New strategy for the lattice evaluation of the leading order hadronic contribution to $(g-2{)}_{\mu }$

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 $Q^2>Q_{\min }^2$, coupled with an $N$-parameter Padé or other representation of the polarization in the interval $0<Q^2<Q_{\min }^2$, for sufficiently high $Q_{\min }^2$ 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 $Q_{\min }^2$ and $N$ can be reduced to well below the 1% level for $Q_{\min }^2$ as low as $0.1{\mathrm{GeV}}^2$ and rather small $N$. For such low $Q_{\min }^2$, both a next-to-next-to-leading-order (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 subpercent-precision determination of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment on the lattice.

Figure 1

$f(Q^2){\widehat{\mathrm{\Pi }}}^{I=1}(Q^2)$ versus $Q^2$ in the low-$Q^2$ region.

Figure 2

The accumulation of the contributions to ${\widehat{a}}_{\mu }^{\mathrm{LO},\mathrm{HVP}}$ as a function of the upper limit, $Q_{\max }^2$, of integration.

Figure 3

The impact of an uncertainty $\delta {\mathrm{\Pi }}^{I=1}(0)=0.001$ in ${\mathrm{\Pi }}^{I=1}(0)$ on ${\widehat{a}}_{\mu }^{\mathrm{LO},\mathrm{HVP}}[Q_{\min }^2,\infty ]$ as a fraction of ${\widehat{a}}_{\mu }^{\mathrm{LO},\mathrm{HVP}}$.

Figure 4

The systematic and statistical components of the error on the evaluation of ${\widehat{a}}_{\mu }^{\mathrm{LO},\mathrm{HVP}}[Q_{\min }^2,2{\mathrm{GeV}}^2]$ by direct trapezoid rule numerical integration, as a fraction of ${\widehat{a}}_{\mu }^{\mathrm{LO},\mathrm{HVP}}$.

Figure 5

Comparison of the exact dispersive model results for ${\widehat{\mathrm{\Pi }}}^{I=1}(Q^2)$ with the Padés constructed from the derivatives of the model with respect to $Q^2$ at $Q^2=0$ in the intervals $0\le Q^2\le 2{\mathrm{GeV}}^2$ (upper panel) and $0\le Q^2\le 0.4{\mathrm{GeV}}^2$ (lower panel).

Figure 6

Deviations of the Padé estimates for ${\widehat{a}}_{\mu }^{\mathrm{LO},\mathrm{HVP}}[Q_{\max }^2]$ as a fraction of ${\widehat{a}}_{\mu }^{\mathrm{LO},\mathrm{HVP}}$ in the intervals $0\le Q_{\max }^2\le 2{\mathrm{GeV}}^2$ (upper panel) and $0\le Q_{\max }^2\le 0.2{\mathrm{GeV}}^2$ (lower panel). Note the difference in scale on the vertical axis.

Figure 7

Fractional errors, relative to the underlying dispersive model input, on the subtracted versions of the $[1,1{]}_H$ and $[2,1{]}_H$ Padés obtained by fitting to the $Q^2=0.10,0.11,\dots ,0.20{\mathrm{GeV}}^2$ dispersive model values. The points with error bars centered at zero indicate the fractional errors on the input data entering the fit.

Figure 8

Comparison of the results of the conformal polynomial representations up to quadratic order with the exact $\tau$-data-based model for ${\widehat{\mathrm{\Pi }}}^{I=1}(Q^2)$.

Figure 9

Comparison of the results of the ${\mathrm{NN}}^{\prime }\mathrm{LO}$ representation (11) and the $\tau$-data-based model for ${\widehat{\mathrm{\Pi }}}^{I=1}(Q^2)$ (solid curve). The dashed line shows the result including the phenomenological term $C^r{Q}^4$, the dotted line the result with the NNNLO contribution $C^r{Q}^4$ removed.

Figure 10

The ${\mathrm{NN}}^{\prime }\mathrm{LO}$ ChPT expectation for the low-$Q^2$ behavior of the integrand for the $I=0$ contribution to $a_{\mu }^{\mathrm{LO},\mathrm{HVP}}$. Also shown, for comparison, is the integrand for the corresponding $I=1$ contribution.