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

This paper demonstrates that proton decay constraints severely limit the size of the M-theory interval in $E_8\times E_8$ heterotic string theory, ruling out the possibility of a micron-scale dark dimension and restricting its length to approximately $10^{-28}$ meters or smaller.

The Gist

Imagine our universe as a giant, multi-layered cake. For a long time, physicists have wondered if there are hidden "layers" (extra dimensions) that we can't see because they are curled up so tightly they are invisible to us.

Recently, a popular idea called the "Dark Dimension" suggested that there might be one extra layer that is surprisingly large—about the size of a human hair (a micron). If this were true, it would explain some of the biggest mysteries in physics, like why gravity is so weak compared to other forces and why the universe is expanding.

However, a new paper by Mario Reig and Ignacio Ruiz argues that this specific "hair-sized" extra dimension cannot exist in the way theorists hoped, at least not if the universe works according to the rules of a specific theory called M-theory.

Here is the breakdown of their argument using simple analogies:

1. The Setup: The "Eleventh Dimension"

In this version of M-theory, our universe is like a sandwich.

• The Bread: Two giant, invisible walls (called "branes") exist at the ends of a hidden space.
• The Filling: The space between these walls is the "eleventh dimension."
• The Rules: All the matter we know (electrons, protons, light) lives on one of the walls. Gravity, however, is the only thing that can travel through the space between the walls.

The "Dark Dimension" idea proposed that this space between the walls could be wide (micron-sized). The authors of this paper wanted to test if this wide space is actually possible.

2. The Problem: The "Proton Leak"

To understand why the space can't be wide, we have to look at the proton.

• A proton is a tiny, stable particle inside every atom. It is the "brick" that holds matter together.
• In many theories of the universe, protons are supposed to be eternal. But in theories that try to unify all forces (Grand Unified Theories), protons can sometimes decay (fall apart) into lighter particles.
• The Catch: The wider the hidden space (the "filling" of the sandwich), the easier it is for the "rules" of the universe to break down, allowing protons to decay much faster.

Think of the hidden dimension as a leaky pipe.

• If the pipe is tiny, the water (protons) stays inside safely.
• If the pipe is huge (micron-sized), the water leaks out so fast that the pipe empties almost instantly.

3. The Evidence: The "Super-Kamiokande" Detector

Scientists have built massive detectors deep underground (like the Super-Kamiokande in Japan) to watch for protons falling apart.

• They have been watching for decades.
• The Result: They haven't seen a single proton decay. This tells us that protons are incredibly stable and must live for at least $10^{34}$ years (a number with 34 zeros).

4. The Verdict: The Dimension Must Be Tiny

Reig and Ruiz did the math to see how wide the hidden space could be before protons would start decaying too fast.

• The Calculation: They found that if the hidden space were even as big as a micron (the "Dark Dimension" size), the protons in our atoms would have decayed billions of years ago. We wouldn't be here.
• The Limit: To keep protons stable, the hidden space must be incredibly small.
  • The paper calculates the maximum size is about $10^{-28}$ meters.
  • To visualize this: If a proton were the size of the Earth, this hidden dimension would be smaller than a single atom. It is effectively zero for all practical purposes.

5. What About "Warping"?

The authors also checked if the space could be "warped" (stretched or squeezed in different places, like a funnel). They found that even with warping, the space still has to be microscopic to prevent protons from decaying.

The Bottom Line

The paper concludes that while the idea of a "Dark Dimension" is fascinating, it is ruled out by the simple fact that protons are still around.

If we ever discover a large extra dimension in the future, it means our current understanding of how the universe is built (specifically the M-theory version with these specific walls) is wrong. We would need a completely different kind of universe where protons don't leak, or where the "bricks" of matter aren't stuck to the walls in the way we currently think.

In short: The universe is stable, so the hidden room between the walls must be a closet, not a hallway.

Technical Summary

Title: The dark dimension, proton decay, and the length of the M-theory interval
Authors: Mario Reig and Ignacio Ruiz
Affiliation: Theoretical Physics Department, CERN

Problem Statement

Recent interest in the "dark dimension" scenario posits the existence of a large, purely gravitational extra dimension (micron-scale, $r \sim 50\,\mu\text{m}$) motivated by Swampland conjectures and the observed value of the cosmological constant. In the strong coupling limit of $E_8 \times E_8$ heterotic string theory (Hořava-Witten M-theory), the eleventh dimension naturally accommodates such a setup, with gauge and matter fields localized on end-of-the-world branes.

However, a large extra dimension implies a low quantum gravity (QG) scale, $\Lambda_{\text{QG}} \sim 10^{9-10}$ GeV. This creates a tension with Grand Unified Theories (GUTs). If the Standard Model (SM) is embedded in a GUT framework (e.g., via $E_8$ branes), the mass of GUT gauge bosons ($M_X$) is bounded by the QG scale. A low QG scale leads to a low $M_X$, which in turn predicts rapid proton decay via dimension-6 operators mediated by these gauge bosons. Current experimental limits on proton lifetime ($\tau_p \gtrsim 2.4 \times 10^{34}$ years) require a GUT scale $M_{\text{GUT}} \gtrsim 2 \times 10^{16}$ GeV. The paper investigates whether the Hořava-Witten M-theory interval can be large enough to serve as a dark dimension while remaining consistent with these proton decay constraints.

Methodology

The authors perform a quantitative analysis of the relationship between the size of the eleventh dimension ($R$), the 11-dimensional Planck mass ($M_{\text{Pl},11}$), and the GUT scale within the framework of heterotic M-theory. The analysis is conducted in three distinct regimes:

1. Full 11d Theory (Warped and Flat):

   • The authors consider a flux compactification on a warped product $X \times S^1/\mathbb{Z}_2$, where $X$ is a Calabi-Yau (CY) threefold.
   • They derive the relation between the 4d Planck mass ($M_{\text{Pl},4}$), the 11d Planck mass, and the physical length of the interval $R$, accounting for warping induced by the instanton number $Q$ (related to the difference in topological charges on the two boundaries).
   • They impose the consistency condition that the Kaluza-Klein (KK) mass scale of GUT gauge bosons must be bounded by the 11d Planck scale ($M_{\text{KK}} \lesssim M_{\text{Pl},11}$) and that the GUT scale must satisfy $M_{\text{GUT}} \lesssim M_{\text{KK}}$ to avoid excessive proton decay.

2. Effective 5d Theory:

   • The authors analyze the system using the domain wall solution in the effective 5d description, valid at energies below $M_{\text{Pl},11}$.
   • They derive bounds on the interval size $R$ by reducing the 5d Einstein-Hilbert action to 4d and applying the same proton decay constraints ($M_{\text{Pl},5} > M_{\text{GUT}}$).

3. M5-branes in the Bulk:

   • The authors extend the analysis to a hypothetical scenario where the SM GUT is realized not on the boundary $E_8$ branes, but on a stack of $n$ M5-branes located at an arbitrary point $x_{\text{M5}}^{11}$ within the interval.
   • They account for the modified warping function and the specific gauge coupling structure arising from the M5-brane worldvolume theory (a 6d $\mathcal{N}=(2,0)$ SCFT reduced to 4d).

Key Contributions and Results

1. Constraints in the Flat Case ($Q=0$):
In the absence of warping (topologically identical boundaries), the relation between the Planck scales and the interval length is derived as $M_{\text{Pl},4}^2 \propto M_{\text{Pl},11}^2 (R/\ell_{11})$. To satisfy the proton decay bound ($M_{\text{GUT}} \gtrsim 2 \times 10^{16}$ GeV), the 11d Planck mass must be sufficiently high. This forces the interval length to be extremely small:
$$R \lesssim 1.4 \times 10^{-12} \, \text{GeV}^{-1} \approx 2.7 \times 10^{-28} \, \text{m}.$$
This result rules out the realization of a micron-size dark dimension in the flat limit.

2. Constraints in the Warped Case ($Q \neq 0$):

• $Q < 0$ (Standard Hořava-Witten): The volume of the CY shrinks along the interval. Classical consistency requires the volume not to vanish, imposing an upper bound on $R$ that is even stricter than the flat case: $R \lesssim 10^{-33}$ m.
• $Q > 0$ (SM on small boundary): The CY volume grows along the interval. While this allows for a larger $R$ classically, the proton decay constraint still imposes a severe upper bound. The bound decreases monotonically as the dimensionless instanton number $\ell_{11}Q$ increases. Even in the limit $\ell_{11}Q \to 0$, the bound recovers the flat case result ($R \lesssim 2.7 \times 10^{-28}$ m).

3. Constraints with M5-branes in the Bulk:
When the SM is realized on M5-branes in the bulk, the gauge coupling depends on the complex structure of the curve wrapped by the branes rather than the CY volume. While this setup relaxes the bound on $R$ by a couple of orders of magnitude compared to the boundary case, the resulting size is still far too small to accommodate a dark dimension ($R \ll 50\,\mu\text{m}$). The authors note that placing M5-branes directly on top of the $E_8$ boundaries leads to small instanton transitions, making the bulk configuration ($0 < x_{\text{M5}}^{11} < \pi$) the only stable option considered, which still yields a negligible $R$.

Significance and Conclusions

The paper establishes a robust, model-independent upper bound on the size of the M-theory interval in the context of $E_8 \times E_8$ heterotic M-theory:
$$R \lesssim \mathcal{O}(10^{-28}) \, \text{m}.$$

Significance:

• Exclusion of Dark Dimension: This bound indicates that the eleventh dimension in standard Hořava-Witten constructions cannot serve as the "dark dimension" (micron-scale) proposed to explain the cosmological constant.
• Incompatibility with Standard GUTs: The results reinforce the argument that if a large extra dimension exists, the Standard Model cannot be realized via standard GUT-like theories where gauge bosons are point-like particles up to the QG scale.
• Alternative Scenarios: The authors conclude that if modifications to Newton's inverse square law are observed at short distances, viable string/M-theory completions of the SM must be non-unified brane models, or involve GUT gauge bosons realized as solitonic strings (as suggested in prior work [26]), or F-theory constructions with 7-branes wrapping holomorphic 4-cycles.

The authors emphasize that these conclusions hold regardless of whether the visible sector resides on the large or small boundary, or if warping is present. Future improvements in proton decay limits (e.g., at Hyper-Kamiokande) would strengthen these bounds by an $\mathcal{O}(1)$ factor.