Multi-level Monte Carlo computation of the hadronic vacuum polarization contribution to $(g_\mu-2)$

TL;DR

This paper introduces a multi-level Monte Carlo method to significantly reduce statistical errors in lattice QCD calculations of the hadronic vacuum polarization contribution to muon g-2, enabling more precise Standard Model predictions.

Contribution

The paper demonstrates the effectiveness of multi-level Monte Carlo integration in reducing variance in lattice QCD computations of the muon anomalous magnetic moment.

Findings

Multi-level Monte Carlo reduces statistical errors exponentially with distance.

The method accelerates convergence of long-distance contributions.

It improves the precision of lattice QCD calculations for g-2.

Abstract

The hadronic contribution to the muon anomalous magnetic moment $a_\mu=(g_\mu-2)/2$ has to be determined at the per-mille level for the Standard Model prediction to match the expected final uncertainty from the ongoing E989 experiment. This is 3 times better than the current precision from the dispersive approach, and 5-15 times smaller than the uncertainty on the purely theoretical determinations from lattice QCD. So far the stumbling-block is the large statistical error in the Monte Carlo evaluation of the required correlation functions which can hardly be tamed by brute force. Here we propose to solve this problem by multi-level Monte Carlo integration, a technique which reduces the variance of correlators exponentially in the distance of the fields. We test our strategy by computing the Hadronic Vacuum Polarization on a lattice with a linear extension of 3 fm, a spacing of 0.065 fm, and a pion mass of 270 MeV. Indeed the two-level integration makes the contribution to the statistical error from long-distances de-facto negligible by accelerating its inverse scaling with the cost of the simulation. These findings establish multi-level Monte Carlo as a solid and efficient method for a precise lattice determination of the hadronic contribution to $a_\mu$. As the approach is applicable to other computations affected by a signal-to-noise ratio problem, it has the potential to unlock many open problems for the nuclear and particle physics community.