N-jettiness for muon jet pairs in electroweak high-energy processes

We study the $N$-jettiness in the electroweak high-energy process for the final muon jet pairs, $e^- e^+ \rightarrow \mu^+ \ \mathrm{jet} + \mu^- \mathrm{jet}$. Compared to QCD, the main difference is that there exist additional gauge nonsinglet contributions in the weak interaction, which make the factorization more elaborate. Especially the nonsinglet contributions arise due to the Block-Nordsieck violation in electroweak processes, which yields the Sudakov logarithms and the rapidity divergence. They change the evolution of the factorized parts considerably in the $N$-jettiness. There are two possible channels, initiated from the gauge bosons $W W \rightarrow \ell_{\mu} \overline{\ell}_{\mu}$, and from the electrons $\ell_e \overline{\ell}_e \rightarrow \ell_{\mu} \overline{\ell}_{\mu}$, where $\ell$ denotes the weak doublet. The latter was discussed previously, and we complete the analysis by studying the first. The factorization for $W W \rightarrow \ell_{\mu} \overline{\ell}_{\mu}$ can be proceeded in a similar way as in the factorization for $\ell_e\overline{\ell}_e \rightarrow \ell_{\mu} \overline{\ell}_{\mu}$, and the result exhibits a rich structure. The new ingredients in this study consist of the $W$ beam functions, and the complex color structure of the soft functions and the hard functions. The resummation of the large logarithms is performed by solving the renormalization group equations with respect to the renormalization scale and the rapidity scale. In the numerical analysis, we confine to the SU(2) weak gauge interaction, and the numerical results are presented for both channels at next-to-leading-logarithmic accuracy including the singlet and the nonsinglet contributions. The nonsinglet contributions turn out to be appreciable in the 2-jettiness.