⚛️ phenomenology

A Casimir bottleneck in primordial large-N baryon formation

This paper demonstrates that in a large- $N$ confining gauge theory, a Casimir-induced bottleneck hinders baryon formation during the early Universe, causing the relic density of stable baryons to be determined at the confinement phase rather than by subsequent annihilation freeze-out, which has significant implications for $SU(N)$ dark matter models.

Original authors: Luca Di Luzio, Samuele Di Valeriano, Enrico Nardi

Published 2026-09-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Luca Di Luzio, Samuele Di Valeriano, Enrico Nardi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the earliest moments of the universe, before stars or galaxies existed, the cosmos was a seething, super-hot soup of fundamental particles. Among these were quarks, the tiny building blocks that usually stick together to form protons and neutrons. In our everyday world, quarks are never found alone; they are permanently bound into groups by a powerful force known as the strong interaction. This force behaves like an unbreakable rubber band: the harder you try to pull quarks apart, the stronger the pull becomes. Eventually, as the universe cooled, this force caused the free-floating quarks to snap together into stable clusters. Most of these clusters formed pairs, creating particles called mesons, while a few formed groups of three, creating the baryons that make up the matter in our bodies.

Physicists have long wondered if this same process could have happened differently in a hypothetical universe governed by a more complex version of this force. Imagine a universe where the rules of the strong interaction are scaled up, involving a much larger number of color charges than the three we observe. In such a scenario, the formation of heavy, stable matter might be drastically altered. If these hypothetical particles were stable and abundant, they could have survived to the present day, potentially explaining the mysterious dark matter that holds galaxies together. Understanding how these particles form is crucial for determining whether they could be the invisible scaffolding of our universe.

A team of researchers has now investigated this possibility by modeling the birth of matter in a universe with a large number of force-carrying charges. They focused on a specific type of theoretical model where the force between particles scales in a predictable way, a principle known as Casimir scaling. By solving a simplified network of differential equations to track how quarks recombine as the universe cools, they discovered a surprising obstacle that prevents the formation of heavy matter. They found that in these large-scale models, the universe acts as a bottleneck, effectively starving the process of building complex particles. Instead of forming the heavy, stable groups needed for dark matter, the quarks overwhelmingly rush to form simple pairs, leaving the heavier clusters virtually non-existent.

The researchers modeled the early universe as a hot plasma where quarks and their antimatter counterparts, antiquarks, were constantly colliding. As the temperature dropped, these particles began to stick together. In our own universe, quarks easily form groups of three to create baryons. However, in the simulated large-scale universe, the path to building these larger groups is blocked by a series of destructive interactions. The study shows that while quarks can easily pair up to form mesons, any attempt to build a larger cluster is immediately thwarted. If a small group of quarks tries to grow, it is far more likely to be broken apart by a collision with an antiquark than to successfully grab another quark to grow larger. This destruction happens so efficiently that the pool of free quarks is rapidly drained into simple pairs before they can ever assemble into the heavy baryons required for dark matter.

This phenomenon creates what the authors call a Casimir bottleneck. It is similar to a traffic jam where cars can easily merge into pairs, but the road ahead is so clogged with collisions that no convoy can ever form. In the simulations, this bottleneck was so severe that for a universe with twelve types of color charges, the number of heavy baryons produced was roughly an order of magnitude smaller than what would be expected from a more complete calculation, and the final yield was suppressed to levels as low as $10^{-12}$ relative to the initial quark density. The researchers found that the formation of these heavy particles is suppressed by factors that grow exponentially with the size of the force. Consequently, the universe ends up filled almost entirely with stable mesons, while the heavy baryons, which were the intended candidates for dark matter, are left with a relic density so low they would be undetectable.

The study also compared this new finding against the traditional way physicists estimate the amount of dark matter. Usually, scientists assume that heavy particles form in equilibrium and then stop annihilating each other as the universe expands, leaving a specific leftover amount. The researchers showed that in these large-scale models, this standard calculation is wrong because the particles never get the chance to form in the first place. The bottleneck prevents the heavy clusters from ever reaching the abundance needed to match the observed amount of dark matter in the universe. Even if the universe contained the right number of heavy particles to explain dark matter, the physics of their formation would have destroyed them before they could survive.

The authors emphasize that their results rely on a simplified model that tracks the most direct steps of particle recombination, but they note that even when accounting for more complex interactions, the suppression remains severe. They conclude that this bottleneck is an intrinsic feature of theories with a large number of force charges. This means that any theory proposing dark matter made of heavy baryons from such a universe must be reconsidered. The universe, it seems, has a strict preference for simplicity when the forces are scaled up, favoring the formation of simple pairs over the complex structures that could have become the dark matter of a parallel reality.

Technical Summary: A Casimir Bottleneck in Primordial Large-N Baryon Formation

Problem Statement
The paper addresses the formation of baryons (B) in the early Universe within the context of a confining $SU(N)$ gauge theory with $N > 3$ . While Standard Model (SM) interactions are governed by $SU(3)_c$ , extensions involving additional $SU(N)$ groups are motivated by axion physics and dark matter (DM) models. In such theories, if $SU(N)$ baryons are stabilized by a conserved baryon number, they constitute viable DM candidates. However, the relic abundance of these baryons depends critically on their formation mechanism during the confinement phase transition (PT).

Previous studies established that $B$ - $\bar{B}$ production or annihilation in mesonic reactions is exponentially suppressed as $e^{-cN}$ . Conversely, in QCD ( $N=3$ ), baryons form directly from the quark-gluon plasma during confinement without significant suppression, leading to a baryon-to-meson abundance ratio comparable to unity. The central question is whether this QCD picture extends to the large- $N$ limit. The authors investigate whether the formation of $SU(N)$ baryons is hindered by a "Casimir bottleneck," where the destruction of small quark clusters outweighs their formation, potentially suppressing the final baryon yield far below predictions based on standard $B$ - $\bar{B}$ annihilation freeze-out scenarios.

Methodology
To model baryon formation, the authors employ two primary theoretical ingredients:

Casimir Scaling: They utilize the hypothesis that the interaction potential between static color sources follows Casimir scaling. The potential is modeled via a Cornell potential $V_{ab}(r) = \hat{C}_{ab}(\alpha/r - \sigma r)$ , where the color-channel-dependent factor $\hat{C}_{ab}$ is determined by the quadratic Casimir operators of the representations involved. This scaling dictates the strength of attractive (bound-state forming) versus repulsive interactions.
Resonance Recombination Model (RRM): Inspired by the QCD RRM, baryon formation is treated as a sequential recombination process. A quark and antiquark recombine into a meson, while baryons form via a stepwise mechanism where quarks add one by one to a cluster.

The authors adopt a simplified "nearest-neighbor" (NN) approximation for the reaction network. They track the evolution of quark clusters $[\Psi_p]_A$ (where $p$ is the number of quarks in the color-antisymmetric representation) interacting with single free quarks or antiquarks. The network includes:

Direct meson formation: $\Psi + \bar{\Psi} \to [\Psi\bar{\Psi}]$ .
Cluster growth: $\Psi + [\Psi_p]_A \to [\Psi_{p+1}]_A$ .
Cluster destruction: $\bar{\Psi} + [\Psi_p]_A \to [\Psi_{p-1}]_A + [\Psi\bar{\Psi}]$ .

The reaction rates are weighted by the square of the relevant Casimir factors ( $\hat{C}^2$ ). Spin factors are included to ensure the antisymmetry of the total wavefunction. The evolution is modeled using coupled Boltzmann-like equations for the dimensionless pseudo-yields $y_p = n_p/\Lambda^3$ , neglecting cosmic expansion during the short hadronization timescale ( $\sim \Lambda^{-1}$ ). The study focuses on a benchmark scenario with one light quark flavor ( $n_f=1$ ) to isolate the Casimir hierarchy effects, acknowledging that a realistic $n_f=3$ model is treated in a companion paper.

Key Contributions and Results
The analysis reveals a fundamental difference between $N=3$ and large- $N$ scenarios:

Casimir Hierarchy: The Casimir factors governing the binding of quarks into mesons ( $\hat{C}_{1,0} \sim O(N)$ ) are significantly larger than those for diquark formation ( $\hat{C}_{1,2} \sim O(1)$ ). Consequently, meson formation proceeds at a rate that dominates over the initial steps of baryon assembly.
The Bottleneck: For small $p$ , the destruction process (where an antiquark hits a $p$ -quark cluster, reducing it to $p-1$ and producing a meson) is enhanced by a large Casimir factor ( $\sim O(N)$ ). This creates a "bottleneck" where small quark clusters are rapidly destroyed before they can grow into larger baryons.
Numerical Suppression:
- For $N=3$ , the simulation reproduces QCD results where the baryon yield reaches several percent of the initial quark density, confirming no significant bottleneck.
- For $N=12$ , the bottleneck is severe. The abundance of free quarks is rapidly depleted by meson formation. The yields of diquarks and three-quark clusters are suppressed by approximately 4 and 5 orders of magnitude, respectively, relative to the initial density. The final baryon pseudo-yield drops to $\sim 10^{-12}$ .
- For $N=15$ , the suppression is even more extreme ( $\sim 10^{-16}$ ).
Comparison with Freeze-out: The authors compare the yield from confinement formation ( $Y_B^{cf}$ ) with the yield from conventional $B$ - $\bar{B}$ annihilation freeze-out ( $Y_B^{fo}$ ). For large $N$ (e.g., $N=12, 15$ ) and confinement scales $\Lambda \gtrsim 20$ GeV, the yield from confinement formation is orders of magnitude lower than the freeze-out prediction. This implies that for large $N$ , baryons are produced in such small numbers that annihilation never becomes efficient; the relic density is set by the formation bottleneck rather than freeze-out.

Significance and Claims
The paper claims that the Casimir bottleneck is an intrinsic feature of large- $N$ $SU(N)$ gauge theories. The authors conclude that:

Relic Density Estimates: Standard estimates for $SU(N)$ dark matter based on $B$ - $\bar{B}$ annihilation freeze-out are likely overestimates for large $N$ . The actual relic abundance is determined during the confinement phase transition and is exponentially suppressed relative to meson production.
Model Dependence: While the $n_f=1$ benchmark is illustrative, the mechanism relies on the Casimir hierarchy, which is independent of the number of flavors. A realistic $n_f=3$ model (discussed in the companion paper) incorporates decay channels for mesons and confirms that the severe suppression of baryon synthesis persists, even if the exact numerical yield varies by an order of magnitude compared to the NN approximation.
Theoretical Limitations: The authors remain modest regarding their quantitative precision, noting that the NN approximation neglects multi-cluster interactions and long-distance string fragmentation effects. However, they assert that the qualitative suppression mechanism is robust and must be accounted for in any phenomenological study of large- $N$ baryonic dark matter.

Technical Summary: A Casimir Bottleneck in Primordial Large-N Baryon Formation

More like this