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 $a_{\mu }^{\pi \mathrm{box}}=-15.9(2)\times {10}^{-11}$. We then construct a suitable input for the ${\gamma }^{*}{\gamma }^{*}\to \pi \pi$ 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 $a_{\mu }^{\pi \mathrm{box}}+a_{\mu ,J=0}^{\pi \pi ,\pi \text{−}\text{pole}\mathrm{LHC}}=-24(1)\times {10}^{-11}$.

• Received 26 January 2017

DOI:https://doi.org/10.1103/PhysRevLett.118.232001

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Published by the American Physical Society