A Model of Annihilogenesis

The Big Picture: Why Are We Here?

Imagine the universe as a giant party that started with a massive explosion (the Big Bang). In a perfect world, this explosion should have created equal amounts of "matter" (the stuff we are made of) and "antimatter" (its evil twin). If that had happened, they would have instantly canceled each other out, leaving behind nothing but empty light.

But we know that didn't happen. We exist. There is a lot more matter than antimatter. Physicists call this the Baryon Asymmetry. The Standard Model of physics (our current rulebook for how particles work) can't explain why one side won. This paper proposes a new story—a new "recipe"—for how the universe managed to save the matter and wipe out the antimatter.

The Setting: A Cosmic Bubble Bath

The story takes place in the very early, hot universe. Imagine the universe is a giant pot of boiling water.

The Phase Transition: Suddenly, the water starts freezing into ice. But it doesn't freeze all at once. Instead, little bubbles of ice (called "true vacuum") start forming and expanding inside the hot water ("false vacuum").
The Walls: The edge where the ice meets the water is the "bubble wall." As these bubbles expand, they sweep through the universe.

The Characters: The Heavy Neutrinos

In this story, we introduce two new characters: heavy, invisible particles called Majorana neutrinos (let's call them χ1 and χ2).

χ1 is the lighter one, the main protagonist.
χ2 is the heavier one, the supporting actor.

Before the ice bubbles form, these particles are light and happy, swimming around freely. But as the ice bubbles expand, something strange happens. Inside the ice, these particles suddenly become extremely heavy.

The Plot: The Great Squeeze

Here is the clever part of the mechanism, which the authors call "Annihilogenesis" (creation through annihilation).

The Trap: As the ice bubbles expand, they act like a giant vacuum cleaner. The heavy particles (χ1) get reflected off the moving ice walls. They can't get into the ice because it's too heavy for them there. So, they get trapped in the shrinking pockets of hot water (false vacuum) left behind between the bubbles.
The Squeeze: As the bubbles collide and merge, these pockets of trapped water get smaller and smaller. The particles are squeezed into a tiny space, and their density skyrockets. It's like squeezing a crowd of people into a tiny elevator; they are packed in tight.
The Crash: Because they are so crowded, the particles start crashing into each other. Instead of just sitting there, they annihilate (destroy each other) in a spectacular 4-way crash. Two particles collide and turn into four new particles (leptons and Higgs bosons).

The Twist: Why Matter Won

Usually, when particles crash and destroy each other, it's a fair fight. One matter particle destroys one antimatter particle, and nothing is left.

But in this model, the crash isn't fair. The authors show that because of a specific interaction with a heavier particle (χ2) that appears briefly in the "loop" of the crash, the universe develops a bias.

Think of it like a coin toss that is slightly weighted.
Every time two χ1 particles crash, the laws of physics (specifically, a violation of "CP symmetry") make it slightly more likely that they produce matter than antimatter.
The paper calculates this bias (called ϵ) and finds it is small (about 1 in a billion), but because there are so many particles crashing in the squeeze, the tiny bias adds up to a huge amount of leftover matter.

The Result: A Baryon Victory

Once the ice bubbles have swallowed the whole universe, the trapped particles are gone (they annihilated). But they left behind a surplus of matter particles.

The Sphaleron: The universe has a magical conversion machine called a "sphaleron" (a complex process involving the weak nuclear force). It takes the leftover lepton asymmetry (the extra matter particles) and converts it into baryon asymmetry (protons and neutrons).
The Outcome: This process successfully creates the exact amount of matter we see in the universe today.

Why This Model is Special

The authors point out a major advantage of their "Annihilogenesis" recipe compared to older theories:

Old Recipe (Thermal Leptogenesis): In previous models, the "weight" of the particles (their mass) was tightly linked to how much matter was created. If the particles were too heavy, the math broke, and you couldn't explain the universe. It was like a strict diet where you could only eat a specific amount of food.
New Recipe (Annihilogenesis): In this model, the "weight" that matters for the creation of matter is the weight the particles had while they were trapped in the shrinking pockets, not their final weight after the universe cooled down.
The Benefit: This breaks the strict link. The authors can use heavier particles and different settings without breaking the math. It relaxes the rules, allowing for a much wider range of possibilities for how our universe could have formed.

Summary

The paper proposes that the universe's matter-antimatter imbalance wasn't caused by a slow decay of heavy particles, but by a cosmic squeeze.

Bubbles of a new state of matter formed.
Heavy particles got trapped in the shrinking gaps between bubbles.
They were squeezed so tightly they crashed into each other.
These crashes were slightly biased toward creating matter over antimatter.
This bias, multiplied by billions of crashes, created the matter-filled universe we live in today.

The authors conclude that this mechanism works well, produces the right amount of matter, and is more flexible than previous theories because it doesn't get stuck on strict mass limits.

1. Problem Statement

The Standard Model (SM) cannot explain the observed baryon asymmetry of the universe (BAU) because the CP violation in the CKM matrix is insufficient, and the electroweak phase transition (EWPT) is a crossover rather than a strong first-order transition, failing to provide the necessary departure from thermal equilibrium.

While Leptogenesis (via the decay of heavy right-handed neutrinos) is a leading candidate, "thermal leptogenesis" faces significant constraints:

Hierarchical Masses: It typically requires hierarchical heavy neutrino masses ( $M_1 \ll M_{2,3}$ ), imposing a lower bound on the lightest neutrino mass $M_1$ (and thus the reheating temperature), which can conflict with constraints from supersymmetry (e.g., gravitino overproduction).
Mass-CP Correlation: In standard thermal leptogenesis, the CP asymmetry parameter $\epsilon$ is bounded from above by the largest light neutrino mass ( $m_{max}$ ). This limits the efficiency of baryogenesis.

The authors propose a mechanism called Annihilogenesis to bypass these constraints. In this scenario, the baryon asymmetry is generated not primarily by the decay of heavy particles, but by the annihilation of particles trapped in collapsing pockets of false vacuum during a strong first-order phase transition (FOPT).

2. Methodology and Model Setup

Theoretical Framework

The model introduces two gauge-singlet, right-handed Majorana fermions, $\chi_1$ and $\chi_2$ , which couple to the SM lepton doublets ( $L$ ) and the Higgs doublet ( $\Phi$ ).

Lagrangian: The new terms include Yukawa couplings $A_{Ba}$ and Majorana mass terms $m_{ab}$ . The authors work in a basis where $m_{ab}$ is diagonal ( $m_{\chi_1} < m_{\chi_2}$ ) and complex phases are isolated in the coupling $A_{12}$ (specifically $A_{21}$ in the notation of the appendix, coupling $\chi_2$ to $L_1$ ).
Phase Transition: An additional scalar singlet $s$ $s$ undergoes a strong FOPT.
- False Vacuum ( $\langle s \rangle = 0$ ): $\chi$ particles have a small residual mass $m_\chi$ .
- True Vacuum ( $\langle s \rangle \neq 0$ ): $\chi$ particles acquire a large mass $M_{\chi}^{in} = y\langle s \rangle + m_\chi$ .
Dynamics: As bubbles of true vacuum expand, $\chi$ particles in the false vacuum are reflected off the bubble walls (due to the large mass jump) and become confined in shrinking "pockets" of false vacuum. Inside these pockets, the density of $\chi$ increases dramatically.

The Mechanism of Asymmetry Generation

The authors focus on the 2 $\to$ 4 annihilation process:
$\chi_1 \chi_1 \to L_1 L_1 \Phi^* \Phi^*$
This process dominates over the standard 1 $\to$ 2 decay ( $\chi_1 \to L \Phi$ ) because the high density in the collapsing pockets enhances the annihilation rate.

CP Violation Source:
The CP asymmetry parameter $\epsilon$ arises from the interference between:

Tree-level diagrams: Mediated by $W$ or $B$ gauge boson exchange.
One-loop diagrams: Involving the heavier neutrino $\chi_2$ in the loop, with both Higgs and gauge boson exchanges.

The complex phase required for CP violation is contained in the Yukawa coupling $A_{12}$ (or $A_{21}$ ).

Computational Approach

Analytical: The authors derive the matrix elements for the tree-level and loop-level processes. They use Cutkosky cutting rules to calculate the imaginary part of the loop amplitude (necessary for the interference term).
Numerical: The phase space integrals for the 2 $\to$ $\to$ 4 process are too complex for analytical evaluation. The authors employ Monte Carlo methods:
- VEGAS: For numerical integration.
- RAMBO: For generating massless final-state momenta uniformly distributed in phase space.
Boltzmann Evolution: They solve the Boltzmann equation for the number density of $\chi_1$ inside the collapsing pockets to determine the final depletion fraction and the resulting asymmetry.

3. Key Contributions

Dominance of Annihilation over Decay: The paper demonstrates that in the context of a strong FOPT with confined particles, the 2 $\to$ 4 annihilation channel ( $\chi_1 \chi_1 \to L L \Phi \Phi$ ) can be the dominant source of lepton asymmetry, replacing the standard 1 $\to$ 2 decay mechanism.
Decoupling of $\epsilon$ and Light Neutrino Masses:
- In standard thermal leptogenesis, $\epsilon \propto m_{max}$ .
- In this model, the CP asymmetry $\epsilon$ depends on the ratio of the heavy neutrino masses ( $m_{\chi_1}/m_{\chi_2}$ ) and the Yukawa couplings.
- Crucially, the mass scale relevant for $\epsilon$ is the residual mass of $\chi_1$ inside the collapsing pockets (before the phase transition completes), whereas the mass scale relevant for light neutrino masses (via the seesaw mechanism) is the post-transition mass $M_{\chi}^{in}$ .
- Result: The upper bound on $|\epsilon|$ imposed by the light neutrino mass constraints in standard leptogenesis does not apply.
Relaxed Constraints: This decoupling relaxes the lower bounds on the right-handed neutrino mass scale and the reheating temperature, allowing for successful baryogenesis in regions of parameter space previously excluded for thermal leptogenesis.

4. Results

CP Asymmetry ( $\epsilon$ ): Numerical evaluation yields a CP asymmetry in the range:
$|\epsilon| \sim 10^{-9} - 10^{-7}$
for $O(1)$ Yukawa couplings. This is sufficient to generate the observed baryon asymmetry.
Baryon Asymmetry ( $Y_{\Delta B}$ ):
- The model successfully reproduces the observed baryon asymmetry ( $Y_{\Delta B} \approx 8 \times 10^{-11}$ ) across a broad region of parameter space.
- Robustness: The final asymmetry is largely insensitive to the bubble wall velocity ( $v_w$ ) and the initial pocket size ( $R_0$ ), provided the annihilation rate is faster than the pocket collapse rate (which holds for $m_{\chi_2}/T \lesssim O(10)$ ).
- The result depends primarily on the mass ratios $m_{\chi_1}/m_{\chi_2}$ and $m_{\chi_2}/T$ .
Light Neutrino Masses: The model allows for the generation of the correct BAU without violating the upper limits on light neutrino masses derived from the seesaw mechanism, as the two mass scales are effectively decoupled.

5. Significance

This work presents a viable alternative to standard thermal leptogenesis that resolves the tension between the required high reheating temperatures for baryogenesis and constraints from supersymmetric models (like gravitino overproduction). By utilizing the dynamics of a strong first-order phase transition and particle confinement, the model:

Relaxes the "Gravitino Problem": It allows for lower reheating temperatures.
Broadens the Landscape: It shows that baryogenesis can occur via scattering/annihilation processes rather than just decays, expanding the theoretical possibilities for early universe cosmology.
Predictive Power: It suggests that the CP violation relevant for the matter-antimatter asymmetry is determined by the dynamics of the phase transition (mass shifts and confinement) rather than being strictly tied to the low-energy neutrino mass spectrum.

The paper concludes that this "Annihilogenesis" mechanism is a robust solution to the baryon asymmetry problem, warranting further study into the interplay between trapped particle dynamics, bubble wall back-reaction, and potential gravitational wave signatures.