Microscopic Structure of Magnetic Bions

TL;DR

The study analyzes confinement in SU(2) gauge theory with massless adjoint fermions on \\mathbb{R}^{3} \\times \\mathbb{S}^{1} by identifying magnetic bions as the dominant nonperturbative objects at small L. Using a traditional instanton framework, the vacuum is reformulated as a bion-anti-bion plasma, enabling a two-loop running analysis of the dual-photon mass M(L) and revealing a special role for n_f=4, where the leading L-dependence cancels and the mass gap can grow with L. The work derives the scales and interactions of the bion constituents, including the monopole-instanton actions, fermion-induced attractions, and the determinants in the bion partition function, and connects these to confinement via a Polyakov-like mechanism in 4D. The results constrain how the mass gap behaves as L grows, insinuating a nuanced picture of the conformal window and suggesting that abelian confinement persists at finite L with an exponentially small string tension under certain infrared fixed-point scenarios. Overall, the analysis clarifies how fermion content shapes topological excitations and confinement in compactified gauge theories, offering analytic insights complementary to lattice studies.

Abstract

Magnetic bions---stable bound states of monopoles and twisted ("Kaluza-Klein") monopoles, carrying two units of magnetic charge---have been shown to be the leading cause of confinement and mass gap in four-dimensional gauge theories with massless adjoint fermions compactified on R**{1,2} x S**1, at least at small S**1. In this paper, we study in detail the bion mechanism and the scales involved for an SU(2) gauge group, using traditional QCD instanton methods. We represent the vacuum functional as the partition function of a bion-anti-bion plasma and obtain the next-to-leading dependence of the mass gap on the S**1 size L at fixed strong-coupling scale Lambda. We find that, at small Lambda x L, the mass gap is an increasing function of L for theories with four massless adjoint Weyl fermions, a case left undetermined by the previous leading-order analysis, and comment on the approach to R**4.

Figures (4)

• Figure 1: The small-$L$, weak-coupling $g(L) \ll 1$, hierarchy of scales in the bion plasma: the constituent monopole core size is $L$, the "Higgs cloud" spreads over $L/g$, the bion size is $L/g^2$, and the typical distance between bions is $L e^{8 \pi^2 \over 3g^2}$.
• Figure 2: Behavior of dual photon mass $M$ and W-boson mass ($\sim 1/L$) with $L$ for $n_f =5$. The behavior to the right of the dotted line at $\Lambda L \sim 1$, is based on the assumed existence of a weakly coupled infrared fixed point. This theory is thus expected to exhibit abelian confinement at any finite $L$ with an exponentially small string tension vanishing in the ${\mathbb R}^4$ limit.
• Figure 3: Behavior of dual photon mass $M$ and W-mass with $L$ for $n_f =4$. The behavior to the right of the dotted line at $\Lambda L \sim 1$ is based on lattice evidence for the existence of a weakly coupled infrared fixed point. If that is the case, the regime of semiclassical abelian confinement is expected to persist at any finite $L$, with an exponentially small string tension vanishing as $L \rightarrow \infty$.
• Figure 4: Behavior of dual photon mass $M$ and W-mass with $L$ for $n_f =2,3$. The behavior to the right of the dotted line at $\Lambda L \sim 1$ assumes that a regime of nonabelian confinement sets in. The precise behavior at large $L$ is not known. The convergence to a common value of order $\Lambda$ is purely conjectural and is drawn similar to the behavior of quantities, analogous to our $m_W$ and $M$, studied in pure Yang-Mills theory on $T^3$ of size $L$ (see GarciaPerez:1993jw, where, as $L$ is changed from $L \Lambda \ll 1$, analytic methods were used, while for $L\Lambda \gg 1$ numerical studies were needed).