Abstract
We show at NNLO that the soft factors for double parton scattering (DPS) for both integrated and unintegrated kinematics, can be presented entirely in the terms of the soft factor for single Drell-Yan process, i.e. the transverse momentum dependent (TMD) soft factor. Using the linearity of the logarithm of TMD soft factor in rapidity divergences, we decompose the DPS soft factor matrices into a product of matrices with rapidity divergences in given sectors, and thus, define individual double parton distributions at NNLO. The rapidity anomalous dimension matrices for double parton distributions are presented in the terms of TMD rapidity anomalous dimension. The analysis is done using the generating function approach to web diagrams. Significant part of the result is obtained from the symmetry properties of web diagrams without referring to explicit expressions or a particular rapidity regularization scheme. Additionally, we present NNLO expression for the web diagram generating function for Wilson lines with two light-like directions.
Highlights
The effects of double parton scattering (DPS), i.e. the scattering with two partons of a hadron participating in the hard subprocess, are usually expected to be small in comparison to a single parton scattering contribution
We show at NNLO that the soft factors for double parton scattering (DPS) for both integrated and unintegrated kinematics, can be presented entirely in the terms of the soft factor for single Drell-Yan process, i.e. the transverse momentum dependent (TMD) soft factor
We have considered the soft factor for the leading order factorization formula of the double parton scattering (DPS) in the perturbative regime (such that ln(b2Q2) is not large, where b is any transverse separation within the soft factor)
Summary
The effects of double parton scattering (DPS), i.e. the scattering with two partons of a hadron participating in the hard subprocess, are usually expected to be small in comparison to a single parton scattering contribution. The structure of perturbative series is exceptionally simplified within the generating function approach for web diagrams, formulated in [13, 14] Within this approach, one should calculate the generating function, which is unique for a given geometry (in the case of TMD-like soft factors, the only important point is two light-like directions). As we demonstrate in this work, the consideration of the generating function at NNLO immediately shows the possibility to present any DPS soft factor in terms of TMD soft factors at this order. It appears that some important results can be obtained without referring to expressions for diagrams. The expressions for basic loop integrals that participate in the generating function are collected in appendix C
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