Rescattering effects in the hadronic-light-by-light contribution to the anomalous magnetic moment of the muon

We present a first model-independent calculation of $\pi \pi$ intermediate states in the hadronic-light-by-light (HLbL) contribution to the anomalous magnetic moment of the muon $(g-2{)}_{\mu }$ that goes beyond the scalar QED pion loop. To this end we combine a recently developed dispersive description of the HLbL tensor with a partial-wave expansion and demonstrate that the known scalar-QED result is recovered after partial-wave resummation. Using dispersive fits to high-statistics data for the pion vector form factor, we provide an evaluation of the full pion box, $a_\mu^{\pi$\text{-box}$}=-15.9(2)\times 10^{-11}$\mathrm{.}Wethenconstructsuitableinputforthe$\gamma^\gamma^\to\pi\pi$ helicity partial waves based on a pion-pole left-hand cut and show that for the dominant charged-pion contribution this representation is consistent with the two-loop chiral prediction and the COMPASS measurement for the pion polarizability. This allows us to reliably estimate $S$-wave rescattering effects to the full pion box and leads to our final estimate for the sum of these two contributions: $a_\mu^{\pi$\text{-box}$} + a_{\mu,J=0}^{\pi\pi,\pi$\text{-pole LHC}$}=-24(1)\times 10^{-11}$.