Neutrinoless double-beta decay of the $\Delta^-$ resonance

This paper presents a first step toward a $\Delta$-full description of neutrinoless double-beta decay by systematically deriving the long-range and short-range contributions to the $\Delta^- \to p e^-e^-$ transition within chiral effective field theory, including predictions for pion-mass dependence and results in the degenerate $\Delta$-nucleon mass limit to facilitate lattice-QCD matching.

The Gist

Imagine the universe as a giant, complex machine where tiny particles are constantly interacting. For decades, physicists have been trying to solve a massive mystery: Are neutrinos their own antiparticles?

If the answer is "yes," it would mean that a fundamental rule of the universe (called "lepton number") can be broken. This would explain why there is more matter than antimatter in the universe and reveal the true nature of neutrino mass. The "smoking gun" for this theory is a rare event called Neutrinoless Double-Beta Decay.

Think of this decay like a double-take in a magic trick. Usually, when two neutrons turn into two protons, they spit out two electrons and two invisible "ghosts" (antineutrinos). But in this special, forbidden version, the ghosts never appear. The two neutrons just vanish, and two protons and two electrons pop out. The neutrinos must have cancelled each other out internally, proving they are their own twins.

The Problem: The "Black Box" of the Nucleus

Detecting this event is incredibly hard because it happens so rarely. To find it, scientists look at heavy atoms (like Xenon or Germanium). But inside an atom, the nucleus is a chaotic crowd of protons and neutrons. Calculating exactly how this "ghost cancellation" happens inside that crowd is like trying to predict the exact path of a single raindrop in a hurricane.

Current calculations are good, but they treat the protons and neutrons as simple, solid balls. However, we know that inside these balls, things get messy. Sometimes, a proton or neutron gets excited and transforms into a heavier, short-lived cousin called the Delta resonance ($\Delta$).

Imagine a proton as a calm, steady drumbeat. The Delta resonance is that same drum, but someone hit it so hard it's vibrating wildly and screaming for a split second before settling back down. Previous theories ignored this "screaming drum," assuming it didn't matter. But this paper argues that ignoring the Delta is like ignoring the bass in a song; you miss a huge part of the rhythm.

The Solution: A New Map with "Delta" Included

The authors of this paper, led by Li-Ping He and Feng-Kun Guo, decided to build a new, more detailed map of this process. They used a sophisticated mathematical toolkit called Chiral Effective Field Theory (think of it as a set of rules for how particles interact at low energies) to include the Delta resonance explicitly.

Here is what they did, broken down into simple concepts:

1. The "Delta" Detour: They calculated what happens when the decay process takes a "detour" through a Delta particle. Instead of just Neutron $\to$ Proton, they looked at Neutron $\to$ Delta $\to$ Proton.
2. The Loop De-Loop: In quantum physics, particles can pop in and out of existence in "loops." The authors found that when the Delta is involved, these loops create some very strange, sharp spikes in the math.
   • Analogy: Imagine driving on a highway. Usually, the road is smooth. But with the Delta included, there are sudden, sharp "speed bumps" (called threshold cusps) and "hairpin turns" (called triangle singularities) that can drastically speed up the process. These spikes might make the decay happen much faster than we thought, making it easier to detect.
3. The "Short-Cut" Fix: Because these loops create mathematical infinities (which don't make sense in the real world), the authors had to invent "counterterms."
   • Analogy: Think of the calculation as a recipe. The loops added too much salt (infinite energy). The authors added a specific "anti-salt" ingredient (counterterms) to balance the flavor perfectly. These ingredients represent unknown physics that future experiments need to measure.

The "Degenerate" Trick

One of the coolest parts of the paper is how they prepared for future experiments.

• The Problem: In the real world, the Delta particle is unstable and decays instantly. This makes it hard to simulate on computers (Lattice QCD) because the math gets messy with complex numbers.
• The Trick: The authors calculated a special "what-if" scenario where the Delta particle has the exact same mass as a proton. In this "degenerate" world, the Delta is stable, and the math becomes clean and simple (purely real numbers).
• Why it matters: This provides a perfect "test case" for supercomputers. If the computer simulation matches the authors' clean math in this special world, scientists can trust the complex calculations for the real world.

The Big Picture

This paper is a crucial step forward. It says: "We can't just ignore the excited, screaming cousins of the proton anymore. They might be the key to unlocking the neutrino mystery."

By including the Delta resonance, the authors have:

• Refined the theoretical prediction for how often this decay should happen.
• Identified specific "spikes" in the data that could enhance the signal.
• Provided a clean, simplified version of the math that computer scientists can use to verify their models.

In short, they have upgraded the blueprint for finding the universe's biggest secret. If we can detect this decay, we will finally know that neutrinos are their own antiparticles, rewriting our understanding of the cosmos.

Technical Summary

1. Problem Statement

The observation of neutrinoless double-beta decay ($0\nu\beta\beta$) is crucial for determining the absolute neutrino mass scale and establishing whether neutrinos are Majorana particles. The fundamental subprocess for nuclear $0\nu\beta\beta$ decay is the transition of two neutrons into two protons and two electrons ($nn \to ppe^-e^-$).

While this transition has been studied within Chiral Effective Field Theory ($\chi$EFT) up to Next-to-Next-to-Leading Order (NNLO), previous calculations explicitly excluded the intermediate $\Delta(1232)$ resonance. Given the strong coupling of the $\Delta$ resonance to pions and nucleons, omitting it may lead to inaccuracies in the nuclear matrix elements and the low-energy constants (LECs). The authors aim to provide the first $\Delta$-full description of the $nn \to ppe^-e^-$ process by investigating the elementary decay $\Delta^- \to p e^- e^-$, which serves as a key building block for the full nuclear transition.

2. Methodology

The authors employ Relativistic Chiral Effective Field Theory ($\chi$EFT) with explicit $\Delta$ degrees of freedom, utilizing the Small Scale Expansion (SSE) scheme where the mass difference $\delta = m_\Delta - m_N$ is counted as $O(p)$.

• Framework: The calculation is based on the Weinberg dimension-five operator, which generates light Majorana neutrino masses and lepton number violation ($\Delta L = 2$).
• Process: They calculate the decay amplitude for $\Delta^-(p_1) \to p(p_2) e^-(q_1) e^-(q_2)$.
• Loop Calculation: The long-range contribution is derived from one-loop Feynman diagrams involving the exchange of light Majorana neutrinos, pions, and nucleons. The authors use dimensional regularization and the $\overline{MS} - 1$ subtraction scheme.
• Renormalization: To handle ultraviolet (UV) divergences arising from the loops, they construct a set of short-range counterterm operators. These operators are built systematically using a bottom-up approach, ensuring chiral invariance and removing redundant structures via Equations of Motion (EOMs).
• Kinematics: The analysis focuses on the kinematic configuration where the two final-state electrons are collinear ($q_1 = q_2$), corresponding to the line $t=u$ in Mandelstam variables. This simplifies the amplitude to two Neutrinoless Transition Form Factors (TFFs), $T$ and $T'$.
• Lattice QCD Matching: To facilitate comparison with future Lattice QCD (LQCD) simulations, the authors calculate the amplitude in the degenerate mass limit ($m_\Delta \to m_N$). In this limit, the $\Delta$ is stable against strong decays, and the amplitude becomes purely real, avoiding the complexities of finite-volume effects associated with unstable resonances.

3. Key Contributions

• First $\Delta$-Full Calculation: This work provides the first systematic derivation of the $0\nu\beta\beta$ matrix element for the $\Delta^- \to p e^- e^-$ transition, extending previous $\Delta$-less $\chi$EFT calculations.
• Counterterm Construction: The authors explicitly construct the necessary contact interactions (counterterms) required to renormalize the one-loop amplitude. They identify the specific LECs ($h_1, h'_1, h_2, \dots$) that absorb UV divergences.
• Singularity Analysis: A detailed investigation of threshold cusps and triangle singularities (Landau singularities) is performed. The authors demonstrate that these kinematic singularities can significantly enhance the decay amplitude, particularly near the physical pion mass.
• LQCD Benchmark: By deriving the amplitude in the degenerate mass limit ($m_\Delta = m_N$), the authors provide a purely real, analytically manageable prediction that can be directly matched with LQCD results, which currently cannot handle the unstable $\Delta$ resonance at physical pion masses.

4. Key Results

• Amplitude Decomposition: The symmetric hadronic amplitude is decomposed into eight Lorentz structures, which reduce to two TFFs ($T$ and $T'$) in the collinear electron limit.
  • $T$ is independent of unknown LECs at the leading order and arises from specific loop diagrams.
  • $T'$ depends on the axial coupling $g_A$ and the $\Delta N \pi$ coupling $g_1$.
• Power Counting: The loop diagrams contribute at $O(p^3)$. The counterterms are constructed to match this order.
• Pion Mass Dependence:
  • The decay amplitude exhibits strong sensitivity to the pion mass ($M_\pi$).
  • Enhancements: Significant enhancements are observed near $M_\pi = \delta$ (the $\Delta$-nucleon mass splitting) due to a triangle singularity involving the intermediate pion, neutrino, and nucleon.
  • Cusps: Threshold cusps appear at $M_\pi = \delta/2$ and $M_\pi = \delta/\sqrt{8}$, arising from the opening of specific intermediate channels.
  • Numerical Impact: While diagrams containing explicit $\Delta$ propagators are numerically subleading over most of the pion mass range, the singularities near the physical point ($M_\pi \approx 140$ MeV) make their contribution non-negligible.
• Degenerate Limit ($m_\Delta = m_N$):
  • In this limit, the imaginary part of the amplitude vanishes.
  • The long-range contribution ($T_{LR}$) is calculated analytically and found to be finite in the chiral limit ($M_\pi \to 0$) due to the cancellation of logarithmic divergences.
  • The short-range contribution ($T_{SR}$) is proportional to the renormalized LEC $h^R_1$, which must be determined by LQCD.

5. Significance

• Theoretical Precision: This work is a critical step toward a high-precision theoretical description of nuclear $0\nu\beta\beta$ decay. Including the $\Delta$ resonance is essential for reducing the theoretical uncertainty in nuclear matrix elements, which currently vary by an order of magnitude.
• Experimental Guidance: The identification of threshold enhancements and singularities suggests that the $\Delta$-exchange mechanism could be more significant than previously estimated, potentially affecting the interpretation of experimental limits on the effective Majorana neutrino mass.
• Bridge to Lattice QCD: The derivation of the real amplitude in the degenerate mass limit provides a concrete, testable benchmark for Lattice QCD. Future LQCD simulations can calculate the relevant LECs ($h^R_1$) in this limit, which can then be used to constrain the full physical amplitude via chiral extrapolation.
• Phenomenology: The study of triangle singularities in this context highlights the importance of kinematic effects in weak decay processes, offering insights that may be applicable to other hadronic weak decays and lepton-number-violating searches.

In summary, this paper establishes the theoretical foundation for including $\Delta$ resonances in neutrinoless double-beta decay calculations, quantifies the impact of kinematic singularities, and provides the necessary tools for matching with Lattice QCD to ultimately improve the sensitivity of $0\nu\beta\beta$ experiments to neutrino properties.