Hadronic vacuum polarization in QCD and its evaluation in Euclidean spacetime

We discuss a new technique to evaluate integrals of QCD Green’s functions in the Euclidean based on their Mellin-Barnes representation. We present as a first application the evaluation of the lowest order hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon $\frac{1}{2}(g_{\mu }-2{)}_{\mathrm{HVP}}\equiv a_{\mu }^{\mathrm{HVP}}$. It is shown that with a precise determination of the slope and curvature of the HVP function at the origin from lattice QCD (LQCD), one can already obtain a result for $a_{\mu }^{\mathrm{HVP}}$ which may serve as a test of the determinations based on experimental measurements of the $e^{+}{e}^{-}$ annihilation cross section into hadrons.

Figure 1

Shape of the function $\mathcal{R}(\tau )$ in the integrand of Eq. (5.14) (first line).

Figure 2

The toy model spectral function below $1{\mathrm{GeV}}^2$.

Figure 3

Mellin transform of the toy model spectral function (including charm).

Figure 4

Mellin transforms of the hadronic spectral function. Red: Toy model without charm. Green: Marichev’s interpolation with only $\mathcal{M}(0)$ as input. Blue: Marichev’s interpolation with $\mathcal{M}(0)$ and $\mathcal{M}(-1)$ as input.

Figure 5

Shape of the functions ${\Re }_0^{1,2}(\tau )$ (green curve) and ${\Re }_{0,-1}^{1,2}(\tau )$ (blue curve).

Figure 6

Mellin transforms corresponding to the Marichev interpolations using as an input the central values of the LQCD results of Ref. [13]. Blue is the first interpolation, and red the second one.

Figure 7

Shape of the functions ${\Re }_0^{1,2}(\tau )$ (blue curve) and ${\Re }_{-1,0}^{1,2}(\tau )$ (red curve) using as an input the central values of the LQCD results of Ref. [13].