Next-to-Next-to-Leading Order Corrections to Three-Jet Observables in Electron-Positron Annihilation

Abstract

I report on a numerical program, which can be used to calculate any infrared safe three-jet observable in electron-positron annihilation to next-to-next-to-leading order in the strong coupling constant αs. The results are compared to a recent calculation by another group. Numerical differences in three colour factors are discussed and explained.PACS numbers: 12.38. Bx, 13.66.Bc, 13.66.Jn, INTRODUCTIONJet observables and event shapes in electron-positron annihilation can be used to extract the value of the strong coupling constant α s [1,2,3]. This applies in particular to three-jet observables, where the leading-order parton process is proportional to α s . In order to extract the numerical value from the LEP data, precise theoretical calculations are necessary, calling for a next-tonext-to-leading order (NNLO) calculation. Due to the large variety of interesting jet observables it is desirable not to perform this calculation for a specific observable, but to set up a computer program, which yields predictions for any infra-red safe observable relevant to the process e + e − → 3 jets. Such a task requires the calculation of the relevant amplitudes up to two loops, a method for the cancellation of infrared divergences and stable and efficient Monte Carlo techniques. For the process e + e − → 2 jets this was done in [4,5,6,7]. In this letter I report on a NNLO calculation for three-jet observables in electron-positron annihilation. Recently another group published results for the NNLO corrections for three-jet observables [8,9,10,11]. In the calculation presented here the methods used are in many parts similar to the ones used in [8,9,10,11], although I will show that in certain points there are important differences. The authors of [8,9,10,11] made major contributions to the development of these methods [5,12,13,14,15].The numerical results of the two calculations are compared. The comparison is facilitated by splitting the NNLO correction term into individually gauge-invariant contributions, such that each contribution is proportional to a specific colour factor. For the NNLO corrections to e + e − → 3 jets there are six different colour factors. In three colour factors the two calculations agree (N −2 c , N f /N c , N 2 f ). They disagree in the remaining three colour factors (N 2 c , N 0 c , N f N c , ). The numerical differences in these colour factors can be traced back to an incomplete cancellation of soft-gluon singularities in the calculation of refs. [8,9,10,11]. These singularities require additional subtraction terms, which are subtracted from the five-parton configuration and added to the four-parton configuration. These subtraction terms have a structure not present in [10] and are related to soft gluons. These terms occur generically in any NNLO calculation with three or more hard coloured partons. GENERAL SET-UPThe perturbative expansion of any infrared-safe observable for the process e + e − → 3 jets can be written up to NNLO asA O gives the LO result, B O the NLO correction and C O the NNLO corr...

Resummation of heavy jet mass and comparison to LEP data

“…This can be compared to the result for thrust, using exactly the same technique, and the same energy aleph data (Table 2 of [11]). Updating this result to include the more recent NNLO distributions [3,4], using the same c S 2 values, Eq. (33), with associated "soft" uncertainty, and restricting to only the aleph data, we find α s (m Z ) = 0.1175 ± 0.0009 (stat) ± 0.0011 (syst) ± 0.0014 (had) ± 0.0016 (pert) ± 0.0006 (soft) = 0.1175 ± 0.0026 (Thrust) .…”

“…Recently, a number of theoretical advances have led to renewed interest in event shapes and the α s measurements. First, the NNLO fixed order Feynman diagrams were calculated [1,2,3,4]. This allowed the prediction of all event shapes to order α 3 s .…”

Third-order QCD corrections to the charged-current structure function

“…Indeed, the second-order partonic cross sections (coefficient functions) for inclusive DIS were completed as early as 1991/2 [4][5][6], while corresponding quantities for jet shapes in e + e − collisions have been presented only very recently [7][8][9]. At large x (with hindsight: x > 10 −2 ) the former quantities are the dominant part of the next-to-next-to-leading order (NNLO) contributions in renormalization-group improved perturbation theory.…”

Double real radiation corrections to gluon scattering at NNLO

“…We will follow the NNLO antenna subtraction method which was derived in [2] for processes involving only (massless) final state partons. This formalism has been applied in the computation of NNLO corrections to three-jet production in electron-positron annihilation [13,14,15,16] and related event shapes [17,18,19,20,21]. It has also been extended at NNLO to include one hadron in the initial state relevant for electronproton scattering [22,23] while in refs.…”