes. Thesesum rules are valid in the kinematic regime where the dipion state has a largeenergy and a low invariant mass, and reproduce analytically the knownlight-cone sum rules for $B\to \rho$ form factors in the limit of$\rho$-dominance with zero width, thus providing a systematics forso-far-unaccounted corrections to $B\to\rho$ transitions. Using data for thepion vector form factor, we estimate finite width-effects and the contributionof excited $\rho$-resonances to the $B\to\pi\pi$ form factors. We find thatthese contributions amount up to $\sim 20\%$ in the small dipion mass regionwhere they can be effectively regarded as a nonresonant ($P$-wave) backgroundto the $B\to\rho$ transition.

$B\to\pi\pi$ Form Factors from Light-Cone Sum Rules with $B$-meson Distribution Amplitudes