pith. sign in

arxiv: 2305.15561 · v1 · pith:APRKLZ5Inew · submitted 2023-05-24 · ❄️ cond-mat.stat-mech · hep-th

Quantum-field multiloop calculations in critical dynamics

Ella Ivanova , Georgii Kalagov , Marina Komarova , Mikhail Nalimov This is my paper

Pith reviewed 2026-05-24 08:49 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-th
keywords critical dynamicsrenormalization groupmultiloop calculationsBorel resummationinstanton analysisperturbation theoryquantum field theoryasymptotic series
0
0 comments X

The pith

Instanton analysis determines higher-order asymptotics of perturbation series in critical dynamics models, enabling Borel resummation like in static theories.

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

The paper establishes that multiloop computations in quantum-field models of critical dynamics produce asymptotic perturbation series that require Borel resummation for physical use. It demonstrates that the instanton method for extracting large-order behavior carries over from static to dynamical models without essential change. This transfer makes it possible to incorporate high-order terms reliably when processing results for critical exponents and scaling functions across a broad class of dynamic field theories.

Core claim

In dynamical critical phenomena the perturbation expansions remain asymptotic, so Borel resummation is needed; the higher-order asymptotics that control the resummation are fixed by instanton analysis, and this analysis applies directly to dynamical models in the same way it does to static ones.

What carries the argument

Instanton analysis applied to the calculation of large-order asymptotics in the perturbation series of dynamical field models.

If this is right

  • Multiloop results for dynamic critical exponents become resumable to the same accuracy level achieved in static models.
  • High-order contributions can be included systematically rather than truncated in calculations of scaling functions.
  • The same resummation framework and error estimates used in static theories become applicable to dynamical observables.
  • A wide class of field models exhibiting asymptotic expansions can be treated uniformly.

Where Pith is reading between the lines

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

  • The method opens the door to quantitative predictions for dynamic critical phenomena in systems where only low-order terms were previously usable.
  • If dynamical corrections to the instanton saddle turn out to be small, the same numerical prefactors and exponential factors can be reused across many models.
  • Validation could come from comparing resummed series against high-precision Monte Carlo data for specific dynamic universality classes.

Load-bearing premise

The instanton analysis technique developed for static theories transfers directly to dynamical models and yields reliable large-order asymptotics without additional dynamical corrections that would change the resummation.

What would settle it

An explicit instanton calculation performed in a concrete dynamical model, such as model A, that produces an asymptotic coefficient differing from the one extracted from the actual high-order terms of the perturbative series.

Figures

Figures reproduced from arXiv: 2305.15561 by Ella Ivanova, Georgii Kalagov, Marina Komarova, Mikhail Nalimov.

Figure 1
Figure 1. Figure 1: The Schwinger - Keldysh contour in the complex [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The second-order contribution to the four-point Green function. Here, the retarded [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The third-order contribution to the four-point Green function. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

The quantum-field renormalization group method is one of the most efficient and powerful tools for studying critical and scaling phenomena in interacting many-particle systems. The multiloop Feynman diagrams underpin the specific implementation of the renormalization group program. In recent years, multiloop computation has had a significant breakthrough in both static and dynamic models of critical behavior. In the paper, we focus on the state-of-the-art computational techniques for critical dynamic diagrams and the results obtained with their help. The generic nature of the evaluated physical observables in a wide class of field models is manifested in the asymptotic character of perturbation expansions. Thus, the Borel resummation of series is required to process multiloop results. Such a procedure also enables one to take high-order contributions into consideration properly. The paper outlines the resummation framework in dynamic models and the circumstances in which it can be useful. An important resummation criterion is the properties of the higher-order asymptotics of the perturbation theory. In static theories, these properties are determined by the method of instanton analysis. A similar approach is applicable in critical dynamics models. We describe the calculation of these asymptotics in dynamical models and present the results of the corresponding resummation.

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

1 major / 2 minor

Summary. The paper reviews recent breakthroughs in multiloop Feynman diagram computations for critical dynamics within the quantum-field renormalization-group framework. It covers state-of-the-art techniques for evaluating dynamic diagrams, emphasizes the asymptotic character of the resulting perturbative series for generic observables, outlines the Borel resummation procedure needed to extract physical information, and describes the application of instanton analysis to determine the large-order asymptotics in dynamical models, claiming that a similar approach to static theories is applicable and yields resummation results.

Significance. If the review faithfully summarizes the cited literature and the instanton-based asymptotics are correctly transferred, the manuscript offers a useful consolidated reference on handling high-order perturbative expansions in critical dynamics, potentially improving estimates of dynamic critical exponents and scaling functions across a range of models.

major comments (1)
  1. [resummation framework and instanton analysis description] The central claim that instanton analysis transfers directly to dynamical models (abstract and resummation section) is load-bearing for the presented resummation results, yet the manuscript does not explicitly demonstrate that the dynamical operator (e.g., ∂_t − ∇² in Model A) contributes no additional terms to the saddle-point action or one-loop determinant around the instanton; without this, the factorial growth rate and prefactors used for Borel resummation could differ from the static case.
minor comments (2)
  1. Notation for the effective action in dynamical models is introduced without a dedicated equation reference, making it harder to trace how the stochastic terms enter the instanton calculation.
  2. The abstract states that 'a similar approach is applicable' but the corresponding section would benefit from a short table comparing the leading instanton asymptotics between a static model and its dynamic counterpart.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment on the resummation framework. We address the point below.

read point-by-point responses

  1. Referee: The central claim that instanton analysis transfers directly to dynamical models (abstract and resummation section) is load-bearing for the presented resummation results, yet the manuscript does not explicitly demonstrate that the dynamical operator (e.g., ∂_t − ∇² in Model A) contributes no additional terms to the saddle-point action or one-loop determinant around the instanton; without this, the factorial growth rate and prefactors used for Borel resummation could differ from the static case.

    Authors: We appreciate the referee drawing attention to the need for an explicit verification of the transfer. The resummation section outlines the instanton analysis adapted to dynamical models and states that the time-dependent part of the operator yields no additional saddle-point or determinant contributions because the relevant instanton solutions remain static. However, the presentation of the intermediate steps is indeed concise. We will revise the section to include a more detailed derivation showing the absence of extra terms from the dynamical operator, thereby making the justification self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: descriptive review of external methods with no self-referential derivations

full rationale

The paper is a review describing multiloop techniques, Borel resummation, and instanton analysis for critical dynamics. It states that a similar approach to static theories is applicable but presents no new derivations, predictions, or equations that reduce by construction to inputs defined within the paper. No fitted parameters are renamed as predictions, no self-citations form load-bearing uniqueness claims, and no ansatzes are smuggled. The text remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No new free parameters, axioms, or invented entities are introduced; the text describes established methods.

pith-pipeline@v0.9.0 · 5748 in / 1020 out tokens · 12280 ms · 2026-05-24T08:49:13.409387+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages

  1. [1]

    On the theory of phase transitions. I.,

    L. Landau, “On the theory of phase transitions. I.,”Phys. Zs. Sowjet., vol. 11, p. 26, 1937

    work page 1937

  2. [2]

    On the theory of phase transitions. II.,

    L. Landau, “On the theory of phase transitions. II.,”Phys. Zs. Sowjet., vol. 547, p. 26, 1937

    work page 1937

  3. [3]

    The renormalization group and theϵ-expansion,

    K. G. Wilson and J. Kogut, “The renormalization group and theϵ-expansion,”Physics Reports, vol. 12, no. 2, pp. 75–199, 1974

    work page 1974

  4. [4]

    A. Z. Patashinskii and V. L. Pokrovskii,Fluctuation Theory of Phase Transitions. Perga- mon Press, 1979

    work page 1979

  5. [5]

    Ma,Modern Theory of Critical Phenomena

    S. Ma,Modern Theory of Critical Phenomena. Routledge, 2000

    work page 2000

  6. [6]

    Zinn-Justin,Quantum Field Theory and Critical Phenomena

    J. Zinn-Justin,Quantum Field Theory and Critical Phenomena. Clarendon Press, 1996

    work page 1996

  7. [7]

    Kleinert,Properties ofϕ4-Theories

    H. Kleinert,Properties ofϕ4-Theories. World Sci., Singapore, 2001

    work page 2001

  8. [8]

    A. N. Vasiliev,The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics. Chapman Hall, 2000

    work page 2000

  9. [9]

    Critical indices from perturbation analysis of the callan-symanzik equation,

    G. A. Baker, B. G. Nickel, and D. I. Meiron, “Critical indices from perturbation analysis of the callan-symanzik equation,”Phys. Rev. B, vol. 17, pp. 1365–1374, Feb 1978

    work page 1978

  10. [10]

    Critical thermodynamics of two-dimensional systems in the five-loop renormalization-group approximation,

    E. V. Orlov and A. I. Sokolov, “Critical thermodynamics of two-dimensional systems in the five-loop renormalization-group approximation,”Physics of the Solid State, vol. 42, pp. 2151–2158, 2000

    work page 2000

  11. [11]

    Critical exponentη in 2d o(n)-symmetric φ4-model up to 6 loops,

    L. T. Adzhemyan, Y. V. Kirienko, and M. V. Kompaniets, “Critical exponentη in 2d o(n)-symmetric φ4-model up to 6 loops,” 2 2016

    work page 2016

  12. [12]

    The nonperturbative functional renormalization group and its applications,

    N.Dupuis, L.Canet, A.Eichhorn, W.Metzner, J.M.Pawlowski, M.Tissier, andN.Wsche- bor, “The nonperturbative functional renormalization group and its applications,”Physics Reports, vol. 910, pp. 1–114, may 2021

    work page 2021

  13. [13]

    V. A. Smirnov,Feynman integral calculus. Springer Berlin Heidelberg, 2006

    work page 2006

  14. [14]

    Sector Decomposition,

    G. Heinrich, “Sector Decomposition,”International Journal of Modern Physics A, vol. 23, no. 10, pp. 1457–1486, 2008. 20

    work page 2008

  15. [15]

    Minimally subtracted six-loop renormalization ofO(n) -symmetric ϕ4 theory and critical exponents,

    M. V. Kompaniets and E. Panzer, “Minimally subtracted six-loop renormalization ofO(n) -symmetric ϕ4 theory and critical exponents,”Physical Review D, vol. 96, p. 036016, aug 2017

    work page 2017

  16. [16]

    The method of uniqueness, a new powerful technique for multiloop calcu- lations,

    D. I. Kazakov, “The method of uniqueness, a new powerful technique for multiloop calcu- lations,”Physics Letters B, vol. 133, no. 6, pp. 406–410, 1983

    work page 1983

  17. [17]

    Infrared R-operation and ultraviolet counterterms in the MS-scheme,

    K. G. Chetyrkin and F. V. Tkachov, “Infrared R-operation and ultraviolet counterterms in the MS-scheme,”Physics Letters B, vol. 114, no. 5, pp. 340–344, 1982

    work page 1982

  18. [18]

    Five-loop renormaliza- tion group calculations in thegϕ4 theory,

    K. G. Chetyrkin, S. G. Gorishny, S. A. Larin, and F. V. Tkachov, “Five-loop renormaliza- tion group calculations in thegϕ4 theory,”Physics Letters B, vol. 132, no. 4-6, pp. 351–354, 1983

    work page 1983

  19. [19]

    Five-loop calculations in thegϕ4 model and the critical indexη,

    K. G. Chetyrkin, A. L. Kataev, and F. V. Tkachov, “Five-loop calculations in thegϕ4 model and the critical indexη,”Physics Letters B, vol. 99, no. 2, pp. 147–150, 1981

    work page 1981

  20. [20]

    K. G. Chetyrkin, A. L. Kataev, and F. V. Tkachov, “Errata,”Physics Letters B, vol. 101, no. 6, pp. 457–458, 1981

    work page 1981

  21. [21]

    Theory of dynamic critical phenomena,

    P. C. Hohenberg and B. I. Halperin, “Theory of dynamic critical phenomena,”Reviews of Modern Physics, vol. 49, p. 435, jul 1977

    work page 1977

  22. [22]

    U. C. T¨ auber,Critical dynamics: A field theory approach to equilibrium and non- equilibrium scaling behavior. Cambridge University Press, 2012

    work page 2012

  23. [23]

    Critical dynamics: a field-theoretical approach,

    Y. Sutou, Y. Saito, S. Shindo, R. Folk, and G. Moser, “Critical dynamics: a field-theoretical approach,”Journal of Physics A: Mathematical and General, vol. 39, p. R207, 2006

    work page 2006

  24. [24]

    Quantum field renormalization group in the theory of fully developed turbulence,

    L. T. Adzhemyan, N. V. Antonov, and A. N. Vasil’ev, “Quantum field renormalization group in the theory of fully developed turbulence,”Physics-Uspekhi, vol. 39, pp. 1193– 1219, 12 1996

    work page 1996

  25. [25]

    Particles and fields in fluid turbulence,

    G. Falkovich, K. Gawedzki, and M. Vergassola, “Particles and fields in fluid turbulence,” Reviews of Modern Physics, vol. 73, p. 913, 11 2001

    work page 2001

  26. [26]

    Onuki,Phase Transition Dynamics

    A. Onuki,Phase Transition Dynamics. Cambridge University Press, 1st edition ed., 2002

    work page 2002

  27. [27]

    P. L. Krapivsky, S. Redner, and E. Ben-Naim,A kinetic view of statistical physics. Cam- bridge University Press, 1 2011

    work page 2011

  28. [28]

    M.Henkel, H.Hinrichsen, S.Lubeck, andM.Pleimling, Non-equilibrium phase transitions., vol. 1. Springer, 2009th edition ed., 2008

    work page 2008

  29. [29]

    Pruessner,Self-organised criticality: Theory, models and characterisation

    G. Pruessner,Self-organised criticality: Theory, models and characterisation. Cambridge University Press, 1 2012

    work page 2012

  30. [30]

    R´ ev´ esz,Random walk in random and non-random environments

    P. R´ ev´ esz,Random walk in random and non-random environments. World Scientific, 3 ed., 2013

    work page 2013

  31. [31]

    Formulation of the theory of turbulence in an incompressible fluid,

    H. W. Wyld, “Formulation of the theory of turbulence in an incompressible fluid,”Annals of Physics, vol. 14, pp. 143–165, 1 1961

    work page 1961

  32. [32]

    Statistical Dynamics of Classical Systems,

    P. C. Martin, E. D. Siggia, and H. A. Rose, “Statistical Dynamics of Classical Systems,” Physical Review A, vol. 8, no. 1, p. 423, 1973. 21

    work page 1973

  33. [33]

    On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties,

    H. K. Janssen, “On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties,”Zeitschrift f¨ ur Physik B Condensed Matter and Quanta, vol. 23, no. 4, pp. 377–380, 1976

    work page 1976

  34. [34]

    Technics of field renormalization and dynamics of critical phenomena,

    C. de Dominicis, “Technics of field renormalization and dynamics of critical phenomena,” J. Phys. (Paris), Colloq, pp. 247–253, 1976

    work page 1976

  35. [35]

    Effective large-scale model of boson gas from microscopic theory,

    J. Honkonen, M. V. Komarova, Y. G. Molotkov, and M. Y. Nalimov, “Effective large-scale model of boson gas from microscopic theory,”Nuclear Physics B, vol. 939, pp. 105–129, 2019

    work page 2019

  36. [36]

    Kinetic Theory of Boson Gas,

    J. Honkonen, M. V. Komarova, Y. G. Molotkov, and M. Y. Nalimov, “Kinetic Theory of Boson Gas,”Theoretical and Mathematical Physics, vol. 200, no. 3, pp. 1360–1373, 2019

    work page 2019

  37. [37]

    Brownian Motion of a Quantum Oscillator,

    J. Schwinger, “Brownian Motion of a Quantum Oscillator,”Journal of Mathematical Physics, vol. 2, no. 2, p. 407, 1961

    work page 1961

  38. [38]

    Diagram technique for nonequilibrium processes,

    L. V. Keldysh, “Diagram technique for nonequilibrium processes,”Soviet Physics JETP, vol. 20, pp. 1018–1026, 1965

    work page 1965

  39. [39]

    Critical dynam- ics of the superfluid phase transition: Multiloop calculation of the microscopic model,

    J. Honkonen, M. Komarova, Y. Molotkov, M. Nalimov, and A. Trenogin, “Critical dynam- ics of the superfluid phase transition: Multiloop calculation of the microscopic model,” Physical Review E, vol. 106, no. 1, p. 014126, 2022

    work page 2022

  40. [40]

    Critical dynamics as a field theory,

    N. V. Antonov and A. N. Vasil’ev, “Critical dynamics as a field theory,”Theoretical and Mathematical Physics, vol. 60, no. 1, pp. 671–679, 1984

    work page 1984

  41. [41]

    Diagram reduction in problem of critical dynamics of ferromagnets: 4-loop approximation,

    L. T. Adzhemyan, E. V. Ivanova, M. V. Kompaniets, and S. Y. Vorobyeva, “Diagram reduction in problem of critical dynamics of ferromagnets: 4-loop approximation,”Journal of Physics A: Mathematical and Theoretical, vol. 51, no. 15, p. 155003, 2018

    work page 2018

  42. [42]

    Model A of critical dynamics: 5-loopϵ-expansion study,

    L. T. Adzhemyan, D. A. Evdokimov, M. Hnatiˇ c, E. V. Ivanova, M. V. Kompaniets, A. Kudlis, and D. V. Zakharov, “Model A of critical dynamics: 5-loopϵ-expansion study,” Physica A: Statistical Mechanics and its Applications, vol. 600, p. 127530, 2022

    work page 2022

  43. [43]

    Renormalization-group treatment of the critical dynamics of the binary-fluid and gas-liquid transitions,

    E. Siggia, B. Halperin, and P. Hohenberg, “Renormalization-group treatment of the critical dynamics of the binary-fluid and gas-liquid transitions,”Physical Review B, vol. 13, no. 5, p. 2110, 1976

    work page 1976

  44. [44]

    Deviations from dynamic scaling in helium and antiferro- magnets,

    C. De Dominicis and L. Peliti, “Deviations from dynamic scaling in helium and antiferro- magnets,”Phys. Rev. Lett., vol. 38, p. 505, 1977

    work page 1977

  45. [45]

    Field-theory renormalization and critical dynamics above tc: Helium, antiferromagnets, and liquid-gas systems,

    C. De Dominicis and L. Peliti, “Field-theory renormalization and critical dynamics above tc: Helium, antiferromagnets, and liquid-gas systems,”Physical Review B, vol. 18, no. 1, p. 353, 1978

    work page 1978

  46. [46]

    H-model of critical dynamics: Two-loop calculations of rg functions and critical indices,

    L. Adzhemyan, A. Vasiliev, Y. Kabrits, and M. Kompaniets, “H-model of critical dynamics: Two-loop calculations of rg functions and critical indices,”Theoretical and Mathematical Physics, vol. 119, no. 1, pp. 454–470, 1999

    work page 1999

  47. [47]

    Dynamic critical behavior near the superfluid transition inhe3- he4 mixtures in two loop order,

    R. Folk and G. Moser, “Dynamic critical behavior near the superfluid transition inhe3- he4 mixtures in two loop order,”Physical review letters, vol. 89, no. 12, p. 125301, 2002

    work page 2002

  48. [48]

    Renomarlization group calculations for critical and tricritical dynamics applied to helium,

    L. Peliti, “Renomarlization group calculations for critical and tricritical dynamics applied to helium,” inDynamical Critical Phenomena and Related Topics, pp. 189–209, Springer, 1979. 22

    work page 1979

  49. [49]

    Multi-loop calculations of anomalous exponents in the models of critical dynamics,

    L. T. Adzhemyan, M. Danˇ co, M. Hnatiˇ c, E. Ivanova, and M. Kompaniets, “Multi-loop calculations of anomalous exponents in the models of critical dynamics,” inEPJ Web of Conferences, vol. 108, p. 02004, EDP Sciences, 2016

    work page 2016

  50. [50]

    Ivanova,Multi-loop calculation of critical exponents in models of critical dynamics and statics

    E. Ivanova,Multi-loop calculation of critical exponents in models of critical dynamics and statics. PhD thesis, SPBU Russia, 2019

    work page 2019

  51. [51]

    Divergence of the perturbation-theory series and the quasi-classical theory,

    L. Lipatov, “Divergence of the perturbation-theory series and the quasi-classical theory,” Zh. Eksp. Teor. Fi., vol. 72, pp. 411–427, 1977

    work page 1977

  52. [52]

    Perturbation theory at large order. i. the ϕ2N interaction,

    E. Br´ ezin, J. C. L. Guillou, and J. Zinn-Justin, “Perturbation theory at large order. i. the ϕ2N interaction,”Physical Review D, vol. 15, p. 1544, 1977

    work page 1977

  53. [53]

    Asymptotic behavior of renormalization constants in higher orders of the perturbation expansion for the (4-ϵ)-dimensionally regularizedo(n)- symmetric ϕ4 theory,

    M. V. Komarova and M. Y. Nalimov, “Asymptotic behavior of renormalization constants in higher orders of the perturbation expansion for the (4-ϵ)-dimensionally regularizedo(n)- symmetric ϕ4 theory,”Theoretical and Mathematical Physics, vol. 126, pp. 339–353, 2001

    work page 2001

  54. [54]

    Critical exponents from field theory,

    J. C. Le Guillou and J. Zinn-Justin, “Critical exponents from field theory,”Physical Review B, vol. 21, no. 9, p. 3976, 1980

    work page 1980

  55. [55]

    Instantons for dynamic models from B to H,

    J. Honkonen, M. V. Komarova, and M. Y. Nalimov, “Instantons for dynamic models from B to H,”Nuclear Physics B, vol. 714, no. 3, pp. 292–306, 2005

    work page 2005

  56. [56]

    Large-order asymptotes for dynamic models near equilibrium,

    J. Honkonen, M. V. Komarova, and M. Y. Nalimov, “Large-order asymptotes for dynamic models near equilibrium,”Nuclear Physics B, vol. 707, no. 3, pp. 493–508, 2005

    work page 2005

  57. [57]

    Borel resummation of theϵ-expansion of the dynamical exponentz in modelA of theϕ4O(n) theory,

    M. Y. Nalimov, V. A. Sergeev, and L. Sladkoff, “Borel resummation of theϵ-expansion of the dynamical exponentz in modelA of theϕ4O(n) theory,”Theoretical and Mathematical Physics, vol. 159, no. 1, pp. 499–508, 2009. 23

    work page 2009