Collider Physics at the Precision Frontier

TL;DR

This article surveys the precision frontier in collider physics with a focus on the Higgs sector, summarizing state-of-the-art fixed-order predictions up to N$^3$LO and beyond, including gluon-fusion, bottom-quark fusion, VH, VBF, and Higgs-pair production. It also provides an in-depth review of modern techniques for multi-loop amplitudes, covering analytic methods (differential equations, IBP, differential geometry) and numerical strategies (sector decomposition, Mellin-Barnes, unitarity, loop-tree duality), and discusses infrared subtraction schemes for real radiation at NNLO and beyond. The work highlights the substantial progress in differential NNLO predictions, the handling of heavy-quark masses, and the development of public tools and grids for fast phenomenology, while acknowledging remaining challenges such as matching to parton showers, multi-scale integrals with elliptic functions, and non-factorisable contributions. Overall, the paper underscores how these advances push the precision frontier toward the experimental capabilities of the HL-LHC and future colliders, enabling tighter tests of the SM and tighter constraints on new physics scenarios. The synthesis of analytic, semi-analytic, and numerical techniques, along with robust subtraction schemes, points to a future where high-precision Higgs phenomenology becomes standard practice across collider processes.

Abstract

The precision frontier in collider physics is being pushed at impressive speed, from both the experimental and the theoretical side. The aim of this review is to give an overview of recent developments in precision calculations within the Standard Model of particle physics, in particular in the Higgs sector. While the first part focuses on phenomenological results, the second part reviews some of the techniques which allowed the rapid progress in the field of precision calculations. The focus is on analytic and semi-numerical techniques for multi-loop amplitudes, however fully numerical methods as well as subtraction schemes for infrared divergent real radiation beyond NLO are also briefly described.

Figures (14)

• Figure 1: Illustration of the heavy top limit (HTL).
• Figure 2: Higgs boson rapidity distribution at different orders in perturbation theory, including scale uncertainties. Figures from Refs. Cieri:2018oms (left) and Dulat:2018bfe (right).
• Figure 3: Differential predictions for the rapidity of the leading photon (left) and the absolute value of the rapidity difference of the two photons (right). Predictions are shown at LO (grey), NLO (green), NNLO (blue), N$^3$LO (red), and for the NNLO prediction rescaled by the inclusive $K_{N3LO}$-factor (orange). The shaded vertical band in the left plot corresponds to the region excluded by the fiducial cuts. Figures from Ref. Chen:2021isd.
• Figure 4: Left: Cumulative cross section as a function of the $p_\perp$-cut at NNLO in the HTL, as well as rescaled by the LO (NLO) full-SM spectrum labelled by EFT-improved$(0)$ (EFT-improved$(1)$). Right: Comparison of cross sections from different production channels: gluon-fusion (green), VBF (red), vector boson associated (blue) and top-quark pair associated (magenta). Figures from Ref. Becker:2020rjp.
• Figure 5: Allowed region in the $C_{t\phi}$-$C_{\phi G}$ plane at 95% confidence level, where $C_{t\phi}$ and $C_{\phi G}$ denote the Wilson coefficients of the top-Higgs and Higgs-gluon operators, respectively. Left: current constraints. Right: Projection for the HL-LHC. The theoretical uncertainties are not included. Figures from Ref. Maltoni:2016yxb.