Annihilation effects in $B\to \pi \pi$ from QCD light-cone sum rules

Using the method of QCD light-cone sum rules, we calculate the $B\to \pi \pi$ hadronic matrix elements with annihilation topology. We obtain a finite result, including the related strong phase. Numerically, the annihilation effects in $B\to \pi \pi$ turn out to be small with respect to the factorizable emission mechanism. Our predictions, together with the earlier sum rule estimates of emission and penguin contributions, are used for the phenomenological analysis of $B\to \pi \pi$ channels. We predict a $\Delta I=1/2$ transition amplitude which significantly differs from this amplitude extracted from the current data.

Figure 1

Different quark topologies in ${\overline{B}}^0\to {\pi }^{+}{\pi }^{-}$; double lines denote the $b$ quark, wavy lines the $W$ boson.

Figure 2

Diagrams corresponding to the emission topology in the OPE of the correlation function (35): (a) factorizable; (b) with nonfactorizable hard gluon (six diagrams); (c) nonfactorizable soft gluon (two diagrams). The solid, double, dashed, wavy lines and the square denote the light quarks, $b$ quark, gluon, external currents, and the weak vertex, respectively. The shaded ovals denote the pion DA’s. The crosses indicate how gluon lines are attached in the other possible diagrams.

Figure 3

Examples of diagrams corresponding to the penguin topology: with (a) hard gluon and (b) soft gluons.

Figure 4

Diagrams corresponding to the annihilation topology in the OPE of the correlation function (35): (a) factorizable; (b), (c) with hard gluon; (d), (e) with soft gluon.

Figure 5

Some of the lowest-order diagrams corresponding to the penguin-annihilation topology.

Figure 6

The first step in the derivation of the sum rule for $B\to \pi \pi$ amplitude in the emission topology: (a) the initial correlation function (only diagrams with emission topology are included) is matched to (b) the hadronic dispersion relation in the pion channel where only the ground-state pion contribution is shown.

Figure 7

The second step in the derivation of the sum rule: (a) the two-pion matrix element calculated in the spacelike region is analytically continued to (b) the same matrix element in the timelike region.

Figure 8

The third step in the derivation of the sum rule: (a) the two-pion matrix element $\prod _{\pi \pi }^{(O,E)}$ after analytic continuation to $(p-q-k{)}^2=m_B^2$ is matched to (b) the hadronic dispersion relation in the $B$-meson channel; the ground-state $B$-meson contribution is proportional to the $B\to \pi \pi$ matrix element in emission topology.

Figure 9

Diagrams used to calculate the pion-pion correlator in the annihilation topology.

Figure 10

Diagrams of the correlation function used to derive LCSR for the pion scalar form factor; $j_0=\overline{d}d$ is the scalar quark current.