The physics of the mass gap

Pure Yang–Mills theory has no classical mass parameter. Perturbation theory describes massless gauge fields. Yet the nonperturbative picture is a spectrum of massive gauge-invariant excitations. Understanding how these statements fit together is the first step toward understanding the mass gap problem.

This volume begins with familiar QFT and follows the physical questions toward the precise mathematical target. The default setting is four spacetime dimensions, zero temperature, and $\theta=0$.

Start with physical states

Entry: two semesters of QFT, with gauge invariance and elementary quantum mechanics. The linked course supplies the spectral and lattice language at the point of use.

Outcome: distinguish a physical vacuum gap from an operator-channel mass, a dynamically generated scale, and a static-source confinement diagnostic.

Gauge redundancy means that the components of a gauge potential do not independently label observable particles. A mass parameter extracted from a gauge-fixed propagator is therefore not automatically a statement about the physical spectrum.

The relevant Hamiltonian acts on a physical Hilbert space. With the vacuum energy set to zero, a gap means

$$H\Omega=0,\qquad \operatorname{spec}(H)\cap(0,\Delta)=\varnothing$$

for some $\Delta>0$. The lightest spectral threshold need not be an isolated eigenvalue in infinite volume. The official problem description makes the construction of the quantum theory and its gap parts of the same task.

Begin with the problem and the states we can observe. These pages distinguish pure Yang–Mills from QCD with dynamical quarks and from theories containing Higgs fields.

A scale is necessary, but not sufficient

The running coupling introduces a reference scale through dimensional transmutation. This explains how a dimensionful quantity can arise in a classically scale-free theory. It does not determine the infrared spectrum by itself.

For example, writing a possible mass as $m=c\Lambda$ leaves the nonperturbative question in the dimensionless coefficient $c$. Establishing that the intended theory exists and that its lowest physical threshold is strictly positive remains substantial work.

Read dimensional transmutation and explore the running-coupling lab. The lab deliberately stops short of interpreting the one-loop divergence as a physical prediction.

Correlations connect the theory to measurements

Let $\psi$ be the state created by a suitable vacuum-subtracted, smeared physical observable. Its Euclidean correlator has a positive spectral representation:

$$C(t)=\langle\psi,e^{-tH}\psi\rangle =\int_0^\infty e^{-Et}\,d\mu_\psi(E).$$

Large Euclidean time favors the lowest energies with nonzero spectral weight. A probe with no overlap with a light state cannot reveal it. This is why an impressive plateau in one channel is not, by itself, a theorem about the whole spectrum.

The sequence Euclidean correlators, operator overlap, and glueball spectroscopy develops this connection. The correlator lab makes the overlap issue visible.

Confinement is a related question

Confinement concerns the behavior of color sources and the realization of appropriate symmetries. A mass gap concerns the spectrum above the vacuum. Relations between them require specified hypotheses, representations, and observables.

Chatterjee’s lattice confinement criterion is a useful example: its correlation-decay and center-symmetry assumptions are part of the result. They must remain attached when the theorem is summarized.

Choose your next step

The core concept pages develop three separate questions with worked examples:

Question	Read next
When does a spectral threshold imply exponential decay, and when is the converse valid?	A gap and correlation decay
Why can screening and Wilson-loop behavior depend on representation and matter content?	Confinement and screening
How does a lattice symmetry channel become a continuum spin assignment?	Glueballs and quantum numbers

• Follow the twelve-lesson course for the complete first route.
• Read the lattice volume to understand the numerical evidence.
• Read construction to understand what remains to be established mathematically.
• Open the glueball benchmark dossier to see how these ideas enter a real calculation.