One loop renormalization of 5D gauge-Yukawa theories

TL;DR

This work analyzes five-dimensional gauge–Yukawa theories compactified on the orbifold $ ext{S}^1/ ext{Z}_2$ to assess their renormalizability and high‑scale behavior. By computing bulk and localized divergences at one loop, the authors show that the divergent structure matches the four‑dimensional case, with power‑law enhancements that can be absorbed into a finite set of bulk and boundary counterterms, and they derive the full localized boundary Lagrangian $oxed{ oxed{ oxed{ ext{L}_4 }}$ in the magic gauge. They establish RGEs for bulk couplings and identify conditions for UV fixed points of gauge, Yukawa, and scalar quartic interactions, arguing that these fixed points can render the theory predictive up to scales well beyond the compactification scale. Localization on the boundary yields a controlled set of 4D‑renormalizable operators, and localized Yukawa interactions can flow to UV fixed points under suitable conditions. Collectively, these results offer a concrete step toward an all‑order proof of renormalizability for gauge–Yukawa theories in five dimensions and provide a framework for constructing UV‑complete 5D models, including aGUTs, while outlining the need for higher‑loop analyses.

Abstract

The common lore dictates that extra dimensional theories loose predictive power at energies just above the compatification scale, due to the power-law running of bulk coupling. We show that five-dimensional gauge-Yukawa theories can be valid up to arbitrarily high scales, provided: 1) A finite number of terms are required to absorb power-law divergences; 2) All power-law running couplings flow to UV fixed points. By explicitly computing bulk and localized divergences for a gauge-Yukawa theory on $\mathcal{S}^1/\mathbb{Z}_2$, we prove the one-loop renormalization properties of Lagrangians containing only interactions that would be renormalizable in four dimensions. The existence of UV fixed points imposes further constraints on the content of the model. Our results provide a consistency check for the high-energy behavior of any 5D theory, and provide a discrimination between UV consistent models and those that can describe only a handful of Kaluza-Klein modes. Hence, we offer the first concrete step towards an all-order proof of `renormalizability' for gauge-Yukawa theories in five dimensions.

Figures (9)

• Figure 1: One loop diagram contributing to the gauge boson self-energy.
• Figure 2: One loop diagrams contributing to the gauge bosons self energy in a pure Yang-Mills theory.
• Figure 3: One loop diagrams contributing to the three gauge boson vertex in a pure Yang Mills theory.
• Figure 4: One loop diagrams contributing to the fermion self energy.
• Figure 5: One loop diagrams contributing to the fermion-gauge boson vertex correction.