Hadronic light-by-light scattering amplitudes from lattice QCD versus dispersive sum rules

The hadronic contribution to the eight forward amplitudes of light-by-light scattering (${\gamma }^{*}{\gamma }^{*}\to {\gamma }^{*}{\gamma }^{*}$) is computed in lattice QCD. Via dispersive sum rules, the amplitudes are compared to a model of the ${\gamma }^{*}{\gamma }^{*}\to \text{hadrons}$ cross sections in which the fusion process is described by hadronic resonances. Our framework can provide an important test for the model estimates of hadronic light-by-light scattering in the anomalous magnetic moment of the muon, $a_{\mu }^{\text{HLbL}}$. Using simple parametrizations of the resonance $M\to {\gamma }^{*}{\gamma }^{*}$ transition form factors, we determine the corresponding monopole and dipole masses by performing a global fit to all eight amplitudes. Together with a previous dedicated calculation of the ${\pi }^0\to {\gamma }^{*}{\gamma }^{*}$ transition form factor, our calculation provides valuable information for phenomenological estimates of $a_{\mu }^{\text{HLbL}}$. The presented calculations are performed in two-flavor QCD with pion masses extending down to 190 MeV at two different lattice spacings. In addition to the fully connected Wick contractions, on two lattice ensembles we also compute the ($2+2$) disconnected class of diagrams, and find that their overall size is compatible with a parameter-free, large-$N$ inspired prediction, where $N$ is the number of colors. Motivated by this observation, we estimate in the same way the disconnected contribution to $a_{\mu }^{\text{HLbL}}$.

Figure 1

The five classes of quark contractions for four-point functions. In this work, we compute the leftmost, fully connected set of contractions, as well as the ($2+2$) class of diagrams (second from the left).

Figure 2

Contribution $(\times {10}^6)$ from ($2+2$) disconnected diagrams to the forward scattering amplitude ${\mathcal{M}}_{TT}$ on ensemble E5, without (left) and with (right) subtraction of the value at $\nu =0$.

Figure 3

Amplitudes ${\overline{\mathcal{M}}}_{TT}$, ${\overline{\mathcal{M}}}_{TT}^{\tau }$, ${\overline{\mathcal{M}}}_{TT}^a$ and ${\overline{\mathcal{M}}}_{LT}$ $(\times {10}^6)$ for the ensemble F6 and for two different values of $Q_1^2$ (left: $Q_1^2=0.039{\mathrm{GeV}}^2$, right: $Q_1^2=0.352{\mathrm{GeV}}^2$). The curves with error bands represent the fit results discussed in Sec. 6.

Figure 4

Amplitudes ${\overline{\mathcal{M}}}_{TL}$, ${\overline{\mathcal{M}}}_{TL}^a$, ${\overline{\mathcal{M}}}_{TL}^{\tau }$ and ${\overline{\mathcal{M}}}_{LL}$ $(\times {10}^6)$ for the ensemble F6 and for two different values of $Q_1^2$ (left: $Q_1^2=0.039{\mathrm{GeV}}^2$, right: $Q_1^2=0.352{\mathrm{GeV}}^2$). The curves with error bands represent the fit results discussed in Sec. 6.

Figure 5

Contribution $(\times {10}^6)$ from ($2+2$) disconnected diagrams to the eight subtracted forward scattering amplitudes on ensemble F6 with one fixed virtuality $Q_1^2=0.352{\mathrm{GeV}}^2$.

Figure 6

The dependence of the amplitudes ${\overline{\mathcal{M}}}_{TT}$, ${\overline{\mathcal{M}}}_{TT}^{\tau }$, ${\overline{\mathcal{M}}}_{TT}^a$ and ${\overline{\mathcal{M}}}_{LT}$ $(\times {10}^6)$ on $\nu$ for two different values of $Q_2^2$, the virtuality $Q_1^2=0.352{\mathrm{GeV}}^2$ being fixed. The results correspond to the lattice ensemble G8. Note that at fixed photon virtualities, the form factors are completely determined. The black line corresponds to the total contribution and each colored line represents a single-meson contribution.

Figure 7

The dependence of the amplitudes ${\overline{\mathcal{M}}}_{TL}$, ${\overline{\mathcal{M}}}_{TL}^a$, ${\overline{\mathcal{M}}}_{TL}^{\tau }$, ${\overline{\mathcal{M}}}_{LL}$ $(\times {10}^6)$ on $\nu$ for two different values of $Q_2^2$, the virtuality $Q_1^2=0.352{\mathrm{GeV}}^2$ being fixed. The results correspond to the lattice ensemble G8. Note that at fixed photon virtualities, the form factors are completely determined. The black line corresponds to the total contribution and each colored line represents a single-meson contribution.

Figure 8

Value of ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ for different dipole masses and form factor normalizations [tensor form factor, helicity $\mathrm{\Lambda }=(0,L)$]. Left: ensemble F7. Right: ensemble G8.

Figure 9

Chiral extrapolations for each monopole and dipole mass, excluding the ensemble E5 with the largest pion mass. The blue point corresponds to the lattice ensemble N6 and gives an indication about discretization effects. Only the statistical error is displayed.

Figure 10

The eight amplitudes $(\times {10}^6)$ for the ensemble G8 and for $Q_1^2=0.352{\mathrm{GeV}}^2$. The curves with error bands represent the fit results discussed in Sec. 6.