Multi-level Monte Carlo computation of the hadronic vacuum polarization contribution to (gμ − 2)

Mattia Dalla Brida,Leonardo Giusti,Tim Harris,Michele Pepe

doi:10.1016/j.physletb.2021.136191

Mattia Dalla Brida, Leonardo Giusti + Show 2 more

Open Access

https://doi.org/10.1016/j.physletb.2021.136191

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Abstract

The hadronic contribution to the muon anomalous magnetic moment aμ=(gμ−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 aμ. 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.

Highlights

The current experimental value of the muon anomalous magnetic moment aμ = 11659208.9(6.3) × 10−10 by the E821 experiment has the remarkable precision of 0.54 parts per million [1], while the on-going E989 experiment at FNAL is expected to reach the astonishing precision of 0.14 ppm by the end of its operation [2] when E34 at J-PARC may be well under way [3]
Supplemented with the operator product expansion (HLbL). This leads to aμ = 11659181.0(4.3) × 10−10 (0.37 ppm) [4], which deviates by 3 − 4 standard deviations from the E821 result, a difference persisting for a decade which may be a hint for a New Physics signal
In order to assess the potential of two-level Monte Carlo integration, we simulate Quantum Chromodynamics (QCD) with two dynamical flavours supplemented by a valence strange quark

Summary

Introduction

Lacking precise purely theoretical computations, the hadronic contributions have been extracted (by assuming the SM) from experimental data via dispersive integrals (HVP & HLbL) and low-energy effective models. The most recent lattice determination of the HVP [12] differs from the dispersive result by more than 3 standard deviations, and generates tensions with the global electroweak fits [13,14,15]. All these facts call for an independent theoretically-sound lattice computation of the hadronic contribution to aμ at the permille level from first principles. We focus on the HVP, but the strategy is general and can be applied to the HLbL, the isospin-breaking and electromagnetic contributions as well

The signal-to-noise problem

Multi-level Monte Carlo

Lattice computation

Results and discussion