The Effective Fine Structure Constant at TESLA Energies

TL;DR

This work refines the hadronic contribution to the running fine-structure constant by combining a data-driven dispersion relation with a theory-driven Adler-function (Euclidean) approach, achieving a more precise determination of $\alpha^{-1}(M_Z^2)$ and illustrating a complementary Euclidean method that reduces reliance on low-energy hadronic data. It demonstrates that the standard evaluation yields $\Delta\alpha^{(5)}_{\rm had}(M_Z^2)=0.027896\pm0.000395$ and $\alpha^{-1}(M_Z^2)=128.907\pm0.054$, while the Adler-function approach gives $\Delta\alpha^{(5)}_{\rm had}(M_Z^2)=0.027730\pm0.000209$ and $\alpha^{-1}(M_Z^2)=128.930\pm0.029$, with a parallel Euclidean result $\Delta\alpha^{(5)}_{\rm had}(-M_Z^2)=0.027685\pm0.000146\pm0.000149$. The paper also discusses the complexities of defining a high-energy running charge due to gauge issues, proposing consistent schemes (on-shell/Abelian photon or BF-MOM) and detailing the $W$, $Z$, and mixing contributions, including a small top-quark effect. Beyond numerical results, it highlights the need for improved low-energy cross-section data and explores future avenues—lattice QCD and experimental programs—to further reduce uncertainties, which is crucial for precision electroweak tests at TESLA energies. Collectively, these insights advance precision predictions for electroweak observables and Higgs-sector constraints at high-energy colliders.

Abstract

We present a new estimate of the hadronic contribution to the shift in the fine structure constant at LEP and TESLA energies and calculate the effective fine structure constant. Substantial progress in a precise determination of this important parameter is a consequence of substantially improved total cross section measurements by the BES II collaboration and an improved theoretical understanding. In the standard approach which relies to a large extend on experimental data we find $Δ\al_{\rm hadrons}^{(5)}(\mz) = 0.027896 \pm 0.000395$ which yields $α^{-1}(\mz) = 128.907 \pm 0.054$. Another approach, using the Adler function as a tool to compare theory and experiment, allows us to to extend the applicability of perturbative QCD in a controlled manner. The result in this case reads $Δα^{(5)}_{\rm had}(M_Z^2) = 0.027730 \pm 0.000209$ and hence $α^{-1}(\mz) = 128.930 \pm 0.029$. At TESLA energies a new problem shows up with the definition of an effective charge. A possible solution of the problem is presented. Prospects for further progress in a precise determination of the effective fine structure constant are discussed.

Figures (6)

• Figure 1: The running of $\alpha$. The "negative" $E$ axis is chosen to indicate space-like momentum transfer. The vertical bars at selected points indicate the uncertainty.
• Figure 2: Recent CMD-2 results CMD.
• Figure 3: Recent BES-II results BES.
• Figure 4: Adler function: theory vs. experiment (a) EJKV98.
• Figure 5: Adler function: theory vs. experiment (b) EJKV98.