First Full Dalitz Plot Measurement in Neutron $β$-Decay using the Nab Spectrometer and Implications for New Physics

This paper presents the first full Dalitz plot measurement of neutron beta-decay phase space using the Nab spectrometer to test the Standard Model and establish new constraints on a hypothesized excited neutron state that could resolve the neutron lifetime discrepancy.

The Gist

Imagine you are a detective trying to solve a mystery about the most fundamental building blocks of our universe. The suspect? The neutron.

For decades, scientists have been trying to measure how long a neutron lives before it decays (breaks apart). But here's the twist: when they measure it one way (by trapping neutrons in a bottle), they get one answer. When they measure it another way (by counting the particles that fly out as neutrons decay in a beam), they get a different answer. It's like if you timed a runner by watching them cross the finish line, but also by counting how many water bottles they dropped, and the two numbers didn't match.

This mismatch is called the "Neutron Lifetime Anomaly." Some physicists think it means there's a "dark decay" happening—a secret, invisible way neutrons disappear that we haven't seen yet. Others think it might be a glitch in our measurements.

Enter the Nab experiment. Think of Nab as a high-tech, super-precise camera and stopwatch combo designed to catch the neutron in the act of breaking apart.

The Setup: A Magnetic Roller Coaster

The experiment takes place at a massive particle accelerator (the Spallation Neutron Source). They shoot a beam of neutrons through a giant magnet. When a neutron decays, it splits into three things:

1. A proton (heavy and slow).
2. An electron (light and fast).
3. An antineutrino (ghost-like and invisible).

The Nab machine is like a magnetic roller coaster track. The magnet guides the charged particles (the proton and electron) along specific paths to detectors at the ends of the track. Because the proton is heavy, it moves slowly. The electron is light, so it zooms ahead. By measuring exactly when they arrive and how much energy they have, scientists can reconstruct the entire event.

The "Dalitz Plot": The Crime Scene Map

The core of this paper is the creation of the first full "Dalitz Plot" for neutron decay.

If you imagine the decay as a 3D explosion, the Dalitz Plot is a 2D map that shows every possible way that explosion could happen.

• The X-axis represents the energy of the electron.
• The Y-axis represents the momentum (speed) of the proton.

In a perfect world, this map would look like a smooth, teardrop-shaped cloud of dots. The edges of this teardrop are the "speed limits" of the universe. Nothing can go faster than the edge allows. If you see dots outside the teardrop, it means something weird is happening—maybe a new particle was created, or a neutron has a secret "excited" state that gives it extra energy.

What Did They Find?

The Nab team took a "commissioning" run (a test run to make sure the machine works) and captured millions of these decays. They mapped them out and compared their real-world map to the computer simulation.

1. The Shape Matches: The "teardrop" shape they saw looked very much like the prediction. This is great news! It means the Standard Model (our current rulebook of physics) is holding up well. The "speed limits" of the universe seem correct.
2. No Ghosts Found (Yet): They looked for any dots outside the teardrop that would suggest a "dark decay" or a secret excited neutron state. They didn't find any significant evidence of this.
3. A New Limit on the "Excited Neutron": There was a theory that neutrons might have a "hyped-up" version (an excited state) that lives longer. If this existed, it would show up as extra energy in the decay. The Nab team measured the maximum energy allowed and said, "If this excited neutron exists, it can't have more than a tiny, tiny amount of extra energy." They effectively closed the door on many versions of this theory.

The Hiccups (Why it's not perfect yet)

The paper is honest about its limitations. It was a test run, and the machine had some "growing pains":

• Broken Pixels: Some parts of the detector were dead (like dead pixels on a phone screen), so they missed some data.
• Noise: The electronics were a bit noisy, making it hard to tell exactly when a particle arrived.
• Calibration: They had to guess a bit about how the machine was reading energy because they didn't have enough calibration data yet.

Because of these issues, they couldn't yet calculate the exact correlation between the electron and the neutrino (which is the main goal for future runs). But they proved the machine works and that the "map" looks right.

The Big Picture

Think of this paper as the first draft of a masterpiece.

• The Goal: To find cracks in the Standard Model that point to "New Physics" (like dark matter or extra dimensions).
• The Result: The Standard Model passed the test so far. The "teardrop" map is clean.
• The Future: Now that they know the machine works, they will fix the broken pixels, reduce the noise, and run the experiment again with much higher precision.

In simple terms: The Nab experiment built a super-precise camera to watch neutrons die. They took the first full picture of the event. The picture looks exactly as the laws of physics predicted, with no obvious signs of "secret" decays. While they didn't find the "smoking gun" for new physics this time, they successfully mapped the crime scene and set strict new rules for what could be hiding in the shadows. The real investigation is just getting started.

Technical Summary

1. Problem Statement and Motivation

Neutron $\beta$-decay is a critical process for testing the Standard Model (SM) of particle physics, specifically the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and the search for physics Beyond the Standard Model (BSM). Two major anomalies drive current research:

• The Cabibbo Angle Anomaly: A tension exists between the SM prediction of CKM unitarity and experimental extractions of the matrix element $V_{ud}$ from superallowed nuclear decays and neutron decay.
• The Neutron Lifetime Discrepancy: There is a $>4\sigma$ difference between neutron lifetime measurements using "disappearance" methods (storing ultracold neutrons) and "appearance" methods (counting decay products in beams).

To resolve these issues, precise measurements of neutron decay correlations are required, specifically the electron-neutrino correlation parameter ($a$) and the Fierz interference term ($b$). Furthermore, exotic explanations for the lifetime discrepancy, such as a hypothesized excited neutron state ($n^*$) that decays via dark channels or emits a gamma ray, require direct kinematic constraints.

Prior to this work, no experiment had simultaneously measured the full range of electron energy ($E_e$) and proton momentum ($p_p$) to reconstruct the complete Dalitz plot (phase space) of neutron $\beta$-decay. Previous experiments typically measured only one variable or narrow slices of phase space.

2. Methodology and Experimental Apparatus

The study utilized the Nab spectrometer located at the Fundamental Neutron Physics Beamline (FNPB) of the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory.

• Spectrometer Design: Nab is an asymmetric magnetic spectrometer designed to capture charged decay products (electrons and protons) from unpolarized neutrons.
  • Magnetic Field: A $\sim1.7$ T field guides particles. A "pinch" region increases the field to $\sim4$ T to reject off-axis protons, creating a sensitive relationship between proton time-of-flight ($t_p$) and momentum.
  • Detection System: Two large-area, highly segmented silicon detectors (127 hexagonal pixels each) are placed at opposite ends of the spectrometer.
    • Upper Detector: Operated at $-30$ kV to accelerate protons, allowing detection of low-energy protons after they pass through a thin dead layer.
    • Lower Detector: Grounded, primarily detecting electrons.
  • Coincidence Measurement: The system detects the fast electron and the slower proton (delayed by $\sim12$ $\mu$s) in coincidence. The electron energy ($E_e$) is reconstructed from the sum of energy deposits, while the proton momentum is derived from its time-of-flight ($t_{pe}$).
• Data Collection: Data was collected during a 3-week commissioning run in Summer 2023. Approximately 101 hours of data were recorded, with 81 hours of "live" time in nominal production mode.
• Calibration: Radioactive sources ($^{113}$Sn, $^{207}$Bi, $^{109}$Cd, $^{148}$Gd) were used to calibrate energy response and gain linearity across the detector pixels.
• Simulation: A detailed Geant4 Monte Carlo simulation modeled the spectrometer's magnetic/electric fields, particle transport, detector response (including dead layers and backscattering), and the specific hardware limitations (e.g., dead pixels).

3. Key Contributions

This work represents the first experimental reconstruction of the full Dalitz plot for neutron $\beta$-decay for electrons with kinetic energy $>100$ keV.

• Full Phase Space Reconstruction: Unlike previous experiments, Nab simultaneously mapped the correlation between electron energy and proton momentum, visualizing the "teardrop" shape of the kinematically allowed phase space.
• Direct Neutron Mass Measurement: By analyzing the kinematic endpoint of the proton momentum distribution, the authors derived a direct measurement of the free neutron mass ($m_n$) independent of nuclear binding energy corrections.
• Constraints on Excited Neutrons: The precise kinematic endpoint was used to place stringent limits on the existence of a hypothesized excited neutron state ($n^*$) proposed to explain the neutron lifetime discrepancy.

4. Key Results

A. The Dalitz Plot

• The measured Dalitz plot (Fig. 6) shows the characteristic "teardrop" shape predicted by theory.
• Agreement: Qualitative agreement with simulation is observed, indicating no severe systematic biases.
• Discrepancies: Quantitative differences exist at the few-percent level, attributed to:
  • Hardware failures: Missing detector pixels reduced coincidence rates and distorted energy reconstruction (particularly for backscattered electrons).
  • Timing jitter: Noise-induced jitter in proton timing ($\sim250$ ns) blurred the time-of-flight distribution.
  • Threshold effects: Imperfect reconstruction near the energy thresholds (10 keV and 100 keV).
• Edge Analysis: The upper and lower edges of the Dalitz plot (corresponding to specific kinematic limits) were compared to simulation. The maximum fractional deviation was $7.8 \times 10^{-3}$ for the upper edge and $5.6 \times 10^{-2}$ for the lower edge.

B. Neutron Mass and Excited State Limits

• Proton Time-of-Flight: The front edge of the proton time-of-flight distribution ($t_{front}$) was fitted. The measured $t_{front}$ was $17 \pm 21$ ns longer than the simulation.
• Neutron Mass: This timing shift corresponds to a momentum measurement of $p_{p,max} = 1188 \pm 13$ keV/$c$. Converting this to mass yields:
  $$m_n = 939.566 \pm 0.012 \text{ MeV}$$
  (Note: A secondary check using the electron energy endpoint yielded $m_n = 939.578 \pm 0.023$ MeV, consistent with the proton method).
• Excited Neutron Constraints: The measurement was used to constrain the energy of a potential de-excitation gamma ray ($\Delta E_\gamma$) from a hypothetical excited neutron.
  • At 90% Confidence Level (C.L.), the available extra energy is limited to $\Delta E_\gamma < 21$ keV (derived from the mass limit).
  • This excludes a significant portion of the parameter space proposed to explain the neutron lifetime discrepancy via excited neutrons with specific lifetimes ($\tau_\gamma$).

5. Significance and Future Outlook

• Validation of Technique: This paper demonstrates the feasibility of using the Nab spectrometer to reconstruct the full kinematic phase space of neutron decay, a prerequisite for high-precision extraction of correlation parameters ($a$, $b$, $A$).
• BSM Physics: The results provide the first direct kinematic limits on excited neutron states using free neutrons, ruling out specific models that attempt to resolve the neutron lifetime anomaly.
• Systematic Understanding: The analysis identifies critical systematic uncertainties (pixel deadness, timing synchronization, noise) that must be addressed in future runs. The authors note that subsequent data sets with pristine detectors and improved electronics will significantly reduce these uncertainties.
• CKM Unitarity: While this specific commissioning run did not extract a final value for the correlation parameter $a$ due to systematic limitations, the methodology and the reconstructed Dalitz plot provide the necessary foundation for future high-precision tests of CKM unitarity and searches for scalar/tensor currents.

In summary, this work marks a milestone in neutron physics by providing the first complete kinematic map of neutron $\beta$-decay, validating the Nab spectrometer's design, and placing new, stringent constraints on exotic neutron states.