Using recently developed methods for the evaluation of five-loop amplitudes in perturbative QCD, corrections of order α 4 s for the non-singlet part of the cross section for electron-positron annihilation into hadrons and for the decay rates of the Z-boson and the τ -lepton into hadrons are evaluated. The new terms lead to a significant stabilization of the perturbative series, to a reduction of the theory uncertainly in the strong coupling constant αs, as extracted from these measurements, and to a small shift of the central value, moving the two central values closer together. The agreement between two values of αs measured at vastly different energies constitutes a striking test of asymptotic freedom. Combining the results from Z and τ decays we find αs(MZ) = 0.1198 ± 0.0015 as one of the most precise and presently only result for the strong coupling constant in order α 4 s . The strong coupling constant α s is one of the three fundamental gauge couplings constants of the Standard Model (SM) of particle physics. Its precise determination is one of the most important aims of particle physics. Experiments at different energies allow to test the predictions for its energy dependence based on the renormalization group equations, the comparison of the results obtained from different processes leads to critical tests of the theory and potentially to the discovery of physics beyond the Standard Model. Last but not least, the convergence of the three gauge coupling constants related by SU(3)xSU(2)xU(1) to a common value, after evolving them to high energies, allows us to draw conclusions about the possibility of embedding the SM in the framework of a Grand Unified Theory.One of the most precise and theoretically safe determination of α s is based on measurements of the cross section for electron-positron annihilation into hadrons. These have been performed in the low-energy region between 2 GeV and 10 GeV and, in particular, at and around the Z resonance at 91.2 GeV. Conceptually closely related is the measurement of the semileptonic decay rate of the τ -lepton, leading to a determination of α s at a scale below 2 GeV.From the theoretical side, in the framework of perturbative QCD, these rates and cross sections are evaluated as inclusive rates into massless quarks and gluons [1,2]. (Power suppressed mass effects are well under control for e + e − -annihilation, both at low energies and around the Z resonance, and for τ decays [3,4,5,6,7,8], and the same applies to mixed QCD and electroweak corrections [9,10]).The ratio R(s) ≡ σ(e + e − → hadrons)/σ(e + e − → µ + µ − ) is expressed through the absorptive part of the correlator of the electromagnetic current j µ :with Q 2 = −q 2 . It is also convenient to introduce the Adler function asWe define the perturbative expansionswhere a s ≡ α s /π and the normalization scale is set to µ 2 = Q 2 or to µ 2 = s for the Euclidian and Minkowskian functions respectively. The results for generic values of µ can be easily recovered with standard RG techniques. Note that the first th...

We analytically compute the five-loop term in the beta function which governs the running of α_{s}-the quark-gluon coupling constant in QCD. The new term leads to a reduction of the theory uncertainty in α_{s} taken at the Z-boson scale as extracted from the τ-lepton decays as well as to new, improved by one more order of perturbation theory, predictions for the effective coupling constants of the standard model Higgs boson to gluons and for its total decay rate to the quark-antiquark pairs.

We compute, for the first time, the order alpha(s)(4) contributions to the Bjorken sum rule for polarized electron-nucleon scattering and to the (nonsinglet) Adler function for the case of a generic color gauge group. We confirm at the same order a (generalized) Crewther relation which provides a strong test of the correctness of our previously obtained results: the QCD Adler function and the five-loop beta function in quenched QED. In particular, the appearance of an irrational contribution proportional to zeta(3) in the latter quantity is confirmed. We obtain the commensurate scale equation relating the effective strong coupling constants as inferred from the Bjorken sum rule and from the Adler function at order alpha(s)(4).

The old "glue-and-cut" symmetry of massless propagators, first established in Ref. [1], leads -after reduction to master integrals is performed -to a host of non-trivial relations between the latter. The relations constrain the master integrals so tightly that they all can be analytically expressed in terms of only few, essentially trivial, watermelon-like integrals. As a consequence we arrive at explicit analytical results for all master integrals appearing in the process of reduction of massless propagators at three and four loops. The transcendental structure of the results suggests a clean explanation of the well-known mystery of the absence of even zetas (ζ 2n ) in the Adler function and other similar functions essentially reducible to massless propagators. Once a reduction of massless propagators at five loops is available, our approach should be also applicable for explicitly performing the corresponding five-loop master integrals.1 The so-called Laporta approach [4-6] seems to be most often utilized but a few other promising methods are being now actively developed [7][8][9][10][11][12].2 At least well-established in practice. See below for an instructive particular example of a class of massless propagators and also [13,14] for an attempt to formalize the concept of the masters integrals and to prove the universality property in general. A related discussion could be found in [15][16][17][18].

We compute the form factors of the photon-quark-anti-quark vertex and the effective vertex of a Higgs-boson and two gluons to three-loop order within massless perturbative quantum chromodynamics. These results provide building blocks for many third-order cross sections. Furthermore, this is the first calculation of complete three-loop vertex corrections.