Lattice QCD Calculation of Hadronic Light-by-Light Scattering

We perform a lattice QCD calculation of the hadronic light-by-light scattering amplitude in a broad kinematical range. At forward kinematics, the results are compared to a phenomenological analysis based on dispersive sum rules for light-by-light scattering. The size of the pion pole contribution is investigated for momenta of typical hadronic size. The presented numerical methods can be used to compute the hadronic light-by-light contribution to the anomalous magnetic moment of the muon. Our calculations are carried out in two-flavor QCD with the pion mass in the range of 270–450 MeV and contain so far only the diagrams with fully connected quark lines.

Figure 1

Four-point function quark contraction topologies. The vertices represent vector currents, and the lines are quark propagators. In this work, we compute only the leftmost, fully connected class of diagrams.

Figure 2

Fully connected quark contractions for the four-point function. Each panel represents two contractions with opposite directions of quark flow. The solid quark lines are computed by using a point-source propagator, the dashed lines by using sequential propagators, and the dotted lines by using double-sequential propagators.

Figure 3

The forward scattering amplitude ${\mathcal{M}}_{TT}$ at a fixed virtuality $Q_1^2=0.377{\mathrm{GeV}}^2$, as a function of the other photon virtuality $Q_2^2$, for different values of $\nu$. The curves represent the predictions based on Eq. (10); see the text for details.

Figure 4

The dependence of the amplitude ${\mathcal{M}}_{TT}$ on $\nu$, both photon virtualities being fixed at $0.377{\mathrm{GeV}}^2$, at three different pion masses. The dashed and dotted curves show the ${\pi }^0$ and ${\pi }^0+{\eta }^{\prime }$ contributions, respectively (there is no $\eta$ meson in two-flavor QCD), the solid curve includes all single-meson and ${\pi }^{+}{\pi }^{-}$ contributions, and the dash-dotted curves additionally include the high-energy contribution for the case of real photons at the physical pion mass.

Figure 5

Two Lorentz contractions of the vector four-point function at nonforward kinematics. For $\lambda =1$ (squares), the pion pole contribution vanishes, while for $\lambda =0$ (circles) it does not. The curves correspond to the ${\pi }^0$ pole contribution in the latter case, using the expression given in Ref. [16] and assuming vector meson dominance for the pion transition form factor.