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Prospects for $K_{L}\to\pi^{0}\nu\bar{\nu}$ , $K_S\to\mu^+\mu^-$ , $K_L\to\pi^0 \ell^+\ell^-$ and $\varepsilon^{\prime}/\varepsilon$ after the new $K^{+}\to\pi^{+}\nu\bar{\nu}$ result from NA62

This paper demonstrates that specific new physics scenarios, such as a $Z'$ model with large complex phases and right-handed couplings, can simultaneously enhance the branching ratios of $K_L\to\pi^0\nu\bar{\nu}$ , $K_S\to\mu^+\mu^-$ , and $K_L\to\pi^0\ell^+\ell^-$ while resolving the $\varepsilon^{\prime}/\varepsilon$ anomaly, all without conflicting with the recent NA62 measurement of $K^{+}\to\pi^{+}\nu\bar{\nu}$ or the $\varepsilon_K$ constraint.

Original authors: Monika Blanke, Andrzej J. Buras, Cristina Lazzeroni, Joel C. Swallow

Published 2026-07-22

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CC BY 4.0

Original authors: Monika Blanke, Andrzej J. Buras, Cristina Lazzeroni, Joel C. Swallow

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

Imagine the universe as a giant, bustling city where tiny particles are the citizens. Most of these citizens follow a strict, well-written rulebook called the Standard Model, which predicts exactly how they should behave, interact, and decay. But sometimes, a few citizens seem to be breaking the rules or acting a little strangely. Physicists are like detectives trying to figure out if these strange behaviors are just glitches in the rulebook or if there is a secret, invisible gang of "New Physics" citizens running around behind the scenes, breaking the laws of the universe in ways we haven't seen yet.

To catch these rule-breakers, scientists look at very rare events, like a specific type of particle called a "kaon" suddenly turning into something else. One of the most famous rulebooks for these kaons involves a process where they turn into a pion and some invisible ghost particles called neutrinos. For a long time, the rulebook predicted exactly how often this should happen. Recently, a team of detectives called NA62 checked the numbers for one version of this event and found it matched the rulebook perfectly. This was a bit of a bummer for those hoping to find the secret gang immediately, because it meant the "left-handed" way of breaking the rules (a specific direction the particles spin) wasn't the culprit. However, the detectives wondered: what if the secret gang is using a different trick? What if they are using a "right-handed" trick, or a mix of both, that we haven't noticed yet? This is the big question this paper tackles.

The authors of this paper, Monika Blanke, Andrzej J. Buras, Cristina Lazzeroni, and Joel C. Swallow, decided to play a game of "what if." They asked: Is it possible that a new, invisible force carrier (which they call a $Z'$ boson, like a new kind of messenger particle) is sneaking around and boosting the rates of other rare kaon decays, even though the NA62 result looks normal?

They found that the answer is a resounding "yes," but with a catch. To make this work, the new force has to be very specific. It needs to have a "complex phase," which is a fancy way of saying it has a hidden twist or a secret angle that makes it behave differently than the standard rules. They also found that this new force needs to have a tiny, almost invisible "right-handed" component to cancel out its own effects on a different, very sensitive measurement called $\epsilon_K$ (which would otherwise scream "New Physics!" if it were too big).

Here is the magic trick they demonstrated: If you set up this specific new force correctly, you can keep the NA62 measurement looking perfectly normal (just like the Standard Model predicts), while simultaneously making other rare decays explode in frequency. Specifically, they showed that the decay $K_L \to \pi^0 \nu \bar{\nu}$ (where a neutral kaon turns into a neutral pion and invisible neutrinos) could be boosted by a factor of ten! This would bring it right up to the theoretical limit known as the "Grossman-Nir bound," which is the absolute maximum speed limit allowed by the laws of physics for this type of event.

At the same time, this same setup would make other rare events, like the kaon turning into two muons ( $K_S \to \mu^+ \mu^-$ ) or a pion and two charged particles ( $K_L \to \pi^0 \ell^+ \ell^-$ ), much more common than we currently expect. It could also fix a long-standing puzzle about the ratio $\epsilon'/\epsilon$ , which measures a specific type of symmetry breaking in the universe. The paper suggests that if this specific $Z'$ scenario is real, we might see these huge increases in the next few years when the KOTO II experiment (looking for the invisible neutrino decay) and LHCb (looking for the muon decays) start collecting more data.

The authors are careful to say they haven't found this new force yet; they have only built a mathematical model showing that it is possible for it to exist without contradicting the recent NA62 results. They used a specific example involving a $Z'$ particle with a mass of 5 TeV to prove their point. They showed that by carefully tuning the "knobs" of this new force—specifically how it talks to left-handed versus right-handed particles and how it twists in time—you can hide the new physics in plain sight for one experiment while making it scream for attention in others.

So, while the recent NA62 result might have seemed like a dead end for finding new physics, this paper suggests it's actually just a red herring. The secret gang might still be there, just wearing a different disguise. The authors urge the scientific community to keep looking, because if their scenario is right, the next few years of experiments could reveal a whole new layer of the universe's rulebook, turning up the volume on these rare decays by a factor of ten or more. It's a hopeful reminder that even when the data looks boring, the universe might just be waiting for us to ask the right question.

Technical Summary: Prospects for $KL \to \pi^0\nu\bar{\nu}$ , $KS \to \mu^+\mu^-$ , $KL \to \pi^0\ell^+\ell^-$ and $\varepsilon'/\varepsilon$ after the new $K^+ \to \pi^+\nu\bar{\nu}$ result from NA62

Problem Statement
The paper addresses the implications of the recent NA62 measurement of the $K^+ \to \pi^+\nu\bar{\nu}$ branching ratio, which is consistent with the Standard Model (SM) prediction, on the prospects for observing New Physics (NP) in other rare kaon decays. Specifically, the authors investigate whether the decay $K_L \to \pi^0\nu\bar{\nu}$ can still be significantly enhanced (potentially by an order of magnitude) despite the SM-like result for $K^+ \to \pi^+\nu\bar{\nu}$ and the absence of NP in $K^0-\bar{K}^0$ mixing ( $\varepsilon_K$ ). The study also examines the correlated prospects for enhancements in $K_S \to \mu^+\mu^-$ , $K_L \to \pi^0\ell^+\ell^-$ , and the direct CP-violation ratio $\varepsilon'/\varepsilon$ , particularly in the context of the Dual QCD prediction which suggests a large anomaly in $\varepsilon'/\varepsilon$ .

Methodology
The authors employ a two-pronged approach:

Model-Independent Analysis: They utilize general formulae for the branching ratios of $K \to \pi\nu\bar{\nu}$ , $K_{L,S} \to \mu^+\mu^-$ , and $K_L \to \pi^0\ell^+\ell^-$ , incorporating both left-handed (LH) and right-handed (RH) quark currents. They analyze the complex function $X_{\text{eff}}$ governing the $K \to \pi\nu\bar{\nu}$ decays, demonstrating how the interplay between the real and imaginary parts of $X_{\text{eff}}$ allows for a large enhancement of $K_L \to \pi^0\nu\bar{\nu}$ while keeping $K^+ \to \pi^+\nu\bar{\nu}$ SM-like. This requires a large weak phase $\beta_X \approx 110^\circ$ .
Specific $Z'$ Scenario: To illustrate a concrete realization of these correlations, the authors analyze a specific $Z'$ model with a new $U(1)'$ gauge symmetry. They impose constraints from $\varepsilon_K$ and $\Delta M_K$ by introducing a specific pattern of flavor-violating couplings: a dominant LH coupling ( $\Delta_{sd}^L$ ) and a highly suppressed RH coupling ( $\Delta_{sd}^R \approx \Delta_{sd}^L / 240$ ). This suppression is necessary to cancel dangerous contributions to $K^0-\bar{K}^0$ mixing via renormalization group effects and chiral enhancements of LR operators. The numerical analysis explores a five-dimensional parameter space (three real and one complex parameter) to satisfy simultaneous constraints from NA62, KOTO, LHCb, and $\varepsilon'/\varepsilon$ .

Key Contributions and Results

Decoupling of $K^+$ and $K_L$ Decays: The paper demonstrates that in the presence of both LH and RH couplings with new CP-violating phases, it is possible to enhance the $K_L \to \pi^0\nu\bar{\nu}$ branching ratio by a factor of $\mathcal{O}(10)$ relative to the SM prediction, even if $K^+ \to \pi^+\nu\bar{\nu}$ remains consistent with the SM. This occurs in a region of parameter space outside the correlations typically found in models with only LH or only RH couplings (which are constrained by $\varepsilon_K$ ).
Simultaneous Enhancements: The authors show that a specific $Z'$ $Z^{'}$ scenario can simultaneously:
- Keep $\varepsilon_K$ and $\Delta M_K$ SM-like.
- Maintain $B(K^+ \to \pi^+\nu\bar{\nu})$ within the NA62 experimental bounds.
- Enhance $B(K_L \to \pi^0\nu\bar{\nu})$ up to the model-independent Grossman-Nir bound.
- Significantly enhance $B(K_S \to \mu^+\mu^-)$ (by factors up to $\sim 100$ in some scenarios) and $B(K_L \to \pi^0\ell^+\ell^-)$ .
- Resolve the potential anomaly in $\varepsilon'/\varepsilon$ (assuming the Dual QCD prediction is correct) by generating a positive NP contribution $\kappa_{\varepsilon'} = \mathcal{O}(1)$ .
Role of Phases and Interference: The analysis highlights that a large complex phase in the $Z'$ coupling is essential for the enhancement of $K_L \to \pi^0\nu\bar{\nu}$ . Furthermore, the paper details how the interference between direct and mixing-induced CP-violating contributions in $K_L \to \pi^0\ell^+\ell^-$ affects the allowed parameter space, noting that negative interference is often required to satisfy current experimental upper bounds while allowing for large enhancements.
Numerical Constraints: The study identifies that the constraint from $K_L \to \mu^+\mu^-$ (specifically the short-distance contribution) limits the real part of the $Z'$ coupling, while the imaginary part drives the enhancements in $K_S \to \mu^+\mu^-$ and $\varepsilon'/\varepsilon$ .

Significance and Claims
The paper claims that the current experimental situation, characterized by a SM-like $K^+ \to \pi^+\nu\bar{\nu}$ and SM-like $\varepsilon_K$ , does not rule out significant NP effects in other rare kaon decays. The authors argue that a detailed, sophisticated SMEFT analysis is premature given current data uncertainties; instead, they provide a "simple" demonstration that large enhancements in $K_L \to \pi^0\nu\bar{\nu}$ , $K_S \to \mu^+\mu^-$ , $K_L \to \pi^0\ell^+\ell^-$ , and $\varepsilon'/\varepsilon$ are theoretically consistent with existing constraints if both LH and RH couplings with new phases are present.

The authors conclude that their work serves as a strong motivation for the continuation of experimental programs, specifically:

The KOTO II experiment's search for $K_L \to \pi^0\nu\bar{\nu}$ and $K_L \to \pi^0\ell^+\ell^-$ .
The LHCb experiment's search for $K_S \to \mu^+\mu^-$ .
Continued theoretical and lattice QCD efforts to refine the SM prediction of $\varepsilon'/\varepsilon$ .

The paper emphasizes that while the specific $Z'$ scenario is an illustration, the general mechanism (requiring both LH and RH couplings) suggests that the "fate" of $K_L \to \pi^0\nu\bar{\nu}$ may differ significantly from $K^+ \to \pi^+\nu\bar{\nu}$ , offering a promising avenue for discovering physics beyond the Standard Model.

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