QCD corrections to inclusive ΔS=1,2\Delta S=1,2 transitions

The talk summarises a calculation of the two-point functions for $\Delta S=1$ current-current and QCD-penguin operators, as well as for the $\Delta S=2$ operator, at the next-to-leading order. The size of the gluonic corrections to current-current operators is large, providing a qualitative understanding of the observed enhancement in $\Delta I=1/2$ transitions. In the $\Delta S=2$ sector the QCD corrections are quite moderate ($\approx -20\%$). This work has been done in collaboration with Antonio Pich.Comment: 3 pages, Invited talk presented at ``QCD'94'', Montpellier, France, July 7 - 13, 1994, hep-ph/yymmnn

Absence of even-integer ζ\zeta-function values in Euclidean physical quantities in QCD

At order $\alpha_s^4$ in perturbative quantum chromodynamics, even-integer $\zeta$-function values are present in Euclidean physical correlation functions like the scalar quark correlation function or the scalar gluonium correlator. We demonstrate that these contributions cancel when the perturbative expansion is expressed in terms of the so-called $C$-scheme coupling $\hat\alpha_s$ which has recently been introduced in Ref. [1]. It is furthermore conjectured that a $\zeta_4$ term should arise in the Adler function at order $\alpha_s^5$ in the $\overline{\rm MS}$-scheme, and that this term is expected to disappear in the $C$-scheme as well.Comment: 5 pages; 2 refs added, version published in Phys. Lett.

alpha_s and the tau hadronic width: fixed-order, contour-improved and higher-order perturbation theory

The determination of $\alpha_s$ from hadronic $\tau$ decays is revisited, with a special emphasis on the question of higher-order perturbative corrections and different possibilities of resumming the perturbative series with the renormalisation group: fixed-order (FOPT) vs. contour-improved perturbation theory (CIPT). The difference between these approaches has evolved into a systematic effect that does not go away as higher orders in the perturbative expansion are added. We attempt to clarify under which circumstances one or the other approach provides a better approximation to the true result. To this end, we propose to describe the Adler function series by a model that includes the exactly known coefficients and theoretical constraints on the large-order behaviour originating from the operator product expansion and the renormalisation group. Within this framework we find that while CIPT is unable to account for the fully resummed series, FOPT smoothly approaches the Borel sum, before the expected divergent behaviour sets in at even higher orders. Employing FOPT up to the fifth order to determine $\alpha_s$ in the \MSb scheme, we obtain $\alpha_s(M_\tau)=0.320 {}^{+0.012}_{-0.007}$, corresponding to $\alpha_s(M_Z) = 0.1185 {}^{+0.0014}_{-0.0009}$. Improving this result by including yet higher orders from our model yields $\alpha_s(M_\tau)=0.316 \pm 0.006$, which after evolution leads to $\alpha_s(M_Z) = 0.1180 \pm 0.0008$. Our results are lower than previous values obtained from $\tau$ decays.Comment: 42 pages, 9 figures; appendix on Adler function in the complex plane added. Version to appear in JHE

Anomalous dimensions of four-quark operators and renormalon structure of mesonic two-point correlators

In this work, we calculate leading-order anomalous dimension matrices for dimension-6 four-quark operators which appear in the operator product expansion of flavour non-diagonal and diagonal vector and axial-vector two-point correlation functions. The infrared renormalon structure corresponding to four-quark operators is reviewed and it is investigated how the eigenvalues of the anomalous dimension matrices influence the singular behaviour of the $u=3$ infrared renormalon pole. It is found that compared to the large-$\beta_0$ approximation where at most quadratic poles are present, in full QCD at $N_f=3$ the most singular pole is more than cubic with an exponent $\kappa\approx 3.2$.Comment: 19 pages, 1 figure; version to appear in JHE