One-loop matching and next-to-next-to-leading log resummation for all partonic2→2processes in QCD

Abstract

The Wilson Coefficients for all 4-parton operators which arise in matching QCD to Soft-Collinear Effective Theory (SCET) are computed at 1-loop. Any dijet observable calculated in SCET beyond leading order will require these results. The Wilson coefficients are separated by spin and color, although most applications will involve only the spin-averaged hard functions. The anomalous dimensions for the Wilson coefficients are given to 2-loop order, and the renormalization group equations are solved explicitly. This will allow for analytical resummation of dijet observables to next-to-next-to-leading logarithmic accuracy. For each channel, there is a natural basis in which the evolution is diagonal in color space. The same basis also diagonalizes the color evolution for the soft function. Even though soft functions required for SCET calculations are observable dependent, it is shown that their renormalization group evolution is almost completely determined by a universal structure. With these results, it will be possible to calculate hadronic event shapes or other dijet observables to next-to-leading order with next-tonext-to-leading log resummation.

NNLO hard functions in massless QCD

“…As a byproduct of our calculation we also evaluated the NLO hard functions, which were previously calculated in [10]. We find agreement with the results in that work, after we account for differences in notation.…”

“…3 Equation (55) in [10] defines a real hard function in the channels with two quarks and two gluons, while we define Hermitian hard functions with complex off diagonal terms. Furthermore, one needs to be careful when applying the crossing relations listed in table 2 of [10] in cases where fermions are switched between the initial and final state, as this necessitates extra minus signs.…”

“…Furthermore, one needs to be careful when applying the crossing relations listed in table 2 of [10] in cases where fermions are switched between the initial and final state, as this necessitates extra minus signs. Finally, there is a minor typo in table 5 of [10], which is related to the four-gluon channel: the color factors listed in the next to the last column of that table apply to the helicities labeled 7 and 8, while the ones listed in the last column of the table apply to the helicities labeled by the numbers 9-16.…”

“…The hard functions are related to the interference of vector components of color-decomposed helicity amplitudes, and are matrices of varying dimension for the different partonic scattering processes -2 × 2 for four-quark processes, 3 × 3 for qg → qg processes, and 9 × 9 for the four-gluon process. The next-to-leading order (NLO) hard functions for such QCD processes were extracted in [10]; these are a necessary ingredient for the resummation of any dijet hadronic process up to next-to-next-to-leading logarithmic (NNLL) accuracy.…”

Factorization and resummation for transverse thrust

“…These operators are built from collinear quark fields χ i and collinear gluon fields A i , and involve one field for each direction. The corresponding operators and their one-loop Wilson coefficients were given in [18], and recently these results were extended to two-loop order in [19]. To keep the following discussion simple, we focus on the hard process qq → q q where the outgoing quarks have a different flavor than the incoming ones.…”