Event shapes in Soft Colinear Effective Theory
Final Report Abstract
In this project we have provided a factorization formula for the thrust distribution in e+e− annihilation which incorporates the previously known O(αs2) and O(αs3) perturbative QCD corrections and summation of large s s logarithms at N3 LL order for the singular terms in the dijet limit where the thrust variable τ = 1 − T is small. The factorization formula used here incorporates a systematic description of nonperturbative effects with a soft function defined in field theory. The soft function describes the dynamics of soft particle radiation at large angles. We have also accounted for bottom mass and QED photon effects for fixed-order contributions as well as for the summation of QED logarithms. With specifically designed τ-dependent profile functions for the renormalization scales the factorization formula can be applied in the peak, tail and far-tail regions of the thrust distribution. It has all nonperturbative effects accounted for up to terms of O(αs ΛQCD /Q), which is parametrically smaller than the remaining perturbative uncertainty (< 2% for Q = mZ) of the thrust distribution predictions in the tail region where we carried out the fits to the experimental data. In the tail region, 2ΛQCD/Q ≪ τ 1/3, the dominant effects of the nonperturbative soft function are encoded in its first moment Ω1 , which is a power correction to the cross section. Fitting to tail data at multiple Qs as we have in this project, the strong coupling αs (mZ ) and the moment Ω1 can be simultaneously determined. An essential ingredient to reduce the theoretical uncertainties to the level of < 2% in the thrust distribution is our use of a short-distance scheme for Ω1, called the R-gap scheme, that induces subtractions related to an O(ΛQCD) renormalon contained in the MS perturbative thrust cross section from large angle soft gluon radiation. The R-gap scheme introduces an additional scale that leads to large logarithms in the subtractions, and we carry out a summation of these additional logarithms with renormalization group equations in the variable R. The R-gap scheme reduces the perturbative uncertainties in our best highest order theory code by roughly a factor of two compared to the pure MS definition, Ω1 , where renormalon effects are not treated. The code we use in this analysis represents the most complete theoretical treatment of thrust existing at this time. As our final result we obtain Eq. (3) αs (mZ ) = 0.1135 ± 0.0011 , Ω1 (R∆ , µ∆ ) = 0.323 ± 0.051 GeV, where αs is defined in the MS scheme, and Ω1 in the R-gap scheme at the reference scales R∆ = µ∆ = 2 GeV. Neglecting the nonperturbative effects incorporated in the soft function, and in particular Ω1 , from the fits gives αs (mZ ) = 0.1241 which exceeds the result by 9%. These results and our result for αs (mZ ) in Eq. (3) are substantially smaller than the results of event shape analyses employing input from Monte Carlo generators to determine nonperturbative effects. We emphasize that using parton-to-hadron level transfer matrices obtained from Monte Carlo generators to incorporate nonperturbative effects is not compatible with a high-order theoretical analysis such as ours, and thus analyses relying on such Monte Carlo input contain systematic errors in the determination of αs from thrust data. We believe Monte Carlo should not be used for hadronization uncertainties in higher order analyses. It is also warranted to apply the high-precision approach using soft-collinear effective theory to other event shape distributions in order to validate the result in Eq. (3). Event shapes that can be clearly treated with similar techniques are: heavy jet mass, the C-parameter, and the angularities. For many of these event shapes it has been proven field theoretically that the same parameter Ω1 describes the leading power corrections in the tail region, although there might be caveats related to the experimental treatment of hadron masses. Thus, one has the potential to extend the analysis done here to include additional data without additional parameters. Finally one can also extend our work to determine the entire nonperturbative soft function, (and not only its first moment Ω1 as we have done in this project), by fitting to data in the peak of the thrust distributions at many Q values. Knowledge of the soft function is of high interest for top quark mass determinations at the future International Linear Collider.
Publications
- “Thrust at N3 LL with Power Corrections and a Precision Global Fit for αs (mZ ),” Phys. Rev. D83 (2011) 074021
R. Abbate, M. Fickinger, A. H. Hoang, V. Mateu, I. W. Stewart
