[Paper Review] Nuclear Matrix Elements for Neutrinoless Double Beta Decay from Lattice QCD

William Detmold, D. Murphy|arXiv (Cornell University)|Nov 13, 2018

Neutrino Physics Research5 references4 citations

TL;DR

This paper presents a first-principles lattice QCD calculation of the long-distance contributions to neutrinoless double beta decay (0νββ) using the π⁻ → π⁺ee⁻ transition as a proxy. By employing advanced FFT-based algorithms to efficiently compute double sums over spacetime points, the authors achieve a computationally feasible method for evaluating the hadronic matrix elements critical for interpreting 0νββ decay experiments, with preliminary results on a 16³×32 lattice at mπ = 420 MeV.

ABSTRACT

While neutrino oscillation experiments have demonstrated that neutrinos have small, nonzero masses, much remains unknown about their properties and decay modes. One potential decay mode --- neutrinoless double beta decay ($0 νββ$) --- is a particularly interesting target of experimental searches, since its observation would imply that the neutrino is a Majorana particle, demonstrate that lepton number conservation is violated in nature, and give further constraints on the neutrino masses and mixing angles. Relating experimental constraints on $0 νββ$ decay rates to the neutrino masses, however, requires theoretical input in the form of non-perturbative nuclear matrix elements which remain difficult to calculate reliably. In this talk we will discuss progress towards first-principles calculations of relevant nuclear matrix elements using lattice QCD and effective field theory techniques, assuming neutrinoless double beta decay mediated by a light Majorana neutrino. We will show preliminary results for the $π^{-} ightarrow π^{+} e^{-} e^{-}$ transition amplitude computed on a $16^{3} imes 32$ domain wall fermion lattice with a pion mass of 420 MeV, and discuss improved methods applicable to general lattice calculations of $0 νββ$ decay amplitudes.

Motivation & Objective

• To develop a first-principles lattice QCD approach to compute nuclear matrix elements for neutrinoless double beta decay (0νββ), a process that would reveal whether neutrinos are Majorana particles and violate lepton number.
• To address the longstanding challenge of reliably calculating non-perturbative nuclear matrix elements M⁰ν, which currently vary by up to 100% across different nuclear models.
• To establish a computationally viable method for evaluating the long-distance contributions to 0νββ decay amplitudes using effective field theory and lattice QCD techniques.
• To benchmark and optimize the double summation over spacetime points in the hadronic matrix element, which is computationally prohibitive with standard methods.

Proposed method

• The study uses the π⁻ → π⁺ee⁻ transition as a minimal proxy for 0νββ decay, enabling first-principles lattice QCD computation of the hadronic matrix element Hαβ(x,y) in Eq. (5).
• Wick contractions are applied to decompose the matrix element into two types of diagrams (Type 1 and Type 2), each requiring evaluation of time-ordered, bilocal matrix elements involving quark propagators.
• The double sum over spacetime points x and y in the hadronic tensor is reduced from O(V²) to O(V log V) using the convolution theorem and fast Fourier transforms (FFTs), enabling feasibility on current computational resources.
• A block Toeplitz matrix structure of the neutrino propagator is exploited to perform efficient convolutions via FFTs, with implementation using the FFTW library on CPUs and cuFFT on GPUs.
• A UV cutoff is applied via a Gaussian regulator in the neutrino propagator, SΛ(x,y), to render the matrix element finite, with Λ = π/a chosen to ensure proper continuum limit.
• The method is benchmarked against explicit double summation and single-sum methods, showing significant performance gains, especially on GPUs, as lattice volume increases.

Experimental results

Research questions

• RQ1 Can lattice QCD be used to compute the long-distance contributions to the 0νββ decay amplitude in a first-principles, ab initio manner?
• RQ2 What computational strategies can reduce the O(V²) cost of double summations over spacetime points in the hadronic matrix element to a feasible O(V log V) scaling?
• RQ3 How do different implementations—CPU vs. GPU, FFT-based vs. direct summation—affect performance and scalability in lattice QCD calculations of 0νββ matrix elements?
• RQ4 To what extent can the π⁻ → π⁺ee⁻ transition serve as a reliable proxy for the full 0νββ decay amplitude in nuclear systems?
• RQ5 How do finite-volume and UV regularization effects impact the reliability and convergence of the computed matrix elements?

Key findings

• The authors successfully computed the π⁻ → π⁺ee⁻ transition amplitude on a 16³×32 domain wall fermion lattice with mπ = 420 MeV, marking a first step toward lattice QCD calculations of 0νββ matrix elements.
• The FFT-based double summation method reduces computational cost from O(V²) to O(V log V), making large-volume calculations feasible on current hardware.
• GPU acceleration with cuFFT provides a significant performance advantage over CPU-based FFTW, especially as lattice volume increases, due to parallelization of multiple FFTs.
• The use of a Gaussian UV regulator SΛ(x,y) with Λ = π/a ensures proper removal of the cutoff in the continuum limit a → 0.
• The method is robustly benchmarked against explicit double summation and single-sum approaches, confirming accuracy and scalability.
• The framework is now being extended to larger lattices (24³×64), multiple pion masses, and inclusion of short-distance contributions, enabling future matching to chiral perturbation theory.