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拓扑振幅的自洽框架及其$SU(N)$分解

A self-consistent framework of topological amplitude and its $SU(N)$ decomposition

Di Wang, Cai-Ping Jia, Fu-Sheng Yu

备注：
62 pages, 3 figures, version published on JHEP
发表信息：
J. High Energ. Phys. 2021, 126 (2021)
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摘要：

我们提出了一个系统的重介子衰变拓扑振幅及其$SU(N)$分解的理论框架。在这个框架中，拓扑振幅被表示为不变张量，并根据树和企鹅四夸克算子，树或企鹅，被插入到其有效的弱顶点，分为树和企鹅算子诱导图。通过将四夸克算符分解为$SU(N)$群的不可约表示，可以从拓扑的张量形式导出$SU(N)$的不可约振幅.以具有$SU(3)_F$对称性的$D\to PP$衰变（$P$表示赝标介子）为例，给出了详细的计算框架.用群论解释了某些拓扑在$SU(3)_F$极限中不独立的事实。结果发现，只有九个独立的拓扑结构，在所有的树和企鹅运营商诱导图的$D\to PP$衰变的标准模型。如果假设一个大的夸克环图，那么大的$ΔA_{CP}$和非常不同的$D^0\to K^+K^-$和$D^0\to π^+π^-$分支分数可以用正常的$U$自旋破缺来解释。此外，我们的框架提供了一个简单而系统的方法来分析的$SU(N)$打破效果。作为例子，我们在我们的框架下重新研究了粲衰变中的线性$SU(3)_F$破缺和高阶$U$ -自旋破缺，结果与文献一致。提出了拓扑分裂和拓扑退化的概念，并用它们来描述无魅力底衰变。我们发现对无粲$B$衰变的$SU(3)_F$分析不同于对$D$衰变的分析，这是因为粲夸克环超出了$SU(3)$对称性，应该在$SU(4)\to SU(3)$的对称性破缺链中进行研究.

英文摘要：

We propose a systematic theoretical framework for the topological amplitudes of the heavy meson decays and their $SU(N)$ decomposition. In the framework, the topological amplitudes are expressed in invariant tensors and classified into tree- and penguin-operator-induced diagrams according to which four-quark operators, tree or penguin, being inserted into their effective weak vertexes. By decomposing the four-quark operators into irreducible representations of $SU(N)$ group, one can derive the $SU(N)$ irreducible amplitudes from the tensor form of the topology. Taking the $D\to PP$ decay ($P$ denoting a pseudoscalar meson) with $SU(3)_F$ symmetry as an example, we show our framework in detail. The fact that some topologies are not independent in the $SU(3)_F$ limit is explained by group theory. It is found that there are only nine independent topologies in all tree- and penguin-operator-induced diagrams contributing to the $D\to PP$ decays in the Standard Model. If a large quark-loop diagram is assumed, the large $ΔA_{CP}$ and the very different $D^0\to K^+K^-$ and $D^0\to π^+π^-$ branching fractions can be explained with a normal $U$-spin breaking. Moreover, our framework provides a simple and systematic way to analyze the $SU(N)$ breaking effects. As examples, the linear $SU(3)_F$ breaking and the high order $U$-spin breaking in charm decays are re-investigated in our framework, which are consistent with literature. We propose the concepts of splitting and degeneracy of topologies, and use them to describe the charm-less bottom decay. We find $SU(3)_F$ analysis for the charm-less $B$ decays is different from the $D$ decays because the charm-quark loop is beyond the $SU(3)$ symmetry and should be investigated in the symmetry breaking chain of $SU(4)\to SU(3)$.

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