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

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.

Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization

“…e + e − → hadrons cross-section data. While the detailed comparison to lattice QCD as well as related observables defines an important path forward [60][61][62][63][64][65][66][67], so does renewed scrutiny of the data-driven approach.…”

Short & intermediate distance HVP contributions to muon g-2: SM (lattice) prediction versus e+e- annihilation data

“…A good agreement (at 1.5 σ level) is instead seen between lattice and data-driven determinations of a HVP,SD µ , which probes the R had (E)-ratio at higher E, where the photon HVP (i.e. ∆α em ) is indeed known (see [11] and refs. therein) to be consistent with electroweak precision tests of the SM.…”

The hadronic running of the electroweak couplings from lattice QCD

“…The first two data points (red symbols) are the lattice results of ref. [7]. Green circles denote results based the data-driven method in, from top to bottom, refs.…”

“…A precise determination of the HVP function and in turn of Δ𝛼 had (𝑞 2 ) can be obtained from experimental hadronic cross-section data encoded in the well-known ratio 𝑅(𝑠) [1][2][3]. Alternatively, the HVP function Π(−𝑄 2 ) at space-like 𝑄 2 > 0 momentum transfers can be computed on the lattice [4][5][6][7]. In combination with the Euclidean split technique [8], this can be used to provide a value of Δ𝛼 had (𝑀 2 𝑍 ) independent of 𝑅(𝑠) data, which is the main result presented here [7].…”

“…Alternatively, the HVP function Π(−𝑄 2 ) at space-like 𝑄 2 > 0 momentum transfers can be computed on the lattice [4][5][6][7]. In combination with the Euclidean split technique [8], this can be used to provide a value of Δ𝛼 had (𝑀 2 𝑍 ) independent of 𝑅(𝑠) data, which is the main result presented here [7]. The interests in computing Π(−𝑄 2 ) on the lattice is further motivated by the connection with the anomalous magnetic moment (𝑔 − 2) of the muon, and the long-standing tension between the muon 𝑔 − 2 theory prediction and experimental results [9].…”