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The form factors of the semileptonic $B\to \pi\pi\ell\bar\nu$ decay arecalculated from QCD light-cone sum rules with the distribution amplitudes ofdipion states. This method is valid in the kinematical region, where thehadronic dipion state has a small invariant mass and simultaneously a largerecoil. The derivation of the sum rules is complicated by the presence of anadditional variable related to the angle between the two pions. In particular,we realize that not all invariant amplitudes in the underlying correlationfunction can be used, some of them generating kinematical singularities in thedispersion relation. The two sum rules that are free from these ambiguities areobtained in the leading twist-2 approximation, predicting the $\bar{B}^0\to\pi^+\pi^0$ form factors $F_{\perp}$ and $F_{\parallel}$ of the vector andaxial $b\to u$ current, respectively. We calculate these form factors at themomentum transfers $0<q^2\lesssim 12 $ GeV$^2$ and at the dipion mass close tothe threshold $4m_\pi^2$. The sum rule results indicate that the contributionsof the higher partial waves to the form factors are suppressed with respect tothe lowest $P$-wave contribution and that the latter is not completelysaturated by the $\rho$-meson term.

Form factors in $B\to \pi\pi\ell\bar\nu_\ell$ from QCD light-cone sum rules