Dispersion relations for hadronic light-by-light and the muon g − 2

¹ Fakultät für Physik, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria
² University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
³ Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA
⁴ Department of Physics,University of California at San Diego, La Jolla, CA 92093, USA

^a e-mail: mprocura@univie.ac.at
^b e-mail: gilberto@itp.unibe.ch
^c e-mail: mhofer@uw.edu
^d e-mail: pstoffer@ucsd.edu

Published online: 27 April 2020

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 ${\alpha }_{\mu }^{\pi \text{-box}}=-15.9(2)\times {10}^{-11}$. 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 ${\alpha }_{\mu }^{\pi \text{-box}}+{\alpha }_{\mu ,J=0}^{\pi \pi ,\pi \text{-pole}\hspace{0.17em}\text{LHC}}=-24(1)\times {10}^{-11}$.