The largest uncertainties in the Standard Model calculation of the anomalous magnetic moment of the muon $(g-2)_\mu$ come from hadronic effects, namely hadronic vacuum polarization (HVP) and hadronic light-by-light (HLbL) contributions. Especially the latter is emerging as a potential roadblock for a more accurate determination of $(g-2)_\mu$.

The main focus here is on a novel dispersive description of the HLbL tensor, which is based on unitarity, analyticity, crossing symmetry, and gauge invariance. This opens up the possibility of a data-driven determination of the HLbL contribution to $(g-2)_\mu$ with the aim of reducing model dependence and achieving a reliable error estimate.

Our dispersive approach defines unambiguously the pion-pole and the pion-box contribution to the HLbL tensor. Using Mandelstam double-spectral representation, we have proven that the pion-box contribution coincides exactly with the one-loop scalar-QED amplitude, multiplied by the appropriate pion vector form factors. Using dispersive fits to high-statistics data for

the pion vector form factor, we obtain $a_\mu^{\pi\text{-box}}=-15.9(2)\times 10^{-11}$. A first model-independent calculation of effects of $\pi\pi$ intermediate states that go beyond the scalar-QED pion loop is also presented. We combine our 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.

After constructing suitable input for the $\gamma^*\gamma^*\to\pi\pi$helicity partial waves based on a pion-pole left-hand cut (LHC), we find 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 $a_\mu^{\pi\text{-box}} + a_{\mu,J=0}^{\pi\pi,\pi\text{-pole LHC}}=-24(1)\times 10^{-11}$.