Hadronic light-by-light scattering in the muon $g-2$: A Dyson-Schwinger equation approach

We determine the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon using the framework of Dyson-Schwinger and Bethe-Salpeter equations of QCD. Our result for the pseudoscalar $({\pi }^0,\eta ,{\eta }^{\prime })$ meson-exchange diagram is commensurate with previous calculations. In our calculation of the quark-loop contribution we improve upon previous approaches by explicitly implementing constraints due to gauge invariance. The impact of transverse contributions, presumably dominated by vector-meson poles, are only estimated at this stage. As a consequence, our value $a_{\mu }^{\mathrm{LBL};\mathrm{quarkloop}}=(136\pm 59)\times {10}^{-11}$ is significantly larger. Taken at face value, this then leads to a revised estimate of the total $a_{\mu }=116591891.0(105.0)\times {10}^{-11}$.

Figure 1

The two classifications of corrections to the photon-muon vertex function are shown: (a) hadronic vacuum polarization contribution to $a_{\mu }$. The vertex is dressed by the vacuum polarization tensor ${\Pi }_{\mu \nu }$; (b) the hadronic light-by-light scattering contribution to $a_{\mu }$.

Figure 2

The hadronic LBL scattering contribution to $a_{\mu }$ and its expansion, using EFT approaches, as a quark-loop part (left), leading pseudoscalar meson-exchange part (middle), and a leading meson-loop part (right), is shown. Note that the quarks here may be interpreted differently to those in Fig. 3.

Figure 3

The hadronic LBL scattering contribution to $a_{\mu }$ and its expansion, using functional methods, as a quark-loop part (left), a ladder-exchange part (middle), and a ladder-ring part (right), is shown. All propagators and vertices are fully dressed, with the ellipsis marks indicating that an infinite number of gluons are resummed.

Figure 4

The expansion of quark-loop contribution to the photon four-point function in terms of planar quark and gluon diagrams (all propagators are fully dressed) is shown.

Figure 5

The hadronic light-by-light scattering contributions to $a_{\mu }$ from the quark loop are shown. There are an additional three diagrams (not shown) in which the quark-spin line is reversed. Principally, these diagrams involve dressed quark propagators and quark-photon vertices.

Figure 6

The ladder-exchange contribution (upper equation) and ring-ladder contribution (lower equation) to the photon four-point amplitude are shown.

Figure 7

The $T$ matrix in the rainbow-ladder approximation is shown. The top diagram shows the series expansion in terms of dressed quarks and gluon, while the middle represents Dyson’s equation. The bottom diagram shows the pole ansatz for the $T$ matrix on-mass-shell.

Figure 8

The pole representation of the ladder-exchange contribution to the photon four-point function is shown.

Figure 9

The Dyson-Schwinger equation for the quark-propagator is shown. Specification of the fully dressed gluon propagator (wiggly line) and quark-gluon vertex (shaded blob) defines the truncation scheme.

Figure 10

The pion-pole part of the LBL contribution to $a_{\mu }$ is shown. The three possible permutations of the photon legs are not shown.

Figure 11

Mass function and wave-function dressing functions corresponding to quark propagators solved with the Maris-Tandy interaction [44] are shown.

Figure 12

The homogeneous Bethe-Salpeter equation for the meson amplitude is shown.

Figure 13

Normalized Bethe-Salpeter amplitudes $F_1$ and $F_2$ of the pion solved with the Maris-Tandy interaction (leading Chebyshev component) [44] are shown.

Figure 14

The inhomogeneous BS equation for the quark-photon vertex, in rainbow-ladder approximation, is shown.

Figure 15

The ${\pi }^0\gamma \gamma$ form factor in impulse approximation is shown. All internal quantities are fully dressed.

Figure 16

The $\pi \to \gamma \gamma$ form factor plotted as a function of the two photon momenta $k_1^2$ and $k_2^2$ is shown. We compare our numerical results in the Dyson-Schwinger/Bethe-Salpeter approach with an ansatz inspired by vector-meson dominance discussed in Ref. 5.