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CP Violation in Charmed Meson Decays into Final States with $\eta'$

This paper derives Standard Model predictions for CP asymmetries in singly-Cabibbo suppressed $D \to P\eta'$ decays using approximate $SU(3)_F$ symmetry with first-order breaking effects, establishing correlations between specific decay modes to guide future experimental tests and the extraction of key theoretical parameters.

Original authors: Carolina Bolognani, Ulrich Nierste, Stefan Schacht, K. Keri Vos

Published 2026-08-19

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CC BY 4.0

Original authors: Carolina Bolognani, Ulrich Nierste, Stefan Schacht, K. Keri Vos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, particles called quarks are the fundamental building blocks of matter, but they are rarely found alone. They are bound together by a force so strong that it behaves differently than gravity or magnetism, creating a complex environment where particles constantly transform into one another. Sometimes, these transformations are not perfectly symmetrical. Just as a mirror image is not identical to the original object, a particle can decay into a set of new particles in a way that is slightly different from how its antimatter twin decays. This subtle difference is known as CP violation. While scientists have observed this phenomenon in other types of particles, it remains a mystery in the realm of charm quarks. Understanding exactly how and why charm particles break this symmetry is crucial because any deviation from the expected rules could reveal the existence of entirely new forces or particles that have not yet been discovered.

A team of researchers has now turned its attention to a specific and previously unexplored corner of this puzzle: the decay of charm mesons into final states containing a particle called the eta-prime. These decays are rare and difficult to study, but they offer a unique window into the underlying mechanics of the strong force. By using a mathematical framework based on the approximate symmetry of the strong force, the researchers have constructed a detailed map of what should happen if the Standard Model of physics is correct. They did not simply guess at the outcomes; they built a systematic prediction that accounts for the known imperfections in the symmetry of nature. Their work provides a clear set of expectations for future experiments, creating a baseline against which new data can be measured to see if the universe behaves exactly as predicted or if it holds a hidden secret.

The researchers focused on four specific ways these charm particles can decay, involving combinations of pions, kaons, and the eta-prime particle. To make sense of the complex interactions, they treated the decay process as a collection of different pathways, or topologies, that the particles can take as they transform. Some of these pathways are straightforward, while others are more suppressed or involve intricate loops of virtual particles. The team realized that these different pathways are not independent; they are deeply connected by the underlying symmetry of the strong force. This connection means that if one decay happens in a certain way, the others must follow a specific pattern. By analyzing the known rates at which these particles decay, the team was able to predict the size of the CP violation for each of the four channels, even though the experimental data for these specific decays is currently very limited.

One of the most significant findings is the prediction of a precise relationship between the CP violation in charged decays and neutral decays. The researchers found that the asymmetry in the decay of a positively charged charm meson into a pion and an eta-prime is directly linked to the asymmetry in the decay of a positively charged strange charm meson into a kaon and an eta-prime. Similarly, the asymmetries in the neutral decay modes are tied together in a strict ratio. These relationships act as a powerful test: if future measurements from major particle physics experiments like LHCb, Belle II, or BESIII show results that break these patterns, it would be a clear sign that the Standard Model is incomplete. The team's work essentially draws a boundary line; any data falling outside this line would indicate the presence of new physics.

The study also highlights a key parameter that controls the size of these effects, which the researchers describe as the ratio between two types of contributions to the decay amplitude. Currently, the available data is not precise enough to pin down this ratio exactly, but the researchers have shown how future measurements can be used to extract it. They predict that the CP asymmetries in these decays will be very small, likely less than one percent, but measurable with high-precision instruments. The fact that the current experimental data for one of the charged modes aligns with their predictions gives confidence in the method, while the large uncertainties in the other modes leave room for discovery. The researchers emphasize that their approach is robust because it relies on the fundamental symmetries of the strong force rather than on unproven theoretical assumptions.

Ultimately, this paper serves as a guide for the next generation of experiments. It transforms a set of unmeasured decays into a coherent story with specific, testable predictions. By establishing these correlations and providing a framework that includes the known breaking of symmetry, the authors have created a tool that allows the scientific community to distinguish between the expected behavior of the Standard Model and the potential signals of new phenomena. As more data is collected, the tightness of these predictions will either confirm our current understanding of the subatomic world or point the way toward a deeper, more complete theory of how matter behaves at its most fundamental level.

Problem Statement
The paper addresses the theoretical prediction of direct CP asymmetries in singly Cabibbo-suppressed (SCS) decays of charmed mesons into final states containing an $\eta'$ meson ( $D \to P\eta'$ , where $P = K, \pi, \eta$ ). While CP violation in the charm sector has been established, particularly in $D^0 \to \pi^+\pi^-$ , the specific decay channels involving $\eta'$ remain largely unexplored experimentally. Theoretical predictions for these modes are complicated by the presence of the $\eta'$ , which is a flavor singlet in the limit of exact $SU(3)_F$ symmetry. This introduces unique topological contributions (such as hairpin diagrams) and mixing effects with the $\eta$ that distinguish the $D \to P\eta'$ system from the more standard $D \to P\eta$ and $D \to PP$ systems. The authors aim to provide Standard Model (SM) predictions for these CP asymmetries and establish correlations (sum rules) between different decay modes to serve as a benchmark for future experimental data from LHCb, Belle II, and BESIII.

Methodology
The analysis extends the authors' previous work on branching ratios [54] by focusing on the CKM-subleading amplitudes ( $A_b$ ), which are responsible for CP violation through interference with the CKM-leading amplitudes ( $A_{sd}$ ). The methodology relies on the approximate $SU(3)_F$ symmetry of QCD, incorporating first-order symmetry-breaking effects systematically.

Topological Decomposition: The authors decompose the CKM-subleading amplitudes into topological diagrams, including tree ( $T$ ), color-suppressed ( $C$ ), annihilation ( $A$ ), exchange ( $E$ ), penguin ( $P$ ), and penguin-annihilation ($PA$) topologies. Crucially, they introduce new topologies specific to the $\eta'$ (singlet) nature, such as hairpin penguin diagrams ( $P_H$ ) and mixed singlet-octet contributions (denoted by subscripts like 18, 81, 11, 88).
Parameter Redefinition: To handle linear dependencies among the hadronic parameters and reduce the number of free parameters, the authors apply a series of redefinitions. These redefinitions absorb specific penguin and hairpin contributions into redefined color-suppressed parameters (e.g., $\hat{C}_{81}$ , $\hat{C}_{18}$ ). This allows for a decoupling of the $D \to P\eta'$ system from $D \to P\eta$ and $D \to PP$ without requiring additional theoretical assumptions about mass dependencies.
Sum Rules: The approximate $SU(3)_F$ symmetry leads to specific relations (sum rules) between the CP asymmetries of different channels. The authors derive both exact $SU(3)_F$ limit sum rules and modified sum rules that account for the breaking of the magnitude of the leading amplitudes while maintaining the symmetry limit for the subleading amplitudes and their relative strong phases.
Statistical Analysis: A frequentist statistical analysis is performed using a $\chi^2$ minimization. The fit incorporates existing experimental branching ratios and the single available measurement of the direct CP asymmetry for $D^+ \to \pi^+\eta'$ . The analysis imposes constraints on the size of $SU(3)_F$ breaking (fixed at $\varepsilon = 30\%$ ) and treats the ratio of the redefined color-suppressed parameter to the tree amplitude ( $|C_{18}/\hat{T}_{18}|$ ) as a key free parameter, bounded by theoretical expectations ( $|C_{18}/\hat{T}_{18}| \leq 3$ ).

Key Contributions

Systematic Framework: The paper provides a complete topological decomposition of CKM-subleading amplitudes for $D \to P\eta'$ decays, explicitly treating the unique contributions of the $\eta'$ flavor singlet and hairpin diagrams.
Derivation of Sum Rules: The authors derive new sum rules correlating the CP asymmetries of charged modes ( $D^+ \to \pi^+\eta'$ vs. $D_s^+ \to K^+\eta'$ ) and neutral modes ( $D^0 \to \pi^0\eta'$ vs. $D^0 \to \eta\eta'$ ). These relations are presented in both their exact $SU(3)_F$ limit forms and modified forms that include partial symmetry-breaking corrections.
Global Fit and Predictions: The paper presents the first global SM predictions for the CP asymmetries in these channels, constrained by current data and theoretical bounds on $SU(3)_F$ breaking.
Parameter Extraction Tool: The authors provide a method to extract the ratio of CKM-subleading to CKM-leading contributions (effectively the penguin-to-tree ratio, though entangled with color-suppressed trees due to redefinitions) directly from future measurements of CP asymmetries.

Results

Correlations: The global fit reveals strong correlations between the CP asymmetries of the charged and neutral decay modes. These correlations are visualized in plots showing the allowed regions for pairs of asymmetries, bounded by the theoretical sum rules.
Numerical Predictions:
- For the measured mode $D^+ \to \pi^+\eta'$ , the theory fit predicts a central value of $(0.40^{+0.19}_{-0.20})\%$ , consistent with the experimental average of $(0.4 \pm 0.2)\%$ .
- For the unmeasured modes, the predictions are:
  - $a_{CP}^{dir}(D^0 \to \pi^0\eta') \in [-0.66, 0.66]\%$
  - $a_{CP}^{dir}(D^0 \to \eta\eta') \in [-0.29, 0.29]\%$
  - $a_{CP}^{dir}(D_s^+ \to K^+\eta') \in [-0.62, 0.36]\%$
Constraint on $C_{18}$ : The current data for $D^+ \to \pi^+\eta'$ implies a weak constraint on the ratio of the redefined color-suppressed parameter to the tree amplitude, favoring $|C_{18}/\hat{T}_{18}| \gtrsim 1$ . This is in tension with the $1/N_c$ power counting expectation of $\sim 1/3$ , though the tension is not statistically significant ( $1.6\sigma$ ) given the current uncertainties.
Sensitivity to New Physics: The paper notes that deviations from the predicted sum rules or correlations would signal New Physics, particularly if such contributions violate the $U$ -spin symmetry relations inherent in the SM predictions.

Significance
The paper claims that $D \to P\eta'$ decays represent "uncharted territory" for CP violation in the charm sector. Its significance lies in providing a robust theoretical framework to interpret future high-precision measurements. By establishing specific correlations and sum rules, the work enables experimentalists to test the consistency of the Standard Model in the charm sector. Furthermore, the ability to extract the ratio of subleading to leading amplitudes from future data offers a pathway to better understand the dynamics of penguin and color-suppressed contributions in charm decays, potentially revealing discrepancies with $1/N_c$ expectations or signs of physics beyond the Standard Model. The authors emphasize that their approach allows for a complete decoupling of the $\eta'$ system from other decay modes, offering a cleaner test of $SU(3)_F$ symmetry breaking effects.

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