Virtual Photon–photon Scattering
Based on analyticity, unitarity, and Lorentz invariance the contribution from hadronic vacuum polarization to the anomalous magnetic moment of the muon is directly related to the cross section of . We review the main difficulties that impede such an approach for light-by-light scattering and identify the required ingredients from experiment. Amongst those, the most critical one is the scattering of two virtual photons into meson pairs. We analyze the analytic structure of the process and show that the usual Muskhelishvili–Omnès representation can be amended in such a way as to remain valid even in the presence of anomalous thresholds.
Keywords: Dispersion relations; anomalous magnetic moment of the muon; Compton scattering; meson–meson interactions.
PACS numbers: 11.55.Fv, 13.40.Em, 13.60.Fz, 13.75.Lb
1 Hadronic Vacuum Polarization
The leading contribution of strong interactions to the anomalous magnetic moment of the muon originates from hadronic intermediate states in the polarization tensor of the photon. By means of gauge invariance, the polarization tensor may be expressed in terms of one single-variable scalar function
Due to analyticity, the renormalized self energy satisfies a subtracted dispersion relation
Unitarity relates the imaginary part to the hadronic cross section
In this way, general principles obeyed by the polarization tensor provide a direct link between its contribution to and observables.
2 Light-by-Light Scattering
2.1 Structure of the Light-by-Light Tensor
No such immediate relation to experiment is known for the light-by-light tensor , describing the scattering process
In contrast to vacuum polarization, there are independent Lorentz structures, cf. Ref. ?, and independent kinematic variables ( Mandelstam variables and virtualities), so that the full amplitude should be expanded in a suitable set of basis functionsbbbAs shown in Ref. ?, gauge invariance for the on-shell photon implies that only the derivative with respect to at is needed for the application in .
In order to write down dispersion relations for the scalar coefficients , the basis functions need to be chosen in such a way that the are free of kinematic singularities and that crossing symmetry, e.g. invariance under , is maintained.
The complicated structure of the light-by-light tensor prohibits a comprehensive analysis of all intermediate states allowed by unitarity. However, the most important states (besides the pseudoscalar meson poles) in the low/intermediate energy region are two-meson reducible. They can be classified according to the analytic structure in the crossed channel as shown in Fig. 1. There are classes of box, triangle, and bulb unitarity diagrams, depending on whether the crossed-channel amplitude involves non-polynomial terms. Such non-polynomial contributions are given by the pion pole and multi-pion exchange, whereas the polynomials for instance include effects due to rescattering. In practice, the multi-pion diagrams may be approximated by resonance exchange, i.e. and / for and pions, respectively. While for and a narrow-width approximation is certainly viable, the effect of the finite width of the is captured through a spectral-function approach that relies on the amplitude for as input.
2.2 Input from Experiment
The experimental ingredients necessary for this program follow from Fig. 1. Diagrams with a pion pole require the pion vector form factor, those with resonance exchange the corresponding transition form factors and the amplitude. This input for the multi-pion diagrams can again be checked for consistency within a framework respecting analyticity and unitarity. The most critical input concerns the polynomial pieces, since they involve the pole-subtracted partial waves for the process . Absent direct experimental information for arbitrary virtualities, e.g. from , these partial waves are again reconstructed dispersively, see Refs. ?, ? for two on-shell photons and Ref. ? for one photon with non-vanishing virtuality. Finally, the dispersion relations for the will involve a contribution of the pion-pole diagram, with a residue determined by the (on-shell) pion transition form factor . In order to eliminate the model-dependence as far as possible, also input for this form factor should fulfill analyticity and unitarity requirements and be backed by data wherever available.
3 Analytic Structure of
In principle, the partial waves for are constrained by a similar set of dispersion relations as derived in Refs. ?, ?, ?. Within a simplified scalar toy example, where the left-hand cut is approximated by the pion pole, one thus obtains the following Muskhelishvili–Omnès representation for the pole-subtracted -wave
with the projection of the pole term
the Omnès function , and -wave
The analytic continuation of this solution in the virtualities in the case that both photons are off-shell is complicated by the occurrence of anomalous thresholds, i.e. the singularities of the logarithm in Eq. (3) located at
In this way, left- and right-hand cut become intertwined, which invalidates the direct derivation of Eq. (3) for large virtualities.
In order to elucidate the role of these anomalous thresholds we first consider the scalar triangle loop function
If , its dispersive representation involves an additional, anomalous piece that emerges because the anomalous branch point’s moving onto the first sheet distorts the integration contour, see Fig. 2 and Ref. ?. The numerical results in Fig. 3 show that the dispersive reconstruction of indeed works for arbitrary virtualities as long as the anomalous contribution is taken into account (upper panel), but that substantial deviations occur in the region of large virtualities if the anomalous piece is ignored (lower panel).
In fact, this procedure to perform the analytic continuation in the for transfers immediately to , the crucial observation being that the integrand of Eq. (3) coincides with the discontinuity of ,
up to a factor , which is independent of and well-defined in the whole complex -plane. Therefore, the full result for becomes merely amended by an additional term that takes care of the anomalous thresholds
We would like to thank Bastian Kubis, Bachir Moussallam, and Sebastian Schneider for numerous useful discussions. Financial support by the Swiss National Science Foundation is gratefully acknowledged. The AEC is supported by the “Innovations- und Kooperationsprojekt C-13” of the “Schweizerische Universitätskonferenz SUK/CRUS.”
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