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

TL;DR

The paper delivers a high-precision, first-principles lattice QCD determination of the hadronic contributions to the running of the electromagnetic coupling and electroweak mixing angle in the space-like region up to Q^2 = 12 GeV^2. Central to the approach is a telescopic window decomposition of the hadronic vacuum polarization, combined with extensive noise-reduction techniques, finite-volume corrections, and a robust chiral–continuum extrapolation across five lattice spacings using CLS N_f = 2+1 ensembles. By integrating lattice results with perturbative QCD via the Euclidean split and leveraging Padé representations, the authors obtain Δα_had^(5)(M_Z^2) = 0.027813(33)_(lat)(35)_(pQCD) and a precise account of the hadronic running of sin^2 θ_W, including the octet–singlet mixing component Π^(0,8). The work reveals persistent tensions with data-driven R-ratio evaluations at low space-like momenta (up to ~7σ near Q^2 ≈ 1 GeV^2), while delivering a total hadronic running with 0.17% relative uncertainty at the Z-pole, thereby strengthening lattice QCD’s role in precision electroweak tests and guiding future improvements in both theory and phenomenology.

Abstract

We present an update of our lattice QCD determination of the hadronic 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 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 $7σ$ at space-like virtualities around $Q^2= 1\ \mathrm{GeV}^2$ between our lattice results for $Δα_{\mathrm{had}}^{(5)}(-Q^2)$ 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 $Δα_{\mathrm{had}}^{(5)}(M_Z^2) = 0.027813(33)_{\mathrm{lat}}(35)_{\mathrm{pQCD}}$, which is more than two times more precise than recent data-driven estimates. Our result deviates slightly, by $1-2σ$, 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 $\barΠ^{(0,8)}$, in good agreement with phenomenological models but with significantly higher precision.

Figures (21)

• Figure 1: Illustration of the integrands for the various contribution entering the computation of the electroweak couplings. Upper left: HV region integrands using the non-subtracted kernel in Eq. \ref{['eq:non_sub_kernel_QSD_QID']}. Upper right: MV region integrands using the non-subtracted kernel in Eq. \ref{['eq:non_sub_kernel_QSD_QID']}. Bottom: LV region integrands from the non-subtracted kernel in Eq. \ref{['eq:non_sub_kernel_QLD']}. Results are shown for physical pion mass ensemble E250 with $a \approx 0.064\ \mathrm{fm}$ at the virtuality $Q^2=9\ \mathrm{GeV}^2$.
• Figure 2: Illustration of noise-reduction techniques for the $\bar{\Pi}^{(3,3)}$ channel in the LV momentum region. Left: integrand contribution on the physical-mass ensemble E250. Black squares correspond to the correlation function obtained using LMA, while the coloured points show the reconstructed integrand from $N_{\pi\pi}$ states. The vertical dotted line marks the transition between LMA and spectroscopy data sets. Right: application of the bounding method on reconstructed correlator. The dashed vertical lines indicates where the correlator is replaced by spectroscopy data. Green and orange points represent the lower and upper bound, respectively, with the yellow band illustrating our final estimate. The dashed horizontal lines indicate the results we would have obtained based solely on the LMA data.
• Figure 3: Illustration of fits to the $\widehat{\Pi}_{\mathrm{sub}}^{(3,3)}(Q^2)$ isovector contribution in the HV region. Specifically, we show results for the SD Euclidean window, where tree-level improvement is applied. Left: continuum limit behaviour for four sets of data based on different improvement schemes and discretizations of the vector current. Each line corresponds to a single fit, with the opacity associated to the weights as given by our model average prescription. Right: chiral approach to the physical pion mass for one of the fits with the highest weight. Data points are projected to $\phi_4^{\mathrm{phys}}$. Coloured lines denote the chiral trajectories at finite lattice spacing, while the grey band shows the dependence on $\phi_2$ in the continuum. Results are shown for $Q^2= 9\ \mathrm{GeV}^2$ and $Q_m^2= 9\ \mathrm{GeV}^2$.
• Figure 4: Same as figure \ref{['fig:isov_high_q_cl']} for the isovector HV contribution $\bar{\Pi}^{(3,3)}_{\mathrm{sub}}(Q^2/16)$ as determined from the subtracted kernel in Eq. \ref{['eq:sub_kernel_QLD']} . Results are shown for $Q^2= 9\ \mathrm{GeV}^2$ and $Q_m^2= 9\ \mathrm{GeV}^2$.
• Figure 5: Same as figure \ref{['fig:isov_high_q_cl']} for the term $\hat{\Delta}_{ls}(Q^2)$ used to extract the HV isoscalar contribution. Results are shown for $Q^2= 9\ \mathrm{GeV}^2$