QCD tests in tau decays with optimized perturbation expansion

The next-to-next-to-leading order perturbative QCD corrections to R_tau and the higher moments of the invariant mass distribution in the hadronic tau decays are considered. The renormalization scheme dependence of these corrections is discussed. The optimized predictions are obtained, using the principle of minimal sensitivity as a guide to select the preferred renormalization scheme. A simplified fit is performed, using R_tau and R^12_tau, to see how the use of the optimized expansion may affect the determination of the alpha_s and the dimension six condensates from the experimental data.Comment: 6 pages in LateX, 4 PostScript figures embedded in text, uses espcrc2.sty (included), to appear in the Proceedings of the Fourth International Workshop on Tau Lepton Physics, 16-19 September 1996, Estes Park, Colorado, U.S.

Resummation of Nonalternating Divergent Perturbative Expansions

A method for the resummation of nonalternating divergent perturbation series is described. The procedure constitutes a generalization of the Borel-Pad\'{e} method. Of crucial importance is a special integration contour in the complex plane. Nonperturbative imaginary contributions can be inferred from the purely real perturbative coefficients. A connection is drawn from the quantum field theoretic problem of resummation to divergent perturbative expansions in other areas of physics.Comment: 5 pages, LaTeX, 2 tables, 1 figure; discussion of the Carleman criterion added; version to appear in Phys. Rev.

Vector and Axial-Vector Spectral Functions and QCD

We present new results of the tau hadronic spectral function analysis using data accumulated by the ALEPH detector at LEP during the years 1991-94. In addition to the vector spectral functions, the axial-vector spectral functions and, separately, the tau --> 3pi nu as well as the tau --> pi 2pi0 nu spectral functions are determined from their respective unfolded, i.e., physical invariant mass spectra. The spectral functions are applied to QCD chiral sum rules in order to extract information about saturation at the tau mass scale. Using the the semi-leptonic tau decay rate for vector and axial-vector currents in addition to spectral moments, we obtain precise measurements of the strong coupling constant alpha_s(M_tau) and the contributing non-perturbative power terms. The evolution to the Z mass yields alpha_s(M_Z) = 0.1219 +/- 0.0019.Comment: 15 pages, 12 figures, LaTex, Talk given at the Fourth International Workshop on Tau Lepton Physics (TAU96), Colorado, September 199

The Gross--Llewellyn Smith Sum Rule in the Analytic Approach to Perturbative QCD

We apply analytic perturbation theory to the Gross--Llewellyn Smith sum rule. We study the $Q^2$ evolution and the renormalization scheme dependence of the analytic three-loop QCD correction to this sum rule, and demonstrate that the results are practically renormalization scheme independent and lead to rather different $Q^2$ evolution than the standard perturbative correction possesses.Comment: 17 pages, 9 eps figures, REVTe

Renormalization scheme dependence and the problem of determination of alpha_s and the condensates from the semileptonic tau decays

The QCD corrections to the moments of the invariant mass distribution in the semileptonic $\tau$ decays are considered. The effect of the renormalization scheme dependence on the fitted values of alpha_s(m^2_tau) and the condensates is discussed, using a simplified approach where the nonperturbative contributions are approximated by the dimension six condensates. The fits in the vector and the axial-vector channel are investigated in the next-to-leading and the next-to-next-to-leading order. The next-to-next-to-leading order results are found to be relatively stable with respect to change of the renormalization scheme. A change from the MS-bar scheme to the minimal sensitivity scheme results in the reduction of the extracted value of alpha_s(m^2_tau) by 0.01.Comment: Some typographical errors have been corrected, including two small misprints in table 1 and table 2 and one in Eq.15. 20 pages Latex, 5 figure

Strong Coupling Constant with Flavour Thresholds at Four Loops in the MS-bar Scheme

We present in analytic form the matching conditions for the strong coupling constant alpha_s^(n_f)(mu) at the flavour thresholds to three loops in the modified minimal-subtraction scheme. Taking into account the recently calculated coefficient beta_3 of the Callan-Symanzik beta function of quantum chromodynamics, we thus derive a four-loop formula for alpha_s^(n_f)(mu) together with appropriate relationships between the asymptotic scale parameters Lambda^(n_f) for different numbers of flavours n_f.Comment: 10 pages (Latex), 3 figures (Postscript

Resummation of the Divergent Perturbation Series for a Hydrogen Atom in an Electric Field

We consider the resummation of the perturbation series describing the energy displacement of a hydrogenic bound state in an electric field (known as the Stark effect or the LoSurdo-Stark effect), which constitutes a divergent formal power series in the electric field strength. The perturbation series exhibits a rich singularity structure in the Borel plane. Resummation methods are presented which appear to lead to consistent results even in problematic cases where isolated singularities or branch cuts are present on the positive and negative real axis in the Borel plane. Two resummation prescriptions are compared: (i) a variant of the Borel-Pade resummation method, with an additional improvement due to utilization of the leading renormalon poles (for a comprehensive discussion of renormalons see [M. Beneke, Phys. Rep. vol. 317, p. 1 (1999)]), and (ii) a contour-improved combination of the Borel method with an analytic continuation by conformal mapping, and Pade approximations in the conformal variable. The singularity structure in the case of the LoSurdo-Stark effect in the complex Borel plane is shown to be similar to (divergent) perturbative expansions in quantum chromodynamics.Comment: 14 pages, RevTeX, 3 tables, 1 figure; numerical accuracy of results enhanced; one section and one appendix added and some minor changes and additions; to appear in phys. rev.

Bilocal expansion of the Borel amplitude and the hadronic tau decay width

The singular part of Borel transform of a QCD amplitude near the infrared renormalon can be expanded in terms of higher order Wilson coefficients of the operators associated with the renormalon. In this paper we observe that this expansion gives nontrivial constraints on the Borel amplitude that can be used to improve the accuracy of the ordinary perturbative expansion of the Borel amplitude. In particular, we consider the Borel transform of the Adler function and its expansion around the first infrared renormalon due to the gluon condensate. Using the next-to-leading order Wilson coefficient of the gluon condensate operator, we obtain an exact constraint on the Borel amplitude at the first IR renormalon. We then extrapolate, using judiciously chosen conformal transformations and Pade approximants, the ordinary perturbative expansion of the Borel amplitude in such a way that this constraint is satisfied. This procedure allows us to predict the $O(\alpha_s^4)$ coefficient of the Adler function, which gives a result consistent with the estimate by Kataev and Starshenko using a completely different method. We then apply this improved Borel amplitude to the tau decay width, and obtain the strong coupling constant $\alpha_s(M_Z) =0.1193 \pm 0.0007_{exp.} \pm 0.0010_{EW+CKM} \pm 0.0009_{meth.} \pm 0.0003_{evol.}$. We then compare this result with those of other resummation methods.Comment: 30 pages, 4 eps-figures, revtex; version as appears in PRD; no major changes; more careful rounding of some number

Renormalization-Scheme Dependence of Pade Summation in QCD

We study the renormalization-scheme (RS) dependence of Pade Approximants (PA's), and compare them with the Principle of Minimal Sensitivity (PMS) and the Effective Charge (ECH) approaches. Although the formulae provided by the PA, PMS and ECH predictions for higher-order terms in a QCD perturbation expansion differ in general, their predictions can be very close numerically for a wide range of renormalization schemes. Using the Bjorken sum rule as a test case, we find that Pade Summation (PS) reduces drastically the RS dependence of the Bjorken effective charge. We use these results to estimate the theoretical error due to the choice of RS in the extraction of $\alpha_s$ from the Bjorken sum rule, and use the available data at $Q^2=3 GeV^2$ to estimate $\alpha_s(M_Z) = 0.117^{+0.004}_{-0.007} \pm 0.002$, where the first error is experimental, and the second is theoretical.Comment: 12 pages (latex), including 6 embedded figures; uses epsfig.st

Results and Techniques of Multi-Loop Calculations

In this review some recent multi-loop results obtained in the framework of perturbative Quantum Chromodynamics (QCD) and Quantum Electrodynamics (QED) are discussed. After reviewing the most advanced techniques used for the computation of renormalization group functions, we consider the decoupling of heavy quarks. In particular, an effective method for the evaluation of the decoupling constants is presented and explicit results are given. Furthermore the connection to observables involving a scalar Higgs boson is worked out in detail. We review the radiative corrections of a Higgs boson into gluons and quarks and present explicit results up to order $\alpha_s^4$ and $\alpha_s^3$, respectively. In this review special emphasis is put on the applications of asymptotic expansions. A method is described which combines expansion terms of different kinematical regions with the help of conformal mapping and Pad\'e approximation. This method allows us to proceed beyond the present scope of exact multi-loop calculations. As far as physical processes are concerned, we review the computation of three-loop current correlators in QCD taking into account the full mass-dependence. In a further application four-loop diagrams are considered which contribute to the order $\alpha^2$ QED corrections to the $\mu$ decay. Finally the calculation of the three-loop relation between the $\bar{\rm MS}$ and on-shell quark mass definitions is presented. To complete the presentation, some technical details are presented in the Appendix, where also explicit analytical results are listed.Comment: 125 pages, to appear in Physics Reports, eqs. (4.11), (4.77), (4.79) and (C.3) corrected, references and note adde