CP violation in $K\toμ^+μ^-$ with and without time dependence through a tagged analysis

This paper proposes a method to extract short-distance information and resolve discrete ambiguities in the Standard Model prediction for $K^0_L \to \mu^+\mu^-$ by analyzing time-integrated and time-dependent CP asymmetries, demonstrating that an LHCb-like experiment could constrain the relevant short-distance amplitude to within 35% of its Standard Model value and resolve the ambiguity at over 3$\sigma$ significance.

The Gist

Imagine the universe as a giant, complex clockwork machine. For decades, physicists have used a specific set of rules, called the "CKM paradigm," to predict how the gears of this machine turn. These rules explain why certain particles behave the way they do, but to be sure the machine is perfect, we need to check it in places we haven't looked closely before.

This paper is about checking one very specific, tiny, and rare gear: a particle called a neutral Kaon (think of it as a short-lived, unstable speck of matter) that occasionally decays into two muons (heavy cousins of electrons).

Here is the breakdown of what the authors are doing, using simple analogies:

1. The Mystery of the "Ghost" Signal

When a neutral Kaon decays into two muons, it's like a magician pulling a rabbit out of a hat. But there's a problem: the "rabbit" (the decay) is made of two parts:

• The Short-Distance Part: This is the "magic trick" itself—the direct, fundamental interaction we want to study to test our rules.
• The Long-Distance Part: This is the "smoke and mirrors"—background noise caused by the particle interacting with its surroundings before it decays.

Currently, the "smoke" is so loud that it's hard to hear the "magic trick." The authors propose a clever way to filter out the noise. They suggest that by measuring how the particle behaves over time and comparing it to other known decays, we can isolate the "short-distance" signal. It's like listening to a specific instrument in an orchestra by knowing exactly when the other instruments are silent.

2. The "Left-Handed vs. Right-Handed" Puzzle

The paper focuses on a specific measurement called CP Asymmetry. Imagine a coin that is supposed to be fair (landing heads or tails 50/50). In the world of these particles, the "coin" is slightly weighted.

• The authors want to know: Is the coin weighted to the left or the right?
• Currently, theory predicts the coin is weighted, but we don't know which direction. This creates a "discrete ambiguity" (a two-way fork in the road).
• The paper argues that if we can measure the sign of this asymmetry (is it positive or negative?), we instantly know which path is correct. This would clear up a major confusion in our predictions for how often these particles decay.

3. The LHCb "Super-Microscope"

To do this, the authors look at the LHCb experiment at CERN (a massive particle collider). They are essentially asking: "Can we build a camera fast enough and smart enough to catch this rare event?"

They simulate two scenarios for the future of this camera:

• Upgrade I: A slightly better camera.
• Upgrade II: A super-camera with a new lens (called the "Upstream Pixel") that can see particles traveling further away from the collision point.

The Analogy: Imagine trying to spot a specific rare bird in a forest.

• Upgrade I lets you see the birds in the trees right next to you.
• Upgrade II gives you a telescope that lets you see birds in the trees 100 meters away, effectively doubling the number of birds you can count.

4. The "Tagging" Trick

To know if a particle is a "Kaon" or an "Anti-Kaon" (its twin), the scientists need a "tag."

• The Analogy: Imagine a dance floor where couples (particles) are created. If you see a partner wearing a red hat (a charged kaon) dancing with someone, you know the partner is a "Kaon." If the partner wears a blue hat, it's an "Anti-Kaon."
• The authors found that at the LHC, it is actually easier to spot these "hats" for Kaons than for other particles (like B-mesons) because the "crowd" around Kaons is less chaotic. This means they can tag the particles with high accuracy, which is crucial for the measurement.

5. The Results: What Can We Expect?

The authors ran thousands of computer simulations to see what would happen if they ran this experiment with the future upgrades.

• The Goal: They want to measure a specific number (related to the "weight" of the coin mentioned earlier) with about 35% precision.
• The Breakthrough: They claim that by the end of the High-Luminosity LHC era (a future phase of the collider), they could solve the "Left vs. Right" ambiguity with more than 99.7% certainty (3 sigma).
• The Catch: This requires "optimistic" conditions. They need to be very good at filtering out background noise (like ignoring the "smoke" in the magician's hat) and they need the new "Upstream Pixel" detector to work perfectly.

Summary

In short, this paper is a proposal and a feasibility study. It says:

"We have a mathematical trick to isolate a rare, fundamental signal from the noise. If we upgrade our particle detector (LHCb) to its next generation, we have a very good chance of catching this rare event, measuring a key number that tests our laws of physics, and finally solving a long-standing puzzle about the direction of 'time-reversal' in these particles."

It does not claim to have done the experiment yet, nor does it suggest immediate medical or technological applications. It is purely about refining our understanding of the fundamental rules of the universe.

Technical Summary

Technical Summary: CP Violation in $K \to \mu^+\mu^-$ with and without Time Dependence through a Tagged Analysis

Problem and Motivation
The CKM paradigm remains the standard framework for describing CP violation, yet independent cross-checks using kaon physics are essential to scrutinize the Standard Model (SM) in uncharted territories. While the rare decay $K^+ \to \pi^+\nu\bar{\nu}$ has recently been observed, the neutral kaon decay $K^0 \to \mu^+\mu^-$ offers a unique probe of short-distance physics. Historically, the theoretical prediction for the branching ratio $\mathcal{B}(K^0_L \to \mu^+\mu^-)$ suffers from a significant discrete ambiguity. This ambiguity arises from the unknown relative sign between the short-distance amplitude and the long-distance contributions (specifically the $K^0_L \to \gamma\gamma$ amplitude), leading to two distinct SM predictions for the decay rate.

Current experimental knowledge of $\mathcal{B}(K^0_L \to \mu^+\mu^-)$ and $\mathcal{B}(K^0_L \to \gamma\gamma)$ is precise, but the short-distance parameters remain obscured by long-distance uncertainties. Furthermore, the time-integrated CP asymmetry, $A_{CP}(K^0 \to \mu^+\mu^-)$, has not yet been measured experimentally. The paper addresses the question of what information can be extracted if time-dependent measurements are out of reach, and whether a tagged analysis at the LHCb experiment can resolve the existing ambiguities.

Methodology
The authors propose a framework that bypasses the challenge of estimating long-distance physics by focusing on observables dictated primarily by short-distance contributions. The analysis relies on the interference effects between $K_L$ and $K_S$ decays to a dimuon pair.

1. Theoretical Framework:

   • The time-dependent decay rate for an initial $K^0$ or $\bar{K}^0$ beam is expressed in terms of four experimental parameters: $C_L$, $C_S$, $C_{Int.}$, and $\phi_0$. These map to four theory parameters: three magnitudes and a relative phase.
   • The key to extracting the short-distance amplitude is the interference term $C_{Int.}$, which is related to the short-distance dominated $\ell=0$ mode of $K_S \to \mu^+\mu^-$.
   • The authors demonstrate that the combination of four observables—$\mathcal{B}(K^0_L \to \gamma\gamma)$, $\mathcal{B}(K^0_L \to \mu^+\mu^-)$, $\mathcal{B}(K^0_S \to \mu^+\mu^-)$, and the time-integrated CP asymmetry $\tilde{A}_{CP}(K^0 \to \mu^+\mu^-)$—fully determines the system's non-vanishing theory parameters.
   • Crucially, the sign of the measured CP asymmetry determines the sign of $\cos \phi_0$, which in turn resolves the discrete ambiguity in the SM prediction for $\mathcal{B}(K^0_L \to \mu^+\mu^-)$ and the sign of the $K^0_L \to \gamma\gamma$ amplitude.

2. Experimental Feasibility Study (LHCb):

   • The study utilizes a fast simulation of the LHCb experiment, considering both Upgrade I and Upgrade II detector configurations.
   • Flavor Tagging: The initial flavor of the $K^0$ is determined by searching for a companion charged kaon in $pp \to K^0 K^- X$ processes (Same Side Kaon tagging). The authors estimate a tagging power ($TP$) of approximately 22%, significantly higher than that achieved for $B_s^0$ mesons due to lower charged kaon multiplicity in the underlying event.
   • Detector Acceptance: Two scenarios are defined:
     • LL (Long Tracks): Only decays inside the Vertex Locator (VELO) volume are used. This corresponds to Upgrade I.
     • LL+DD (Long + Downstream): Decays downstream of the VELO are reconstructed using a proposed Upstream Pixel (UP) detector. This scenario, relevant for Upgrade II, potentially doubles the sample size and improves mass resolution.
   • Background Suppression: The primary background is $K^0_S \to \pi^+\pi^-$ where pions decay in flight to muons. The authors propose using impact parameter cuts to suppress this background, estimating that the background can be reduced by one to two orders of magnitude with a manageable loss in signal efficiency.
   • Sensitivity Analysis: The sensitivity to the short-distance parameter combination $|A^2 \lambda^5 \bar{\eta}|$ is extracted via an unbinned extended maximum likelihood fit to the decay-time distributions of tagged $K^0$ and $\bar{K}^0$ decays. The analysis assumes the branching ratio $\mathcal{B}(K^0_S \to \mu^+\mu^-)$ can be constrained with $\mathcal{O}(16\%)$ precision.

Key Contributions and Results

• Resolution of Discrete Ambiguity: The paper establishes that measuring the sign of the time-integrated CP asymmetry, $\tilde{A}_{CP}$, is sufficient to determine the relative sign between short- and long-distance amplitudes. This eliminates the two-fold ambiguity in the SM prediction for $\mathcal{B}(K^0_L \to \mu^+\mu^-)$.
• Short-Distance Constraints: The authors derive that the combined measurement of $\tilde{A}_{CP}$ and $\mathcal{B}(K^0_S \to \mu^+\mu^-)$ allows for the extraction of the CKM parameter combination $(\lambda^5 A^2 \bar{\eta})^2$.
• LHCb Sensitivity Projections:
  • In an optimistic scenario (Upgrade II, $TP \approx 22\%$, significant background suppression, and inclusion of downstream tracks), the LHCb experiment could constrain the short-distance amplitude proportional to $|A^2 \lambda^5 \bar{\eta}|$ to approximately 35% of its SM value.
  • The same setup is projected to resolve the sign of the CP asymmetry (and thus the sign of the $K^0_L \to \gamma\gamma$ amplitude) with a significance of greater than $3\sigma$ by the end of the High-Luminosity LHC era.
  • The effective yield ($Y_{eff}$) required to achieve these sensitivities is estimated to be around 300 events.

Significance and Claims
The paper claims that a tagged analysis of $K^0 \to \mu^+\mu^-$ at LHCb offers a unique opportunity to perform a direct measurement of the CP-violating parameter $|\bar{\eta}|$ with competitive precision, complementary to efforts in $B$ physics and other kaon channels like $K^0_L \to \pi^0 \nu \bar{\nu}$.

The authors emphasize that the measurement of the CP asymmetry sign is a critical "smoking gun" for the SM prediction of $\mathcal{B}(K^0_L \to \mu^+\mu^-)$, as it removes the theoretical ambiguity currently limiting the interpretation of this decay rate. While the study relies on optimistic assumptions regarding background rejection and the performance of the future Upstream Pixel detector, the authors argue that even less optimistic scenarios would yield very interesting levels of precision. The work serves as a call for dedicated studies of the experimental requirements to realize this analysis within the LHCb program.