Abstract
We discuss the structure of rapidity divergences that are presented in the soft factors of transverse momentum dependent (TMD) factorization theorems. To provide the discussion on the most general level we consider soft factors for multi-parton scattering. We show that the rapidity divergences are result of the gluon exchanges with the distant transverse plane, and are structurally equivalent to the ultraviolet divergences. It allows to formulate and to prove the renormalization theorem for rapidity divergences. The proof is made with the help the conformal transformation which maps rapidity divergences to ultraviolet divergences. The theorem is the systematic form of the factorization of rapidity divergences, which is required for the definition of TMD parton distributions. In particular, the definition of multi parton distributions is presented. The equivalence of ultraviolet and rapidity divergences leads to the exact relation between soft and rapidity anomalous dimensions. Using this relation we derive the rapidity anomalous dimension at the three-loop order.
Highlights
Soft factors are the inherent part of the modern factorization theorems, and have the common structure of vacuum matrix elements of a product of Wilson lines
We have considered the structure of rapidity divergences of the multi-parton scattering (MPS) soft factor
We have proven the renormalization theorem for rapidity divergences (RTRD) for MPS soft factors and discussed some of its consequences
Summary
Soft factors are the inherent part of the modern factorization theorems, and have the common structure of vacuum matrix elements of a product of Wilson lines. The structure of the exponentiated diagrams for the TMD soft factor is rather simple and gives some intuition how the cancellation of higher-order divergences takes place. This intuition is supported by NNLO calculation. To access the problem of rapidity divergences on a more general level, we study the multi-parton scattering (MPS) factorization and its soft factor. Some additional materials are collected in the set of appendices
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