Pith. sign in

REVIEW 3 major objections 4 minor 220 references

Two-Loop Turbulent Helical Magnetohydrodynamics: Large-Scale Dynamo and Energy Spectrum

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Helical MHD turbulence generates its own large-scale magnetic field

desk verdict Solid two-loop technical extension, but the headline spectral slope has a factor-of-two mismatch between abstract and body and rests on a reverse-engineered coefficient, so treat the quantitative claims as provisional. read the letter →

arxiv 2506.20578 v1 pith:GOBES7NG submitted 2025-06-25 physics.plasm-ph cond-mat.stat-mechphysics.flu-dyn

Michal Hnatič , Tomáš Lučivjanský , Lukáš Mižišin , Yurii Molotkov , Andrei Ovsiannikov This is my paper · ORCID
classification physics.plasm-phcond-mat.stat-mechphysics.flu-dyn MSC 76W0576F0576F5582B28 PACS 52.30.Cv47.27.-i47.65.-d
keywords helicalmagnetohydrodynamicsturbulentdynamorenormalizationgrouptwo-loopcorrectionsenergyspectrumspontaneoussymmetrybreakingkinetichelicityequipartition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that at two-loop order the field-theoretic renormalization group for helical, incompressible magnetohydrodynamic turbulence explains the turbulent dynamo as spontaneous symmetry breaking: the zero-field state $\langle b\rangle=0$ is destabilized by mass-like 'curl' loop corrections, and the system relaxes to a new ground state with a nonzero large-scale field $B$ whose magnitude is fixed by exactly cancelling those corrections. It also claims that in this dynamo regime the magnetic energy spectrum steepens relative to the Kolmogorov velocity spectrum, from a $k^{-11/3}$ form to $k^{-11/3+\gamma_{3*}}$ with $\gamma_{3*} = -0.319\epsilon + (0.0556 - 0.4202\rho^2)\epsilon^2$, where $\rho$ is the helicity degree; hence helicity controls the slope and kinetic-magnetic equipartition is broken. The result matters because it derives a dynamo and a spectral prediction from the MHD equations themselves rather than from phenomenological closures, and it gives a two-loop quantitative target for simulations and liquid-metal experiments.

What carries the argument

The load-bearing machinery is the shifted action (78) with the new propagator matrix (79), whose central object is the function $\xi(\omega,k) = -\omega^2 - i\omega(1+u)\nu_0 k^2 + u\nu_0^2 k^4 + (B_0\cdot k)^2$, the dispersion law of damped Alfv\'en waves. Loop diagrams are evaluated by frequency integration using the stable-polynomial pole structure (Hermite-Biehler), with 'dangerous poles' of the form $\omega_2 + \omega_1(p) - \omega_2(p-q)$ that the paper argues cancel in the full two-loop self-energy; tensor integrals are scalarized ad hoc by decomposing into the metric and the $B_0$ direction. The two-loop analysis enumerates 488 diagrams for $\Sigma_{b'b}$, classified into types F, IR, P, L, and S; the cancellation condition (96) fixing $B_0$, together with $\Lambda$-renormalization and the counterterm $\delta h = -2 J_S^{[\Lambda]}$, produces the renormalized field $B$. This machinery is what turns the presence of curl terms from an instability into a dynamical selection of $\langle b\rangle\neq 0$.

What would settle it

Evaluate one of the post-shift two-loop diagrams of $\Gamma_{v'v}$ or $\Gamma_{b'v}$ (e.g., one of the eleven $D_S$ diagrams in Fig. 7) by closing the frequency contour and checking whether the residue at the pole $\omega_2 + \omega_1(p) - \omega_2(p-q)$ leaves a step-function or $\sqrt{D(p)}$ contribution; if it does, the claimed cancellation fails and the two-loop field $B$ and spectral correction collapse.

Watch

Extended reading notes

Core claim

The paper's central claim is that a two-loop renormalized analysis of the model defined by the stochastic MHD action (10) yields a stable turbulent dynamo regime: after the shift $b\to b+B$, the dangerous linear-in-momentum 'curl' term $\rho\nu_0 h_0\, b'\cdot(\nabla\times b)$ in the magnetic response function can be cancelled order by order, and the renormalized magnitude of the spontaneously generated field is $B = 16\pi\sqrt{u}(1+u)\nu|h|/g$ times a two-loop factor given in Eq. (147). With $B$ fixed this way, no new instability appears: Goldstone-like corrections to Alfv\'en waves decay or grow only polynomially against an exponential damping. The magnetic two-point function then acquires a critical dimension different from the velocity field's, so the magnetic energy spectrum is predicted to scale as $E_b(k)\sim k^{-11/3+\gamma_{3*}}$ at $d=3$, $\epsilon=2$, with $\gamma_{3*} = -0.319\epsilon + (0.0556 - 0.4202\rho^2)\epsilon^2$; the abstract states this slope as $-11/3 + 2\gamma_{b\star}$ with $\gamma_{b\star} = -0.1039 - 0.4202\rho^2$. This breaks equipartition because the velocity spectrum retains the exact Kolmogorov slope $-11/3$.

Load-bearing premise

The calculation assumes that the 'dangerous poles' arising in two-loop frequency integrals cancel across all 488 diagrams of the shifted theory, even though the authors verified the cancellation directly only for the $\Gamma_{b'b}$ diagrams and state they believe it holds for all others.

Editorial extensions

If this is right

  • A nonzero mean magnetic field of the magnitude in Eq. (147) is a genuine prediction of the turbulent state, not an imposed boundary condition; its direction is arbitrary, signalling spontaneous breaking of $SO(3)$ to $SO(2)$.
  • The magnetic energy spectrum is predicted to be steeper than Kolmogorov by the helicity-dependent amount $\gamma_{3*}$; in 3D at $\epsilon=2$ the exponent correction is negative and grows with $|\rho|$, so stronger chirality means a steeper magnetic spectrum.
  • Kinetic and magnetic energy spectra no longer coincide in the dynamo regime, so equipartition is violated even at the level of critical scaling.
  • The linearized dynamics around the new ground state contains Goldstone-like corrections to Alfv\'en waves; at two-loop order these still decay exponentially (times polynomial factors) and introduce no additional instability.
  • Helicity alone, without magnetic noise in the forcing, is sufficient to drive the dynamo, and the two-loop correction to the field amplitude is small for the physically relevant inverse Prandtl numbers listed in Table IV.

Reading between the lines

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

  • If the dangerous-pole cancellation holds generally, the same shifted-propagator technique could be applied to the $\alpha$-effect, computing the turbulent electromotive-force coefficient from the $\langle v\otimes b\rangle$ correlator at two loops, which the paper lists as future work.
  • The prediction is testable in numerical simulations and liquid-sodium experiments: measure the magnetic spectral slope relative to the kinetic slope for flows with controlled kinetic helicity; the slope difference should grow as $\rho^2$ with coefficient about $-1.68$ at $\epsilon=2$.
  • The dimension-dependence found in Eq. (111) suggests the stabilization-by-mean-field mechanism is special to $d=3$; in other dimensions the exotic term itself destabilizes, which could explain why the dynamo is observed in 3D geometry.
  • The factorization pattern of the frequency integrals (results independent of $\sqrt{D(p)}$) hints at a hidden algebraic structure in this class of non-Feynman integrals; proving it generally would reduce the computational cost of any higher-loop dynamo calculation.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper presents a two-loop field-theoretic renormalization-group analysis of incompressible, helical MHD turbulence, extending an earlier one-loop treatment of the turbulent dynamo regime. The central mechanism is an infrared-unstable, momentum-linear 'curl' term generated in the magnetic response function; the authors propose two ways to stabilize the theory: a kinematic-regime cancellation by a bare curl-like parameter, and a dynamo regime in which the magnetic field is shifted by a spontaneously generated large-scale mean field B. The main quantitative claims are a two-loop expression for B in Eq. (147), the absence of additional instabilities from the newly generated anisotropic terms, and a magnetic energy spectrum steeper than Kolmogorov, with a helicity-dependent slope quoted in the abstract. The manuscript contains a large diagrammatic calculation, including a classification of 488 two-loop self-energy diagrams, explicit frequency-integration techniques, tensor-reduction appendices, and numerical evaluation of two-loop coefficients.

Significance. If the central claims hold, this would be a substantial step beyond the one-loop treatment of the helical MHD dynamo: it would provide a two-loop renormalized mean-field amplitude, a concrete mechanism for the Goldstone-like corrections to Alfvén waves, and a quantitative prediction that helicity modifies the magnetic spectral slope and breaks energy equipartition. The paper is commendable for its explicit and detailed technical apparatus: the diagram classification in Sec. V B, the appendices with integrands and tensor reduction, the numerical Monte Carlo evaluation of the two-loop coefficients in Table IV, and the transparent enumeration of the regularization and renormalization steps. The significance is, however, conditional: the headline spectral prediction is internally inconsistent between the abstract and Sec. VI B, and it rests on a two-loop coefficient that is reverse-engineered from earlier work rather than derived here.

major comments (3)
  1. [Abstract; Sec. VI B; Eq. (56)] The abstract's advertised magnetic spectral slope is inconsistent with the derivation in Sec. VI B by a factor of two. Sec. VI B states that 2Δ_b = -11/3 + γ_3⋆, so with Eq. (56) at ε=2 the magnetic exponent is -11/3 - 0.4156 - 1.6808ρ². The abstract, however, quotes -11/3 + 2γ_b⋆ with γ_b⋆ = -0.1039 - 0.4202ρ², i.e. -11/3 - 0.2078 - 0.8404ρ². These two expressions differ by a factor of two, and the abstract's γ_b⋆ is not the γ_3⋆/2 introduced in Sec. VI B. The abstract and Sec. VI B must be reconciled before the quantitative prediction can be assessed.
  2. [Table III, footnote b; Eqs. (54) and (56)] The load-bearing two-loop coefficient z_21^(2)(u⋆,ρ) is not derived in this paper. Footnote b of Table III states that the value is 'reverse-engineered' from Table I of [167] because the appendix of [167] lacks the relevant data. This coefficient enters u⋆^(2) = 0.0138 + 0.0312ρ² in Eq. (54), and through Eq. (56) it produces the helicity-dependent term -0.4202ρ² ε² in γ_3⋆ that drives the abstract's spectral steepening and equipartition breaking. An undocumented inversion of published fixed-point data is not a substitute for a two-loop calculation; a sign error or a scheme-dependent mapping in this inversion would directly change the central result. The authors should either provide an independent derivation of z_21^(2) within the present framework or clearly and prominently label the ρ²-dependent slope as an unverified input from the literature.
  3. [Sec. IV B 1; Sec. V B] The paper explicitly states that the cancellation of the 'dangerous' poles of the form (84) was verified directly for the Γ^{b′b} diagrams and that the authors believe the property applies to all two-loop diagrams. Since the two-loop mean-field result in Eq. (147) is built from the 488 diagrams contributing to Σ^{b′b}, the exact scope of the verification matters: which of the Type F, IR, P, L, and S classes in Sec. V B were actually checked? If any sector remains unchecked, the diagrammatic technique and the central result for B are not fully established. The authors should either provide a direct check for all 488 diagrams or state precisely which diagrams remain unverified and why the cancellation is nevertheless expected to hold.
minor comments (4)
  1. [Sec. III B] The sentence 'to account for the emergence of the mean field B and its estimation in [155], and its estimation in [155]' contains a duplicated phrase and should be rewritten.
  2. [Sec. II D] Eq. (56) is presented as 'found to be [174]' without further derivation; given that this expression is a central input to the spectral prediction, the provenance and any assumptions behind it should be stated more prominently, perhaps in the introduction or the conclusion.
  3. [Sec. V C 1] The sentence 'Note that, by construction, h < 0 and h ∝ g' appears twice in the same paragraph; one occurrence should be deleted.
  4. [Table IV] The numerical entries for c_2^[B](u,0) are quoted to one significant digit without error estimates; since these values enter the two-loop correction in Eq. (147), a short statement of the numerical uncertainty or the precision of the VEGAS integration would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new two-loop mean-field calculation is self-contained, and the spectral-slope input from prior work is disclosed external support, not a self-referential reduction.

full rationale

The paper's principal new derivation is the two-loop correction to the spontaneously generated field B, Eq. (147). This is obtained from an explicit diagram-by-diagram calculation in the shifted theory (Sec. V): the 488 diagrams are classified (Types F, IR, P, L, S), the dangerous-pole cancellation is checked for Gamma_b'b diagrams, and the two-loop coefficient c2^[B](u,0) is evaluated numerically (Table IV). The stabilization condition (96) determines B by requiring the curl contribution to vanish; that is a standard self-consistency (tadpole) condition, not a definition of the predicted spectrum. The spectral exponent in the abstract is not derived in this paper: Eq. (56) for gamma_3* is explicitly quoted from [174], and gamma_b* = gamma_3*/2 enters through the standard RG relation (158)-(160). Reliance on a prior same-group result is self-citation, but it is not circular because the quoted coefficient is not produced by the present derivation and no equation here reduces to its own input. The reverse-engineered z_21^(2) in Table III is disclosed as such and is used as an input; the paper does not pretend to derive it. The unproven belief that dangerous poles cancel for all 488 diagrams is an explicitly flagged limitation, a correctness risk, not circularity. There is therefore no circular step meeting the standard of Eq. X = Eq. Y by construction.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The model uses standard MHD parameters plus the introduced curl coefficient h, whose finite value is not fixed by the dynamics. A key borrowed numerical input is z_21^(2). No new particles or forces are invented.

free parameters (2)
  • h (curl-term coefficient)
    The coefficient of the added curl operator rho nu h b' . (curl b) in (28)/(120) absorbs the Lambda-divergent curl terms; its finite part is not predicted by the dynamics and sets the scale of B via Eq. (147).
  • z_21^(2) (two-loop residue) = -2.25613e-5 + 3.40506e-7 rho^2 (from Table III)
    Presented as a known result, but reverse-engineered from prior papers (Table III footnote b); it directly enters gamma_3* and hence the headline slope.
assumptions (3)
  • domain assumption Standard perturbative RG and MS scheme are valid for fully developed turbulence.
    The whole two-loop calculation relies on the epsilon-expansion and the existence of an IR-stable kinetic fixed point, cited from prior work (Sec. II D).
  • domain assumption Cancellation of dangerous poles in two-loop frequency integrals holds for all diagrams.
    Verified only for Gamma_b'b; assumed for all 488 diagrams contributing to Sigma_b'b (Sec. IV B 1).
  • domain assumption K41 hypotheses apply to MHD turbulence.
    Stated in Sec. II A as a basic assumption for the inertial range behavior.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Two-Loop Turbulent Helical Magnetohydrodynamics: Large-Scale Dynamo and Energy Spectrum." pith.science (2026). https://pith.science/paper/GOBES7NG

@misc{pith2026250620578,
  author       = {Pith},
  title        = {Pith review of: Two-Loop Turbulent Helical Magnetohydrodynamics: Large-Scale Dynamo and Energy Spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GOBES7NG}},
  note         = {Machine review of arXiv:2506.20578}
}
abstract

We present a two-loop field-theoretic analysis of incompressible helical magnetohydrodynamics (MHD) in fully developed stationary turbulence. A key feature of helical MHD is the appearance of an infrared-unstable ``mass-like'' term in the loop diagrams of the magnetic response function. Physically, this term corresponds to the relevant perturbation of the Joule damping, proportional to $\boldsymbol{\nabla} \times \boldsymbol{b}$ ($\boldsymbol{b} =$ magnetic field). Its presence destabilizes the trivial ground state $\langle \boldsymbol{b} \rangle = 0$ and forces us to look for a mechanism for stabilizing the system. We show that such stabilization can be achieved in two ways: (i) by introducing into induction equation an external mass-like parameter that precisely cancels these dangerous loop corrections (kinematic regime), or (ii) via spontaneous breaking of the rotational symmetry, leading to a new ground state with nonzero large-scale magnetic field (turbulent dynamo regime). For the latter case, we study the two-loop correction to the spontaneously generated magnetic field and demonstrate that Goldstone-like corrections to Alfv\'en modes along with some other anisotropic structures arise. Our results also confirm that the emergent mean magnetic field leads to a steeper slope of the magnetic energy spectrum, $-11/3 + 2\gamma_{b\star}$ (with $\gamma_{b\star} = -0.1039 - 0.4202\rho^2$, for $|\rho| \leqslant 1$ as the degree of helicity), compared to the Kolmogorov velocity spectrum of $-11/3$, thereby breaking equipartition.

Figures

Figures reproduced from arXiv: 2506.20578 by the authors.

Figure 3
Figure 3. FIG. 3. Two-loop Γ [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Graphical representation of all independent propaga [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. FIG. 5. Behavior of [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figures from the paper (3 more)
Figure 6
Figure 6. Figure 6: FIG. 6. The general structure of the linear in external mo [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Two-loop type S diagrams along with the [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Two topologies of two-loop Σ [PITH_FULL_IMAGE:figures/full_fig_p042_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

220 extracted references · 71 canonical work pages

  1. [167]

    N. V. Antonov and M. M. Kostenko, Phys. Rev. E90, 063016 (2014)

    work page 2014

  2. [1]

    physical

    Exact one-loop approximation Before proceeding to two-loop estimates, we have to first refine the one-loop result for the curl term part of self-energy correction Σ b′b. In [155], only the result accurate up toO(ϵ 0) is given. Since the curl terms in Σb′b originate at the one-loop level, they start to correct the propagator lines (79) only in higher loops...

    2023

  3. [2]

    leg-fixed

    Description of ancillary files This article is supplemented by a comprehensive dataset in the form of human-readable text files provided in Wolfram Mathematica [213] (.wl) format. Specifically, the Supplemental Material includes: (i) FilesExportSunSet.wlandExportTriangles.wl, which contain all necessary information about the integrals for each topology: t...

  4. [3]

    L. D. Landau and E. M. Lifshitz,Fluid Mechanics, 2nd ed. (Pergamon, Oxford, 1987)

    1987

  5. [4]

    T. G. Cowling,Magnetohydrodynamics(Interscience, 1957)

    1957

  6. [5]

    P. A. Davidson,An Introduction to Magnetohydrody- namics(Cambridge University Press, 2001)

    2001

  7. [6]

    H. K. Moffatt,Magnetic field generation in electrically conducting fluids(Cambridge University Press, 1978)

    1978

  8. [7]

    F. M. White,Fluid Mechanics, 7th ed. (McGraw-Hill, New York, 2011)

    2011

Show all 220 references
  1. [8]

    E. M. Lifshitz and L. P. Pitaevskii,Physical Kinetics, 1st ed. (Pergamon Press, 1981)

    1981

  2. [9]

    P. M. Bellan,Fundamentals of Plasma Physics(Cam- bridge University Press, 2006)

    2006

  3. [10]

    Goedbloed and S

    J. Goedbloed and S. Poedts,Principles of Magneto- hydrodynamics: With Applications to Laboratory and Astrophysical Plasmas(Cambridge University Press, 2004)

    2004

  4. [11]

    Hattori, M

    K. Hattori, M. Hongo, and X.-G. Huang, Symmetry14, 1851 (2022)

    2022

  5. [12]

    Brekis, A

    A. Brekis, A. Alemany, O. Alemany, and A. Montisci, Sustainability13, 13498 (2021)

    2021

  6. [13]

    J. C. Dom ´ ınguez-Lozoya, S. Cuevas, D. R. Dom ´ ınguez, R. ´Avalos Z´ u˜ niga, and E. Ramos, Sustainability13, 4641 (2021)

    2021

  7. [14]

    Alemany, A

    A. Alemany, A. Brekis, and A. Montisci, Sustainability 15, 12619 (2023)

    2023

  8. [15]

    Xu, Matter Radiat

    Y. Xu, Matter Radiat. Extremes1, 192 (2016)

    2016

  9. [16]

    S. J. P. Pamela, et al., Phys. Plasmas27, 102510 (2020)

    2020

  10. [17]

    Hoelzl, et al., Nucl

    M. Hoelzl, et al., Nucl. Fusion64(2024)

    2024

  11. [18]

    T. J. Rhodes, S. Smolentsev, and M. Abdou, Phys. Plas- mas30, 057101 (2018)

    2018

  12. [19]

    Smolentsev, Fluids6, 110 (2021)

    S. Smolentsev, Fluids6, 110 (2021)

    2021

  13. [20]

    Mistrangelo, L

    C. Mistrangelo, L. B¨ uhler, C. Alberghi, S. Bassini, L. Candido, C. Courtessole, A. Tassone, F. R. Urgorri, and O. Zikanov, Energies14, 6640 (2021)

    2021

  14. [21]

    J. C. R. Hunt and R. Moreau, J. Fluid Mech.78, 261 (1976)

    1976

  15. [22]

    V. S. Beskin, Phys.-Uspekhi53, 1199–1233 (2010)

    2010

  16. [23]

    S. R. Cranmer and A. R. Winebarger, Annu. Rev. As- tron. Astrophys.57, 157 (2019)

    2019

  17. [24]

    L. M. Voigt and A. C. Fabian, Mon. Not. R. Astron. Soc.347, 1130 (2004)

    2004

  18. [25]

    S. A. Balbus and C. S. Reynolds, Astrophys. J.681, L65–L68 (2008)

    2008

  19. [26]

    L. M. Voigt and A. C. Fabian, Astrophys. J.703, 96– (2009)

    2009

  20. [27]

    Yan, (2021), arXiv:2109.07847

    H. Yan, (2021), arXiv:2109.07847

    2021 arXiv

  21. [28]

    N. E. Engelbrecht, F. Effenberger, M. V. Florinski, S. Potgieter, D. Ruffolo, R. Chhiber, A. V. Usmanov, J. S. Rankin, and P. L. Els, Space Sci. Rev.218(2022)

    2022

  22. [29]

    Gupta, IAU Colloquia177, 539–544 (2000)

    Y. Gupta, IAU Colloquia177, 539–544 (2000)

    2000

  23. [30]

    Falceta-Gon¸ calves, G

    D. Falceta-Gon¸ calves, G. Kowal, E. Falgarone, and A. C.-L. Chian, Nonlinear Process. Geophys.21, 587–604 (2014)

    2014

  24. [31]

    Popova and A

    E. Popova and A. Lazarian, Fluids8, 142 (2023)

    2023

  25. [32]

    Stefani, A

    F. Stefani, A. Gailitis, and G. Gerbeth, J. Appl. Math. Mech.88, 930 (2008)

    2008

  26. [33]

    Mistrangelo, L

    C. Mistrangelo, L. B¨ uhler, S. Smolentsev, V. Kl¨ uber, I. Maione, and J. Aubert, Fusion Eng. Des.173, 112795 (2021)

    2021

  27. [34]

    Beresnyak and A

    A. Beresnyak and A. Lazarian,Turbulence in Magneto- hydrodynamics(Berlin/Boston: De Gruyter, 2019)

    2019

  28. [35]

    Alfv´ en and K

    H. Alfv´ en and K. Falthammar,Cosmical Eletrodynam- ics, 2nd ed. (Oxford University Press, 1963)

    1963

  29. [36]

    G. D. Fleishman and I. N. Toptygin,Cosmic Electrody- namics(Springer New York, 2013)

    2013

  30. [37]

    Wald,Advanced Classical Electromagnetism(Prince- ton University Press, 2022)

    R. Wald,Advanced Classical Electromagnetism(Prince- ton University Press, 2022)

    2022

  31. [38]

    S. I. Vainshtein, I. B. Zeldovich, and A. A. Ruz- maikin,The turbulent dynamo in astrophysics [in Rus- sian](Nauka, Moscow, 1980)

    1980

  32. [39]

    S. I. Vainshtein,Magnetic Fields in Space [in Russian] (Nauka, Moscow, 1983)

    1983

  33. [40]

    I. B. Zeldovich, A. A. Ruzmaikin, and D. D. Sokoloff, Magnetic fields in astrophysics(Gordon and Breach, 1983)

    1983

  34. [41]

    Krause and K.-H

    F. Krause and K.-H. R¨ adler,Mean-Field Magnetohydro- dynamics and Dynamo Theory(Pergamon, 1980)

    1980

  35. [42]

    Ruediger, L

    G. Ruediger, L. Kitchatinov, and R. Hollerbach,Mag- netic Processes in Astrophysics: Theory, Simulations, Experiments(Wiley-VCH, 2013)

    2013

  36. [43]

    Shukurov and K

    A. Shukurov and K. Subramanian,Astrophysical mag- netic fields: from galaxies to the early universe(Cam- bridge University Press, 2022)

    2022

  37. [44]

    Moffatt and E

    H. Moffatt and E. Dormy,Self-Exciting Fluid Dynamos (Cambridge University Press, 2019)

    2019

  38. [45]

    Tobias, J

    S. Tobias, J. Fluid Mech.912(2021)

    2021

  39. [46]

    Tobias, Philos

    S. Tobias, Philos. Trans. R. Soc. A43, 2741 (2002)

    2002

  40. [47]

    C. A. Jones, Annu. Rev. Fluid Mech.43, 583 (2011)

    2011

  41. [48]

    P. H. Roberts and E. M. King, Rep Prog Phys.76, 096801 (2013)

    2013

  42. [49]

    S. Sur, D. R. G. Schleicher, R. Banerjee, C. Federrath, and R. S. Klessen, Astrophys. J.721, L134 (2010)

    2010

  43. [50]

    R. D. Blandford and D. G. Payne, Mon. Not. R. Astron. Soc.199, 883 (1982)

    1982

  44. [51]

    D. L. Meier, S. Koide, and Y. Uchida, Science291, 84 (2001)

    2001

  45. [52]

    Hartmann,Accretion processes in star formation, 2nd ed

    L. Hartmann,Accretion processes in star formation, 2nd ed. (Cambridge University Press, 2009)

    2009

  46. [53]

    J. E. Pringle, Annu. Rev. Astron. Astrophys.19, 137 (1981)

    1981

  47. [54]

    Larmor, Rep

    J. Larmor, Rep. Brit. Assoc. Adv. Sci.113, 159 (1919)

    1919

  48. [55]

    T. G. Cowling, Mon. Not. Roy. Astr. Soc.94, 39 (1934)

    1934

  49. [56]

    Bullard and H

    E. Bullard and H. Gellman, Philos. Trans. R. Soc. A 247, 213 (1954)

    1954

  50. [57]

    Y. B. Zel’dovich, Zh. Exp. Teor. Fiz. SSSR31, 154 (1956)

    1956

  51. [58]

    Backus, Ann

    G. Backus, Ann. Phys.4, 372 (1958)

    1958

  52. [59]

    E. N. Parker, Astrophys. J.122, 293 (1955)

    1955

  53. [60]

    Elsasser, SciAm198, 44 (1958)

    W. Elsasser, SciAm198, 44 (1958)

    1958

  54. [61]

    Braginsky, Soviet Physics JETP20, 726 (1964)

    S. Braginsky, Soviet Physics JETP20, 726 (1964)

    1964

  55. [62]

    Braginsky, Soviet Physics JETP20, 1462 (1964)

    S. Braginsky, Soviet Physics JETP20, 1462 (1964)

    1964

  56. [63]

    Braginsky, Geomagnetism and Aeronomy4, 572 (1964)

    S. Braginsky, Geomagnetism and Aeronomy4, 572 (1964)

    1964

  57. [64]

    Braginsky, Geomagnetism and Aeronomy4, 698 (1964)

    S. Braginsky, Geomagnetism and Aeronomy4, 698 (1964)

    1964

  58. [65]

    Steenbeck, F

    M. Steenbeck, F. Krause, and K.-H. R¨ adler, Z. Natur- forsch.21, 369 (1966)

    1966

  59. [66]

    G. O. Roberts and E. C. Bullard, Philos. Trans. R. Soc. A271, 411 (1972)

    1972

  60. [67]

    Y. B. Ponomarenko, J. Appl. Mech. Tech. Phys.14, 775 44 (1973)

    1973

  61. [68]

    Alemany, P

    R. Alemany, P. L. Moreau, P. L. Sulem, and U. Frisch, J. de M´ ecanique18, 277 (1979)

    1979

  62. [69]

    S. F. Selyuto, Magnetohydrodynamics3, 55 (1984)

    1984

  63. [70]

    Alfv´ en, Nature150, 405 (1942)

    H. Alfv´ en, Nature150, 405 (1942)

    1942

  64. [71]

    Eyink, J

    Gregory L. Eyink, J. Math. Phys.50, 083102 (2009)

    2009

  65. [72]

    H. K. Moffatt, J. Fluid Mech.35, 117 (1969)

    1969

  66. [73]

    Berger and G

    M. Berger and G. Field, J. Fluid Mech.147, 133 (1984)

    1984

  67. [74]

    Biskamp,Magnetic Reconnection in Plasmas(Cam- bridge University Press, 2000)

    D. Biskamp,Magnetic Reconnection in Plasmas(Cam- bridge University Press, 2000)

    2000

  68. [75]

    Priest and T

    E. Priest and T. Forbes,Magnetic Reconnection: MHD Theory and Applications(Cambridge University Press, 2000)

    2000

  69. [76]

    Yamada, R

    M. Yamada, R. Kulsrud, and H. Ji, Rev. Mod. Phys. 82, 603 (2010)

    2010

  70. [77]

    Childress and A

    S. Childress and A. Gilbert,Stretch, Twist, Fold: The Fast Dynamo(Springer-Verlag, 1995)

    1995

  71. [78]

    Bullard, Math

    E. Bullard, Math. Proc. Camb. Philos. Soc.51, 744 (1955)

    1955

  72. [79]

    Brandenburg, J

    A. Brandenburg, J. Plasma Phys.84(2018)

    2018

  73. [80]

    Brandenburg and K

    A. Brandenburg and K. Subramanian, Phys. Rep.417, 1 (2005)

    2005

  74. [81]

    Rincon, F

    F. Rincon, F. Califano, A. A. Schekochihin, and V. F., Proc. Natl. Acad. Sci. USA113, 3950 (2016)

    2016

  75. [82]

    Brandenburg and E

    A. Brandenburg and E. Ntormousi, Annu. Rev. Astron. Astrophys.61, 561 (2023)

    2023

  76. [83]

    B. E. Goldstein, E. J. Smith, A. Balogh, T. S. Horbury, M. L. Goldstein, and D. A. Roberts, Geophys. Res. Lett. 22, 3393 (1995)

    1995

  77. [84]

    J. W. Armstrong, B. J. Rickett, and S. R. Spangler, Astrophys. J.443, 209 (1995)

    1995

  78. [85]

    Chepurnov and A

    A. Chepurnov and A. Lazarian, Astrophys. J.710, 853 (2010)

    2010

  79. [86]

    C. C. Finlay, et al., Geophys. J. Int.183, 1216 (2010)

    2010

  80. [87]

    C. L. Johnson, M. E. Purucker, H. Korth, B. J. Ander- son, R. M. Winslow, M. M. H. Al Asad, J. A. Slavin, I. I. Alexeev, R. J. Phillips, M. T. Zuber, and S. C. Solomon, J. Geophys. Res. E117(2012)

    2012

  81. [88]

    K. M. Moore, H. Cao, J. Bloxham, D. J. Stevenson, J. E. P. Connerney, and S. J. Bolton, Nat. Astron.3, 730 (2019)

    2019

  82. [89]

    J. E. P. Connerney, S. Kotsiaros, R. J. Oliversen, J. R. Espley, J. L. Joergensen, P. S. Joergensen, J. M. G. Merayo, M. Herceg, J. Bloxham, K. M. Moore, S. J. Bolton, and S. M. Levin, Geophys. Res. Lett.45, 2590 (2018)

    2018

  83. [90]

    P. H. Roberts and A. M. Soward, Annu. Rev. Fluid Mech.4, 117 (1972)

    1972

  84. [91]

    Dormy and A

    E. Dormy and A. M. E. Soward,Mathematical Aspects of Natural Dynamos, 1st ed. (Chapman and Hall/CRC, 2007)

    2007

  85. [92]

    S. A. Balbus and J. F. Hawley, Astrophys. J.376, 214 (1991)

    1991

  86. [93]

    S. A. Balbus and J. F. Hawley, Astrophys. J.376, 223 (1991)

    1991

  87. [94]

    D. R. Sisan, N. Mujica, W. A. Tillotson, Y.-M. Huang, W. Dorland, A. B. Hassam, T. M. Antonsen, and D. P. Lathrop, Phys. Rev. Lett.93, 114502 (2004)

    2004

  88. [95]

    Kawazura and S

    Y. Kawazura and S. S. Kimura, Sci. Adv.10(2024)

    2024

  89. [96]

    Leidi, G., Andrassy, R., Higl, J., Edelmann, P. V. F., and R¨ opke, F. K., A&A679, A132 (2023)

    2023

  90. [97]

    A. A. Schekochihin, S. C. Cowley, S. F. Taylor, J. L. Maron, and J. C. McWilliams, Astrophys. J.612, 276 (2004)

    2004

  91. [98]

    Beresnyak, Astrophys

    A. Beresnyak, Astrophys. J. Lett.784, L20 (2014)

    2014

  92. [99]

    Miesch, W

    M. Miesch, W. Matthaeus, A. Brandenburg, A. Pet- rosyan, A. Pouquet, C. Cambon, F. Jenko, D. Uzden- sky, J. Stone, S. Tobias, J. Toomre, and M. Velli, Space Sci. Rev.194, 97 (2015)

    2015

  93. [100]

    Kupka and H

    F. Kupka and H. Muthsam, Living Rev. Comput. As- trophys.3, 1 (2017)

    2017

  94. [101]

    Grete, D

    P. Grete, D. G. Vlaykov, W. Schmidt, and D. R. G. Schleicher, Phys. Rev. E95, 033206 (2017)

    2017

  95. [102]

    M. S. Miesch and J. Toomre, Annu. Rev. Fluid Mech. 41, 317 (2009)

    2009

  96. [103]

    Meheut, S

    H. Meheut, S. Fromang, G. Lesur, M. Joos, and P.- Y. Longaretti, A&A579, A117 (2015), program repos- itory:https://ipag.osug.fr/ ~lesurg/snoopy.html

    2015

  97. [104]

    Open Source Softw.6, 2807 (2021)

    Brandenburg , et al., J. Open Source Softw.6, 2807 (2021)

    2021

  98. [105]

    M. D. Nornberg, E. J. Spence, R. D. Kendrick, C. M. Jacobson, and C. B. Forest, Phys. Rev. Lett.97, 044503 (2006)

    2006

  99. [106]

    Monchaux, M

    R. Monchaux, M. Berhanu, M. Bourgoin, M. Moulin, P. Odier, J.-F. Pinton, R. Volk, S. Fauve, N. Mordant, F. P´ etr´ elis, A. Chiffaudel, F. Daviaud, B. Dubrulle, C. Gasquet, L. Mari´ e, and F. Ravelet, Phys. Rev. Lett. 98, 044502 (2007)

    2007

  100. [107]

    Brito, T

    D. Brito, T. Alboussi` ere, P. Cardin, N. Gagni` ere, D. Jault, P. La Rizza, J.-P. Masson, H.-C. Nataf, and D. Schmitt, Phys. Rev. E83, 066310 (2011)

    2011

  101. [108]

    Rahbarnia, B

    K. Rahbarnia, B. P. Brown, M. M. Clark, E. J. Kaplan, M. D. Nornberg, A. M. Rasmus, N. Z. Taylor, C. B. Forest, F. Jenko, A. Limone, J.-F. Pinton, N. Plihon, and G. Verhille, Astrophys. J.759(2012)

    2012

  102. [109]

    Gailitis, O

    A. Gailitis, O. Lielausis, E. Platacis, G. Gerbeth, and F. Stefani, Rev. Mod. Phys.74, 973 (2002)

    2002

  103. [110]

    P. S. Iroshnikov, Astron. Zh.40, 742 (1963)

    1963

  104. [111]

    R. H. Kraichnan, Phys. Fluids8, 1385 (1965)

    1965

  105. [112]

    Dobrowolny, A

    M. Dobrowolny, A. Mangeney, and P. Veltri, Phys. Rev. Lett.45, 144 (1980)

    1980

  106. [113]

    Goldreich and S

    P. Goldreich and S. Sridhar., Astrophys. J.438, 763 (1995)

    1995

  107. [114]

    Briard and T

    A. Briard and T. Gomez, J. Plasma Phys.84, 905840110 (2018)

    2018

  108. [115]

    Boldyrev, Phys

    S. Boldyrev, Phys. Rev. Lett.96, 115002 (2006)

    2006

  109. [116]

    O. G. Chkhetiani, A. Eidelman, and E. Golbraikh, Non- lin. Processes Geophys.13, 613 (2006)

    2006

  110. [117]

    Pouquet and P

    A. Pouquet and P. D. Mininni, Phil. Trans. R. Soc. A. 368, 1635 (2010)

    2010

  111. [118]

    J. C. Fuchs, J. Math. Phys.32, 1703 (1991)

    1991

  112. [119]

    Faraco, S

    D. Faraco, S. Lindberg, and L. Sz´ ekelyhidi, Geophys. Astrophys. Fluid Dyn.116, 237 (1991)

    1991

  113. [120]

    Barbu and G

    V. Barbu and G. Da Prato, Appl. Math. Optim56, 145 (2007)

    2007

  114. [121]

    Chueshov and A

    I. Chueshov and A. Millet, Appl. Math. Optim61, 379 (2010)

    2010

  115. [122]

    Da Prato and J

    G. Da Prato and J. Zabczyk,Stochastic Equations in In- finite Dimensions, 2nd ed. (Cambridge University Press, 2014)

    2014

  116. [123]

    D. J. Amit and V. Mart ´ ın-Mayor,Field Theory, the Renormalization Group, and Critical Phenomena, 3rd ed. (WORLD SCIENTIFIC, 2005)

    2005

  117. [124]

    Zinn-Justin,Quantum Field Theory and Critical Phe- nomena, 5th ed

    J. Zinn-Justin,Quantum Field Theory and Critical Phe- nomena, 5th ed. (Oxford University Press, 2021)

    2021

  118. [125]

    Zinn-Justin,Phase Transitions and Renormalization Group(Oxford University Press, 2007)

    J. Zinn-Justin,Phase Transitions and Renormalization Group(Oxford University Press, 2007)

    2007

  119. [126]

    Frisch,Turbulence: The Legacy of A

    U. Frisch,Turbulence: The Legacy of A. N. Kolmogorov 45 (Cambridge University Press, 1995)

    1995

  120. [127]

    W. D. McComb,The physics of fluid turbulence(Oxford University Press, 1990)

    1990

  121. [128]

    A. N. Vasil’ev,The Field Theoretic Renormalization Group in Critical Behavior Theory And Stochastic Dy- namics(Chapman & Hall/CRC, 2004)

    2004

  122. [129]

    U. C. T¨ auber,Critical Dynamics: A Field Theory Ap- proach to Equilibrium and Non-equilibrium Scaling Be- havior(Cambridge University Press, 2014)

    2014

  123. [130]

    Brissaud, U

    A. Brissaud, U. Frisch, J. Leorat, M. Lesieur, and A. Mazure, Phys. Fluids16, 1366 (1973)

    1973

  124. [131]

    Frisch, A

    U. Frisch, A. Pouquet, J. L´ eorat, and A. Mazure, J. Fluid Mech68, 769 (1975)

    1975

  125. [132]

    Pouquet, U

    A. Pouquet, U. Frisch, and J. L´ eorat, J. Fluid Mech77, 321 (1976)

    1976

  126. [133]

    Pouquet, J.-D

    A. Pouquet, J.-D. Fournier, and P. L. Sulem, J. Physique Lett.39, 199 (1978)

    1978

  127. [134]

    J. D. Fournier and U. Frisch, Phys. Rev. A28, 1000 (1983)

    1983

  128. [135]

    Forster, D

    D. Forster, D. R. Nelson, and M. J. Stephen, Phys. Rev. Lett.36, 867 (2006)

    2006

  129. [136]

    A. N. V. L. Ts. Adzhemyan, N. V. Antonov, Physics–Uspekhi39, 1193 (1996)

    1996

  130. [137]

    Eyink, J

    G. Eyink, J. Stat. Phys.83, 955 (1996)

    1996

  131. [138]

    Yakhot and S

    V. Yakhot and S. Orszag, J. Sci. Comput.1, 3–51 (1986)

    1986

  132. [139]

    Canet, B

    L. Canet, B. Delamotte, and N. Wschebor, Phys. Rev. E93, 063101 (2016)

    2016

  133. [140]

    L. T. Adzhemyan, N. V. Antonov, P. B. Gol’din, T. L. Kim, and M. V. Kompaniets, J. Phys. A41, 495002 (2008)

    2008

  134. [141]

    L. T. Adzhemyan, N. V. Antonov, P. B. Gol’din, and M. V. Kompaniets, J. Phys. A46, 135002 (2013)

    2013

  135. [142]

    L. T. Adzhemyan, D. A. Evdokimov, and M. V. Kom- paniets, Nucl. Phys. B1008, 116716 (2024)

    2024

  136. [143]

    L. T. Adzhemyan and Y. Kirienko, (2024), arXiv:2406.14575

    2024 arXiv

  137. [144]

    Bykov, Theor

    D. Bykov, Theor. Math. Phys.208, 993 (2021), arXiv:2106.15598 [hep-th]

    2021 arXiv

  138. [145]

    Affleck, D

    I. Affleck, D. Bykov, and K. Wamer, Phys. Rept.953, 1 (2022), arXiv:2101.11638 [hep-th]

    2022 arXiv

  139. [146]

    Bykov and V

    D. Bykov and V. Krivorol, (2024), arXiv:2407.20423 [hep-th]

    2024 arXiv

  140. [147]

    Bykov and V

    D. Bykov and V. Krivorol, Adv. Theor. Math. Phys.28, 3 (2024), arXiv:2306.04555 [hep-th]

    2024 arXiv

  141. [148]

    Krivorol, On First-Order GLSM for Sigma Models, in International Workshop Supersymmetries and Quantum Symmetries(2025) arXiv:2502.07612 [hep-th]

    V. Krivorol, On First-Order GLSM for Sigma Models, in International Workshop Supersymmetries and Quantum Symmetries(2025) arXiv:2502.07612 [hep-th]

    2025

  142. [149]

    L. T. Adzhemyan, J. Honkonen, T. Kim, and L. Slad- koff, Phys. Rev. E71, 056311 (2005)

    2005

  143. [150]

    A. M. Polyakov, Phys. Rev. E52, 6183 (1995)

    1995

  144. [151]

    Gurarie and A

    V. Gurarie and A. Migdal, Phys. Rev. E54, 4908 (1996)

    1996

  145. [152]

    Balkovsky, G

    E. Balkovsky, G. Falkovich, I. Kolokolov, and V. Lebe- dev, Phys. Rev. Lett.78, 1452 (1997)

    1997

  146. [153]

    Fleischer and P

    J. Fleischer and P. Diamond, Phys. Lett. A283, 237 (2001)

    2001

  147. [154]

    Grafke, R

    T. Grafke, R. Grauer, and T. Sch¨ afer, J. Phys. A: Math. Theor.48, 333001 (2015)

    2015

  148. [155]

    L. T. Adzhemyan, A. N. Vasil’ev, and M. Gnatich, Theor. Math. Phys.64, 777 (1985)

    1985

  149. [156]

    L. T. Adzhemyan, M. Hnatich, D. Horv´ ath, and M. Stehl ´ ık, Int. J. Mod. Phys. B09, 3401 (1995)

    1995

  150. [157]

    L. T. Adzhemyan, A. N. Vasil’ev, and M. Gnatich, Theor. Math. Phys.72, 940 (1987)

    1987

  151. [158]

    Coleman and E

    S. Coleman and E. Weinberg, Phys. Rev. D7, 1888 (1973)

    1973

  152. [159]

    A. P. Kazantsev, Sov. Phys. JETP26, 1031 (1968)

    1968

  153. [160]

    R. H. Kraichnan, Phys. Fluids11, 945 (1968)

    1968

  154. [161]

    N. V. Antonov and N. M. Gulitskiy (Springer Berlin Heidelberg, 2012) pp. 128–135

    2012

  155. [162]

    N. V. Antonov and N. M. Gulitskiy, Phys. Rev. E85, 128 (2012)

    2012

  156. [163]

    N. V. Antonov and N. M. Gulitskiy, Phys. Rev. E87, 039902 (2013)

    2013

  157. [164]

    Jurˇ ciˇ sinov´ a, M

    E. Jurˇ ciˇ sinov´ a, M. Jurˇ ciˇ sin, and R. Remeck´ y, Phys. Rev. E84, 046311 (2011)

    2011

  158. [165]

    N. V. Antonov and M. M. Kostenko, Phys. Rev. E92, 053013 (2015)

    2015

  159. [166]

    Jurˇ ciˇ sinov´ a, M

    E. Jurˇ ciˇ sinov´ a, M. Jurˇ ciˇ sin, M. Menkyna, and R. Re- meck´ y, Phys. Rev. E104, 015101 (2021)

    2021

  160. [168]

    N. V. Antonov and M. M. Kostenko, J. Math. Sci.257, 425 (2021)

    2021

  161. [169]

    Hnatiˇ c and P

    M. Hnatiˇ c and P. Zalom, Phys. Rev. E94, 053113 (2014)

    2014

  162. [170]

    N. V. Antonov and N. M. Gulitskiy, Phys. Rev. E92, 043018 (2015)

    2015

  163. [171]

    Jurˇ ciˇ sinov´ a, M

    E. Jurˇ ciˇ sinov´ a, M. Jurˇ ciˇ sin, and P. Zalom, Phys. Rev. E89, 043023 (2014)

    2014

  164. [172]

    Jurˇ ciˇ sinov´ a, M

    E. Jurˇ ciˇ sinov´ a, M. Jurˇ ciˇ sin, and R. Remeck´ y, Phys. Rev. E93, 033106 (2016)

    2016

  165. [173]

    N. V. Antonov, N. M. Gulitskii, M. M. Kostenko, and T. Luˇ civjansky, Theor. Math. Phys.200, 1294 (2019)

    2019

  166. [174]

    N. V. Antonov, J. Phys. A: Math. Gen.39, 7825 (2006)

    2006

  167. [175]

    P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977)

    1977

  168. [176]

    Hnatiˇ c, T

    M. Hnatiˇ c, T. Luˇ civjansk´ y, L. Miˇ ziˇ sin, Y. Molotkov, and A. Ovsiannikov, Universe10, 240 (2024)

    2024

  169. [177]

    Adzhemyan, L. Ts, N. V. Antonov, and A. N. Vasil’ev., Zh. Eksp. Teor. Fiz95, 1272 (1989)

    1989

  170. [178]

    L. T. Adzhemyan, N. Antonov, M. Kompaniets, and A. Vasil’ev, Int. J. Mod. Phys. B17, 2137 (2003)

    2003

  171. [179]

    Adzhemyan, A

    L. Adzhemyan, A. N. Vasiliev, and N. V. Antonov,The Field Theoretic Renormalization Group in Fully Devel- oped Turbulence(Gordon & Breach, 1999)

    1999

  172. [180]

    P. C. Martin, E. D. Siggia, and H. A. Rose, Phys. Rev. A8, 423 (1973)

    1973

  173. [181]

    Janssen, Z

    H. Janssen, Z. Phys. B23, 377 (1976)

    1976

  174. [182]

    De Dominicis, J

    C. De Dominicis, J. Phys. Colloq. Fr.37, 247 (1976)

    1976

  175. [183]

    L. D. Faddeev and A. A. Slavnov,Gauge Fields. Intro- duction to Quantum Theory, 2nd ed. (Addison-Wesley, 1991)

    1991

  176. [184]

    Vasil’ev,Functional Methods in Quantum Field The- ory and Statistical Physics(Gordon and Breach, 1998)

    A. Vasil’ev,Functional Methods in Quantum Field The- ory and Statistical Physics(Gordon and Breach, 1998)

    1998

  177. [185]

    Wyld, Ann

    H. Wyld, Ann. Phys.14, 143 (1961)

    1961

  178. [186]

    Hnatiˇ c, J

    M. Hnatiˇ c, J. Honkonen, and T. Luˇ civjansk´ y, Symmetry 11, 1193 (2019)

    2019

  179. [187]

    E. V. Ivashkevich, J. Phys. A Math. Gen.30, L525 (1997)

    1997

  180. [188]

    Berera and D

    A. Berera and D. Hochberg, Phys. Rev. Lett.99, 254501 (2007)

    2007

  181. [189]

    Berera and D

    A. Berera and D. Hochberg, Nucl. Phys. B814, 522 (2009)

    2009

  182. [190]

    Collins,Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion(Cambridge University Press, 1984)

    J. Collins,Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion(Cambridge University Press, 1984). 46

    1984

  183. [191]

    E. R. Speer,Generalized Feynman Amplitudes(Prince- ton University Press, 1969)

    1969

  184. [192]

    Honkonen and M

    J. Honkonen and M. Y. Nalimov, Z. Phys. B99, 297 (1995)

    1995

  185. [193]

    Adzhemyan, A

    L. Adzhemyan, A. Vasil’ev, and M. Gnatich, Theor. Math. Phys.58, 47 (1984)

    1984

  186. [194]

    Jurˇ ciˇ sinov´ a, M

    E. Jurˇ ciˇ sinov´ a, M. Jurˇ ciˇ sin, and R. Remeck´ y, Phys. Rev. E79, 046319 (2009), Erratum in Phys. Rev. E,86, 049902 (2012)

    2009

  187. [195]

    Biskamp,Magnetohydrodynamic Turbulence(Cam- bridge University Press, 2003)

    D. Biskamp,Magnetohydrodynamic Turbulence(Cam- bridge University Press, 2003)

    2003

  188. [196]

    S. S. Moiseev, R. Z. Sagdeev, A. V. Tur, G. A. Khomenko, and V. V. Yanovskii, Sov. Phys. JETP58, 1149 (1983)

    1983

  189. [197]

    S. S. Moiseev, R. Z. Sagdeev, A. V. Tur, G. A. Khomenko, , and A. M. Shukurov, Soviet Phys. Dokl. 28, 926 (1983)

    1983

  190. [198]

    Ivanov, V

    M. Ivanov, V. Gal’burt, and V. Fortov, JETP Lett.63, 813 (1996)

    1996

  191. [199]

    Cheng and L.-F

    T.-P. Cheng and L.-F. Li,Gauge Theory of Elementary Particle Physics(Oxford University Press, 1984)

    1984

  192. [200]

    Iliopoulos, C

    J. Iliopoulos, C. Itzykson, and A. Martin, Rev. Mod. Phys.47, 165 (1975)

    1975

  193. [201]

    Fukuda, M

    R. Fukuda, M. Komachiya, S. Yokojima, Y. Suzuki, K. Okumura, and T. Inagaki, Prog. Theor. Phys.121, 1 (1995)

    1995

  194. [202]

    B. S. DeWitt,The global approach to quantum field the- ory. Vol. 1, 2(Oxford University Press, 2003)

    2003

  195. [203]

    Kleinert,Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed

    H. Kleinert,Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed. (WORLD SCIENTIFIC, 2009)

    2009

  196. [204]

    Cooper and J

    F. Cooper and J. F. Dawson, Ann. Phys.365, 118 (2016)

    2016

  197. [205]

    Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states, Adv

    H. Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states, Adv. Phys.49, 815 (2000)

    2000

  198. [206]

    Henkel, H

    M. Henkel, H. Hinrichsen, and S. L¨ ubeck,Non- Equilibrium Phase Transitions, Vol. I: Absorbing Phase Transitions (Springer, Dordrecht, The Netherlands, 2008)

    2008

  199. [207]

    L. M. B. C. Campos, Phys. Plasmas6, 57 (1999)

    1999

  200. [208]

    Weinzierl,Feynman Integrals(Springer Cham, 2022)

    S. Weinzierl,Feynman Integrals(Springer Cham, 2022)

    2022

  201. [209]

    B. Y. Levin,Distribution of Zeros of Entire Functions (AMS, 1964)

    1964

  202. [210]

    Davydychev, Phys

    A. Davydychev, Phys. Lett. B263, 107 (1991)

    1991

  203. [211]

    O. V. Tarasov, Phys. Rev. D54, 6479 (1996)

    1996

  204. [212]

    O. V. Tarasov, Nucl. Phys. B502, 455 (1997)

    1997

  205. [213]

    Passarino and M

    G. Passarino and M. Veltman, Nucl. Phys. B160, 151 (1979)

    1979

  206. [214]

    C.-B. Kim, J. Turbul.11, N19 (2010)

    2010

  207. [215]

    Wolfram Research, Inc., Mathematica, Version 14.1 (2024), Champaign, IL

    2024

  208. [216]

    Peter Lepage, J

    G. Peter Lepage, J. Comput. Phys.27, 192 (1978)

    1978

  209. [217]

    B. G. Nickel, D. I. Meiron and G. A. Baker, Jr., (1977), (unpublished)

    1977

  210. [218]

    Batkovich, Y

    D. Batkovich, Y. Kirienko, M. Kompaniets, and S. Novikov, (2014), program repository:https:// gitlab.com/mkompan/graph_state, arXiv:1409.8227

    2014 arXiv

  211. [219]

    M. V. Kompaniets and E. Panzer, Phys. Rev. D96, 036016 (2017)

    2017

  212. [220]

    L. T. Adzhemyan, M. Hnatiˇ c, E. V. Ivanova, M. V. Kompaniets, T. Luˇ civjansk´ y, and L. Miˇ ziˇ sin, Phys. Rev. E107, 064138 (2023)

    2023

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.