Hadronic Light-by-Light Scattering Contribution to the Muon Anomalous Magnetic Moment from Lattice QCD

The most compelling possibility for a new law of nature beyond the four fundamental forces comprising the standard model of high-energy physics is the discrepancy between measurements and calculations of the muon anomalous magnetic moment. Until now a key part of the calculation, the hadronic light-by-light contribution, has only been accessible from models of QCD, the quantum description of the strong force, whose accuracy at the required level may be questioned. A first principles calculation with systematically improvable errors is needed, along with the upcoming experiments, to decisively settle the matter. For the first time, the form factor that yields the light-by-light scattering contribution to the muon anomalous magnetic moment is computed in such a framework, lattice $\mathrm{QCD}+\mathrm{QED}$ and QED. A nonperturbative treatment of QED is used and checked against perturbation theory. The hadronic contribution is calculated for unphysical quark and muon masses, and only the diagram with a single quark loop is computed for which statistically significant signals are obtained. Initial results are promising, and the prospect for a complete calculation with physical masses and controlled errors is discussed.

Figure 1

Two classes of diagrams contributing to $a_{\mu }(\mathrm{HLbL})$. On the left, all QED vertices lie on a single quark loop, The right diagram is one of six diagrams where QED vertices are distributed over two (three, or four) quark loops. We refer to these as (quark) connected and disconnected diagrams, respectively.

Figure 2

Perturbative expansion of the first term in Eq. (1) with respect to QED. The symbols ${⟨,⟩}_{\mathrm{QCD}+\mathrm{q}\text{−}\mathrm{QED}}$ and $⟨,{⟩}_{\mathrm{q}\text{−}\mathrm{QED}}$ represent the average over $\mathrm{QCD}+\mathrm{QED}$ configurations ($U^{\mathrm{QCD}}$, $A^{\mathrm{QED}}$) and that over $A^{\mathrm{QED}}$, respectively. Terms represented by the ellipsis contain four or more internal photons, and so their orders are higher than ${\alpha }^3$.

Figure 3

The muon’s magnetic form factor in units of $(\alpha /\pi {)}^3$ from light-by-light scattering, evaluated at the lowest nontrivial lattice momentum $2\pi /L$. Results for several symmetric source-sink separations are shown; the quark loop is the same for each and corresponds to $m_{\pi }=329\mathrm{MeV}$ (circles). Also shown is the pure QED result (squares, triangle).

Figure 4

The muon’s magnetic form factor in units of $(\alpha /\pi {)}^3$ from hadronic light-by-light scattering. $m_{\pi }=329\mathrm{MeV}$; $t_{\text{sep}}=10$.