Development of a simulation and analysis framework for NνDEx experiment

This paper presents a comprehensive simulation and analysis framework for the N$\nu$DEx experiment that integrates theoretical ion mobility calculations, detailed detector modeling, and advanced machine learning techniques to validate the experimental workflow and optimize signal-background separation for neutrinoless double beta decay searches.

The Gist

Imagine you are a detective trying to solve the most elusive crime in the universe: finding a ghost that shouldn't exist. This "ghost" is a hypothetical particle decay called neutrinoless double beta decay. If scientists can find it, it proves that neutrinos are their own antiparticles, which would rewrite the laws of physics.

The NνDEx experiment is the high-tech crime scene investigation team built to catch this ghost. This paper describes the digital twin (a computer simulation) the team built to test their detective tools before they ever turn on the real machine.

Here is a breakdown of how they built this simulation, using everyday analogies:

1. The Crime Scene: A High-Pressure Gas Room

The experiment takes place in a giant, pressurized tank filled with a special gas (Selenium Hexafluoride, or SeF₆).

• The Analogy: Think of this gas as a thick, invisible fog. When a "ghost" (the decay) happens, it shoots out two electrons. These electrons are like fast runners sprinting through the fog.
• The Twist: In normal air, electrons zip away instantly. But in this special gas, the electrons get "stuck" to gas molecules, turning into heavy, slow-moving negative ions. It's like the runners suddenly put on heavy backpacks and start walking instead of running.

2. The Mystery of the Two Walkers

The scientists suspected that the gas might create two different types of these heavy walkers (ions): one slightly lighter and one slightly heavier.

• The Analogy: Imagine two people walking down a hallway. One is wearing sneakers (SeF₅⁻), and the other is wearing heavy boots (SeF₆⁻). Even though they start at the same time, the one in boots arrives at the finish line a tiny bit later.
• The Simulation: The team used advanced computer chemistry (like a virtual molecular lab) to calculate exactly how fast each "walker" moves. They found the "sneaker" walker moves at a speed of 0.444 and the "boot" walker at 0.430 (in specific units). This tiny speed difference is the key to solving the mystery.

3. The Camera System: The "Topmetal" Sensor

At the bottom of the tank is a giant floor covered in 10,000 tiny, super-sensitive cameras (pixels).

• The Analogy: When the slow walkers finally reach the floor, they drop their backpacks (charge). The cameras catch this drop and record a "ping."
• The Challenge: Because the walkers are slow, the "ping" takes a long time to settle. The team had to simulate how their custom-made camera chip (Topmetal-S) would react to these slow, lingering signals, ensuring they wouldn't get confused by electronic noise (static).

4. Reconstructing the 3D Movie

This is the most clever part of the simulation.

• The Problem: The cameras only see the 2D floor (X and Y coordinates). They don't know how high up in the room the event happened (the Z coordinate).
• The Solution: Because the two types of walkers arrive at slightly different times, the computer can calculate the height!
  • The Analogy: Imagine you hear a siren. If you know the speed of the siren and the time delay between two different sounds, you can figure out exactly how far away the ambulance is.
  • The Math: The computer measures the time gap between the "sneaker" arrival and the "boot" arrival. Using their known speeds, it calculates the exact 3D position of the event. This allows them to create a 3D movie of the electron's path.

5. Spotting the Ghost vs. the Fake

The real goal is to distinguish the "Ghost" (the rare signal) from "Fakes" (background noise like radiation from the tank walls).

• The Clue:
  • The Ghost (Signal): Shoots out two electrons. In the 3D movie, this looks like a path with two heavy "blobs" (stops) at the ends, like a dumbbell.
  • The Fake (Background): Usually shoots out one electron. This looks like a path with only one blob at the end.
• The AI Detective: The team taught a computer algorithm (a "Boosted Decision Tree") to look at these 3D movies. It learned to spot the dumbbell shape (two blobs) and ignore the single-line shapes.
  • The Result: The simulation showed that this AI could successfully separate the real signal from the background noise with high accuracy, proving the detector design will work.

Summary

The paper essentially says: "We built a super-accurate virtual version of our experiment. We calculated how the gas particles move, simulated how the sensors see them, and used math to turn 2D data into 3D movies. Our AI detective learned to spot the rare 'ghost' events in these movies. The simulation proves our real-life experiment is ready to go!"

This framework is the blueprint that ensures when they finally build the real machine in the deep underground lab, they will know exactly what to look for.

Technical Summary

1. Problem Statement

The NνDEx experiment aims to search for neutrinoless double beta decay ($0\nu\beta\beta$) in the isotope $^{82}\text{Se}$ using a high-pressure (10 bar) $^{82}\text{SeF}_6$ gas Time Projection Chamber (TPC). The experiment faces several critical challenges:

• Ion Mobility Characterization: Unlike electron-based TPCs, NνDEx uses negative ions ($SeF_6^-$) formed by electron attachment to avoid signal fluctuations from avalanches. However, the specific ion species formed in high-pressure $SeF_6$ and their reduced mobilities were not fully characterized, creating uncertainty in 3D track reconstruction.
• 3D Reconstruction: While the readout plane provides precise 2D ($x, y$) coordinates, determining the absolute $z$-coordinate (drift distance) is essential for defining a fiducial volume to reject surface backgrounds. This requires distinguishing between different ion species based on drift time.
• Signal vs. Background Separation: The experiment must distinguish $0\nu\beta\beta$ events (two electrons) from background events, primarily single-electron tracks from $\gamma$-ray interactions and the irreducible two-neutrino double beta decay ($2\nu\beta\beta$).

2. Methodology

The authors developed a comprehensive simulation and analysis framework integrating four core components:

A. Theoretical Calculation of Ion Mobility

• Molecular Modeling: Used Density Functional Theory (DFT) (B3LYP functional, 6-31G(d) basis set) via Gaussian16 to optimize the geometry and calculate the polarizability of candidate ions: $SeF_5^-$ and $SeF_6^-$.
• Mobility Calculation: Applied the Two-Temperature Theory and the Mason-Schamp expression using the IMoS software. The Trajectory-Diffuse Hard Sphere Scattering (TDHSS) method was used to calculate collision cross-sections and reduced mobilities ($K_0$).
• Validation: Validated the calculation method by computing mobilities for $SF_5^-$ and $SF_6^-$ in $SF_6$ gas, comparing results with experimental data to ensure an accuracy within ~3%.

B. Detector Simulation Architecture

The framework orchestrates data flow between four major software libraries:

1. Event Generation (Geant4 & BxDecay0): Simulates primary particle decays ($0\nu\beta\beta$, $2\nu\beta\beta$, and $\gamma$ backgrounds) and their interactions within the detector materials.
2. Field Calculation (COMSOL): Models the TPC geometry (cathode, focusing plane, anode with ~10,000 Topmetal-S pixels) to compute electric and weighting fields. The design includes a focusing plane to guide ions to the anode, achieving a field ratio of ~125:1 (focusing vs. drift) for 100% charge collection.
3. Charge Transport & Signal Induction (Garfield++): Simulates ion drift, diffusion, and signal induction using the Shockley-Ramo theorem. It incorporates the calculated ion mobilities and the specific response of the Topmetal-S readout chip (modeled via a unipolar shaping function).
4. Data Management (ROOT): Handles the storage of true particle information and reconstructed waveforms.

C. Track Reconstruction & Analysis

• 3D Reconstruction: Utilizes the drift time difference ($\Delta t$) between the two assumed ion species ($SeF_5^-$ and $SeF_6^-$) to calculate the absolute $z$-position using the formula:
  $$z = \frac{v_{SeF_5^-} \cdot v_{SeF_6^-}}{v_{SeF_5^-} - v_{SeF_6^-}} \cdot \Delta t$$
• Graph Construction: A Breadth-First Search (BFS) algorithm connects hit nodes (spatial coordinates + energy) into continuous tracks based on a spatial adjacency criterion (12 mm).
• Topological Analysis: Extracts variables such as track length, number of hits, total energy, and "blob" energies (energy deposits at track endpoints corresponding to Bragg peaks).
• Classification: Uses a Boosted Decision Tree (BDT) via TMVA to classify events based on the six topological variables.

3. Key Contributions

• First-Principles Mobility Data: Provided the first theoretical calculation of reduced mobilities for $SeF_5^-$ and $SeF_6^-$ in $SeF_6$ gas, yielding values of 0.444 and 0.430 cm$^2$V$^{-1}$s$^{-1}$ respectively, with a stable ratio (~1.03) crucial for $z$-reconstruction.
• Integrated Simulation Framework: Successfully linked molecular physics (DFT), macroscopic detector physics (COMSOL), microscopic transport (Garfield++), and event generation (Geant4) into a single workflow.
• Dual-Ion Reconstruction Strategy: Demonstrated a viable method for absolute $z$-position reconstruction without external calibration, relying on the natural separation of ion species.
• Signal-Background Discrimination: Established a topological analysis pipeline capable of distinguishing $0\nu\beta\beta$ (two-blob) events from single-electron backgrounds.

4. Results

• Mobility Validation: The calculated mobilities for sulfur-fluoride ions ($SF_5^-, SF_6^-$) matched experimental results with ~3% relative error, validating the methodology for $SeF_6$.
• Reconstruction Performance:
  • $z$-Position: Linear fit between true and reconstructed $z$-positions showed a gradient of 1.0, confirming the validity of the dual-ion drift time method.
  • Energy Resolution: With an Equivalent Noise Charge (ENC) of 40 $e^-$, the reconstructed energy resolution (FWHM) was 1.41%, meeting the experiment's target of ~1%.
• Background Rejection:
  • Topological Variables: $0\nu\beta\beta$ events typically show two high-energy blobs, while single-electron backgrounds show one high-energy blob and one low-energy tail.
  • BDT Performance: For a signal efficiency of 75%, the framework achieved a background rejection factor of 84%. At 90% signal efficiency, rejection was 68%.
• Waveform Simulation: Successfully simulated voltage waveforms showing that while individual ion contributions overlap due to long shaping times, drift time differences allow for species separation over sufficient drift lengths.

5. Significance

This work provides a robust, end-to-end simulation tool essential for the design and sensitivity studies of the NνDEx experiment.

• Design Optimization: The framework allows for the optimization of electric field configurations and detector geometry before construction.
• Sensitivity Projections: By validating the ability to reconstruct 3D tracks and discriminate backgrounds, the study supports the projected sensitivity of NνDEx-100 to reach a half-life limit of $4 \times 10^{25}$ years (natural gas) or $4 \times 10^{26}$ years (enriched gas) after 5 years of operation.
• Future Physics: The successful demonstration of the dual-ion reconstruction technique opens new avenues for high-pressure gas TPCs using electronegative gases, potentially applicable to other isotopes and experiments (e.g., $^{136}\text{Xe}$ or $^{130}\text{Te}$).

The paper concludes that the framework successfully validates the complete experimental workflow, serving as a critical foundation for the upcoming NνDEx physics program. Future work will focus on refining gas parameters through dedicated measurements and optimizing the tracking algorithms.