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arxiv: 2605.02878 · v1 · submitted 2026-05-04 · ✦ hep-th · hep-lat· hep-ph

Recognition: 3 theorem links

· Lean Theorem

Finite-temperature operator basis on mathbb{R}³ times S¹ for SMEFT

Joydeep Chakrabortty , Bruno Siqueira Eduardo , Siddhartha Karmakar , Philipp Schicho
Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:21 UTC · model grok-4.3

classification ✦ hep-th hep-lathep-ph
keywords SMEFTfinite temperatureoperator basisHilbert seriesimaginary time formalismthermal operatorseffective field theory
0
0 comments X

The pith

The first complete non-redundant operator basis for SMEFT at finite temperature is derived on R^3 × S^1 using the Hilbert series method.

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

This paper develops a systematic way to list all effective operators in the Standard Model Effective Field Theory when temperature is finite. It applies the Hilbert series technique to the space-time geometry R^3 times a circle, which represents imaginary time at finite temperature. Only spatial directions are used for integration by parts and equations of motion to remove redundant operators. The resulting basis includes new operators that exist solely due to the finite temperature and disappear in the zero-temperature limit. This construction matters for accurately modeling thermal effects in beyond-Standard-Model physics, such as early-universe processes.

Core claim

We present the first complete non-redundant operator basis for the Standard Model Effective Field Theory (SMEFT) at finite temperature, using the imaginary-time formalism on R^3 × S^1. All effective operators up to dimension six are classified, with spatial integration-by-parts and equations-of-motion constraints imposed. Intrinsically thermal operators at dimensions five and six are identified that vanish at zero temperature, and the basis is expressed in terms of static three-dimensional and zero-temperature SMEFT operators.

What carries the argument

Hilbert series method on the manifold R^3 × S^1 with spatial-only integration-by-parts and equations-of-motion constraints.

If this is right

  • The basis enables consistent inclusion of finite-temperature effects in SMEFT calculations up to dimension six.
  • Operators can be matched to three-dimensional effective theories for thermal analyses.
  • Additional constraints like vanishing curl of fields can be applied to reduce the basis further.
  • The framework extends naturally to higher mass dimensions and other symmetry groups.

Where Pith is reading between the lines

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

  • Applications to cosmology could include better modeling of electroweak phase transitions at finite temperature.
  • Independent verification at low dimensions could test if the spatial constraints suffice for completeness.
  • Extension to include full four-dimensional IBP might reveal additional relations or operators.

Load-bearing premise

The Hilbert series method applied to R^3 × S^1 with spatial integration-by-parts and equations-of-motion constraints produces a complete and non-redundant basis.

What would settle it

Counting the number of independent operators at dimension six and checking if it matches the sum of zero-temperature SMEFT operators plus the identified thermal ones; a discrepancy would falsify completeness.

Figures

Figures reproduced from arXiv: 2605.02878 by Bruno Siqueira Eduardo, Joydeep Chakrabortty, Philipp Schicho, Siddhartha Karmakar.

Figure 1
Figure 1. Figure 1: Schematic workflow of the finite-temperature Hilbert series construction. view at source ↗
Figure 2
Figure 2. Figure 2: Total number of independent operators Nops for bosonic SMEFT at different mass dimensions: a comparison among (i) T = 0, (ii) T > 0 with CI, and (iii) T > 0 with CII. B. Growth of the number of independent operators Here, we display the total number of independent SMEFT operators, up to mass dimension ten, at finite temperature, and compare them with the zero-temperature SMEFT operator basis [45]. To be on… view at source ↗
read the original abstract

We present the first complete non-redundant operator basis for the Standard Model Effective Field Theory (SMEFT) at finite temperature, using the imaginary-time formalism. By employing the Hilbert series method on the space-time manifold $\mathbb{R}^3 \times S^1$, we classify all effective operators up to dimension-six. In constructing the basis, we consistently impose integration-by-parts and equations-of-motion constraints along spatial directions. We further analyze the impact of additional constraints, including the vanishing of the curl of the electric and magnetic fields and gauge choices for the temporal components on an operator basis. We also express them in terms of static three-dimensional spatial and zero-temperature SMEFT operators. At dimension five and six, we identify intrinsically thermal operators that vanish in zero temperature. Our framework is fully general and extends to arbitrary mass dimension and compact connected internal symmetry groups.

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 constructs the first complete non-redundant operator basis for SMEFT at finite temperature via the Hilbert series on the manifold R^3 x S^1 in the imaginary-time formalism. Operators up to dimension six are classified by imposing spatial-only integration-by-parts and equations-of-motion constraints, supplemented by analysis of curl(E/B)=0 and temporal gauge choices. The resulting basis is expressed in terms of static 3D and zero-temperature SMEFT operators, with intrinsically thermal operators (vanishing at T=0) identified at dimensions five and six. The framework is claimed to be general for arbitrary mass dimension and compact connected internal symmetry groups.

Significance. If the completeness and non-redundancy claims hold, the work supplies a systematic and usable operator basis for finite-temperature SMEFT calculations, directly relevant to cosmological applications such as electroweak phase transitions and thermal relic calculations. The explicit identification of intrinsically thermal operators and the mapping to established zero-temperature bases are practical strengths. The Hilbert-series approach on a compactified spacetime provides a reproducible counting method that extends naturally to higher dimensions.

major comments (1)
  1. [§3] §3 (Hilbert series on R^3 x S^1 and constraint implementation): The central claim that spatial-only IBP, EOM, curl(E/B)=0, and temporal gauge choices produce a complete non-redundant basis is load-bearing. The S^1 compactification introduces periodicity and Matsubara frequencies whose associated relations (involving time derivatives or full spacetime EOM) are not automatically eliminated by spatial constraints alone. If any such relation exists among the dimension-5 or -6 operators, the listed intrinsically thermal operators may contain hidden redundancies or the basis may be incomplete when matched to 3D static operators. An explicit verification or proof that no additional thermal redundancies arise would be required to secure the claim.
minor comments (2)
  1. The abstract states that operators are expressed in terms of static 3D and zero-T SMEFT operators, but a compact summary table listing the mapping for the new thermal operators at dim 5 and 6 would improve readability and allow immediate cross-checks.
  2. Notation for the additional constraints (curl vanishing and gauge choices) is introduced without a dedicated equation reference; adding an explicit equation label when these are first imposed would clarify their precise implementation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for recognizing the potential utility of our finite-temperature SMEFT operator basis. We address the single major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (Hilbert series on R^3 x S^1 and constraint implementation): The central claim that spatial-only IBP, EOM, curl(E/B)=0, and temporal gauge choices produce a complete non-redundant basis is load-bearing. The S^1 compactification introduces periodicity and Matsubara frequencies whose associated relations (involving time derivatives or full spacetime EOM) are not automatically eliminated by spatial constraints alone. If any such relation exists among the dimension-5 or -6 operators, the listed intrinsically thermal operators may contain hidden redundancies or the basis may be incomplete when matched to 3D static operators. An explicit verification or proof that no additional thermal redundancies arise would be required to secure the claim.

    Authors: The Hilbert series is evaluated directly on the manifold R^3 × S^1, so the compactification, periodicity, and Matsubara structure are built into the character integrals and the representation theory from the beginning; operators are generated only after these geometric features are imposed. Spatial IBP and EOM are the correct reductions because a full spacetime derivative would mix distinct Matsubara modes and violate the static/thermal separation used in thermal EFT matching. We have performed an explicit cross-check at dimensions 5 and 6 by enumerating all candidate operators, imposing the listed constraints, and matching the resulting basis both to the Warsaw basis (at T=0) and to the known three-dimensional static EFT operators; the intrinsically thermal operators that survive are linearly independent and no further relations appear. This verification is already implicit in the tables and counting of the manuscript; we will add a short paragraph in §3 making the matching explicit. The same construction applies unchanged at higher dimension because the Hilbert-series algorithm itself does not depend on the mass dimension. revision: partial

Circularity Check

0 steps flagged

Hilbert series classification on R^3 x S^1 uses standard method with no circular reductions

full rationale

The derivation applies the established Hilbert series to R^3 x S^1 and reduces via spatial IBP/EOM plus gauge checks; the resulting basis (including dim-5/6 thermal operators) is generated directly from the manifold and constraints without self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations. The central claim remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The classification rests on standard mathematical and EFT assumptions without new free parameters or postulated entities.

axioms (2)
  • domain assumption Hilbert series method correctly enumerates operators on the manifold R^3 x S^1
    Invoked to classify all effective operators up to dimension six.
  • domain assumption Integration-by-parts and equations-of-motion constraints are imposed only along spatial directions
    Used to remove redundancies while preserving thermal structure.

pith-pipeline@v0.9.0 · 5462 in / 1288 out tokens · 27340 ms · 2026-05-08T18:21:06.108980+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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matches
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supports
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extends
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uses
The paper appears to rely on the theorem as machinery.
contradicts
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unclear
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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Matching higher-dimensional operators at finite temperature for general models

    hep-ph 2026-05 conditional novelty 8.0

    The authors automate matching of generic 3D dimension-five and -six operators for arbitrary models, implemented in an extension of DRalgo with public code and examples for scalar-Yukawa, hot QCD, and the full Standard Model.

Reference graph

Works this paper leans on

108 extracted references · 84 canonical work pages · cited by 1 Pith paper · 4 internal anchors

  1. [1]

    Matsubara,A New approach to quantum statistical mechanics,Prog

    T. Matsubara,A New approach to quantum statistical mechanics,Prog. Theor. Phys.14(1955) 351

    1955

  2. [2]

    P. H. Ginsparg,First Order and Second Order Phase Transitions in Gauge Theories at Finite Temperature,Nucl. Phys. B170(1980) 388

    1980

  3. [3]

    Appelquist and R

    T. Appelquist and R. D. Pisarski,High-Temperature Yang-Mills Theories and Three- Dimensional Quantum Chromodynamics,Phys. Rev. D23(1981) 2305. 33

    1981

  4. [4]

    Susskind,Lattice Models of Quark Confinement at High Temperature,Phys

    L. Susskind,Lattice Models of Quark Confinement at High Temperature,Phys. Rev. D20(1979) 2610

    1979

  5. [5]

    Weiss,The Effective Potential for the Order Parameter of Gauge Theories at Finite Tem- perature,Phys

    N. Weiss,The Effective Potential for the Order Parameter of Gauge Theories at Finite Tem- perature,Phys. Rev. D24(1981) 475

    1981

  6. [6]

    Generic Rules for High Temperature Dimensional Reduction and Their Application to the Standard Model

    K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov,Generic rules for high temperature dimensional reduction and their application to the standard model,Nucl. Phys. B 458(1996) 90 [hep-ph/9508379]

    work page Pith review arXiv 1996

  7. [7]

    A. I. Bochkarev, S. Y. Khlebnikov, and M. E. Shaposhnikov,Sphalerons and Baryogenesis: Electroweak CP Violation at High Temperatures,Nucl. Phys. B329(1990) 493

    1990

  8. [8]

    Laine, M

    M. Laine, M. Meyer, and G. Nardini,Thermal phase transition with full 2-loop effective potential, Nucl. Phys. B920(2017) 565 [1702.07479]

    work page arXiv 2017

  9. [9]

    Croon, O

    D. Croon, O. Gould, P. Schicho, T. V. I. Tenkanen, and G. White,Theoretical uncertainties for cosmological first-order phase transitions,JHEP04(2021) 055 [2009.10080]

    work page arXiv 2021

  10. [10]

    Ekstedt, P

    A. Ekstedt, P. Schicho, and T. V. I. Tenkanen,DRalgo: A package for effective field theory ap- proach for thermal phase transitions,Comput. Phys. Commun.288(2023) 108725 [2205.08815]

    work page arXiv 2023

  11. [11]

    Ekstedt, P

    A. Ekstedt, P. Schicho, and T. V. I. Tenkanen,Cosmological phase transitions at three loops: The final verdict on perturbation theory,Phys. Rev. D110(2024) 096006 [2405.18349]

    work page arXiv 2024

  12. [12]

    Bernardo, P

    F. Bernardo, P. Klose, P. Schicho, and T. V. I. Tenkanen,Higher-dimensional operators at finite temperature affect gravitational-wave predictions,JHEP08(2025) 109 [2503.18904]

    work page arXiv 2025

  13. [13]

    Matchotter: An Automated Tool for Dimensional Reduction at Finite Temperature

    J. Fuentes-Mart´ ın, J. L´ opez Miras, and A. Moreno-S´ anchez,Matchotter: An Automated Tool for Dimensional Reduction at Finite Temperature,[2604.21972]

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [14]

    O(g) plasma effects in jet quenching

    S. Caron-Huot,O(g) plasma effects in jet quenching,Phys. Rev. D79(2009) 065039 [0811.1603]

    work page Pith review arXiv 2009

  15. [15]

    A lattice study of the jet quenching parameter

    M. Panero, K. Rummukainen, and A. Sch¨ afer,Lattice Study of the Jet Quenching Parameter, Phys. Rev. Lett.112(2014) 162001 [1307.5850]

    work page Pith review arXiv 2014

  16. [16]

    D’Onofrio, A

    M. D’Onofrio, A. Kurkela, and G. D. Moore,Renormalization of Null Wilson Lines in EQCD, JHEP03(2014) 125 [1401.7951]

    work page arXiv 2014

  17. [17]

    Ghiglieri, J

    J. Ghiglieri, J. Hong, A. Kurkela, E. Lu, G. D. Moore, and D. Teaney,Next-to-leading or- der thermal photon production in a weakly coupled quark-gluon plasma,JHEP05(2013) 010 [1302.5970]

    work page arXiv 2013

  18. [18]

    Ghiglieri and G

    J. Ghiglieri and G. D. Moore,Low Mass Thermal Dilepton Production at NLO in a Weakly Coupled Quark-Gluon Plasma,JHEP12(2014) 029 [1410.4203]

    work page arXiv 2014

  19. [19]

    Ghiglieri and M

    J. Ghiglieri and M. Laine,Neutrino dynamics below the electroweak crossover,JCAP07(2016) 015 [1605.07720]

    work page arXiv 2016

  20. [20]

    Comparison of 4d and 3d Lattice Results for the Electroweak Phase Transition

    M. Laine,Comparison of 4-D and 3-D lattice results for the electroweak phase transition,Phys. Lett. B385(1996) 249 [hep-lat/9604011]. 34

    work page Pith review arXiv 1996

  21. [21]

    The renormalized gauge coupling and non-perturbative tests of dimensional reduction

    M. Laine,The Renormalized gauge coupling and nonperturbative tests of dimensional reduction, JHEP06(1999) 020 [hep-ph/9903513]

    work page Pith review arXiv 1999

  22. [22]

    Chala, J

    M. Chala, J. C. Criado, L. Gil, and J. L. Miras,Higher-order-operator corrections to phase- transition parameters in dimensional reduction,JHEP10(2024) 025 [2406.02667]

    work page arXiv 2024

  23. [23]

    The High-Temperature Limit of the SM(EFT)

    M. Chala and G. Guedes,The high-temperature limit of the SM(EFT),JHEP07(2025) 085 [2503.20016]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  24. [24]

    Bernardo, M

    F. Bernardo, M. Chala, L. Gil, and P. Schicho,Hard thermal contributions to phase transition observables at NNLO,[2602.06962]

    work page arXiv

  25. [25]

    Weinberg,Baryon and Lepton Nonconserving Processes,Phys

    S. Weinberg,Baryon and Lepton Nonconserving Processes,Phys. Rev. Lett.43(1979) 1566

    1979

  26. [26]

    Buchmuller and D

    W. Buchmuller and D. Wyler,Effective Lagrangian Analysis of New Interactions and Flavor Conservation,Nucl. Phys. B268(1986) 621

    1986

  27. [27]

    Dimension-Six Terms in the Standard Model Lagrangian

    B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek,Dimension-Six Terms in the Standard Model Lagrangian,JHEP10(2010) 085 [1008.4884]

    work page internal anchor Pith review arXiv 2010

  28. [28]

    Aebischer, A

    J. Aebischer, A. J. Buras, and J. Kumar,SMEFT ATLAS: The Landscape Beyond the Standard Model,[2507.05926]

    work page arXiv

  29. [29]

    Soft thermal contributions to 3-loop gauge coupling

    M. Laine, P. Schicho, and Y. Schr¨ oder,Soft thermal contributions to 3-loop gauge coupling, JHEP05(2018) 037 [1803.08689]

    work page Pith review arXiv 2018

  30. [30]

    G. D. Moore,Fermion determinant and the sphaleron bound,Phys. Rev. D53(1996) 5906 [hep-ph/9508405]

    work page Pith review arXiv 1996

  31. [31]

    Chala, M

    M. Chala, M. C. Fiore, and L. Gil,Hot news on the phase-structure of the SMEFT,[2507.16905]

    work page arXiv

  32. [32]

    H. A. Weldon,Covariant Calculations at Finite Temperature: The Relativistic Plasma,Phys. Rev. D26(1982) 1394

    1982

  33. [33]

    J. S. Schwinger,On gauge invariance and vacuum polarization,Phys. Rev.82(1951) 664

    1951

  34. [34]

    B. S. DeWitt,Quantum Field Theory in Curved Space-Time,Phys. Rept.19(1975) 295

    1975

  35. [35]

    D. I. D’yakonov, V. Y. Petrov, and A. V. Yung,Quasiclassical expansion in an external Yang- Mills field and the approximate calculation of functional determinants,Sov. J. Nucl. Phys.39:1 (1984)

    1984

  36. [36]

    A New Dimensionally Reduced Effective Action for QCD at High Temperature

    S. Chapman,A New dimensionally reduced effective action for QCD at high temperature,Phys. Rev. D50(1994) 5308 [hep-ph/9407313]

    work page Pith review arXiv 1994

  37. [37]

    The thermal heat kernel expansion and the one-loop effective action of QCD at finite temperature

    E. Megias, E. Ruiz Arriola, and L. L. Salcedo,The Thermal heat kernel expansion and the one loop effective action of QCD at finite temperature,Phys. Rev. D69(2004) 116003 [hep-ph/0312133]

    work page Pith review arXiv 2004

  38. [38]

    Chakrabortty and S

    J. Chakrabortty and S. Mohanty,One Loop Thermal Effective Action,Nucl. Phys. B1020 (2025) 117165 [2411.14146]. 35

    work page arXiv 2025

  39. [39]

    Balui, T

    D. Balui, T. Biswas, J. Chakrabortty, D. Dey, C. Englert, and S. Mohanty,Gauge choices, infrared pitfalls, and thermal effects in effective potentials,Phys. Rev. D112(2025) 056022 [2507.22706]

    work page arXiv 2025

  40. [40]

    Balui, J

    D. Balui, J. Chakrabortty, D. Dey, and S. Mohanty,Gauge invariant effective potential,Phys. Rev. D111(2025) 085032 [2502.17156]

    work page arXiv 2025

  41. [41]

    High Temperature Dimensional Reduction and Parity Violation

    K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov,High temperature dimen- sional reduction and parity violation,Phys. Lett. B423(1998) 137 [hep-ph/9710538]

    work page Pith review arXiv 1998

  42. [42]

    Electroweak phase diagram at finite lepton number density

    A. Gynther,Electroweak phase diagram at finite lepton number density,Phys. Rev. D68(2003) 016001 [hep-ph/0303019]

    work page Pith review arXiv 2003

  43. [43]

    Lehman and A

    L. Lehman and A. Martin,Hilbert Series for Constructing Lagrangians: expanding the phe- nomenologist’s toolbox,Phys. Rev. D91(2015) 105014 [1503.07537]

    work page arXiv 2015

  44. [44]

    Hilbert series and operator bases with derivatives in effective field theories

    B. Henning, X. Lu, T. Melia, and H. Murayama,Hilbert series and operator bases with deriva- tives in effective field theories,Commun. Math. Phys.347(2016) 363 [1507.07240]

    work page Pith review arXiv 2016

  45. [45]

    Henning, X

    B. Henning, X. Lu, T. Melia, and H. Murayama,2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimension operators in the SM EFT,JHEP08(2017) 016 [1512.03433]

    work page arXiv 2017

  46. [46]

    Henning, X

    B. Henning, X. Lu, T. Melia, and H. Murayama,Operator bases,S-matrices, and their partition functions,JHEP10(2017) 199 [1706.08520]

    work page arXiv 2017

  47. [47]

    Lehman and A

    L. Lehman and A. Martin,Low-derivative operators of the Standard Model effective field theory via Hilbert series methods,JHEP02(2016) 081 [1510.00372]

    work page arXiv 2016

  48. [48]

    Das Bakshi, J

    Anisha, S. Das Bakshi, J. Chakrabortty, and S. Prakash,Hilbert Series and Plethystics: Paving the path towards 2HDM- and MLRSM-EFT,JHEP09(2019) 035 [1905.11047]

    work page arXiv 2019

  49. [49]

    Ruhdorfer, J

    M. Ruhdorfer, J. Serra, and A. Weiler,Effective Field Theory of Gravity to All Orders,JHEP 05(2020) 083 [1908.08050]

    work page arXiv 2020

  50. [50]

    Delgado, A

    A. Delgado, A. Martin, and R. Wang,Constructing operator basis in supersymmetry: a Hilbert series approach,JHEP04(2023) 097 [2212.02551]

    work page arXiv 2023

  51. [51]

    Grojean, J

    C. Grojean, J. Kley, and C.-Y. Yao,Hilbert series for ALP EFTs,JHEP11(2023) 196 [2307.08563]

    work page arXiv 2023

  52. [52]

    Alonso and S

    R. Alonso and S. U. Rahaman,Counting and building operators in theories with hidden symme- tries and application to HEFT,JHEP08(2025) 071 [2412.09463]

    work page arXiv 2025

  53. [53]

    Lagrangian

    U. Banerjee, J. Chakrabortty, S. Prakash, and S. U. Rahaman,Characters and group invariant polynomials of (super)fields: road to “Lagrangian”,Eur. Phys. J. C80(2020) 938 [2004.12830]

    work page arXiv 2020

  54. [54]

    C. B. Marinissen, R. Rahn, and W. J. Waalewijn,..., 83106786, 114382724, 1509048322, 2343463290, 27410087742, ... efficient Hilbert series for effective theories,Phys. Lett. B808 (2020) 135632 [2004.09521]

    work page arXiv 2020

  55. [55]

    Melia and S

    T. Melia and S. Pal,EFT Asymptotics: the Growth of Operator Degeneracy,SciPost Phys.10 (2021) 104 [2010.08560]. 36

    work page arXiv 2021

  56. [56]

    Kondo, H

    D. Kondo, H. Murayama, and R. Okabe,23, 381, 6242, 103268, 1743183, . . . : Hilbert series for CP-violating operators in SMEFT,JHEP03(2023) 107 [2212.02413]

    work page arXiv 2023

  57. [57]

    Delgado, A

    A. Delgado, A. Martin, and R. Wang,Counting operators in N = 1 supersymmetric gauge theories,JHEP07(2023) 081 [2305.01736]

    work page arXiv 2023

  58. [58]

    Sun, Y.-N

    H. Sun, Y.-N. Wang, and J.-H. Yu,Chiral effective field theories for strong and weak dynamics, JHEP08(2025) 185 [2501.14018]

    work page arXiv 2025

  59. [59]

    Kobach and S

    A. Kobach and S. Pal,Conformal Structure of the Heavy Particle EFT Operator Basis,Phys. Lett. B783(2018) 311 [1804.01534]

    work page arXiv 2018

  60. [60]

    Li, Y.-N

    Y.-K. Li, Y.-N. Wang, and J.-H. Yu,Systematic Operator Construction for Non-relativistic Effective Field Theories: Hilbert Series versus Young Tensor,[2602.12263]

    work page arXiv

  61. [61]

    Beckers and V

    J. Beckers and V. Hussin,SPINORS AND RELATED TENSORS INVARIANT UNDER E(3) AND ITS SUBGROUPS,J. Phys. A14(1981) 317

    1981

  62. [62]

    Y. A. Sitenko,Path integral formalism for finite-temperature field theory and generation of chiral currents,[2303.02145]

    work page arXiv

  63. [63]

    M. A. Aguilar and M. Socolovsky,The Universal Covering Group of U(n) and Projective Rep- resentations,[math-ph/9911028]

    work page arXiv

  64. [64]

    C. P. Bacry H and R. J. L,Group-theoretical analysis of elementary particles in an external electromagnetic field: I. The relativistic particle in a constant and uniform field,Nuovo Cimento 67(1970) 267

    1970

  65. [65]

    M. J. Douglas and J. Repka,Embeddings of the Euclidean algebrae(3)intosl(4,C)and re- strictions of irreducible representations ofsl(4,C),Journal of Mathematical Physics52(2011) 013504

    2011

  66. [66]

    A. M. Polyakov,Compact Gauge Fields and the Infrared Catastrophe,Phys. Lett. B59(1975) 82

    1975

  67. [67]

    V. N. Popov and S. A. Fedotov,The functional-integration method and diagram technique for spin systems,Sov. Phys. JETP67(1988) 535

    1988

  68. [68]

    Prokof’ev and B

    N. Prokof’ev and B. Svistunov,From Popov-Fedotov trick to universal fermionization,Phys. Rev. B84(2011) 073102 [1103.3730]

    work page arXiv 2011

  69. [69]

    F. J. Moral-Gamez and L. L. Salcedo,Derivative expansion of the heat kernel at finite temper- ature,Phys. Rev. D85(2012) 045019 [1110.6300]

    work page arXiv 2012

  70. [70]

    D. J. Gross, R. D. Pisarski, and L. G. Yaffe,QCD and Instantons at Finite Temperature,Rev. Mod. Phys.53(1981) 43

    1981

  71. [71]

    R. D. Pisarski,Quark gluon plasma as a condensate of SU(3) Wilson lines,Phys. Rev. D62 (2000) 111501 [hep-ph/0006205]

    work page arXiv 2000

  72. [72]

    R. D. Pisarski,Notes on the deconfining phase transition,[hep-ph/0203271]. 37

    work page internal anchor Pith review arXiv

  73. [73]

    The Phase Diagram of Three-Dimensional SU(3) + Adjoint Higgs Theory

    K. Kajantie, M. Laine, A. Rajantie, K. Rummukainen, and M. Tsypin,The Phase diagram of three-dimensional SU(3) + adjoint Higgs theory,JHEP11(1998) 011 [hep-lat/9811004]

    work page Pith review arXiv 1998

  74. [74]

    Z(3)-symmetric effective theory for SU(3) Yang-Mills theory at high temperature

    A. Vuorinen and L. G. Yaffe,Z(3)-symmetric effective theory for SU(3) Yang-Mills theory at high temperature,Phys. Rev. D74(2006) 025011 [hep-ph/0604100]

    work page Pith review arXiv 2006

  75. [75]

    Kurkela,Framework for non-perturbative analysis of a Z(3)-symmetric effective theory of finite temperature QCD,Phys

    A. Kurkela,Framework for non-perturbative analysis of a Z(3)-symmetric effective theory of finite temperature QCD,Phys. Rev. D76(2007) 094507 [0704.1416]

    work page arXiv 2007

  76. [76]

    Langelage, S

    J. Langelage, S. Lottini, and O. Philipsen,Centre symmetric 3d effective actions for thermal SU(N) Yang-Mills from strong coupling series,JHEP02(2011) 057 [1010.0951]

    work page arXiv 2011

  77. [77]

    Bergner, J

    G. Bergner, J. Langelage, and O. Philipsen,Effective lattice Polyakov loop theory vs. full SU(3) Yang-Mills at finite temperature,JHEP03(2014) 039 [1312.7823]

    work page arXiv 2014

  78. [78]

    Fukushima and V

    K. Fukushima and V. Skokov,Polyakov loop modeling for hot QCD,Prog. Part. Nucl. Phys.96 (2017) 154 [1705.00718]

    work page arXiv 2017

  79. [79]

    Perturbative Thermal QCD: Formalism and Applications,

    J. Ghiglieri, A. Kurkela, M. Strickland, and A. Vuorinen,Perturbative Thermal QCD: Formalism and Applications,Phys. Rept.880(2020) 1 [2002.10188]

    work page arXiv 2020

  80. [80]

    Bernardo, R

    F. Bernardo, R. G. Reinle, and P. Schicho,Higher-dimensional operators at finite temperature for general models,forthcoming (2026)

    2026

Showing first 80 references.