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We reanalyze the factorization theorems for Drell-Yan process and for deepinelastic scattering near threshold, as constructed in the framework of thesoft-collinear effective theory (SCET), from a new, consistent perspective. Inorder to formulate the factorization near threshold in SCET, we should includean additional degree of freedom with small energy, collinear to the beamdirection. The corresponding collinear-soft mode is included to describe theparton distribution function (PDF) near threshold. The soft function ismodified by subtracting the contribution of the collinear-soft modes in orderto avoid double counting on the overlap region. As a result, the proper softfunction becomes infrared finite, and all the factorized parts are free ofrapidity divergence. Furthermore, the separation of the relevant scales in eachfactorized part becomes manifest. We apply the same idea to the dihadronproduction in $e^+ e^-$ annihilation near threshold, and show that theresultant soft function is also free of infrared and rapidity divergences.

Threshold Factorization Redux