Muon trident process at far-forward LHC detectors

This paper investigates the feasibility of observing the muon trident process, including the first potential detection of electromagnetic $\tau^+\tau^-$ pair production and QED bound states, at the LHC's far-forward FASER$\nu$ and FASER$\nu$2 detectors through $\mu$-tungsten collisions.

The Gist

Imagine the Large Hadron Collider (LHC) as a massive, high-speed particle racetrack. Usually, we watch the main event: protons smashing into protons in the center of the ring. But this paper is about looking at the "backstage" of the race, specifically at the very far end of the track where a tiny, specialized camera (called a detector) is waiting to catch particles that fly straight ahead.

Here is the story of what the authors are investigating, explained simply:

1. The Setup: The "Muon Mailman"

In the chaos of proton collisions, a huge number of muons (heavy cousins of electrons) are created. They zip forward like a high-speed mail delivery service, heading straight toward the far-forward detectors (FASER$\nu$ and FASER$\nu2$).

Usually, these muons just pass through the detector's layers of tungsten (a very heavy metal) like ghosts. But sometimes, something interesting happens.

2. The Main Event: The "Muon Trident"

The paper studies a rare event called the Muon Trident Process.

• The Analogy: Imagine a muon is a fast-moving billiard ball. It zooms toward a heavy tungsten nucleus (which is like a giant, stationary bowling ball).
• The Collision: As the muon passes the tungsten, it doesn't just bounce off. Instead, the intense electromagnetic force acts like a whip. The muon "whips" the vacuum of space, and suddenly, two new particles pop into existence out of thin air.
• The Result: The original muon keeps going (but slightly slower), and now there are three particles leaving the scene: the original muon plus the new pair. It looks like a trident (a three-pronged spear), hence the name.

The authors calculated how often this happens when the new pair is:

• Electrons ($e^+e^-$): Very common. Like catching raindrops in a bucket.
• Muons ($\mu^+\mu^-$): Less common. Like catching snowflakes.
• Taus ($\tau^+\tau^-$): Extremely rare. Like catching a specific, rare bird that only flies at high speeds.

Why does this matter?
The authors predict that the FASER$\nu$ detector will see billions of electron pairs and thousands of muon pairs. But the big news is about the Taus. They predict that for the first time ever, we might actually see a muon create a pair of Taus. This would be a historic first observation of this specific type of "magic trick" in physics.

3. The Special Case: The "Exotic Couples" (QED Bound States)

The paper also looks at a weirder possibility. What if the two new particles created don't just fly apart, but immediately grab onto each other to form a temporary "couple"?

• Positronium: An electron and a positron holding hands. The authors predict we will see millions of these.
• True Muonium: A muon and an anti-muon holding hands. This is the "holy grail" of this study. It has never been seen before.
  • The Challenge: These couples are very fragile. They break apart almost instantly.
  • The Prediction: The current detector (FASER$\nu$) might be too small to catch enough of them (less than one event). However, the upgraded detector (FASER$\nu2$), which will be much larger and run longer, might finally catch about 60 of these "True Muonium" couples.

4. The "Why" and the "How"

• The "Why": By counting how many of these events happen, physicists can test if our current laws of physics (the Standard Model) are perfect. If the numbers don't match the predictions, it could mean there is "New Physics" hiding in the shadows. It also helps us understand how muons lose energy when traveling through materials (like lead or tungsten), which is crucial for designing future particle accelerators.
• The "How": The authors used powerful computer simulations (like a virtual physics lab) to calculate the odds. They looked at the math of how light and matter interact at the highest energies possible.

Summary in a Nutshell

This paper is a "menu" for what the FASER$\nu$ detectors will see in the coming years.

1. Electron pairs: We will see a flood of them (billions).
2. Muon pairs: We will see a steady stream (thousands).
3. Tau pairs: We might finally see the first-ever examples of this rare event.
4. True Muonium: The upgraded detector might finally catch the elusive "muon couple" that has never been observed before.

It's like saying, "We know the main race is exciting, but if you look at the very edge of the track with a super-magnifying glass, you might just see a brand-new species of butterfly that no one has ever seen before."

Technical Summary

Here is a detailed technical summary of the paper "Muon trident process at far-forward LHC detectors."

1. Problem Statement and Motivation

The paper investigates the muon trident process, defined as the electromagnetic production of a dilepton pair ($l^+l^-$) via muon-ion scattering ($\mu^\pm + A \to \mu^\pm + A + l^+ + l^-$). While the neutrino trident process has been extensively studied theoretically and recently observed by the FASER and SND@LHC collaborations, the muon trident process remains largely unexplored experimentally, particularly regarding the production of heavy lepton pairs ($\tau^+\tau^-$) and QED bound states.

The motivation stems from:

• LHC Forward Physics: The Large Hadron Collider (LHC) generates a high-intensity, collimated flux of forward muons that reach far-forward detectors like FASER$\nu$ and its proposed upgrade, FASER$\nu$2.
• Standard Model Tests: Studying this process allows for precision tests of Quantum Electrodynamics (QED) and constraints on nuclear effects in a new kinematic range.
• New Physics Searches: Deviations in cross-sections could signal Beyond Standard Model (BSM) physics.
• Unobserved States: The paper aims to predict the feasibility of observing True Muonium ($(\mu^+\mu^-)_S$), a bound state of a muon and antimuon, which has never been observed despite decades of theoretical effort.

2. Methodology

The authors employ a theoretical framework combining perturbative QED calculations with Monte Carlo simulations tailored for the specific geometry and luminosity of LHC forward detectors.

• Formalism:

  • Process: The analysis focuses on coherent scattering off a Tungsten (W) nucleus, where the nucleus remains intact. The cross-section scales with $Z^2$. Incoherent scattering (off individual nucleons) is neglected as it is subleading ($\propto Z$).
  • Diagrams: The calculation includes leading-order electromagnetic diagrams:
    1. Bethe-Heitler (BH): Photon-photon fusion.
    2. Bremsstrahlung: Radiation from the incoming muon.
    • Note: Nucleus bremsstrahlung is neglected. Interference between BH and bremsstrahlung is absent in total cross-sections due to charge-conjugation invariance but present in differential distributions.
  • Bound States: For QED bound states (True Muonium and Positronium), the authors use the Equivalent Photon Approximation (EPA) to describe photon emission by the incident muon, followed by photonuclear fusion.

• Simulation Tools:

  • A modified version of a Monte Carlo generator (originally for neutrino tridents) was adapted for muon-tungsten interactions.
  • Nuclear Form Factors: Implemented using the Fourier transform of the Woods-Saxon nuclear charge distribution for Tungsten.
  • Matrix Elements: Calculated exactly using FeynCalc without approximations, accounting for all lepton masses ($e, \mu, \tau$).
  • Flux Data: Utilized muon fluxes simulated with FLUKA for the forward direction of ATLAS, provided in the LHAPDF format.

• Experimental Parameters:

  • Detectors: FASER$\nu$ (Run 3, 250 fb$^{-1}$, 1.1 tons W) and FASER$\nu$2 (HL-LHC, 3 ab$^{-1}$, ~20 tons W).
  • Target: Tungsten target length set to 50 cm for momentum reconstruction purposes.
  • Kinematics: Incident muons with energy $E_\mu > 10$ GeV.

3. Key Contributions

1. First Comprehensive Prediction for Muon Trident at LHC: The paper provides the first detailed estimates of total and differential cross-sections for $\mu W \to \mu W l^+l^-$ at LHC energies.
2. Feasibility of $\tau^+\tau^-$ Observation: It demonstrates that the production of tau pairs in muon scattering, previously unobserved, is feasible at FASER$\nu$ during Run 3.
3. QED Bound State Production: The study extends the analysis to the production of True Muonium ($(\mu^+\mu^-)_S$) and Positronium ($(\mu^+e^-)_S$), predicting event rates for these exotic states.
4. Background Discrimination: The authors analyze kinematic variables (invariant mass and opening angle) to distinguish the produced dilepton pair from the outgoing scattered muon, a crucial step for experimental identification.

4. Key Results

A. Cross-Sections and Event Rates

• Cross-Section Hierarchy: The cross-section for $e^+e^-$ production is $\sim 10^5$ times larger than $\mu^+\mu^-$, which is $\sim 10^3$ times larger than $\tau^+\tau^-$ at TeV energies.
• FASER$\nu$ (Run 3, 250 fb$^{-1}$):
  • $e^+e^-$: $\sim 4.1 \times 10^{10}$ events.
  • $\mu^+\mu^-$: $\sim 2.6 \times 10^5$ events.
  • $\tau^+\tau^-$: $\sim 22$ events (increasing to $\sim 20$ with a 10 GeV energy cut).
  • Significance: The observation of $\tau^+\tau^-$ pairs is feasible, marking a potential first-time observation of this process at the LHC.
• FASER$\nu$2 (HL-LHC, 3 ab$^{-1}$):
  • Event rates increase by a factor of $\sim 150$.
  • $\tau^+\tau^-$: $\sim 3,250$ events, allowing for detailed differential studies.

B. Kinematic Distributions

• Energy Loss:
  • $e^+e^-$ pairs are produced with low energy ($< 1$ GeV) in the rest frame; the incident muon loses a small fraction of energy.
  • $\tau^+\tau^-$ pairs require TeV-scale incident muons, resulting in significant energy loss for the incident muon.
• Distinguishing Pairs:
  • The produced dilepton pair ($\mu^+\mu^-$) has a lower invariant mass and a larger opening angle compared to the pair formed by the scattered muon and the produced muon ($\mu^\mp_{f}\mu^\pm$). This distinction is vital for separating the signal from the "muon + outgoing muon" background.

C. QED Bound States

• True Muonium ($(\mu^+\mu^-)_S$):
  • The cross-section is suppressed by factors of $10^4$to$10^6$compared to open$\mu^+\mu^-$ pairs.
  • FASER$\nu$: $< 1$ event (unobservable).
  • FASER$\nu$2: $\sim 57$ events (unbound) or $\sim 30$ (with 10 GeV cut).
  • Conclusion: Observation of True Muonium is only feasible at FASER$\nu$2. The signal is characterized by a resonance peak at $W \approx 2m_\mu$ and highly collimated muon pairs.
• Positronium ($(\mu^+e^-)_S$):
  • Predicted to be abundant: $\sim 1.5 \times 10^5$ events at FASER$\nu$ and $\sim 2.2 \times 10^7$ at FASER$\nu$2.

5. Significance and Conclusion

This paper establishes that far-forward LHC detectors are powerful tools for studying electromagnetic muon interactions.

• Standard Model Constraints: The high statistics for $e^+e^-$ and $\mu^+\mu^-$ production will allow for precise constraints on cross-sections, aiding in the calculation of electromagnetic energy loss for muons in matter.
• Discovery Potential: The prediction of $\sim 20$ $\tau^+\tau^-$ events in Run 3 suggests that FASER$\nu$ could achieve the first observation of electromagnetic $\tau$-pair production in muon scattering.
• Exotic Physics: The study highlights FASER$\nu$2 as the first facility capable of potentially observing True Muonium, a long-sought QED bound state.

The authors conclude that while the current FASER$\nu$ detector can study light lepton pairs and potentially observe tau pairs, the upgraded FASER$\nu$2 is required for the discovery of True Muonium and for high-precision differential measurements of the trident process.