Abstract
The Lagrangian for a non-abelian gauge theory with an $SU(N_{\! c})$ symmetry and a linear covariant gauge fixing is constructed in eight dimensions. The renormalization group functions are computed at one loop with the special cases of $N_{\! c}$ $=$ $2$ and $3$ treated separately. By computing the critical exponents derived from these in the large $N_{\! f}$ expansion at the Wilson-Fisher fixed point it is shown that the Lagrangian is in the same universality class as the two dimensional non-abelian Thirring model and Quantum Chromodynamics (QCD). As the eight dimensional Lagrangian contains new quartic gluon operators not present in four dimensional QCD, we compute in parallel the mixing matrix of four dimensional dimension $8$ operators in pure Yang-Mills theory.
Highlights
Non-Abelian gauge theories are established as the core quantum field theories which govern the particles of nature through the Standard Model
One sector, which is known as quantum chromodynamics (QCD), describes the strong force between fundamental quarks and gluons which leads to the binding of these quanta into the mesons and hadrons seen in Nature
At high energy quarks and gluons become effectively free particles due to the property of asymptotic freedom, [1,2]. While this attribute is essential to developing a field theoretic formalism which allows us to extract meaningful information from experimental data, it has an implicit sense that at lower energies quarks and gluons can never be treated as distinct particles in the same spirit as a free electron in quantum electrodynamics (QED) which is an Abelian gauge theory
Summary
This involved a more intricate Lagrangian but the connection of the two loop renormalization group functions with the universal d-dimensional large Nf critical exponents was verified This reinforced the remarkable connection with the non-Abelian Thirring model in that the six dimensional theory has quintic and sextic gluon selfinteractions in addition to cubic and quartic structures which are the only ones present in four dimensions. Given that dimension 8 operators are of interest in four dimensional effective field theories of QCD having renormalization group function data in the eight dimensional non-Abelian gauge theory for SUðNcÞ, where Nc is the number of colors, is an additional motivation for future studies.
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