Analysis of exclusive $k_T$ jet algorithms in electron-positron annihilation

We study the factorization of the dijet cross section in $e^+ e^-$ annihilation using the generalized exclusive jet algorithm which includes the cone-type, the JADE, the $k_T$, the anti-$k_T$ and the Cambridge/Aachen jet algorithms as special cases. In order to probe the characteristics of the jet algorithms in a unified way, we consider the generalized $k_T$ jet algorithm with an arbitrary weight of the energies, in which various types of the $k_T$-type algorithms are included for specific values of the parameter. We show that the jet algorithm respects the factorization property for the parameter $\alpha <2$. The factorized jet function and the soft function are well defined and infrared safe for all the jet algorithms except the $k_T$ algorithm. The $k_T$ algorithm ($\alpha=2$) breaks the factorization since the jet and the soft functions are infrared divergent and are not defined for $\alpha=2$, though the dijet cross section is infrared finite. In the jet algorithms which enable factorization, we give a phenomenological analysis using the resummed and the fixed-order results.