Hadronic light-by-light scattering and the pion polarizability

We compute the charged pion loop contribution to the light-by-light scattering amplitude for off-shell photons in chiral perturbation theory through next-to-leading order (NLO). We show that for small photon virtualities ($k^2\ll m_{\pi }^2$) the NLO contributions are relatively more important due to an accidental numerical suppression of the LO terms. This behavior is consistent with previous calculations of the hadronic light-by-light contribution to the muon anomalous magnetic moment, $a_{\mu }^{\mathrm{HLBL}}$, whose leading order value receives $\mathcal{O}(1)$ corrections from models incorporating some of the NLO physics. We also show that models employed thus far for the charged pion loop contribution to $a_{\mu }^{\mathrm{HLBL}}$ are not fully consistent with low-momentum behavior implied by quantum chromodynamics, having omitted potentially significant contributions from the pion polarizability.

Figure 1

Representative diagrams for charged pion loop contributions to the LO and NLO to LBL amplitude.