Asymptotic Freedom and Confinement

The QED coupling grows with energy: a bare charge is screened by virtual electron-positron pairs, so probing at shorter distance penetrates the screening cloud and sees a larger effective charge, as derived for the running of $\alpha$. QCD does the opposite. The gluon self-coupling adds a contribution of the opposite sign that overwhelms quark screening, and the effective color charge falls at short distance. This lesson establishes that sign through the one-loop beta function, follows the coupling down to the scale where it diverges, and interprets the strong-coupling end as confinement.

Throughout, $\hslash =c=1$, ${\alpha }_s=g_s^2/4\pi$, and $N_c=3$ colors with $n_f$ active quark flavors.

The one-loop beta function

The coupling depends on the scale $\mu$ at which the theory is probed. The dependence is governed by the beta function, defined as the logarithmic derivative of the coupling,

with the one-loop coefficient¹

The two terms have opposite physical origins.

• The $-\frac{4}{3}{T}_F{n}_f$ term comes from quark loops on the gluon propagator. Like the electron loop in QED, a virtual $q\bar{q}$ pair screens the color charge and drives the coupling up at short distance. Alone, it would give a positive beta function, as in QED.
• The $+\frac{11}{3}{C}_A$ term comes from gluon loops, available only because the gluon carries color. It antiscreens: the cloud of virtual gluons spreads color charge outward rather than screening it, so a short-distance probe sees less charge. This term has no QED counterpart and it dominates.

For QCD with $n_f\le 6$, the gluon term wins and $b_0>0$, so the beta function is negative. The condition for the antiscreening to survive is

comfortably satisfied by the six quark flavors of nature.

The running coupling and the QCD scale

Solving $\beta (g_s)=-b_0{g}_s^3/(4\pi {)}^2$ and rewriting in terms of ${\alpha }_s$ gives the one-loop running

The coupling decreases logarithmically as $Q^2$ increases — asymptotic freedom made quantitative. It is convenient to absorb the reference scale into a single dimensionful constant ${\Lambda }_{\text{QCD}}$, defined as the scale at which the extrapolated one-loop coupling diverges. In terms of it,

A classical Lagrangian with massless quarks contains no scale at all; the quantum theory manufactures one, ${\Lambda }_{\text{QCD}}\approx 200\text{–}300$ MeV, through the running of the coupling. This is dimensional transmutation: the dimensionless bare coupling is traded for a dimensionful scale.² For $Q\gg {\Lambda }_{\text{QCD}}$ the coupling is small and perturbation theory in ${\alpha }_s$ converges; for $Q\sim {\Lambda }_{\text{QCD}}$ it is of order one and perturbation theory fails, which is precisely the hadronic scale.

The measured coupling runs from ${\alpha }_s\approx 0.12$ at the $Z$ mass down toward ${\alpha }_s\approx 0.09$ at the TeV scale, and the world data over three orders of magnitude in $Q$ lie on the single predicted curve — one of the most direct confirmations of the theory.³

Confinement and the color flux tube

Asymptotic freedom is a statement about short distances; confinement is its long-distance counterpart. As the separation between a quark and an antiquark grows, ${\alpha }_s$ grows, and the field between them does not spread out like a Coulomb field. The gluon self-interaction pulls the chromoelectric field lines into a narrow flux tube of roughly constant cross section, so the field energy grows linearly with length rather than falling off. The static potential between a heavy quark and antiquark is

a short-distance Coulomb term with the color factor $C_F=\frac{4}{3}$, plus a linear term with string tension $\sigma \approx 0.9$ GeV/fm $\approx (0.42\text{GeV}{)}^2$ on dimensional grounds fixed by ${\Lambda }_{\text{QCD}}$.⁴ Separating the pair costs energy without bound, so an isolated quark cannot be produced: the only finite-energy states are color singlets. This is the dynamical content behind the color-singlet rule stated in the quark model.

String breaking

The linear rise does not continue forever. Once the energy stored in the tube exceeds twice the mass of the lightest quark, $\sigma r>2m_q$, it becomes energetically favorable to create a new $q\bar{q}$ pair from the vacuum. The pair snaps the string, and each end caps off into a color-singlet hadron. Pulling a quark out of a proton does not liberate it; it produces a jet of hadrons, because the energy invested in separation is converted into new quark-antiquark pairs.

Two regimes, one theory

The negative beta function ties the two faces of QCD together. At high energy the coupling is small: quarks and gluons are nearly free, and cross sections are computed in perturbation theory, the domain of deep inelastic scattering and jets. At low energy the coupling is large: color is confined, the spectrum is hadrons, and quantitative predictions require lattice methods rather than diagrams. The crossover is the single scale ${\Lambda }_{\text{QCD}}$, generated from a classically scale-free theory by the running of the coupling.

The following table contrasts the two limits.