Measurement of the Strong Coupling Constant and the Vector and Axial-Vector Spectral Functions in Hadronic Tau Decays

TL;DR

The paper measures vector and axial-vector spectral functions in hadronic tau decays with the OPAL detector, extracting α_s via moments of R_τ within the Operator Product Expansion. By applying perturbative schemes (CIPT, FOPT, RCPT) and allowing dimension-6/8 operators and the gluon condensate to vary, it achieves a precise α_s(m_Z^2) around 0.122, while highlighting scheme-dependent differences in the perturbative treatment. The study also tests the running of α_s, examines non-perturbative corrections, and saturates QCD sum rules at the tau mass, including a derivation of the pion polarizability α_E. Overall, the results reinforce the applicability of OPE-based QCD analyses to tau decays and provide a robust cross-check against other determinations of the strong coupling.

Abstract

The spectral functions of the vector current and the axial-vector current have been measured in hadronic tau decays using the OPAL detector at LEP. Within the framework of the Operator Product Expansion a simultaneous determination of the strong coupling constant alpha_s, the non-perturbative operators of dimension 6 and 8 and of the gluon condensate has been performed. Different perturbative descriptions have been compared to the data. The Contour Improved Fixed Order Perturbation Theory gives alpha_s(mtau**2) = 0.348 +- 0.009 +- 0.019 at the tau-mass scale and alpha_s(mz**2) = 0.1219 +- 0.0010 +- 0.0017 at the Z-mass scale. The values obtained for alpha_s(mz**2) using Fixed Order Perturbation Theory or Renormalon Chain Resummation are 2.3% and 4.1% smaller, respectively. The running of the strong coupling between s_0 ~1.3 GeV**2 and s_0 = mtau**2 has been tested from direct fits to the integrated differential hadronic decay rate R_tau. A test of the saturation of QCD sum rules at the tau-mass scale has been performed.

Figures (11)

• Figure 1: The $\mathrm{\gamma}$$\mathrm{\gamma}$-mass in the $\mathrm{\pi}$$\mathrm{\pi^0}$ channel for decays with two reconstructed photons with a minimal energy of $\mathit{0.5\,\mathit{GeV}}$. OPAL data is shown as data points; the total Monte Carlo prediction is given by the open histogram and the shaded histogram denotes the $\mathit{\tau}$ and non-$\mathit{\tau}$ background.
• Figure 2: The measured $\mathit{s_\mathit{meas}}$ spectra for 1-prong decays. Plots (a) and (b) are the $\mathit{\mathrm{\pi} \mathrm{\pi^0}}$ channel, (c) and (d) are the $\mathit{\mathrm{\pi} 2\mathrm{\pi^0}}$ and $\mathit{\mathrm{\pi} 3\mathrm{\pi^0}}$ modes, respectively. The points denote OPAL data (statistical errors only). The open histograms show the fitted spectra after the regularized unfolding, refolded into detector space. The background contributions from simultaneously unfolded channels (correlated background) are shown as light grey areas while the background from other sources (uncorrelated background) is represented in dark grey.
• Figure 3: The measured $\mathit{s_\mathit{meas}}$ spectra for 3-prong decays. Plot (a) is the $\mathit{3\mathrm{\pi}}$ channel, (b) and (c) are the $\mathit{3\mathrm{\pi} \mathrm{\pi^0}}$ and $\mathit{3\mathrm{\pi} 2\mathrm{\pi^0}}$ modes, respectively. The points denote OPAL data (statistical errors only). The open histograms show the fitted spectra after the regularized unfolding, refolded into detector space. The background contributions from simultaneously unfolded channels (correlated background) are shown as light grey areas while the background from other sources (uncorrelated background) is represented in dark grey.
• Figure 4: The unfolded $\mathit{s_\mathit{true}}$ spectra. Shown are the three vector channels (left) and the three axial-vector channels (right) together with the Monte Carlo prediction. There are strong correlations between the data points due to the unfolding. The plots (a),(d),(e) are the unfolded spectra of plots (a),(c),(d) in figure \ref{['fig:dataQ2-1pr']} and the plots (b),(c),(f) are the unfolded spectra of the plots (a),(b),(c) in figure \ref{['fig:dataQ2-3pr']}. The error bars include statistical and systematic uncertainties.
• Figure 5: The vector and axial-vector spectral functions. Shown are the sums of all contributing channels as data points (upper two plots). Some exclusive contributions are shown as shaded areas. The naıve parton model prediction is shown as dashed line, while the solid line depicts the perturbative, massless QCD prediction for $\mathit{\alpha_s(m_\mathrm{Z}^2) = 0.122}$. The error bars include statistical and systematic uncertainties. The pion pole is subtracted from the axial-vector spectrum. The lower plot shows the correlations of the two spectral functions in continuous gray-levels from white to black which correspond to the correlations in percent from $\mathit{-100\,\%}$ to $\mathit{+100\,\%}$. The contour lines are drawn in equidistant steps of $\mathit{20\,\%}$.