A dispersive approach to the CP conserving $K\to\pi\ell^+\ell^-$ radiative decays

This paper employs a dispersive approach constrained by analyticity, unitarity, and recent Khuri-Treiman solutions for $K\to3\pi$ amplitudes to derive a minimal two-parameter representation of the form factors governing CP-conserving $K\to\pi\ell^+\ell^-$ decays, successfully reproducing experimental data and determining previously unknown signs and amplitude components.

The Gist

Imagine the universe as a giant, complex machine where tiny particles called Kaons (K) are like delicate, short-lived sparks. Occasionally, these sparks decay, breaking apart into other particles, including a Pion (π) and a pair of leptons (either electrons or muons, denoted as ℓ⁺ℓ⁻).

This paper is about figuring out exactly how these sparks break apart. Specifically, the authors are trying to solve a mystery regarding the "shape" of the energy distribution in these decays. They want to know the precise mathematical rules (called form factors) that govern this process, which are crucial for understanding the fundamental laws of physics, including why the universe has more matter than antimatter (CP violation).

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Blind" Map

Previously, scientists tried to map this decay using a "best guess" model (called the B1L model). It was like trying to draw a map of a city using only a few street signs and some guesses about traffic. It worked okay, but it had too many "free parameters"—basically, too many knobs you could turn to make the math fit the data. This made it hard to be sure if the theory was actually correct or just lucky.

The authors wanted a rigorous map. They wanted to use the fundamental laws of physics—specifically Analyticity (the idea that the map must be smooth and continuous, no sudden jumps) and Unitarity (the idea that probability is conserved; particles don't just vanish or appear out of nowhere).

2. The Solution: The "Dispersive" Approach

Instead of guessing, the authors used a technique called a Dispersive Representation.

• The Analogy: Imagine you are trying to reconstruct a song you only heard a few notes of.
  • The Old Way: You guess the rest of the melody based on what sounds "nice."
  • The New Way (Dispersive): You know the song must follow the laws of music theory (harmony, rhythm). You use those strict rules to mathematically "fill in" the missing parts. You don't guess; you calculate what must be there based on the rules.

The authors applied this to the Kaon decay. They realized that if you look at two specific combinations of the decay rates (let's call them Combination A and Combination B), the laws of physics force them to behave in a very specific way at high energies. This allowed them to strip away the guesswork and reduce the number of "knobs" they needed to turn down to just two.

3. The Secret Ingredient: The "Ghost" Amplitude

To make their map accurate, they needed a specific piece of data: how Kaons turn into three Pions (K → 3π).

• The Analogy: Think of the Kaon decay as a bridge. To cross the bridge, you need to know the strength of the pillars. One pillar is well-known (the "Delta I = 3/2" part), but the other pillar (the "Delta I = 1/2" part) is hidden in the fog.
• The Breakthrough: The authors used a sophisticated mathematical tool (the Khuri-Treiman equations) to look at the "foggy" pillar. They realized that even though this part is hard to see in the actual decay data, it leaves a fingerprint on the "Combination A" and "Combination B" they were studying.
• The Result: By analyzing the data they did have, they could mathematically deduce the value of this hidden "ghost" amplitude. They found that this hidden part is actually quite significant when you look at the decay from a different angle (scattering), even if it's hard to see in the direct decay.

4. The Findings: Signs and Directions

The paper solves two major puzzles:

• Puzzle 1: The Sign of the Charged Decay (W+)
  The authors looked at the charged Kaon decay (K⁺ → π⁺ℓ⁺ℓ⁻). They tested two possibilities: Is the underlying value positive or negative?

  • The Result: They found that a positive value creates a shape that contradicts experimental data (it would look like a hill going down, but the data shows a hill going up). Therefore, the value must be negative. This is a definitive answer that previous models couldn't give with certainty.

• Puzzle 2: The Sign of the Neutral Decay (WS)
  The neutral Kaon decay (Kₛ → π⁰ℓ⁺ℓ⁻) is much rarer and harder to measure. The authors showed that the current data isn't precise enough to tell if the value is positive or negative yet. However, they proved that if we get better data in the future, the shape of the energy curve will reveal the sign. They also showed that the charged and neutral decays are "coupled" (like two dancers holding hands); knowing one helps constrain the other.

5. The "Resonance" Effect

The authors also accounted for "resonances."

• The Analogy: Imagine driving a car. Sometimes the road has a bump (a resonance) that makes the car bounce. In particle physics, particles like the rho (ρ) and K-star (K)* act like these bumps.
• The authors showed that these "bumps" are essential. If you ignore them, your map is wrong. They included the effects of these bumps (and even heavier, invisible bumps like the omega and phi particles) to ensure their map was accurate. They found that while the heavier bumps matter, the lighter ones (rho and K-star) are the main drivers of the shape.

Summary

In short, this paper is a mathematical masterclass in reconstruction.

1. They stopped guessing and started using the strict laws of physics (Analyticity and Unitarity) to build a model.
2. They reduced the model to the absolute minimum number of variables needed.
3. They used the model to prove that the charged Kaon decay has a negative sign (ruling out the positive possibility).
4. They showed how to find a "hidden" part of the neutral decay that was previously unknown.
5. They demonstrated that with better future data, we can finally pin down the exact nature of these rare decays, which is a crucial step in understanding the fundamental asymmetry of our universe.

It's like taking a blurry, noisy photograph of a crime scene, using the laws of physics to sharpen the image, and finally being able to read the license plate of the getaway car.

Technical Summary

Here is a detailed technical summary of the paper "A dispersive approach to the CP conserving $K \to \pi \ell^+ \ell^-$ radiative decays" by Bernard, Descotes-Genon, Knecht, and Moussallam.

1. Problem Statement

The paper addresses the theoretical description of the CP-conserving rare radiative decay modes $K^+ \to \pi^+ \ell^+ \ell^-$ and $K_S \to \pi^0 \ell^+ \ell^-$ (where $\ell = e, \mu$). These decays are governed by form factors $W_+(s)$ and $W_S(s)$, where $s$ is the dilepton invariant mass squared.

• Current Limitations: Previous analyses relied on the "Beyond One-Loop" (B1L) model, which uses a quadratic expansion in terms of Dalitz parameters ($X, Y$) derived from $K \to 3\pi$ data. This approach introduces four free parameters ($a_\pm, b_\pm$) and lacks a rigorous foundation in the fundamental non-perturbative properties of QCD (analyticity and unitarity).
• The Challenge: Determining the magnitude and phase of these amplitudes is crucial for calculating the indirect CP-violating contribution to the ultra-rare decay $K_L \to \pi^0 \ell^+ \ell^-$. Current lattice QCD results are not yet precise enough to fix the relative phase. Furthermore, the sign of the interference term and the specific values of the parameters $a_+$ and $a_S$ remain ambiguous.
• Goal: To develop a model-independent dispersive framework that reduces the number of free parameters, incorporates the correct analytic structure (including resonances like $\rho(770)$ and $K^*(892)$), and utilizes recent advances in $K \to 3\pi$ amplitudes.

2. Methodology

The authors construct a dispersive representation of the form factors based on analyticity and unitarity.

A. Decomposition into Isospin Channels

To handle the coupling between charged and neutral modes, the authors define two linear combinations corresponding to $K\pi$ isospin states:

• $W^{[1/2]}(s) \equiv 2W_+(s) - W_S(s)$ (Isospin $I=1/2$)
• $W^{[3/2]}(s) \equiv W_+(s) + W_S(s)$ (Isospin $I=3/2$)

This diagonalizes the $K\pi \to K\pi$ scattering matrix, allowing the unitarity relations to be written in terms of elastic phase shifts $\delta_1^I(s)$.

B. Dispersive Representations (Muskhelishvili-Omnès)

The authors derive integral equations for these combinations:

1. $W^{[3/2]}(s)$: Based on asymptotic behavior analysis (Operator Product Expansion), this combination is shown to satisfy a once-subtracted (or unsubtracted, depending on the specific asymptotic fall-off assumption) dispersion relation. The authors argue it requires no subtraction constants (or effectively one determined by the $\pi\pi$ cut) because the $I=3/2$ component of the local operator $O_{7V}$ vanishes.
2. $W^{[1/2]}(s)$: This combination requires one subtraction. The subtraction constant is identified with the physical value at $s=0$, related to the parameter $a_+$.

The resulting system of coupled equations (Eqs. 21 and 25) expresses $W_+(s)$ and $W_S(s)$ in terms of:

• The $\pi\pi$ discontinuity (driven by the pion vector form factor $F_V^{\pi\pi}$ and $K\pi \to \pi\pi$ amplitudes).
• The $K\pi$ discontinuity (driven by $K\pi$ elastic phase shifts and the vector form factor $f_+^{K\pi}$).
• Two free parameters: $a_+$ and $a_S$ (values of the form factors at $s=0$).

C. Inputs and Extrapolation

• $K \to 3\pi$ Amplitudes: The authors utilize a set of 11 independent solutions to the Khuri-Treiman (KT) integral equations (from a previous work, Ref. [23]). These solutions provide a rigorous analytic continuation of $K \to 3\pi$ amplitudes from the physical decay region into the resonant $K\pi \to \pi\pi$ scattering region (up to $\sim 1.3$ GeV).
• Unknown Parameter $\tilde{\mu}_1$: The KT solution for the $K_S \to \pi^+\pi^-\pi^0$ amplitude contains a parameter $\tilde{\mu}_1$ corresponding to the $\Delta I = 1/2$ component. This is chirally suppressed in the physical Dalitz plot and undetermined by current data. The authors show that the dispersive relations allow $\tilde{\mu}_1$ to be determined once $a_+$ and $a_S$ are fixed.
• Resonances: The framework explicitly includes the $\rho(770)$ and $K^*(892)$ resonances via the Omnès functions. Contributions from heavier resonances ($\omega, \phi$) and higher mass states are estimated using chiral-resonance Lagrangians and Regge models, respectively.

3. Key Contributions

1. Minimal Parameter Set: The authors reduce the description of $K \to \pi \ell^+ \ell^-$ to a system dependent on only two parameters ($a_+$ and $a_S$), compared to the four parameters in the B1L model.
2. Coupled System: They demonstrate that the $K^+$ and $K_S$ amplitudes are coupled via $K\pi \to K\pi$ scattering. This means information from the charged mode ($K^+$) can constrain the neutral mode ($K_S$) and vice versa.
3. Determination of $\tilde{\mu}_1$: They derive a linear relation (Eq. 37) between the unknown $\Delta I = 1/2$ parameter $\tilde{\mu}_1$ and the sum $a_+ + a_S$. This allows the extraction of the previously undetermined isospin component of the $K_S \to 3\pi$ amplitude.
4. Sign Determination: The approach provides a rigorous way to determine the sign of the form factors, which is critical for calculating CP-violating interference terms.

4. Results

• Sign of $a_+$: The analysis strongly disfavors a positive value for $a_+$. A positive $a_+$ results in a decreasing $|W_+(s)|^2$ with energy, which contradicts experimental data (NA62, NA48, E865). The data favors $a_+ < 0$ (specifically $a_+ \approx -0.575$).
• Sign of $a_S$:
  • The branching fractions for $K_S \to \pi^0 \ell^+ \ell^-$ are consistent with both positive and negative $a_S$ (approx. $\pm 1.1$).
  • However, the energy dependence of $|W_+(s)|^2$ is sensitive to the sign of $a_S$. The best fit to $K^+$ data is achieved when $a_S > 0$ (specifically $a_S \approx +1.06$).
  • Including contributions from $I=0$ resonances ($\omega, \phi$) slightly modifies the slopes but does not change the preference for $a_S > 0$ in the $W_+$ channel.
• Branching Fractions: Using the preferred parameters ($a_+ = -0.575, a_S = +1.06$), the calculated branching fractions are:
  • $BF(K^+ \to \pi^+ e^+ e^-) = (3.01 \pm 0.05) \times 10^{-7}$ (Matches PDG: $3.00 \pm 0.09$)
  • $BF(K^+ \to \pi^+ \mu^+ \mu^-) = (8.89 \pm 0.24) \times 10^{-8}$ (Matches PDG: $9.17 \pm 0.14$)
• Comparison with B1L: The dispersive amplitude agrees with the B1L model in the real part below the $\pi\pi$ threshold but shows a significantly larger imaginary part (by a factor of $\sim 2$) in the physical region, attributed to $K \to 3\pi$ loops.

5. Significance

• Theoretical Rigor: This work moves beyond phenomenological fits by grounding the description in the fundamental principles of QCD (unitarity and analyticity).
• CP Violation Constraints: By fixing the sign of $a_+$ and constraining $a_S$, the paper provides essential inputs for calculating the indirect CP-violating contribution to $K_L \to \pi^0 \ell^+ \ell^-$. This is a "golden mode" for testing the Standard Model and searching for New Physics.
• Future Prospects: The authors note that while current data cannot uniquely fix the sign of $a_S$ solely from $K_S$ branching ratios, a combined fit of $|W_+|^2$ and $|W_S|^2$ with improved experimental statistics (expected from LHCb and NA62) could definitively determine the sign of $a_S$ and the value of $\tilde{\mu}_1$.
• Methodological Advance: The successful application of Khuri-Treiman equations to extrapolate $K \to 3\pi$ amplitudes into the scattering region demonstrates a powerful tool for analyzing rare decays where direct experimental data is scarce.