Abstract
We reanalyze the factorization theorems for 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 $e^+ e^-$ annihilation near threshold, and show that the resultant soft function is also free of infrared and rapidity divergences.
Highlights
Factorization theorems in which high-energy processes are divided into hard, collinear, and soft parts are essential in providing precise theoretical predictions
It is well established in full quantum chromodynamics (QCD) that the structure functions FDY for DY process and F1 for deep inelastic scattering (DIS) can be schematically written in a factorized form as [1,2]
In Refs. [13,14], we have suggested how some of the divergences can be transferred from the soft part to the collinear part to make the soft function IR finite, while the collinear part reproduces the parton distribution function (PDF) near threshold
Summary
Factorization theorems in which high-energy processes are divided into hard, collinear, and soft parts are essential in providing precise theoretical predictions. The issues on the factorization near threshold in SCET are summarized as follows: First, since the incoming active partons take almost all the hadron momenta, the emission of additional collinear particles is prohibited. The invariant masses of the collinear particles and the soft particles are different and there is no reason for the rapidity divergence to cancel in the sum of the collinear and the soft parts. The main points of our paper are to identify the new degrees of freedom, to incorporate them in the definition of the PDF and the soft functions, to calculate the perturbative corrections at order αs, and to show that we obtain the proper factorization near threshold.
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