Pith. sign in

REVIEW 2 major objections 3 minor 2 cited by

DoFun 3.0: Functional equations in Mathematica

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read DoFun 3.0 now derives composite-operator correlation equations in Mathematica.

desk verdict A solid, honest software update whose composite-operator feature is demonstrated within its documented limits; the main weakness is the lack of independent verification of the worked example. read the letter →

arxiv 1908.02760 v2 pith:B2RIL62O submitted 2019-08-07 hep-ph cond-mat.otherhep-th

Markus Q. Huber , Anton K. Cyrol , Jan M. Pawlowski This is my paper · ORCID
classification hep-phcond-mat.otherhep-th PACS 11.10.-z03.70.+k11.15.Tk
keywords Dyson-SchwingerequationsfunctionalrenormalizationgroupcorrelationfunctionscompositeoperatorsquantumfieldtheoryFeynmandiagramsMathematicapackagesymbolicderivation
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

This paper presents version 3.0 of DoFun, a Mathematica package that derives Dyson-Schwinger equations, functional renormalization group flow equations, and, newly in this version, correlation functions of composite operators. The central claim is that all three types of equations can be obtained from a user-supplied action and field list in symbolic Feynman-diagram form, then converted to algebraic expressions for further computation. The new composite-operator feature rests on the identity that any full correlation function can be expressed by acting with the operator on fields shifted by propagator times a field derivative, $\langle F(\varphi)\rangle = F(\Phi_i + D_{ij}^J \, \delta/\delta\Phi_j)$. A worked example derives the two-point function of the gluonic energy-momentum tensor up to two loops, including connected and one-particle-irreducible (1PI) extraction.

What carries the argument

The central mechanism is the replacement identity $\langle F(\varphi)\rangle = F(\Phi_i + D_{ij}^J \, \delta/\delta\Phi_j)$, which turns a full correlation function of any operator into a sequence of functional derivatives acting on dressed propagators and vertices. The package represents a composite operator as an auxiliary contracted object $C$ that behaves like a vertex, so the same differentiation and diagram-generation code used for DSEs and flow equations handles operator correlation functions. Three derivative rules, including $\delta/\delta\Phi_i\, D_{jk}^J = -\epsilon^i_{jm}\, D_{jm}^J\, \Gamma_{imn}^J\, D_{nk}^J$, carry all propagator and vertex derivatives, with the sign function $\epsilon$ encoding Grassmann field anticommutation.

What would settle it

Take a two-loop DSE or composite-operator equation in a theory with mixed boson-fermion propagators, derive it by hand, and run DoFun's identifyGraphs: if a mixed-propagator diagram is not recognized as identical to its hand-derived counterpart or is dropped after 1PI extraction, the stated limitation is confirmed.

Watch

Extended reading notes

Core claim

DoFun 3.0 claims to automate the derivation of DSEs, functional RGEs, and composite-operator correlation functions from a symbolic action, producing output that can be plotted as Feynman diagrams and translated into algebraic expressions. The composite-operator derivation writes the operator as a contracted vertex-like object and then applies the replacement identity of Eq. (21), with the number of loops in the final correlation function ranging up to $n-2$ for an $n$-field operator. In the energy-momentum-tensor example, the package generates 72 diagrams, reduces them by symmetry, keeps connected diagrams, and extracts 1PI diagrams, yielding a compact two-loop expression. The authors state that the symbolic and algebraic results are correct, while the automated identification of identical diagrams is known to work reliably only up to two loops and can fail for mixed propagators.

Load-bearing premise

The automated recognition and classification of Feynman diagrams must correctly identify all generated diagrams, but the paper itself states that this identification only works reliably up to two loops and can fail when mixed propagators appear.

Editorial extensions

If this is right

  • Users can derive DSEs and flow equations from a symbolic action without enumerating Feynman diagrams by hand.
  • Composite-operator correlation functions, such as those of the energy-momentum tensor, become accessible through the same automated pipeline.
  • The symbolic output can be converted to algebraic expressions suitable for trace evaluation or numerical computation.
  • Explicit field typing removes ambiguity for complex scalar fields and improves sign handling with left-derivatives.
  • New diagram-classification tools let users select by loop number, connectedness, 1PI property, or named diagram type.

Reading between the lines

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

  • The same auxiliary-vertex trick likely extends to composite operators with more than two fields, but the two-loop limit on graph identification is the practical bottleneck for higher-loop operator equations.
  • Applying the method to fermionic bound-state operators would require carefully rechecking the Grassmann sign conventions, a task the paper's left-derivative setup makes tractable.
  • If the graph-isomorphism step were made more robust, the package could handle three-loop operator equations and theories with mixed propagators without manual diagram identification.
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

2 major / 3 minor

Summary. This paper presents version 3.0 of DoFun, a Mathematica package for deriving Dyson-Schwinger equations, functional renormalization group equations, and—new in this version—correlation functions of composite operators. The authors describe installation and basic workflow, the explicit field-type handling introduced in this version, the derivation algorithms based on the standard master equations (the DSE master equation, the Wetterich equation, and the replacement identity Eq. (21)), and a set of new tools such as diagram identification, connected/1PI extraction, and canonical ordering. The main new-feature demonstration is the two-loop correlation function of the spatial, traceless energy-momentum tensor in Yang-Mills theory, Eq. (26), obtained through doCO, getConnected, identifyGraphs, get1PI, and getAE, with diagrammatic results in Figs. 1–2. Appendices list changes from DoFun 2 and document known limitations of diagram identification and plotting.

Significance. If the implementation is correct, the new composite-operator functionality extends a widely used, publicly available tool (GPLv3, with a public git repository) and provides a nontrivial worked example relevant to transport calculations. The algorithms are parameter-free implementations of established functional identities, so there is no circularity or fitting. However, the paper does not provide an independent check of the new feature's output: no algebraic result is displayed, no test suite or verification notebook is referenced, and Appendix C concedes that diagram identification is reliable only up to two loops and can fail for mixed propagators. This makes the unverified two-loop example the main risk to the paper's central claim.

major comments (2)
  1. [§3.4, Figs. 1–2 and Appendix C] The two-loop composite-operator result, which is the advertised new feature of DoFun 3.0, is presented only as a symbolic diagrammatic expression, with the algebraic translation described only schematically around In[19]–In[20]. Appendix C states that identifyGraphs works only up to two loops and can fail for mixed propagators, yet the paper asserts without further evidence that “the symbolic and algebraic results, though, are correct.” This assertion is load-bearing: if identifyGraphs merges distinct diagrams or misassigns symmetry factors, or if get1PI drops a nonvanishing diagram, then Eq. (26) and Figs. 1–2 are wrong. Please add a reproducibility artifact—for example, a notebook that runs the full pipeline of Sec. 3.4 and checks the number of diagrams, their symmetry factors, and the 1PI truncation—or provide a low-order algebraic expression verified against an independent manual derivation. Please also state explicitly that the two-loop example lies within the reliability regime of identifyGraphs and clarify whether get1PI has any analogous limitation.
  2. [§3.4, In[8] and Eq. (29)] The example uses the Yang-Mills action without ghost fields, stating that ghosts “do not contribute in this case.” This is not obvious: even though the composite operator πij depends only on the gluon field, ghost loops can enter connected multi-loop diagrams through ghost-gluon vertices, and no color or BRST argument is given for their vanishing at this order. If ghost diagrams were nonvanishing, the displayed result would be incomplete. Please provide the missing justification or repeat the example with ghosts included to demonstrate that doCO treats them correctly.
minor comments (3)
  1. [§2, In[3]–In[4]] The typeset code samples for setFields and the action contain brace structures that are easy to misread and may have unbalanced delimiters when copied verbatim; please check that the displayed input matches the code in the repository.
  2. [§3.4, In[14]] The sentence “Of the originally 72 diagrams, many of which are identical, though, now 63 remain” is grammatically confusing; clarify whether the 63 are before or after summing identical diagrams.
  3. [Appendix C] The limitation section would be more useful if it stated explicitly which functions are affected by the two-loop and mixed-propagator restrictions, and whether getConnected and get1PI are free of the same restrictions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is a software implementation of externally established master equations, and its self-citations are not load-bearing.

full rationale

The derivation chain in DoFun 3.0 starts from externally established master equations: the DSE master equation from a total derivative (Eq. (15)), the Wetterich equation (Eq. (18)) attributed to Wetterich [72], and the composite-operator identity (Eq. (21)) taken from Pawlowski's review [8]. The composite-operator result, the paper's new feature, is obtained by substituting the operator into Eq. (21) and representing the operator as an auxiliary vertex C contracted with ordinary fields (Eqs. (23)-(24)); no fitted parameter, no output-dependent normalization, and no target equation is inserted as an input. Self-citations to DoFun 2 [20] and DoDSE [71] are used for implementation details and background, not to justify the central result, and the new functionality is presented with its own derivation and explicit Mathematica steps. Appendix C's admission that diagram identification only works up to two loops and can fail for mixed propagators is a correctness and robustness caveat about the example's reliability, not a circularity: it concerns whether the automated code misclassifies diagrams, not whether the input already contains the output. The symbolic and algebraic results are claimed correct but this is an unverified software-verification issue, which is outside the circularity categories. No step in the paper is equivalent to its input by construction, no prediction is a renamed fit, and no load-bearing claim rests solely on a self-citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented physical entities appear. The paper relies on standard QFT identities and master equations from the literature. The auxiliary field FF is a bookkeeping device, not a physical entity.

assumptions (4)
  • domain assumption Legendre transform defines the effective action and correlation functions (Eq. 1).
    Used throughout Sec. 3 to define propagators and vertices; standard in quantum field theory.
  • domain assumption Wetterich equation for the effective average action (Eq. 18).
    Used as master equation for flow equations in Sec. 3.3; cited to Ref. [72].
  • domain assumption Master identity for composite operator correlation functions (Eq. 21).
    Used in Sec. 3.4 to derive the energy-momentum tensor correlation function; cited to Ref. [8].
  • standard math Functional derivative rules (Eq. 10).
    Used to apply derivatives in DSE, RGE, and composite-operator derivations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of DoFun 3.0: Functional equations in Mathematica." pith.science (2026). https://pith.science/paper/B2RIL62O

@misc{pith2026190802760,
  author       = {Pith},
  title        = {Pith review of: DoFun 3.0: Functional equations in Mathematica},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2RIL62O}},
  note         = {Machine review of arXiv:1908.02760}
}
read the original abstract

We present version 3.0 of the Mathematica package DoFun for the derivation of functional equations. In this version, the derivation of equations for correlation functions of composite operators was added. In the update, the general workflow was slightly modified taking into account experience with the previous version. In addition, various tools were included to improve the usage experience and the code was partially restructured for easier maintenance.

Figures

Figures reproduced from arXiv: 1908.02760 by the authors.

Figure 1
Figure 1. The correlation function Eq. (26) up to two loops. Red, continuous lines are gluons, the triangle represents [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. The correlation function Eq. (26) up to two loops with some diagrams discarded that vanish due to color [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Examples for three-loop contributions to [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Functional renormalization of QCD in $1 + 1$ dimensions: four-fermion interactions from quark-gluon dynamics

    hep-ph 2024-12 conditional novelty 7.0 of 10

    First functional renormalization group study of two-dimensional QCD with a mass-like regulator, producing flow equations for the gauge coupling, quark mass, and a Fierz-complete set of four-fermion interactions.

  2. A beginner's guide to functional methods in particle physics

    hep-ph 2025-10 accept novelty 1.0 of 10

    A pedagogical review showing how Dyson-Schwinger, 3PI, and Bethe-Salpeter equations can be chained together to compute glueball masses in pure Yang-Mills theory, matching lattice QCD.

Reference graph

Works this paper leans on

74 extracted references · 26 canonical work pages · cited by 2 Pith papers

  1. [8]

    J. M. Pawlowski, Annals Phys. 322 (2007) 2831–2915. doi: 10.1016/j.aop.2007.01.007. arXiv:hep-th/0512261

    arXiv 2007

  2. [19]

    Benedetti, K

    D. Benedetti, K. Groh, P. F. Machado, F. Saueressig, JHEP 06 (2011) 079. doi: 10.1007/ JHEP06(2011)079. arXiv:1012.3081

    arXiv 2011

  3. [20]

    M. Q. Huber, J. Braun, Comput.Phys.Commun. 183 (2012) 1290–1320. doi: 10.1016/j.cpc. 2012.01.014. arXiv:1102.5307

    arXiv 2012

  4. [1]

    Wolfram, The Mathematica Book, Wolfram Media and Cambridge University Press, 2004

    S. Wolfram, The Mathematica Book, Wolfram Media and Cambridge University Press, 2004

    work page 2004

  5. [2]

    Harlander, M

    R. Harlander, M. Steinhauser, Prog. Part. Nucl. Phys. 43 (1999) 167–228. doi: 10.1016/ S0146-6410(99)00095-2. arXiv:hep-ph/9812357

    arXiv 1999

  6. [3]

    Precision Calculations for Future Colliders

    U. Baur, Int. J. Mod. Phys. E17 (2008) 826–844. doi: 10.1142/S0218301308010192. arXiv:hep-ph/0701164

    work page Pith review arXiv 2008

  7. [4]

    Track 3: Computations in theoretical physics -- techniques and methods

    G. Luisoni, S. Poslavsky, Y. Schroder, J. Phys. Conf. Ser. 762 (2016) 012077. doi: 10.1088/ 1742-6596/762/1/012077. arXiv:1604.03370. 12

    work page Pith review arXiv 2016

  8. [5]

    Berges, N

    J. Berges, N. Tetradis, C. Wetterich, Phys. Rept. 363 (2002) 223–386. doi: 10.1016/ S0370-1573(01)00098-9. arXiv:hep-ph/0005122

    arXiv 2002

Show all 74 references
  1. [6]

    C. D. Roberts, S. M. Schmidt, Prog. Part. Nucl. Phys. 45 (2000) S1–S103. doi: 10.1016/ S0146-6410(00)90011-5. arXiv:nucl-th/0005064

    2000 arXiv

  2. [7]

    Alkofer, L

    R. Alkofer, L. von Smekal, Phys. Rept. 353 (2001) 281. doi: 10.1016/S0370-1573(01) 00010-2. arXiv:hep-ph/0007355

    2001 arXiv

  3. [9]

    C. S. Fischer, J. Phys. G32 (2006) R253–R291. doi: 10.1088/0954-3899/32/8/R02. arXiv:hep-ph/0605173

    2006 arXiv

  4. [10]

    Gies, Lect.Notes Phys

    H. Gies, Lect.Notes Phys. 852 (2012) 287–348. doi: 10.1007/978-3-642-27320-9_6 . arXiv:hep-ph/0611146

    2012 arXiv

  5. [11]

    Schaefer, J

    B.-J. Schaefer, J. Wambach, Phys. Part. Nucl. 39 (2008) 1025–1032. doi: 10.1134/ S1063779608070083. arXiv:hep-ph/0611191

    2008 arXiv

  6. [12]

    Binosi, J

    D. Binosi, J. Papavassiliou, Phys. Rept. 479 (2009) 1–152. doi: 10.1016/j.physrep.2009.05

    2009 doi

  7. [13]

    Braun, J

    J. Braun, J. Phys. G39 (2012) 033001. doi: 10.1088/0954-3899/39/3/033001. arXiv:1108.4449

    2012 arXiv

  8. [14]

    Maas, Phys.Rept

    A. Maas, Phys.Rept. 524 (2013) 203–300. doi: 10.1016/j.physrep.2012.11.002. arXiv:1106.3942

    2013 arXiv

  9. [15]

    Eichmann, H

    G. Eichmann, H. Sanchis-Alepuz, R. Williams, R. Alkofer, C. S. Fischer, Prog. Part. Nucl. Phys. 91 (2016) 1–100. doi: 10.1016/j.ppnp.2016.07.001. arXiv:1606.09602

    2016 arXiv

  10. [16]

    Sanchis-Alepuz, R

    H. Sanchis-Alepuz, R. Williams, Comput. Phys. Commun. 232 (2018) 1–21. doi: 10.1016/j. cpc.2018.05.020. arXiv:1710.04903

    2018 arXiv

  11. [17]

    M. Q. Huber (2018). arXiv:1808.05227

    2018 arXiv

  12. [18]

    Alkofer, M

    R. Alkofer, M. Q. Huber, K. Schwenzer, Phys. Rev. D81 (2010) 105010. doi: http://link. aps.org/doi/10.1103/PhysRevD.81.105010. arXiv:0801.2762

    2010 arXiv

  13. [21]

    Fischbacher, F

    T. Fischbacher, F. Synatschke-Czerwonka, Comput. Phys. Commun. 184 (2013) 1931–1945. doi:10.1016/j.cpc.2013.03.002. arXiv:1202.5984

    2013 arXiv

  14. [22]

    M. Q. Huber, M. Mitter, Comput.Phys.Commun. 183 (2012) 2441–2457. doi: 10.1016/j.cpc. 2012.05.019. arXiv:1112.5622

    2012 arXiv

  15. [23]

    A. K. Cyrol, M. Mitter, N. Strodthoff, Comput. Phys. Commun. 219 (2017) 346–352. doi: 10. 1016/j.cpc.2017.05.024. arXiv:1610.09331

    2017 arXiv

  16. [24]

    van Ritbergen, A

    T. van Ritbergen, A. N. Schellekens, J. A. M. Vermaseren, Int. J. Mod. Phys. A14 (1999) 41–96. doi:10.1142/S0217751X99000038. arXiv:hep-ph/9802376

    1999 arXiv

  17. [25]

    J. A. M. Vermaseren (2000). arXiv:math-ph/0010025

    2000 arXiv

  18. [26]

    Kuipers, T

    J. Kuipers, T. Ueda, J. A. M. Vermaseren, J. Vollinga, Comput. Phys. Commun. 184 (2013) 1453–1467. doi:10.1016/j.cpc.2012.12.028. arXiv:1203.6543

    2013 arXiv

  19. [27]

    Kuipers, T

    J. Kuipers, T. Ueda, J. A. M. Vermaseren, Comput. Phys. Commun. 189 (2015) 1–19. doi: 10. 1016/j.cpc.2014.08.008. arXiv:1310.7007. 13

    2015 arXiv

  20. [28]

    Ruijl, T

    B. Ruijl, T. Ueda, J. Vermaseren (2017). arXiv:1707.06453

    2017 arXiv

  21. [29]

    Wiebusch, Comput

    M. Wiebusch, Comput. Phys. Commun. 195 (2015) 172–190. doi: 10.1016/j.cpc.2015.04

    2015 doi

  22. [30]

    Mertig, M

    R. Mertig, M. Bohm, A. Denner, Comput. Phys. Commun. 64 (1991) 345–359. doi: 10.1016/ 0010-4655(91)90130-D

    1991

  23. [31]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, F. Orellana, Comput. Phys. Commun. 207 (2016) 432–444. doi:10. 1016/j.cpc.2016.06.008. arXiv:1601.01167

    2016 arXiv

  24. [32]

    Shtabovenko, Comput

    V. Shtabovenko, Comput. Phys. Commun. 218 (2017) 48–65. doi: 10.1016/j.cpc.2017.04

    2017 doi

  25. [33]

    Alkofer, M

    R. Alkofer, M. Q. Huber, K. Schwenzer, Eur. Phys. J. C62 (2009) 761–781. doi: 10.1140/ epjc/s10052-009-1066-3 . arXiv:0812.4045

    2009 arXiv

  26. [34]

    M. Q. Huber, K. Schwenzer, R. Alkofer, Eur. Phys. J. C68 (2010) 581–600. doi: 10.1140/ epjc/s10052-010-1371-x . arXiv:0904.1873

    2010 arXiv

  27. [35]

    M. Q. Huber, R. Alkofer, S. P. Sorella, Phys. Rev. D81 (2010) 065003. doi:10.1103/PhysRevD. 81.065003. arXiv:0910.5604

    2010 arXiv

  28. [36]

    Fister, R

    L. Fister, R. Alkofer, K. Schwenzer, Phys. Lett. B688 (2010) 237–243. doi: 10.1016/j. physletb.2010.04.001. arXiv:1003.1668

    2010 arXiv

  29. [37]

    Macher, A

    V. Macher, A. Maas, R. Alkofer, Int. J. Mod. Phys. A27 (2012) 1250098. doi: 10.1142/ S0217751X12500984. arXiv:1106.5381

    2012 arXiv

  30. [38]

    Alkofer, R

    N. Alkofer, R. Alkofer, Phys. Lett. B702 (2011) 158–163. doi: 10.1016/j.physletb.2011.06

    2011 doi

  31. [39]

    M. Q. Huber, A. Maas, L. von Smekal, JHEP 1211 (2012) 035. doi:10.1007/JHEP11(2012)035. arXiv:1207.0222

    2012 arXiv

  32. [40]

    M. Q. Huber, L. von Smekal, JHEP 1304 (2013) 149. doi: 10.1007/JHEP04(2013)149. arXiv:1211.6092

    2013 arXiv

  33. [41]

    A. Blum, M. Q. Huber, M. Mitter, L. von Smekal, Phys. Rev. D 89 (2014) 061703(R). doi: 10. 1103/PhysRevD.89.061703. arXiv:1401.0713

    2014 arXiv

  34. [42]

    Braun, L

    J. Braun, L. Fister, J. M. Pawlowski, F. Rennecke, Phys. Rev. D94 (2016) 034016. doi: 10. 1103/PhysRevD.94.034016. arXiv:1412.1045

    2016 arXiv

  35. [43]

    Mitter, J

    M. Mitter, J. M. Pawlowski, N. Strodthoff, Phys. Rev. D91 (2015) 054035. doi: 10.1103/ PhysRevD.91.054035. arXiv:1411.7978

    2015 arXiv

  36. [44]

    M. Q. Huber, D. R. Campagnari, H. Reinhardt, Phys.Rev. D91 (2015) 025014. doi: 10.1103/ PhysRevD.91.025014. arXiv:1410.4766

    2015 arXiv

  37. [45]

    A. K. Cyrol, M. Q. Huber, L. von Smekal, Eur. Phys. J. C75 (2015) 102. doi: 10.1140/epjc/ s10052-015-3312-1 . arXiv:1408.5409

    2015 arXiv

  38. [46]

    M. Q. Huber, L. von Smekal, JHEP 1406 (2014) 015. doi: 10.1007/JHEP06(2014)015. arXiv:1404.3642

    2014 arXiv

  39. [47]

    Rennecke, Phys

    F. Rennecke, Phys. Rev. D92 (2015) 076012. doi: 10.1103/PhysRevD.92.076012. arXiv:1504.03585

    2015 arXiv

  40. [48]

    M. Q. Huber, Phys. Rev. D91 (2015) 085018. doi: 10.1103/PhysRevD.91.085018. arXiv:1502.04057

    2015 arXiv

  41. [49]

    M. Q. Huber, Phys. Rev. D93 (2016) 085033. doi: 10.1103/PhysRevD.93.085033. arXiv:1602.02038. 14

    2016 arXiv

  42. [50]

    A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowski, N. Strodthoff, Phys. Rev. D94 (2016) 054005. doi:10.1103/PhysRevD.94.054005. arXiv:1605.01856

    2016 arXiv

  43. [51]

    M. Q. Huber, Eur. Phys. J. C77 (2017) 733. doi: 10.1140/epjc/s10052-017-5310-y . arXiv:1709.05848

    2017 arXiv

  44. [52]

    A. K. Cyrol, M. Mitter, J. M. Pawlowski, N. Strodthoff, Phys. Rev. D97 (2018) 054006. doi:10.1103/PhysRevD.97.054006. arXiv:1706.06326

    2018 arXiv

  45. [53]

    Corell, A

    L. Corell, A. K. Cyrol, M. Mitter, J. M. Pawlowski, N. Strodthoff, SciPost Phys. 5 (2018) 066. doi:10.21468/SciPostPhys.5.6.066. arXiv:1803.10092

    2018 arXiv

  46. [54]

    M. Q. Huber, EPJ Web Conf. 137 (2017) 07009. doi: 10.1051/epjconf/201713707009. arXiv:1611.06136

    2017

  47. [55]

    A. K. Cyrol, M. Mitter, J. M. Pawlowski, N. Strodthoff, Phys. Rev. D97 (2018) 054015. doi:10.1103/PhysRevD.97.054015. arXiv:1708.03482

    2018 arXiv

  48. [56]

    Contant, M

    R. Contant, M. Q. Huber, Phys. Rev. D96 (2017) 074002. doi: 10.1103/PhysRevD.96.074002. arXiv:1706.00943

    2017 arXiv

  49. [57]

    Leonhardt, M

    M. Leonhardt, M. Pospiech, B. Schallmo, J. Braun, C. Drischler, K. Hebeler, A. Schwenk (2019). arXiv:1907.05814

    2019 arXiv

  50. [58]

    Braun, M

    J. Braun, M. Leonhardt, M. Pospiech (2019). arXiv:1909.06298

    2019 arXiv

  51. [59]

    Hajizadeh, M

    O. Hajizadeh, M. Q. Huber, A. Maas, J. M. Pawlowski (2019). arXiv:1909.12727

    2019 arXiv

  52. [60]

    Contant, M

    R. Contant, M. Q. Huber (2019). arXiv:1909.12796

    2019 arXiv

  53. [61]

    Strodthoff, Phys

    N. Strodthoff, Phys. Rev. D95 (2017) 076002. doi: 10.1103/PhysRevD.95.076002. arXiv:1611.05036

    2017 arXiv

  54. [62]

    J. M. Pawlowski, N. Strodthoff, N. Wink, Phys. Rev. D98 (2018) 074008. doi: 10.1103/ PhysRevD.98.074008. arXiv:1711.07444

    2018 arXiv

  55. [63]

    Braun, M

    J. Braun, M. Leonhardt, M. Pospiech, Phys. Rev. D96 (2017) 076003. doi:10.1103/PhysRevD. 96.076003. arXiv:1705.00074

    2017 arXiv

  56. [64]

    Braun, M

    J. Braun, M. Leonhardt, M. Pospiech, Phys. Rev. D97 (2018) 076010. doi:10.1103/PhysRevD. 97.076010. arXiv:1801.08338

    2018 arXiv

  57. [65]

    J. Eser, F. Divotgey, M. Mitter, D. H. Rischke, Phys. Rev. D98 (2018) 014024. doi: 10.1103/ PhysRevD.98.014024. arXiv:1804.01787

    2018 arXiv

  58. [66]

    Alkofer, A

    R. Alkofer, A. Maas, W. A. Mian, M. Mitter, J. Par´ ıs-L´ opez, J. M. Pawlowski, N. Wink, Phys. Rev. D99 (2019) 054029. doi: 10.1103/PhysRevD.99.054029. arXiv:1810.07955

    2019 arXiv

  59. [67]

    Divotgey, J

    F. Divotgey, J. Eser, M. Mitter, Phys. Rev. D99 (2019) 054023. doi: 10.1103/PhysRevD.99. 054023. arXiv:1901.02472

    2019 arXiv

  60. [68]

    T. Denz, J. M. Pawlowski, M. Reichert, Eur. Phys. J. C78 (2018) 336. doi: 10.1140/epjc/ s10052-018-5806-0 . arXiv:1612.07315

    2018 arXiv

  61. [69]

    Janssen, H

    L. Janssen, H. Gies, Phys. Rev. D86 (2012) 105007. doi: 10.1103/PhysRevD.86.105007. arXiv:1208.3327

    2012 arXiv

  62. [70]

    Brizuela, J

    D. Brizuela, J. M. Martin-Garcia, G. A. Mena Marugan, Gen. Rel. Grav. 41 (2009) 2415–2431. doi:10.1007/s10714-009-0773-2 . arXiv:0807.0824

    2009 arXiv

  63. [71]

    Alkofer, M

    R. Alkofer, M. Q. Huber, K. Schwenzer, Comput. Phys. Commun. 180 (2009) 965–976. doi:10. 1016/j.cpc.2008.12.009. arXiv:0808.2939

    2009 arXiv

  64. [72]

    Wetterich, Phys

    C. Wetterich, Phys. Lett. B301 (1993) 90–94. doi: 10.1016/0370-2693(93)90726-X

    1993 doi

  65. [73]

    R. Kubo, J. Phys. Soc. Jap. 12 (1957) 570–586. doi: 10.1143/JPSJ.12.570

    1957 doi

  66. [74]

    Haas, Spectral functions in finite temperature SU(3) gauge theory and appli- cations to transport phenomena, 2014

    M. Haas, Spectral functions in finite temperature SU(3) gauge theory and appli- cations to transport phenomena, 2014. URL: http://archiv.ub.uni-heidelberg.de/ volltextserver/17875/, Ph.D. Thesis, University of Heidelberg. 15

    2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.