Z Boson Radiative Decay $Z\to \mu^+ \mu^- \gamma$ at the LHC

This paper presents a precision Standard Model analysis and new physics projections for the radiative decay $Z \to \mu^+\mu^-\gamma$ at the LHC, demonstrating its potential to measure the branching ratio with sub-percentage accuracy and to probe axion-like particles and anomalous $U(1)_X$ gauge bosons with enhanced sensitivity down to couplings of $\mathcal{O}(10^{-3})$.

The Gist

Imagine the Z boson as a massive, unstable celebrity at a crowded party (the Large Hadron Collider, or LHC). Usually, this celebrity decays instantly into a pair of muons (heavy cousins of electrons), which is like the celebrity leaving the party with their two best friends. This is a very common, boring event that happens millions of times.

However, sometimes, just as the celebrity is leaving, they trip over a photon (a particle of light) and accidentally drop it. This rare event is called $Z \to \mu^+\mu^-\gamma$ (Z boson decaying into two muons and a photon).

This paper is like a team of detectives (physicists) using a super-powered camera to study this specific "trip and drop" moment. They have two main goals:

1. Check the rulebook: Does this rare trip happen exactly as the Standard Model (the current rulebook of physics) predicts?
2. Look for ghosts: Is there something invisible hiding in the background that causes the celebrity to trip in a weird way?

Here is a breakdown of their investigation using simple analogies:

1. The "Rulebook" Check (Standard Model)

The detectives first asked: "How often does this trip happen?"

• The Old Camera: In the past (at a collider called LEP), they didn't have enough photos to be sure. They could only say, "It happens less than X times."
• The New Super-Camera: The LHC has taken billions of photos of the Z boson. With this massive dataset, the team calculated that they can now measure this "trip rate" with incredible precision—down to less than 1% error.
• The Result: They looked at old photos (Run-1 data) and found the rate is about 3.34 out of every 10,000 Z bosons. This matches the rulebook perfectly. It's like checking a clock and finding it ticking exactly as expected.

2. The "Ghost" Hunters (New Physics)

Now, the detectives asked: "What if the celebrity isn't just tripping? What if they are being pushed by an invisible force?"

They looked for two specific types of "invisible pushers":

A. The "Axion-Like Particle" (The Invisible Ghost)

Imagine the Z boson doesn't just drop a photon. Instead, it drops a ghost (an Axion-Like Particle, or ALP) that immediately turns into the two muons.

• The Clue: In a normal trip, the photon and the muons are scattered randomly. But if a ghost is involved, the two muons would huddle together at a very specific distance, forming a perfect "resonance" (like two singers hitting the exact same note).
• The Search: The team simulated millions of scenarios. They found that if these ghosts exist, the LHC is sensitive enough to spot them even if they are very weak or very light. This is a new way to hunt for ghosts that previous searches (which looked for light particles turning into two photons) might have missed.

B. The "Dark Force" (The Invisible String)

Imagine a new, invisible string (a Dark Force) that only tugs on muons.

• The Clue: The Z boson decays into a photon and a heavy "dark particle" (X), which then snaps into two muons.
• The Result: This scenario creates a similar "huddle" of muons. The team showed that the LHC can detect this invisible string even if it's incredibly weak (about 1,000 times weaker than the forces we usually see).

3. Why This Matters

Think of the LHC as a giant microscope. Usually, we look at the "big" things (like the Higgs boson). But this paper says, "Let's look at the tiny, rare scratches on the lens."

• Precision: By measuring this rare decay so precisely, they can test if the "Rulebook" of physics has any tiny errors.
• Sensitivity: Because they are looking at a "clean" signal (muons and photons are easy to spot), they can detect new physics that is too weak to be seen in "messy" collisions.

The Bottom Line

This paper is a roadmap for the future.

1. Now: We can measure this rare event with sub-percent precision.
2. Soon (HL-LHC): With more data, we will be even more precise.
3. The Goal: If the measurements ever deviate from the "Rulebook," it won't just be a small error; it will be the first crack in the foundation of our understanding of the universe, pointing us toward new particles (like ALPs) or new forces (Dark Forces) that we have never seen before.

In short: They are checking the Z boson's "trip" to see if the universe is behaving exactly as predicted, or if there are invisible ghosts and strings pulling the strings behind the scenes.

Technical Summary

1. Problem Statement

The radiative decay of the Z boson into a muon pair and a photon ($Z \to \mu^+\mu^-\gamma$) is a Standard Model (SM) process dominated by final-state radiation (FSR). While theoretically well-understood, it has historically been difficult to measure with high precision due to:

• Infrared Divergences: The process contains large infrared logarithms requiring resummation or careful phase-space cuts.
• Limited Statistics: Previous measurements at LEP ($e^+e^-$ colliders) yielded only upper limits on the branching ratio due to low Z boson production rates.
• Detector Effects: The measurement is highly sensitive to detector resolution, photon identification, and kinematic cuts (specifically the separation between the photon and muons).

Furthermore, this channel serves as a unique probe for Beyond the Standard Model (BSM) physics, specifically:

• Axion-Like Particles (ALPs): Pseudoscalar particles that could mediate $Z \to a\gamma \to \mu^+\mu^-\gamma$.
• Anomalous Dark Forces: Vector bosons ($X$) coupled to muons, leading to $Z \to X\gamma \to \mu^+\mu^-\gamma$.

The paper addresses the need for a precise SM prediction to establish a baseline and explores the sensitivity of the Large Hadron Collider (LHC) and High-Luminosity LHC (HL-LHC) to these new physics scenarios.

2. Methodology

The authors employed a multi-stage simulation and analysis framework:

• Simulation Tools:
  • MadGraph5_aMC@NLO: Used for generating parton-level signal and background events, including higher-order corrections and jet merging.
  • Pythia8: Used for parton showering and hadronization.
  • Delphes3: Used for fast detector simulation to model ATLAS/CMS performance (Run-2 and HL-LHC benchmarks).
• Signal and Background Modeling:
  • Signal: $pp \to Z \to \mu^+\mu^-\gamma$.
  • Backgrounds: Dominated by the Drell-Yan process ($pp \to \mu^+\mu^-\gamma$) and diboson production ($WW\gamma, ZZ\gamma, WZ\gamma$) where neutrinos or extra muons are missed.
• Kinematic Selections (Fiducial Cuts):
  • To avoid soft/collinear divergences and stay in the fixed-order regime, strict cuts were applied:
    • Photon $p_T > 20$ GeV (Run-2/HL-LHC).
    • Muon $p_T$ thresholds (e.g., $p_{T1} > 20$ GeV, $p_{T2} > 10$ GeV for Run-2).
    • Angular separation $\Delta R(\mu, \gamma) > 0.2$.
    • Invariant mass window $80 < M_{\mu\mu\gamma} < 100$ GeV to isolate the Z pole.
• BSM Scenarios:
  • ALPs: Simulated via effective Lagrangian couplings to gauge bosons and leptons. The analysis looked for a resonance in the dimuon invariant mass ($M_{\mu\mu}$) spectrum.
  • Anomalous Dark Force ($U(1)_X$): Modeled as a vector boson coupled to right-handed muons. The $Z \to X\gamma$ amplitude is generated via a fermion loop (triangle anomaly), which does not decouple even for heavy fermions.

3. Key Contributions

A. Standard Model Precision Analysis

• Fiducial Branching Ratio Extraction: The authors extracted the fiducial branching ratio from existing Run-1 (7 TeV) CMS data, obtaining:
  $$\text{Br}_{\text{fid}}(Z \to \mu\mu\gamma) = (3.34 \pm 0.016) \times 10^{-4}$$
  The uncertainty is primarily statistical.
• Run-2 and HL-LHC Projections:
  • Run-2 (13 TeV, 140 fb$^{-1}$): Projected statistical precision of 0.26% on the fiducial branching ratio.
  • HL-LHC (14 TeV, 3 ab$^{-1}$): Projected statistical precision of 0.043%.
  • The study highlights that while statistical precision is sub-percent, systematic uncertainties (PDFs, luminosity) currently dominate, suggesting the ratio $\text{Br}(Z \to \mu\mu\gamma)/\text{Br}(Z \to \mu\mu)$ as a robust observable.

B. New Physics Sensitivity

• ALP Search: The paper demonstrates that the $Z \to a\gamma \to \mu^+\mu^-\gamma$ channel provides a clean resonance search in the $M_{\mu\mu}$ spectrum.
  • The signal topology (photon back-to-back with the dimuon system) differs significantly from the SM background (collinear FSR), aiding discrimination.
  • The analysis covers ALP masses ($m_a$) from 5 GeV to 90 GeV.
• Anomalous Dark Force Search:
  • Investigated a $U(1)_X$ gauge boson coupled to muons.
  • The longitudinal component of the massive vector boson is enhanced by $m_Z^2/m_X^2$, making the channel sensitive to low masses.
  • The study covers $m_X$ from 1 GeV to 90 GeV.

4. Results

• SM Measurement: The analysis confirms that $Z \to \mu^+\mu^-\gamma$ is a statistically powerful channel at the LHC. The Run-2 analysis can measure the branching ratio with $\sim 0.3\%$ precision, improving to $\sim 0.04\%$ at HL-LHC.
• ALP Constraints:
  • The $Z \to \mu\mu\gamma$ channel offers superior sensitivity to ALPs compared to diphoton searches in specific coupling regimes (e.g., when $c_\mu \gg c_{\gamma\gamma}$).
  • For $m_a \sim 10-50$ GeV, the projected limits on the ALP decay constant $f_a$ reach $10^3 - 10^4$ GeV, competitive with or better than existing $e^+e^-$ and diphoton resonance searches.
• Dark Force Constraints:
  • The channel probes the dark gauge coupling $g_X$ down to $O(10^{-3})$ for masses $m_X < 5$ GeV.
  • This extends the reach of current collider constraints significantly in the low-mass region, outperforming existing $Z' \to \mu\mu$ searches in certain mass ranges.
• Background Discrimination: The study confirms that the dominant background is the SM radiative decay itself. However, the distinct kinematic features of BSM resonances (narrow peaks in $M_{\mu\mu}$ and different $\Delta R$ distributions) allow for effective signal extraction.

5. Significance

• Precision Electroweak Physics: This work establishes $Z \to \mu^+\mu^-\gamma$ as a high-precision test of the electroweak sector. Achieving sub-percent precision allows for stringent tests of SM predictions and potential indirect detection of new physics.
• Complementary BSM Probe: The radiative Z decay provides a unique "leptophilic" probe. Unlike diphoton searches which are sensitive to ALPs coupling to gauge bosons, this channel is sensitive to ALPs and dark forces coupling directly to leptons.
• Low-Mass Reach: The analysis demonstrates that the LHC can effectively probe light new particles ($m < 5$ GeV) in the dimuon channel, a region often obscured by hadronic backgrounds but accessible here due to the clean Z boson tag and high luminosity.
• Experimental Guidance: By providing detailed cutflows, efficiency estimates, and systematic considerations, the paper offers a roadmap for experimental collaborations (ATLAS/CMS) to perform dedicated measurements of this rare decay mode using Run-2 and HL-LHC data.

In conclusion, the paper argues that the radiative decay $Z \to \mu^+\mu^-\gamma$ is a "golden channel" for both precision SM tests and the discovery of leptophilic new physics, with the HL-LHC poised to push sensitivity to unprecedented levels.