QCD Analysis of Hadronic $τ$ Decays Revisited

TL;DR

This work reassesses the perturbative QCD description of hadronic $\tau$ decays by comparing exact $\mathcal{O}(\alpha_s^3)$ results with Le Diberder–Pich resummation and with renormalon-chain resummation, and by formulating a model-independent treatment of power corrections. It demonstrates that perturbative uncertainties in spectral moments are at the few-percent level and that the Le Diberder–Pich approach does not substantially improve this, highlighting fundamental limits in extracting $\alpha_s(m_\tau^2)$ (uncertainty around $\delta\alpha_s(m_\tau^2) \approx 0.05$) and the gluon condensate from τ-decay data. It shows that infrared renormalon ambiguities can be absorbed into redefinitions of nonperturbative parameters within the OPE, and proposes two weighted spectral integrals, $D_2$ and $D_4$, to test the absence of dimension-two operators and to measure the gluon condensate. The findings imply that previous analyses underestimated theoretical uncertainties and that robust conclusions require careful handling of both perturbative asymptotics and power corrections, with specific strategies focusing on testing SVZ assumptions and extracting the gluon condensate rather than pursuing higher-dimension condensates.

Abstract

The calculation of perturbative corrections to the spectral moments observable in hadronic $τ$ decays is reconsidered. The exact order-$α_s^3$ results and the resummation procedure of Le~Diberder and Pich are compared with a partial resummation of the perturbative series based on the analysis of so-called renormalon chains. The perturbative analysis is complemented by a model-independent description of power corrections. For the contributions of dimension four and six in the OPE, it is demonstrated how infrared renormalon ambiguities in the definition of perturbation theory can be absorbed by a redefinition of nonperturbative parameters. We find that previous determinations of QCD parameters from a measurement of spectral moments in $τ$ decays have underestimated the theoretical uncertainties. Given the present understanding of the asymptotic behaviour of perturbation theory, the running coupling constant can be measured at best with a theoretical uncertainty $δα_s(m_τ^2)\simeq 0.05$, and the gluon condensate with an uncertainty of order its magnitude. Two weighted integrals of the hadronic spectral function are constructed, which can be used to test the absence of dimension-two operators and to measure directly the gluon condensate.

Figures (5)

• Figure 1: Renormalon-chain contribution to the correlator $D(-q^2)$. The shaded bubble represents a self-energy insertion on the gluon propagator. We note that bubble summation is not a gauge-invariant procedure in a non-abelian theory; the figure is meant as an illustration only.
• Figure 2: Distribution function $w_D(\tau)$ versus $\ln\tau$.
• Figure 3: Functions $W_k(\tau)$ versus $\ln\tau$, for $k=0$ (solid line), $k=1$ (dashed line), $k=2$ (dash-dotted line) and $k=3$ (dotted line). The asymptotic result for $k\to\infty$ is shown as the thin solid line.
• Figure 4: Examples of leading contributions of four-quark condensates in the large-$n_f$ limit. The first diagram is of order $n_f\alpha_s$, the second of order $(n_f\alpha_s)^2$.
• Figure 5: Different perturbative approximations for the quantity $\delta_{\rm pert}$: resummation in the large-$\beta_0$ limit (solid line), exact order-$\alpha_s^3$ result in the $\overline{\rm MS}$ scheme (dashed line), resummation of Le Diberder and Pich (dash-dotted line).