Momentum Measurement of Charged Particles in FASER's Emulsion Detector at the LHC

This paper presents and validates a method for measuring the momenta of charged particles in the FASER$\nu$ emulsion detector using multiple Coulomb scattering, demonstrating its effectiveness from a few GeV to several TeV through Geant4 simulations and test beam data.

The Gist

The Big Picture: Catching Ghosts in a Maze

Imagine the Large Hadron Collider (LHC) as a massive, high-speed train station where protons (tiny particles) crash into each other at nearly the speed of light. These crashes create a shower of new particles, including neutrinos.

Neutrinos are the "ghosts" of the particle world. They are so light and shy that they can pass through entire planets without hitting anything. To catch them, the FASER experiment built a detector called FASERν right in the path of these ghosts, far down the tunnel from the crash site.

Inside this detector is a special "camera" made of nuclear emulsion. Think of this not as a digital camera, but as a stack of 730 ultra-thin photographic films sandwiched between heavy tungsten plates. When a charged particle (like a muon, which is a heavy cousin of an electron) zips through, it leaves a tiny trail of silver grains, like a snail leaving a slime trail on a glass window.

The Problem: We can see the trail, but we don't know how fast the snail was going. In particle physics, knowing the speed (momentum) is crucial to understanding what happened during the crash.

The Solution: This paper describes a clever new way to measure that speed by watching how much the particle "wobbles" as it travels through the detector.

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The Method: The "Wobble" Test (Multiple Coulomb Scattering)

Imagine you are walking through a crowded hallway.

• If you are a heavy, slow person (low momentum), you might bump into people, get pushed around, and your path will be very zig-zaggy.
• If you are a super-fast, heavy sprinter (high momentum), you will plow through the crowd. You might get nudged slightly, but you'll mostly keep going in a straight line.

In the FASER detector, the "crowd" is the tungsten plates. As a particle passes through, it bounces off atomic nuclei. This is called Multiple Coulomb Scattering (MCS).

• Low Energy Particles: Wobble a lot.
• High Energy Particles: Wobble very little.

The scientists realized that by measuring exactly how much the particle's path deviates from a straight line, they can calculate its speed.

The "Coordinate Method": A Game of Connect-the-Dots

The paper introduces a specific math trick called the Coordinate Method. Here is how it works, using an analogy:

Imagine you are trying to guess how fast a car is driving by looking at its tire tracks on a muddy road.

1. You look at the track at point A.
2. You look at the track at point B (a bit further down).
3. You draw a straight line connecting A and B and predict where the car should be at point C.
4. You look at the actual track at point C.

If the car is driving fast and straight, the actual track at C will be very close to your prediction. If the car is wobbling, the actual track will be far off.

The FASER team does this mathematically across 100 layers of the detector. They calculate the "second difference"—essentially, how much the path curves compared to a straight line.

• Big curve? Slow particle.
• Tiny curve? Fast particle.

Because the emulsion detector is so precise (it can see movements smaller than a human hair), they can detect even the tiniest wobbles of very fast particles.

The Proof: The "Test Drive"

To make sure their math wasn't broken, the team took their detector to a "test track" at CERN. They shot beams of muons at it with known speeds: 100, 200, and 300 GeV (Giga-electronvolts).

• The Result: The detector's "guess" of the speed matched the "real" speed almost perfectly.
• The Accuracy: They could measure the speed with about 20-23% accuracy. In the world of particle physics, where measuring something that fast is incredibly hard, this is like hitting a bullseye from a mile away.

The Real Deal: Catching TeV-Scale Ghosts

Finally, they applied this method to real data from the LHC. They looked at "background muons" (particles that weren't the main target but were recorded by the detector).

These particles were moving at TeV speeds (1,000 times faster than the test beams).

• The team looked at how much these particles wobbled.
• They calculated the speed based on the wobble.
• The Result: The calculated speeds matched what they expected based on the angle of the particles.

This proves that their method works even for the fastest particles in the universe, opening the door to studying neutrinos with unprecedented detail.

Why Does This Matter?

Think of the FASER experiment as a crime scene investigation.

• The Crash: The LHC collision.
• The Evidence: The neutrinos and muons.
• The Detective Work: Measuring their speed.

If you don't know how fast the suspect was moving, you can't figure out how the crime happened. By mastering this "wobble measurement," the FASER team can now reconstruct the energy of neutrino interactions with high precision. This helps physicists understand the fundamental laws of the universe, potentially revealing new physics beyond what we currently know.

In a Nutshell

The FASER team built a super-precise camera that sees how much particles wiggle as they pass through heavy metal. By measuring the wiggle, they can tell how fast the particles are going, from slow-moving ones to the fastest things in the universe. They tested it on a track, it worked perfectly, and now they are using it to solve the mysteries of the LHC.

Technical Summary

1. Problem Statement

The FASER experiment at the Large Hadron Collider (LHC) aims to study high-energy neutrino interactions in the TeV energy range. A critical requirement for analyzing these interactions is the precise measurement of the momenta of charged particles (muons and hadrons) produced in neutrino collisions.

• Challenge: Traditional magnetic spectrometers are not feasible for the FASERν emulsion detector due to its dense, passive structure (tungsten plates interleaved with nuclear emulsion films).
• Limitation of Existing Methods: Previous emulsion experiments (e.g., OPERA, DONuT) used Multiple Coulomb Scattering (MCS) for momentum measurement but were generally limited to lower energies (up to ~100 GeV) due to spatial resolution constraints and limited tracking lengths.
• Goal: Develop and validate a momentum measurement method capable of determining particle momenta from a few GeV up to several TeV using the unique properties of the FASERν detector (sub-micron spatial resolution and long tracking length).

2. Methodology

The paper proposes and validates a momentum measurement technique based on the Coordinate Method of Multiple Coulomb Scattering (MCS).

• Physical Principle: As charged particles traverse matter, they undergo small-angle deflections due to elastic scattering off nuclei. The root-mean-square (RMS) of these angular deviations is inversely proportional to the particle's momentum ($p$).
• The Coordinate Method: Instead of measuring angles directly, this method analyzes the second difference ($s$) of the particle's trajectory coordinates across multiple detector layers.
  • The position displacement $s_i$ is calculated using coordinates from three plates: $i$, $i+n_{cell}$, and $i+2n_{cell}$.
  • The RMS of these displacements ($s_{RMS}$) is modeled as a combination of the scattering term (dependent on momentum) and the position resolution error ($\sigma_{pos}$).
  • The relationship is governed by the Highland formula, where $s_{RMS} \propto 1/p$.
• Fitting Procedure:
  • The $s_{RMS}$ values are calculated for varying cell lengths ($n_{cell}$).
  • A fit is performed using the equation: $s_{RMS} = \sqrt{(s_{RMS}^{MCS}(p, n_{cell}))^2 + 6\sigma_{pos}^2}$.
  • The momentum ($p$) and position resolution ($\sigma_{pos}$) are extracted as fitting parameters.
• Optimization: The study determines the optimal maximum cell length ($n_{cell}^{max}$) to balance statistical independence with sensitivity to high-momentum scattering.

3. Key Contributions

1. Extension of MCS to TeV Scale: Successfully demonstrates that the coordinate MCS method can be extended to measure momenta up to 3 TeV, a regime previously inaccessible to emulsion detectors.
2. Optimization of Analysis Parameters: Identifies $n_{cell}^{max} = 24$ (using 100 plates total) as the optimal configuration, offering the best trade-off between resolution, relative offset (bias), and success rate.
3. Experimental Validation: Validates the method using real-world data from a dedicated muon test beam at CERN's SPS (H8 beamline) with momenta of 100, 200, and 300 GeV.
4. Real-World Application: Applies the method to background muons recorded by the FASERν detector during LHC Run 3, successfully reconstructing TeV-scale momenta consistent with angular spread expectations.

4. Results

A. Monte Carlo (MC) Simulations

• Range: Simulated muons from 10 GeV to 3 TeV.
• Performance:
  • Resolution: Achieved a momentum resolution of approximately 17% (theoretical limit with perfect alignment) and ~35% at 2 TeV with realistic detector resolution ($\sigma_{pos} = 0.3 \mu m$).
  • Linearity: The method maintains good linearity ($p_{rec} \approx p_{true}$) even above 1 TeV.
  • Success Rate: With $n_{cell}^{max}=24$, the success rate (fraction of tracks reconstructed below 7 TeV) remains high (>90%) across the entire range.

B. Test Beam Validation (CERN SPS)

• Setup: A 100-plate emulsion detector module was exposed to 100, 200, and 300 GeV muon beams.
• Data-MC Agreement: The reconstructed momentum distributions ($p_{rec}$) and inverse distributions ($1/p_{rec}$) showed excellent agreement between experimental data and MC simulations.
• Measured Resolution:
  • 100 GeV: $20.7 \pm 0.6\%$
  • 200 GeV: $22.7 \pm 0.6\%$
  • 300 GeV: $23.2 \pm 0.4\%$
• Bias: The central reconstructed momentum ($p_{center}$) matched the beam momentum within uncertainties (total uncertainty ~4.5%, dominated by thickness variations).

C. LHC Data Application

• Dataset: Background muons from 9.5 fb$^{-1}$ of 13.6 TeV $pp$ collisions.
• Observation: The reconstructed momentum distribution for tracks penetrating >100 plates showed a peak around 1 TeV.
• Consistency: This result is consistent with an independent estimation of muon momentum ($1290^{+150}_{-130}$ GeV) derived from the angular spread of the tracks caused by MCS in the upstream rock.

5. Significance

• Enabling Neutrino Physics: This method provides the necessary kinematic tool to fully reconstruct neutrino interaction events at the LHC, allowing for precise measurements of neutrino cross-sections and fluxes in the TeV regime.
• Detector Capability: It proves that nuclear emulsion detectors, traditionally limited to low-energy or short-track applications, can serve as high-precision momentum spectrometers for multi-TeV particles without magnetic fields.
• Future Prospects: The successful validation motivates future cross-checks using matched tracks between the FASERν emulsion detector and the downstream FASER magnetic spectrometer, which will further refine the momentum scale and reduce systematic uncertainties.

In conclusion, the paper establishes a robust, validated framework for measuring charged particle momenta in the FASERν detector, bridging the gap between GeV and TeV scales and unlocking new physics potential for collider neutrino studies.