Precision lattice calculation of the hadronic contribution to the running of the electroweak gauge couplings

Abstract

We present an update of our lattice QCD determination of the hadronic contribution to the running of the electromagnetic coupling, $\Delta\alpha_{\mathrm{had}}^{(5)}(-Q^2)$, and of the electroweak mixing angle in the space-like momentum region up to $Q^2=12\ \mathrm{GeV}^2$. The calculation is based on CLS ensembles with $N_f=2+1$ flavours of $O(a)$-improved Wilson fermions, covering five lattice spacings between $0.039$ and $0.085$ fm and a range of pion masses, including the physical point. A refined analysis employing a telescopic window strategy allows for a clean separation of systematic effects across Euclidean distance scales. Statistical precision is further enhanced through low-mode averaging, combined with a spectral reconstruction of the vector-vector correlator at long distances on the most chiral ensembles. We confirm significant tensions of up to $7\sigma$ at space-like virtualities around $Q^2= 1\ \mathrm{GeV}^2$ between our lattice results for $\Delta\alpha_{\mathrm{had}}^{(5)}(-Q^2)$ and the corresponding data-driven estimates based on $e^+e^-$ cross section data. Combining our lattice data with perturbative QCD via the Euclidean split technique, we obtain at the $Z$-pole $\Delta\alpha_{\mathrm{had}}^{(5)}(M_Z^2) = 0.027813(33)_{\mathrm{lat}}(35)_{\mathrm{pQCD}}$, which is more than two times more precise than recent data-driven estimates. Our result deviates slightly, by $1-2\sigma$, from the value produced by global electroweak fits. For the electroweak mixing angle, we present the hadronic contribution to its running and provide a precise determination of the octet-singlet mixing component $\bar\Pi^{(0,8)}$, in good agreement with phenomenological models but with significantly higher precision.