Abstract
We study N-jettiness in electroweak processes at extreme high energies, in which the mass of the weak gauge bosons can be regarded as small. The description of the scattering process such as e−e+ → μ−μ+ + X is similar to QCD. The incoming leptons emit initial-state radiation and the resultant particles, highly off-shell, participate in the hard scattering, which are expressed by the beam functions. After the hard scattering, the final- state leptons or leptonic jets are observed, described by the fragmenting jet functions or the jet functions respectively. At present, electroweak processes are prevailed by the processes induced by the strong interaction, but they will be relevant at future e−e+ colliders at high energy. The main difference between QCD and electroweak processes is that the initial- and final-state particles should appear in the form of hadrons, that is, color singlets in QCD, while there can be weak nonsinglets as well in electroweak interactions. We analyze the factorization theorems for the N-jettiness in e−e+ → μ−μ+ + X, and compute the factorized parts to next-to-leading logarithmic accuracy. To simplify the comparison with QCD, we only consider the SU(2)W gauge interaction, and the extension to the Standard Model is straightforward. Put it in a different way, it corresponds to an imaginary world in which colored particles can be observed in QCD, and the richer structure of effective theories is probed. Various nonzero nonsinglet matrix elements are interwoven to produce the factorized results, in contrast to QCD in which there are only contributions from the singlets. Another distinct feature is that the rapidity divergence is prevalent in the contributions from weak nonsinglets due to the different group theory factors between the real and virtual corrections. We verify that the rapidity divergence cancels in all the contributions with a different number of nonsinglet channels. We also consider the renormalization group evolution of each factorized part to resum large logarithms, which are distinct from QCD.
Highlights
The understanding of high-energy scattering has reached a state of the art with the advent of the effective theories such as soft-collinear effective theory (SCET) [1,2,3,4]
We study N -jettiness in electroweak processes at extreme high energies, in which the mass of the weak gauge bosons can be regarded as small
Depending on the observables of interest, the phase space is divided by the definite properties of the specific modes and the factorization theorem is constructed according to how the modes or the phase spaces are organized
Summary
The understanding of high-energy scattering has reached a state of the art with the advent of the effective theories such as soft-collinear effective theory (SCET) [1,2,3,4]. Regardless of SCETI or SCETII, the rapidity divergence is not cancelled in each sector when the weak charges of the initial or final states are specified. The purposes of this paper are to analyze the structure of the factorization in the weak processes, and to resum large logarithms by probing the structure of the divergences including the rapidity divergence from the nonsinglet contributions. We consider the total N -jettiness TN , which is given by TN = i Ti. For T2 Q, SCET can be applied to the process e−e+ → μ−μ+ + X, but we should determine which effective theories are to be employed, depending on the hierarchy of the scales in the collinear and soft momenta and the magnitude of the jettiness.
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