Towards a dispersive determination of the η and η′ transition form factors

Abstract

Abstract.We discuss status and prospects of a dispersive analysis of the η and η ′ transition form factors. Particular focus is put on the various pieces of experimental information that serve as input to such a calculation. These can help improve on the precision of an evaluation of the η and η ′ pole contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon. Hadronic light-by-light scattering and the anomalous magnetic moment of the muonThere is a by now long-standing discrepancy between the experimental determination of the anomalous magnetic moment of the muon as measured by the BNL E821 experiment [1], and its calculation within the Standard Model [2], with both experiment and theory affected by a comparable uncertainty. With two different proposals to further improve on the accuracy of the measurement well under way [3,4], it is of utmost importance to also improve the accuracy of the theoretical Standard-Model prediction before a persistent discrepancy of potentially increased significance can be interpreted in terms of physics beyond the Standard Model. The Standard-Model uncertainty is entirely dominated by hadronic contributions. While the dominant hadronic vacuum polarization can be expressed, via a dispersion relation, in terms of measurable e + e − → hadrons total cross sections, which therefore can be further improved upon by a strong experimental effort to determine many of the exclusive channels contributing therein with yet improved precision [5], the situation for the α QED -suppressed hadronic light-by-light scattering is less straightforward, relying until recently to a much larger extent on modeling [6], with badly controlled errors.This presentation is part of an effort to analyze also the hadronic light-by-light scattering tensor using dispersion theory [7,8]. Dispersion theory makes maximal use of analyticity and unitarity, two fundamental consequences of relativistic quantum field theories that mathematically incorporate the principles of causality and probability conservation. The various (in principle infinitely many different) contributions to hadronic light-by-light scattering are organized in terms of their analytic structure, according to the cuts and poles in the different energy variables that are dictated by the above principles. This has the advantage that all contributions will be given in terms of on-shell form factors and scattering amplitudes, hence observables that can to a large extent again be related to experimentally accessible quantities [9]. An ordering principle will be to consider intermediate

Dispersion relation for hadronic light-by-light scattering: pion pole

“…As the largest individual piece, our determination of the pion-pole contribution to a µ is a critical step towards a complete data-driven evaluation of HLbL scattering [48][49][50][51][52][53]. Moreover, the strategies developed here regarding the incorporation of high-energy constraints will facilitate similar studies of the η and η TFFs [178][179][180][181][182], thus paving the way towards a fully data-driven determination of all light pseudoscalar-meson-pole contributions to HLbL scattering in (g − 2) µ . Here, r = M 2 π 0 /m 2 µ , Λ is a UV cutoff, in ChPT to be identified with the scale of chiral symmetry breaking Λ χ ∼ 4πF π , the IR scale µ should be identified with M π 0 [12], χ(Λ) is a LEC that renormalizes the 1-loop ChPT expression for π 0 → e + e − , and C(Λ) subsumes all terms not enhanced by a logarithm.…”

Asymptotic behavior of meson transition form factors

“…In fact, with the contribution of various hadronic intermediate states organized in terms of dispersion relations [22][23][24][25][26][27], the pseudoscalar poles are completely determined by the respective TFFs. For the pion, the TFF has, in turn, been reconstructed from dispersion relations [28][29][30][31][32][33][34], leading to a result for the pion-pole contribution in perfect agreement with calculations using Canterbury approximants [35], lattice QCD [36], and Dyson-Schwinger equations [37,38], and a similar program exists for the η, η poles [39][40][41][42][43]. In either case, asymptotic constraints on the TFF are critical in controlling the high-energy part of the g − 2 integral.…”

Effects of longitudinal short-distance constraints on the hadronic light-by-light contribution to the muon $$\mathbf {g-2}$$

“…The dispersive formalism for the singly-virtual η/η TFF has been established[55] and progress has been made towards the determination of the doubly-virtual isovector contribution[56,57].…”