$η^{(\prime)}\toπ^+π^-l^+l^-$ decays in the NJL model

Using the Nambu--Jona-Lasinio model, this paper calculates the branching ratios for the dilepton anomalous decays $\eta^{(\prime)}\to\pi^+\pi^-l^+l^-$ by determining low-energy parameters through experimental constraints and $\eta$-$\eta^\prime$ mixing schemes, ultimately showing that the theoretical predictions align fully with available experimental data.

The Gist

Imagine the subatomic world as a bustling, chaotic city where particles are constantly colliding, transforming, and dancing. In this city, there are two very famous "celebrities": the Eta ($\eta$) and the Eta-prime ($\eta'$) mesons. They are heavy, unstable particles that don't like to stay still; they quickly decay (break apart) into lighter particles.

This paper is like a detective story where physicists try to predict exactly how these celebrities break apart, specifically when they split into a pair of pions (like a mother and child) and a pair of leptons (like a tiny electron or muon couple).

Here is the breakdown of the story using simple analogies:

1. The Mystery: The "Lepton" Leak

Usually, when an Eta meson decays, it might spit out a photon (a particle of light). But sometimes, that photon is so energetic it instantly turns into a pair of charged particles (an electron and a positron, or a muon and an antimuon). This is the $\eta \to \pi^+\pi^- l^+l^-$ decay.

The scientists wanted to calculate the probability of this happening. Think of it like trying to predict the exact odds of a specific, rare dance move in a crowded ballroom. To do this, they used a theoretical tool called the NJL Model (Nambu-Jona-Lasinio).

The NJL Model Analogy:
Imagine the NJL model as a giant, complex simulation game. In this game, the "players" are quarks (the building blocks of protons and neutrons). The model simulates how these quarks interact to form mesons. It's like a video game physics engine that tries to recreate the rules of the universe from the bottom up.

2. The Problem: The Missing "Glue"

When the scientists ran their simulation, they hit a snag. The math worked perfectly for some parts of the decay, but there was a mysterious "glue" holding the pieces together that the standard rules couldn't calculate.

In the language of the paper, this is a parameter called $\delta$ (delta).

• The Analogy: Imagine you are baking a cake (the decay). You know exactly how much flour (quarks) and sugar (energy) you need. But the cake keeps collapsing. You realize there's a secret ingredient—maybe a specific type of yeast—that makes the cake rise perfectly. The standard recipe doesn't tell you how much yeast to use; it just says "add some."
• In physics terms, this "yeast" is a result of a weird quantum effect called a triangle diagram. It's a loop of quarks that creates a tiny "surface term" (a mathematical glitch that actually has real physical consequences). Because of how the math works, the model can't predict the exact amount of this "yeast" from first principles alone.

3. The Solution: Using the "Real World" to Calibrate

Since the model couldn't predict the amount of "yeast" ($\delta$) on its own, the scientists had to look at the real world to calibrate their simulation.

They looked at a simpler, related decay: $\eta \to \pi^+\pi^- \gamma$ (where the photon doesn't turn into leptons).

• The Analogy: It's like a chef who can't figure out the exact amount of salt needed for a new soup. So, they taste a similar, well-known soup (the photon decay) that has already been tasted by thousands of people (experimental data). By seeing how salty the known soup is, they can deduce exactly how much salt to put in the new soup.

By matching their simulation to the real-world data of the photon decay, they could finally pin down the value of the mysterious $\delta$.

4. The Connection: The "Slope" and the "Mix"

Once they knew the "yeast" amount, they discovered a beautiful, simple relationship between two numbers:

1. $\delta$ (Delta): The "yeast" (the anomaly/glitch).
2. $\alpha$ (Alpha): The "slope" (how the decay probability changes as the particles move faster).

The Analogy:
Imagine a ramp.

• $\delta$ is the angle of the ramp.
• $\alpha$ is how steep the ramp feels to a rolling ball.
  The paper proves that if you know the angle ($\delta$), you can mathematically calculate the steepness ($\alpha$) without measuring it again. This is a huge deal because it links two seemingly different aspects of the decay into one simple rule.

They also had to deal with $\eta$-$\eta'$ mixing.

• The Analogy: Think of the Eta and Eta-prime as two twins who look very similar but have different personalities. They are actually "mixtures" of two different underlying states (like a smoothie made of strawberry and banana). Sometimes the strawberry flavor dominates; sometimes the banana does. The paper shows that how these twins mix affects the "yeast" amount, and they tested different mixing recipes to see which one matched reality.

5. The Grand Finale: Predicting the Rare Decay

With the "yeast" calibrated and the "mixing" understood, the scientists finally ran the simulation for the rare dilepton decay (the one with the electron/muon pair).

• The Result: Their predictions matched the experimental data almost perfectly.
• Why it matters: Previous theories (like the "Hidden Local Symmetry" approach) were like a map that was missing a few key roads. They predicted the decay would happen, but the numbers were off by a factor of two. The NJL model, with its "yeast" correction, drew the map correctly.

Summary in One Sentence

The paper shows that by using a detailed simulation of quark interactions and calibrating it with real-world data from a simpler decay, physicists can finally accurately predict the odds of a very rare, complex particle breakup, proving that a tiny, previously misunderstood quantum "glitch" is actually the key to the whole puzzle.

The Takeaway:
Science isn't just about knowing all the rules; sometimes it's about realizing there's a hidden variable (the "yeast") that you have to measure in the real world to make your theory work. Once you find it, everything clicks into place.

Technical Summary

Here is a detailed technical summary of the paper "η(′) →π+π−l+l− decays in the NJL model" by Volkov et al.

1. Problem Statement

The paper addresses the theoretical description of anomalous decays of the $\eta$ and $\eta'$ mesons into a pion pair and a dilepton pair ($\eta^{(\prime)} \to \pi^+\pi^- l^+l^-$, where $l = e, \mu$). While the related radiative decays $\eta^{(\prime)} \to \pi^+\pi^-\gamma$ have been studied extensively, the dilepton modes offer a unique probe into the hadronic structure of the amplitude under conditions of non-zero photon virtuality ($q^2 \neq 0$).

Previous theoretical approaches, such as Chiral Perturbation Theory (ChPT) combined with dispersion relations or hidden local symmetry (HLS) models, often assume specific constraints (e.g., vanishing low-energy constants in the chiral limit or strict adherence to the KSFR relation). The authors argue that these approaches may be insufficient because recent experimental data suggests that explicit SU(3) flavor symmetry breaking plays a more significant role in the anomalous sector than previously accounted for, particularly in the contact terms of the effective Lagrangian.

2. Methodology

The authors utilize the Nambu–Jona-Lasinio (NJL) model with $U(3) \times U(3)$ symmetry, explicitly including vector ($\rho$) and axial-vector ($a_1, f_1$) mesons. The calculation proceeds through the following steps:

• Amplitude Construction: The decay amplitude is constructed by combining the anomalous Wess-Zumino-Witten (WZW) term with meson loop diagrams. The process involves two main topological contributions:
  1. Box Anomaly: A direct contact term involving the photon and three pseudoscalars (VAAA-type).
  2. Triangle Anomaly: A quark loop diagram involving intermediate vector/axial-vector mesons.
• Handling PA Mixing: A critical technical step involves eliminating off-diagonal pseudoscalar-axial vector (PA) transitions (e.g., $\pi-a_1$ and $\eta-f_1$). The authors employ a $U(1)_{em}$ gauge-covariant transformation to remove these mixings. This procedure generates surface terms arising from the finite but formally linearly divergent triangle quark diagrams.
• Low-Energy Parameters: These surface terms introduce two reaction-specific low-energy parameters, $\delta$ (for $\eta$) and $\delta'$ (for $\eta'$), which modify the polynomial part of the decay amplitude.
• Mixing Schemes: The study investigates the impact of different $\eta-\eta'$ mixing schemes on the results, comparing:
  • A one-angle scheme (single mixing angle $\theta$).
  • A two-angle scheme (mixing angles $\theta_8, \theta_0$ and decay constants $f_8, f_0$).
• Virtual Photon Extension: The amplitude for $\eta^{(\prime)} \to \pi^+\pi^-\gamma$ is extended to the dilepton case by attaching a virtual photon propagator ($1/q^2$) and the lepton current, incorporating the Vector Meson Dominance (VMD) form factor$F_V(s_{\pi\pi})$for the$\rho^0$ contribution.

3. Key Contributions and Theoretical Insights

A. The Origin and Role of $\delta^{(\prime)}$

The paper establishes that the parameter $\delta^{(\prime)}$ is a residual effect of the surface terms generated by PA mixing elimination.

• In the NJL model, $\delta^{(\prime)}$ is not zero even in the chiral limit (though it vanishes strictly in the limit of exact SU(3) symmetry).
• The value of $\delta^{(\prime)}$ is arbitrary within the model due to the shift ambiguity of the triangle integral. Therefore, it cannot be calculated purely from first principles in the NJL framework but must be fixed by experimental data.

B. The $\delta^{(\prime)}$ - $\alpha^{(\prime)}$ Relationship

A major theoretical contribution is the derivation of a direct relationship between the contact term parameter $\delta^{(\prime)}$ and the slope parameter $\alpha^{(\prime)}$ (which describes the energy dependence of the amplitude):
$$\alpha^{(\prime)} = \frac{1}{m_\rho^2} \left[ \frac{3}{a(1+\delta^{(\prime)})} - 1 \right]$$
where $a \approx 1.87$.

• This formula implies that the slope $\alpha^{(\prime)}$ contains crucial information about explicit SU(3) symmetry breaking.
• If one assumes $\delta^{(\prime)} = 0$ (as in some HLS models), the predicted slope $\alpha$ is roughly half the phenomenological value. The inclusion of $\delta^{(\prime)} \neq 0$ resolves this discrepancy.

C. Impact of Mixing Schemes

The authors demonstrate that the extracted values of $\delta$ and $\alpha$ depend on the chosen $\eta-\eta'$ mixing scheme.

• One-angle scheme: Yields consistent results with experimental decay widths when $\delta \approx -0.1$ to $-0.15$.
• Two-angle scheme: Also yields consistent results but highlights that for $\eta' \to \pi^+\pi^-\gamma$, a linear polynomial approximation for the amplitude may be insufficient, suggesting the need for quadratic terms or higher-order corrections (e.g., $a_2(1320)$ contributions).

4. Results

A. Parameter Determination

By fitting the NJL model predictions to the experimental decay widths $\Gamma(\eta^{(\prime)} \to \pi^+\pi^-\gamma)$ and the measured slope parameters $\alpha^{(\prime)}$ from WASA-at-COSY, KLOE, and CRYSTAL BARREL collaborations, the authors determine:

• $\delta \approx -0.1$ to $-0.25$
• $\delta' \approx -0.23$ to $-0.39$
• These values lead to slope parameters $\alpha \approx 1.3 - 1.5 \, \text{GeV}^{-2}$ and $\alpha' \approx 2.1 - 2.7 \, \text{GeV}^{-2}$, which are in excellent agreement with experimental data.

B. Branching Ratios for Dilepton Decays

Using the determined parameters, the authors calculate the branching ratios for the dilepton modes:

• $\eta \to \pi^+\pi^- e^+e^-$: $2.69(15) \times 10^{-4}$(Matches PDG:$2.68(11) \times 10^{-4}$).
• $\eta \to \pi^+\pi^- \mu^+\mu^-$: $7.26(46) \times 10^{-9}$.
• $\eta' \to \pi^+\pi^- e^+e^-$: $2.20(10) \times 10^{-3}$(Matches PDG:$2.42(10) \times 10^{-3}$).
• $\eta' \to \pi^+\pi^- \mu^+\mu^-$: $1.90(26) \times 10^{-5}$(Matches PDG:$1.9(4) \times 10^{-5}$).

The results are consistent with both experimental data and alternative theoretical approaches (ChPT + unitarization), validating the NJL model's structural description of the hadronic form factor at non-zero $q^2$.

5. Significance

• Validation of NJL Model: The study confirms that the NJL model, when properly accounting for PA mixing and surface terms, can accurately describe complex anomalous decays involving virtual photons, a regime where many effective theories struggle.
• Resolution of Slope Discrepancy: The paper provides a theoretical justification for the large experimental values of the slope parameter $\alpha$ by linking it to the non-zero contact term parameter $\delta$, which arises from explicit SU(3) symmetry breaking.
• Phenomenological Benchmark: The calculated branching ratios serve as precise benchmarks for future experimental searches, particularly for the rare $\eta \to \pi^+\pi^-\mu^+\mu^-$ decay, where the theoretical upper bound is significantly lower than previous PDG averages.
• Insight into Anomalies: The work clarifies the interplay between box and triangle anomalies in the presence of vector mesons, showing that the balance between them is distorted by SU(3) breaking, a feature captured naturally by the NJL bosonization procedure.

In conclusion, the paper successfully extends the NJL model's predictive power to semileptonic anomalous decays, demonstrating that the model's structural parameters, once fixed by radiative decay data, accurately predict dilepton branching ratios without free parameters.