Bubble wall velocity from Kadanoff-Baym equations: fluid dynamics and microscopic interactions

This paper establishes a first-principles framework based on non-local Kadanoff-Baym equations that unifies macroscopic fluid dynamics and microscopic particle interactions to systematically determine bubble wall velocity, revealing a linear friction force from $2\rightarrow 2$ scattering that prevents runaway bubbles in the ballistic regime.

The Gist

The Big Picture: The Cosmic Bubble Race

Imagine the early universe as a giant, super-hot soup. As it cooled down, it underwent a "phase transition," similar to water turning into ice. But instead of freezing all at once, it started forming bubbles of the new "ice" phase inside the old "water" phase.

These bubbles expand, pushing the old soup out of the way. The speed at which the wall of these bubbles expands is crucial. If the wall moves too fast (a "runaway" bubble), it creates a different kind of cosmic signal (gravitational waves) and might ruin the conditions needed to create the matter that makes up our universe today.

The big question the authors are trying to answer is: How fast do these bubbles actually go?

To find the speed, you have to look at a tug-of-war:

1. The Push: The energy difference between the inside and outside of the bubble pushes the wall forward.
2. The Drag (Friction): The particles in the soup (the plasma) hit the bubble wall and slow it down.

The Problem: Two Different Maps

For a long time, physicists have used two different methods to calculate this "drag," and they didn't agree with each other. It was like trying to navigate a city using two different maps that gave conflicting directions.

• Method A (The Fluid Map): This treats the soup like a continuous fluid (like water in a river). It calculates drag based on how the fluid flows around the bubble. It predicts that at very high speeds, the drag stops increasing, allowing the bubble to accelerate forever (a "runaway" bubble).
• Method B (The Microscopic Map): This treats the soup as individual particles (like billiard balls) hitting the wall. It predicts that at very high speeds, the drag gets stronger and stronger, eventually stopping the bubble from running away.

The paper argues that Method B is missing a piece of the puzzle, and Method A is missing another. They are inconsistent because they treat the interaction between the bubble wall and the particles differently.

The Solution: The "Background Field" Trick

The authors introduce a new, unified framework based on Quantum Field Theory (the rules that govern how particles and forces interact).

Think of the bubble wall not just as a physical barrier, but as a shifting landscape. As a particle moves through the wall, its "mass" (its heaviness) changes because the environment around it is changing.

In standard physics, when particles collide, they usually conserve momentum (like two billiard balls hitting each other; the total bounce is the same). However, because the bubble wall is a shifting landscape, momentum is not perfectly conserved in the direction the wall is moving. It's like a car driving over a speed bump that is also moving; the car loses some forward momentum to the bump in a way that standard calculations miss.

The authors show that if you account for this "shifting landscape" correctly:

1. You get the drag from the fluid flow (Method A).
2. You also get the extra drag from the particles scattering off the changing mass (Method B).
3. Crucially: When you combine them, the total drag increases with speed. This means bubbles cannot run away forever. They will eventually hit a "terminal velocity" (a top speed) and stop accelerating.

The New Discovery: The "2-to-2" Collision

The paper also looked at a specific type of particle collision that previous studies often ignored: 2-to-2 scattering.

• 1-to-1: A particle hits the wall and bounces off (or changes mass).
• 1-to-2: A particle hits the wall and splits into two.
• 2-to-2: Two particles collide with each other right at the wall and bounce off in new directions.

The authors calculated the friction caused by these 2-to-2 collisions. They found that this specific type of interaction creates a new kind of drag that grows linearly with the bubble's speed.

The Analogy: Imagine a crowd of people (particles) trying to push a giant door (the bubble wall).

• The old view said that if the door moves fast enough, the people just slip past it, and the door runs away.
• The new view says that as the door moves fast, the people start bumping into each other right against the door (2-to-2 collisions). These collisions create a massive pile-up of pressure that acts like a brake, ensuring the door never runs away, even if there are no other forces holding it back.

The Conclusion

The authors have built a "master equation" (based on the Kadanoff-Baym equations) that unifies the fluid and microscopic views.

1. It fixes the math: It shows why the two previous methods disagreed and combines them into one consistent picture.
2. It stops the runaway: It proves that due to these microscopic interactions (especially the 2-to-2 collisions), bubble walls in the early universe likely reached a steady speed rather than accelerating to the speed of light forever.
3. Why it matters: This changes how we predict the "sound" of the early universe (gravitational waves) and how we understand the creation of matter. If the bubbles don't run away, the signals we look for today will look different than previously thought.

In short, the paper provides a more complete and accurate rulebook for how cosmic bubbles move, showing that the "friction" of the universe is stronger and more complex than we previously realized.

Technical Summary

Technical Summary: Bubble Wall Velocity from Kadanoff-Baym Equations

Problem Statement
The determination of the bubble wall velocity ($v_w$) during a first-order cosmological phase transition is critical for predicting the viability of electroweak baryogenesis (EWBG) and the spectrum of gravitational waves (GWs). Existing theoretical approaches to calculating the friction pressure that opposes the bubble wall expansion are inconsistent. The "fluid method" assumes local thermal equilibrium and momentum conservation in microscopic interactions, yielding a friction force that peaks near the Jouguet velocity but fails to prevent "runaway" bubbles at ultra-relativistic speeds. Conversely, the "microscopic method," often employing a ballistic approximation ($v_w \sim 1$), incorporates multi-particle interactions (such as $1 \to 2$ splitting) and finds a friction force proportional to the Lorentz factor $\gamma_w$, which prevents runaway behavior. However, these two methods have historically been treated as separate regimes with no unified framework, leading to a discrepancy in the predicted friction pressure structure (a single peak vs. a two-peak structure).

Methodology
The authors develop a first-principles, systematic framework to resolve this inconsistency by unifying macroscopic fluid dynamics and microscopic interactions. The methodology proceeds in three stages:

1. Background Field Quantum Field Theory (BQFT): The authors utilize BQFT to derive local Boltzmann equations that explicitly account for the spacetime-varying background field of the bubble wall. By quantizing fields in this background, they demonstrate that the collision terms in the Boltzmann equation naturally incorporate momentum non-conservation in the direction normal to the wall ($z$-direction). This non-conservation arises because the background field breaks translational invariance, allowing for friction forces associated with particle mass variations and multi-particle splitting processes ($1 \to 2$).
2. Derivation from Kadanoff-Baym Equations (KBE): To establish a rigorous foundation, the authors derive the modified BQFT Boltzmann equation from the non-equilibrium Kadanoff-Baym equations within the Closed Time Path (CTP) formalism. They perform a systematic expansion in the wall gradient parameter $\epsilon_{wall}$. Crucially, they identify that while standard perturbative expansions fail to capture momentum non-conservation, a specific treatment involving the resummation of derivatives acting on $\delta$-functions within the collision terms recovers the non-local momentum structures. This demonstrates that the BQFT results emerge naturally from the more general KBE framework under well-motivated approximations.
3. Application to Scattering Processes: The authors apply this unified framework to compute the friction pressure arising from $2 \to 2$ scattering processes in a scalar field theory (specifically a Higgs portal model with light and heavy scalars), a contribution previously unexplored in this context.

Key Contributions and Results

• Resolution of Inconsistency: The paper provides a unified equation (the modified Boltzmann equation derived from KBEs) that simultaneously incorporates the classical friction from mass variation (fluid method) and the microscopic friction from multi-particle interactions (splitting and scattering). This framework predicts a friction pressure profile with two peaks: one near the Jouguet velocity (from fluid dynamics) and one near $v_w \sim 1$ (from microscopic interactions), as illustrated in the paper's Figure 1.
• Microscopic Origin of Splitting Friction: The authors prove that the friction force associated with $1 \to 2$ splitting processes, previously calculated using BQFT amplitudes, is exactly generated by the collision terms in the Boltzmann equation when the background field dependence is correctly treated.
• $2 \to 2$ Scattering Friction: A novel result is the calculation of the friction pressure arising from $2 \to 2$ scattering processes. The authors find that in a pure scalar theory with significant mass differences between fields, this process generates a friction force that scales linearly with the Lorentz factor $\gamma_w$ ($F_{fric} \propto \gamma_w$).
• Prevention of Runaway Bubbles: The linear dependence of the $2 \to 2$ friction force on $\gamma_w$ implies that runaway bubbles (where the wall accelerates indefinitely) are precluded even in the absence of gauge fields, provided the phase transition involves scalars with significant mass splitting. This contrasts with previous soft emission studies which suggested runaway behavior persists without gauge interactions.

Significance
The paper claims to establish the first systematic, first-principles framework for determining bubble wall velocity that self-consistently integrates the physics of both fluid dynamics and microscopic particle interactions. By deriving the relevant Boltzmann equations from the Kadanoff-Baym equations, the authors provide a theoretically robust basis for calculating friction pressure that was previously lacking. This framework is essential for accurately assessing the gravitational wave signatures and baryogenesis potential of first-order phase transitions. The authors note that while they have derived the governing equations and computed specific friction components, the practical challenge of solving the full non-local kinetic equation for $v_w$ in specific BSM models remains a task for future work, referencing recent efforts in that direction.