Lattice Calculation of Short-Range Contributions to Neutrinoless Double-Beta Decay $\pi^-\to\pi^+ ee$ at Physical Pion Mass

This paper presents a lattice QCD calculation of the $\pi^- \to \pi^+ ee$ matrix elements at the physical pion mass using domain wall fermions, employing a novel method to subtract around-the-world effects and nonperturbative renormalization to reduce uncertainties and reconcile discrepancies in previous studies of neutrinoless double-beta decay short-range contributions.

The Gist

The Big Picture: Solving a Cosmic Mystery

Imagine the universe is a giant puzzle, and one of the missing pieces is the nature of neutrinos. These are tiny, ghost-like particles that zip through everything. Scientists want to know: Are they "Dirac" fermions (like regular electrons, where a particle and its antiparticle are distinct) or "Majorana" fermions (where a particle is its own antiparticle)?

The only way to solve this mystery is to watch for a very rare event called neutrinoless double-beta decay. It's like watching two atoms spontaneously change into different atoms and spit out two electrons, but without releasing any neutrinos. If we see this happen, it proves neutrinos are their own antiparticles.

The Problem: A Noisy Signal

To predict if this event will happen, physicists have to do some heavy math. They break the calculation into two parts:

1. The Long-Distance Part: Like a whisper traveling across a room.
2. The Short-Distance Part: Like a shout happening right next to your ear.

This paper focuses on the Short-Distance Part. Specifically, they are calculating how two pions (particles made of quarks) interact to produce two electrons. Think of this as measuring the "loudness" of that shout.

The Conflict: Two different teams of scientists had previously tried to measure this "loudness" using supercomputers (called Lattice QCD). However, their results disagreed by a factor of two. It was like one team saying the shout was 60 decibels, and the other saying it was 120 decibels. This huge disagreement made it hard to trust the predictions for the neutrino mystery.

The Solution: A New Way to Clean Up the Data

The authors of this paper decided to run their own experiment to settle the score. They used a massive supercomputer to simulate the subatomic world. But they faced a specific technical problem: "Around-the-World Effects."

The Analogy: Imagine you are recording a conversation in a small, echoey room with a circular wall. If you clap your hands, the sound travels forward, hits the wall, wraps around the room, and comes back to you from behind. In the computer simulation, the "room" is the grid of space-time. Because the grid is finite, the particles can travel all the way around the loop and interfere with the measurement, creating a confusing "echo" that ruins the data.

The Innovation: Previous methods tried to guess how to cancel out these echoes. This team invented a new subtraction method.

• Instead of guessing, they isolated the "echo" signal directly from the data.
• They calculated exactly how strong the echo was and subtracted it from the main signal.
• The Result: The "noise" vanished, leaving a clean, stable signal (a "plateau") that they could trust.

The Verification: Checking the Ruler

To make sure their new method wasn't broken, they checked their work against a known standard. They calculated a specific value (called a "bag parameter") that had been measured by other teams before.

• Their result matched the trusted standard perfectly.
• When they compared their result to the team that had the "factor of two" difference, they found their numbers were exactly double the other team's numbers.
• The Conclusion: It turns out the other team likely used a slightly different "ruler" (normalization convention) for their measurements. Once you account for that difference, the data points actually line up very well. The authors' method confirms their calculation is correct and resolves the confusion.

The Final Result

The team successfully calculated the "short-range" contribution to the neutrinoless double-beta decay process with much higher precision than before.

• They removed the "echoes" (around-the-world effects) that were messing up previous data.
• They used two different mathematical "lenses" (renormalization schemes) to ensure their math was solid.
• They provided a definitive, high-precision number that helps physicists predict whether we will eventually see this rare decay in real-life experiments.

In short: They built a better microscope, cleaned up the static noise, and confirmed that the previous disagreement was just a matter of using different measuring tapes. Now, the scientific community has a reliable number to help solve the mystery of neutrino mass.

Technical Summary

1. Problem Statement

Neutrinoless double-beta ($0\nu\beta\beta$) decay is a critical process for determining whether neutrinos are Majorana fermions and for probing Lepton Number Violation (LNV). The decay rate depends on nuclear matrix elements (NMEs). In the Standard Model Effective Field Theory (SMEFT), short-range contributions to $0\nu\beta\beta$ arise from dimension-9 operators. These contributions are mediated by the process $\pi^- \to \pi^+ ee$ via pion exchange between nucleons.

The Core Issue: Previous lattice QCD calculations of the short-range matrix elements for $\pi^- \to \pi^+ ee$ (specifically Refs. [30] and [31]) reported significantly discrepant results. For the bag parameter $B_\pi^{MS}$ at $\mu=3$ GeV, Ref. [30] reported $\approx 0.42$, while Ref. [31] reported $\approx 0.20$—a discrepancy of roughly a factor of two. This inconsistency hinders reliable theoretical predictions for $0\nu\beta\beta$ decay rates. Furthermore, previous studies suffered from:

• Systematic Uncertainties: Significant "around-the-world" (ATW) effects due to pion backward propagation in periodic temporal lattices, which spoil the extraction of ground-state matrix elements.
• Chiral Extrapolation: Some studies relied on unphysical pion masses, requiring extrapolation to the physical point, introducing additional model dependence.

2. Methodology

The authors performed a high-precision lattice QCD calculation using Domain Wall Fermion (DWF) ensembles generated by the RBC/UKQCD Collaboration.

A. Lattice Setup

• Ensembles: Used $N_f = 2+1$ ensembles, including two with physical pion masses (48I and 64I) with different lattice spacings ($a^{-1} \approx 1.73$ and $2.36$ GeV) to enable a direct continuum extrapolation. Unphysical mass ensembles were also used for cross-checks.
• Operators: Calculated matrix elements for five dimension-9 four-quark operators ($O_1, O_2, O_3, O'_1, O'_2$) relevant to the $\pi^- \to \pi^+ ee$ transition.

B. Mitigation of "Around-the-World" (ATW) Effects

A major innovation of this work is a new method to handle ATW effects, which arise when pions propagate across the periodic temporal boundary, contaminating the correlation functions.

• Analysis: The authors analyzed the temporal structure of three-point correlation functions, identifying contributions from the desired diagram (A) and ATW diagrams (B, C, D).
• New Subtraction Method: Instead of relying on ratio methods (R1 or R2) that only partially suppress ATW effects, the authors proposed a direct subtraction technique.
  • They identified time regions where the ATW contribution from a single pion crossing the boundary (Diagram B) dominates.
  • They reconstructed the ATW contribution directly from the lattice data using specific time separations ($t_{\pi\pi}, t_\pi$).
  • Formula: The corrected matrix element is obtained by subtracting the reconstructed ATW term:
    $$O^{(sub)}_i(t) = N_\pi^{-2} e^{2m_\pi t} \left[ C^i_3(t,t) - 2 C^i_3(t_{\pi\pi}, T - t_{\pi\pi} - t_\pi) e^{-m_\pi(T-t_\pi-2t_{\pi\pi})} \right]$$
  • This method successfully restored stable plateaus in the effective mass plots, significantly reducing systematic errors compared to previous ratio methods.

C. Renormalization

• Scheme: Non-perturbative renormalization was performed using the RI/SMOM (Rome-Southampton) method with non-exceptional kinematics.
• Schemes: Two schemes were employed to estimate perturbative truncation errors: $(\gamma_\mu, \gamma_\mu)$ and $(\not{q}, \not{q})$.
• Matching: The results were converted to the $\overline{MS}$ scheme at $\mu = 3$ GeV using one-loop matching coefficients. The difference between the two RI/SMOM schemes was used to estimate the systematic uncertainty from perturbative truncation.

3. Key Results

A. Bare Matrix Elements

The study extracted bare matrix elements $\langle \pi^+ | O_i | \pi^- \rangle$ at the physical pion mass.

• Cross-Check with Bag Parameters: The authors verified their normalization by comparing the matrix element of $O_3$ (related to the neutral meson bag parameter $B_K$) with previous FLAG-reviewed results. The results agreed perfectly with Ref. [38] (which used the same normalization convention) but were approximately twice as large as the results in Ref. [31].
• Normalization Discrepancy: The ratio of the authors' results to Ref. [31] is consistently $\approx 2.0$ for all five operators. This suggests a global normalization factor error in Ref. [31], rather than a physics discrepancy.

B. Renormalized Matrix Elements and Bag Parameter

The final results in the $\overline{MS}$ scheme ($\mu = 3$ GeV) are:

• Bag Parameter: $B_\pi^{MS}(3 \text{ GeV}) = 0.3769(24)_{stat}(76)_{PT}(1)_L$.
  • This value is consistent with Ref. [30] ($0.421$) within uncertainties but significantly different from Ref. [31] ($0.197$).
  • The ratio to Ref. [31] is $1.91(18)$, confirming the factor-of-2 normalization issue.
• Continuum Extrapolation: Using the physical mass ensembles (48I and 64I), the authors performed a direct continuum extrapolation, eliminating uncertainties associated with chiral extrapolation.
• Uncertainty Reduction: The total uncertainties in this work are significantly reduced compared to previous studies due to:
  1. Direct calculation at physical pion mass.
  2. Improved statistical precision.
  3. Effective removal of ATW systematic errors.
  4. Robust estimation of perturbative truncation errors.

4. Significance and Conclusion

• Resolution of Discrepancy: This work provides an independent cross-check that strongly supports the normalization used in Ref. [30] and the FLAG review, while identifying a likely normalization error in Ref. [31]. The factor-of-2 discrepancy is resolved as a normalization convention issue rather than a fundamental physics disagreement.
• Methodological Advancement: The proposed subtraction method for ATW effects is a robust technique that can be applied to other lattice calculations involving light mesons in periodic volumes, improving the reliability of future $0\nu\beta\beta$ matrix element calculations.
• Impact on Physics: By providing precise, continuum-extrapolated matrix elements at physical pion masses, this work reduces the non-perturbative QCD uncertainties in the theoretical prediction of $0\nu\beta\beta$ decay rates. This is crucial for interpreting experimental results from next-generation experiments (e.g., LEGEND, nEXO, CUPID) to determine the nature of neutrino mass and the scale of new physics.

In summary, this paper establishes a new benchmark for short-range contributions to neutrinoless double-beta decay, resolving previous inconsistencies through rigorous methodology, physical mass simulations, and a novel treatment of finite-volume artifacts.