A perturbative approach to $B$ decays into two $π$ mesons

TL;DR

This work applies the Botts-Sterman modified perturbative approach to $B$ decays into two pions, explicitly including transverse momenta and Sudakov resummation to regulate end-point regions. It derives a factorized decay amplitude with a leading-order hard kernel $T_H$, meson wave functions, and a Sudakov factor $S$, and analyzes short-distance corrections via the effective Hamiltonian with Wilson coefficients $C_i(\mu)$. Numerically, the predicted branching ratios for $\bar{B}^0\to\pi^+\pi^-$ and related channels are of order $10^{-6}$ to $10^{-8}$ with sizable but controlled uncertainties, and the perturbative contributions are below current experimental limits; the authors also compute $B\to\pi$ transition form factors and compare with other approaches, noting notable normalization differences. The study concludes that the modified perturbative framework yields self-consistent, largely perturbative estimates for these decays, albeit with significant sensitivity to CKM parameters and meson distribution amplitudes, and suggests that soft dynamics may still play an important role in $B\to\pi\pi$ decays. The methodology, including explicit convolution formulas and Sudakov treatment, provides a framework for evaluating other light-m meson final states in exclusive $B$ decays.

Abstract

The modified perturbative approach in which transverse degrees of freedom as well as Sudakov suppressions are taken into account, is applied to $B$ decays into two $π$ mesons. The influence of various model parameters (CKM matrix elements, $B$ decay constant, mesonic wave functions) on the results as well as short distance corrections to the weak Hamiltonian are discussed in some detail. The perturbative contributions to the $B$ decays yield branching ratios of the order of $10^{-7}\;-\;10^{-6}$ which values are well below the upper limit for the $\bar{B}^0\toπ^+π^-$ branching ratio as measured by CLEO.

Figures (6)

• Figure 1: The basic graph for the decay of a $B$ meson into two pions. The circle stands for the effective weak Hamiltonian. The quark momenta are specified.
• Figure 2: Lowest order Feynman graphs for $B$ decays into two pions. The internal quark and gluon momenta are indicated.
• Figure 3: Saturation of the perturbative contribution: $\Omega_b$ vs. $\alpha_c$ (see text). Solid (dashed) line represents the perturbative contribution with (without) the Sudakov factor.
• Figure 4: The annihilation topology.
• Figure 5: The $B$-$\pi$ transition form factors vs. $\eta$. The solid (dashed) and dash-dotted (dotted) lines represent the results for the form factor $F_+$ ($F_{-}$) obtained with the AS and CZ wave functions respectively.