Modern Physics Letters A

A general, and very basic introduction to QCD sum rules is presented, with emphasis on recent issues to be described at length in other papers in this issue. Collectively, these papers constitute the proceedings of the International Workshop on Determination of the Fundamental Parameters of QCD, Singapore, March 2013.

Starting from the divergent character of the perturbative expansions in QCD and using the technique of series acceleration by the conformal mappings of the Borel plane, I define a novel, non-power perturbative expansion for the Adler function, which simultaneously implements renormalisation-group summation and has a tamed large-order behaviour. The new expansion functions, which replace the standard powers of the coupling, are singular at the origin of the coupling plane and have divergent perturbative expansions, resembling the expanded function itself. Confronting the new perturbative expansions with the standard ones on specific models investigated recently in the literature, I show that they approximate in an impressive way the exact Adler function and the spectral function moments. Applied to the τ hadronic width, the contour-improved and the renormalisation-group summed non-power expansions in the scheme lead to the prediction , which translates to .

We determine the strong coupling constant α_s from the τ hadronic width using a renormalization group summed (RGS) expansion of the QCD Adler function. The main theoretical uncertainty in the extraction of α_s is due to the manner in which renormalization group invariance is implemented, and the as yet uncalculated higher order terms in the QCD perturbative series. We show that new expansion exhibits good renormalization group improvement and the behavior of the series is similar to that of the standard CIPT expansion. The value of the strong coupling in scheme obtained with the RGS expansion is . The convergence properties of the new expansion can be improved by Borel transformation and analytic continuation in the Borel plane. This is discussed elsewhere in these issues.

The frozen QCD coupling is a parameter often used as an effective fixed coupling. It is supposed to mimic both the running coupling effects and the lack of knowledge of α_s in the infrared region. Usually the value of the frozen coupling is fixed from the analysis of the experimental data. A novel way to define such coupling(s) independently of the experiments is presented. We argue that there are different frozen couplings which are used in the double-logarithmic (DL) and single-logarithmic (SL) approximations. They also differ for space- and time-like arguments. Our estimates are in a good agreement with the results available in the literature.

A determination of the strong coupling α_s at rather low energies is possible through the analysis of hadronic decays of the τ lepton. In turn, the low energy necessitates sufficient control over perturbative QCD corrections, the nonperturbative condensate contributions in the framework of the operator product expansion (OPE), as well as corrections going beyond the OPE, the duality violations (DVs). Perturbative QCD uncertainties arise from open questions regarding the renormalization group resummation of the series. The fit quantities are moment integrals of the τ spectral function data in a certain energy window and care should be taken to have good perturbative behavior of those moments as well as control over higher-dimensional operator corrections. Furthermore, all parameters occurring in the theoretical description should be extracted from fits to the data in a self-consistent manner.

The BaBar and Belle experiments together recorded a total sample of some 3 ×10⁹ tau lepton decays between the years 2000 and 2010. This enabled many new and improved measurements of tau hadronic decays to be made, and resulted in lower limits for branching fractions to rare and forbidden hadronic modes, such as those mediated by second-class weak currents which are sensitive to the u–d quark mass difference. Better measurements of strange decays have contributed to more precise data on the CKM matrix element, |V_us|. However, only limited progress has been made in measurements that feed into knowledge of the value of the strong coupling constant, α_s, and the strange quark mass. This is largely due to the dominance of systematic uncertainties on branching fractions and shapes of mass spectra (spectral functions) for some of the important hadronic tau decay modes.

We review the status of R measurements in the energy range below 12 GeV, where recently a major progress has been achieved recently. These measurements are phenomenologically motivated by their impact on precision QCD and on electroweak parameters. We also report on improved results which can be expected in near future from presently running or upcoming facilities.