Charged-current quasielastic-like neutrino scattering from $^{12}$C in the coherent density fluctuation model with two-nucleon emission

This paper calculates charged-current quasielastic-like neutrino and antineutrino scattering cross-sections on $^{12}$C using the coherent density fluctuation model with a relativistic effective mass and two-nucleon emission, providing predictions for accelerator experiments like MiniBooNE, T2K, and MINERvA while analyzing the axial form factor $C^A_5(0)$ and momentum transfer dependencies.

The Gist

Imagine you are trying to understand how a ghost (a neutrino) passes through a crowded dance floor (an atomic nucleus).

Neutrinos are the ultimate ghosts: they have no electric charge, almost no mass, and they rarely interact with anything. But when they do bump into a nucleus, they can knock a dancer (a nucleon, like a proton or neutron) off the floor. Physicists call this "scattering." By studying exactly how the dancers fly off, scientists can learn about the ghost's properties and the rules of the dance floor itself.

However, the dance floor isn't empty. The dancers are holding hands, pushing each other, and moving in complex groups. This makes predicting the outcome incredibly difficult.

This paper is a new, sophisticated guidebook for predicting exactly what happens when these ghostly neutrinos hit a specific dance floor: the Carbon-12 nucleus.

Here is the breakdown of their work, using simple analogies:

1. The Old Map vs. The New GPS

For a long time, physicists used a simple map called the Relativistic Fermi Gas (RFG) model.

• The Analogy: Imagine the dance floor as a giant, empty warehouse where everyone is just running in straight lines at random speeds. It's a good start, but it ignores the fact that people actually bump into each other, hold hands, and form clusters.
• The Problem: Real dance floors (nuclei) are crowded and chaotic. The old map failed to predict the results of recent experiments (like MiniBooNE, T2K, and MINERvA) because it didn't account for the "crowd dynamics."

2. The New Approach: The "Coherent Density Fluctuation" Model (CDFMM*)

The authors of this paper used a much more advanced GPS called the CDFMM model*.

• The Analogy: Instead of seeing the dancers as isolated runners, this model sees the dance floor as a fluid, breathing entity. It accounts for the fact that the density of dancers changes from the center of the floor to the edges.
• The "Effective Mass" Trick: In this model, the authors realized that inside the crowded nucleus, a nucleon doesn't feel like its normal self. It feels "heavier" or "lighter" due to the pressure of the crowd. They adjusted the math to treat the nucleons as having an effective mass (about 80% of their normal weight). This tiny adjustment was the key to making the predictions match reality.

3. The "Two-Person" Dance (Two-Nucleon Emission)

In the old days, physicists thought a neutrino would hit one dancer, and that dancer would fly off alone.

• The Reality: Sometimes, the neutrino hits a pair of dancers who are holding hands. The impact knocks both of them off the floor at the same time.
• The Paper's Contribution: This paper specifically calculates the probability of this "two-person" ejection (called 2p-2h or two-particle/two-hole emission). They found that ignoring this "double-knockout" is a big mistake. It accounts for about 20–30% of the total action in some experiments. If you don't count the double-knockouts, your math is off.

4. The "Delta" Mystery

When a neutrino hits a nucleon, sometimes it doesn't just knock it off; it excites it into a temporary, heavier state called a Delta resonance (think of it as a dancer getting a sudden burst of energy and spinning wildly before settling down).

• The Question: There is a specific number in the math (called $C_A^5(0)$) that determines how strong this "spin" is. Some scientists thought it was 1.2; others thought it was 0.89.
• The Finding: The authors tested both numbers against real data. They found that the 1.2 value fits the experimental data much better. This helps settle a debate in the physics community about how to handle these "spinning" particles.

5. The Results: A Perfect Match?

The authors took their new, complex model (which includes the "breathing" nucleus, the "effective mass," and the "two-person knockouts") and compared it against data from three major real-world experiments:

• MiniBooNE (USA)
• T2K (Japan)
• MINERvA (USA)

The Verdict: Their model worked beautifully. It matched the experimental data across almost all angles and energies.

• One Caveat: The model works best when the neutrino hits with a lot of energy (high momentum). When the hit is very gentle (low momentum), the "dance floor" behaves in ways that are too messy for this specific type of math, and the predictions get a little wobbly. But for the high-energy crashes that matter most for neutrino oscillation studies, the model is spot on.

Why Does This Matter?

Neutrinos are the key to understanding the universe, from why there is more matter than antimatter to how stars explode. But to study them, we need to know exactly how they interact with the detectors (which are often made of Carbon or Argon).

If our math for how neutrinos hit nuclei is wrong, our measurements of neutrino properties will be wrong. This paper provides a better, more accurate rulebook for those interactions. It's like upgrading from a rough sketch of a city to a high-definition 3D map, ensuring that when we send our "ghosts" through the "dance floor," we know exactly where they will end up.

In short: The authors built a smarter, more realistic simulation of how neutrinos smash into atomic nuclei, proving that you can't ignore the "crowd effects" or the "double-knockouts" if you want to understand the universe.

Technical Summary

1. Problem Statement

The accurate modeling of neutrino-nucleus interactions is a critical bottleneck for modern neutrino oscillation experiments (e.g., MiniBooNE, T2K, MINERvA). These experiments rely on precise measurements of oscillation parameters and CP-violating phases, which require a robust understanding of the underlying interaction cross-sections.

• Challenges: Standard models like the Relativistic Fermi Gas (RFG) fail to account for complex nuclear effects such as medium modifications, finite-size effects, nucleon-nucleon (NN) correlations, and final-state interactions.
• Specific Gap: While previous scaling models (SuSA, SuSAM) improved descriptions of electron scattering, there is a need for a unified framework that incorporates relativistic effective mass effects and two-nucleon emission (2p-2h) processes specifically within the context of charged-current (CC) quasielastic (QE) neutrino scattering on Carbon-12 ($^{12}\text{C}$). Furthermore, the treatment of the $\Delta$-resonance current and the axial form factor at zero momentum transfer, $C^A_5(0)$, introduces significant uncertainties in Meson Exchange Current (MEC) calculations.

2. Methodology

The authors employ the Coherent Density Fluctuation Model with Relativistic Effective Mass (CDFMM)*. This approach extends the Coherent Density Fluctuation Model (CDFM) by integrating dynamics from the Relativistic Mean Field (RMF) theory.

A. Theoretical Framework (CDFMM*)

• Scaling Function: The model utilizes a scaling function $f^{QE}(\psi^*)$ derived from the CDFM. This function is constructed by convoluting the RFG scaling function with a weight function $|F(x)|^2$, which is related to the nuclear density distribution $\rho(r)$.
• Relativistic Effective Mass: A key innovation is the inclusion of a relativistic effective mass $m^*_N = 0.8 m_N$ (where $M^* = 0.8$). This parameter accounts for dynamical relativistic effects (scalar and vector potentials) inherent in the RMF model, which enhances the transverse response of the nucleus.
• Scaling Variable: A new scaling variable $\psi^*$ is used, derived from the RMF model of nuclear matter, replacing the standard RFG variables.

B. Inclusion of Two-Nucleon Emission (MEC)

• 2p-2h Contributions: The model explicitly includes contributions from two-particle–two-hole (2p-2h) excitations, which are crucial for explaining the "dip" region between the QE peak and the $\Delta$-resonance.
• RMF-based Parametrization: Unlike previous works that used RFG-based MEC calculations, this study utilizes a semi-empirical parametrization (from Ref. [35]) adapted for the RMF framework. This ensures consistency by applying the same effective mass ($M^*=0.8$) to both the one-body (QE) and two-body (MEC) sectors.
• $\Delta$-Current Treatment: The study investigates the axial form factor value for $\Delta$ excitation at zero momentum transfer, $C^A_5(0)$. It compares two values: the widely used $C^A_5(0) = 1.2$ and the revised value $C^A_5(0) = 0.89$ (from Ref. [51]). The $\Delta$-current is treated in vacuum, though the authors note that nuclear medium modifications to the $\Delta$ mass are negligible for the analyzed experiments.

C. Kinematic Handling

• The authors address the breakdown of the scaling approximation at low momentum transfer ($q \leq 0.4$ GeV/c). They separate calculations into two regions:
  • $q > 0.4$ GeV/c: Treated with the CDFMM* scaling approach.
  • $q \leq 0.4$ GeV/c: Treated with a distinct approach acknowledging finite-size effects, though the scaling model is still applied with caution in this regime for comparison.

3. Key Contributions

1. Unified RMF-based MEC: First application of 2p-2h MEC contributions derived from the RMF model (using $M^*=0.8$) to neutrino scattering, replacing the inconsistent RFG-based MEC used in prior CDFM studies.
2. Systematic Analysis of $C^A_5(0)$: A detailed evaluation of how the choice of the axial form factor ($1.2$ vs. $0.89$) impacts the total cross-section, providing a green band of uncertainty in the results.
3. Comprehensive Kinematic Coverage: A thorough evaluation of the model across different momentum transfer regions ($q > 0.4$ and $q \leq 0.4$ GeV/c) for three major experiments.
4. Validation on Multiple Experiments: The model is tested against flux-averaged double-differential cross-section data from MiniBooNE, T2K, and MINERvA for both neutrinos and antineutrinos on $^{12}\text{C}$.

4. Results

The CDFMM* + MEC model demonstrates excellent agreement with experimental data across most kinematical situations.

• MiniBooNE:

  • The model (CDFM/QE/+MEC) reproduces the double-differential cross-sections well.
  • The 2p-2h MEC contribution accounts for approximately 20–25% of the total response at the peak.
  • Results using $C^A_5(0) = 1.2$ align more closely with data than those using $0.89$.
  • Deviations are observed only at extremely forward angles ($\cos \theta_\mu \approx 0.95$) and very low $q$, where the scaling approximation is known to be less reliable due to finite-size effects.

• T2K:

  • The T2K flux is narrower and peaked at lower energies ($\sim 0.7$ GeV), resulting in a smaller relative 2p-2h contribution (~10%, up to 25% at forward angles) compared to MiniBooNE.
  • The CDFMM* + MEC model aligns very well with T2K data for all scattering angles.
  • Detailed analysis of the function $\zeta(E_\nu)$ confirms that the model correctly handles the interplay between neutrino energy bins and momentum transfer constraints.

• MINERvA:

  • The model successfully describes data for both neutrino and antineutrino scattering on hydrocarbon (CH) targets.
  • The 2p-2h MEC contribution is significant, reaching 30–50% of the total response depending on kinematics.
  • The inclusion of the effective mass and consistent MEC treatment resolves discrepancies seen in pure QE models.

• Antineutrino vs. Neutrino:

  • In antineutrino scattering, the relative strength of the 2p-2h MEC increases for backward scattering angles due to destructive interference between transverse ($T$) and $T'$ channels, making the cross-section more sensitive to nuclear effects.

5. Significance and Conclusion

• Validation of CDFMM:* The study confirms that the CDFMM* model, which incorporates relativistic effective mass and NN correlations via the CDFM, is a robust tool for predicting neutrino-nucleus cross-sections.
• Importance of Consistency: The results highlight the necessity of using a consistent effective mass ($M^*=0.8$) for both QE and MEC sectors. Using RFG-based MEC with CDFM QE leads to inconsistencies; the RMF-based MEC parametrization significantly improves the description of data.
• Form Factor Constraints: The data favors the traditional value of $C^A_5(0) = 1.2$ over the revised $0.89$ for the $\Delta$-current in this specific context, though the uncertainty band is provided for future refinement.
• Future Applicability: The approach is easily extendable to heavier nuclei (e.g., $^{40}\text{Ar}$ for MicroBooNE/DUNE), making it a valuable tool for the next generation of neutrino oscillation experiments. The model successfully bridges the gap between nuclear matter theory and finite nuclear systems, providing a reliable theoretical baseline for extracting oscillation parameters.