Relativistic distorted-wave analysis of the missing-energy spectrum measured with monochromatic $ν_μ$-$^{12}$C interactions at JSNS$^{2}$

This paper analyzes the missing-energy spectrum from monochromatic neutrino-$^{12}$C interactions measured by the JSNS$^2$ collaboration using a relativistic distorted-wave approach with an improved spectral function, while discussing the roles of nuclear recoil, final-state interactions, and neutrino event generators in describing low-energy nuclear effects.

The Gist

Imagine you are trying to figure out what's inside a sealed, black box by throwing a single, perfectly timed ping-pong ball at it and watching what bounces back.

That is essentially what the JSNS2 experiment did. They fired a very specific, "monochromatic" beam of neutrinos (ghostly particles that rarely interact) at a target made of Carbon-12 atoms. Because the neutrinos came from a very specific type of decay (Kaon decays at rest), they all had the exact same energy. This is like firing ping-pong balls that all have the exact same speed, making it much easier to predict what should happen.

When a neutrino hits a neutron inside a Carbon atom, it turns that neutron into a proton and shoots it out, leaving behind a "residual" nucleus (the rest of the atom). The scientists measured the energy of the particles that came out to calculate something called "missing energy." Think of missing energy as the "receipt" of what happened inside the box. If the math doesn't add up, the difference is the "missing" part, which tells us about the internal structure of the atom.

The Problem: The Receipt Was Blurry

The JSNS2 team measured the shape of this "missing energy" spectrum, but they didn't know the exact total number of hits (normalization). More importantly, the data looked a bit different from what standard computer models predicted. The peaks were in the wrong places, and there was energy showing up where it theoretically shouldn't be.

The Solution: A New Way to Look at the Atom

The authors of this paper (Franco-Patino and colleagues) decided to build a better model to explain the data. They used a "Relativistic Distorted-Wave" approach. Here is the analogy:

1. The Old Map (Mean Field): Imagine the nucleus as a calm, quiet library where every book (neutron) sits perfectly still on a specific shelf. This is the "Mean Field" model. It's simple, but in reality, the library is chaotic. Books are jostling, and some are in a hurry.
2. The New Map (Spectral Function): The authors updated their map using data from electron-scattering experiments. Instead of books sitting perfectly still, they realized the "neutron books" have a range of energies and positions. They created a new "profile" that accounts for the fact that neutrons are correlated and moving, not just sitting in neat rows.
3. The Distorted Path (Distorted Waves): When the proton is knocked out of the nucleus, it doesn't fly out in a straight line like a bullet in a vacuum. It has to push through the "crowd" of other nucleons. The authors modeled this as the proton's path being "distorted" or bent by the nuclear environment, much like a runner trying to sprint through a dense crowd of people.

The Three Big Surprises

The paper highlights three main factors that explain why the data looked the way it did:

1. The Recoil of the Heavy Nucleus (The Bowling Ball Effect)
When the neutrino hits a neutron, the neutron turns into a proton and flies out. But the rest of the atom (the residual nucleus) also has to move to conserve momentum. It's like a heavy bowling ball rolling backward when you throw a bowling pin forward.

• The Finding: The JSNS2 detector might not have "seen" this heavy, slow-moving bowling ball because it moves too slowly to create a signal. The authors showed that if you include this recoil in your math, the theoretical curve shifts and matches the experimental data much better. If you ignore it, the peak of the energy distribution is in the wrong spot.

2. The "Ghost" Neutrons (The Escape Artist)
Sometimes, the proton that gets knocked out doesn't just leave; it bumps into other neutrons inside the nucleus before escaping. This can knock a second neutron out of the atom.

• The Finding: Neutrons are invisible to the detector (they don't leave a light trail). If a neutron escapes, the detector thinks less energy was used than actually was. This makes the "missing energy" calculation look higher than it really is. The authors used a simulation (NuWro) to show that this "neutron emission" shifts the data toward higher missing energies, which helps explain why the experimental data has a "tail" stretching out to the right.

3. The Blurry Camera (Energy Smearing)
The detector isn't perfect; it's like a camera with a slightly shaky hand. The energy measurements have a small margin of error (about 2.5%).

• The Finding: Theoretically, there should be zero events below a certain energy threshold (the point where you need enough energy to break a neutron loose). But the data showed events below this line. The authors showed that if you "smear" the theoretical data to account for the detector's slight blurriness, the theory suddenly fits the data perfectly, filling in that "forbidden" zone.

The Conclusion

The paper concludes that to understand these tiny interactions, we can't just use simple, static models. We need to account for:

• The messy, correlated motion of neutrons inside the nucleus.
• The fact that the heavy leftover nucleus recoils (even if we can't see it).
• The fact that protons might knock out invisible neutrons on their way out.
• The fact that our detectors are slightly blurry.

By combining all these effects, the authors created a much more accurate picture of what happens when a neutrino hits a Carbon atom. This is crucial for future experiments, like those trying to measure neutrino oscillations, because if we don't understand the "target" (the nucleus) perfectly, we can't accurately measure the "projectile" (the neutrino).

In short: The universe is messier than our simple equations suggest, and by adding a few more realistic details (like recoil and invisible neutrons), the math finally matches the reality.

Technical Summary

Here is a detailed technical summary of the paper "Relativistic distorted-wave analysis of the missing-energy spectrum measured with monochromatic $\nu_\mu$–$^{12}$C interactions at JSNS2".

1. Problem Statement

The paper addresses the challenge of interpreting low-energy neutrino-nucleus interactions, specifically focusing on the missing-energy spectrum of neutrons in Carbon-12 ($^{12}$C) measured by the JSNS2 collaboration.

• Context: JSNS2 utilizes a monochromatic muon neutrino beam ($E_\nu = 235.5$ MeV) produced by Kaon Decays At Rest (KDAR). This provides a clean probe for studying nuclear effects (initial and final state interactions) without the energy spectrum complications of standard accelerator beams.
• The Challenge: The JSNS2 data provides only the shape of the cross-section versus missing energy ($E_m$), not the absolute normalization. Furthermore, there are ambiguities in how the "visible energy" is defined experimentally (specifically regarding the kinetic energy of the recoiling residual nucleus) and how nuclear correlations and final-state interactions (FSI) are modeled.
• Goal: To theoretically reproduce the measured missing-energy distribution using a rigorous relativistic framework to understand the role of nuclear structure, recoil effects, and inelastic scattering (neutron emission).

2. Methodology

The authors employ a Relativistic Distorted-Wave Impulse Approximation (RDWIA) framework, enhanced with specific parameterizations for the nuclear ground state and final-state interactions.

A. Kinematics and Formalism

• Process: Charged-current (CC) quasielastic scattering: $\nu_\mu + ^{12}\text{C} \to \mu^- + p + ^{11}\text{C}$.
• Missing Energy ($E_m$): Defined as $E_m = E_\nu - E_\mu - T_p - T_{recoil} - \xi$. The paper rigorously treats the recoil kinetic energy ($T_B$) of the residual nucleus, which introduces complex angular dependencies in the energy conservation delta function.
• Hadronic Tensor: Calculated using the RDWIA approach, summing over occupied single-particle states ($\kappa$) with relativistic spinors for bound and scattered nucleons.

B. Nuclear Ground State Modeling

• Base Model: Relativistic Mean Field (RMF) using the NLSH potential to solve the Dirac equation for bound nucleons.
• Spectral Function Parameterization: To move beyond the simple mean-field approximation (which yields discrete Dirac delta functions), the authors introduce a new parameterization of the spectral function $\rho_\kappa(E_m)$ based on high-resolution $(e, e'p)$ data (Ref. [7]). This includes three components:
  1. p-shell: Modeled by three Dirac deltas (ground state $1p_{3/2}$and excited states$1p_{1/2}, 1p_{3/2}$).
  2. s-shell: Modeled by an asymmetric Maxwell-Boltzmann distribution (instead of a Gaussian) to better fit the large width observed in experiments and avoid unphysical contributions below the emission threshold.
  3. Background: An exponential tail representing short-range correlations (SRC) at high missing energies ($E_m > 26$ MeV).

C. Final-State Interactions (FSI)

Three distinct approaches are compared to model the interaction of the outgoing proton with the residual nucleus:

1. ED-RMF: Uses a real, energy-dependent relativistic mean-field potential (same as the ground state).
2. ROP (Relativistic Optical Potential): Uses a complex optical potential to account for absorption and inelastic channels.
3. ED-RMF + NuWro: Uses the ED-RMF wave function as input for the NuWro intranuclear cascade generator. This simulates re-scattering processes where the primary proton interacts with other nucleons, potentially leading to neutron emission (e.g., $1p1n$ final states).

D. Detector Effects

• Energy Smearing: A Gaussian smearing ($\sigma = 2.5\%$ of visible energy) is applied to theoretical predictions to mimic the JSNS2 detector resolution.
• Recoil Treatment: Theoretical predictions are generated both with and without including the residual nucleus recoil energy in the definition of visible energy ($E_{vis}$), to test the sensitivity of the results to this experimental ambiguity.

3. Key Contributions

• New Spectral Function Parameterization: The introduction of an asymmetric Maxwell-Boltzmann distribution for the s-shell and a specific background term for SRCs, optimized to match exclusive $(e, e'p)$ data, provides a more realistic description of the $^{12}$C initial state than standard mean-field models.
• Systematic Analysis of Recoil: The paper quantifies the impact of the residual nucleus recoil on the missing-energy spectrum, demonstrating that neglecting it shifts the theoretical peak by 1–5 MeV.
• Benchmarking Event Generators: By coupling the RDWIA formalism with the NuWro cascade model, the authors isolate the specific impact of neutron emission (induced by proton re-scattering) on the missing-energy spectrum.
• Resolution of Peak Discrepancies: The study offers a potential explanation for the mismatch between JSNS2 data and standard event generators regarding the position of the main peak, attributing it to the treatment of nuclear recoil and detector thresholds.

4. Results

• Spectral Shape: The theoretical distribution is dominated by the p-shell contribution (peaking around $E_m \approx 18.7$ MeV), followed by a broad s-shell contribution (peaking around $E_m \approx 37$ MeV). The background (SRC) and Meson Exchange Currents (MEC) contribute significantly less.
• Impact of Recoil:
  • With Recoil: Theoretical peaks align better with the experimental data.
  • Without Recoil: The spectrum shifts to higher missing energies. The p-shell strength moves into the $20 < E_m < 25$ MeV bin. The authors suggest that if the JSNS2 detector does not detect the low-energy recoil of the heavy nucleus, the "no-recoil" scenario might explain why experimental data appears shifted compared to generator predictions that ignore recoil.
• Impact of Neutron Emission (NuWro):
  • Including the NuWro cascade (simulating neutron emission) causes a migration of events from the p-shell region ($E_m < 22$ MeV) to higher missing energies ($E_m > 30$ MeV).
  • This results in an overestimation of the cross-section at high missing energies compared to the data, suggesting that current cascade models may overestimate inelastic neutron emission at these low energies ($T_p < 50$ MeV).
• Threshold Behavior: The pure RDWIA models predict zero cross-section below the neutron emission threshold ($18.7$ MeV). However, experimental data shows non-zero values. Applying the 2.5% energy smearing fills this region, significantly improving agreement with the data.
• Cross-Section Magnitude: The ROP model predicts a total cross-section approximately 27% smaller than the ED-RMF model due to flux reduction in the elastic channel caused by the imaginary part of the optical potential.

5. Significance

• Validation of Low-Energy Nuclear Physics: The work demonstrates that accurate modeling of neutrino interactions at $\sim 235$ MeV requires going beyond simple plane-wave approximations and incorporating detailed spectral functions and recoil kinematics.
• Clarification of Experimental Ambiguities: The paper highlights a critical ambiguity in the JSNS2 analysis: the treatment of the residual nucleus recoil. It suggests that the detector's inability to see the recoil energy (or the generator's failure to account for it) is a primary source of discrepancy in the peak position.
• Limitations of Cascade Models: The results indicate that semi-classical cascade models (like NuWro) may struggle to accurately describe inelastic final states (specifically neutron emission) at low nucleon energies, where quantum effects are significant.
• Future Directions: The authors emphasize the need for absolute cross-section measurements (not just shapes) and further theoretical investigation into low-energy collective excitations and the validity of semi-classical FSI treatments in this energy regime.

In conclusion, this paper provides a sophisticated relativistic analysis of the first monochromatic neutrino missing-energy data, offering a refined theoretical framework that accounts for nuclear correlations, recoil, and detector resolution, while identifying specific areas where current event generators and experimental interpretations require improvement.