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arxiv: 2606.17485 · v1 · pith:DZWRAEBQnew · submitted 2026-06-16 · ✦ hep-ph · astro-ph.CO· gr-qc· hep-th

Fourier-Preconditioned Path Deformations for Multi-Field Vacuum Tunnelling

Suriyah R. Kannagi , Aadarsh Singh , Sudhir K Vempati This is my paper

Pith reviewed 2026-06-27 00:32 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COgr-qchep-th
keywords multi-field vacuum tunnellingbounce solutionsFourier sine modespath deformationvacuum decaypreconditioninghigh-dimensional potentialsautomatic differentiation
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The pith

A straight-line path plus a handful of endpoint-vanishing Fourier sine modes reproduces multi-field bounce actions to sub-percent accuracy and accelerates existing solvers by up to 90 percent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an endpoint-safe Fourier representation of the tunnelling path in multi-field scalar potentials. The path is written as the straight-line segment between the false and true vacua plus a small number of sine-mode deformations that automatically vanish at the endpoints. This converts the path search into a finite-dimensional optimisation problem solved by automatic differentiation. On standard benchmarks the resulting actions agree with FindBounce, OptiBounce and CosmoTransitions at the sub-percent level up to 50 fields while using only modest numbers of modes. When supplied as an initial guess the same Fourier paths reduce the number of subsequent deformation steps and cut runtime in high-dimensional cases by as much as 90 percent.

Core claim

The tunnelling path in field space can be expressed as a straight-line interpolation between the false and true vacua plus sine-mode deformations that vanish at the endpoints; the resulting finite-dimensional variational problem, when solved by automatic differentiation, yields bounce actions that match established numerical codes at the sub-percent level on smooth benchmark potentials for up to 50 fields and simultaneously supplies geometric information that accelerates those same codes when used as an initialiser.

What carries the argument

Endpoint-safe Fourier sine-mode deformations added to the straight-line interpolation between vacua; these modes reduce the infinite-dimensional path problem to a low-dimensional coefficient optimisation while preserving fixed endpoints.

If this is right

  • The Fourier ansatz reproduces known bounce actions to sub-percent accuracy on the OptiBounce benchmark for 3–20 fields and on random-coefficient families up to 50 fields.
  • Only a modest number of sine modes suffices for smooth potentials, outperforming or matching other endpoint-safe bases.
  • When used as an initial path the Fourier representation reduces the number of deformation steps required by CosmoTransitions.
  • In FindBounce point-injection tests the same initial paths produce runtime improvements reaching 90 percent in the highest-dimensional cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same low-mode basis could be paired with machine-learned potential surrogates to explore still larger field spaces without recomputing the full bounce each time.
  • Replacing the single straight-line segment with a piecewise linear backbone would allow the method to handle potentials containing multiple intermediate minima.
  • In cosmological scans over first-order phase transitions the reduced runtime could make multi-scalar models with 20–50 fields routinely accessible.

Load-bearing premise

The true tunnelling path stays close to a straight line plus a modest number of low-frequency sine modes that vanish at the ends, provided the potential is smooth enough that higher modes remain negligible.

What would settle it

Run the method on a potential whose bounce path contains sharp turns or high-frequency oscillations and check whether the computed action deviates by more than a few percent from a fully converged high-resolution numerical result.

read the original abstract

We present an endpoint-safe Fourier method for multi-field vacuum tunnelling. The field-space tunnelling path is written as a straight-line interpolation between the false and true vacua, plus sine-mode deformations that vanish at the endpoints. This gives a finite-dimensional path optimisation problem, which we implement using automatic differentiation in the JAX numerical framework. The method is studied both as a standalone variational ansatz for curved tunnelling paths and as a preconditioner for existing bounce solvers. On the OptiBounce benchmark potential for $N_\phi=3,\ldots,20$ and on a nested random-coefficient potential family up to $N_\phi=50$, the Fourier result agrees with FindBounce, OptiBounce, and CosmoTransitions at the sub-percent level in the regular benchmark cases, while requiring only a modest number of modes. We also compare several endpoint-safe basis families and find that Fourier sine modes provide a robust default for smooth tunnelling paths. When used as an initialiser, the Fourier path supplies useful geometric information to existing solvers before the final bounce calculation is carried out. In the CosmoTransitions tests, this reduces the number of steps in subsequent path deformation, while in the FindBounce point-injection tests, it gives large runtime improvements in the high-dimensional cases up to 90 $\%$. These results suggest that endpoint-safe Fourier paths provide a useful bridge between simple analytic path ans\"atze and fully numerical multi-field bounce algorithms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper presents an endpoint-safe Fourier method for multi-field vacuum tunnelling. The tunnelling path is expressed as a straight-line interpolation between the false and true vacua plus sine-mode deformations that vanish at the endpoints. This finite-dimensional optimisation problem is solved using automatic differentiation in JAX. The method is tested as a standalone variational ansatz and as a preconditioner for existing solvers on benchmark potentials with up to 50 fields, showing sub-percent agreement with FindBounce, OptiBounce, and CosmoTransitions, and runtime gains up to 90% when used as an initialiser.

Significance. This method offers a computationally efficient way to approximate curved tunnelling paths in high-dimensional field spaces, which is relevant for vacuum decay calculations in particle physics and cosmology. The direct comparisons with multiple established codes and the exploration of different basis families provide a solid foundation for the claims. The JAX implementation facilitates reproducibility and extension. If the numerical results hold under closer scrutiny, it could serve as a useful tool for initializing more expensive solvers in multi-field models.

minor comments (3)
  1. [Results] The manuscript would benefit from including convergence tests or plots showing how the action and path accuracy depend on the number of Fourier modes for the reported benchmarks.
  2. [Abstract and §3] Specify the typical number of modes used for different field dimensions N_φ and the criterion for choosing them.
  3. A table summarizing the runtime improvements across different N_φ in the FindBounce tests would enhance the presentation of the speedup results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript, the assessment of its significance, and the recommendation for minor revision. No major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces a numerical variational ansatz (straight-line path plus finite Fourier sine deformations vanishing at endpoints) implemented in JAX and benchmarks it directly against three independent external codes (FindBounce, OptiBounce, CosmoTransitions) on OptiBounce and random-coefficient potentials up to N_φ=50. Reported sub-percent agreement on action and path, plus runtime gains when used as initializer, are empirical outcomes of these external comparisons rather than quantities defined by the authors' own parameters or prior self-citations. No load-bearing step reduces by construction to an internal fit, self-citation chain, or ansatz smuggled via citation; the method's scope is explicitly conditioned on smooth paths where low-mode sine deformations suffice, and this is tested rather than assumed as a theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard completeness of the Fourier sine basis for functions vanishing at endpoints and on the assumption that the bounce action is sufficiently smooth; no new free parameters are fitted and no new physical entities are postulated.

axioms (1)
  • standard math Fourier sine series on [0,1] with zero boundary conditions can represent smooth functions vanishing at the endpoints to arbitrary accuracy
    Invoked when the path is written as straight-line interpolation plus sine-mode deformations.

pith-pipeline@v0.9.1-grok · 5805 in / 1385 out tokens · 38526 ms · 2026-06-27T00:32:16.505219+00:00 · methodology

discussion (0)

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Reference graph

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