Factorization and Regularization by Dimensional Reduction

TL;DR

Signier and Stöckinger address a long-standing factorization puzzle in dimensional reduction (DRED) for QCD processes, focusing on gg→ttbar at NLO. They reinterpret the 4-dimensional gluon $G$ as a sum of a $D$-dimensional gluon $g$ and an $(4-D)$-dimensional scalar $\phi$, treating $g$ and $\phi$ as independent partons. The collinear limit then becomes a linear combination of LO processes with initial states $g$ or $\phi$, with coefficients given by universal splitting functions $P_{i\to jk}$, restoring factorization in DRED. They derive the relevant splitting functions, demonstrate that the apparent non-factorizing terms are absorbed into these contributions, and show that cross sections can be computed consistently in either DREG or DRED via appropriate subtraction terms. The work clarifies the role of $\epsilon$-scalars in factorization and provides a systematic framework applicable to higher orders and to SUSY-relevant calculations.

Abstract

Since an old observation by Beenakker et al, the evaluation of QCD processes in dimensional reduction has repeatedly led to terms that seem to violate the QCD factorization theorem. We reconsider the example of the process gg->ttbar and show that the factorization problem can be completely resolved. A natural interpretation of the seemingly non-factorizing terms is found, and they are rewritten in a systematic and factorized form. The key to the solution is that the D- and (4-D)-dimensional parts of the 4-dimensional gluon have to be regarded as independent partons.

Figures (2)

• Figure 1: Generic structure of (a) LO diagrams, (b) NLO real correction diagrams, (c) NLO diagrams giving rise to a collinear divergence for $k_3\to(1-x)k_2$.
• Figure 2: Tree level diagrams for the four splitting processes involving $g$ and $\phi$.