Direct measurement of \(\alpha_{\rm QED}(m_{\rm Z}^{2})\) at the FCC-ee

Abstract

When the measurements from the FCC-ee become available, an improved determination of the standard-model ”input” parameters will be needed to fully exploit the new precision data towards either constraining or fitting the parameters of beyond-the-standard-model theories. Among these input parameters is the electromagnetic coupling constant estimated at the Z mass scale, \(\alpha_{\rm QED}(m^{2}_{\rm Z})\). The measurement of the muon forward-backward asymmetry at the FCC-ee, just below and just above the Z pole, can be used to make a direct determination of \(\alpha_{\rm QED}(m^{2}_{\rm Z})\) with an accuracy deemed adequate for an optimal use of the FCC-ee precision data.

The design study of the Future Circular Colliders (FCC) in a 100-km ring in the Geneva area has started at CERN at the beginning of 2014, as an option for post-LHC particle accelerators. The study has an emphasis on proton-proton and electron-positron high-energy frontier machines (FCC 2015). In the current plans, the first step of the FCC physics programme would exploit a high-luminosity \({\rm e^{+}e^{-}}\) collider called FCC-ee, with centre-of-mass energies ranging from the Z pole to the \({\rm t\bar{t}}\) threshold and beyond. A first look at the physics case of the FCC-ee can be found in Ref. (Bicer 2014).

In this first look, an estimate of the achievable precision on a number of Z-pole observables was inferred and used in a global electroweak fit to set constraints on weakly-coupled new physics up to a scale of 100 TeV (Ellis 2015). These constraints were obtained under two assumptions: (i) the precision of the pertaining theoretical calculations will match the expected experimental accuracy by the time of the FCC-ee startup; and (ii) the determination of standard-model input parameters – four masses: \(m_{\rm Z}\), \(m_{\rm W}\), \(m_{\rm top}\), \(m_{\rm Higgs}\); and three coupling constants: \(\alpha_{s}(m^{2}_{\rm Z})\), \(G_{\rm F}\), \(\alpha_{\rm QED}(m^{2}_{\rm Z})\) – will improve in order not to be the limiting factors to the constraining power of the fit. The determinations of the Higgs boson mass from the LHC data (Aad 2015) and of the Fermi constant from the muon lifetime measurement (Tishchenko 2013) are already sufficient for this purpose. It is argued in Refs. (Bicer 2014, d’Enterria 2015) that the FCC-ee can adequately improve the determination of the other three masses and of the strong coupling constant by one order of magnitude or more: the experimental precision targets for the FCC-ee are 100 keV for the Z-boson mass, 500 keV for the W-boson mass, 10 MeV for the top-quark mass, and 0.0001 for the strong coupling constant. (The FCC-ee also aims at reducing the Higgs boson mass uncertainty down to 8 MeV.)

No mention was made, however, of a way to improve the determination of the electromagnetic coupling constant evaluated at the Z mass, and it was simply assumed that a factor \(5\) improvement with respect to today’s uncertainty – down to \(2\times 10^{-5}\) – could be achieved by the time of the FCC-ee startup. Today, \(\alpha_{\rm QED}(m^{2}_{\rm Z})\) is determined from \(\alpha_{\rm QED}(0)\) (itself known with an accuracy of \(10^{-10}\)) with the running coupling constant formula:

\begin{equation} \alpha_{\rm QED}(m^{2}_{\rm Z})=\frac{\alpha_{\rm QED}(0)}{1-\Delta\alpha_{\ell}(m^{2}_{\rm Z})-\Delta\alpha^{(5)}_{\rm had}(m^{2}_{\rm Z})}.\\ \end{equation}Its uncertainty is dominated by the experimental determination of the hadronic vacuum polarization, \(\Delta\alpha^{(5)}_{\rm had}(m^{2}_{\rm Z})\), obtained from the dispersion integral:

\begin{equation} \Delta\alpha^{(5)}_{\rm had}(m^{2}_{\rm Z})=\frac{\alpha m^{2}_{\rm Z}}{3\pi}\int_{4m_{\pi}^{2}}^{\infty}\frac{R_{\gamma}(s)}{s(m^{2}_{\rm Z}-s)}ds,\\ \end{equation}where \(R_{\gamma}(s)\) is the hadronic cross section \(\sigma^{0}({\rm e^{+}e^{-}}\to\gamma^{\ast}\to{\rm hadrons})\) at a given centre-of-mass energy \(\sqrt{s}\), normalized to the muon pair cross section at the same centre-of-mass energy. At small values of \(\sqrt{s}\), typically up to 5 GeV, and in the \(\Upsilon\) resonance region from 9.6 to 13 GeV, the evaluation of the dispersion integral relies on the measurements made with low-energy \({\rm e^{+}e^{-}}\) data accumulated by the KLOE, CMD-2/SND, BaBar, Belle, CLEO and BES experiments. The most recent re-evaluation (Davier 2011, Davier 2012) gives \(\Delta\alpha^{(5)}_{\rm had}(m^{2}_{\rm Z})=(275.7\pm 1.0)\times 10^{-4}\), which leads to

\begin{equation} \label{eq:alphaQED}\alpha_{\rm QED}^{-1}(m_{\rm Z}^{2})=128.952\pm 0.014,\\ \end{equation}corresponding to a relative uncertainty on the electromagnetic coupling constant, \(\Delta\alpha/\alpha\), of \(1.1\times 10^{-4}\). It is hoped that future low-energy \({\rm e^{+}e^{-}}\) data collected by the BES III and VEPP-2000 colliders will improve this figure to \(5\times 10^{-5}\) or better (Jegerlehner 2011).