A Hybrid Strategy for the Lattice Evaluation of the Leading Order Hadronic Contribution to $(g-2)_μ$

Abstract: The leading-order hadronic contribution to the muon anomalous magentic moment, $a_\mu^{\rm LO,HVP}$, can be expressed as an integral over Euclidean $Q^2$ of the vacuum polarization function. We point out that a simple trapezoid-rule numerical integration of the current lattice data is good enough to produce a result with a less-than-$1\%$ error for the contribution from the interval above $Q^2\gtrsim 0.1-0.2\ \mathrm{GeV}^2$. This leaves the interval below this value of $Q^2$ as the one to focus on in the future. In order to achieve an accurate result also in this lower window $Q^2\lesssim 0.1-0.2\ \mathrm{GeV}^2$, we indicate the usefulness of three possible tools. These are: Pad\'{e} Approximants, polynomials in a conformal variable and a NNLO Chiral Perturbation Theory representation supplemented by a $Q^4$ term. The combination of the numerical integration in the upper $Q^2$ interval together with the use of these tools in the lower $Q^2$ interval provides a hybrid strategy which looks promising as a means of reaching the desired goal on the lattice of a sub-percent precision in the hadronic vacuum polarization contribution to the muon anomalous magnetic moment.