Abstract
In this paper we make a further step towards a dispersive description of the hadronic light-by-light (HLbL) tensor, which should ultimately lead to a data-driven evaluation of its contribution to $(g-2)_\mu$. We first provide a Lorentz decomposition of the HLbL tensor performed according to the general recipe by Bardeen, Tung, and Tarrach, generalizing and extending our previous approach, which was constructed in terms of a basis of helicity amplitudes. Such a tensor decomposition has several advantages: the role of gauge invariance and crossing symmetry becomes fully transparent; the scalar coefficient functions are free of kinematic singularities and zeros, and thus fulfill a Mandelstam double-dispersive representation; and the explicit relation for the HLbL contribution to $(g-2)_\mu$ in terms of the coefficient functions simplifies substantially. We demonstrate explicitly that the dispersive approach defines both the pion-pole and the pion-loop contribution unambiguously and in a model-independent way. The pion loop, dispersively defined as pion-box topology, is proven to coincide exactly with the one-loop scalar QED amplitude, multiplied by the appropriate pion vector form factors.
Highlights
The anomalous magnetic moment of the muon, aμ = (g − 2)μ/2, is one of the very rare quantities in particle physics where a significant discrepancy between its experimental determination and the Standard-Model evaluation still persists
In this paper we make a further step towards a dispersive description of the hadronic light-by-light (HLbL) tensor, which should lead to a data-driven evaluation of its contribution to (g − 2)μ
The uncertainty on the theory side is dominated by hadronic contributions, see e.g. [3,4,5], the largest error arising from hadronic vacuum polarization (HVP)
Summary
Since HVP is straightforwardly related to the total e+e− hadronic cross section via a dispersion relation, the evaluation of this contribution is expected to become more accurate within the few years [6] thanks to upcoming improvements in the experimental input, reducing further the present sub-percent accuracy will be challenging This implies that hadronic light-by-light (HLbL) scattering is likely to soon dominate the theory uncertainty in aμ.. Our dispersive formalism is based on the fundamental principles of unitarity, analyticity, crossing symmetry, and gauge invariance The derivation of such a systematic framework becomes challenging due to the fact that HLbL scattering is described by a hadronic four-point function whose properties are significantly more complicated than those of the two-point function entering HVP.
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