Soft and Collinear Functions for the Standard Model

TL;DR

The paper develops a comprehensive SCET-based framework to compute electroweak radiative corrections in high-energy Standard Model scattering by deriving universal soft and collinear functions for all SM particles. It provides explicit running and matching formulas, plus detailed soft-function matrices for fermion scattering, gauge-boson pair production, single W/Z production, and gluon scattering, including custodial SU(2) breaking and Yukawa effects. The results rely on a three-stage EFT approach—matching at $\mu_h$, running down to $\mu_l \sim M_Z$, then continuing with QCD+QED to $\mu_f$—with analytic expressions for anomalous dimensions and low-scale matching, and an appendix giving analytic integration of the SCET anomalous dimension up to the three-loop cusp. Together, these ingredients enable RG-improved, process-specific predictions for high-energy SM cross sections within a consistent EFT framework.

Abstract

Radiative corrections to high energy scattering processes were given previously in terms of universal soft and collinear functions. This paper gives the collinear functions for all standard model particles, the general form of the soft function, and explicit expressions for the soft functions for fermion-fermion scattering, longitudinal and transverse gauge boson production, single W/Z production, and associated Higgs production. An interesting subtlety in the use of the Goldstone boson equivalence theorem for longitudinal W+ production is discussed.

Figures (12)

• Figure 1: Plot of the collinear functions against $\bar{n} \cdot p$ for (a) lower panel: $u_L$ (dotted green), $u_R$ (dotted cyan), $t_L$ (dashed red), $t_R$ (dashed blue) (b) upper panel: $d_L$ (dotted green), $d_R$ (dotted cyan), $b_L$ (dashed red), $b_R$ (dashed blue).
• Figure 2: Plot of the collinear functions against $\bar{n} \cdot p$ for $u_L$ (dotted green), $u_R$ (dotted cyan), $d_L$ (dashed red), and $d_R$ (dashed blue).
• Figure 3: Plot of the collinear functions against $\bar{n} \cdot p$ for $\nu_L$ (dashed blue), $e_L$ (dotted red), and $e_R$ (solid red).
• Figure 4: Plot of the collinear functions due to running from $M_Z$ to $\mu_f$ against $\bar{n} \cdot p$ for electrons with $\mu_f=30$ GeV (solid red) and $\mu_f=50$ GeV (dashed blue) are shown in the upper panel. The QCD correction for quarks with $\mu_f=30$ GeV (solid red) and $\mu_f=50$ GeV (dashed blue) and gluons with $\mu_f=30$ GeV (dotted green) and $\mu_f=50$ GeV (dot-dashed cyan) are shown in the lower panel.
• Figure 5: Plot of the collinear functions against $\bar{n} \cdot p$ for $W \to \gamma$ (solid red), $B \to \gamma$ (dotted red) and gluons (dashed blue).