Power Corrections in Charmless Nonleptonic B-Decays: Annihilation is Factorizable and Real

TL;DR

This paper uses soft-collinear effective theory (SCET) to classify LambdaQCD/mb power corrections in charmless nonleptonic B decays and to show that leading annihilation amplitudes are real, with complex strong phases arising only at higher orders. It provides a complete factorization framework for both local and chirally enhanced annihilation, including a full basis of SCET$_{II}$ operators and a real, convergent convolution structure with twist-2 and twist-3 meson distributions. The authors show that nonperturbative strong phases are suppressed unless the intermediate scale is poorly behaved, and they quantify annihilation’s small but non-negligible role in penguin amplitudes for modes like B→Kπ and B→KK. Simple numerical modeling within this framework yields annihilation contributions at the 10–15% level of the penguin amplitude, consistent with power-counting expectations and offering constraints on prior phenomenological approaches. Overall, the work provides a rigorous, scale-separated description of annihilation effects and their weak-phase content, aiding interpretation of CP asymmetries and guiding future fits to data.

Abstract

We classify LambdaQCD/mb power corrections to nonleptonic B-> M1 M2 decays, where M1 and M2 are charmless non-isosinglet mesons. Using recent developments in soft-collinear effective theory, we prove that the leading contributions to annihilation amplitudes of O[alphas(mb) LambdaQCD/mb] are real. The leading annihilation amplitudes depend on twist-2 and twist-3 three parton distributions. A complex nonperturbative parameter from annihilation first appears at O[alphas^2(sqrt{Lambda mb}) LambdaQCD/mb]. ``Chirally enhanced'' contributions are also factorizable and real at lowest order. Thus, incalculable strong phases are suppressed in annihilation amplitudes, unless the alphas(sqrt{Lambda mb}) expansion breaks down. Modeling the distribution functions, we find that (11 +- 9)% and (15 +- 11)% of the absolute value of the measured B-> K- pi+ and B-> K- K0 penguin amplitudes come from annihilation. This is consistent with the expected size of power corrections.

Figures (3)

• Figure 1: Three types of factorization contributions to annihilation amplitudes which are the same order in $\eta=\Lambda_{\rm QCD}/m_b$. a) shows $Q_i^{(4)}$ which has $\ge 1$ hard gluon and factorizes at the scale $m_b$. The rapidity parameter, $\zeta= p^-/p^+$, controls the MS-factorization between soft momenta ($B$), $n$-collinear momenta ($M_2$), and ${\bar{n}}$-collinear momenta ($M_1$). b) shows the time-ordered product $Q_i^{(2)}{\cal L}^{(1)}_{\xi q}$, which involves factorization at $m_b$ and $\sqrt{m_b\Lambda}$. c) shows the time-ordered product $Q_i^{(1)} [{\cal L}^{(1)}_{\xi q}]^3$, which factorizes at the scale $\sqrt{m_b\Lambda}$ and does not need a hard gluon. Graphs a) and b) are of order $\alpha_s(\mu_h)$, while c) is $\alpha_s(\mu_i)^2$.
• Figure 2: Tree level annihilation graphs for $B\to M_1 M_2$ decays. Here soft, $n$, ${\bar{n}}$ denote quarks that are soft, $n$-collinear, and ${\bar{n}}$-collinear respectively.
• Figure 3: Graphs which generate a strong phase in lowest order matching of ${\rm SCET}_{\rm I}$ operators onto ${\rm SCET}_{\rm II}$: a) has a $Q^{(1)}$, two ${\cal L}_{\xi_{n_1} q}^{(1)}$, and one ${\cal L}_{\xi_{n_2} q}^{(1)}$ and contributes to the annihilation amplitude at ${\cal O}(\alpha_s^2(\mu_i))$; and b) has a $Q^{(1)}$, one ${\cal L}_{\xi_{n_1} q}^{(1)}$, and one ${\cal L}_{\xi_{n_2} \xi_{n_2}}^{(2)}$ and contributes to non-annihilation amplitudes at ${\cal O}(\alpha_s(\mu_i))$. Dashed quark lines are $n_1$ or $n_2$ collinear, and solid quark lines are soft.