QCD in a Finite Volume

TL;DR

The work surveys QCD-like non-abelian gauge theories in finite volumes, leveraging a Hamiltonian framework to access non-perturbative dynamics while carefully handling gauge fixing and topology. It develops the zero-momentum, abelian vacuum valley on the three-torus, analyzes boundary identifications, and constructs an effective Hamiltonian whose spectrum matches lattice results in intermediate volumes; it also treats massless quarks, twisted boundary conditions, and supersymmetric extensions. The treatise then advances through instantons and sphalerons on toroidal and spherical geometries, connects finite-volume observables to infinite-volume physics via Lüscher's formalism and dualities, and culminates with a discussion of large-volume behavior, Goldstone modes, and dualities, while outlining key open problems for future work.

Abstract

We will review our understanding of non-abelian gauge theories in finite physical volumes. It allows one in a reliable way to trace some of the non-perturbative dynamics. The role of gauge fixing ambiguities related to large field fluctuations is an important lesson that can be learned. The hamiltonian formalism is the main tool, partly because semiclassical techniques are simply inadequate once the coupling becomes strong. Using periodic boundary conditions, continuum results can be compared to those on the lattice. Results in a spherical finite volume will be discussed as well.

Figures (20)

• Figure 1: Sketch of the fundamental (shaded) and Gribov regions. The dotted lines indicate the boundary identifications.
• Figure 2: Sketch of a singular boundary point due to a bifurcation of the norm functional. It can be used to show that there are homotopically trivial gauge copies inside the Gribov horizon ( H).
• Figure 3: A two dimensional slice of the vacuum valley along the $(C_1,C_2)$ plane. The fat square give the Gribov horizon, the grey square is the fundamental domain. The dots at the Gribov horizon are Gribov copies of the origin.
• Figure 4: Sketch of the potential for the torus. Shown are two vacuum valleys, related to each other by a gauge transformation $h_1$, with winding number $\nu(h_1)=1$. The induced one-loop effective potential, of height $3.210/L$, has degenerate minima related to each other by the anti-periodic gauge transformations $h_{(k)}$. The classical barrier, separating the two valleys, has a height $72.605/Lg^2$.
• Figure 5: The top figures show $L E(g)$ for the relevant (positive parity) representations. The dotted lines are without the two loop correction included. The dashed curve denotes the barrier height $L E_{sph}(g)$. The bottom figures show $z(g)=L m(g)=L(E(g)-E_0(g))$, and the dashed line is at $z=2\pi$.