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REVIEW 1 major objections 5 minor 67 references

In the large-field regime, the dominant vacuum-decay bounce in any multi-scalar theory is a radial one, so the leading tunneling rate reduces to an effective quartic coupling λ_eff = 4 min V(θ).

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-04 01:01 UTC pith:TP3BPOLE

load-bearing objection Useful and mostly sound; the dominance proof has a repairable gap, so it needs a revision before publication. the 1 major comments →

arxiv 2608.00216 v1 pith:TP3BPOLE submitted 2026-07-31 hep-ph

Large-Field Vacuum Decay in General Multi-Scalar Theories

Giorgio Busoni , Florian Goertz , Navneet Krishnan , Rickson Wielian This is my paper
classification hep-ph
keywords false vacuum decayvacuum metastabilitymulti-scalar field theoryradial bounceeffective quartic couplingbiquadratic potential2HDM+a model3-3-1 model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes a general shortcut for false-vacuum decay in theories with many scalar fields. In the large-field regime, where the quartic part of the potential dominates, it shows that the bounce of least action always moves along a single radial line in field space; any non-radial bounce contributes to the decay rate with an exponentially suppressed factor. As a result, the leading-order tunneling rate is captured by the single-field formula B = 8π²/(3|λ_eff|), with λ_eff obtained by minimizing the quartic potential over all angular directions. For the common case of biquadratic potentials, λ_eff becomes a simple expression in terms of matrix inverses. A reader should care because this turns a difficult multi-field boundary-value problem into a minimization problem, and the paper shows that in two realistic models the resulting vacuum-stability constraints often dominate over current experimental bounds.

Core claim

The central claim is that, when the instability scale is far above the electroweak scale, the dominant vacuum-decay bounce is radial: all fields move along a fixed line through the origin, and this radial bounce gives the exact leading contribution to the Euclidean action. The paper proves this by bounding the action of any non-radial bounce from below by the action of the radial path at the angular minimum θ*, so non-radial solutions are exponentially suppressed in the decay rate. The resulting action is B = 8π²/(3|λ_eff|), where λ_eff = 4 V(θ*) and V(θ*) is the minimum of the quartic potential on the unit sphere. In biquadratic theories, minimizing V(θ) reduces to a quadratic program, and

What carries the argument

The central objects are the radial-line ansatz and the effective quartic coupling. After the standard rescaling ψ = ρφ and t = ln ρ, the bounce equation becomes a conservative Hamiltonian system; because the quartic potential is homogeneous, it splits into radial and angular parts, and angular minima θ* are invariant lines for the dynamics. The action integral is then bounded below by the path evaluated at the minimum of V(θ), which proves radial-bounce dominance. For biquadratic potentials the angular minimization becomes a constrained quadratic minimization in the simplex; the solution is u = Λ_K^{-1} 1 / (1^T Λ_K^{-1} 1) for each subset K, and the paper provides closed forms for up to thr

Load-bearing premise

The calculation assumes the quartic part of the potential dominates and that the instability scale is much larger than the electroweak scale, so that mass and cubic terms can be dropped from the bounce equation.

What would settle it

Run a numerical bounce solver on a two-field potential with a negative quartic direction and a mass term that is not negligible at the bounce scale; if the least-action bounce deviates from the radial line and gives a Euclidean action smaller than B = 8π²/(3|λ_eff|), the leading-order claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Vacuum-stability computations in multi-scalar models reduce to minimizing an effective quartic coupling; no multi-field numerical bounce solver is needed at leading order.
  • For biquadratic potentials, all possible subsets of fields must be checked, because a single-field direction can appear stable while a multi-field direction yields an unstable vacuum.
  • In the 2HDM+a and 3-3-1 benchmarks here, high-scale vacuum-stability constraints are typically more restrictive than current experimental bounds, effectively limiting large Yukawa couplings to about 1 or less.
  • Small mass splittings of order 10 GeV can dramatically change vacuum stability, offering a region where otherwise-excluded parameters become viable.

Where Pith is reading between the lines

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

  • A direct extension would be to apply the same angular-minimization logic to potentials with mild non-quartic terms, treating cubic and mass terms as perturbations whose effect on λ_eff is calculable order by order.
  • The proof's reliance on homogeneity suggests the result may transfer to any scale-invariant (or conformal) limit of the effective potential, not only strictly quartic truncations.
  • Near-degenerate angular minima are a natural place to test the approximation: if two angular directions give almost equal V(θ), non-radial mixing may become less suppressed and the radial-only answer may need refinement.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper addresses false-vacuum decay in multi-scalar theories in the large-field limit, where the scalar potential is dominated by its quartic part. Using the Fubini substitution, the authors argue that the exponentially dominant bounce is a one-dimensional radial bounce along the angular direction that minimizes the quartic potential. The leading action is then B = 8π²/(3|λ_eff|) with λ_eff = 4 min_θ V(θ). For biquadratic potentials, the minimization is reduced to solving linear equations, including boundary and non-invertible cases, with explicit formulas up to three fields. The formalism is applied to the 2HDM+a and 3-3-1 models, with parameter scans for vacuum stability combined with perturbativity, unitarity, boundedness-from-below, and selected experimental constraints.

Significance. If the central theorem holds, the paper gives a substantial practical simplification: multi-field vacuum decay at leading order is reduced to a single angular minimization, avoiding numerical solution of multi-field bounce equations. The explicit treatment of biquadratic potentials, including boundary and non-invertible cases, is useful, and the two worked examples demonstrate the method. The central proof, however, has a real gap: the key inequality (13) is not valid in a physically relevant regime. The claim is plausible and likely repairable, but the manuscript as written does not rigorously establish its main result.

major comments (1)
  1. [II, Eq. (13)] The first inequality in (13) is used to prove radial-bounce dominance, but it is not valid pointwise. For ψ > ψ_* ≡ [-2V(θ*)]^{-1/2}, the radicand at θ* is negative and the square root is undefined, while a physical bounce can have V(θ) > V(θ*) and a turning point at ψ_max = [-2V(θ_max)]^{-1/2} > ψ_*. Thus the comparison to θ* cannot be made in this region. This is a proof gap in the central claim, not a typographical issue. The manuscript needs a rigorous argument controlling the region beyond ψ_*, e.g. by splitting at ψ_* and using H=0 to show that any excursion beyond ψ_* costs at least the radial action.
minor comments (5)
  1. [Table I] The header is ambiguous: the column printed as "1 4 λeff" should be "λeff/4" (or the entries should be multiplied accordingly). Please clarify so that the dim=1 entry is immediately consistent with Eq. (10).
  2. [III, around Eq. (21)] The treatment of non-invertible Λ_K is terse. Please spell out why a null vector of Λ_K can always be used to reach the boundary of the simplex without changing λeff, and state any non-negativity conditions on the null vector components.
  3. [IV.B, lemma before Eq. (29)] The submatrix lemma is too compressed. In particular, the statement "since det ΛK ≠ 0, we cannot have C=K" is not obvious and needs a fuller proof or a reference. This lemma is used to limit the enumeration of bounces, so it should be easy to follow.
  4. [V, Eqs. (46) and (47)] The matrices Λ_NT and R are hard to decode because of line breaks and inline fractions. Please typeset them with explicit matrix entries, using e.g. ζ12ζ13/ζ23, so that the transformation to the standard simplex is unambiguous.
  5. [II and throughout] The assumption ΛI ≫ v is stated but not quantified. A sentence estimating the size of the neglected mass and cubic terms at the bounce scale would help readers understand the regime of validity of Eq. (6).

Circularity Check

0 steps flagged

No significant circularity: the central λ_eff minimization is derived from the Euclidean action, not fitted to the decay rate it predicts.

full rationale

The central derivation is self-contained. Starting from the Euclidean action and bounce equation (1)-(2), the paper assumes quartic dominance, applies Fubini's substitution to obtain the conservative form (8), and uses the homogeneity of V^(4) to separate radial and angular variables. The radial bounce action then follows from the one-dimensional Fubini solution, giving B = 8π²/(3|λ_eff|) with λ_eff = 4 V(θ*) minimized over angles, Eq. (10). The biquadratic reduction in Section III is a direct constrained-minimization calculation, Eq. (16)-(21), not a fit. No parameter is tuned to reproduce the predicted tunneling rate; the example applications choose benchmark points but do not fit λ_eff to decay data. Self-citations appear only in phenomenological contexts (e.g., STU analysis along the lines of Ref. [37]) and are not load-bearing for the theorem. The skeptical concern about the pointwise bound in Eq. (13) is a formal-completeness or correctness issue, not circularity: even if the proof needs repair, the claimed result is not equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The central derivation is self-contained and introduces no free parameters or invented entities. The only 'chosen' numbers are the cutoff scales and benchmark model parameters in the phenomenological applications, which do not enter the main theorem.

free parameters (2)
  • High-scale validity cutoff Λ = 10^10 GeV (2HDM+a); 10^12 GeV (3-3-1)
    Chosen arbitrarily to define the scale up to which the models must be valid; affects the displayed exclusion regions but not the central formula.
  • Benchmark model parameters in scans = e.g., mH=600 GeV, tanβ=2, λ1=0.6 (2HDM+a); mH=10 TeV, vχ=20 TeV (3-3-1)
    Chosen by hand to illustrate typical behavior; vacuum decay constraints depend on these inputs in the applications.
axioms (6)
  • standard math Coleman-Callan semiclassical decay rate Γ/L³ = A e^{-B}
    Used throughout Section I as the starting point for the tunneling rate.
  • domain assumption The bounce solution is O(4)-symmetric
    Assumed in Eq. (2) and standard in the literature; not proven in this paper.
  • domain assumption Quartic part of the potential dominates and false vacuum is at zero field (Λ_I >> v)
    Section II states this assumption; all subsequent formulas depend on it.
  • standard math Fubini substitution and single-field bounce solution B=8π²/(3|λ|)
    Used to derive Eq. (6) and (8); a known result.
  • standard math Bounce path satisfies H=0 due to boundary conditions
    Used to rewrite the action as an integral over the path; Eq. (12).
  • domain assumption RG running of couplings (notably two-loop for 3-3-1 using SARAH)
    The applications rely on perturbative RG evolution; this is standard but not derived.

pith-pipeline@v1.3.0-alltime-deepseek · 186 in / 23468 out tokens · 267053 ms · 2026-08-04T01:01:14.817535+00:00 · methodology

0 comments
read the original abstract

Many theories beyond the standard model exhibit multiple scalar particles. Such multi-scalar theories can in principle host lower-energy vacua, and thus predict that our universe has a finite lifetime due to false vacuum decay. This scenario cannot be ruled out a priori as even the standard model's electroweak vacuum has been shown to be metastable; however, for theoretical consistency, we still require that the model does not predict a lifetime much smaller than the age of the universe. The calculation of these tunneling rates at leading order for multi-scalar theories typically includes numerical approaches, or approximations which are frequently not analytically controlled. In this article we show that, in the large-field regime, a one-dimensional radial bounce always produces the exact dominant contribution to the leading order tunneling rate, with corrections being exponentially suppressed. This allows us to write simple analytical expressions to calculate the tunneling rate in multi-scalar theories, in terms of an effective quartic coupling $\lambda_\text{eff}$. For theories with biquadratic scalar potentials, we also derive straightforward analytical expressions for $\lambda_\text{eff}$ in terms of the original theory's couplings. Finally, we provide example applications of our results to study the vacuum stability of the 2HDM+a model and the 3-3-1 model.

Figures

Figures reproduced from arXiv: 2608.00216 by Florian Goertz, Giorgio Busoni, Navneet Krishnan, Rickson Wielian.

Figure 1
Figure 1. Figure 1: FIG. 1. Example bounce solutions for the two-scalar model of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Parameter scans of the 2HDM+a model phase diagram (a) in the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Parameter scans of the 2HDM+a model phase diagram (a) in the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Parameter scans of the 3-3-1 model phase diagram (a) in the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Running of [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

discussion (0)

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Reference graph

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