Reconciling Euclidean Instantons with Real Time in False Vacuum Decay ✛ ✎ ✦
Context
False vacuum decay (quantum tunneling from a metastable state) plays a key role in cosmology, including the eternal inflation scenario proposed by Alan Guth and the stability problem of the Higgs vacuum in the Standard Model. Traditionally, Euclidean instantons have been used to calculate decay probabilities, but real-time lattice simulations have systematically predicted much faster decay. This discrepancy undermined confidence in the semiclassical formalism and motivated a search for resolution. Additional impetus comes from upcoming analog experiments with Bose-Einstein condensates, as well as future observations of gravitational waves from relic phase transitions and new data on the cosmic microwave background.
Methods
The authors performed numerical simulations of a one-dimensional relativistic quantum scalar field on a lattice using a stochastic semiclassical method (truncated Wigner approximation). Initial fluctuations in the metastable vacuum were set by a Gaussian ensemble with an effective mass, and evolution followed classical equations of motion derived from the principle of least action. To extract nucleating bubble profiles, a procedure of boosting to the rest frame and averaging over 500 realizations was applied. Renormalization effects due to short-wavelength modes were accounted for by constructing an effective potential via Gaussian resummation and Fourier series expansion, fitting a single free parameter—the difference in vacuum energy densities.
Results
Measured ensemble bubble profiles (red curves in Fig. 2) reveal a clear dependence on the ultraviolet cutoff: as the number of short-wavelength modes increases, the field amplitude at the bubble center decreases. This is direct evidence of renormalization's impact on nucleation. The bare Coleman instanton (dashed black curves) systematically overestimates this amplitude, and the classical bubble studied by Andrei Linde for thermally activated decay (dashed gray) also gives inflated values at high cutoffs. Meanwhile, the Coleman solution in the effective potential with the optimal energy-density difference parameter (solid black curves) perfectly describes the data, improving χ² by two orders of magnitude. The classical profile in the effective potential underperforms, especially in the tails. Decay rates computed via Euclidean action for the same effective potentials (Table I) agree with those extracted from simulations to within an order-of-unity factor, naturally explained by the absence of the one-loop determinant and nonequilibrium effects.
Implications
The work proves that real-time lattice simulations and Euclidean instantons are equivalent semiclassical descriptions of false vacuum decay, provided that the renormalized effective potential is carefully accounted for. This lifts a years-long tension and legitimizes the use of lattice methods for predictions in cosmology and analog experiments. Conceptually, the result indicates that tunneling of a collective coordinate is statistically equivalent to the classical evolution of vacuum fluctuations.
Future development
Future studies should test whether the agreement holds for a broader class of potentials and in 2+1 and 3+1 dimensions. Of particular interest is the direct simulation of ultracold atom experiments, where precisely the same potentials are realized, enabling the first laboratory observation of nonperturbative decay dynamics to be compared with theory.
Impact
The results will directly impact the interpretation of future gravitational-wave signals from relic phase transitions and the refinement of constraints on vacuum stability in particle physics.
Next steps
Immediate next steps include calculating the fluctuation determinant to refine the pre-exponential factor and studying bubble clustering and oscillon precursors, important for cosmological observables.
Key open problems
The research is directly connected to the unsolved problem of Higgs vacuum stability in the Standard Model, as well as quantum tunneling in the context of the cosmological constant and dark energy.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The authors numerically modeled false vacuum decay in a one-dimensional relativistic scalar field using a stochastic semiclassical method. They showed that incorporating renormalization through the effective potential brings the measured bubble profiles into agreement with Coleman's Euclidean instanton, improving χ² by two orders of magnitude.
Where its strength is
The work demonstrates that renormalization of short-wavelength modes significantly affects the shape of nucleating bubbles and offers a concrete method for reconciling lattice simulations with semiclassics.
Where it could go next
- It might be interesting to compare the obtained profiles with results from analog experiments on Bose–Einstein condensates, where bubble nucleation in a ring geometry has recently been observed.based on: False vacuum in a ring of atoms arXiv:2512.04637
- A related work studied false vacuum decay on a two-dimensional lattice and found a nontrivial shape of nucleating regions—a pair of rings instead of a ball; perhaps your effective potential method could help clarify the origin of such a structure.based on: Vacuum cracks not with a bubble but with a pair of rings arXiv:2606.16024
- In our archive there is a full analysis of a study where quantum boiling of the false vacuum was modeled using a stochastic approach similar to yours; comparing methods and results could clarify the role of the ultraviolet cutoff.based on: Quantum Boiling of the False Vacuum arXiv:2607.01994
Worth trying
- Apply your method for constructing the effective potential to two-dimensional simulations to check whether Coleman's instanton in the effective potential describes the observed bubble profiles.
- Use your ensemble averaging procedure to analyze data from an analog experiment with a Bose condensate and compare with theoretical predictions.
- Investigate how decay rates change when one-loop corrections to the effective action are included, and compare with lattice simulation results.
Why this may matter
Reconciling lattice simulations with the semiclassical formalism strengthens confidence in predictions for false vacuum decay rates, which is important for cosmological scenarios and for assessing the stability of the Higgs vacuum.
Nearby works in our archive
- False vacuum in a ring of atomssimilarity 0.62
- Vacuum cracks not with a bubble but with a pair of ringssimilarity 0.62
- Quantum Boiling of the False Vacuumsimilarity 0.62
- The Universe May Have Switched to a New Energy Statesimilarity 0.62
- The Cosmos Hides a Hidden Phase: Pseudoentropy Sees More Sharplysimilarity 0.61
- How to Ignite the Big Bangsimilarity 0.61
The closest work in the archive sits at 0.62 by meaning — the usual level for our corpus.
If the Higgs vacuum of our Universe were metastable, its decay could happen at any moment at near-light speed, and we would never know about the approaching catastrophe because signals cannot outrun the bubble wall.
The theme of false vacuum decay is dramatically portrayed in Geoffrey Landis's short story 'Vacuum States' and in the sci-fi film 'Interstellar,' where vacuum collapse becomes a threat to humanity.
Key numbers
- Bare Euclidean exponent B: 36.82
- Renormalized exponents (ncut=128..1024): from -9.66 to -4.15
- Measured exponents (ncut=128..1024): from -7.50 to -4.55
- Potential parameters (m0, λ, φ0): 1, 1.6, 1.8
- Number of lattice points: 4096