Perturbative heavy quark contributions to the anomalous magnetic moment of the muon

We discuss a method for calculating the heavy quark vacuum polarisation contribution to the muon anomalous magnetic moment, $a_\mu$, using perturbative QCD. This approach is independent of $e^+e^-$ cross-section data allowing a fully theoretical evaluation of these contributions. We confirm an existing result at lower orders in $\alpha_s$ and state a new explicit analytic formula which includes terms up to $\mathcal{O}\left(\alpha_s^3\right)$. Numerically the charm quark contribution to $a_\mu$ is found to be $a_\mu^c = (14.5 \pm 0.2)\times 10^{-10}$ and the bottom contributes $a_\mu^b = (0.302 \pm 0.002) \times 10^{-10} $. Our uncertainty estimates include both parametric uncertainties from $\hat{m}_q(\hat{m}_q)$ and $\alpha_s(\hat{m}_q)$, and theoretical uncertainties in the perturbative expansion. Comparison is made between these results and alternative approaches such as lattice QCD or those based on a dispersion relation and cross-section data.