We give a factorization formula for the e + e − thrust distribution dσ/dτ with τ = 1 − T based on soft-collinear effective theory. The result is applicable for all τ , i.e. in the peak, tail, and fartail regions. The formula includes O(α 3 s ) fixed-order QCD results, resummation of singular partonic α j s ln k (τ )/τ terms with N 3 LL accuracy, hadronization effects from fitting a universal nonperturbative soft function defined in field theory, bottom quark mass effects, QED corrections, and the dominant top mass dependent terms from the axial anomaly. We do not rely on Monte Carlo generators to determine nonperturbative effects since they are not compatible with higher order perturbative analyses. Instead our treatment is based on fitting nonperturbative matrix elements in field theory, which are moments Ωi of a nonperturbative soft function. We present a global analysis of all available thrust data measured at center-of-mass energies Q = 35 to 207 GeV in the tail region, where a two parameter fit to αs(mZ) and the first moment Ω1 suffices. We use a short distance scheme to define Ω1, called the R-gap scheme, thus ensuring that the perturbative dσ/dτ does not suffer from an O(ΛQCD) renormalon ambiguity. We find αs(mZ) = 0.1135 ± (0.0002)expt ± (0.0005) hadr ± (0.0009)pert, with χ 2 /dof = 0.91, where the displayed 1-sigma errors are the total experimental error, the hadronization uncertainty, and the perturbative theory uncertainty, respectively. The hadronization uncertainty in αs is significantly decreased compared to earlier analyses by our two parameter fit, which determines Ω1 = 0.323 GeV with 16% uncertainty.

We consider cumulant moments (cumulants) of the thrust distribution using predictions of the full spectrum for thrust including O(α 3 s ) fixed order results, resummation of singular N 3 LL logarithmic contributions, and a class of leading power corrections in a renormalon-free scheme. From a global fit to the first thrust moment we extract the strong coupling and the leading power correction matrix element Ω1. We obtain αs(mZ) = 0.1140 ± (0.0004)exp ± (0.0013) hadr ± (0.0007)pert, where the 1-σ uncertainties are experimental, from hadronization (related to Ω1) and perturbative, respectively, and Ω1 = 0.377 ± (0.044)exp ± (0.039)pert GeV. The n-th thrust cumulants for n ≥ 2 are completely insensitive to Ω1, and therefore a good instrument for extracting information on higher order power corrections, Ω ′ n /Q n , from moment data. We find (Ω ′ 2 ) 1/2 = 0.74 ± (0.11)exp ± (0.09)pert GeV.

We study the collinear splitting functions needed for next-to-next-to-leading order calculations of jet production in the vacuum and in dense QCD matter. These splitting functions describe the probability of a parton to evolve into three-parton final state and are generalizations of the traditional DGLAP splitting kernels to a higher perturbative order. Of particular interest are the angular distributions of such splitting functions, which can elucidate the significance of multiple parton branching for jet observables and guide the construction of parton shower Monte Carlo generators. We find that to O(α 2 s ) both the vacuum and the in-medium collinear splitting functions are neither angular ordered nor anti-angular ordered. Specifically, in dense QCD matter they retain the characteristic broad angular distribution already found in the O(α s ) result.

Using an approach based on Soft Collinear Effective Theory (SCET) and Heavy Quark Effective Theory (HQET) we determine the b-quark fragmentation function from electron-positron annihilation data at the Z-boson peak at next-to-next-to leading order with next-to-next-to leading log resummation of DGLAP logarithms, and next-to-next-tonext-to leading log resummation of endpoint logarithms. This analysis improves, by one order, the previous extraction of the b-quark fragmentation function. We find that while the addition of the next order in the calculation does not much shift the extracted form of the fragmentation function, it does reduce theoretical errors indicating that the expansion is converging. Using an approach based on effective field theory allows us to systematically control theoretical errors. While the fits of theory to data are generally good, the fits seem to be hinting that higher order correction from HQET may be needed to explain the b-quark fragmentation function at smaller values of momentum fraction.

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