The hadronic running of the electromagnetic coupling and the electroweak mixing angle from lattice QCD

Abstract

Abstract We compute the hadronic running of the electromagnetic and weak couplings in lattice QCD with Nf = 2 + 1 flavors of $$ \mathcal{O} $$ O (a) improved Wilson fermions. Using two different discretizations of the vector current, we compute the quark-connected and –disconnected contributions to the hadronic vacuum polarization (HVP) functions $$ \overline{\varPi} $$ Π ¯ γγ and $$ \overline{\varPi} $$ Π ¯ γZ for Euclidean squared momenta Q2 ≤ 7 GeV2. Gauge field ensembles at four values of the lattice spacing and several values of the pion mass, including its physical value, are used to extrapolate the results to the physical point. The ability to perform an exact flavor decomposition allows us to present the most precise determination to date of the SU(3)-flavor-suppressed HVP function $$ \overline{\varPi} $$ Π ¯ 08 that enters the running of sin2θW. Our results for $$ \overline{\varPi} $$ Π ¯ γγ, $$ \overline{\varPi} $$ Π ¯ γZ and $$ \overline{\varPi} $$ Π ¯ 08 are presented in terms of rational functions for continuous values of Q2 below 7 GeV2. We observe a tension of up to 3.5 standard deviation between our lattice results for $$ \Delta {\alpha}_{\mathrm{had}}^{(5)} $$ ∆ α had 5 (−Q2) and estimates based on the R-ratio for space-like momenta in the range 3–7 GeV2. The tension is, however, strongly diminished when translating our result to the Z pole, by employing the Euclidean split technique and perturbative QCD, which yields $$ \Delta {\alpha}_{\mathrm{had}}^{(5)}\left({M}_Z^2\right) $$ ∆ α had 5 M Z 2 = 0.02773(15) and agrees with results based on the R-ratio within the quoted uncertainties.