Diogo Boito

Phys. Rev. D 114, 059902 (2026)

We compare the isospin-one, vector-current hadronic vacuum polarization (HVP) obtained from isospin-symmetric lattice QCD with that obtained from a dispersive representation employing inclusive hadronic τ decay data corrected for isospin breaking. We consider the subtracted HVP evaluated at squared Euclidean momenta ranging from 0.5 GeV 2 to 12 GeV 2 , together with the light-quark-connected HVP contribution to the muon anomalous magnetic moment and the short-, intermediate- and long-distance RBC/UKQCD window components thereof. Dispersive contributions from the region of hadronic invariant masses above the τ mass are evaluated using perturbative QCD. We also consider dispersive determinations using τ data only for contributions from two-pion, or two-pion and four-pion, modes, and evaluating the remaining contributions using exclusive-mode e + e − → hadrons cross sections up to about 2 GeV, lessening the dependence on perturbation theory. We find generally good agreement between lattice and τ -based results. However, a comparison of τ -based window-quantity contributions for the two four-pion modes to expectations for those contributions based on the Pais relations and e + e − four-pion cross sections, reveals significant differences for the 2 π − π + π 0 mode.

In this paper we carefully assess the theory prediction for R ( s ) below charm threshold, R u d s , and address tensions with the existing data, notably with the 2021 BES-III results. We analyze the uncertainty of the perturbative quantum chromodynamics description in the light of renormalons making use of the large- β 0 limit and the renormalon-free gluon-condensate scheme. We provide a reliable estimate of the duality violation contributions; we show they are sizable up to 2.5 GeV and improve the agreement between theory and data, but are negligible for higher energies. We then combine the available experimental data for R u d s and find the datasets to be mutually compatible. Finally, we compare theory and data, both locally and in their contributions to the anomalous magnetic moment of the muon. Theory is compatible with the combined data but discrepancies with the BES-III data reach more than 3 σ .

Tau-based finite-energy sum-rule (FESR) analyses often assume that scales s_0\sim m_\tau^2 s 0 ∼ m τ 2 are large enough that (i) integrated duality violations (DVs) can be neglected, and (ii) contributions from non-perturbative OPE condensates of dimension D D scale as (\Lambda_{QCD}/m_\tau )^D ( Λ Q C D / m τ ) D , allowing the OPE series to be truncated at low dimension. The latter assumption is not necessarily valid since the OPE series is not convergent, while the former is open to question given experimental results for the electromagnetic, I=1 I = 1 vector ( V V ), I=1 I = 1 axial vector ( A A ) and I=1 I = 1 V+A V + A current spectral functions, which show DV oscillations with amplitudes comparable in size to the corresponding \alpha_s α s -dependent perturbative contributions at s\sim2-3 s ∼ 2 − 3 GeV ^2 2 . Here, we discuss recently introduced new tools for assessing the numerical relevance of omitted higher- D D OPE contributions. Applying these to the “truncated OPE” strategy used in Refs.[M. Davier et al., Eur. Phys. J. C 74, 2803 (2014); A. Pich and A. Rodríguez-Sánchez, Phys. Rev. D 94, 034027 (2016)] and earlier work by the same authors, we find that this strategy fails to yield reliable results for the strong coupling from hadronic \tau τ decays.

We perform a precise extraction of the QCD coupling at the \tau τ -mass scale, \alpha_s(m_\tau) α s ( m τ ) , from a new vector isovector spectral function which combines ALEPH and OPAL distributions for the dominant channels, \tau\to\pi \pi^0\nu_\tau τ → π π 0 ν τ , \tau\to 3\pi \pi^0\nu_\tau τ → 3 π π 0 ν τ and \tau\to \pi 3\pi^0\nu_\tau τ → π 3 π 0 ν τ , with estimates of sub-leading contributions obtained from electroproduction cross-sections using CVC, as well as BaBar results for \tau \to K^-K^0\nu_\tau τ → K − K 0 ν τ . The fully inclusive spectral function thus obtained is entirely based on experimental data, without Monte Carlo input. From this new data set, we obtain \alpha_s(m_\tau)=0.3077±0.0075 α s ( m τ ) = 0.3077 ± 0.0075 , which corresponds to \alpha_s(m_Z)=0.1171±0.0010 α s ( m Z ) = 0.1171 ± 0.0010 . This analysis can be improved on the experimental side with new measurements of the dominant \pi\pi^0 π π 0 , \pi 3\pi^0 π 3 π 0 , and 3\pi \pi^0 3 π π 0 \tau τ decay modes.

Nonperturbative effects have a small but nontrivial impact on the determination of the strong coupling from hadronic τ decay data. Several approaches have been proposed to take these into account, the two most important of which are the “truncated Operator Product Expansion (OPE)” approach and “Duality Violation (DV)-model” approach. Recently, Pich and Rodríguez-Sánchez have raised a number of criticisms of the latter approach, including, most notably, claims of the existence of (i) a supposed instability with respect to variations of the model for incorporating quark-hadron duality violations, and (ii) an alleged redundancy in the fitting strategy employed in the DV-model approach. In this paper, we investigate these criticisms and show they fail to survive more detailed scrutiny of the mathematical or numerical arguments that underpin them. We also show that, while the redundancy claim does not apply to the DV-model approach, it does, in fact, apply to the truncated OPE approach. In particular, the α s value determined in the latter turns out to derive purely from perturbation theory, with no role played by the nonperturbative condensates determined in the rest of the analysis. This leads to the conclusion that a revision of the conventional understanding of what is learned from truncated OPE analyses is necessary and that only very limited self-consistency checks are possible within this framework. These observations raise new, nontrivial issues for the truncated OPE approach.

In previous work we have combined the π − π 0 , 2 π − π + π 0 , and π − 3 π 0 spectral data obtained from hadronic τ decays measured by the ALEPH and OPAL experiments, together with electroproduction data for several of the subleading hadronic modes and data for the K K ¯ mode to construct an inclusive nonstrange vector spectral function entirely based on experimental data, with no Monte-Carlo generated input. In this paper, we include, for the first time, the Belle τ → π − π 0 ν τ high-statistics decay data to construct a new inclusive nonstrange vector spectral function that combines more of the world’s available data. As no Belle data are at present available for the two 4 π modes, this requires a revised data analysis in comparison with our previous work. From the resulting new spectral function, we obtain a new determination of the strong coupling, α s , using our previously developed strategy based on finite-energy sum rules. We find, at the Z mass scale, α s ( m Z 2 ) = 0.1159 ( 14 ) . We discuss the smaller central value and larger error of our new result compared to our previous result, showing the shifts to be due mainly to significant changes in updated HFLAV results for the π − 3 π 0 decay mode.