Beta Functions of Orbifold Theories and the Hierarchy Problem

TL;DR

The paper assesses whether non-supersymmetric orbifold projections of ${\cal N}=4$ SU($N$) gauge theory can yield conformal theories at finite $N$ that address the hierarchy problem. By computing the one-loop beta functions for Yukawa and quartic scalar couplings, it finds zeros near the large-$N$ SUSY point but none that guarantee exact fixed points when all operators are considered, and it shows that quadratic divergences do not cancel at subleading order in $1/N$. The resulting scalar mass is $m^2_{\phi}=\frac{3g^2}{N}\frac{M^2}{16\pi^2}$ at the putative conformal point, implying $N$ must be enormous (\(\sim 10^{28}\)) to avoid fine-tuning. Additionally, RG flows induce new operators and the two-loop gauge beta function is positive, suggesting no perturbative UV fixed point for these models. Overall, the work casts significant doubt on the naturalness of embedding the Standard Model in finite-$N$ orbifold theories without further mechanisms or a broader operator analysis.

Abstract

We examine a class of gauge theories obtained by projecting out certain fields from an N=4 supersymmetric SU(N) gauge theory. These theories are non-supersymmetric and in the large N limit are known to be conformal. Recently it was proposed that the hierarchy problem could be solved by embedding the standard model in a theory of this kind with finite N. In order to check this claim one must find the conformal points of the theory. To do this we calculate the one-loop beta functions for the Yukawa and quartic scalar couplings. We find that with the beta functions set to zero the one-loop quadratic divergences are not canceled at sub-leading order in N; thus the hierarchy between the weak scale and the Planck scale is not stabilized unless N is of the order 10^28 or larger. We also find that at sub-leading orders in N renormalization induces new interactions, which were not present in the original Lagrangian.

Figures (6)

• Figure 1: The group theory Feynman diagrams for the Yukawa couplings and the quartic scalar couplings of the ${\cal N}=4$ theory.
• Figure 2: Contributions to the $\beta$ function of the quartic scalar couplings of the fields $\phi_1^a \phi^{\dagger b}_1 \phi_2^c \phi^{\dagger d}_2$ in the ${\cal N}=4$ theory. The ordering of the fields in the above diagrams is clockwise, with $\phi_1^a$ in the upper left corner. The meaning of the above group theory diagrams is explained in Fig. \ref{['fig:rules']}.
• Figure 3: Contributions to the $\beta$ function of the quartic scalar couplings of the fields $\phi_1^a \phi^{\dagger b}_1 \phi_1^c \phi^{\dagger d}_1$ in the ${\cal N}=4$ theory. The ordering of the fields in the above diagrams is clockwise, with $\phi_1^a$ in the upper left corner. The meaning of the above group theory diagrams is explained in Fig. \ref{['fig:rules']}.
• Figure 4: The diagrammatic representation of the $SU(N)$ group theory identities needed to show that the ${\cal N}=4$$\beta$ functions of the quartic couplings do indeed vanish. The first line gives the decomposition of the "gluon box diagram" in terms of a complete set of tensors, the second line is the Jacobi identity, while the third line is an identity relating different combinations of the $d$ and $f$ tensors. A single unconnected line corresponds to $\delta^a_b$. These results are taken from Cvitanovic.
• Figure 5: The large $N$ rules for adjoints.