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The dark dimension, proton decay, and the length of the M-theory interval

Mario Reig, Ignacio Ruiz

TL;DR

This work asks whether a micron-scale dark dimension can arise in the strong-coupling limit of heterotic M-theory. It combines full 11d warped flux-compactification analyses with an effective 5d description to bound the M-theory interval length from proton-decay constraints, using parameters like the instanton number $Q$ and warp factors. The main finding is a robust upper bound $R \lesssim \mathcal{O}(10^{-28})$ m (with possible tightening under warping), which rules out the dark-dimension realization in this class of GUT-like models. Consequently, if short-distance deviations from Newtonian gravity are observed, viable string completions would likely have to be non-GUT-like (e.g., solitonic GUTs, F-theory GUTs, or non-unified brane setups). Hyper-Kamiokande prospects could further strengthen the bound, highlighting a strong tension between dark dimensions and proton-decay constraints in these frameworks.

Abstract

The existence of a large extra dimension in which only gravity propagates would have spectacular consequences for cosmology and laboratory experiments. In the strong coupling limit of the $E_8\times E_8$ heterotic string theory, the gauge and matter fields live at the end of the eleventh dimension, which becomes a natural candidate for a micron-size \textit{dark dimension}. In this work, however, we show that the length of the M-theory interval is severely constrained by proton decay searches. Our results indicate that in such constructions the size of the eleventh dimension is $R\lesssim \mathcal{O}(10^{-28})$ meters.

The dark dimension, proton decay, and the length of the M-theory interval

TL;DR

This work asks whether a micron-scale dark dimension can arise in the strong-coupling limit of heterotic M-theory. It combines full 11d warped flux-compactification analyses with an effective 5d description to bound the M-theory interval length from proton-decay constraints, using parameters like the instanton number and warp factors. The main finding is a robust upper bound m (with possible tightening under warping), which rules out the dark-dimension realization in this class of GUT-like models. Consequently, if short-distance deviations from Newtonian gravity are observed, viable string completions would likely have to be non-GUT-like (e.g., solitonic GUTs, F-theory GUTs, or non-unified brane setups). Hyper-Kamiokande prospects could further strengthen the bound, highlighting a strong tension between dark dimensions and proton-decay constraints in these frameworks.

Abstract

The existence of a large extra dimension in which only gravity propagates would have spectacular consequences for cosmology and laboratory experiments. In the strong coupling limit of the heterotic string theory, the gauge and matter fields live at the end of the eleventh dimension, which becomes a natural candidate for a micron-size \textit{dark dimension}. In this work, however, we show that the length of the M-theory interval is severely constrained by proton decay searches. Our results indicate that in such constructions the size of the eleventh dimension is meters.

Paper Structure

This paper contains 7 sections, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proton decay in heterotic M-theory
  3. Estimates in the full 11d theory
  4. The flat case: $Q=0$
  5. The warped case: $Q\neq 0$
  6. Estimates in the effective 5d theory
  7. Conclusions

Figures (1)

  • Figure 1: Allowed region for the length of the $S^1/\mathbb{Z}_2$ interval for $Q>0$, as a function of the dimensionless $\ell_{11} Q$ instanton number controlling the warping of the interval. The upper bound comes from imposing that the mass of the KK modes of the GUT gauge bosons is sufficiently heavy to avoid rapid proton decay, that is imposing $g_{\rm GUT}M_{\rm GUT}\lesssim M_{\rm KK}\lesssim M_{\rm Pl, 11}$ (see text for details). Any region above the solid line is excluded by the current limit from Super-K Super-Kamiokande:2020wjk. For $\ell_{11} Q\to 0^+$ the bound results in the unwarped $Q=0$ case, \ref{['eq.bound q0']}.