Abstract
We classify the weakly interacting fixed points of general gauge theories coupled to matter and explain how the competition between gauge and matter fluctuations gives rise to a rich spectrum of high- and low-energy fixed points. The pivotal role played by Yukawa couplings is emphasised. Necessary and sufficient conditions for asymptotic safety of gauge theories are also derived, in conjunction with strict no go theorems. Implications for phase diagrams of gauge theories and physics beyond the Standard Model are indicated.
Highlights
Fixed points of the renormalisation group play an important role in quantum field theory and particle physics [1,2]
It was discovered that gauge theories can develop interacting ultraviolet (UV) fixed points [8], a scenario known as asymptotic safety
We hope that insights into the inner working of asymptotic safety at weak coupling will offer clues for mechanisms of asymptotic safety at strong coupling [11,12]
Summary
For any Lie algebra, the highest weight of irreducible representations with the smallest quadratic Casimir operator must be one of the fundamental weights k (with k ∈ {1, . . . , n}), whose components are defined as. Inserting (7) into (6), and denoting by G the weight metric of the gauge group G, we find the quadratic Casimir operator in terms of the fixed index k as It remains to identify the minima of (9) with respect to k for the four classical and the five exceptional Lie algebras separately, following the Cartan classification, starting with the rank-n classical Lie algebras An, Bn, Cn and Dn [19]. This implies that global minima may be achieved for integer values of k within the interval (1, n) With this in mind, and after evaluating all possible cases, the final result for the smallest quadratic Casimir operator for the classical Lie algebras is found to be nn+2 min. This degeneracy is due to the fact that the Dynkin diagram for D4 possesses a three-fold symmetry, and there
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