Four loop anomalous dimension of the second moment of the non-singlet twist-2 operator in QCD

We present the result of a calculation for the first even moment of the non-singlet four-loop anomalous dimension of Wilson twist-2 operators in QCD with full color and flavor structures.Comment: 8 pages, comparison with prediction from Pade approximation (hep-ph/9905310) adde

The 2009 World Average of αs\alpha_s

Measurements of $\alpha_s$, the coupling strength of the Strong Interaction between quarks and gluons, are summarised and an updated value of the world average of $\alpha_s (M_Z)$ is derived. Building up on previous reviews, special emphasis is laid on the most recent determinations of $\alpha_s$. These are obtained from $\tau$-decays, from global fits of electroweak precision data and from measurements of the proton structure function \F_2, which are based on perturbative QCD calculations up to $O(\alpha_s^4)$; from hadronic event shapes and jet production in \epem annihilation, based on $O(\alpha_s^3)$ QCD; from jet production in deep inelastic scattering and from $\Upsilon$ decays, based on $O(\alpha_s^2)$ QCD; and from heavy quarkonia based on unquenched QCD lattice calculations. Applying pragmatic methods to deal with possibly underestimated errors and/or unknown correlations, the world average value of $\alpha_s (M_Z)$ results in $\alpha_s (M_Z) = 0.1184 \pm 0.0007$. The measured values of $\alpha_s (Q)$, covering energy scales from Q \equiv \mtau = 1.78 GeV to 209 GeV, exactly follow the energy dependence predicted by QCD and therefore significantly test the concept af Asymptotic Freedom.Comment: 14 pages, 7 figure

Fractured water injection wells: Pressure transient analysis

AbstractIn this paper we study the pressure drop in a hydraulic fracture after shut-in of a water injection well. The pressure transient behavior depends on fracture closure, lateral stress, rock elasticity and fracture fluid leak-off. Under the assumption that horizontal cross-sections of a vertical fracture do not depend on the vertical variable, we formulate a mathematical model which allows for determination of both pore pressure and elastic rock displacements jointly with the fracture aperture and fracture fluid pressure. An analytical consideration is performed for the case of an ideal very long fracture with the same aperture along its full length. In the general case, fracture closure is analyzed numerically