Confinement, Screening, and Mass Gaps in Controlled Regimes

Confinement, screening, and a spectral mass gap are distinct properties of gauge theories. This chapter compares their observables, develops mechanisms in compact Abelian and small-circle regimes, and explains what must be established before extending a controlled example to a more general theory.

Enter this chapter

Begin with Confinement Definitions and Their Non-Equivalence to state exactly which confinement proposition is being tested.

Helpful background. Review Coulomb, Higgs, and Confining Regimes and Gauge-Phase Diagnostics from Line Operators for gauge-invariant phase and line-operator diagnostics.

Topics

1. Confinement Definitions and Their Non-Equivalence — Separates inequivalent uses of confinement.
2. Line Operators, Screening, and Generalized-Symmetry Diagnostics — Tests which genuine probes dynamical matter can screen.
3. Static Potentials, Flux Tubes, and Effective Strings — Extracts static energies and long-string behavior.
4. Compact U(1) in 2+1 Dimensions and the Monopole Plasma — Derives a confinement mechanism in a dilute monopole regime.
5. Small-Circle Abelianization, Bions, and Adiabatic Continuity — States the assumptions behind weakly coupled compactification.
6. Proposed Confinement Mechanisms and Discriminating Observables — Connects proposed mechanisms to discriminating observables.
7. The Non-Abelian Mass Gap as a Spectral Statement — Defines a gap in the physical spectrum.
8. Confinement Evidence, Mechanisms, and Open Problems — Separates controlled results from wider unresolved claims.
9. Low-Dimensional Confinement and Screening Laboratories — Uses lower-dimensional examples to test supposed equivalences.

How the ideas fit together

A gap does not imply an unscreened confining force. The massless Schwinger model has a massive gauge-invariant excitation while dynamical charged matter screens external probes. Conversely, pure two-dimensional Yang–Mills theory has an area law without local propagating gauge particles. These examples show why line behavior and spectral statements require separate definitions. Controlled monopole or small-circle calculations supply mechanisms in their own regimes; extending them needs additional dynamical arguments, not just the same vocabulary. Coleman 1985, ch. 8.

Where to go next

Continue to Large-N Dynamics to fix the index family, normalization, and quantities held constant.

References

• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 8. DOI.