Can Theta/N Dependence for Gluodynamics be Compatible with 2 pi Periodicity in Theta ?

TL;DR

This work resolves the tension between $\theta/N$-type dependence and the expected $2\pi$ periodicity in $\theta$ by showing that finite-volume, multi-sector vacua restore the correct periodic structure when the thermodynamic limit is taken carefully. Through explicit analyses of the $N_f$-flavor Schwinger model and a four-dimensional gluodynamics effective Lagrangian, the authors derive a multi-branch vacuum spectrum with energies $E_{l}(\theta) = -E\cos(-\frac{q}{p}\theta + \frac{2\pi}{p}l)$, which under $\theta \to \theta + 2\pi$ cycles the branches and preserves overall periodicity. They demonstrate that cusp singularities and vacuum doubling occur at certain $\theta$ values (Dashen phenomenon) and that a proper summation over sectors yields the correct $2\pi$ periodicity for any rational $\xi$ (and hence for physically relevant $N_c$), with the small-$\theta$ behavior matching a $\theta/N$ pattern. The framework, which relies on an integer-valued Lagrange multiplier enforcing topological charge quantization and an improved, single-valued effective potential, is stated to apply to arbitrary gauge groups and may have implications for domain walls and finite-volume phenomena in gauge theories.

Abstract

In a number of field theoretical models the vacuum angle θenters physics in the combination θ/N, where N stands generically for the number of colors or flavors, in an apparent contradiction with the expected 2 πperiodicity in θ. We argue that a resolution of this puzzle is related to the existence of a number of different θdependent sectors in a finite volume formulation, which can not be seen in the naive thermodynamic limit V -> \infty. It is shown that, when the limit V -> \infty is properly defined, physics is always 2 πperiodic in θfor any integer, and even rational, values of N, with vacuum doubling at certain values of θ. We demonstrate this phenomenon in both the multi-flavor Schwinger model with the bosonization technique, and four-dimensional gluodynamics with the effective Lagrangian method. The proposed mechanism works for an arbitrary gauge group.