Hidden symmetries induced by a canonical transformation and gauge structure of compactified Yang-Mills theories

TL;DR

This paper reveals hidden symmetries in various Yang-Mills theories through canonical transformations, showing that gauge invariances like SGTs and NSGTs are more general and not limited to compactified models, with implications for understanding gauge structure and symmetry breaking.

Contribution

The work introduces a notion of hidden symmetry based on canonical transformations, applying it to different gauge systems to unify their gauge invariances and analyze their classical equivalence.

Findings

Hidden symmetries exist in multiple gauge theories beyond compactified scenarios.

Canonical transformations map objects with well-defined gauge transformation laws to those under a subgroup.

In spontaneous symmetry breaking, SGTs correspond to the unbroken group, while NSGTs relate to broken generators.

Abstract

Compactified Yang-Mills theories with one universal extra dimension were found [arXiv:1008.4638] to exhibit two types of gauge invariances: the standard gauge transformations (SGTs) and the nonstandard gauge transformations (NSGTs). In the present work we show that these transformations are not exclusive to compactified scenarios. Introducing a notion of hidden symmetry, based on the fundamental concept of canonical transformation, we analyse three different gauge systems, each of which is mapped to a certain effective theory that is invariant under the so-called SGTs and NSGTs. The systems under discussion are: (i) four dimensional pure $SU(3)$ Yang-Mills theory, (ii) four dimensional $SU(3)$ Yang-Mills with spontaneous symmetry breaking, and (iii) pure Yang-Mills theory with one universal compact extra dimension. The canonical transformation, that induces the notion of hidden symmetry, maps objects with well defined transformation laws under a gauge group $G$ to well defined objects under a non-trivial subgroup $H \subset G$. In the case where spontaneous symmetry breaking is present, the set of SGTs corresponds to the group into which the original gauge group is broken into, whereas the NSGTs are associated to the broken generators and can be used to define the unitary gauge. For the system (iii), the SGTs coincide with the gauge group $ SU(N,\cal{M}^{4}) $, whereas the NSGTs do not form a group; in this system the 'fundamental' theory and the effective one are shown to be classically equivalent.