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Abstract

This paper studies multijet cross sections in the kinematic limit where one or more of the jets are unresolved. We study the asymptotic expansion of the cross section by using the method of regions. We explain in large generality how to identify the relevant phase space regions and derive the leading-power approximation of the phase space. The leading-power phase space factorizes into a hard phase space and a radiation phase space for each collinear sector and the soft sector. Using the infrared factorization properties of squared matrix elements at leading power, we derive a factorization formula for generic resolution variables describing the exclusive n-jet limit in terms of fully differential beam, jet, and soft functions. We show how the factorization formula can be simplified for specific resolution variables. We regularize rapidity divergences in the beam and jet functions using a time-like reference vector, a method we call the ``$z_N$-prescription'', and demonstrate how the associated zero-bin contributions combine with the soft function to define a soft subtracted function free of rapidity divergences and suitable for numerical evaluation. The $z_N$-prescription can be used both for $\mathrm{SCET}_{\mathrm{II}}$- and $\mathrm{SCET}_{\mathrm{I}}$-type resolution variables. As an application of our framework, we derive factorization formulas for several transverse-momentum-like variables, and we present a variable that factorizes into simple cumulant functions to all orders in perturbation theory.