Threshold factorization redux

Abstract

We reanalyze the factorization theorems for the Drell-Yan process and for deep inelastic scattering near threshold, as constructed in the framework of the soft-collinear effective theory (SCET), from a new, consistent perspective. In order to formulate the factorization near threshold in SCET, we should include an additional degree of freedom with small energy, collinear to the beam direction. The corresponding collinear-soft mode is included to describe the parton distribution function (PDF) near threshold. The soft function is modified by subtracting the contribution of the collinear-soft modes in order to avoid double counting on the overlap region. As a result, the proper soft function becomes infrared finite, and all the factorized parts are free of rapidity divergence. Furthermore, the separation of the relevant scales in each factorized part becomes manifest. We apply the same idea to the dihadron production in e þ e − annihilation near threshold, and show that the resultant soft function is also free of infrared and rapidity divergences.

Leading-logarithmic threshold resummation of the Drell-Yan process at next-to-leading power

“…[∆ LP ](z, µ)| NLP = (1 − z)∆ LP (z, µ) , = −4 Γ cusp (α s )(1 − z)∆ LP (z, µ) . (B 17). Note that this result is consistent with the structure of the formal expansions (B.3) of ∆ (N)LP : Since the right-hand side contributes to the µ-evolution of ∆ NLP , it should possess a formal series expansion in terms of ln m (1 − z), m ≥ 1.…”

“…The above classification of regions refers to the 'standard' treatment of factorization near threshold, which neglects momentum regions which lead to scaleless integrals in dimensional regularization. If one aims at the correct identification of ultraviolet and infrared singularities, the situation is more subtle, as discussed in[17]. The whole effect of the additional collinear-soft region introduced in this reference is to convert the IR singularities in the soft function of the standard treatment into UV singularities, ultimately leading to the same factorization formula at leading power.…”