Grand Unification at Intermediate Mass Scales through Extra Dimensions

TL;DR

This work proposes that adding extra spacetime dimensions lowers the grand unification scale by inducing power-law (indeed exponential-like) running of gauge and Yukawa couplings via Kaluza–Klein towers. The framework preserves gauge coupling unification at an intermediate scale $M'_{ m GUT}$, while suppressing proton decay through higher-dimensional selection rules and orbifold fixed-point dynamics; it also offers mechanisms to address the fermion mass hierarchy, including brane- and string-theory–based constructions. The approach extends to non-supersymmetric theories and outlines string/D-brane embeddings, with collider and cosmological implications that could render GUT-scale physics accessible at present or future experiments. Together, these results present a coherent pathway to integrate grand unification with large-radius extra dimensions and string-theoretic ideas, transforming the phenomenology and testability of high-scale physics.

Abstract

One of the drawbacks of conventional grand unification scenarios has been that the unification scale is too high to permit direct exploration. In this paper, we show that the unification scale can be significantly lowered (perhaps even to the TeV scale) through the appearance of extra spacetime dimensions. Such extra dimensions are a natural consequence of string theories with large-radius compactifications. We show that extra spacetime dimensions naturally lead to gauge coupling unification at intermediate mass scales, and moreover may provide a natural mechanism for explaining the fermion mass hierarchy by permitting the fermion masses to evolve with a power-law dependence on the mass scale. We also show that proton-decay constraints may be satisfied in our scenario due to the higher-dimensional cancellation of proton-decay amplitudes to all orders in perturbation theory. Finally, we extend these results by considering theories without supersymmetry; experimental collider signatures; and embeddings into string theory. The latter also enables us to develop several novel methods of explaining the fermion mass hierarchy via $D$-branes. Our results therefore suggest a new approach towards understanding the physics of grand unification as well as the phenomenology of large-radius string compactifications.

Figures (16)

• Figure 1: Unification of gauge couplings in the presence of extra spacetime dimensions. We consider four representative cases: $\mu_0 = 10^{5}$ GeV (top left), $\mu_0 = 10^{8}$ GeV (top right), $\mu_0 = 10^{11}$ GeV (bottom left), and $\mu_0 = 10^{15}$ GeV (bottom right). In each case we have taken $\delta=1$ and $\eta=0$.
• Figure 2: Unification of gauge couplings in the presence of extra spacetime dimensions. Here we fix $\mu_0= 10^{12}$ GeV, $\delta=1$, and we vary $\eta$. For this value of $\mu_0$, we see that the unification remains perturbative for all $\eta$.
• Figure 3: The ratio of the unification scale $M'_{\rm GUT}$ to the scale $\mu_0$ at which $\delta$ extra spacetime dimensions appear, as a function of $\mu_0$. This ratio describes the size of the energy range over which our effective higher-dimensional field theory is meant to apply. This curve is independent of the value of $\eta$. The limit of the usual four-dimensional MSSM is indicated with a dot.
• Figure 4: The unified coupling $(\alpha'_{\rm GUT})^{-1}$ as a function of the unification scale $M'_{\rm GUT}$, for $\eta=0,1,2,3$. This curve is independent of the number of extra spacetime dimensions, and the limit of the usual four-dimensional MSSM is indicated with a dot. Note that the unified coupling is independent of the unification scale for $\eta=1$, while it is weaker than the MSSM value for $\eta=0$ and stronger if $\eta>1$. It is also clear from this figure that there are natural lower bounds on the possible radii of extra dimensions in the $\eta=2,3$ cases if we wish the couplings to unify before diverging. These bounds correspond to $M'_{\rm GUT}\hbox{$\;\stackrel{>}{\sim}\;$} 100$ TeV and $M'_{\rm GUT} \hbox{$\;\stackrel{>}{\sim}\;$} 3\times 10^{10}$ GeV for $\eta=2,3$ respectively.
• Figure 5: Typical diagrams that can mediate proton decay. Here the external lines correspond to the MSSM (s)quarks and leptons, while the internal $\Psi$ fields correspond to $X$-bosons (as in the diagram on the left) and Higgsino fields (as in the diagram on the right). We choose the wavefunctions of the $\Psi$ fields to be odd $(-)$ functions of the extra spacetime coordinates. With this choice, all vertices between the $\Psi$ fields and the chiral MSSM fermions vanish identically in the "minimal" $\eta=0$ scenario, and all possible proton-decay diagrams vanish to all orders in perturbation theory.