Using Dimensional Reduction for Hadronic Collisions

TL;DR

The paper demonstrates that dimensional reduction (DRED) can be consistently applied to hadronic collisions at next-to-leading order, clarifying infrared singularities and factorization across multiple regularization schemes. It distinguishes two versions, dred and fdH, and derives transition rules between dred, cdr, hv, and fdH, ensuring that hadronic cross sections remain finite and scheme-independent when combined with standard PDFs, such as those in the MSbar scheme. The authors show how to handle internal vs external gluons via the hat/tilde decomposition, compute real and virtual corrections, and construct collinear counterterms within DRED. The work provides practical guidance for implementing DRED in NLO calculations, highlighting its compatibility with supersymmetry and its potential to simplify computations by leveraging established PDFs and factorization formalisms.

Abstract

We discuss how to apply regularization by dimensional reduction for computing hadronic cross sections at next-to-leading order. We analyze the infrared singularity structure, demonstrate that there are no problems with factorization, and show how to use dimensional reduction in conjunction with standard parton distribution functions. We clarify that different versions of dimensional reduction with different infrared and factorization behaviour have been used in the literature. Finally, we give transition rules for translating the various parts of next-to-leading order cross sections from dimensional reduction to other regularization schemes.

Figures (6)

• Figure 1: Gluon splitting into two collinear gluons in the four schemes, indicating the appropriate treatment of each gluon.
• Figure 2: Illustration of Eq. (\ref{['outColl']}) for the case where $a_i$ is an outgoing splitting gluon. The sum over all relevant real processes, $\sum_{a_k}$, gives rise to two contributions. The one on the left (right) results in the $N_c$ ($N_F$) part of $\gamma_{{\rm RS}*}(g)$.
• Figure 3: Illustration of Eq. (\ref{['inColl']}) for an incoming splitting gluon. The sum over all relevant real processes, $\sum_{a_k}$, gives rise to three terms.
• Figure 4: Illustration of Eq. (\ref{['outCollDRED']}) for the case of an outgoing splitting gluon. The sum over all relevant real processes, $\sum_{a_k}$, together with the sum due to the split ${\breve{a}}_i\in\{{\hat{g}},{\tilde{g}}\}$ gives rise to four terms, resulting in the $N_c$ and $N_F$ parts of $\gamma_{{\rm DRED}}({\hat{g}})$ and $\gamma_{{\rm DRED}}({\tilde{g}})$ respectively. Gluons $g$ and (anti)quarks are drawn as usual. Dashed lines represent ${\tilde{g}}$ and ${\hat{g}}$ is represented by a zigzag line.
• Figure 5: Illustration of Eq. (\ref{['inCollDRED']}) for an incoming splitting gluon with partons represented as in Figure \ref{['fig:outB']}. The sum over all relevant real processes, $\sum_{a_k}$, together with the sum due to the split ${\breve{a}}_{(1k)}\in\{{\hat{g}},{\tilde{g}}\}$ gives rise to four terms.