To $d$, or not to $d$: Recent developments and comparisons of regularization schemes

TL;DR

The paper surveys and compares regularization schemes for handling UV and IR divergences in higher-order quantum field theory calculations, unifying traditional dimensional schemes (CDR/HV, FDH, DRED) with newer four- and six-dimensional reformulations (FDF, SDF) and non-dimensional approaches. It emphasizes the role of evanescent couplings and epsilon-scalars, demonstrates scheme dependence at intermediate steps through concrete NLO examples, and shows that physical observables are scheme-independent when all contributions are properly combined. It introduces four-dimensional formulations (FDF) and six-dimensional formalisms (SDF) to enable unitarity-based and automated calculations while preserving four-dimensional external states, and discusses wave-function and spinor representations needed for these approaches. The article also discusses the connections between schemes, transition rules, and the implications for NNLO automation, highlighting practical tools like GoSam and the broader goal of establishing robust, complementary methods to traditional dimensional regularization.

Abstract

We give an introduction to several regularization schemes that deal with ultraviolet and infrared singularities appearing in higher-order computations in quantum field theories. Comparing the computation of simple quantities in the various schemes, we point out similarities and differences between them.

Figures (5)

• Figure 1: Diagrams contributing to the electron self-energy at the one- and two-loop level including a quasi ${d}$-dimensional photon (solid wavy line) and a quasi $n_\epsilon$-dimensional $\epsilon$-scalar (dashed wavy line), respectively. The insertion of a coupling counterterm is denoted by a cross. The $\epsilon$-scalar diagrams only exist in fdh and dred.
• Figure 2: Tree-level diagrams contributing to the process $e^{+}e^{-}\!\to\!\gamma^{*}\!\to\!q\bar{q}$. The interaction is mediated by a photon $\gamma$ (left) and an $\epsilon$-scalar photon $\tilde{\gamma}$ (right), respectively. The left diagram is present in all considered schemes, whereas the right one only exists in dred.
• Figure 3: Virtual diagrams for $e^{+}e^{-}\!\to\!\gamma^{*}\!\to\!q\bar{q}$ including a gluon $g$ or an $\epsilon$-scalar $\tilde{g}$. In cdr and hv, only the first diagram contributes, whereas in fdh also the second diagram is present. In dred, all diagrams contribute.
• Figure 4: Counterterm diagram for $e^{+}e^{-}\!\to\!\gamma^{*}\!\to\!q\bar{q}$ which only contributes in dred.
• Figure 5: Real diagrams for $e^+ e^- \to q\bar{q} g$ and $e^+ e^- \to q\bar{q} \tilde{g}$. In cdr and hv there is only the first diagram, whereas in fdh also the second diagram is present. In dred, all diagrams contribute. An analogous diagram where the gluon couples to the other quark leg is understood.