The hadronic contribution to the running of the electroweak gauge couplings

Abstract

We present an updated determination of the hadronic vacuum polarization contribution to the running of the electromagnetic coupling $Δα_{\mathrm{had}}^{(5)}(-Q^2)$, and of the electroweak mixing angle in the space-like momentum range up to $12 \ \mathrm{GeV}^2$. Using $N_f=2+1$ CLS ensembles at five values of the lattice spacing and several pion masses, including the physical point, we achieve a significantly enhanced precision over our previous result. A refined analysis strategy based on telescopic series and a new family of kernel functions enables a clean separation of distinct Euclidean regions, disentangling strong cutoff effects at short distances from the pronounced chiral dependence at larger ones. Employing the Euclidean split technique, we convert our lattice results into an ab initio estimate of $Δα_{\mathrm{had}}^{(5)}(M_Z^2)$. A comparison with results from other lattice calculations and phenomenology is performed. We also analyze improvement scenarios required to match the projected precision of future electroweak measurements at next-generation colliders.

Figures (3)

• Figure 1: Illustration of the fits to the isovector contribution $\bar{\Pi}^{(3,3)}_{\mathrm{sub}}(Q^2/16)$ in the LV region. Left: continuum extrapolation for different improvement schemes and vector-current discretizations. Lines correspond to individual fits weighted by the model averaging procedure. Right: chiral extrapolation to the physical pion mass for the highest-weight fit, showing finite-$a$ chiral trajectories and the continuum dependence on $\phi_2$. Results are shown for $Q^2=9~\mathrm{GeV}^2$.
• Figure 2: Summary of results for the hadronic running of the electromagnetic coupling. Left: ratios of lattice and phenomenological determinations of $\Delta\alpha_{\mathrm{had}}^{(5)}(-Q^2)$ to our central values; the orange band shows our total uncertainty, including the bottom-quark contribution. Right: comparison of determinations of $\Delta\alpha_{\mathrm{had}}^{(5)}(M_Z^2)$. Our main result (AdlerPy) and the two pQCDAdler variants are shown together with the Mainz 2022 result Ce:2022eix, dispersive evaluations based on the $R$-ratio (green), and values from global EW fits (blue).
• Figure 3: Projected total uncertainty $\sigma_{\mathrm{tot}}$ in the $(Q_0^2,f_{\mathrm{lat}})$ plane. Contours show the total error (in units of $10^{-5}$) as a function of the matching scale and of a global lattice improvement factor that rescales the current lattice uncertainty. The black cross indicates the uncertainty achieved in this work.