Hadronic vacuum polarization and muon $g-2$ from magnetic susceptibilities on the lattice

We present and test a new method to compute the hadronic vacuum polarization function in lattice simulations. This can then be used, e.g., to determine the leading hadronic contribution to the anomalous magnetic moment of the muon. The method is based on computing susceptibilities with respect to external electromagnetic plane wave fields and allows for a precision determination of both the connected and the disconnected contributions to the vacuum polarization. We demonstrate that the statistical errors obtained with our method are much smaller than those quoted in previous lattice studies, primarily due to a very effective suppression of the errors of the disconnected terms. These turn out to vanish within small errors, enabling us to quote an upper limit. We also comment on the accuracy of the vacuum polarization function determined from present experimental $R$-ratio data.

Figure 1

The renormalized vacuum polarization determined from the experimental $R$ ratio [29], performing the integral (7) up to $s=s_{\max }=(2\mathrm{GeV}{)}^2$, where three quark flavors are active. Also indicated is the result of the integral supplemented by three-flavor perturbation theory for $s>(2\mathrm{GeV}{)}^2$.

Figure 2

The low-momentum region of the oscillatory susceptibilities as measured on the ${24}^3\times 32$ configurations at $\beta =3.45$. The curves correspond to polynomial- and Padé-type extrapolations of $2{\chi }_p$ to $p=0$. The direct determination ${\chi }_0$ is shifted horizontally to the left for better visibility. Also included are results obtained using random wall sources, displaced horizontally to the right.

Figure 3

Magnetic susceptibility with respect to a homogeneous background as a function of the logarithm of the lattice spacing ($a_0=0.29\mathrm{fm}$), using three different approaches (the generalized integral method [48], the finite difference method [58, 59] and data generated in this study using the half-half method [56]). Also included are a comparison to $\mathcal{O}(g^2)$ perturbation theory and a parametrization via a rational Ansatz.

Figure 4

Vacuum polarization via magnetic susceptibilities in the low-momentum region. The data include both connected and disconnected contributions.

Figure 5

Subtracted vacuum polarization with independent determinations of $\mathrm{\Pi }(p^2)$ and $\mathrm{\Pi }(0)$. The data include both connected and disconnected contributions. The solid red line indicates the experimental result (cf. Fig. 1) and the dotted line the three-loop perturbative prediction (see the text).

Figure 6

Statistical error of the total (connected plus disconnected) $\mathrm{\Pi }(p^2=0.03{\mathrm{GeV}}^2)$ as a function of the number of inversions. Compared are the results obtained from oscillatory susceptibilities, using point sources and random wall sources. In addition, the error of the connected oscillatory susceptibility alone is shown. Note the logarithmic scale.

Figure 7

Disconnected contribution to $\mathrm{\Pi }(p^2)$ as a function of $p^2$ for our five lattice spacings.

Figure 8

The statistical error of the vacuum polarization at low momenta around $p^2=0.03{\mathrm{GeV}}^2$ for several lattice studies in the literature and for the present work (shaded area). Open points denote the error of the unsubtracted $\mathrm{\Pi }(p^2)$, while full symbols indicate that of the renormalized ${\mathrm{\Pi }}_{\mathrm{R}}(p^2)$. Studies involving only the connected contribution are indicated in yellow, while those also taking into account the disconnected terms are indicated in blue. The determination using the experimental $R$ ratio is also included for comparison (solid green point).