Perturbative QCD

TL;DR

This chapter introduces perturbative QCD as a framework for predicting high-energy cross sections by separating short-distance hard scattering from long-distance hadronic structure via factorisation, and then expanding amplitudes and cross sections in the strong coupling $\alpha_s$. It covers dimensional regularisation, the running of $\alpha_s$ through the $\beta$-function, and the treatment of ultraviolet and infrared divergences, including real- and virtual-emission cancellations and subtraction schemes. It also discusses jets and event shapes as key observables, and outlines how theory uncertainties are estimated through scale variation and beyond, as well as the current state of the art in multi-loop amplitudes and NNLO/NNNLO methods. The discussion highlights the balance between perturbative calculations and non-perturbative inputs like PDFs, fragmentation, and power corrections, and points to future directions in resummation, parton showers, and electroweak effects for collider phenomenology.

Abstract

We give an introduction to perturbative Quantum Chromodynamics, focusing on a pedagogical description of concepts and methods to calculate cross sections measured at high energy colliders. After introducing basic concepts that allow for a perturbative expansion, such as factorisation and asymptotic freedom, we introduce loop integrals and the treatment of ultraviolet and infrared divergences in QCD. The definition of jets and event shape observables is also discussed. Finally, we give a brief overview of the current state of the art.

Figures (17)

• Figure 1: Tree-level diagrams (respectively of $t-$, $u-$ and $s$-channel type) contributing to the LO $q\bar{q} \to gg$ amplitude. The $s$-channel type diagram features a three-gluon vertex and is thus a consequence of the non-Abelian nature of QCD.
• Figure 2: One-loop two-point function ("bubble").
• Figure 3: Experimental determination from the CMS collaboration of the strong coupling $\alpha_s$ as a function of the scale $Q$. Figure taken from Ref. CMS:2024hwr.
• Figure 4: One-loop diagrams contributing to the correction of the gluon propagator and the running of $\alpha_s$. They comprise the first $\beta$-function coefficient, $b_0$. The quark loop contribution is proportional to $n_f$, the number of active flavours.
• Figure 5: Labelling for the planar two-loop box example with $p_4$ off-shell.