Long-distance Contributions to Neutrinoless Double Beta Decay $\pi^- \to\pi^+ e e$

TL;DR

This paper calculates the long-distance contributions to the neutrinoless double beta decay process involving pions using lattice QCD, providing essential inputs for understanding lepton number violation and neutrino properties.

Contribution

It presents the first lattice QCD calculation of the $ ext{pi}^- o ext{pi}^+ e e$ transition amplitude at physical pion mass, including finite-volume effects and deriving relevant low energy constants.

Findings

Calculated the low energy constant $g_ u^{ extpi extpi}$ with controlled uncertainties.

Demonstrated the use of the infinite-volume reconstruction method for this decay process.

Provided lattice QCD inputs for chiral perturbation theory related to neutrinoless double beta decay.

Abstract

Neutrinoless double beta decay, if detected, would prove that neutrinos are Majorana fermions and provide the direct evidence for lepton number violation. If such decay would exist in nature, then $\pi^-\pi^-\to ee$ and $\pi^-\to\pi^+ ee$ (or equivalently $\pi^-e^+\to\pi^+ e^-$) are the two simplest processes accessible via first-principle lattice QCD calculations. In this work, we calculate the long-distance contributions to the $\pi^-\to\pi^+ee$ transition amplitude using four ensembles at the physical pion mass with various volumes and lattice spacings. We adopt the infinite-volume reconstruction method to control the finite-volume effects arising from the (almost) massless neutrino. Providing the lattice QCD inputs for chiral perturbation theory, we obtain the low energy constant $g_\nu^{\pi\pi}(m_\rho)=-10.89(28)_\text{stat}(74)_\text{sys}$, which is close to $g_\nu^{\pi\pi}(m_\rho)=-11.96(31)_\text{stat}$ determined from the crossed-channel $\pi^-\pi^-\to ee$ decay.