String Unification, Higher-Level Gauge Symmetries, and Exotic Hypercharge Normalizations

TL;DR

This work tackles the tension between the string unification scale and MSSM gauge-coupling unification by exploring how higher-level gauge symmetries and non-standard hypercharge normalizations can align the couplings at the string scale. It combines a renormalization-group analysis of Kač-Moody levels with explicit hypercharge embeddings in the MSSM, and a modular-invariance/charge-integrality framework (via Schellekens’ simple currents) to constrain allowed level configurations and hypercolor scenarios. The authors show that, within a broad class, $k_Y\ge 5/3$ is required for realistic minimal embeddings, but higher twists and higher non-Abelian levels can yield isolated embeddings with $k_Y<5/3$; they also identify specific hypercolor groups that can confine fractionally charged states if such values are realized. Collectively, the results provide a structured path to constructing string models that achieve string-scale unification with MSSM content, while highlighting the sensitivity to low-energy inputs and model-dependent threshold and non-MSSM effects.

Abstract

We explore the extent to which string theories with higher-level gauge symmetries and non-standard hypercharge normalizations can reconcile the discrepancy between the string unification scale and the GUT scale extrapolated from the Minimal Supersymmetric Standard Model (MSSM). We determine the phenomenologically allowed regions of (k_Y,k_2,k_3) parameter space, and investigate the proposal that there might exist string models with exotic hypercharge normalizations k_Y which are less than their usual value k_Y=5/3. For a broad class of heterotic string models (encompassing most realistic string models which have been constructed), we prove that k_Y >= 5/3. Beyond this class, however, we show that there exist consistent MSSM embeddings which lead to k_Y < 5/3. We also consider the constraints imposed on k_Y by demanding charge integrality of all unconfined string states, and show that only a limited set of hypercolor confining groups and corresponding values of k_Y are possible.

Figures (4)

• Figure 1: Dependence of unification scale $M_{\rm string}$ on the chosen value of $r\equiv k_Y/k_2$. The different curves correspond to different values of $\sin^2\theta_W(M_Z)$, with the lower curve arising for greater values.
• Figure 2: Values of $(r\equiv k_Y/k_2,r'\equiv k_3/k_2)$ yielding the experimentally observed low-energy couplings. The different curves correspond to different values of the couplings, with the higher curves arising for smaller values of $\sin^2\theta_W(M_Z)$ and $\alpha_{\rm strong}(M_Z)$, and the lower curves arising for greater values. Different points on any single curve correspond to different unification scales.
• Figure 3: Solutions to the transcendental equation (\ref{['transcendental']}) for different values of $k_Y+k_2$, with the two-loop, Yukawa, and scheme-conversion corrections included (upper curve), and omitted (lower curve).
• Figure 4: Dependence of $k_2$ on $r\equiv k_Y/k_2$. The absolute size of the Kač-Moody levels is set by the two-loop-corrected self-consistency constraint (\ref{['transcendental']}), in conjunction with the constraints from the low-energy value of $\sin^2\theta_W(M_Z)$. The different curves correspond to different values of $\sin^2\theta_W(M_Z)$, with the lower/left curves arising for greater values.