Analytic bounces in d dimensions

Antonio Amariti

REVIEW

Analytic bounces in d dimensions

Antonio Amariti

High Energy Physics - Theory · Submitted 29 Sep 2020 · 22 references

Pith's verdict: Solid extension of exact bounces to general d, with standard derivations and thin-wall checks; the smooth-quadratic branch selection is the one real gap.

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arXiv ID2009.14102 (version 1)

Pith IDHMUPNHT7, Pith's permanent record of this paper and its review

CitationAntonio Amariti. "Analytic bounces in d dimensions." arXiv:2009.14102 (2020). https://pith.science/paper/HMUPNHT7

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Editor's note

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The paper is basically sound. It extends the known 4d exact bounce solutions for triangular, square, and quadratic potentials to general spacetime dimension, and the genuinely new parts are the explicit general-d formulas for the square barrier and the quadratic cases, plus the uniform thin-wall checks. The ODE matching is standard, and the reductions to the thin-wall action (2.6) are done carefully, including the 3d/4d/6d triangular appendices. The analytic continuation to Lorentzian time with damped oscillations is a nice extra. This is honest, useful work, not a new framework but a real extension.

The soft spots are real but manageable. First, Eq. (4.6) as printed is dimensionally inconsistent; it needs a factor of the integration constant from (4.5). The final formula (4.7) survives, so it is a typo, but it should be fixed. Second, and more important, is the branch selection in the smooth quadratic case. Equation (5.25) has infinitely many roots, and the paper picks R1^(1) based on numerical plots for d=3,...,6. That is existence evidence, not a proof, and Eq. (5.26) depends on that choice. If a higher root were admissible or had lower action, the claimed bounce action would be wrong. I think the higher roots probably overshoot into the true vacuum and are inadmissible, but the paper needs a stronger argument or at least numerical checks for d>6. The abstract also overstates 'an analytic formula in each case'—the smooth-quadratic formula is implicit in a Bessel root, not closed-form in elementary functions.

For a metastability or vacuum-decay audience, this is a useful reference with exact formulas and clear derivations. The concerns are addressable corrections, not fatal flaws. I would send it to peer review; a competent referee can check the algebra, verify the thin-wall limits, and push on the branch question. My verdict is conditional: with the typo fixed and the branch selection either proved or more thoroughly checked, it is publishable.

Referee report

Significance

If fully established, the closed forms for the triangular first case and the square barrier would be useful reference results for false-vacuum decay in d-dimensional scalar theories, generalizing known 4d findings and providing concrete limits that numerical codes can test. The thin-wall checks in d=3,4,6 and the ODE matching give independent support to those explicit formulas. However, the completeness claim in the abstract is substantially stronger than what is demonstrated: one case is left implicit, one case is explicitly numerical, and the smooth-quadratic first case depends on an unproved branch-selection assumption. The branch-selection gap in Section 5.2.1 is load-bearing for the claimed decay exponent and is the main technical obstacle to acceptance.

Major objections (2)

[Section 5.2.1, Eqs. (5.25)-(5.26), Fig. 5] The selection of the root R1^(1) is not established. Equation (5.25) admits infinitely many roots R1^(j), and the manuscript chooses R1^(1) based on the numerical statement, for d=3,...,6 only, that it is the unique root for which φ(r)<φ1 in the region r<R1^(1). The physical bounce action is the action of the admissible bounce of least action, yet no proof is given that in all dimensions and for all parameter values R1^(1) is admissible, nor is it shown that no other admissible root has smaller action. Since Eq. (5.26) is evaluated at R1^(1), the claimed analytic tunneling exponent for the smooth-quadratic potential is not supported. The text itself describes this as 'numerical evidence'; the proof (or a precise restriction of the claim) is required.
[Abstract; Sections 3.2, 5.1, 5.2.2] The abstract's statement that an analytic formula is provided 'in each case' overstates the contents. In Section 3.2 the bounce action (3.10) is expressed in terms of R± that solve the implicit system (3.9), which the text states cannot be solved for generic d. In Section 5.1, Eq. (5.11) depends on RT determined only implicitly by the transcendental equation (5.9). In Section 5.2.2, the manuscript explicitly states that Eqs. (5.32)-(5.33) 'cannot be solved analytically and one should study them numerically.' These cases are not closed-form analytic formulas of the same status as Eqs. (3.5) and (4.7); the abstract and conclusions should distinguish explicit, implicit, and numerical cases.

Minor objections (9)

[Eq. (2.2)] The right-hand side should be V'(φ(r)), not V'(φ(r))'.
[Eq. (4.6)] As written, Eq. (4.6) has an incorrect dimension: the right-hand side is not proportional to Δφ unless multiplied by a radius factor. The matching of the constant terms in (4.4) gives Δφ = sqrt(2)/(d-2)(R_- sqrt(ΔV_-) - R_+ sqrt(ΔV_+)). The final formula (4.7) can be rederived from (4.4)-(4.5), so this is a typographical issue, but it should be corrected.
[Eq. (5.3)] The displayed equation is not the modified Bessel equation as written; it should read r^2 y''(r) + r y'(r) - (m^2 r^2 + (d/2 - 1)^2) y(r) = 0.
[Eq. (5.9)] The right-hand side of Eq. (5.9) has the wrong sign in the denominator: consistency with Eq. (5.12) requires φ_- - φ_T rather than φ_+ - φ_T.
[Section 5.1.1] The text says that RT corresponds to 'the solution of (5.17)', but the defining equation is Eq. (5.9); Eq. (5.17) is the large-RT expansion used in the thin-wall check.
[Figure 2 caption] The caption's assignment of the first and second cases to the regions below and above the curve is inverted relative to the body text; the text places the second case (true vacuum reached at finite time) below the curve, whereas the caption assigns that region to the first case.
[Figure 5 caption] The caption lists j = -1,...,4 while the text defines j = 1,...,∞; this should be corrected.
[Eq. (5.29)] The expression for a_+ has the wrong Bessel-function argument: it should be K_{d/2-1}(m_+ R_1), not K_{d/2-1}(m_- R_2).
[General] The paper should state explicitly that all formulas assume d > 2, since expressions such as Eq. (3.5) contain (d-2) in denominators and the d = 2 case would require separate treatment.

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The paper

The paper claims

This paper derives analytic false-vacuum decay exponents for scalar potentials in arbitrary spacetime dimension d, extending known four-dimensional results.

The reading

This paper derives closed-form expressions for the Euclidean bounce action describing false-vacuum decay in a scalar field theory, for triangular, square, and two quadratic potentials in arbitrary spacetime dimension $d$. Previously, exact analytic results of this type were available only in four dimensions. The new formulas give the decay exponent directly and reduce to the standard thin-wall approximation when the two vacua are nearly degenerate. The paper also shows that, when the field does not reach the true vacuum in finite Euclidean time, analytic continuation to Lorentzian time produces damped oscillations around the true vacuum.

What carries the argument

The load-bearing object is the Euclidean bounce $\varphi(r)$: an $O(d)$-symmetric solution of $\varphi'' + \frac{d-1}{r}\varphi' = V'(\varphi)$, with $\varphi(r)\to \varphi_+$ as $r\to\infty$ and $\varphi'(0)=0$. For each potential the equation is integrable in elementary or special functions — power laws for the linear and square-potential branches, and modified Bessel functions $I_{d/2-1}(mr)$, $K_{d/2-1}(mr)$ for the quadratic branches. The bounce action is then the action functional evaluated on this solution minus the false-vacuum background; the only non-explicit ingredient is a matching radius ($R_T$ or $R_1$) fixed by requiring the field and its derivative to be continuous at the junction between branches. For the smooth quadratic potential an infinite family of such radii solves the junction condition, and the paper selects the smallest one, $R_1^{(1)}$.

Extended reading notes

Core claim

The central discovery is that for the triangular potential, the square potential, the 'volcanic' piecewise-quadratic potential, and the smooth quadratic potential with a quadratic cap, the spherically symmetric Euclidean equation of motion can be solved in closed form in any dimension $d$. The bounce actions are Eqs. (3.5), (4.7), (5.11), and (5.26). In the degenerate-vacuum limit each formula reduces to the thin-wall result (2.6). Where the true vacuum is not reached at finite Euclidean radius, the Lorentzian continuation of the solution is a damped oscillation about the true vacuum with amplitude suppressed as $\tau^{-d/2}$.

Load-bearing premise

The physical bounce in the smooth quadratic case is assumed to be the smallest-radius solution $R_1^{(1)}$ of equation (5.25), supported by numerical evidence in $d=3,\ldots,6$; if a different branch $R_1^{(j)}$ has smaller action in some dimension, formula (5.26) would not give the decay exponent.

Editorial extensions

If this is right

False-vacuum decay exponents for triangular, square, and quadratic potentials are now known in closed form for any spacetime dimension $d$.
All derived formulas reduce to the thin-wall approximation in the degenerate-vacuum limit, providing a consistency check across $d$.
For bounces that do not reach the true vacuum at finite Euclidean time, the Lorentzian-continued field oscillates with damping $\tau^{-d/2}$ around the true vacuum.
The infinite family of candidate solutions for the smooth quadratic potential is resolved by selecting the branch with the smallest radius, $R_1^{(1)}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same Bessel-function reduction should be applicable to other potentials with quadratic asymptotics, for instance adding a small cubic term, giving controlled corrections to these exact actions.
The closed-form actions in $d>4$ provide a quantitative way to test numerical bounce codes in regimes where no exact benchmarks previously existed.
The branch-selection pattern for the smooth quadratic potential suggests that, when multiple Euler-Lagrange solutions exist, the minimal-action bounce may generally be the one with the smallest turning-point radius; testing this in other multi-branch potentials would clarify its scope.
A direct comparison of the exact formulas with the thin-wall prediction at small but finite energy gap would quantify how quickly the thin-wall approximation sets in as $d$ grows, an extension not pursued in the paper.

Original abstract

We study the Euclidean bounce action interpolating between a false and a true vacuum for a scalar field theory with various types of potential. We focus on the cases of a triangular, a square and a quadratic barrier, where the bounce action has already been computed analytically in four dimensions. We generalize the result to d dimensions, providing an analytic formula in each case. Furthermore we show that our results reduce to the ones computed from the thin wall approximation, when the true and the false vacuum are close in energy. When the true vacuum cannot be reached in a finite amount of Euclidean time we study the damped oscillations of the solution by analytical continuation to Lorentzian spacetime.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.45 · human participants 0.01 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the Euclidean bounce action interpolating between a false and a true vacuum for a scalar field theory with various types of potential. We focus on the cases of a triangular, a square and a quadratic barrier, where the bounce action has already been computed analytically in four dimensions. We generalize the result to d dimensions, providing an analytic formula in each case. Furthermore we show that our results reduce to the ones computed from the thin wall approximation, when the true and the false vacuum are close in energy. When the true vacuum cannot be reached in a finite amount of Euclidean time we study the damped oscillations of the solution by analytical continuation to Lorentzian spacetime.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the Euclidean bounce action interpolating between a false and a true vacuum for a scalar field theory with various types of potential.	contribution 0.89	0.91	0.06	asserted 0.91
s1	We focus on the cases of a triangular, a square and a quadratic barrier, where the bounce action has already been computed analytically in four dimensions.	background 0.82	0.18	0.05	asserted 0.98
s2	We generalize the result to d dimensions, providing an analytic formula in each case.	contribution 0.70	0.96	0.68	asserted 1.00
s3	Furthermore we show that our results reduce to the ones computed from the thin wall approximation, when the true and the false vacuum are close in energy.	result 0.85	0.91	0.15	asserted 1.00
s4	When the true vacuum cannot be reached in a finite amount of Euclidean time we study the damped oscillations of the solution by analytical continuation to Lorentzian spacetime.	contribution 0.83	0.92	0.09	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (22)

[13]

M. J. Duncan and L. G. Jensen, Exact tunneling solutions in scalar ﬁeld theory , Phys. Lett. B291 (1992) 109

View on Pith 1992
[16]

G. Pastras, Exact Tunneling Solutions in Minkowski Spacetime and a Candidate for Dark Energy, JHEP 08 (2013) 075 [ 1102.4567]

View on Pith Pith review arXiv 2013
[1]

I. Kobzarev, L. Okun and M. Voloshin, Bubbles in Metastable Vacuum , Sov. J. Nucl. Phys. 20 (1975) 644. – 22 –

View on Pith 1975
[2]

S. R. Coleman, V. Glaser and A. Martin, Action Minima Among Solutions to a Class of Euclidean Scalar Field Equations , Commun. Math. Phys. 58 (1978) 211

View on Pith 1978
[3]

J. Callan, Curtis G. and S. R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D 16 (1977) 1762

View on Pith 1977
[4]

S. R. Coleman, The Fate of the False Vacuum. 1. Semiclassical Theory , Phys. Rev. D15 (1977) 2929

View on Pith 1977
[5]

S. R. Coleman and F. De Luccia, Gravitational Eﬀects on and of Vacuum Decay , Phys. Rev. D21 (1980) 3305

View on Pith 1980
[6]

F. C. Adams, General solutions for tunneling of scalar ﬁelds with quartic potentials , Phys. Rev. D 48 (1993) 2800 [ hep-ph/9302321]

View on Pith Pith review arXiv 1993
[7]

U. Sarid, Tools for tunneling, Phys. Rev. D 58 (1998) 085017 [ hep-ph/9804308]

View on Pith Pith review arXiv 1998
[8]

J. R. Espinosa, A Fresh Look at the Calculation of Tunneling Actions , JCAP 1807 (2018) 036 [ 1805.03680]

View on Pith Pith review arXiv 2018
[9]

R. Jinno, Machine learning for bounce calculation , 1805.12153

View on Pith Pith review arXiv
[10]

V. Guada, A. Maiezza and M. Nemevek, Multiﬁeld Polygonal Bounces , Phys. Rev. D99 (2019) 056020 [ 1803.02227]

View on Pith Pith review arXiv 2019
[11]

V. Guada, Multiﬁeld False Vacuum Decay: Polygonal Bounce , Acta Phys. Polon. B50 (2019) 1859

View on Pith 2019
[12]

V. Guada, M. Nemevek and M. Pintar, FindBounce: package for multi-ﬁeld bounce actions, Comput. Phys. Commun. 256 (2020) 107480 [ 2002.00881]

View on Pith Pith review arXiv 2020
[14]

K. Dutta, C. Hector, T. Konstandin, P. M. Vaudrevange and A. Westphal, Validity of the kink approximation to the tunneling action , Phys. Rev. D86 (2012) 123517 [1202.2721]

View on Pith Pith review arXiv 2012
[15]

K. A. Intriligator, N. Seiberg and D. Shih, Dynamical SUSY breaking in meta-stable vacua, JHEP 04 (2006) 021 [ hep-th/0602239]

View on Pith Pith review arXiv 2006
[17]

K. Dutta, P. M. Vaudrevange and A. Westphal, An Exact Tunneling Solution in a Simple Realistic Landscape, Class. Quant. Grav. 29 (2012) 065011 [ 1102.4742]

View on Pith Pith review arXiv 2012
[18]

K. Dutta, P. M. Vaudrevange and A. Westphal, The Overshoot Problem in Inﬂation after Tunneling, JCAP 1201 (2012) 026 [ 1109.5182]

View on Pith Pith review arXiv 2012
[19]

K. Dutta, C. Hector, P. M. Vaudrevange and A. Westphal, More Exact Tunneling Solutions in Scalar Field Theory , Phys. Lett. B708 (2012) 309 [ 1110.2380]

View on Pith Pith review arXiv 2012
[20]

A. Amariti and M. Siani, R-symmetry and supersymmetry breaking in 3D WZ models, JHEP 08 (2009) 055 [ 0905.4725]. – 23 –

View on Pith Pith review arXiv 2009
[21]

A. Aslanov, An elegant exact solutions for the emdenfowler equations of the ﬁrst kind, Mathematical Methods in The Applied Sciences 39 (2016) 1039

View on Pith 2016
[22]

A. Giveon, D. Kutasov and O. Lunin, Spontaneous SUSY Breaking in Various Dimensions, Nucl. Phys. B822 (2009) 127 [ 0904.2175]. – 24 –

View on Pith Pith review arXiv 2009

How the literature uses it

5 arXiv papers cite this in 6 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 3 sentences without a confident label

Uses its method or tool 1 paper

The Euclidean action is estimated using the triangular approximation [ref] S_4^trg = 32π^2/31+c/(√(1+c)-1)^4 (Δϕ_+^4/ΔV_+) , where ΔV_+ ∼M^4 , Δϕ_+ ∼f , ΔV_- ∼g^4ϕ_c^4/λ , Δϕ_- ∼ϕ_c , c ≡ΔV_-/ΔV_+ Δϕ_+/ Δϕ_- ≈g^4/λ (ϕ_c/M)^4 f/ϕ_c = g Λ^3 f/λM^4 .

Attracting the Electroweak Scale to a Tachyonic Trap · Quantum tunneling · 1.00

Cites it as background 1 paper

Also notable is the triangular potential with two piecewise linear segments that was solved exactly in $D=4$ [ref] , see also [ref] .

Analytic thin wall false vacuum decay rate · Introduction · 0.95

What it cites · 18 sentences without a confident label

Builds on 1 paper

The asymptotic formula is [equation] such that [equation] where the oscillatory behavior is damped by the $τ^-d/2$ term, generalizing to $d$-dimensions the discussion of [ref] .

Exact Tunneling Solutions in Minkowski Spacetime and a Candidate for Dark Energy · Damped oscillations · 1.00

Cites as support 1 paper

It has been shown in [ref] that ([ref]) is actually always satisfied with the potential given in formula ([ref]).

Exact Tunneling Solutions in Minkowski Spacetime and a Candidate for Dark Energy · The interpolating solution · 0.96

Cites as background 4 papers

Non singular 4d quadratic potentials have been studied in [ref] while cubic and quartic cases have been further discussed in [ref] .

Exact Tunneling Solutions in Minkowski Spacetime and a Candidate for Dark Energy · Introduction · 0.86
Non singular 4d quadratic potentials have been studied in [ref] while cubic and quartic cases have been further discussed in [ref] .

An Exact Tunneling Solution in a Simple Realistic Landscape · Introduction · 0.86
Non singular 4d quadratic potentials have been studied in [ref] while cubic and quartic cases have been further discussed in [ref] .

The Overshoot Problem in Inflation after Tunneling · Introduction · 0.86

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [13]

M. J. Duncan and L. G. Jensen, Exact tunneling solutions in scalar ﬁeld theory , Phys. Lett. B291 (1992) 109

View on Pith 1992

[2] [16]

G. Pastras, Exact Tunneling Solutions in Minkowski Spacetime and a Candidate for Dark Energy, JHEP 08 (2013) 075 [ 1102.4567]

View on Pith Pith review arXiv 2013

[3] [1]

I. Kobzarev, L. Okun and M. Voloshin, Bubbles in Metastable Vacuum , Sov. J. Nucl. Phys. 20 (1975) 644. – 22 –

View on Pith 1975

[4] [2]

S. R. Coleman, V. Glaser and A. Martin, Action Minima Among Solutions to a Class of Euclidean Scalar Field Equations , Commun. Math. Phys. 58 (1978) 211

View on Pith 1978

[5] [3]

J. Callan, Curtis G. and S. R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D 16 (1977) 1762

View on Pith 1977

[6] [4]

S. R. Coleman, The Fate of the False Vacuum. 1. Semiclassical Theory , Phys. Rev. D15 (1977) 2929

View on Pith 1977

[7] [5]

S. R. Coleman and F. De Luccia, Gravitational Eﬀects on and of Vacuum Decay , Phys. Rev. D21 (1980) 3305

View on Pith 1980

[8] [6]

F. C. Adams, General solutions for tunneling of scalar ﬁelds with quartic potentials , Phys. Rev. D 48 (1993) 2800 [ hep-ph/9302321]

View on Pith Pith review arXiv 1993

[9] [7]

U. Sarid, Tools for tunneling, Phys. Rev. D 58 (1998) 085017 [ hep-ph/9804308]

View on Pith Pith review arXiv 1998

[10] [8]

J. R. Espinosa, A Fresh Look at the Calculation of Tunneling Actions , JCAP 1807 (2018) 036 [ 1805.03680]

View on Pith Pith review arXiv 2018

[11] [9]

R. Jinno, Machine learning for bounce calculation , 1805.12153

View on Pith Pith review arXiv

[12] [10]

V. Guada, A. Maiezza and M. Nemevek, Multiﬁeld Polygonal Bounces , Phys. Rev. D99 (2019) 056020 [ 1803.02227]

View on Pith Pith review arXiv 2019

[13] [11]

V. Guada, Multiﬁeld False Vacuum Decay: Polygonal Bounce , Acta Phys. Polon. B50 (2019) 1859

View on Pith 2019

[14] [12]

V. Guada, M. Nemevek and M. Pintar, FindBounce: package for multi-ﬁeld bounce actions, Comput. Phys. Commun. 256 (2020) 107480 [ 2002.00881]

View on Pith Pith review arXiv 2020

[15] [14]

K. Dutta, C. Hector, T. Konstandin, P. M. Vaudrevange and A. Westphal, Validity of the kink approximation to the tunneling action , Phys. Rev. D86 (2012) 123517 [1202.2721]

View on Pith Pith review arXiv 2012

[16] [15]

K. A. Intriligator, N. Seiberg and D. Shih, Dynamical SUSY breaking in meta-stable vacua, JHEP 04 (2006) 021 [ hep-th/0602239]

View on Pith Pith review arXiv 2006

[17] [17]

K. Dutta, P. M. Vaudrevange and A. Westphal, An Exact Tunneling Solution in a Simple Realistic Landscape, Class. Quant. Grav. 29 (2012) 065011 [ 1102.4742]

View on Pith Pith review arXiv 2012

[18] [18]

K. Dutta, P. M. Vaudrevange and A. Westphal, The Overshoot Problem in Inﬂation after Tunneling, JCAP 1201 (2012) 026 [ 1109.5182]

View on Pith Pith review arXiv 2012

[19] [19]

K. Dutta, C. Hector, P. M. Vaudrevange and A. Westphal, More Exact Tunneling Solutions in Scalar Field Theory , Phys. Lett. B708 (2012) 309 [ 1110.2380]

View on Pith Pith review arXiv 2012

[20] [20]

A. Amariti and M. Siani, R-symmetry and supersymmetry breaking in 3D WZ models, JHEP 08 (2009) 055 [ 0905.4725]. – 23 –

View on Pith Pith review arXiv 2009

[21] [21]

A. Aslanov, An elegant exact solutions for the emdenfowler equations of the ﬁrst kind, Mathematical Methods in The Applied Sciences 39 (2016) 1039

View on Pith 2016

[22] [22]

A. Giveon, D. Kutasov and O. Lunin, Spontaneous SUSY Breaking in Various Dimensions, Nucl. Phys. B822 (2009) 127 [ 0904.2175]. – 24 –

View on Pith Pith review arXiv 2009