Precise theoretical prediction on branching fractions and polarizations of $D \to V V$ decays

Abstract: We present a precise and systematic analysis of $D \to V V$ decays within the factorization-assisted topological-amplitude (FAT) approach, where $D$ denotes the set ${D^0, \, D^+,\, D<sup>+_s}$ and $V$ represents the vector mesons $ρ, K^*, ω$, and $φ$. Given the limited current experimental data, the FAT approach serves as a available phenomenological framework for predicting charmed meson decays to both vector mesons. In this framework, incorporating flavor SU(3) symmetry breaking effects, we can express nonfactorizable contributions of different modes as a minimal set of universal parameters globally fitted to experimental data. Utilizing 36 experimental data points for $D \to VV$ decays, we precisely extract 10 nonfactorizable parameters associated with the $C$ and $E$ topological diagrams with $χ<sup>2/\mathrm{d.o.f.}=8.43$. We find that a large strong phase in the longitude $E$ amplitude cause strong destructive interference with the $C$ longitudinal component, yielding $f_\parallel &gt;f_L $, contrary to the naive factorization predictions. Additionally, for modes processing exclusively by the $E$ diagram, the amplitude hierarchy $|S|&lt;|D|$ leads to a $D$-wave branching fraction larger than that of the $S$-wave. This explains recent observations that contradict $S$-wave dominance predictions. The predicted branching fractions and polarizations for 28 decay modes are consistent with existing experimental data. Unobserved modes, especially those with branching fractions of order $10<sup>{-3}\sim10<sup>{-2}$, the $D$-wave dominated modes, and modes exhibiting $f_\parallel &gt;f_L $, await measurement by BESIII, STCF, Belle II and LHCb.

• The paper demonstrates that the FAT approach yields precise predictions for D→VV decay branching fractions and polarizations by incorporating universal nonfactorizable parameters.
• Methodologically, it analyzes decay amplitudes via a minimal parameter set, explicitly accounts for SU(3) breaking, and fits 36 experimental measurements.
• Key results include revealing nontrivial polarization hierarchies and inverted partial wave structures, challenging conventional naive factorization assumptions.

Precise Predictions of Branching Fractions and Polarizations in $D \to VV$ Decays via the FAT Approach

Introduction

The theoretical description of hadronic charm meson decays, particularly $D \to VV$ ($V$ denotes a vector meson), represents a stringent test of the interplay between weak dynamics, nonperturbative QCD effects, and flavor SU(3) breaking. This study delivers a systematic and precision analysis of $D \to VV$ decay amplitudes, branching fractions, and polarization observables within the Factorization-Assisted Topological amplitude (FAT) framework. The analysis confronts the limitations of naive factorization and conventional topological amplitude methods by extracting a minimal set of universal nonfactorizable parameters that effectively encode strong interaction effects, with explicit SU(3) breaking implemented in all relevant hadronic objects.

Theoretical Framework: The FAT Approach

The FAT approach parameterizes hadronic decay amplitudes in terms of topological diagrams ($T$, $C$, $E$, $A$), corresponding to color-favored tree, color-suppressed tree, $W$-exchange, and $W$-annihilation contributions, respectively. Unlike conventional diagrammatic analyses reliant on SU(3) symmetry and large numbers of free parameters, the FAT method systematically factors out all process-dependent hadronic matrix elements—decay constants and transition form factors—and attributes SU(3) breaking deterministically. The residual nonfactorizable contributions for each topology and polarization channel are constrained to be universal parameters, globally fitted to available data.

The $D \to VV$0 amplitudes are retained in a factorizable form with a single scale-dependent effective Wilson coefficient $D \to VV$1, where $D \to VV$2 is the only free parameter for all color-favored tree amplitudes. For $D \to VV$3 and $D \to VV$4, the analysis introduces complex-valued universal parameters ($D \to VV$5 for magnitudes and $D \to VV$6 for strong phases). The $D \to VV$7 contributions are neglected as they are found to be statistically insignificant for the measured data set.

A central feature is the handling of polarization structure. All amplitudes are computed for longitudinal ($D \to VV$8), parallel, and perpendicular polarization states, and translated into transversity ($D \to VV$9) and partial wave ($V$0, $V$1, $V$2) bases for comparison with experimental measurements.

Data Inputs, Parameter Extraction, and Fit Quality

The analysis employs 36 experimental measurements on $V$3 branching fractions and partial wave components, including full and partial rates for numerous Cabibbo-favored, singly Cabibbo-suppressed, and doubly Cabibbo-suppressed channels. All decay constants and masses are taken from the latest Particle Data Group evaluation, and $V$4 form factors are assigned using the parametrization of Wirbel, Stech, and Bauer, supplemented with a conservative uncertainty budget.

A comprehensive fit yields 11 independent parameters: one factorization scale $V$5, and real and imaginary parts (modulus and strong phase) for the $V$6 and $V$7 topologies (with $V$8 and all $V$9 amplitudes neglected due to their smallness). The resulting fit achieves $D \to VV$0, and the nonfactorizable parameters are determined at high precision (excepting one poorly constrained $D \to VV$1 strong phase).

Amplitude Hierarchies and Polarization Patterns

The fit reveals several nontrivial amplitude and polarization hierarchies:

• Nonfactorizable Color-Suppressed Enhancement: $D \to VV$2. The extracted magnitude for the $D \to VV$3 amplitude is found to be comparable to or even larger than the factorizable $D \to VV$4 amplitude in the longitudinal channel. This underscores the significant role of nonfactorizable effects, which are not properly treated in naive or pure factorization models.
• Polarization Anomalies: The hierarchy $D \to VV$5 holds for $D \to VV$6 and $D \to VV$7 topologies, in direct contradiction to naive factorization, which would expect $D \to VV$8. This indicates that the FAT method is essential to explain the observed polarization phenomena.
• Partial Wave Structure: In modes with significant $D \to VV$9 contributions, particularly those with $T$0–$T$1 and $T$2–$T$3 interference, the traditionally expected $T$4 pattern is inverted, yielding $T$5. This provides a natural explanation of experimental $T$6-wave dominance seen in modes such as $T$7.
• Strong Phase Interference: Large strong phase differences, especially in the $T$8 amplitude, drive the observed destructive and constructive interferences crucial for reproducing both branching fraction and polarization data.

Predictions for Branching Fractions and Polarizations

The framework delivers precise predictions for 28 $T$9 decay modes, including full and partial wave branching fractions and longitudinal/transverse polarization fractions. The results for total rates are systematically consistent with experiment (within uncertainties), except for two $C$0-wave modes dominated by the omitted $C$1 amplitude. Notable features include:

• For decays mediated by only $C$2 or $C$3 ($C$4), $C$5 is always dominant, in line with theoretical expectations for tree-driven modes.
• Modes with $C$6–$C$7 and $C$8–$C$9 interference may exhibit $E$0, a property not reproducible by naive or broken SU(3) diagrammatic approaches.
• For $E$1, the predicted $E$2 matches experimental observations ($E$3).
• In modes governed by the $E$4 amplitude, destructive interference, arising from large strong phase differences, suppresses the $E$5-wave, pushing the $E$6-wave partial width above the $E$7-wave, and elevates $E$8 above $E$9.

Channels with predicted large $A$0-wave fractions, as well as unmeasured branching ratios in the $A$1–$A$2 range and modes with $A$3, are identified as targets for future measurement by BESIII, STCF, Belle II, and LHCb.

Implications and Outlook

The FAT formalism applied here delivers a precision approach for quantitatively connecting theory and experiment in the $A$4 sector. The findings underscore the necessity of including both universal nonfactorizable parameters and explicit SU(3) breaking in describing charm hadronic decays. The evidence for sizable nonfactorizable color-suppressed amplitudes, nontrivial strong phase structure, and inverted partial wave hierarchies now places robust demands on future QCD-based or lattice calculations to connect the extracted parameters with fundamental strong interaction dynamics.

Future improvements will hinge on enhanced experimental precision (particularly for partial wave and polarization measurements in poorly constrained modes) and on extending the framework to accommodate potential subleading topologies as more data become available. High-statistics data samples at next-generation flavor experiments will further enable scrutiny of small amplitude effects and the validation of the underlying universality assumptions of the FAT approach.

Conclusion

This work provides a comprehensive, precision analysis of the $A$5 decay sector incorporating both factorizable and nonfactorizable QCD dynamics with rigorous treatment of SU(3) breaking. The FAT formalism, with a minimal and universal parameter set, successfully accounts for observed branching ratios and polarization patterns, including features that contradict naive theoretical expectations. The results highlight the critical role of strong phase interference and nonfactorizable effects in charm decays and set a new standard for phenomenological analysis in this sector. Upcoming experimental measurements will further test the robustness and universality of the proposed framework and sharpen the theoretical understanding of nonleptonic weak decays in the charm sector.

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Reference: "Precise theoretical prediction on branching fractions and polarizations of $A$6 decays" (2604.01008).