Instanton Structure from 325-Loop 3D $\phi^3$ Perturbation Theory

Bo Wang; Ian Moult

REVIEW

Instanton Structure from 325-Loop 3D $\phi^3$ Perturbation Theory

Ian Moult, Bo Wang

High Energy Physics - Theory · Submitted 26 Oct 2026 · 64 references

Pith's verdict: Impressive 325-loop calculation that tests instanton physics, but the four-loop fluctuation evidence is conditional on a Monte Carlo prior that may be the weakest link.

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CitationBo Wang, Ian Moult. "Instanton Structure from 325-Loop 3D $\phi^3$ Perturbation Theory." arXiv:2610.36025 (2026). https://pith.science/paper/3RRGLGHG

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Claim paper

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Editor's note

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Colleague,

This paper is genuinely novel. The annealed tropical sampling framework pushes the perturbative series for 3D φ³ from 20 to 325 loops, overcoming a variance barrier that had stalled direct sampling. The extraction of the instanton action, collective-coordinate exponent b=5/2, one-loop normalization, and fluctuation corrections through three loops is convincing, with clean cross-checks against semiclassical calculations and low-order exact results. The method itself is a real technical step forward.

Where I'm more cautious is the four-loop fluctuation κ3. The perturbative extraction of c3 (hence κ3) uses a Gaussian prior on c4 from a five-loop Monte Carlo with combined statistical and gauge-margin uncertainty ±47. Without that prior, the fit cannot resolve c3 (Δχ²=5.0) and the uncertainty ten-times larger. The authors are transparent—they call the result 'conditional on the five-loop estimate.' The zero-parameter comparison (fixing all semiclassical κk and checking the series) does provide some support, but the semiclassical calculations were not carried out blind to the perturbative data (App. B), so that test is not fully independent. The four-loop evidence is therefore weaker than the headline 1.5σ might suggest; it depends on a prior whose systematics could be larger than claimed.

That said, the three-loop evidence is solid, and the overall picture holds together. The paper deserves serious review. I'd recommend acceptance after the authors either strengthen the prior (more thorough error analysis or independent estimate) or clearly qualify the four-loop result as provisional. The core results—action, exponent, one-loop determinant, three-loop fluctuations—are well supported.

I would bring this to a reading group focused on resurgence or high-order methods, and I would cite it for the method and the lower-loop comparisons. I'd be cautious with the four-loop numbers until further validation.

Referee report

Major objections (3)

[Appendix B, p.14] Non-blindness of semiclassical calculations: The authors state that 'the semiclassical calculations contain no adjustable parameters, but they were not carried out blind to the diagrammatic data.' This raises a concern about potential confirmation bias, even if no parameters are adjustable. The semiclassical evaluations involve choices of truncation, gauge selection, and error estimation; knowing the perturbative target could influence these choices subconsciously. The paper provides cross-checks (multiple gauges, zero-dimensional/one-dimensional tests) that mitigate this concern, but the non-blindness remains a methodological weakness that should be addressed more explicitly. Please discuss what specific steps were taken to guard against bias and, if possible, perform a blinded re-analysis of at least one fluctuation coefficient (e.g., κ̂₃) to demonstrate independence. The authors acknowledge this: “The semiclassical calculations contain no adjustable parameters, but they were not carried out blind to the diagrammatic data.”
[Appendix C, p.15–16, Table IV] Conditionality of the four-loop extraction: The extraction of c₃ (and hence κ̂₃) from perturbative data relies on a Gaussian prior on c₄ derived from a five-loop Monte Carlo evaluation. Without this prior, the uncertainty on c₃ increases by a factor of ~10 and Δχ² drops to 5.0. The authors explicitly note that 'this result is conditional on the five-loop estimate.' While the zero-parameter comparison (with all κ̂ fixed) provides independent strong evidence for the four-loop term (χ²=57.4 vs 190.6 with κ̂₃=0), the claimed 'extraction' of κ̂₃ is not as robust as for the lower loops. The discussion in the main text—particularly in the abstract and the concluding paragraph—could give the impression that all four fluctuation corrections are extracted with equal reliability. I recommend that the main text clearly distinguish between the unconditional evidence (the zero-parameter test) and the conditional extraction, and that the abstract be amended to reflect that the four-loop extraction is conditional on the five-loop prior. The authors acknowledge this: “It is compatible if in addition|pull|≤2, the fit hasp≥0.01, and the result is stable underL min = 35, 40 and one more remainder term.”
[Appendix C, p.16, slope test] The slope test for d_L/s_L is stated to be conditional because the Hill index of the influence functions falls below 2. This conditionality appears to affect only the slope coefficients (e₂, e₃) and does not impact the central instanton parameters. However, the paper should briefly clarify in the main text that the slope test is less robust than the action and exponent determinations, to avoid overstatement.

Minor objections (8)

[Abstract] The phrase 'non-perturbative phenomenon' should be changed to 'non-perturbative phenomena' (plural).
[p.5, Discussion paragraph] 'allowing us ot overcome' contains a typo: 'ot' should be 'to'.
[p.5, line after Eq. (6)] 'the variance still increase' should be 'the variance still increases'.
[p.6, Fig. 4 caption] The caption refers to 'diagram sum' but the figure label is 'diagrams'. Please make consistent.
[p.7, Table I] The column headings could be clearer: 'Perturbation' is used as a column label, but it refers to the perturbative extraction. Consider 'Extracted from series' or 'Perturbative extraction'.
[Appendix A, Eq. (A6)] The definition of the weight w in terms of u variables is given; but the reader might benefit from a brief reminder that u_e = 1 - e^{-α_e} and that the Jacobian is accounted for in the measure. This is already clear in the main text, but a one-line note would help.
[Appendix B, p.12, Eq. (B6)] The summation over l in the partial-wave expression: it may be useful to note that the sum is truncated in practice; the text later mentions 'summed to l=11' for the hardest graphs, but a forward reference would improve readability.
[References] Reference [33] is a whimsical note ('Yale's age'); while this is harmless, it is unconventional for a journal article. Consider moving or removing.

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The paper

The paper claims

325-loop perturbative series in 3D φ³ theory determines instanton parameters with 0.03% precision, confirming semi-classical predictions within 1.5σ.

The reading

This paper uses a new annealed importance sampling method for tropical field theory to compute the perturbative two-point function of massive three-dimensional φ³ theory through 325 loops. From these coefficients, the authors extract the instanton action, the factorial-growth exponent, the one-loop normalization, and fluctuation corrections up to four loops. All extracted parameters agree with independent semi-classical calculations within 1.5σ, demonstrating that perturbation theory at high orders can serve as a quantitative probe of non-perturbative instanton structure.

What carries the argument

The annealed importance sampling framework on the moduli space of tropical curves, which overcomes the exponential weight-variance barrier that stalls direct tropical sampling. It uses a sequence of intermediate distributions interpolating between the exactly sampled tropical measure and the physical integrand, enabling precise Monte Carlo estimates of perturbative coefficients up to 325 loops. The loop-insertion identity further allows efficient extraction of consecutive ratios, yielding complete coefficients with full covariance.

Extended reading notes

Core claim

The perturbative series for the two-point function in massive 3D φ³ theory, computed to 325 loops using an annealed tropical sampling technique, encodes the instanton sector with such precision that the instanton action S₀ = 174.6409 is determined to 0.03%, the collective-coordinate exponent b = 5/2 is confirmed, the one-loop normalization Cₛ = 85.1315 is extracted, and the fluctuation corrections κ₁, κ₂, κ₃ match semi-classical calculations. This shows that the large-order divergence of perturbation theory faithfully reproduces the non-perturbative instanton contribution, without any adjustable parameters.

Load-bearing premise

The semi-classical calculations were not carried out blind to the diagrammatic data, so the agreement at four loops depends on an auxiliary five-loop Monte Carlo prior, meaning the comparison is not fully independent.

Editorial extensions

If this is right

If correct, this work establishes that high-order perturbation theory in interacting field theories can be practically computed to hundreds of loops, providing direct access to non-perturbative sectors.
The method extends to other three-dimensional theories (φ⁴, O(N) models) and potentially to renormalizable theories like QED and QCD after handling subdivergences and fermions.
The extracted instanton parameters can be used to continue the perturbative series across Stokes barriers, enabling analytic continuation to critical points such as the Lee–Yang edge singularity.
The agreement between perturbative and semi-classical calculations validates the resurgence framework as a practical tool in quantum field theory beyond toy models.

Reading between the lines

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

The annealed importance sampling technique could be adapted to compute high-order perturbative coefficients in other field theories where factorial growth of diagrams is a bottleneck, such as four-dimensional gauge theories, provided renormalization and numerator handling are incorporated.
The discrepancy in four-loop fluctuation corrections (1.5σ) may hint at either underestimated errors in the semi-classical calculations or the need for higher-loop corrections beyond four loops; a dedicated blind comparison would strengthen confidence.
The ability to determine instanton parameters to sub-percent precision from perturbation alone suggests that the same strategy could be used to measure non-perturbative data in theories where direct semi-classical calculations are unavailable, such as theories with multiple instantons or renormalon contributions.

Original abstract

The high order behavior of perturbation theory provides insight into non-perturbative phenomenon in quantum field theory. While extensively studied in simplified models, high order behavior is challenging to access in more realistic field theories due to the complexity of the standard Feynman diagrammatic expansion. Building on recent progress in global sampling on the moduli space of tropical curves, we introduce an annealed importance sampling framework for tropical field theory. Our approach overcomes the exponential weight-variance barrier, allowing precise numerical calculations to high orders. We apply our technique to compute the perturbative series for the two-point function in the massive three-dimensional $\phi^3$ theory through 325 loops. These coefficients determine the instanton action in $\phi^3$ theory to 0.03\% precision, confirm the collective-coordinate exponent, and extract the one-loop determinant and the instanton fluctuation corrections through four loops, in remarkable agreement with direct semiclassical calculations. Our approach is applicable to a variety of related field theories, facilitating the quantitative exploration of their high loop behavior.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-28: 1 request, 5,627 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.94 · mentions code or data availability 0.03.

The abstract, sentence by sentence

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

The high order behavior of perturbation theory provides insight into non-perturbative phenomenon in quantum field theory. While extensively studied in simplified models, high order behavior is challenging to access in more realistic field theories due to the complexity of the standard Feynman diagrammatic expansion. Building on recent progress in global sampling on the moduli space of tropical curves, we introduce an annealed importance sampling framework for tropical field theory. Our approach overcomes the exponential weight-variance barrier, allowing precise numerical calculations to high orders. We apply our technique to compute the perturbative series for the two-point function in the massive three-dimensional $\phi^3$ theory through 325 loops. These coefficients determine the instanton action in $\phi^3$ theory to 0.03\% precision, confirm the collective-coordinate exponent, and extract the one-loop determinant and the instanton fluctuation corrections through four loops, in remarkable agreement with direct semiclassical calculations. Our approach is applicable to a variety of related field theories, facilitating the quantitative exploration of their high loop behavior.

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	The high order behavior of perturbation theory provides insight into non-perturbative phenomenon in quantum field theory.	background 1.00	0.05	0.03	asserted 0.95
s1	While extensively studied in simplified models, high order behavior is challenging to access in more realistic field theories due to the complexity of the standard Feynman diagrammatic expansion.	gap 0.55	0.04	0.02	asserted 1.00
s2	Building on recent progress in global sampling on the moduli space of tropical curves, we introduce an annealed importance sampling framework for tropical field theory.	contribution 0.51	0.88	0.05	asserted 1.00
s3	Our approach overcomes the exponential weight-variance barrier, allowing precise numerical calculations to high orders.	contribution 0.47	0.86	0.10	asserted 1.00
s4	We apply our technique to compute the perturbative series for the two-point function in the massive three-dimensional $\phi^3$ theory through 325 loops.	contribution 0.81	0.97	0.12	asserted 1.00
s5	These coefficients determine the instanton action in $\phi^3$ theory to 0.03\% precision, confirm the collective-coordinate exponent, and extract the one-loop determinant and the instanton fluctuation corrections through four loops, in remarkable agreement with direct semiclassical calculations.	result 1.00	0.95	0.79	asserted 1.00
s6	Our approach is applicable to a variety of related field theories, facilitating the quantitative exploration of their high loop behavior.	future work 0.84	0.38	0.08	asserted 0.94

What kind of paper the classifier reports this is: method or tool 0.98 · evidence from simulation or computation 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

not run

S sections
T body sentences
C citations
R cross-paper relations

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

Figures

**Figure 1.** FIG. 1. The 1PI two-point graphs through two loops. (a)

**Figure 2.** FIG. 2. Annealed tropical sampling. (a) Distribution of

**Figure 5.** FIG. 5. Borel plane of the self-energy, with

**Figure 6.** FIG. 6. Tropical structures of the kite, with the labels of Fig.

**Figure 7.** FIG. 7. Global tropical sampling. (a) The recursion (

**Figure 8.** FIG. 8. Annealed sampling. (a) Distribution of ln

**Figure 9.** FIG. 9. Feynman rules for the fluctuations around the instanton, in the spectral gauge. (a) The propagator. (b) The cubic

**Figure 10.** FIG. 10. The five two-loop vacuum graphs, with their symmetry factors

Showing 9 of the paper's 10 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.36025 by the authors.

References

Works cited (64)

[1]

All 1PI diagrams with two or more external legs are then UV and infrared finite form 0 >0. We setm 0 = 1 and expand the self-energy Σ(p), defined by the full 2 (a) (b) (c) α1 α2 α3 α4 α5 a b (d) α1 α3 α1α3∈U (e) α2 α3 α5 α2α3α5∈UR (f) FIG. 1. The 1PI two-point graphs through two loops. (a) the one-loop bubble, (b) the two-loop kite, (c) the two-loop self-...

View on Pith 1944
[2]

C. M. Bender and T. T. Wu, Anharmonic oscillator, Phys. Rev.184, 1231 (1969)

View on Pith 1969
[3]

C. M. Bender and T. T. Wu, Anharmonic oscillator. II. A Study of perturbation theory in large order, Phys. Rev. D7, 1620 (1973)

View on Pith 1973
[4]

L. N. Lipatov, Divergence of the perturbation-theory se- ries and the quasi-classical theory, Sov. Phys. JETP45, 216 (1977), [Zh. Eksp. Teor. Fiz. 72, 411 (1977)]

View on Pith 1977
[5]

E. Br´ ezin, J. C. Le Guillou, and J. Zinn-Justin, Pertur- bation Theory at Large Order. I. Theϕ 2N Interaction, Phys. Rev. D15, 1544 (1977)

View on Pith 1977
[6]

E. Br´ ezin, J. C. Le Guillou, and J. Zinn-Justin, Pertur- bation Theory at Large Order. II. Role of the Vacuum Instability, Phys. Rev. D15, 1558 (1977)

View on Pith 1977
[7]

J. C. Le Guillou and J. Zinn-Justin, eds.,Large-Order Be- haviour of Perturbation Theory, Current Physics–Sources and Comments, Vol. 7 (North-Holland, Amsterdam, 1990)

View on Pith 1990
[8]

G. ’t Hooft, Can We Make Sense Out of Quantum Chro- modynamics?, Subnucl. Ser.15, 943 (1979)

View on Pith 1979
[9]

M. Beneke, Renormalons, Phys. Rept.317, 1 (1999), arXiv:hep-ph/9807443

View on Pith Pith review arXiv 1999
[10]

I. Aniceto, G. Ba¸ sar, and R. Schiappa, A Primer on Resurgent Transseries and Their Asymptotics, Phys. Rept.809, 1 (2019), arXiv:1802.10441 [hep-th]

View on Pith Pith review arXiv 2019
[11]

M. Mari˜ no,Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory (Cambridge University Press, 2015)

View on Pith 2015
[12]

G. V. Dunne and M. ¨Unsal, Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model, JHEP 2012(11), 170, arXiv:1210.2423 [hep-th]

View on Pith Pith review arXiv 2012
[13]

P. C. Argyres and M. ¨Unsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, in- stanton, bion, and renormalon effects, JHEP2012(08), 063, arXiv:1206.1890 [hep-th]

View on Pith Pith review arXiv
[14]

J. Zinn-Justin and U. D. Jentschura, Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions, Annals Phys.313, 197 (2004), arXiv:quant-ph/0501136

View on Pith Pith review arXiv 2004
[15]

J. Zinn-Justin and U. D. Jentschura, Multi-instantons and exact results II: Specific cases, higher-order ef- fects, and numerical calculations, Annals Phys.313, 269 (2004), arXiv:quant-ph/0501137

View on Pith Pith review arXiv 2004
[16]

M. A. Escobar-Ruiz, E. Shuryak, and A. V. Turbiner, Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential, Phys. Rev. D92, 025046 (2015), [Erratum: Phys. Rev. D 92, 089902 (2015)], arXiv:1501.03993 [hep-th]

View on Pith Pith review arXiv 2015
[17]

L. T. Giorgini, U. D. Jentschura, E. M. Malatesta, G. Parisi, T. Rizzo, and J. Zinn-Justin, Correlation func- tions of the anharmonic oscillator: Numerical verification of two-loop corrections to the large-order behavior, Phys. Rev. D105, 105012 (2022), arXiv:2111.12765 [hep-th]

View on Pith Pith review arXiv 2022
[18]

M. Mari˜ no and T. Reis, Renormalons in integrable field theories, JHEP2020(04), 160, arXiv:1909.12134 [hep- th]

View on Pith Pith review arXiv 1909
[19]

M. Mari˜ no, R. Schiappa, and M. Weiss, Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings, Commun. Num. Theor. Phys.2, 349 (2008), arXiv:0711.1954 [hep-th]

View on Pith Pith review arXiv 2008
[20]

M. Borinsky, G. V. Dunne, and M. Meynig, Semiclas- sical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations:ϕ 3 QFT in 6 Dimensions, SIGMA17, 087 (2021), arXiv:2104.00593 [hep-th]

View on Pith Pith review arXiv 2021
[21]

M. Borinsky and D. Broadhurst, Resonant resurgent asymptotics from quantum field theory, Nucl. Phys. B 981, 115861 (2022), arXiv:2202.01513 [hep-th]

View on Pith Pith review arXiv 2022
[22]

P.-H. Balduf and E. Panzer, Renormalized tropical field theory (2025), arXiv:2512.21091 [math-ph] 7

View on Pith arXiv 2025
[23]

G. S. Bali, C. Bauer, and A. Pineda, Perturbative ex- pansion of the plaquette toO(α 35) in four-dimensional SU(3) gauge theory, Phys. Rev. D89, 054505 (2014), arXiv:1401.7999 [hep-ph]

View on Pith Pith review arXiv 2014
[24]

G. S. Bali, C. Bauer, and A. Pineda, Model- independent determination of the gluon condensate in four-dimensional SU(3) gauge theory, Phys. Rev. Lett. 113, 092001 (2014), arXiv:1403.6477 [hep-ph]

View on Pith Pith review arXiv 2014
[25]

J. C. Le Guillou and J. Zinn-Justin, Critical Exponents from Field Theory, Phys. Rev. B21, 3976 (1980)

View on Pith 1980
[26]

M. Serone, G. Spada, and G. Villadoro,λϕ 4 theory — Part I. The symmetric phase beyond NNNNNNNNLO, JHEP2018(08), 148, arXiv:1805.05882 [hep-th]

View on Pith Pith review arXiv
[27]

N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamon- don, and H. Thomas, All loop scattering for all multiplic- ity, JHEP09, 033, arXiv:2311.09284 [hep-th]

View on Pith Pith review arXiv
[28]

N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamon- don, and H. Thomas, All loop scattering as a counting problem, JHEP08, 194, arXiv:2309.15913 [hep-th]

View on Pith Pith review arXiv
[29]

M. Borinsky, Tropicalized quantum field theory and global tropical sampling, Ann. Henri Poincar´ e 10.1007/s00023-026-01695-y (2026), arXiv:2508.14263 [math-ph]

View on Pith Pith review arXiv 2026
[30]

G. Salvatori, All loop scattering as a sampling problem, JHEP2026(02), 196, arXiv:2503.07707 [hep-th]

View on Pith Pith review arXiv
[31]

R. M. Neal, Annealed importance sampling, Stat. Com- put.11, 125 (2001)

View on Pith 2001
[32]

C. Jarzynski, Nonequilibrium Equality for Free En- ergy Differences, Phys. Rev. Lett.78, 2690 (1997), arXiv:cond-mat/9610209

View on Pith Pith review arXiv 1997
[33]

M. Borinsky, J. Radkovski, and S. R. Kannagi, Tropical QFT and the sine-Gordon model (2026), arXiv:2610.12653 [hep-th]

View on Pith Pith review arXiv 2026
[34]

This particular order was chosen to match the age of Yale. Higher loops can be achieved as desired

View on Pith
[35]

M. Borinsky, J. A. Gracey, M. V. Kompaniets, and O. Schnetz, Five-loop renormalization ofϕ 3 theory with applications to the Lee-Yang edge singularity and per- colation theory, Phys. Rev. D103, 116024 (2021), arXiv:2103.16224 [hep-th]

View on Pith Pith review arXiv 2021
[36]

O. Schnetz,ϕ 3 theory at six loops, Phys. Rev. D112, 016028 (2025), arXiv:2505.15485 [hep-th]

View on Pith Pith review arXiv 2025
[37]

A. Houghton, J. S. Reeve, and D. J. Wallace, High-order behavior inϕ 3 field theories and the percolation problem, Phys. Rev. B17, 2956 (1978)

View on Pith 1978
[38]

A. J. McKane, Vacuum Instability in Scalar Field Theo- ries, Nucl. Phys. B152, 166 (1979)

View on Pith 1979
[39]

E. M. Malatesta, G. Parisi, and T. Rizzo, Two-loop cor- rections to large order behavior ofφ 4 theory, Nucl. Phys. B922, 293 (2017), arXiv:1704.04458 [cond-mat.stat- mech]

View on Pith Pith review arXiv 2017
[40]

L. T. Giorgini, U. D. Jentschura, E. M. Malatesta, T. Rizzo, and J. Zinn-Justin, Instantons inϕ 4 theo- ries: Transseries, virial theorems, and numerical aspects, Phys. Rev. D110, 036003 (2024), arXiv:2405.18191 [hep- th]

View on Pith Pith review arXiv 2024
[41]

M. E. Fisher, Yang-Lee Edge Singularity andϕ 3 Field Theory, Phys. Rev. Lett.40, 1610 (1978)

View on Pith 1978
[42]

J. A. Gracey, Six loop critical exponent analysis for Lee- Yang and percolation theory, Phys. Rev. D112, 105019 (2025), arXiv:2510.05723 [hep-th]

View on Pith arXiv 2025
[43]

C. Bogner and S. Weinzierl, Feynman graph polynomials, Int. J. Mod. Phys. A25, 2585 (2010), arXiv:1002.3458 [hep-ph]

View on Pith Pith review arXiv 2010
[44]

A. K. Rajantie, Feynman diagrams to three loops in three-dimensional field theory, Nucl. Phys. B480, 729 (1996), [Erratum: Nucl. Phys. B 513, 761–762 (1998)], arXiv:hep-ph/9606216

View on Pith Pith review arXiv 1996
[45]

A. I. Davydychev and J. B. Tausk, Two loop selfenergy diagrams with different masses and the momentum ex- pansion, Nucl. Phys. B397, 123 (1993)

View on Pith 1993
[46]

P. Cvitanovi´ c, B. E. Lautrup, and R. B. Pearson, The Number and Weights of Feynman Diagrams, Phys. Rev. D18, 1939 (1978)

View on Pith 1978
[47]

M. Borinsky, Tropical Monte Carlo quadrature for Feyn- man integrals, Ann. Inst. H. Poincare D Comb. Phys. Interact.10, 635 (2023), arXiv:2008.12310 [math-ph]

View on Pith Pith review arXiv 2023
[48]

I. Moult and B. Wang, Tropical Fan Sampler: interactive diagram, global and annealed tropical sampling,https: //boyuuwang-oss.github.io/tropical-fan-sampler/ (2026)

View on Pith 2026
[49]

B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys.68, 209 (1979)

View on Pith 1979
[50]

B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry of pos- itive solutions of nonlinear elliptic equations inR n, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Stud., Vol. 7a, edited by L. Nachbin (Aca- demic Press, New York, 1981) pp. 369–402

View on Pith 1981
[51]

M. K. Kwong, Uniqueness of positive solutions of ∆u− u+u p = 0 inR n, Arch. Ration. Mech. Anal.105, 243 (1989)

View on Pith 1989
[52]

S. R. Coleman, The Fate of the False Vacuum. 1. Semi- classical Theory, Phys. Rev. D15, 2929 (1977), [Erratum: Phys. Rev. D 16, 1248 (1977)]

View on Pith 1977
[53]

C. G. Callan, Jr. and S. R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D16, 1762 (1977)

View on Pith 1977
[54]

M. Dine, R. G. Leigh, P. Y. Huet, A. D. Linde, and D. A. Linde, Towards the theory of the electroweak phase transition, Phys. Rev. D46, 550 (1992), arXiv:hep- ph/9203203

View on Pith 1992
[55]

I. M. Gel’fand and A. M. Yaglom, Integration in func- tional spaces and its applications in quantum physics, J. Math. Phys.1, 48 (1960)

View on Pith 1960
[56]

G. V. Dunne and K. Kirsten, Functional determinants for radial operators, J. Phys. A39, 11915 (2006), arXiv:hep- th/0607066

View on Pith 2006
[57]

G. V. Dunne and H. Min, Beyond the thin-wall approx- imation: Precise numerical computation of prefactors in false vacuum decay, Phys. Rev. D72, 125004 (2005), arXiv:hep-th/0511156

View on Pith Pith review arXiv 2005
[58]

A. Ekstedt, O. Gould, and J. Hirvonen, BubbleDet: a Python package to compute functional determi- nants for bubble nucleation, JHEP2023(12), 056, arXiv:2308.15652 [hep-ph]

View on Pith Pith review arXiv
[59]

O. Gould, A. Kormu, and D. J. Weir, Nonper- turbative test of nucleation calculations for strong phase transitions, Phys. Rev. D111, L051901 (2025), arXiv:2404.01876 [hep-th]

View on Pith arXiv 2025
[60]

A. J. McKane, Perturbation expansions at large order: Results for scalar field theories revisited, J. Phys. A52, 055401 (2019), arXiv:1807.00656 [hep-th]

View on Pith Pith review arXiv 2019
[61]

J. Zinn-Justin,Quantum Field Theory and Critical Phe- nomena, 4th ed., International Series of Monographs on Physics, Vol. 113 (Oxford University Press, Oxford, 2002)

View on Pith 2002
[62]

C. N. Yang and T. D. Lee, Statistical theory of equations of state and phase transitions. 1. Theory of condensation, 8 Phys. Rev.87, 404 (1952)

View on Pith 1952
[63]

T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. 2. Lattice gas and Ising model, Phys. Rev.87, 410 (1952)

View on Pith 1952
[64]

U. D. Jentschura, A. Surzhykov, and J. Zinn-Justin, Multi-instantons and exact results. III: Unification of even and odd anharmonic oscillators, Annals Phys.325, 1135 (2010), arXiv:1001.3910 [math-ph] Supplemental Materials Appendix A: T ropical methods With the measure d 3k/π3/2 per loop, anL-loop graphGwithE= 3L−1 internal edges contributesy LΣG(p) to t...

View on Pith Pith review arXiv 2010

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] [1]

All 1PI diagrams with two or more external legs are then UV and infrared finite form 0 >0. We setm 0 = 1 and expand the self-energy Σ(p), defined by the full 2 (a) (b) (c) α1 α2 α3 α4 α5 a b (d) α1 α3 α1α3∈U (e) α2 α3 α5 α2α3α5∈UR (f) FIG. 1. The 1PI two-point graphs through two loops. (a) the one-loop bubble, (b) the two-loop kite, (c) the two-loop self-...

View on Pith 1944

[2] [2]

C. M. Bender and T. T. Wu, Anharmonic oscillator, Phys. Rev.184, 1231 (1969)

View on Pith 1969

[3] [3]

C. M. Bender and T. T. Wu, Anharmonic oscillator. II. A Study of perturbation theory in large order, Phys. Rev. D7, 1620 (1973)

View on Pith 1973

[4] [4]

L. N. Lipatov, Divergence of the perturbation-theory se- ries and the quasi-classical theory, Sov. Phys. JETP45, 216 (1977), [Zh. Eksp. Teor. Fiz. 72, 411 (1977)]

View on Pith 1977

[5] [5]

E. Br´ ezin, J. C. Le Guillou, and J. Zinn-Justin, Pertur- bation Theory at Large Order. I. Theϕ 2N Interaction, Phys. Rev. D15, 1544 (1977)

View on Pith 1977

[6] [6]

E. Br´ ezin, J. C. Le Guillou, and J. Zinn-Justin, Pertur- bation Theory at Large Order. II. Role of the Vacuum Instability, Phys. Rev. D15, 1558 (1977)

View on Pith 1977

[7] [7]

J. C. Le Guillou and J. Zinn-Justin, eds.,Large-Order Be- haviour of Perturbation Theory, Current Physics–Sources and Comments, Vol. 7 (North-Holland, Amsterdam, 1990)

View on Pith 1990

[8] [8]

G. ’t Hooft, Can We Make Sense Out of Quantum Chro- modynamics?, Subnucl. Ser.15, 943 (1979)

View on Pith 1979

[9] [9]

M. Beneke, Renormalons, Phys. Rept.317, 1 (1999), arXiv:hep-ph/9807443

View on Pith Pith review arXiv 1999

[10] [10]

I. Aniceto, G. Ba¸ sar, and R. Schiappa, A Primer on Resurgent Transseries and Their Asymptotics, Phys. Rept.809, 1 (2019), arXiv:1802.10441 [hep-th]

View on Pith Pith review arXiv 2019

[11] [11]

M. Mari˜ no,Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory (Cambridge University Press, 2015)

View on Pith 2015

[12] [12]

G. V. Dunne and M. ¨Unsal, Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model, JHEP 2012(11), 170, arXiv:1210.2423 [hep-th]

View on Pith Pith review arXiv 2012

[13] [13]

P. C. Argyres and M. ¨Unsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, in- stanton, bion, and renormalon effects, JHEP2012(08), 063, arXiv:1206.1890 [hep-th]

View on Pith Pith review arXiv

[14] [14]

J. Zinn-Justin and U. D. Jentschura, Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions, Annals Phys.313, 197 (2004), arXiv:quant-ph/0501136

View on Pith Pith review arXiv 2004

[15] [15]

J. Zinn-Justin and U. D. Jentschura, Multi-instantons and exact results II: Specific cases, higher-order ef- fects, and numerical calculations, Annals Phys.313, 269 (2004), arXiv:quant-ph/0501137

View on Pith Pith review arXiv 2004

[16] [16]

M. A. Escobar-Ruiz, E. Shuryak, and A. V. Turbiner, Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential, Phys. Rev. D92, 025046 (2015), [Erratum: Phys. Rev. D 92, 089902 (2015)], arXiv:1501.03993 [hep-th]

View on Pith Pith review arXiv 2015

[17] [17]

L. T. Giorgini, U. D. Jentschura, E. M. Malatesta, G. Parisi, T. Rizzo, and J. Zinn-Justin, Correlation func- tions of the anharmonic oscillator: Numerical verification of two-loop corrections to the large-order behavior, Phys. Rev. D105, 105012 (2022), arXiv:2111.12765 [hep-th]

View on Pith Pith review arXiv 2022

[18] [18]

M. Mari˜ no and T. Reis, Renormalons in integrable field theories, JHEP2020(04), 160, arXiv:1909.12134 [hep- th]

View on Pith Pith review arXiv 1909

[19] [19]

M. Mari˜ no, R. Schiappa, and M. Weiss, Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings, Commun. Num. Theor. Phys.2, 349 (2008), arXiv:0711.1954 [hep-th]

View on Pith Pith review arXiv 2008

[20] [20]

M. Borinsky, G. V. Dunne, and M. Meynig, Semiclas- sical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations:ϕ 3 QFT in 6 Dimensions, SIGMA17, 087 (2021), arXiv:2104.00593 [hep-th]

View on Pith Pith review arXiv 2021

[21] [21]

M. Borinsky and D. Broadhurst, Resonant resurgent asymptotics from quantum field theory, Nucl. Phys. B 981, 115861 (2022), arXiv:2202.01513 [hep-th]

View on Pith Pith review arXiv 2022

[22] [22]

P.-H. Balduf and E. Panzer, Renormalized tropical field theory (2025), arXiv:2512.21091 [math-ph] 7

View on Pith arXiv 2025

[23] [23]

G. S. Bali, C. Bauer, and A. Pineda, Perturbative ex- pansion of the plaquette toO(α 35) in four-dimensional SU(3) gauge theory, Phys. Rev. D89, 054505 (2014), arXiv:1401.7999 [hep-ph]

View on Pith Pith review arXiv 2014

[24] [24]

G. S. Bali, C. Bauer, and A. Pineda, Model- independent determination of the gluon condensate in four-dimensional SU(3) gauge theory, Phys. Rev. Lett. 113, 092001 (2014), arXiv:1403.6477 [hep-ph]

View on Pith Pith review arXiv 2014

[25] [25]

J. C. Le Guillou and J. Zinn-Justin, Critical Exponents from Field Theory, Phys. Rev. B21, 3976 (1980)

View on Pith 1980

[26] [26]

M. Serone, G. Spada, and G. Villadoro,λϕ 4 theory — Part I. The symmetric phase beyond NNNNNNNNLO, JHEP2018(08), 148, arXiv:1805.05882 [hep-th]

View on Pith Pith review arXiv

[27] [27]

N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamon- don, and H. Thomas, All loop scattering for all multiplic- ity, JHEP09, 033, arXiv:2311.09284 [hep-th]

View on Pith Pith review arXiv

[28] [28]

N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamon- don, and H. Thomas, All loop scattering as a counting problem, JHEP08, 194, arXiv:2309.15913 [hep-th]

View on Pith Pith review arXiv

[29] [29]

M. Borinsky, Tropicalized quantum field theory and global tropical sampling, Ann. Henri Poincar´ e 10.1007/s00023-026-01695-y (2026), arXiv:2508.14263 [math-ph]

View on Pith Pith review arXiv 2026

[30] [30]

G. Salvatori, All loop scattering as a sampling problem, JHEP2026(02), 196, arXiv:2503.07707 [hep-th]

View on Pith Pith review arXiv

[31] [31]

R. M. Neal, Annealed importance sampling, Stat. Com- put.11, 125 (2001)

View on Pith 2001

[32] [32]

C. Jarzynski, Nonequilibrium Equality for Free En- ergy Differences, Phys. Rev. Lett.78, 2690 (1997), arXiv:cond-mat/9610209

View on Pith Pith review arXiv 1997

[33] [33]

M. Borinsky, J. Radkovski, and S. R. Kannagi, Tropical QFT and the sine-Gordon model (2026), arXiv:2610.12653 [hep-th]

View on Pith Pith review arXiv 2026

[34] [34]

This particular order was chosen to match the age of Yale. Higher loops can be achieved as desired

View on Pith

[35] [35]

M. Borinsky, J. A. Gracey, M. V. Kompaniets, and O. Schnetz, Five-loop renormalization ofϕ 3 theory with applications to the Lee-Yang edge singularity and per- colation theory, Phys. Rev. D103, 116024 (2021), arXiv:2103.16224 [hep-th]

View on Pith Pith review arXiv 2021

[36] [36]

O. Schnetz,ϕ 3 theory at six loops, Phys. Rev. D112, 016028 (2025), arXiv:2505.15485 [hep-th]

View on Pith Pith review arXiv 2025

[37] [37]

A. Houghton, J. S. Reeve, and D. J. Wallace, High-order behavior inϕ 3 field theories and the percolation problem, Phys. Rev. B17, 2956 (1978)

View on Pith 1978

[38] [38]

A. J. McKane, Vacuum Instability in Scalar Field Theo- ries, Nucl. Phys. B152, 166 (1979)

View on Pith 1979

[39] [39]

E. M. Malatesta, G. Parisi, and T. Rizzo, Two-loop cor- rections to large order behavior ofφ 4 theory, Nucl. Phys. B922, 293 (2017), arXiv:1704.04458 [cond-mat.stat- mech]

View on Pith Pith review arXiv 2017

[40] [40]

L. T. Giorgini, U. D. Jentschura, E. M. Malatesta, T. Rizzo, and J. Zinn-Justin, Instantons inϕ 4 theo- ries: Transseries, virial theorems, and numerical aspects, Phys. Rev. D110, 036003 (2024), arXiv:2405.18191 [hep- th]

View on Pith Pith review arXiv 2024

[41] [41]

M. E. Fisher, Yang-Lee Edge Singularity andϕ 3 Field Theory, Phys. Rev. Lett.40, 1610 (1978)

View on Pith 1978

[42] [42]

J. A. Gracey, Six loop critical exponent analysis for Lee- Yang and percolation theory, Phys. Rev. D112, 105019 (2025), arXiv:2510.05723 [hep-th]

View on Pith arXiv 2025

[43] [43]

C. Bogner and S. Weinzierl, Feynman graph polynomials, Int. J. Mod. Phys. A25, 2585 (2010), arXiv:1002.3458 [hep-ph]

View on Pith Pith review arXiv 2010

[44] [44]

A. K. Rajantie, Feynman diagrams to three loops in three-dimensional field theory, Nucl. Phys. B480, 729 (1996), [Erratum: Nucl. Phys. B 513, 761–762 (1998)], arXiv:hep-ph/9606216

View on Pith Pith review arXiv 1996

[45] [45]

A. I. Davydychev and J. B. Tausk, Two loop selfenergy diagrams with different masses and the momentum ex- pansion, Nucl. Phys. B397, 123 (1993)

View on Pith 1993

[46] [46]

P. Cvitanovi´ c, B. E. Lautrup, and R. B. Pearson, The Number and Weights of Feynman Diagrams, Phys. Rev. D18, 1939 (1978)

View on Pith 1978

[47] [47]

M. Borinsky, Tropical Monte Carlo quadrature for Feyn- man integrals, Ann. Inst. H. Poincare D Comb. Phys. Interact.10, 635 (2023), arXiv:2008.12310 [math-ph]

View on Pith Pith review arXiv 2023

[48] [48]

I. Moult and B. Wang, Tropical Fan Sampler: interactive diagram, global and annealed tropical sampling,https: //boyuuwang-oss.github.io/tropical-fan-sampler/ (2026)

View on Pith 2026

[49] [49]

B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys.68, 209 (1979)

View on Pith 1979

[50] [50]

B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry of pos- itive solutions of nonlinear elliptic equations inR n, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Stud., Vol. 7a, edited by L. Nachbin (Aca- demic Press, New York, 1981) pp. 369–402

View on Pith 1981

[51] [51]

M. K. Kwong, Uniqueness of positive solutions of ∆u− u+u p = 0 inR n, Arch. Ration. Mech. Anal.105, 243 (1989)

View on Pith 1989

[52] [52]

S. R. Coleman, The Fate of the False Vacuum. 1. Semi- classical Theory, Phys. Rev. D15, 2929 (1977), [Erratum: Phys. Rev. D 16, 1248 (1977)]

View on Pith 1977

[53] [53]

C. G. Callan, Jr. and S. R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D16, 1762 (1977)

View on Pith 1977

[54] [54]

M. Dine, R. G. Leigh, P. Y. Huet, A. D. Linde, and D. A. Linde, Towards the theory of the electroweak phase transition, Phys. Rev. D46, 550 (1992), arXiv:hep- ph/9203203

View on Pith 1992

[55] [55]

I. M. Gel’fand and A. M. Yaglom, Integration in func- tional spaces and its applications in quantum physics, J. Math. Phys.1, 48 (1960)

View on Pith 1960

[56] [56]

G. V. Dunne and K. Kirsten, Functional determinants for radial operators, J. Phys. A39, 11915 (2006), arXiv:hep- th/0607066

View on Pith 2006

[57] [57]

G. V. Dunne and H. Min, Beyond the thin-wall approx- imation: Precise numerical computation of prefactors in false vacuum decay, Phys. Rev. D72, 125004 (2005), arXiv:hep-th/0511156

View on Pith Pith review arXiv 2005

[58] [58]

A. Ekstedt, O. Gould, and J. Hirvonen, BubbleDet: a Python package to compute functional determi- nants for bubble nucleation, JHEP2023(12), 056, arXiv:2308.15652 [hep-ph]

View on Pith Pith review arXiv

[59] [59]

O. Gould, A. Kormu, and D. J. Weir, Nonper- turbative test of nucleation calculations for strong phase transitions, Phys. Rev. D111, L051901 (2025), arXiv:2404.01876 [hep-th]

View on Pith arXiv 2025

[60] [60]

A. J. McKane, Perturbation expansions at large order: Results for scalar field theories revisited, J. Phys. A52, 055401 (2019), arXiv:1807.00656 [hep-th]

View on Pith Pith review arXiv 2019

[61] [61]

J. Zinn-Justin,Quantum Field Theory and Critical Phe- nomena, 4th ed., International Series of Monographs on Physics, Vol. 113 (Oxford University Press, Oxford, 2002)

View on Pith 2002

[62] [62]

C. N. Yang and T. D. Lee, Statistical theory of equations of state and phase transitions. 1. Theory of condensation, 8 Phys. Rev.87, 404 (1952)

View on Pith 1952

[63] [63]

T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. 2. Lattice gas and Ising model, Phys. Rev.87, 410 (1952)

View on Pith 1952

[64] [64]

U. D. Jentschura, A. Surzhykov, and J. Zinn-Justin, Multi-instantons and exact results. III: Unification of even and odd anharmonic oscillators, Annals Phys.325, 1135 (2010), arXiv:1001.3910 [math-ph] Supplemental Materials Appendix A: T ropical methods With the measure d 3k/π3/2 per loop, anL-loop graphGwithE= 3L−1 internal edges contributesy LΣG(p) to t...

View on Pith Pith review arXiv 2010