SCET approach to regularization-scheme dependence of QCD amplitudes

TL;DR

The paper tackles the problem of regularization-scheme dependence in massless QCD amplitudes, focusing on FDH and DRED and their consistency with conventional methods up to NNLO. It develops a soft-collinear effective theory approach that links IR structure to scheme-dependent anomalous dimensions ($\gamma_{\text{cusp}}$, $\gamma_q$, $\gamma_g$) through jet and soft functions, enabling explicit transition rules between schemes (CDR, HV, FDH, DRED). The authors compute the soft and jet functions at NNLO in FDH/DRED, derive the epsilon-scalar anomalous dimension $\bar{\gamma}_{\epsilon}$, and provide an alternative determination from the $\epsilon$-scalar form factor, with cross-checks against explicit two-loop amplitudes. The results establish that hard, soft, and jet functions are separately scheme-independent, allowing cross-scheme calculations and facilitating fully differential NNLO predictions by combining components computed in different schemes, thus broadening computational flexibility and reliability in precision QCD.

Abstract

We investigate the regularization-scheme dependence of scattering amplitudes in massless QCD and find that the four-dimensional helicity scheme (FDH) and dimensional reduction (DRED) are consistent at least up to NNLO in the perturbative expansion if renormalization is done appropriately. Scheme dependence is shown to be deeply linked to the structure of UV and IR singularities. We use jet and soft functions defined in soft-collinear effective theory (SCET) to efficiently extract the relevant anomalous dimensions in the different schemes. This result allows us to construct transition rules for scattering amplitudes between different schemes (CDR, HV, FDH, DRED) up to NNLO in massless QCD. We also show by explicit calculation that the hard, soft and jet functions in SCET are regularization-scheme independent.

Figures (6)

• Figure 1: Feynman rules for the emission of one and two gluons from a Wilson line. Figure taken from Becher:2014oda.
• Figure 2: Selected non-zero Feynman diagrams contributing to the one-loop and two-loop soft functions. A complete list of diagrams can be found in Belitsky:1998tc. Double lines indicate the direction of Wilson lines while the red vertical cut indicates on-shell partons. The scheme dependence originates from the diagram $D_2$. Diagrams $D_2$, $D_3$ and $D_4$ represent double real soft emissions while diagram $D_5$ represents a single virtual-real emission.
• Figure 3: Examples of two-loop diagrams contributing to the quark jet function. Gluons emitted from the crossed circles originate from the Wilson lines. Diagram (a) contributes in cdr and fdh, whereas diagram (b) with two $\epsilon$-scalars contributes only in fdh.
• Figure 4: Sample two-loop diagrams contributing to the gluon jet function. Diagram (a) is present both in cdr and fdh, diagram (b) including an $\epsilon$-scalar contributes only in fdh.
• Figure 5: Sample two-loop diagrams contributing to the $\epsilon$-scalar jet function both. Diagram (a) is proportional to $\alpha_s\alpha_e$ whereas diagram (b) is $\sim\alpha^2_{4\epsilon}$