First-order phase transitions

• Bubbles of stable phase nucleate in metastable phase
• Bubbles then expand and collide
  • Out-of-equilibrium: can facilitate baryogenesis
  • Collisions produce gravitational waves

[Sketch: Anna Kormu]

LISA: "Astrophysics" signals

LISA: Stochastic background?

[qualitative curve, sketched on]

[what BSM physics might there be?]
Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Dimensional reduction
$\Downarrow \mathcal{L}_{3\mathrm{d}}$
Phase transition parameters
from lattice simulations
$\Downarrow \alpha, \beta, T_N, v_\mathrm{w}, \ldots$
Real time cosmological simulations
$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background
[what would we see as a result?]

Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Dimensional reduction
$\Downarrow \mathcal{L}_{3\mathrm{d}}$
Phase transition parameters
from lattice simulations
$\Downarrow \alpha, \beta, T_N, v_\mathrm{w}, \ldots$
Real time cosmological simulations
$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background

Phase transition parameters connect particle physics and cosmology

• $\alpha$, the phase transition strength ($\sim$ latent heat)
• $T_N$, the temperature at which bubbles nucleate
• $\beta$, the inverse phase transition duration
  ($\sim$ peak nucleation rate)
• $v_\mathrm{w}$, the speed at which bubbles expand

Our focus: nucleation rate

• We compute the bubble nucleation rate $\Gamma$
• Earlier work doing this on the lattice focussed on radiatively induced transitions
  hep-ph/0009132; hep-lat/0103036; arXiv:2205.07238
• Current BSM phenomenological interest is in stronger transitions with tree-level barriers, like ours: $$\begin{align} \mathscr{L} & = -\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-V(\varphi) - J_1 \varphi - J_2 \varphi^2, \\ V(\varphi) &= \sigma \varphi +\frac{1}{2}m^2\varphi^2+\frac{1}{3!}g\varphi^3+\frac{1}{4!}\lambda\varphi^4 \end{align}$$

Bubbles in the lab

• Another motivation: test classical (non-relativistic) nucleation theory in the laboratory:
  • A-B transition in $^{3}\mathrm{He}$ arXiv:2401.07878
  • Ferromagnetic superfluids arXiv:2305.05225
  • Proposed: ultracold atomic gases
    arXiv:1408.1163; arXiv:2212.03621; arXiv:2307.02549
• Hints (e.g. from $^{3}\mathrm{He}$) that theory not totally consistent with experiment
• Lattice simulations provide a third path between nucleation theory and analogue experiments

How to compute the nucleation rate

1. Use multicanonical simulations to generate order parameter histogram
2. Calculate probability of critical bubble $P_\mathrm{c}$ relative to metastable phase

How to compute the nucleation rate

3. Evolve critical bubble configurations forward and backward in a heat bath

How to compute the nucleation rate

4. Compute fraction $\mathbf{d}$ of configurations that tunnel relative to crossings of $\theta_\mathrm{c}$

$$\mathbf{d} = \frac{\delta_{\mathrm{tunnel}}}{N_{\mathrm{crossings}}}$$ $\delta_{\text{tunnel}}=1$ if bubble tunnels, 0 otherwise

Starting
⬋ point

Critical bubble

Stable phase

Metastable phase

How to compute nucleation rate

5. Determine (analytically) rate of change of order parameter across transition surface
   ($\sim$ set of all critical bubbles in configuration space) $$\langle \text{flux} \rangle = \left< \left|\frac{\Delta \theta}{\Delta t} \right|_{\theta_\mathrm{c}} \right> = \sqrt{\frac{8}{\pi \mathcal{V}} (\theta_\mathrm{c} + A^2)}$$

Then the nucleation rate is $\Gamma \mathcal{V} \approx P_\mathrm{c} \langle \text{flux} \rangle \frac{\langle \mathbf{d} \rangle}{2}$

(assuming that $\langle \text{flux} \rangle$ consists of short-range fluctuations
and $\mathbf{d}$ long-range fluctuations hep-lat/0103036 )

Picking a good order parameter

• Tried two order parameters $\theta = \overline{\phi}$; $\theta' = \overline{\phi^2} - 2A \overline{\phi}$
Metastable peak can be broadened by bulk (not bubble) fluctuations - hiding critical bubble
[Sketch: Anna Kormu]
⬋ Bulk fluctuations?
• We found $\theta'$ helped a lot (see arXiv:2205.07238)

Main results: nucleation rate

• $a\to 0$ with $\smash{L\lambda_3 = 42}$

⬋ Naive order parameter

• Up to $\smash{60^3}$ at $\smash{a\lambda_3 = 1.5}$

Reweighting; comparison with PT

20% discrepancy in $\log \; \Gamma$ $\Rightarrow$ $10^{\text{several}}$ discrepancy in $\Gamma$

Key points

• Testing nucleation theory is important for particle physics, cosmology (and lab experiments).
• Our potential (with tree level barrier) shows a significant discrepancy with analytical nucleation theory.
• Need a good [quasi]order parameter to suppress bulk fluctuations in multicanonical simulations.