Probing CP and flavor violation in neutral kaon decays with ALPs

This paper analyzes neutral kaon decays into axion-like particles (ALPs), demonstrating that comparing three-body ($K_L \to \pi\pi a$) and two-body ($K_L \to \pi^0 a$) channels—while accounting for often-neglected weak-interaction contributions—provides a powerful probe of CP violation and flavor-violating ALP couplings in both minimal and non-minimal scenarios.

The Gist

Imagine the universe as a giant, high-stakes detective story. For decades, physicists have been trying to solve two major mysteries: Why does matter exist at all? (which requires a violation of "CP symmetry") and Why do particles change their identities so rarely? (which requires a violation of "Flavor symmetry").

In the Standard Model (our current rulebook for physics), these two mysteries are tightly linked. You can't have one without the other. But what if there are new, invisible players in the game? Enter ALPs (Axion-Like Particles). Think of ALPs as "ghosts" that might be hiding in the shadows, potentially solving the mystery of dark matter or the strong CP problem.

This paper is a guidebook for detectives (experimentalists) on how to catch these ghosts using Neutral Kaons (a specific type of unstable particle) as their trap.

Here is the breakdown of their investigation, using some everyday analogies:

1. The Setup: The Two-Door Mystery

The researchers are looking at how a Neutral Kaon ($K_L$) decays. It's like a magician pulling a rabbit out of a hat, but the magician has two different hats:

• Hat A (The Two-Body Decay): The Kaon turns into a neutral pion and an ALP ($K_L \to \pi^0 a$).
• Hat B (The Three-Body Decay): The Kaon turns into two pions and an ALP ($K_L \to \pi\pi a$).

The Twist:

• To pull a rabbit out of Hat A, the magician needs to break both the Flavor rules and the CP rules. It's a very strict, difficult trick.
• To pull a rabbit out of Hat B, the magician only needs to break the Flavor rules. The CP rules can stay intact.

The Detective's Insight:
If you see the magician doing the easy trick (Hat B) way more often than the hard trick (Hat A), it tells you something profound about the magician's secret technique. It suggests that the "ghost" (the ALP) has a specific relationship with the laws of physics that we didn't expect. By comparing the rate of Hat B to Hat A, scientists can figure out if the new physics is "Minimal" (playing by the old rules) or "Maximal" (breaking all the rules).

2. The Hidden Mechanic: The Weak Force

For a long time, physicists thought the Kaon just directly swapped a quark for an ALP. But this paper points out a sneaky mechanic: The Weak Force.

Imagine the Kaon is a car.

• Old View: The driver (the ALP) just jumps into the car and drives away.
• New View: The driver actually calls a tow truck (the Weak Force) to move the car, and then jumps in.

The authors realized that this "tow truck" method is often ignored in calculations for the three-body decay (Hat B). When they included it, they found that in many scenarios, the "tow truck" method is actually the dominant way the decay happens. This changes the math significantly, sometimes making the three-body decay much more common than previously thought.

3. The "Grossman-Nir" Safety Net

Usually, physicists have a safety net called the Grossman-Nir bound. It's like a speed limit sign. It says: "The rate of the neutral Kaon decay (Hat A or B) cannot be faster than the rate of the charged Kaon decay."

Why? Because charged Kaons don't need to break the CP rules, so they are generally faster and easier to produce. If you see a neutral Kaon decay happening faster than the charged one, it usually means your math is wrong or you've missed something.

The Paper's Surprise:
The authors found that under very specific conditions (when the ALP is almost the same mass as a pion), this speed limit sign gets blurry. The "tow truck" mechanism can cause a resonance—like pushing a swing at exactly the right moment to make it go super high. This allows the neutral Kaon to decay faster than the charged Kaon, temporarily breaking the usual rules. This is a huge deal because it opens up a new "blind spot" where ALPs could be hiding right under our noses.

4. The Hunt: Where to Look?

The paper concludes by telling experimentalists where to look next:

• The "Long-Lived" Ghosts: If the ALP is light and travels far before disappearing, current experiments (like KOTO and E391a) might have missed it because they were looking for the "hard trick" (Hat A). They need to start looking harder for the "easy trick" (Hat B).
• The "Prompt" Ghosts: If the ALP is heavier and decays immediately into photons (light), the paper suggests that the three-body decay channel is actually a better detector than the two-body one in certain mass ranges.

The Big Picture Analogy

Imagine you are trying to find a specific type of bird in a forest.

• Old Strategy: You only look for the bird singing a very complex, rare song (Two-body decay). You assume if you don't hear it, the bird isn't there.
• New Strategy (This Paper): You realize the bird also makes a simple, common chirp (Three-body decay) that you were ignoring. Furthermore, you discover that the wind (Weak Force) sometimes carries the chirp much further than you thought.
• The Result: You realize that by listening for the simple chirp, you might actually find the bird more easily than by listening for the complex song. And if you hear the chirp louder than the complex song, you know you've found a very special, rare species of bird that breaks the usual rules of the forest.

In short: This paper provides a new, more sensitive map for hunting Axion-Like Particles. It tells us that by looking at the "messier" three-particle decays of Kaons, we might find new physics that the "cleaner" two-particle decays are hiding from us.

Technical Summary

1. Problem Statement

The paper addresses the search for Axion-Like Particles (ALPs) in neutral kaon decays, specifically focusing on the interplay between Flavor Violation (FV) and CP Violation (CPV).

• Context: While charged kaon decays ($K^+ \to \pi^+ a$) are sensitive to FV, they are largely insensitive to the CP properties of the new physics. Neutral kaon decays ($K_L$) offer a unique probe because $K_L$ is an approximate CP-odd eigenstate.
• The Gap: Previous analyses of neutral kaon decays involving ALPs often focused on two-body decays ($K_L \to \pi^0 a$) or neglected specific contributions in three-body decays ($K_L \to \pi\pi a$).
• Key Question: Can the ratio of three-body to two-body decay rates serve as a probe for the CP nature of the underlying UV theory? Furthermore, under what conditions can the three-body decay dominate despite the phase-space suppression?

2. Methodology

The authors employ a systematic Effective Field Theory (EFT) approach combined with Chiral Perturbation Theory ($\chi$PT).

• UV to IR Matching: They start with a general dimension-5 ALP EFT at a high scale $\Lambda$ (above the electroweak scale) containing couplings to quarks, gluons, and electroweak bosons. They run these couplings down to the QCD scale, accounting for radiative corrections (RGEs) that generate off-diagonal flavor-violating couplings even in Minimal Flavor Violation (MFV) scenarios.
• Chiral Perturbation Theory: The theory is matched to $\chi$PT at the low-energy scale. The authors calculate decay amplitudes for:
  • Two-body: $K_L \to \pi^0 a$
  • Three-body: $K_L \to \pi^0 \pi^0 a$ and $K_L \to \pi^+ \pi^- a$
• Basis Independence & Contact Terms: A critical methodological innovation is the rigorous treatment of naively factorizable contributions. The authors demonstrate that calculating three-body amplitudes requires including diagrams where the decay is mediated by the short-lived $K_S$ (or charged pions for the charged mode). These diagrams cancel unphysical, basis-dependent contact terms arising from the field redefinitions needed to remove the gluon coupling. This ensures the final results depend only on physical combinations of couplings.
• Symmetry Analysis: They utilize spurion analysis under Parity (P) and CP symmetries to classify which couplings (real vs. imaginary, vector vs. axial) contribute to specific decay modes.
  • $K_L \to \pi^0 a$ requires both FV and CPV.
  • $K_L \to \pi\pi a$ requires FV but can proceed via CP-conserving couplings.

3. Key Contributions

1. Re-evaluation of Weak-Interaction Contributions: The paper emphasizes that "indirect" contributions, where flavor violation arises from SM weak interactions combined with flavor-preserving ALP couplings, are often dominant in three-body decays. These were frequently neglected in recent literature.
2. Cancellation of Unphysical Terms: They provide the first complete calculation of the three-body amplitudes that explicitly includes the $K_S$-mediated diagrams required to cancel basis-dependent artifacts, ensuring the physical observables are well-defined.
3. CPV Probes via Rate Ratios: They define the ratios $R_0 = \Gamma(K_L \to \pi^0 \pi^0 a) / \Gamma(K_L \to \pi^0 a)$ and $R_\pm = \Gamma(K_L \to \pi^+ \pi^- a) / \Gamma(K_L \to \pi^0 a)$. They show these ratios are sensitive to the phase of the flavor-violating coupling, allowing discrimination between CP-conserving and CP-violating UV completions.
4. MFV Scenarios: They map out specific hierarchies in Minimal Flavor Violation (MFV) scenarios where the three-body decay rate can dominate the two-body rate, overcoming the expected $\mathcal{O}(10^{-3})$ phase-space suppression.

4. Key Results

A. Decay Amplitudes and Symmetries

• Two-body ($K_L \to \pi^0 a$): The amplitude is proportional to $\text{Im}(f_M) - i\varepsilon \text{Re}(f_M)$. It vanishes if the coupling is purely real (CP conserving) in the absence of the SM $\varepsilon$ parameter.
• Three-body ($K_L \to \pi\pi a$): The amplitude is dominated by the real part of the axial coupling (or the weak parameter $N_8$) and does not require CP violation to proceed.
• Role of $K_S$ Mediation: The inclusion of the $K_S$ propagator diagram is essential to cancel the unphysical dependence on the arbitrary field redefinition parameters ($\delta_q$), rendering the amplitude basis-independent.

B. Rate Hierarchies

The authors identify three distinct regimes based on the source of flavor violation:

1. Maximal Flavor Violation (Non-MFV):

   • If the new physics coupling $\kappa_{FV}$ is large, the ratio $R \sim I(m_a) \times |\frac{\text{CP-conserving}}{\text{CP-violating}}|^2$.
   • If the coupling is CP-conserving ($\theta_\kappa \sim 0$), the two-body decay is suppressed by $|\varepsilon|^2$, leading to $R \sim I(m_a) / |\varepsilon|^2 \sim 10^2 - 10^3$.
   • If the coupling is CP-violating ($\theta_\kappa \sim \pi/2$), the two-body decay is unsuppressed, and $R \sim I(m_a) \sim 10^{-3}$.

2. Minimal Flavor Violation (MFV):

   • Here, FV is driven by SM Yukawas. The leading order MFV expansion does not introduce new CP phases.
   • Scenario 1 (Direct Mediation): If $\kappa^{MFV}_L$ dominates, the ratio is determined by the CKM phase: $R \sim (\text{Re } V_{td} / \text{Im } V_{td})^2 I(m_a) \approx 10 \times I(m_a)$.
   • Scenario 2 (Indirect Mediation): If the process is mediated by SM weak currents ($N_8$) combined with flavor-diagonal ALP couplings (e.g., ALP couples only to gluons or $W$ bosons), the three-body decay can be enhanced.
     • For ALPs coupling to $W$ bosons ($\tilde{c}_{WW} \neq 0$), the ratio can reach $R \sim 1600 \times I(m_a)$.
     • For ALPs coupling to hypercharge ($\tilde{c}_{BB} \neq 0$), the ratio can reach $R \sim 1.6 \times 10^5 \times I(m_a)$.
   • Result: In these specific MFV cases, the three-body decay dominates the two-body decay despite phase-space suppression.

3. Grossman-Nir (GN) Bound Violation:

   • The GN bound relates neutral kaon decays to charged kaon decays ($K^+ \to \pi^+ a$). Usually, $K^+$ decays provide stronger bounds because they do not require CPV.
   • However, the authors show that near the pion mass ($m_a \approx m_\pi$), resonance effects in the three-body decay (specifically from the $N_8$ term) can cause the ratio $\Gamma(K_L \to \pi^0 \pi^0 a) / \Gamma(K^+ \to \pi^+ a)$ to spike significantly, potentially violating the naive expectation that charged decays always dominate.

C. Phenomenological Implications

• Long-lived ALPs: For $m_a \lesssim 0.1$ GeV, current bounds on $K_L \to \pi^0 \pi^0 X$ (E391a) are weaker than $K^+ \to \pi^+ X$ (NA62). However, if the three-body decay dominates (as in specific MFV scenarios), improving the $K_L \to \pi^0 \pi^0 X$ bound by $\mathcal{O}(10^3)$ could surpass the charged kaon constraints.
• Promptly Decaying ALPs: For $m_a \gtrsim 0.1$ GeV, the paper recasts experimental limits from KOTO, NA48, and E949. They find that for certain models (e.g., ALP coupling to gluons), the neutral three-body decay channel provides the strongest constraints, particularly in the region near the pion mass where the two-body decay is suppressed.

5. Significance

• Theoretical Rigor: The paper corrects a common oversight in ALP phenomenology by demonstrating that three-body kaon decays require the inclusion of specific weak-interaction diagrams to be physically consistent.
• New Discovery Channels: It identifies a class of MFV models where the three-body decay is the dominant production mode for ALPs in neutral kaons, challenging the assumption that two-body decays are always the primary search channel.
• CP Diagnostics: The ratio of three-body to two-body rates offers a unique "smoking gun" to distinguish between CP-conserving and CP-violating new physics, a capability not available in charged kaon decays.
• Experimental Guidance: The results suggest that experiments like KOTO (and future upgrades) should prioritize the search for $K_L \to \pi^0 \pi^0 a$ and $K_L \to \pi^+ \pi^- a$, as these channels may offer sensitivity to ALP couplings that are invisible to charged kaon searches or standard two-body searches.