Reconciling Grand Unification with Strings by Anisotropic Compactifications

TL;DR

Reconciling Grand Unification with Strings by Anisotropic Compactifications analyzes gauge coupling unification in heterotic string models with anisotropic orbifolds. It uses an effective 5D SU(6) orbifold GUT picture and an EFT RG running framework that includes a light exotic scale $M_{ ext{ex}}$, a compactification scale $M_{ ext{C}}$, and a string scale $M_{ ext{S}}$, demonstrating that unification generically requires $M_{ ext{ex}} \ll M_{ ext{C}}$ and a sizable $M_{ ext{S}}$. A comprehensive scan yields hundreds of EFT solutions (e.g., 252 versions for Model 2; 82 shared with Model 1), with about 48 surviving current proton decay bounds and a subset potentially testable by next-generation experiments; representative cases yield $M_{ ext{S}} \sim 5.5\times 10^{17}$ GeV, $M_{ ext{C}} \sim 8.2\times 10^{15}$ GeV, and $M_{ ext{ex}} \sim 8.2\times 10^{13}$ GeV. In Model 1A a consistent $F=0$ vacuum exists under tuned singlet VEVs, providing a concrete unification scenario, while Model 2 lacks a simple string vacuum with $F=0$ and requires further fine-tuning. The work demonstrates that string-derived anisotropic orbifolds can realize gauge coupling unification and delineates experimental implications for proton decay and KK-scale thresholds.

Abstract

We analyze gauge coupling unification in the context of heterotic strings on anisotropic orbifolds. This construction is very much analogous to effective 5 dimensional orbifold GUT field theories. Our analysis assumes three fundamental scales, the string scale, $\mstring$, a compactification scale, $\mc$, and a mass scale for some of the vector-like exotics, $\mex$; the other exotics are assumed to get mass at $\mstring$. In the particular models analyzed, we show that gauge coupling unification is not possible with $\mex = \mc$ and in fact we require $\mex \ll \mc \sim 3 \times 10^{16}$ GeV. We find that about 10% of the parameter space has a proton lifetime (from dimension 6 gauge exchange) $10^{33} {\rm yr} \lesssimτ(p\to π^0e^+) \lesssim 10^{36} {\rm yr}$. The other 80% of the parameter space gives proton lifetimes below Super-K bounds. The next generation of proton decay experiments should be sensitive to the remaining parameter space.

Figures (5)

• Figure 1: The geometry of the compact dimensions.
• Figure 2: Setup of the 5d orbifold GUT, where the 5th dimension ($e_5$) is large compared to the other compact dimensions.
• Figure 3: Histogram of solutions with $M_{\textsc{s}} > {M_{\textsc{c}}} \gtrsim {M_{\textsc{ex}}}$, showing the models which are excluded by Super-K bounds (darker green) and those which are potentially accessible in a next generation proton decay experiment (lighter green). Of 252 total solutions, 48 are not experimentally ruled out by the current experimental bound, and most of the remaining parameter space can be eliminated in the next generation of proposed proton decay searches.
• Figure 4: An example of the type of gauge coupling evolution we see in these models, versus the typical behavior in the MSSM. The "tail" is due to the power law running of the couplings when towers of Kaluza-Klein modes are involved. Unification in this model occurs at $M_{\textsc{s}} \simeq 5.5\times 10^{17} {\rm ~GeV~}$, with a compactification scale of ${M_{\textsc{c}}} \simeq 8.2 \times 10^{15} {\rm ~GeV~}$, and an exotic mass scale of ${M_{\textsc{ex}}} \simeq 8.2 \times 10^{13} {\rm ~GeV~}$.
• Figure 5: Here we show the correlation between the hierarchies in the problem. Quite generally, a small value of $\alpha_{\textsc{string}}^{-1}$ requires a large hierarchy between the compactification scale and the exotic scale. Again we show the excluded (darker green) and possibly testable (lighter green) models. The exact relationship between the ratio of $M_{\textsc{s}}/{M_{\textsc{c}}}$ and the proton lifetime is given in Appendix \ref{['sec:proton-decay']}. In particular, note the "nice" models (black diamonds) in the large red box, characterized by moderate hierarchies between all scales. These models are collected in Table \ref{['tab:interesting_models']}. Finally, note the one point in the small red box---this model is described in Section \ref{['sec:top_down']}.