B -> M1 M2: Factorization, Charming Penguins, Strong Phases, and Polarization

Abstract

Using the soft-collinear effective theory we derive the factorization theorem for the decays B-> M1 M2 with M{1,2}= pi,K, rho,K*, at leading order in Lambda/E_M and Lambda/mb. The results derived here apply even if alpha_s(E_M Lambda) is not perturbative, and we prove that the physics sensitive to the E*Lambda scale is the same in B-> M1 M2 and B-> M form factors. We argue that c-cbar penguins could give long-distance effects at leading order. Decays to two transversely polarized vector mesons are discussed. Analyzing B-> pi pi we find predictions for B^0 -> pi^0 pi^0 and |V_{ub}| f_+(0) as a function of gamma.

Figures (5)

• Figure 1: Example of long distance charming penguins. The $mv$ gluons are nonperturbative and LO soft gluons are exchanged by the $b$, $c$, $\bar{c}$ and spectator quark which is not shown.
• Figure 2: Example of a long distance light quark penguin which matches onto a power suppressed operator. The $\overline q$ goes in the ${\bar{n}}$ direction, the $q$ goes in the $n$ direction, the broken $u$ quark line is soft or collinear and the $\bar{u}$ and gluon remain hard.
• Figure 3: Factorization of $B\to M M^{\prime}$ in SCET.
• Figure 4: Constraints on the triangle of tree amplitudes $T/T_c - T_n/T_c = 1$ from current world averaged data on $B\to \pi\pi$. The shaded regions show the two $1$-$\sigma$ regions for $\gamma=64^\circ$ including the error correlation between $|t|$ and $|t_n|$. The central values for $\gamma=54^\circ$ and $\gamma=74^\circ$ are also shown.
• Figure 5: Model independent results for $\zeta^{B\pi}$, $\zeta_J^{B\pi}$, and the $B\to \pi$ form factor $f_+(q^2=0)$ as a function of $\gamma$. The shaded bands show the $1$-$\sigma$ errors propagated from the $B\to\pi\pi$ data.