A New Strategy for the Lattice Evaluation of the Leading Order Hadronic Contribution to $(g-2)_\mu$

TL;DR

This paper proposes a hybrid lattice QCD strategy combining trapezoid-rule integration and Padé approximants to accurately evaluate the hadronic vacuum polarization contribution to $(g-2)_$ with systematic errors below 1%.

Contribution

It introduces a novel hybrid approach using simple integration and rational function representations to reduce systematic errors in lattice evaluations of the muon anomalous magnetic moment.

Findings

Systematic errors can be reduced below 1% for $Q^2_{min}$ as low as 0.1 GeV$^2$.

Both NNLO chiral and conformal polynomial representations are sufficiently accurate at low $Q^2$.

The strategy shows promise for sub-percent precision lattice calculations of the hadronic vacuum polarization.

Abstract

A reliable evaluation of the integral giving the hadronic vacuum polarization contribution to the muon anomalous magnetic moment should be possible using a simple trapezoid-rule integration of lattice data for the subtracted electromagnetic current polarization function in the Euclidean momentum interval $Q^2>Q^2_{min}$, coupled with an $N$-parameter Pad\'e or other representation of the polarization in the interval $0<Q^2<Q^2_{min}$, for sufficiently high $Q^2_{min}$ and sufficiently large $N$. Using a physically motivated model for the $I=1$ polarization, and the covariance matrix from a recent lattice simulation to generate associated fake "lattice data," we show that systematic errors associated with the choices of $Q^2_{min}$ and $N$ can be reduced to well below the 1% level for $Q^2_{min}$ as low as 0.1 GeV$^2$ and rather small $N$. For such low $Q^2_{min}$, both an NNLO chiral representation with one additional NNNLO term and a low-order polynomial expansion employing a conformally transformed variable also provide representations sufficiently accurate to reach this precision for the low-$Q^2$ contribution. Combined with standard techniques for reducing other sources of error on the lattice determination, this hybrid strategy thus looks to provide a promising approach to reaching the goal of a sub-percent precision determination of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment on the lattice.