Improved analysis of rare $Z$-boson decays into a heavy vector quarkonium plus lepton pair

This paper improves theoretical predictions for rare $Z$-boson decays into heavy vector quarkonium and lepton pairs by including all relevant Standard Model tree-level diagrams, revealing that while fragmentation dominates charmonium channels, additional diagrams increase bottomonium branching fractions by 4–9%, and confirming that forward-backward asymmetries vanish in the Standard Model to provide a baseline for future experimental tests.

The Gist

Imagine the Z-boson as a massive, unstable "super-parent" particle living in the subatomic world. It's so heavy and energetic that it loves to break apart into smaller pieces. Usually, it splits into pairs of invisible ghosts (neutrinos) or familiar particles like electrons. But sometimes, in very rare cases, it tries to give birth to something exotic: a heavy, glowing "quarkonium" (a tight ball of heavy quarks, like a miniature atom) plus a pair of lepton twins (electrons or muons).

This paper is like a team of theoretical detectives (Li Ang and Dao-Neng Gao) who have decided to re-examine the blueprints of how this rare event happens. They want to make sure their predictions are perfect before the next generation of giant particle colliders (like the FCC-ee or High-Luminosity LHC) starts taking pictures of these events.

Here is the breakdown of their investigation using simple analogies:

1. The Main Event: The "Fragmentation" Shortcut

Previously, physicists thought the Z-boson decayed in one specific, dominant way. They imagined the Z-boson first splitting into a pair of leptons and a "virtual photon" (a fleeting, invisible flash of light). This virtual photon then instantly "fragments" or condenses into the heavy quarkonium ball.

• The Analogy: Think of the Z-boson as a magician. It throws a ball (the virtual photon) into the air. As the ball falls, it magically transforms into a glowing lantern (the quarkonium) while two assistants (the leptons) fly out the other side.
• The Old View: Scientists thought this was the only trick the magician knew. They assumed all other tricks were so weak they didn't matter.

2. The New Discovery: The "Hidden" Paths

The authors of this paper said, "Wait a minute, let's look at every possible way this could happen, not just the most obvious one." They drew out all the other Feynman diagrams (the blueprints of particle interactions).

• The Analogy: Imagine the magician has a secret back door. Instead of throwing the ball and letting it transform, he might first hand the ball to a heavy assistant (the quark pair), who then juggles it and passes it to the leptons.
• The Finding:
  • For "Charmonium" (Lighter heavy balls like J/Ψ): The old "shortcut" (fragmentation) is so overwhelmingly powerful that the secret back doors are practically invisible. The old predictions were almost perfect.
  • For "Bottomonium" (Heavier, denser balls like Υ): The secret back doors are actually significant! When they included these extra paths, the predicted rate of these decays went up by 4% to 9%. It's like realizing that while the main stage show is the star, the backstage crew actually adds a noticeable amount of energy to the performance.

3. The "Symmetry" Test: The Perfectly Balanced Scale

One of the most interesting things the team looked at was the direction the particles fly. Do the leptons prefer to fly forward or backward relative to the Z-boson's original path?

• The Analogy: Imagine a perfectly balanced seesaw. If you sit on one side, it tips. If the physics is "Standard Model" (the current rules of the universe), the seesaw is perfectly balanced. The leptons fly forward and backward with zero preference. The distribution is perfectly symmetrical.
• The Result: In the Standard Model, the "Forward-Backward Asymmetry" is exactly zero. If you see a tilt in the future experiments, it means the rules of the universe have changed.

4. The "New Physics" Hunt: Looking for the Tilt

Why does this matter? Because if we see a tilt (a non-zero asymmetry), it's a smoking gun for New Physics.

• The Analogy: Imagine the Standard Model is a perfectly smooth, flat road. The authors calculated that on this road, a car (the lepton) should drive straight. However, they also calculated what would happen if there were hidden "potholes" or "winds" (New Physics, specifically something called "anomalous couplings") that push the car to the left or right.
• The Prediction: They found that if these "potholes" exist, they could create a tiny, measurable tilt (an asymmetry) of about 0.06% to 1%. It's a tiny number, but with the massive number of Z-bosons future machines will produce, we might finally see the road isn't as flat as we thought.

5. The Big Picture: Why Wait?

The paper concludes that while we have seen some of these decays (like the J/Ψ case), we haven't seen the others yet.

• The Outlook: Future machines will be like super-powered cameras taking billions of photos of Z-bosons. Because the authors have now provided a much more precise "map" of what should happen (including those 4-9% corrections for the heavy balls), experimentalists can compare their photos against this map.
  • If the photos match the map perfectly: The Standard Model is confirmed again.
  • If the photos show a tilt or a different rate: We have discovered a crack in the foundation of physics, pointing to new particles or forces we haven't met yet.

In summary: This paper is a meticulous "quality control" check. It says, "We used to think the heavy ball decays were simple. We checked the math again, found some small but important extra steps for the heaviest balls, and confirmed that the universe should be perfectly symmetrical in this process. If you see it break symmetry in the future, you've found something new!"

Technical Summary

Here is a detailed technical summary of the paper "Improved analysis of rare Z-boson decays into a heavy vector quarkonium plus lepton pair" (USTC-ICTS/PCFT-25-60).

1. Problem Statement

The paper addresses the theoretical prediction of rare exclusive decays of the Z boson into a heavy vector quarkonium ($V$) and a lepton pair ($\ell^+\ell^-$), denoted as $Z \to V \ell^+\ell^-$, where $V$ includes $J/\Psi$, $\Psi(2S)$, and $\Upsilon(nS)$ ($n=1,2,3$), and $\ell = e, \mu$.

• Context: While the Standard Model (SM) predicts these decays are dominated by the electromagnetic fragmentation process ($Z \to \gamma^* \ell^+\ell^-$ followed by $\gamma^* \to V$), previous literature (Refs. [13-15]) assumed that other tree-level diagrams (specifically $Z \to V \gamma^* \to V \ell^+\ell^-$) were suppressed by a factor of $m_V^2/m_Z^2$ and could be neglected.
• Motivation: With the accumulation of large Z-boson event samples at future facilities (HL-LHC, CEPC, FCC-ee), precise theoretical predictions are required to test the SM and search for New Physics (NP). The authors aim to perform a complete, systematic calculation of all tree-level SM contributions to determine if the "negligible" diagrams actually impact branching fractions or angular distributions, particularly for bottomonium states.

2. Methodology

The authors perform a complete tree-level calculation within the Standard Model and extend the analysis to include potential New Physics effects via anomalous couplings.

A. Standard Model Calculation

1. Feynman Diagrams: The study includes all relevant tree-level diagrams:
   • Fig. 1 (Dominant): Electromagnetic fragmentation ($Z \to \gamma^* \ell^+\ell^-$, $\gamma^* \to V$). This includes contributions where the virtual photon is emitted from the lepton line or the Z boson.
   • Fig. 2 (Sub-dominant): $Z \to V \gamma^* \to V \ell^+\ell^-$, where the Z couples to the heavy quark pair ($Q\bar{Q}$), which forms the quarkonium, and a virtual photon is emitted from the quark line.
   • Fig. 3 (Higher Order in Coupling): $Z \to \ell^+\ell^- Z^*$ followed by $Z^* \to V$. Although formally of order $g_Z^3$, it is included for completeness.
2. Amplitude Derivation:
   • The amplitudes are derived using the neutral current Lagrangian.
   • Hadronization:
     • For the fragmentation diagram (Fig. 1), the hadronic matrix element is parameterized by the decay constant $f_V$, extracted from the measured width $\Gamma(V \to e^+e^-)$.
     • For the $Z \to V \gamma^*$ diagram (Fig. 2), the authors employ the Non-Relativistic Color-Singlet Model (NRCSM). The heavy quark pair is projected into the quarkonium state using the wave function at the origin, $\psi_V(0)$.
   • Total Amplitude: $M = M_1 + M_2 + M_3$, where $M_1$ includes the photon and virtual Z contributions to the fragmentation channel, $M_2$ is the $Z \to V \gamma^*$ contribution, and $M_3$ is the $Z \to \ell^+\ell^- Z^*$ contribution.
3. Differential Rates: The differential decay rate $d^2\Gamma / ds d\cos\theta$ is calculated, where $s$ is the dilepton invariant mass squared and $\theta$ is the angle between the Z boson and the lepton in the dilepton rest frame.
4. Symmetry Analysis: The authors analytically prove that in the SM, the differential rate is an even function of $\cos\theta$, implying the Forward-Backward Asymmetry ($A_{FB}$) is strictly zero.

B. Beyond the Standard Model (BSM) Analysis

• Anomalous Couplings: The authors introduce CP-violating anomalous neutral triple gauge couplings (NTGC) via an effective $Z\gamma G^*$ vertex (where $G=Z, \gamma$).
• Mechanism: These effective vertices (specifically the dimension-6 $h^G_1$ terms) generate a new amplitude that interferes with the SM amplitude.
• Observable: This interference introduces a linear term in $\cos\theta$, resulting in a non-zero Forward-Backward Asymmetry ($A_{FB}$), which serves as a probe for CP violation and New Physics.

3. Key Contributions

1. Complete Tree-Level Calculation: Unlike previous studies that neglected non-fragmentation diagrams, this paper calculates the full set of tree-level SM contributions.
2. Quantification of Suppression Factors: The study rigorously quantifies the suppression of non-fragmentation diagrams:
   • Charmonium ($J/\Psi, \Psi(2S)$): The fragmentation contribution saturates the total rate. Non-fragmentation diagrams contribute less than 1% and are negligible.
   • Bottomonium ($\Upsilon(nS)$): The non-fragmentation diagrams are not negligible. They increase the branching fractions by 4% to 9%.
3. Theoretical Uncertainty Estimation: The authors estimate theoretical errors arising from uncalculated QCD corrections ($\alpha_s$) and relativistic corrections ($v^2$) within the NRQCD framework, showing that uncertainties for bottomonium are well-controlled.
4. CP-Violation Probe: The paper establishes that $A_{FB} = 0$ is a robust SM prediction for these decays. Any measured non-zero $A_{FB}$ would be a clear signal of CP-violating New Physics.

4. Key Results

Branching Fractions (SM)

Using experimental inputs for $\Gamma(V \to e^+e^-)$ and $\psi_V(0)$, the predicted branching ratios are:

• Charmonium:
  • $B(Z \to J/\Psi \ell^+\ell^-) = (7.78 \pm 0.14) \times 10^{-7}$
  • $B(Z \to \Psi(2S) \ell^+\ell^-) = (2.40 \pm 0.04) \times 10^{-7}$
  • Note: These are consistent with previous estimates and experimental observations (CMS).
• Bottomonium:
  • $B(Z \to \Upsilon(1S) e^+e^-) = (2.18 \pm 0.04) \times 10^{-8}$
  • $B(Z \to \Upsilon(1S) \mu^+\mu^-) = (2.13 \pm 0.04) \times 10^{-8}$
  • Correction: The inclusion of Fig. 2 diagrams increases these values by +6% (electrons) and +4% (muons) compared to fragmentation-only calculations.
  • Similar enhancements (+5% to +9%) are found for $\Upsilon(2S)$ and $\Upsilon(3S)$.

Distributions

• Invariant Mass ($m_{\ell\ell}$): For bottomonium, the inclusion of Fig. 2 creates a distinct peak in the low invariant mass region ($m_{\ell\ell} < 15$ GeV) due to the virtual photon pole.
• Angular Distribution: The distribution is symmetric ($\cos\theta \leftrightarrow -\cos\theta$) in the SM, confirming $A_{FB} = 0$.

New Physics Sensitivity

• Using current experimental bounds on anomalous couplings ($h^Z_1$), the integrated forward-backward asymmetry could reach:
  • $A_{FB} \approx 6 \times 10^{-4}$ (full phase space).
  • $A_{FB} \approx 2 \times 10^{-3}$ (with kinematic cuts to enhance sensitivity).
• While small, these values are potentially accessible to high-luminosity future colliders (e.g., FCC-ee with $\sim 10^{12}$ Z events).

5. Significance

• Precision Physics: The paper provides the necessary "clean" theoretical baseline for future high-statistics experiments. For charmonium, the predictions are essentially free of non-perturbative QCD uncertainties. For bottomonium, the 4-9% correction is crucial for precision tests.
• New Physics Discovery: The identification of $A_{FB}$ as a null test in the SM makes it a powerful observable. A non-zero measurement would directly indicate CP-violating interactions beyond the SM, specifically anomalous triple gauge couplings.
• Experimental Guidance: The results guide experimentalists at the HL-LHC, CEPC, and FCC-ee to focus on these specific decay channels, particularly the bottomonium modes where sub-leading SM diagrams matter, and to design analyses sensitive to angular asymmetries.