Abstract
The largest uncertainties in the Standard Model calculation of the anomalous magnetic moment of the muon (g−2)µ come from hadronic effects, namely hadronic vacuum polarization (HVP) and hadronic lightby-light (HLbL) contributions. Especially the latter is emerging as a potential roadblock for a more accurate determination of (g−2)µ. 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)µ 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 $ \alpha _\mu ^{\pi {\rm{ - box}}} = - 15.9(2) \times {10^{ - 11}} $. A first model-independent calculation of effects of ππ 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 γ*γ* → ππ 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 $ \alpha _\mu ^{\pi {\rm{ - box}}} + \alpha _{\mu ,J = 0}^{\pi \pi ,\pi {\rm{ - pole}}\,{\rm{LHC}}} = - 24(1) \times {10^{ - 11}} $.
Highlights
The anomalous magnetic moment of the muon (g − 2)μ has been measured [1] and computed to very high precision of about 0.5 ppm
The largest uncertainties in the Standard Model calculation of the anomalous magnetic moment of the muon (g − 2)μ come from hadronic effects, namely hadronic vacuum polarization (HVP) and hadronic lightby-light (HLbL) contributions
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
Summary
The anomalous magnetic moment of the muon (g − 2)μ has been measured [1] and computed to very high precision of about 0.5 ppm (see e.g. [2]). If forthcoming data from e+e− experiments and/or progress in lattice calculations help reduce uncertainties in the HVP, the subleading HLbL contribution would dominate the theory error. We have constructed a generating set of Lorentz structures for the HLbL tensor that is free of kinematic singularities and zeros. This simplifies significantly the calculation of the HLbL contribution to (g − 2)μ. We present a numerical evaluation of the pion box using a form factor fit to high-statistics data, in turn using a dispersive representation to analytically continue the time-like data into the space-like region required for the (g − 2)μ integral and show that this contribution can be calculated with negligible uncertainties. Our calculation settles the role of the pion polarizability, which enters at next-toleading order in the chiral expansion of the HLbL amplitude [31,32,33] and has been suspected to produce sizable corrections in [32]
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