Particle productions during collisions of highly boosted bubble walls

Technical Summary: Particle Productions during Collisions of Highly Boosted Bubble Walls

Problem Statement
This paper investigates the production of particles with masses significantly exceeding the characteristic energy scale of a cosmological first-order phase transition (FOPT). In scenarios involving runaway bubbles, the Lorentz factor of the bubble walls ($\gamma_w$) at the time of collision can be extremely large, concentrating energy in highly localized regions. While such collisions are known to generate gravitational waves and potentially drive baryogenesis or leptogenesis, the mechanism for producing superheavy particles (candidates for superheavy dark matter) remains poorly understood. Specifically, previous treatments of the ultraviolet (UV) behavior of the field spectrum during bubble collisions yielded conflicting results regarding the scaling of the production rate, particularly for elastic collisions.

Methodology
The authors employ a combined analytical and numerical approach to determine the high-energy spectrum of the scalar order-parameter field $\phi$ during bubble wall collisions in $(1+1)$ and $(3+1)$ dimensions.

1. Analytical Framework:

   • The dynamics of highly boosted bubble walls are described using the "trapping equation," a simplified equation of motion for the field profile $h(s)$ dependent on the proper time $s = \sqrt{t^2 - x^2}$.
   • The field configuration is decomposed into pre-collision and post-collision regimes, distinguishing between inelastic collisions (where the field oscillates around the true vacuum) and elastic collisions (where it oscillates around the false vacuum).
   • The authors derive the Fourier-space profile $\tilde{\phi}(\chi)$, where $\chi = \omega^2 - k^2$ is the invariant mass squared. They perform a systematic asymptotic expansion of the integral representation of $\tilde{\phi}$ in the deep UV regime ($\chi \gg M_h^2$, where $M_h$ is the effective mass of the field).
   • Crucially, the analysis utilizes integration by parts and the properties of the modified Bessel function $K_0$ to isolate the leading-order behavior, demonstrating that the result depends primarily on the discontinuity in the second derivative of the field at the moment of collision.

2. Numerical Simulation:

   • To verify the analytical predictions, the authors solve the trapping equation numerically for a quartic potential.
   • They address the challenge of spectral leakage caused by truncating the integration domain at a finite $s=L$. A sharp cutoff introduces unphysical $\chi^{-1}$ tails that obscure the physical $\chi^{-2}$ behavior.
   • To mitigate this, they implement an $i\epsilon$ prescription, effectively multiplying the integrand by a smooth exponential window function. This suppresses the artificial boundary contribution while preserving the physical UV scaling.
   • High-precision simulations are conducted across multiple grid resolutions to ensure convergence and validate the analytical scaling over a broad UV range.

3. Extension to (3+1) Dimensions:

   • The analysis is generalized to Minkowski spacetime by decomposing the field into smooth components and characteristic functions representing bubble domains.
   • The singular part of the source term $\partial^2 V'(\phi)$ is identified as arising from the intersection of bubble walls.
   • Geometric correction factors are derived to account for finite bubble radii, comparing the results to the parallel-wall approximation.

Key Results

• Universal UV Scaling: The paper establishes a universal result for the Fourier-space profile of the scalar field in the high-energy regime ($\chi \gg M_h^2$):
  $$\tilde{\phi}(\chi) = -\frac{2V'(2v_\phi)}{\chi^2} + \mathcal{O}(\chi^{-3})$$
  where $v_\phi$ is the vacuum expectation value and $V'$ is the derivative of the potential. This scaling holds for both elastic and inelastic collisions.
• Spectral Density: Consequently, the spectral density scales as $F(\chi) \propto [V'(2v_\phi)]^2 \chi^{-4}$. This differs from previous literature (specifically Ref. [9] for elastic collisions) which predicted a $\chi^{-2}$ scaling for elastic cases. The discrepancy is attributed to the assumption in prior works that the field instantaneously relaxes to a uniform value, whereas the correct treatment accounts for the finite timescale of relaxation and the resulting discontinuity in the second derivative.
• Heavy-Particle Production Rates: The authors derive analytical production rates for general heavy-particle thresholds and specifically for fermion pairs. The production is found to be localized near the instant of collision and depends on the scalar potential only through the term $V'(2v_\phi)$.
• Numerical Verification: High-precision numerical simulations confirm the $\chi^{-2}$ scaling of $\tilde{\phi}(\chi)$ and the $\chi^{-4}$ scaling of the spectral density over a wide range of energies, validating the analytical derivation and the effectiveness of the $i\epsilon$ prescription in suppressing spectral leakage.
• Geometric Suppression: In $(3+1)$ dimensions, accounting for the finite radius of bubbles introduces an order-one suppression factor (approximately $0.65$) relative to the parallel-wall approximation when calculating the number density per unit volume.

Significance and Claims
The authors claim that their results revise the standard understanding of ultraviolet scaling in bubble collision scenarios. By correcting the spectral density scaling from $\chi^{-2}$ to $\chi^{-4}$ for elastic collisions, the paper suggests that previous estimates of superheavy dark matter production and baryogenesis yields based on bubble collisions may need significant re-evaluation. The work provides a robust, universal framework for calculating particle production rates that applies to a broad class of potentials, provided the produced particle mass is much larger than the characteristic scale of the phase transition. The findings have direct implications for models of superheavy dark matter production and baryogenesis driven by highly boosted bubble collisions.