The QCD Running Coupling

TL;DR

The paper analyzes the QCD running coupling $α_s$ across all scales, detailing its quantum-origin scale dependence and the resulting UV perturbative and IR nonperturbative regimes. It surveys UV determinations via perturbative QCD and experimental inputs, and IR treatments through effective charges, holographic QCD, and Dyson–Schwinger/Lattice approaches, highlighting a convergence toward a universal IR coupling. It emphasizes the current sub-percent precision on $α_s(M_Z)$ and the necessity of higher-order pQCD and nonperturbative insights, while arguing for a single, process-independent coupling that coherently describes hadronic physics from short to long distances. The work underscores the practical impact on precision tests of the Standard Model and the understanding of confinement, mass generation, and hadron structure through a unified coupling framework across all energy scales.

Abstract

We describe the coupling of the strong force. Denoted as $α_s$, it sets the strength of that force, just as $G$ or $α$ specify the strength of the gravity and electromagnetism. Its value depends on the scale at which phenomena are observed. In this chapter, we will explain the nature of the coupling, the quantum origin of its scale dependence, and the crucial consequences this entails for quantum chromodynamics, the gauge theory for the strong force. We describe the theories for the calculation of $α_s$, using the perturbative method at high-momentum scales (equivalently, short-distance scales) and nonperturbative approaches at low-momentum scales (equivalently, long-distance scales). We also present the experimental determinations of $α_s$ at both short and long distance scales.

Figures (10)

• Figure 1: The QCD coupling, $\alpha_s$, in function of momentum scale $Q$. The points are experimental data. The red line is simple fit using the 1-loop perturbative formula $\propto 1/\ln(Q^2/\Lambda^2)$, a Fermi-Dirac function for the scale $Q_r$=$b/(e^{(Q^2-c)/d} +1)$ regularizing the divergence at $Q=\Lambda$, and another Fermi-Dirac function $T_r$=$(1+(\pi-1)/(e^{(Q-f)/g}+1))$ to smooth the low-to-high $Q$ transition. The parameters values yielding $\chi^2/n.d.f.=1.00$ are: $a=1.56$, $\Lambda=0.246$ GeV, $b=0.808$ GeV, $c=0.11$ GeV$^2$, $d=0.20$ GeV$^2$, $f=$1.29 GeV and $g=$0.59 GeV.
• Figure 2: The inverse square law of forces, and where it fails. For static sources, a classical force follows $\vec{F}=\frac{\mathcal{A} c_1 c_2}{r^2}e^{-m|\vec{r}|}(1+m|\vec{r}|) \vec{u}_r$, where $\mathcal{A}$ is the force coupling, $c_1$ and $c_2$ are the source charges (masses for gravity, electric charge for electricity, color for QCD, and weak isospin for the weak force), $\vec{r}$ is the vector linking the two sources, $\vec{u}_r\equiv \vec{r}/|\vec{r}|$, and $m$ is the field mass, set to $m=0$ in this chapter. Classically, the inverse square law, i.e. the $1/r^2$ factor, stems from the force flux that freely spreads in 3D space (left panel). The straight black lines are the field lines, the blue square represents the unit area, distant from the source by $r$, through which the flux is measured. The force equals the flux going through the area. The red sphere symbolizes one of the sources (the other is the test particle and therefore not shown). In QFT, the $1/r^2$ dependence originates from the Fourier transform of the gauge boson propagator (dashed line in middle panel) in the Born approximation; $1/r^2 \propto\mathcal{F}(1/q^2)$ with $q$ the boson 4-momentum. The $1/r^2$ law fails at short distances because of additional $r-$dependences from quantum loops, as illustrated in the magnified area in the left panel. The straight lines now depict the trajectories of the gauge bosons rather than field lines. Fermion loops are shown in red. The $1/r^2$ law also fails for strongly interacting non-linear theories because the self-interaction of the field prevents its free spreading ( viz propagation) in space. QCD is the prototypical example of such phenomenon (right panel). In the renormalization process, the additional scale dependence from quantum loops is folded into the definition of the coupling rather than modifying the gauge boson propagator, thereby making the coupling scale-dependent. This is the origin of its running.
• Figure 3: Short distance QED (panel a-top graph) and QCD (all panels) processes making $\bm \alpha$ and $\bm {\alpha_s}$ to run. (a): Vacuum polarization; (b): Quark self-energy; (c): Vertex corrections. Other graphs exist, see Muta:1998vi for the list up to next-to-leading order (NLO).
• Figure 4: Left: screening of the QED electric charge by quantum loops. Positrons tend to be nearer to the bare negative charge (horizontal line) than electrons. Right: anti-screening of QCD color charges. The color-charged gluons spatially spread the color initially carried by the quark: the initially red quark changes first into a blue quark when its red color is carried away by a gluon. Then, the quark becomes green after another gluon emission, etc. (Anti-blue is symbolized by yellow, and anti-green by magenta.) The spatial dilution of the initial red charge suppresses interaction occurring via high-resolution anti-red gluons, while lower-resolution gluons still couple since they do not resolve the quark from the gluons carrying away the color. Thus, $\alpha_s$ grows weaker at shorter distances. (Figure from Ref. Deur:2023dzc.)
• Figure 5: Panel (a): 1$^{\rm st}$ order quark--quark scattering: classical result (no quantum loops) leading to the $1/r^2$ law. Panels (b), (c) and (d): gluon propagator with a quark, gluon and ghost loops, respectively. (Figure from Ref. Deur:2023dzc.)