\textit{Ab initio} calculations of first-forbidden $\beta$ transitions in the reactor antineutrino anomaly

This paper presents the first \textit{ab initio} calculations of key first-forbidden $\beta$ transitions using chiral nuclear forces, demonstrating that incorporating these microscopic shape factors into reactor antineutrino spectrum models yields a pronounced enhancement around 5 MeV that may partially explain the observed "5 MeV bump" and the reactor antineutrino anomaly.

The Gist

The Big Mystery: The "Missing" Antineutrinos

Imagine you are running a massive factory (a nuclear power plant) that produces a specific type of invisible particle called an antineutrino. For decades, scientists have had a very precise recipe (a mathematical model called the "HM model") to predict exactly how many of these particles the factory should produce and what their "energy levels" (speeds) should be.

However, when scientists actually went to the factory to count them, they found two weird things:

1. The Shortage: They found about 6% fewer particles than the recipe predicted. This became known as the "Reactor Antineutrino Anomaly."
2. The Bump: In the middle of the energy spectrum (around 5 million electron volts, or "5 MeV"), the actual data showed a huge, unexpected hill or "bump" that the recipe didn't predict.

Scientists spent years wondering: Is the recipe wrong? Is there a ghost particle hiding? Or are we just missing a step in the cooking process?

The Old Recipe: A "Good Enough" Approximation

The old recipe (the HM model) was like a simplified cooking guide. It treated almost all the nuclear reactions happening in the fuel as "allowed" transitions. Think of an "allowed" transition as a smooth, straight highway where a car (an electron) drives from point A to point B without any obstacles.

However, in reality, about 30% of these reactions are "forbidden" transitions. These aren't illegal; they just mean the car has to take a bumpy, winding dirt road with potholes and detours. The old recipe ignored these dirt roads, assuming they were just like the highway. The authors of this paper suspected that ignoring the dirt roads was causing the recipe to be wrong.

The New Approach: Building the Car from Scratch

The authors (a team of physicists from China) decided to stop using the simplified recipe. Instead, they wanted to build the car from scratch using first principles (what they call ab initio calculations).

• The Ingredients: They started with the most fundamental forces known in nature (chiral forces) that hold protons and neutrons together.
• The Construction: They used a powerful mathematical method (MBPT) to figure out how these forces work inside the specific atoms (like Uranium-235) used in reactors.
• The Result: They didn't just guess; they calculated the exact behavior of 20 specific "forbidden" transitions that are the biggest contributors to the reactor's output.

Think of it like this: Instead of assuming all cars drive at 60 mph, they went out, measured the suspension, the engine, and the tires of the specific cars taking the dirt roads, and calculated exactly how fast they would actually go.

The Discovery: The Shape of the Road Matters

When they calculated the "shape factors" (which describe how the energy is distributed on these dirt roads), they found something surprising: The roads were not straight.

• The Old View: The energy distribution was a flat line (like a highway).
• The New View: The energy distribution was a curve (like a hill or a valley).

Specifically, for the "forbidden" transitions, the shape of the curve sloped downward. In physics terms, this means the electrons (the cars) were losing a bit more energy than expected. But here is the magic trick: Energy is conserved. If the electron loses energy, the antineutrino (the passenger) gains it.

Solving the "5 MeV Bump"

This is where the mystery gets solved.

Because the "forbidden" transitions sloped downward, they pushed more energy into the antineutrinos. When the authors added this new, detailed calculation to the total reactor spectrum, they saw a massive increase in the number of antineutrinos around the 5 MeV mark.

• Before: The recipe predicted a flat line at 5 MeV.
• After: The new calculation showed a distinct "bump" right where the experiments had been seeing one.

It turns out the "5 MeV bump" wasn't a mystery or a sign of new physics; it was just the result of the "dirt roads" (forbidden transitions) being treated more accurately. The bump was there all along; the old recipe just couldn't see it.

The Conclusion

This paper is a major step forward because:

1. It's the first time anyone has done this level of "from-scratch" calculation for these specific forbidden transitions.
2. It explains the anomaly: By treating the "forbidden" transitions correctly, the predicted spectrum now matches the experimental data much better, especially around the famous 5 MeV bump.
3. It rules out ghosts: It suggests we don't need to invent a "sterile neutrino" (a ghost particle) to explain the missing data; we just needed better math for the particles we already know exist.

In short: The scientists realized the reactor wasn't leaking particles or hiding secrets. They just realized that the "forbidden" paths the particles take are bumpier and more complex than anyone thought, and those bumps create the "5 MeV bump" we see in the data.

Technical Summary

1. Problem Statement

The paper addresses two interconnected challenges in nuclear physics and neutrino phenomenology:

• The Reactor Antineutrino Anomaly (RAA): Short-baseline reactor experiments have observed a deficit in the detected antineutrino flux (~6%) compared to theoretical predictions based on the Huber-Mueller (HM) model. Additionally, a spectral distortion known as the "5 MeV bump" (an excess of events around 5 MeV) has been observed in the 4–7 MeV energy range.
• Theoretical Limitations: The standard HM model relies on converting measured electron spectra to antineutrino spectra. A critical weakness in this approach is the treatment of forbidden $\beta$ transitions. In the HM model, first-forbidden transitions (which contribute ~30% of the total spectrum and dominate above 4 MeV) are approximated as "allowed" transitions. This simplification ignores the complex nuclear structure and multipole operator interplay, potentially introducing significant uncertainties in the predicted flux and spectral shape.
• Lack of Ab Initio Data: Prior to this work, there were no ab initio (first-principles) calculations for first-forbidden $\beta$ transitions. Existing studies often relied on phenomenological models requiring empirical "quenching" factors to match data.

2. Methodology

The authors employed a rigorous $ab$ $initio$ framework based on chiral effective field theory ($\chi$EFT) to derive both the nuclear Hamiltonian and the transition operators self-consistently.

• Nuclear Forces: The calculations started from chiral two-nucleon (2NF) and three-nucleon (3NF) forces. The 2NF was truncated at N4LO and 3NF at N2LO, using local and non-local regulators. These forces were evolved using the Similarity Renormalization Group (SRG) to a momentum resolution scale of $\lambda = 2.0 \text{ fm}^{-1}$.
• Many-Body Framework:
  • Basis: A Hartree-Fock basis was constructed using a harmonic oscillator potential with 15 major shells ($e_{max}=14$) and optimized frequency $\hbar\omega = 16 \text{ MeV}$.
  • Effective Hamiltonian & Operators: Using Many-Body Perturbation Theory (MBPT), the authors derived valence-space effective Hamiltonians and effective $\beta$-decay transition operators. This was achieved via the $\hat{Q}$-box and $\hat{\Theta}$-box folded diagram formalism, decoupling a small valence space from the full Hilbert space without empirical parameters.
  • Valence Space: The calculations utilized a specific valence space: $\pi\{0f_{5/2}, 1p_{3/2}, 1p_{1/2}, 0g_{9/2}\}$ and $\nu\{0g_{7/2}, 1d_{5/2}, 1d_{3/2}, 2s_{1/2}\}$.
  • Diagonalization: The resulting effective Hamiltonians were diagonalized using the large-scale shell-model code KSHELL.
• Transition Operators: The formalism explicitly calculated the shape factors $C(W)$ for first-forbidden transitions. This involved computing nuclear matrix elements for various multipole operators (vector and axial-vector) and accounting for Coulomb corrections and finite nuclear size effects.
• Spectrum Reconstruction: The calculated shape factors and branching ratios for 20 dominant transitions were combined with cumulative fission yields for $^{235}\text{U}$ (from the ENDF database) to reconstruct the antineutrino spectrum.

3. Key Contributions

• First Ab Initio Calculation: This is the first study to perform ab initio calculations for first-forbidden $\beta$ transitions, moving beyond phenomenological models.
• Self-Consistent Derivation: The work derives both the effective Hamiltonian and the forbidden transition operators from the same underlying chiral nuclear forces, ensuring consistency.
• No Quenching Factors: The calculations reproduce experimental data without introducing empirical quenching factors, which are typically required in phenomenological shell-model calculations to correct for missing correlations.
• Microscopic Shape Factors: The authors provide microscopic shape factors that deviate significantly from the "allowed" approximation (unity), demonstrating that the energy dependence of forbidden transitions is non-trivial.

4. Results

• Log $ft$ Values: The authors calculated 20 dominant first-forbidden transitions relevant to reactor spectra.
  • $\Delta J = 0$: Calculated values show reasonable agreement with experimental data.
  • $\Delta J = 1$: Good agreement, though some cases show slight overestimation (~1.2 units).
  • $\Delta J = 2$: Systematic overestimation observed. The authors attribute this to the neglect of two-body currents (2BCs), which are more critical for unique $\Delta J = 2$ transitions that depend on a single operator.
• Shape Factors: The calculated normalized shape factors for forbidden transitions deviate significantly from unity (the allowed approximation).
  • $\Delta J = 0$ transitions show parabolic structures.
  • $\Delta J = 2$ transitions exhibit simple parabolic shapes due to dependence on a single operator ($AF^0_{211}$).
  • These deviations imply that treating forbidden transitions as allowed introduces substantial bias in spectral distributions.
• Impact on $^{235}\text{U}$ Antineutrino Spectrum:
  • Incorporating the microscopic shape factors resulted in a ~6% enhancement in the predicted antineutrino flux around 5 MeV.
  • Below 4 MeV, the spectrum showed a parabolic deviation from HM predictions, improving agreement with experimental data (e.g., Daya Bay).
  • The enhancement at 5 MeV is attributed to the downward-sloping shape factors of forbidden transitions, which redistribute flux toward lower electron energies (and thus higher antineutrino energies).

5. Significance

• Mitigating the "5 MeV Bump": The study provides a partial explanation for the "5 MeV bump" observed in reactor experiments. By explicitly treating first-forbidden transitions, the theoretical prediction moves closer to experimental observations, reducing the discrepancy without invoking new physics (such as sterile neutrinos) as the sole explanation.
• Validation of Ab Initio Methods: The work demonstrates that modern ab initio methods (MBPT with chiral forces) are capable of accurately describing complex weak interaction observables in medium-mass nuclei, a crucial step for nuclear astrophysics and neutrino physics.
• Future Directions: The paper highlights the importance of including two-body currents (2BCs) in future calculations to resolve the remaining discrepancies in $\Delta J = 2$ transitions and further refine the antineutrino spectrum predictions.

In conclusion, this paper establishes that the simplified treatment of forbidden transitions in standard reactor models is a major source of uncertainty. By applying rigorous ab initio methods, the authors demonstrate that explicit treatment of these transitions significantly alters the predicted antineutrino spectrum, offering a compelling nuclear-physics-based explanation for the reactor antineutrino anomaly and the 5 MeV bump.