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Neutrinoless double beta decay, if detected, would prove that neutrinos areMajorana 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 processesaccessible via first-principle lattice QCD calculations. In this work, wecalculate the long-distance contributions to the $\pi^-\to\pi^+ee$ transitionamplitude using four ensembles at the physical pion mass with various volumesand lattice spacings. We adopt the infinite-volume reconstruction method tocontrol the finite-volume effects arising from the (almost) massless neutrino.Providing the lattice QCD inputs for chiral perturbation theory, we obtain thelow energy constant$g_\nu^{\pi\pi}(m_\rho)=-10.89(28)_\text{stat}(74)_\text{sys}$, which is closeto $g_\nu^{\pi\pi}(m_\rho)=-11.96(31)_\text{stat}$ determined from thecrossed-channel $\pi^-\pi^-\to ee$ decay.

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