Abstract
The largest uncertainties in the Standard Model calculation of the anomalous magnetic moment of the muon (g – 2)μ come from hadronic effects, and in a few years the subleading hadronic light-by-light (HLbL) contribution might dominate the theory error. We present a 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 [see formula in PDF]. 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 [see formula in PDF].
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, and in a few years the subleading hadronic light-by-light (HLbL) contribution might dominate the theory error
We present a 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]). In present calculations of the HLbL contribution, systematic errors are difficult, if not impossible, to quantify, due to model dependence. 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 spacelike region required for the (g − 2)μ integral and show that this contribution can be calculated with negligible uncertainties. We present a first numerical evaluation of S -wave ππ-rescattering effects, which unitarize the pion-pole contribution to γ∗γ∗ → ππ. This constitutes the first step towards a full treatment of the γ∗γ∗ → ππ partial waves [20,21,22]. Our calculation settles the role of the pion polarizability, which enters at next-to-leading order in the chiral expansion of the HLbL amplitude [23,24,25] and has been suspected to produce sizable corrections in [24]
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