[Paper Review] SU(3) Analysis of Annihilation Contributions and CP Violating Relations in $B o PP$ Decays

Xiao-Gang He|arXiv (Cornell University)|Oct 19, 1998

Particle physics theoretical and experimental studies19 citations

TL;DR

This paper uses SU(3) flavor symmetry to test the smallness of tree-level annihilation contributions in $B \to PP$ decays, particularly in $B^{-} \to \pi^{-}\bar{K}^{0}$, and derives model-independent CP-violating rate difference relations without assuming small annihilation effects. The key contribution is a set of SU(3)-symmetric relations that provide robust, dynamic model-independent tests for the Standard Model of CP violation and the validity of $\gamma$-angle measurement methods.

ABSTRACT

Several methods proposed to measure the angle $γ$ in the KM unitarity triangle assumed that the tree contribution to $B^- o π^-\bar K^0 $ is purely due to annihilation contributions and is negligibly small. This assumption has to be tested in order to have a correct interpretation of the experimental data. In this paper we show that using SU(3) symmetry, the smallness of the tree contribution can be tested in a dynamic model independent way. We also derive several relations between CP violating rate differences for $B o P P$ decays without assuming the smallness of the annihilation contributions. These relations provide important tests for the Standard Model of CP violation.

Motivation & Objective

• To test the assumption that tree-level amplitudes in $B^{-} \to \pi^{-}\bar{K}^{0}$ are dominated by annihilation contributions and are negligibly small, as required by several $\gamma$-angle measurement methods.
• To provide a dynamic model-independent method to test the smallness of annihilation contributions using SU(3) symmetry.
• To derive CP-violating rate difference relations for $B \to PP$ decays without assuming small annihilation contributions, enabling new tests of the Standard Model.
• To assess the validity of SU(3) symmetry in $B$ decays by comparing predictions with factorization estimates and experimental data, particularly for $B \to D\pi$ and $B \to DK$ decays.

Proposed method

• Utilizes SU(3) flavor symmetry to express $B \to PP$ decay amplitudes in terms of a small set of invariant amplitudes, decomposed into irreducible representations ($\bar{3}$, $6$, $\overline{15}$).
• Applies the effective Hamiltonian formalism with Wilson coefficients and matrix elements, separating tree ($T(q)$) and penguin ($P(q)$) contributions.
• Derives relations between CP-violating rate differences by exploiting SU(3) symmetry, assuming only flavor symmetry and no assumptions on annihilation size.
• Uses factorization approximation as a guide to estimate SU(3) breaking effects, though emphasizing that results may still be closer to the SU(3) limit than estimated.
• Compares theoretical SU(3)-symmetric predictions with experimental branching ratios (e.g., $Br(B^{-} \to D^{0}K^{-})/Br(B^{-} \to D^{0}\pi^{-})$) to test the robustness of SU(3) relations.
• Analyzes the $I=3/2$ tree amplitude relation between $B^{-} \to \pi^{-}\pi^{0}$ and $B^{-} \to \pi^{0}K^{-}$, showing that experimental values may align better with SU(3) limits than factorization estimates.

Experimental results

Research questions

• RQ1 Can the smallness of the tree amplitude in $B^{-} \to \pi^{-}\bar{K}^{0}$ be tested independently of dynamic models using SU(3) symmetry?
• RQ2 Are the CP-violating rate difference relations between $B \to PP$ decays valid without assuming small annihilation contributions?
• RQ3 How do SU(3) breaking effects, estimated via factorization, compare with actual experimental data for $B \to D\pi$ and $B \to DK$ decays?
• RQ4 Can SU(3) symmetry relations provide more accurate predictions than factorization-based estimates for branching ratio ratios?
• RQ5 To what extent do experimental results on $B \to PP$ decays support the SU(3) limit, especially in $\gamma$-angle determination methods?

Key findings

• The paper derives model-independent SU(3) relations for CP-violating rate differences in $B \to PP$ decays, such as $\Delta(\bar{B}^{0} \to \pi^{+}\pi^{-}) \approx -\frac{f_{\pi}^{2}}{f_{K}^{2}}\Delta(\bar{B}^{0} \to \pi^{+}K^{-})$, which are valid regardless of annihilation contribution size.
• The relation $\Delta(B_{s} \to K^{+}K^{-}) \approx -\frac{f_{K}^{2}}{f_{\pi}^{2}}\Delta(B_{s} \to \pi^{-}K^{+})$ is derived, providing a testable prediction for $B_s$ decays.
• The assumption that $A(B^{-} \to \pi^{-}\bar{K}^{0}) = \bar{A}(B^{+} \to K^{0}\pi^{+})$ requires not only small annihilation contributions but also that the tree amplitude is purely annihilation-mediated, a condition that can be tested experimentally.
• The branching ratio of $B^{-} \to K^{-}K^{0}$ is shown to be a sensitive probe for testing the smallness of the tree amplitude in $B^{-} \to \pi^{-}\bar{K}^{0}$, as it is dominated by the same tree amplitude.
• Experimental data on $B^{-} \to D^{0}K^{-}$ and $B^{-} \to D^{0}\pi^{-}$ decays show a ratio closer to the SU(3) limit ($R \approx 0.049$) than to the factorization estimate ($R \approx 0.07$), suggesting SU(3) relations may be more accurate than expected.
• The $I=3/2$ tree amplitude relation $A^{T}_{3/2} \approx (f_{K}^{2}/f_{\pi}^{2})|V_{us}/V_{ud}|^{2}A^{T}$ may be less accurate than the SU(3) limit, implying caution is needed when using such relations to extract $\gamma$.