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

Abstract

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 $\Delta A_{CP}$ and the very different $D^0\to K^{+}{K}^{-}$ and $D^0\to {\pi }^{+}{\pi }^{-}$ 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 $SU(4)\to SU(3)$.

arXiv comment62 pages, 3 figures, version published on JHEP

References in corpus (85)

• Weak Decays Beyond Leading Logarithms
• Soft-Collinear Factorization in Effective Field Theory
• QCD Factorization for $B\to \pi \pi$ Decays: Strong Phases and CP Violation in the Heavy Quark Limit
• QCD factorization for exclusive, non-leptonic B meson decays: General arguments and the case of heavy-light final states
• QCD factorization for B->PP and B->PV decays
• QCD Factorization in B -> pi K, pi pi Decays and Extraction of Wolfenstein Parameters
• Fat penguins and imaginary penguins in perturbative QCD
• Decays of $B$ Mesons to Two Light Pseudoscalars
• A proof of factorization for B -> D pi
• Weak Coupling Phase from Decays of Charged B Mesons to $\pi K$ and $\pi \pi$
• Broken SU(3) Symmetry in Two-Body B Decays
• Branching ratios and direct CP asymmetries in $D\to PP$ decays
• Nonleptonic charmless B decays: factorization vs perturbative QCD
• Discovery Potentials of Doubly Charmed Baryons
• $B\to \pi \rho$, $\pi \omega$ Decays in Perturbative QCD Approach
• Two-body hadronic charmed meson decays
• Weak Decays of Doubly Heavy Baryons: SU(3) Analysis
• Test Flavor SU(3) symmetry in Exclusive ${\Lambda }_c$ decays
• CP Violation and Flavor SU(3) Breaking in D-meson Decays
• A consistent picture for large penguins in D -> pi+ pi-, K+ K-
• Weak Decays of Doubly Heavy Baryons: Multi-body Decay Channels
• SU(3) symmetry breaking and CP violation in D -> PP decays
• SU(3) Sum Rules for Charm Decay
• Updating B --> PP, VP decays in the framework of flavor symmetry
• SU(3)-Flavor Anatomy of Non-Leptonic Charm Decays
• Revisiting CP violation in $D\to PP$ and $VP$ decays
• Anti-triplet charmed baryon decays with SU(3) Flavor Symmetry
• Topological amplitudes in $D$ decays to two pseudoscalars: a global analysis with linear $SU(3{)}_F$ breaking
• Repercussions of Flavour Symmetry Breaking on CP Violation in D-Meson Decays
• Singly Cabibbo-suppressed hadronic decays of ${\Lambda }_c^{+}$
• The Emergence of the $\Delta U=0$ Rule in Charm Physics
• Branching ratios and direct CP asymmetries in $D\to PV$ decays
• Charmed Baryon Weak Decays with SU(3) Flavor Symmetry
• D-meson lifetimes within the heavy quark expansion
• Analysis of Two-Body Decays of Charmed Baryons Using the Quark-Diagram Scheme
• Asymmetries of anti-triplet charmed baryon decays
• Non-leptonic two-body weak decays of ${\Lambda }_c(2286)$
• SU(3) Decomposition of Two-Body B Decay Amplitudes
• On $SU(3{)}_F$ Breaking through Final State Interactions and CP Asymmetries in $D\to PP$ Decays
• Semileptonic decays of anti-triplet charmed baryons
• A global analysis of two-body D to VP decays within the framework of flavor symmetry
• $\Delta A_{CP}$ within the Standard Model and beyond
• Analysis of Charmless Two-body B decays in Factorization Assisted Topological Amplitude Approach
• Charmed meson decays to two pseudoscalars
• Three-body charmed baryon Decays with SU(3) flavor symmetry
• U-spin breaking in CP asymmetries in B decays
• A diagrammatic analysis of two-body charmed baryon decays with flavor symmetry
• Sum Rules of Charm CP Asymmetries beyond the SU(3)${}_F$ Limit
• Analysis of Two-body Charmed $B$ Meson Decays in Factorization-Assisted Topological-Amplitude Approach
• Possible resolution of the B -> pi pi, pi K puzzles
• A global $SU(3)/U(3)$ flavor symmetry analysis for $B\to PP$ with $\eta -{\eta }^{\prime }$ Mixing
• Implications of the LHCb discovery of CP violation in charm decays
• Singly Cabibbo suppressed decays of ${\Lambda }_c^{+}$ with SU(3) flavor symmetry
• SU(3) symmetry breaking in charmed baryon decays
• Flavor SU(3) Topological Diagram and Irreducible Representation Amplitudes for Heavy Meson Charmless Hadronic Decays: Mismatch and Equivalence
• Nonleptonic Two-Body Decays of D Mesons in Broken SU(3)
• Weak Decays of Triply Heavy Baryons
• $K_S^0-K_L^0$ Asymmetries and $CP$ Violation in Charmed Baryon Decays into Neutral Kaons
• Decays of Charmed Mesons to PV Final States
• Charmless $B_{(s)}\to VV$ Decays in Factorization-Assisted Topological-Amplitude Approach
• Flavor SU(3) symmetry and QCD factorization in $B\to PP$ and $PV$ decays
• Weak Decays of Stable Doubly Heavy Tetraquark States
• Flavor symmetry and decays of charmed mesons to pairs of light pseudoscalars
• Sum rules for $CP$ asymmetries of charmed baryon decays in the $SU(3{)}_F$ limit
• U-Spin Sum Rules for CP Asymmetries of Three-Body Charmed Baryon Decays
• Charmed Baryon Weak Decays with Vector Mesons
• Charmed baryon decays in $SU(3{)}_F$ symmetry
• $CP$ asymmetries in charm decays into neutral kaons
• Glauber gluons in spectator amplitudes for $B\to \pi M$ decays
• Measuring gamma with B -> K pi pi and B -> K K Kbar Decays
• Up-down asymmetries of charmed baryon three-body decays
• General Analysis of U-Spin Breaking in B Decays
• Extraction of the CP-violating phase $\gamma$ using $B\to K\pi \pi$ and $B\to KK\bar{K}$ decays
• Fragmentation-fraction ratio $f_{{\Xi }_b}/f_{{\Lambda }_b}$ in $b$- and $c$-baryon decays
• $K_S^0-K_L^0$ asymmetries in $D$-meson decays
• Addendum to "High order U-spin breaking: A precise amplitude relation in D0 decays"
• SU(3)${}_F$ Analysis for Beauty Baryon Decays
• Non-leptonic beauty baryon decays and $CP$-asymmetries based on $SU(3)$-Flavor analysis
• Weak Decays of Stable Open-bottom Tetraquark by SU(3) Symmetry Analysis
• Weak Decays of Triply Heavy Tetraquarks $b\bar{c}b\bar{q}$
• SU(3) in D decays: From $30\%$ symmetry breaking to $10^{-4}$ precision
• Weak Decays of Doubly-Heavy Tetraquarks $b\bar{c}q\bar{q}$
• Beauty baryon non-leptonic decays into decuplet baryons and $CP$-asymmetries based on $SU(3)$-Flavor analysis
• Extraction of the CKM phase $\gamma$ using charmless 3-body decays of $B$ mesons
• Glauber gluons in annihilation amplitudes for heavy meson decays

Cited by in corpus (10)

• Search for time-dependent $CP$ violation in $D^0\to K^{+}{K}^{-}$ and $D^0\to {\pi }^{+}{\pi }^{-}$ decays
• Topological amplitudes of charmed baryon decays in the $SU(3{)}_F$ limit
• From topological amplitude to rescattering dynamics in doubly charmed baryon decays
• Topological amplitudes of bottom baryon decays in the $SU(3{)}_F$ limit
• Isospin sum rules for the nonleptonic $B$ decays
• Study of CP Violation in $D^{\pm }\to K^{\ast }(892{)}^0{\pi }^{\pm }+{\bar{K}}^{\ast }(892{)}^0{\pi }^{\pm }\to K_{S,L}^0{\pi }^0{\pi }^{\pm }$ Decays
• A Review of $CP$ Violation Measurements in Charm at LHCb
• Topological diagrams of doubly charmed baryon decays in the $SU(3{)}_F$ limit
• $U$-spin sum rules for two-body decays of bottom baryons
• Estimating QCD-factorization amplitudes through $SU(3)$ symmetry in $B\to PP$ decays