Disordered System Approaches to the Yang-Mills Vacuum

Abstract

Two novel dual descriptions of d = 4 U(Nc) Yang-Mills theory (YM) are constructed and studied in this dissertation. We consider a network theory inspired by Budczies-Zirnbauer model (BZ), which will be abbreviated as BZN, and a continuum field theory, Dirac-Yang- Mills model (DYM). In either BZN or DYM, the dual theory is obtained by integrating out the original gluon degrees of freedom, which leads to a strongly-disordered system of some auxiliary matter fields. We examine the possibilities of applying a modern method, superbosonization (SuB) formula for disordered systems, in the investigation of the dual theories. In the first project, we reformulate BZN using Gaussian integral representation, and derive a master action for gluons and auxiliary matter fields, both of which live on the links of a lattice. The dual description, dual-BZN, is derived using Cayley parametrisation and a gauge-averaging trick, and the resulting dual action is a large−Nc series of color-neutral composite operators. However, using SuB for a direct replacement of these operators by some supermatrix-valued fields is not possible due to the rank-deficiency in the boson-boson sector of the supermatrix. The rank-deficiency is a result of the universality condition Nf ≥ Nc, which is necessary for BZN to flow to YM in its continuum-limit. In the second project, we study both sides of DYM: the induced Yang-Mills (IYM) and its dual (dual-IYM). The theory of dual-IYM describes a system of massive Dirac bosons and Dirac fermions constrained by a zero-current condition (ZC). A beautiful connection between gluon condensates in IYM and matter condensates in dual-IYM inspires a low-energy effective theory (dual-EFT). We discover the relevant dual symmetry groups and assemble a Lagrangian for dual-EFT in analogy with the chiral perturbation theory. Furthermore, we explore the ZC solution space and find out dual-IYM contains all Lorentz-types components, which suggests an energy- hierarchy scheme where dual-EFT is included as the low-energy sector of dual-IYM. Dual-IYM is color gauge-invariant. However, Witten’s bosonization method leads to a divergent effective action for the external field, and hence it is difficult to derive an action for some color-neutral dual-field. An attempt to directly transform the composite super-meson to the dual-field by SuB also fails because of rank-deficiency. In the absence of successful color-neutralisation, we proceed to explore some physical aspects of dual-BZN and dual-IYM. For dual-BZN, the masses and interaction strength of the composite operators are identified. We briefly examine the dual symmetry group and the saddle-point solutions, and point out a challenge to a semi-classical approximation due to the universality condition. For dual-IYM, we present two possible applications for YM mass gap and quark confinement. Furthermore, we explain a possibility of a large−Nc analysis, which might lead to a description of dual-IYM as a gravitational theory and/or a nonlinear sigma model