Experimental Tests of Factorization in Charmless Non-Leptonic Two-Body B Decays

TL;DR

This study tests the factorization approach for charmless two-body B decays (B → PP, PV, VV) using a NLO QCD-improved effective Hamiltonian and a factorization ansatz. By classifying decays into five $N_c$-sensitivity classes, it shows that $B^+ o K^+ oldsymbol{ exteta'}$, $B^0 o K^0 oldsymbol{ exteta'}$, $B^0 o K^+oldsymbol{ extpi^-}$, $B^+ o oldsymbol{ extpi^+}K^0$, and similar modes are penguin-dominated (Class-IV) and largely $N_c$-stable, aligning with data, while PV and VV modes fall into Class-V or Class-III with greater sensitivity. It introduces ratios of branching fractions that cancel a_i dependence to test factorization and determine form factors, and proposes measurements to extract the effective coefficients $a_1,a_2,a_4,a_6,a_9$, thereby constraining CKM parameters and CKM phenomenology. The work emphasizes where annihilation and final-state interactions could matter and outlines a program to use non-leptonic B decays to sharpen CKM constraints and test QCD dynamics in hadronic decays.

Abstract

Using a theoretical framework based on the next-to-leading order QCD-improved effective Hamiltonian and a factorization Ansatz for the hadronic matrix elements of the four-quark operators, we reassess branching fractions in two-body non-leptonic decays $B \to PP, PV, VV$, involving the lowest lying light pseudoscalar $(P)$ and vector $(V)$ mesons in the standard model. Using the sensitivity of the decay rates on the effective number of colors, $N_c$, as a criterion of theoretical predictivity, we classify all the current-current (tree) and penguin transitions in five different classes. The recently measured charmless two-body $B \to PP$ decays $(B^+ \to K^+ η^\prime, B^0 \to K^0 η^\prime, B^0 \to K^+π^-, B^+ \to π^+ K^0$ and charge conjugates) are dominated by the $N_c$-stable QCD penguins (class-IV transitions) and their estimates are consistent with data. The measured charmless $B \to PV$ $(B^+ \to ωK^+, ~B^+ \to ωh^+)$ and $B\to VV$ transition $(B \to φK^*)$, on the other hand, belong to the penguin (class-V) and tree (class-III) transitions. The class-V penguin transitions are in general more difficult to predict. We propose a number of tests of the factorization framework in terms of the ratios of branching ratios for some selected $B \to h_1 h_2$ decays involving light hadrons $h_1$ and $h_2$, which depend only moderately on the form factors. We also propose a set of measurements to determine the effective coefficients of the current-current and QCD penguin operators. The potential impact of $B \to h_1 h_2$ decays on the CKM phenomenology is emphasized by analyzing a number of decay rates in the factorization framework.

Figures (5)

• Figure 9: $y_1=\cos\delta_1 \cos \alpha$ as a function of $z_1$ in the factorization approach. The dotted, dashed-dotted and dashed curves correspond to $N_c=\infty$ and $\vert V_{ub}/V_{cb} \vert =0.11$, $N_c=3$ and $\vert V_{ub}/V_{cb} \vert =0.08$ and $N_c=2$ and $\vert V_{ub}/V_{cb} \vert =0.06$, yielding in the BSW model the values $S_1=2.07$, $S_1=0.94$ and $S_1=0.59$, respectively. The two vertical lines indicate the bounds on $z_1$ from our model and the CKM unitarity fits $0.08<z_1<0.50$.
• Figure 10: $y_{12}=\cos\delta_{12} \cos \gamma$ as a function of $z_{12}$ in the factorization approach. The dotted, dashed-dotted and dashed curves correspond to $N_c=\infty$ and $\vert V_{ub}/V_{cb} \vert =0.11$, $N_c=3$ and $\vert V_{ub}/V_{cb} \vert =0.08$, and $N_c=2$ and $\vert V_{ub}/V_{cb} \vert =0.06$, yielding in the BSW model the values $S_{12}=0.46$, $S_{12}=0.91$ and $S_{12}=1.12$, respectively. The two vertical lines indicate the bounds on $z_{12}$ from our model and the CKM factors discussed in the text, yielding $0.15<z_{12}<0.29$.
• Figure 11: $S_{12}$ as a function of $\cos \gamma$ in the factorization approach. The dotted, dashed-dotted and dashed curves correspond to $N_c=\infty$, $N_c=3$ and $N_c=2$, respectively. The horizontal lines are the CLEO $(\pm 1\sigma)$ measurements of $S_{12}$. The two vertical lines correspond to $32^\circ < \gamma < 122^\circ$.
• Figure 12: $y_{13}=\cos\delta_{13} \cos \gamma$ as a function of $z_{13}$ in the factorization approach. The dotted, dashed-dotted and dashed curves correspond to $N_c=\infty$ and $\vert V_{ub}/V_{cb} \vert =0.11$, $N_c=3$ and $\vert V_{ub}/V_{cb} \vert =0.08$, and $N_c=2$ and $\vert V_{ub}/V_{cb} \vert =0.06$, yielding in the BSW model the values $S_{13}=0.49$, $S_{13}=0.95$ and $S_{13}=1.37$, respectively. The two vertical lines indicate the bounds on $z_{13}$ from our model and the CKM factors $0.30<z_{13}<0.60$.
• Figure 13: $S_{13}=S_{15}$ as a function of $\cos \gamma$. The dotted, dashed-dotted and dashed lines correspond to results with $N_c=\infty$, $N_c=3$ and $N_c=2$, respectively. The two vertical lines correspond to $32^\circ < \gamma < 122^\circ$.