The Quest for the <em>η</em> and <em>η'</em> Transition Form Factors

Abstract

An important precision test of the Standard Model of Particle Physics is posed by the prediction of the anomalous magentic moment of the muon. Currently, a deviation of about 4.2<em>σ</em> persists between the experimental average of this quantity and its prediction, the error of which is dominated by hadronic uncertainties. This thesis is aimed at constructing a dispersive data-driven representation of the <em>η</em> and <em>η'</em> transition form factors and subsequently obtaining their pole contributions in the hadronic light-by-light scattering part in the prediction of the muon <em>g-2</em>. <br /> A representation of the singly-virtual <em>η'</em> transition form factor is constructed in the time-like regime. This representation is obtained by a multi-channel formalism dispersively combining data for <em>η'→π⁺π⁻γ</em> and the pion vector form factor. Furthermore, it consistently takes the isospin-breaking <em>ρ-ω</em>-mixing effect in both the isovector as well as the isoscalar channel into account. <br /> A step into the doubly-virtual regime of the <em>η</em> transition form factor is taken by analyzing data for the reaction <em>e⁺e⁻→π⁺π^-η</em>. It is demonstrated that the <em>a₂</em> tensor meson provides a natural breaking mechanism of the commonly applied factorization ansatz in the doubly-virtual form factor. <br /> The model of factorization breaking through the left-hand-cut contribution of the <em>a₂</em> by crossed-channel effects is further employed in the construction of the <em>η</em> and <em>η'</em> transition form factors starting from the decay <em>η'→2(π⁺π^-)</em>. In this decay the <em>a₂</em> left-hand-cut contribution leads to an inhomogeneous Omnès problem, whose solution is numerically challenging. A solution strategy through path deformation in the corresponding dispersion integral is followed. Subsequently, the further application of dispersion relations leads to representations of <em>η⁽'⁾→γ^*γ^*</em>. <br /> The pole contributions of pseudoscalar mesons to the hadronic light-by-light scattering part in the muon <em>g-2</em> are solely determined from the corresponding transition form factors. The general strategy in case of the <em>η</em> and <em>η'</em> mesons is outlined in terms of a simplified form factor model excluding factorization-breaking effects. It is demonstrated that the supplementation of the form factor’s dispersive part by contributions motivated by the fulfillment of the corresponding asymptotic limits leads to a description with quantifiable and controlled sources of uncertainties. The corresponding pole contributions and their uncertainty estimates were extracted by carrying out the <em>g-2</em> loop integrals. The strategy outlined can also be applied for the full-fledged model including factorization-breaking effects.