Dispersion relations for hadronic light-by-light scattering in triangle kinematics

Abstract

We present a new strategy for the dispersive evaluation of the hadronic light-by-light contribution to the anomalous magnetic moment of the muon aμ. The new approach directly applies in the kinematic limit relevant for aμ: one of the photons is treated as an external electromagnetic field with vanishing momentum, so that the kinematics corresponds to a triangle. We derive expressions for the relevant single-particle intermediate states, as well as the tensor decompositions of the two-pion sub-processes that appear in addition to those needed in the established dispersive approach. The existing approach is based on a set of dispersion relations for the hadronic light-by-light tensor in four-point kinematics. At present it is not known how to consistently include in this framework resonant intermediate states of spin 2 or larger, due to the appearance of kinematic singularities that can be traced back to the redundancy of the tensor decomposition. We show that our new approach circumvents this problem and enables dispersion relations in the limit of triangle kinematics that are manifestly free from kinematic singularities, paving the way towards a data-driven evaluation of all relevant exclusive hadronic intermediate states.

Theory of g-2

“…In order to include higher partial waves as well as tensor-meson resonances, the dispersive framework itself needs to be extended. Reference [60] proposed a solution to this problem, establishing dispersion relations not for fourpoint kinematics, but directly in the kinematic limit (8). This modified framework necessitates the reconstruction of further sub-processes [61], but when combined with the established dispersive framework, it promises to enable a complete dispersive evaluation of HLbL.…”

Puzzles in the hadronic contributions to the muon anomalous magnetic moment

“…The uncertainty on HLbL is now dominated on the one hand by the contributions of hadronic resonances in the (1-2) GeV range, which were estimated using hadronic models, on the other hand by the matching to short-distance constraints (SDCs) that follow from the operator-product expansion (OPE). Therefore, since the publication of the WP, efforts were directed towards an improved evaluation of these sub-dominant contributions, addressing scalar resonances beyond the 𝑓 0 (500) [10], axial-vector resonances [11][12][13][14], tensor resonances and 𝐷-waves [15,16], which require a modification of the dispersive framework itself [17], as well as the SDCs on HLbL [12,[18][19][20]. Work is in progress to combine all these developments, together with improvements of the 𝜂, 𝜂 ′ pole contributions [21][22][23][24], into a full data-driven evaluation of 𝑎 HLbL 𝜇 .…”

Puzzles in the hadronic contributions to the muon anomalous magnetic moment

“…A solution to this problem was proposed in Ref. [59], establishing dispersion relations not for four-point kinematics, but directly in the kinematic limit (8). This modified framework requires the reconstruction of further sub-processes [60], but in combination with the established dispersive framework it promises to enable a complete dispersive evaluation of HLbL.…”