CP asymmetries in $D\to K^0_{S,L}P$ and $D\to K^0_{S,L}V$ decays

This paper derives time-dependent and time-integrated CP asymmetry formulas for $D\to K^0_{S,L}P$ and $D\to K^0_{S,L}V$ decays by incorporating $D^0-\overline D^0$ mixing and $K^0_L$ modes, uses a global fit of branching fractions to extract hadronic parameters, and demonstrates that these effects can reach $\mathcal{O}(10^{-3})$ while mitigating theoretical tensions in specific $D^0$ decay channels.

The Gist

Imagine the universe as a giant, chaotic dance floor where particles are the dancers. For a long time, physicists thought this dance was perfectly symmetrical: if you played the movie of a particle's life backward, it would look exactly the same as playing it forward. This is called "CP symmetry."

However, we know the universe isn't perfectly symmetrical. There is a slight "tilt" in the dance, a preference for matter over antimatter. This tilt is called CP violation, and it's crucial for explaining why we exist at all.

This paper focuses on a specific group of dancers: the D mesons (heavy particles made of a charm quark) and their interactions with neutral kaons (lighter particles that can change their identity). Here is a simple breakdown of what the authors did and what they found.

The Setup: A Dance with a Twist

When a D meson decays (dies), it often turns into a neutral kaon and another particle. The tricky part is that neutral kaons are "shape-shifters." They can exist as a "K-zero" or an "anti-K-zero," and they constantly oscillate between the two, like a coin spinning in the air before landing.

Usually, physicists look at two ways a D meson can decay:

1. The Favorite Move (Cabibbo-Favored): The most likely, natural way the dance happens.
2. The Rare Move (Doubly Cabibbo-Suppressed): A very unlikely, awkward way the dance happens.

In the past, scientists mostly looked at the "Favorite Move." But this paper argues that to see the full picture of the "tilt" (CP violation), you have to watch what happens when the Favorite Move and the Rare Move happen at the same time and interfere with each other. It's like two different dance routines overlapping; the interference creates a new, unique rhythm that reveals hidden secrets.

The New Discoveries

The authors, Ying-Xin Lai and Di Wang, did three main things:

1. They Wrote a New "Instruction Manual" (Formulas)
They created new mathematical formulas to calculate the "asymmetry" (the tilt) in these decays. Crucially, they included two things that previous studies often ignored or simplified:

• The D meson's own mixing: Just like the kaon, the D meson can also oscillate between particle and antiparticle. They added this to the mix.
• The "Long-Lived" Kaon: Neutral kaons come in two flavors: a short-lived one ($K_S$) and a long-lived one ($K_L$). Previous studies often focused only on the short-lived one. This paper treats both equally, providing a more complete view.

2. They Tuned the "Dance Moves" (Global Fit)
To make their predictions accurate, they had to figure out the exact "strength" and "timing" (phases) of the different decay moves. They used a method called the "topological diagram approach," which is like breaking a complex dance down into basic steps (like a spin, a jump, or a slide).
They looked at a massive amount of experimental data (branching fractions) to "tune" these steps. The result? Their tuned model fits the real-world data very well, resolving some previous disagreements (tensions) between theory and experiment, specifically for decays involving omega ($\omega$) and phi ($\phi$) particles.

3. They Found a Hidden "Tilt" (The $A_{int}$ Effect)
The most exciting finding is a specific type of CP violation they call $A_{int}$.

• The Old View: Scientists thought the main source of the tilt came from the kaon mixing itself (the coin spinning).
• The New View: The authors found that the interference between the "Favorite Move" and the "Rare Move," combined with the kaon mixing, creates a new source of tilt.
• The Magnitude: This new effect is surprisingly large—about 1 in 1,000 ($10^{-3}$). While that sounds small, in the world of particle physics, it's a huge signal, much larger than the "direct" tilt from the D meson itself.

What This Means for the Future

The paper doesn't claim to solve the mystery of the universe's existence right now, but it provides a clearer map for future explorers.

• The "Difference" Test: The authors suggest a specific experiment: compare the tilt in the decay of a $D^+$ meson into a kaon and a pion, versus a $D_s^+$ meson into a kaon and a kaon.
• The Goal: By looking at the difference between these two, scientists can cancel out the background noise (like the kaon's own mixing) and isolate this new, interesting "interference tilt."
• The Venue: They predict that the massive particle detectors at LHCb (in Europe) and Belle II (in Japan) will be able to measure this difference very soon.

In a Nutshell

Think of the universe as a song. For years, we only listened to the main melody (the most common decays). This paper says, "Wait, listen to the background harmonies and the rare notes too." When you listen to the whole song, including the rare notes and the way the instruments mix, you hear a new, distinct rhythm (the $A_{int}$ effect) that explains the music much better than before. This new rhythm is loud enough that the next generation of particle detectors should be able to hear it clearly.

Technical Summary

Technical Summary: CP Asymmetries in $D \to K^0_{S,L}P$ and $D \to K^0_{S,L}V$ Decays

Problem Statement
CP violation (CPV) in the charm sector is a critical probe for understanding matter-antimatter asymmetry and searching for physics beyond the Standard Model (SM). While CPV in singly Cabibbo-suppressed (SCS) $D$ meson decays is theoretically challenging due to large uncertainties in penguin diagrams, $D$ meson decays into neutral kaons ($K^0$) offer a unique environment. These modes involve the simultaneous presence of Cabibbo-favored (CF) and doubly Cabibbo-suppressed (DCS) amplitudes, alongside final-state neutral kaon mixing ($K^0 - \bar{K}^0$). Previous theoretical analyses have largely focused on the dominant CPV arising from $K^0 - \bar{K}^0$ mixing, often neglecting the interference between CF/DCS amplitudes and mixing, as well as the effects of $D^0 - \bar{D}^0$ mixing and $K^0_L$ modes. Furthermore, discrepancies exist between theoretical predictions and experimental data regarding $K^0_S - K^0_L$ asymmetries in specific decay channels.

Methodology
The authors employ a topological diagram approach to analyze CP asymmetries in $D \to K^0_{S,L}P$ (pseudoscalar) and $D \to K^0_{S,L}V$ (vector) decays. The methodology proceeds in three main stages:

1. Formalism Development: The paper derives rigorous formulas for both time-dependent and time-integrated CP asymmetries. Crucially, this formalism incorporates:

   • The interference between CF and DCS amplitudes.
   • Neutral kaon mixing ($K^0 - \bar{K}^0$).
   • $D^0 - \bar{D}^0$ mixing effects for neutral $D$ meson decays.
   • The distinct treatment of $K^0_L$ modes, utilizing a non-unitary transformation to define left-eigenvectors in the presence of CP violation.
   • A new CP-violating term, $A_{int}^{CP}$, arising from the interference between the mother decay (charm) and daughter mixing (kaon).

2. Global Fit of Hadronic Parameters: To determine the hadronic parameters governing these asymmetries, the authors perform a global fit of branching fractions for Cabibbo-favored and doubly Cabibbo-suppressed $D \to PP$ and $D \to PV$ decays.

   • The fit includes 17 branching fractions for $D \to PP$ modes to extract 7 parameters (magnitudes and phases of $T, C, E, A$ amplitudes).
   • It includes 26 branching fractions for $D \to PV$ modes to extract 15 parameters.
   • Partial $SU(3)_F$ breaking effects are included via a factor $f_K/f_\pi \approx 1.193$ in DCS amplitudes.
   • SCS modes are excluded from the fit to avoid large $SU(3)$ breaking uncertainties associated with penguin topologies.

3. Numerical Prediction: Using the extracted parameters, the authors calculate the ratios of DCS to CF amplitudes ($r_f$) and relative strong phases ($\delta_f$) for various neutral kaon modes. These are then used to predict time-dependent and time-integrated CP asymmetries, including the $K^0_S - K^0_L$ asymmetry ($R$).

Key Contributions and Results

• Derivation of New CP-Violating Effects: The paper explicitly derives the contribution of $A_{int}^{CP}$, a term resulting from the interference between CF/DCS amplitudes and neutral kaon mixing. The authors demonstrate that this effect can reach the order of $O(10^{-3})$ in specific modes, significantly larger than the direct CP asymmetry ($A_{dir}^{CP}$), which is typically $O(10^{-5})$.
• Inclusion of $D^0 - \bar{D}^0$ Mixing and $K^0_L$ Modes: For the first time in this context, the formalism accounts for $D^0 - \bar{D}^0$ mixing effects in the time-integrated asymmetries and provides a consistent framework for $K^0_L$ final states, showing that the signs of certain CPV terms flip between $K^0_S$ and $K^0_L$ modes.
• Resolution of Theoretical Tensions: The global fit results mitigate the tension between theoretical predictions and experimental data for the $K^0_S - K^0_L$ asymmetries in the $D^0 \to K^0_{S,L}\omega$ and $D^0 \to K^0_{S,L}\phi$ modes. The authors' predictions align more closely with experimental data compared to previous approaches like the factorization-assisted topological-amplitude (FAT) and $SU(3)_F$ symmetric topological diagram (STD) methods.
• Quantitative Predictions:
  • In modes such as $D^+ \to K^0_S \pi^+$, $D_s^+ \to K^0_S K^+$, $D^0 \to K^0_S \rho^0$, and $D^0 \to K^0_S \phi$, the $A_{int}^{CP}$ term is found to be of the order of $10^{-3}$.
  • The total CP asymmetry is dominated by the $K^0 - \bar{K}^0$ mixing term ($A_{K^0}^{CP} \approx -3.23 \times 10^{-3}$), but the interference terms provide measurable deviations.
  • Two solutions for the strong phases ($S1$ and $S2$) are presented, yielding different signs for the interference terms, both of which are consistent with current experimental data for $D^+ \to K^0_S \pi^+$.

Significance and Claims
The paper claims that the interference between Cabibbo-favored and doubly Cabibbo-suppressed amplitudes with neutral kaon mixing constitutes a significant source of CP violation in charm decays, potentially reaching $O(10^{-3})$. This challenges the assumption that direct CPV in charm is negligible in these channels.

The authors propose that the difference in CP asymmetries between the $D^+ \to K^0_S \pi^+$ and $D_s^+ \to K^0_S K^+$ modes, denoted as $\Delta A_{CP}(\pi^+, K^+)$, serves as a robust observable. This observable largely cancels systematic asymmetries related to particle selection and production. The paper asserts that the magnitude of this difference is within the reach of current and future experiments, specifically LHCb and Belle II, which can utilize favorable time intervals to measure these effects and thereby determine the sign of the strong phases.

Ultimately, the work provides a comprehensive theoretical framework that integrates $D$-meson mixing and $K^0_L$ modes, offering improved predictions that resolve existing discrepancies in $K^0_S - K^0_L$ asymmetry data and identifying specific channels where new CP-violating effects are most prominent.