Timelike-helicity $B\to \pi \pi$ form factor from light-cone sum rules with dipion distribution amplitudes

We complete the set of QCD light-cone sum rules for $B\to \pi \pi$ transition form factors, deriving a new sum rule for the timelike-helicity form factor $F_t$ in terms of dipion distribution amplitudes. This sum rule, in the leading twist-two approximation, is directly related to the pion vector form factor. Employing a relation between $F_t$ and other $B\to \pi \pi$ form factors, we obtain also the longitudinal-helicity form factor $F_0$. In this way, all four (axial-)vector $B\to \pi \pi$ form factors are predicted from light-cone sum rules with dipion distribution amplitudes. These results are valid for small dipion masses with large momentum.

Figure 1

LCSR predictions for the absolute values of the timelike- and longitudinal-helicity $P$-wave $B\to \pi \pi$ form factors $\sqrt{q^2}{F}_{t,0}^{(\ell =1)}(q^2,k^2=0.1{\mathrm{GeV}}^2)$. Solid (dashed) curves are the central values with the nonasymptotic (asymptotic) dipion DAs. Shaded bands show the estimated theoretical uncertainties.

Figure 2

Absolute value of the timelike-helicity $P$-wave $B\to \pi \pi$ form factor $\sqrt{q^2}{F}_t^{(\ell =1)}(q^2,k^2)$ at $q^2=0$ obtained from LCSR with the asymptotic dipion DA, using the data [10] on the pion form factor.