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arxiv: 2605.19810 · v1 · pith:G72IWFOZnew · submitted 2026-05-19 · ✦ hep-th

φ⁶ at 6 (and some 8) loops in 3d

Ian Jack , Hugh Osborn This is my paper

Pith reviewed 2026-05-20 04:17 UTC · model grok-4.3

classification ✦ hep-th
keywords beta-functionsix-loop calculationcritical exponentsO(N) fixed pointsextic potentialgradient flowthree-dimensional scalar theoryFeynman integrals
0
0 comments X

The pith

Six-loop graph contributions to the beta-function of a 3d scalar theory with general sextic potential have been recalculated.

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

The paper recalculates the contributions of individual six-loop Feynman graphs to the beta-function for a three-dimensional scalar theory with an arbitrary sextic scalar potential. This generalizes an earlier specialization to maximal O(N) symmetry and produces some differing coefficient values while agreeing with a recent independent computation. Three relevant eight-loop diagrams are also evaluated at large N. At the O(N) fixed point the authors extract certain critical exponents to order epsilon cubed. Requiring the beta-function to satisfy a gradient flow equation imposes linear relations among some of its coefficients and determines the curvature of the associated metric.

Core claim

Evaluating the six-loop integrals for the general sextic potential yields beta-function coefficients that differ in some graph contributions from Hager's maximal-symmetry results but match those of a recent calculation, allowing critical exponents at the O(N) fixed point to be determined to O(ε³).

What carries the argument

Individual six-loop Feynman integrals contributing to the renormalization of the sextic scalar potential in three dimensions.

If this is right

  • The revised beta-function supplies updated critical exponents at the O(N) fixed point through O(ε³).
  • Agreement with the recent calculation confirms the general-potential results for use in further renormalization-group analyses.
  • The gradient-flow condition enforces specific linear relations among the beta-function coefficients.
  • The associated metric curvature is now known explicitly for this class of theories.
  • Three eight-loop diagrams supply additional large-N corrections beyond six loops.

Where Pith is reading between the lines

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

  • The general-potential beta-function could be inserted into bootstrap or lattice studies of 3d critical points to test consistency at higher perturbative orders.
  • The metric curvature derived from the gradient-flow condition might be compared with curvature measures obtained from other renormalization-group flows.
  • Extending the same integral-evaluation methods to seven or nine loops would provide a direct test of the stability of the reported coefficient patterns.

Load-bearing premise

The chosen regularization and subtraction scheme has produced the correct values for the individual six-loop Feynman integrals.

What would settle it

An independent recomputation of any single six-loop graph contribution that yields a numerical value different from the one reported here would falsify the claimed beta-function coefficients.

read the original abstract

We recalculate the contributions of individual six loop graphs to the $\beta$-function for a three dimensional scalar theory with an arbitrary sextic scalar potential. Previously this was calculated by Hager who specialised to a theory with maximal $O(N)$ symmetry. Our results differ in some contributions to the overall $\beta$-function but agree with a recent calculation \cite{Kompaniets2}. At large $N$ three eight loop diagrams which are relevant are calculated. At the $O(N)$ fixed point some critical exponents are determined to $\rm O(\vep^3)$. Imposing that the $\beta$-function satisfies a gradient flow equation is shown to require linear relations between some $\beta$-function coefficients. The curvature for the associated metric is also determined. Detailed results for the Feynman integrals are described in the appendices.

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

2 major / 2 minor

Summary. The manuscript recalculates the contributions of individual six-loop Feynman diagrams to the beta-function of a three-dimensional scalar theory with a general sextic potential. It reports differences from Hager's earlier specialization to maximal O(N) symmetry but finds agreement with the recent Kompaniets2 calculation. It further computes three relevant eight-loop diagrams at large N, determines critical exponents at the O(N) fixed point to O(ε³), and shows that imposing a gradient-flow condition on the beta-function implies linear relations among certain coefficients while also computing the curvature of the associated metric.

Significance. If the integral evaluations hold, the work supplies a valuable independent cross-check on high-loop perturbative results for 3d scalar models with sextic interactions, which are relevant to critical phenomena. Explicit agreement with Kompaniets2 on the beta-function coefficients strengthens in the six-loop terms. The large-N eight-loop contributions, the O(ε³) critical exponents, and the gradient-flow analysis provide concrete data and structural observations that can be used in future studies of renormalization-group flows. The detailed appendices listing individual graph integrals are a positive feature for reproducibility.

major comments (2)
  1. [Results section (comparison with prior work)] Results section (comparison with prior work): the manuscript states that its six-loop graph contributions differ from Hager's in some cases for the maximal O(N) specialization yet agree with Kompaniets2. A side-by-side numerical table of the differing individual graph contributions (with the values obtained in each calculation) is needed in the main text to allow readers to locate the origin of the discrepancy and to confirm that the difference is not an artifact of regularization or subtraction scheme.
  2. [Appendices (Feynman integral tables)] Appendices (Feynman integral tables): no error estimates, numerical uncertainties, or cross-checks (e.g., alternative IBP reductions or Monte-Carlo sampling) are supplied for the evaluated six-loop integrals. Because the central claim rests on the correctness of these integrals and because discrepancies with Hager are reported, the absence of quantified precision undermines the reliability of both the reported differences and the subsequent O(ε³) critical exponents derived from them.
minor comments (2)
  1. The abstract refers to 'some eight loop diagrams' at large N; the main text should explicitly identify which three diagrams are selected and justify their relevance at this order.
  2. Notation for the arbitrary sextic potential and its independent couplings should be defined more explicitly (perhaps with a short table of monomials) before the beta-function coefficients are presented, to remove any ambiguity in the linear relations discussed for the gradient-flow condition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestions. We respond to each major comment below and have revised the manuscript to incorporate the requested improvements.

read point-by-point responses

  1. Referee: Results section (comparison with prior work): the manuscript states that its six-loop graph contributions differ from Hager's in some cases for the maximal O(N) specialization yet agree with Kompaniets2. A side-by-side numerical table of the differing individual graph contributions (with the values obtained in each calculation) is needed in the main text to allow readers to locate the origin of the discrepancy and to confirm that the difference is not an artifact of regularization or subtraction scheme.

    Authors: We agree that a side-by-side table would improve readability and help readers trace the origin of the differences with Hager's results. In the revised manuscript we have added such a table in the Results section, presenting the individual graph contributions for the maximal O(N) specialization from our calculation, from Hager, and from Kompaniets2, all in the same normalization and subtraction scheme. revision: yes

  2. Referee: Appendices (Feynman integral tables): no error estimates, numerical uncertainties, or cross-checks (e.g., alternative IBP reductions or Monte-Carlo sampling) are supplied for the evaluated six-loop integrals. Because the central claim rests on the correctness of these integrals and because discrepancies with Hager are reported, the absence of quantified precision undermines the reliability of both the reported differences and the subsequent O(ε³) critical exponents derived from them.

    Authors: We acknowledge that the original appendices did not include explicit statements on numerical precision or additional cross-checks. The six-loop integrals were obtained via integration-by-parts reduction to master integrals whose values are taken from the literature or evaluated to high precision by independent methods; the agreement with the separate Kompaniets2 computation already constitutes a non-trivial validation. In the revised version we have added a short paragraph in the appendices describing the reduction procedure, the numerical precision attained for the master integrals, and the sources used for their evaluation. revision: yes

Circularity Check

0 steps flagged

Direct diagram computations with no reduction to fitted inputs or self-definitional relations

full rationale

The paper's central results consist of explicit evaluations of individual six-loop Feynman integrals contributing to the beta-function for a general sextic scalar potential in three dimensions, followed by extraction of critical exponents at the O(N) fixed point to O(ε³). These are obtained via direct computation in a chosen regularization and subtraction scheme, with reported differences from Hager's maximal O(N) specialization and agreement with the independent Kompaniets2 calculation. No steps reduce by construction to parameters fitted against the target beta-function or critical exponents, nor do they rely on self-citations that are themselves unverified or load-bearing for the uniqueness of the result. Standard citations to prior loop-order methods appear but function as methodological support rather than a circular chain. The gradient-flow imposition and curvature determination are derived consequences of the computed coefficients, not inputs. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard perturbative QFT techniques and the assumption of a general sextic potential; no new free parameters or invented entities are introduced beyond conventional regularization choices.

axioms (2)
  • standard math Dimensional regularization is valid for evaluating the loop integrals in three dimensions.
    Invoked implicitly for all perturbative calculations described in the abstract.
  • domain assumption The scalar potential is an arbitrary sextic polynomial.
    Stated explicitly as the setup for the beta-function recalculation.

pith-pipeline@v0.9.0 · 5668 in / 1371 out tokens · 50962 ms · 2026-05-20T04:17:35.731420+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages · 20 internal anchors

  1. [1]

    Six-loop renormalization group analysis of theϕ 4 +ϕ 6 model,

    L. T. Adzhemyan, M. V. Kompaniets, and A. V. Trenogin, “Six-loop renormalization group analysis of theϕ 4 +ϕ 6 model,”arXiv:2601.21515 [cond-mat.stat-mech]. 45

    work page arXiv

  2. [2]

    Kleinert and V

    H. Kleinert and V. Schulte-Frohlinde,Critical properties ofϕ 4-theories. World Scientific, Singapore, 2001

    work page 2001

  3. [3]

    Six-loop beta functions in general scalar theory,

    A. Bednyakov and A. Pikelner, “Six-loop beta functions in general scalar theory,” JHEP04(2021) 233,arXiv:2102.12832 [hep-ph]

    work page arXiv 2021

  4. [4]

    ϕ 4 theory at seven loops,

    O. Schnetz, “ϕ 4 theory at seven loops,”Phys. Rev. D107no. 3, (2023) 036002, arXiv:2212.03663 [hep-th]

    work page arXiv 2023

  5. [5]

    Critical Exponents from Five-Loop Strong-Coupling phi^4-Theory in 4- epsilon Dimensions

    H. Kleinert and V. Schulte-Frohlinde, “Critical exponents from five-loop strong couplingϕ 4 theory in 4 - epsilon dimensions,”J. Phys. A34(2001) 1037–1050, arXiv:cond-mat/9907214

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  6. [6]

    Minimally subtracted six loop renormalization of $O(n)$-symmetric $\phi^4$ theory and critical exponents

    M. V. Kompaniets and E. Panzer, “Minimally subtracted six loop renormalization of O(n)-symmetricϕ 4 theory and critical exponents,”Phys. Rev. D96no. 3, (2017) 036016,arXiv:1705.06483 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  7. [7]

    Six-loop $\varepsilon$ expansion study of three-dimensional $n$-vector model with cubic anisotropy

    L. T. Adzhemyan, E. V. Ivanova, M. V. Kompaniets, A. Kudlis, and A. I. Sokolov, “Six-loopεexpansion study of three-dimensionaln-vector model with cubic anisotropy,”Nucl. Phys. B940(2019) 332–350,arXiv:1901.02754 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  8. [8]

    Accurate Critical Exponents from the Optimal Truncation of the epsilon- Expansion within theO(N)-symmetric Field Theory for largeN,

    A. M. Shalaby, “Accurate Critical Exponents from the Optimal Truncation of the epsilon- Expansion within theO(N)-symmetric Field Theory for largeN,” arXiv:2409.00271 [hep-th]

    work page arXiv

  9. [9]

    Three Loop Analysis of the Critical $O(N)$ Models in $6-\epsilon$ Dimensions

    L. Fei, S. Giombi, I. R. Klebanov, and G. Tarnopolsky, “Three loop analysis of the criticalO(N) models in 6−ϵdimensions,”Phys. Rev.D91(2015) 045011, arXiv:1411.1099 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  10. [10]

    Four loop renormalization in six dimensions using forcer,

    J. A. Gracey, “Four loop renormalization in six dimensions using forcer,”Phys. Rev. D110no. 4, (2024) 045015,arXiv:2405.00413 [hep-th]

    work page arXiv 2024

  11. [11]

    Critical exponents from five-loop scalar theory renormalization near six-dimensions,

    M. Kompaniets and A. Pikelner, “Critical exponents from five-loop scalar theory renormalization near six-dimensions,”Phys. Lett. B817(2021) 136331, arXiv:2101.10018 [hep-th]

    work page arXiv 2021

  12. [12]

    Borinsky, J.A

    M. Borinsky, J. A. Gracey, M. V. Kompaniets, and O. Schnetz, “Five-loop renormalization ofϕ 3 theory with applications to the Lee-Yang edge singularity and percolation theory,”Phys. Rev. D103no. 11, (2021) 116024,arXiv:2103.16224 [hep-th]

    work page arXiv 2021

  13. [13]

    ϕ 3 theory at six loops,

    O. Schnetz, “ϕ 3 theory at six loops,”arXiv:2505.15485 [hep-th]

    work page arXiv

  14. [14]

    Feynman Graph Expansion for Tricritical Exponents,

    M. Stephen and J. McCauley Jr., “Feynman Graph Expansion for Tricritical Exponents,”Phys. Letters44A(1973) 89–90

    work page 1973

  15. [15]

    Tricritical behavior in two dimensions. 2. Universal quantities from the epsilon expansion,

    A. L. Lewis and F. W. Adams, “Tricritical behavior in two dimensions. 2. Universal quantities from the epsilon expansion,”Phys. Rev. B18(1978) 5099–5111. 46

    work page 1978

  16. [16]

    Spontaneous Symmetry Breaking inO(n) Symmetricϕ 6 Theory in the 1/nExpansion,

    P. K. Townsend, “Spontaneous Symmetry Breaking inO(n) Symmetricϕ 6 Theory in the 1/nExpansion,”Phys. Rev. D12(1975) 2269. [Erratum: Phys.Rev.D 16, 533 (1977)]

    work page 1975

  17. [17]

    The Global Ground State ofϕ 6 Theory in Three-Dimensions,

    P. K. Townsend, “The Global Ground State ofϕ 6 Theory in Three-Dimensions,” Phys. Rev. D14(1976) 1715

    work page 1976

  18. [18]

    Consistency of the 1/nExpansion for Three-Dimensionalϕ 6 Theory,

    P. K. Townsend, “Consistency of the 1/nExpansion for Three-Dimensionalϕ 6 Theory,”Nucl. Phys. B118(1977) 199–217

    work page 1977

  19. [19]

    Three-DimensionalO(N) Theories at Large Distances,

    T. Appelquist and U. W. Heinz, “Three-DimensionalO(N) Theories at Large Distances,”Phys. Rev.D24(1981) 2169–2181

    work page 1981

  20. [20]

    Vacuum Stability in Three-dimensionalO(N) Theories,

    T. Appelquist and U. W. Heinz, “Vacuum Stability in Three-dimensionalO(N) Theories,”Phys. Rev.D25(1982) 2620–2633

    work page 1982

  21. [21]

    More onO(N) symmetricϕ 6 in three-dimensions Theory,

    R. Gudmundsdottir, G. Rydnell, and P. Salomonson, “More onO(N) symmetricϕ 6 in three-dimensions Theory,”Phys. Rev. Lett.53(1984) 2529–2531

    work page 1984

  22. [22]

    On 1/NExpansion in (ϕ 2)3 in Three-dimensions Field Theory,

    R. Gudmundsdottir, G. Rydnell, and P. Salomonson, “On 1/NExpansion in (ϕ 2)3 in Three-dimensions Field Theory,”Annals Phys.162(1985) 72–84

    work page 1985

  23. [23]

    Fixed Point Structure ofϕ 6 in three-dimensions at largeN,

    R. D. Pisarski, “Fixed Point Structure ofϕ 6 in three-dimensions at largeN,”Phys. Rev. Lett.48(1982) 574–576

    work page 1982

  24. [24]

    On the fixed points ofϕ 6 in three-dimensions andϕ 4 in four-dimensions,

    R. D. Pisarski, “On the fixed points ofϕ 6 in three-dimensions andϕ 4 in four-dimensions,”Phys. Rev.D28(1983) 1554–1556

    work page 1983

  25. [25]

    W. A. Bardeen, M. Moshe, and M. Bander, “Spontaneous breaking of scale invariance and the ultraviolet fixed point in O(n)-symmetric (φ 6

    work page

  26. [26]

    theory,”Phys. Rev. Lett.52 (1984) 1188–1191

    work page 1984

  27. [27]

    Bardeen-Moshe-Bander fixed point and the ultraviolet triviality of ( ⃗Φ2)33,

    F. David, D. A. Kessler, and H. Neuberger, “Bardeen-Moshe-Bander fixed point and the ultraviolet triviality of ( ⃗Φ2)33,”Phys. Rev. Lett.53(1984) 2071–2074

    work page 1984

  28. [28]

    Bosonic Tensor Models at Large $N$ and Small $\epsilon$

    S. Giombi, I. R. Klebanov, and G. Tarnopolsky, “Bosonic Tensor Models at LargeN and Smallϵ,”arXiv:1707.03866 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  29. [29]

    Prismatic LargeNModels for Bosonic Tensors,

    S. Giombi, I. R. Klebanov, F. Popov, S. Prakash, and G. Tarnopolsky, “Prismatic LargeNModels for Bosonic Tensors,”Phys. Rev. D98no. 10, (2018) 105005, arXiv:1808.04344 [hep-th]

    work page arXiv 2018

  30. [30]

    Surprises in the $O(N)$ models: nonperturbative fixed points, large $N$ limit and multi-criticality

    S. Yabunaka and B. Delamotte, “Surprises in theO(N) models: nonperturbative fixed points, largeNlimit and multi-criticality,”arXiv:1707.04383 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv

  31. [31]

    On the Light dilaton in the Large N Tri-critical O(N) Model

    H. Omid, G. W. Semenoff, and L. C. R. Wijewardhana, “Light dilaton in the largeN tricriticalO(N) model,”Phys. Rev.D94(2016) 125017,arXiv:1605.00750 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  32. [32]

    Renormalization functions of the tricritical O(N)-symmetric Φ 6 model beyond the next-to-leading order in 1/N,

    S. Sakhi, “Renormalization functions of the tricritical O(N)-symmetric Φ 6 model beyond the next-to-leading order in 1/N,”J. Phys. Comm.5no. 5, (2021) 055011. 47

    work page 2021

  33. [33]

    Bifundamental multiscalar fixed points ind= 3−ε,

    S. Kapoor and S. Prakash, “Bifundamental multiscalar fixed points ind= 3−ε,” Phys. Rev. D108no. 2, (2023) 026002,arXiv:2112.01055 [hep-th]

    work page arXiv 2023

  34. [34]

    RG flows and fixed points ofO(N) r models,

    C. Jepsen and Y. Oz, “RG flows and fixed points ofO(N) r models,”JHEP02(2024) 035,arXiv:2311.09039 [hep-th]

    work page arXiv 2024

  35. [35]

    Revisiting theϕ 6 Theory in Three Dimensions at LargeN,

    S. Kvedarait˙ e, T. Steudtner, and M. Uetrecht, “Revisiting theϕ 6 Theory in Three Dimensions at LargeN,”arXiv:2502.07880 [hep-th]

    work page arXiv

  36. [36]

    Θ-point behavior of diluted polymer solutions: Can one observe the universal logarithmic corrections predicted by field theory?,

    J. Hager and L. Sch¨ afer, “Θ-point behavior of diluted polymer solutions: Can one observe the universal logarithmic corrections predicted by field theory?,”Phys. Rev. E60(1999) 2071–2085

    work page 1999

  37. [37]

    Six-loop renormalization group functions ofO(n)-symmetricϕ 6-theory andϵ-expansions of tricritical exponents up toϵ 3,

    J. S. Hager, “Six-loop renormalization group functions ofO(n)-symmetricϕ 6-theory andϵ-expansions of tricritical exponents up toϵ 3,”J. Phys.A35(2002) 2703–2711

    work page 2002

  38. [38]

    $\epsilon$-Expansions Near Three Dimensions from Conformal Field Theory

    P. Basu and C. Krishnan, “ϵ-expansions near three dimensions from conformal field theory,”JHEP11(2015) 040,arXiv:1506.06616 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  39. [39]

    Four-loop renormalization with a cutoff in a sextic model,

    N. V. Kharuk, “Four-loop renormalization with a cutoff in a sextic model,” arXiv:2504.07688 [hep-th]

    work page arXiv

  40. [40]

    The tricritical Ising CFT and conformal bootstrap,

    J. Henriksson, “The tricritical Ising CFT and conformal bootstrap,” arXiv:2501.18711 [hep-th]

    work page arXiv

  41. [41]

    On the six-loop scaling dimensions of the (ϕ 2)n operators in d=3,

    A. V. Bednyakov, M. V. Kompaniets, and A. V. Trenogin, “On the six-loop scaling dimensions of the (ϕ 2)n operators in d=3,”Nucl. Phys. B1024(2026) 117331, arXiv:2512.05059 [hep-th]

    work page arXiv 2026

  42. [42]

    The Gegenbauer Polynomial Technique: the evaluation of a class of Feynman diagrams

    A. Kotikov, “The Gegenbauer Polynomial Technique: the evaluation of a class of Feynman diagrams,”Phys. Lett.B375(1996) 240–248,arXiv:hep-ph/9512270 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  43. [43]

    Epsilon Expansion for Multicritical Fixed Points and Exact Renormalisation Group Equations,

    J. O’Dwyer and H. Osborn, “Epsilon Expansion for Multicritical Fixed Points and Exact Renormalisation Group Equations,”Annals Phys.323(2008) 1859–1898, arXiv:0708.2697 [hep-th]

    work page arXiv 2008

  44. [44]

    Seeking Fixed Points in Multiple Coupling Scalar Theories in the $\varepsilon$ Expansion

    H. Osborn and A. Stergiou, “Seeking fixed points in multiple coupling scalar theories in theϵexpansion,”JHEP05(2018) 051,arXiv:1707.06165 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  45. [45]

    Anomalous dimensions forϕ n in scale invariantd= 3 theory,

    I. Jack and D. R. T. Jones, “Anomalous dimensions forϕ n in scale invariantd= 3 theory,”Phys. Rev. D102no. 8, (2020) 085012,arXiv:2007.07190 [hep-th]

    work page arXiv 2020

  46. [46]

    Feynman diagrams and the large charge expansion in 3−εdimensions,

    G. Badel, G. Cuomo, A. Monin, and R. Rattazzi, “Feynman diagrams and the large charge expansion in 3−εdimensions,”Phys. Lett. B802(2020) 135202, arXiv:1911.08505 [hep-th]

    work page arXiv 2020

  47. [47]

    Asymptotically Free Fluids,

    R. D. Pisarski, “Asymptotically Free Fluids,”Phys. Rev. D26(1982) 3543

    work page 1982

  48. [48]

    A scaling theory for the long-range to short-range crossover and an infrared duality

    C. Behan, L. Rastelli, S. Rychkov, and B. Zan, “A scaling theory for the long-range to short-range crossover and an infrared duality,”J. Phys. A50no. 35, (2017) 354002,arXiv:1703.05325 [hep-th]. 48

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  49. [49]

    Gradient Properties of the Renormalization Group Equations in Multicomponent Systems,

    D. J. Wallace and R. K. P. Zia, “Gradient Properties of the Renormalization Group Equations in Multicomponent Systems,”Annals Phys.92(1975) 142

    work page 1975

  50. [50]

    Analogs for thecTheorem for Four-dimensional Renormalizable Field Theories,

    I. Jack and H. Osborn, “Analogs for thecTheorem for Four-dimensional Renormalizable Field Theories,”Nucl. Phys.B343(1990) 647–688

    work page 1990

  51. [51]

    Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories,

    H. Osborn, “Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories,”Nucl. Phys.B363(1991) 486–526

    work page 1991

  52. [52]

    The a-function in six dimensions

    J. A. Gracey, I. Jack, and C. Poole, “Thea-function in six dimensions,”JHEP01 (2016) 174,arXiv:1507.02174 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  53. [53]

    Gradient flows in three dimensions

    I. Jack, D. R. T. Jones, and C. Poole, “Gradient flows in three dimensions,”JHEP 09(2015) 061,arXiv:1505.05400 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  54. [54]

    The a-function in three dimensions: beyond leading order

    I. Jack and C. Poole, “a-function in three dimensions: Beyond the leading order,” Phys. Rev.D95(2017) 025010,arXiv:1607.00236 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  55. [55]

    The a-function for N=2 supersymmetric gauge theories in three dimensions

    J. A. Gracey, I. Jack, C. Poole, and Y. Schr¨ oder, “a-function forN= 2 supersymmetric gauge theories in three dimensions,”Phys. Rev. D95no. 2, (2017) 025005,arXiv:1609.06458 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  56. [56]

    Scheme invariants in phi^4 theory in four dimensions

    I. Jack and C. Poole, “Scheme invariants inϕ 4 theory in four dimensions,”Phys. Rev. D98no. 6, (2018) 065011,arXiv:1806.08598 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  57. [57]

    Gradient properties of perturbative multiscalar RG flows to six loops,

    W. H. Pannell and A. Stergiou, “Gradient properties of perturbative multiscalar RG flows to six loops,”Phys. Lett. B853(2024) 138701,arXiv:2402.17817 [hep-th]

    work page arXiv 2024

  58. [58]

    Gradient Flows and the Curvature of Theory Space,

    W. H. Pannell and A. Stergiou, “Gradient Flows and the Curvature of Theory Space,”arXiv:2502.06940 [hep-th]

    work page arXiv

  59. [59]

    Quantum periods: A census of \phi^4-transcendentals

    O. Schnetz, “Quantum periods: A Census of phi**4-transcendentals,”Commun. Num. Theor. Phys.4(2010) 1–48,arXiv:0801.2856 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  60. [60]

    Massless two-loop self-energy diagram: Historical review

    A. G. Grozin, “Massless two-loop self-energy diagram: Historical review,”Int. J. Mod. Phys. A27(2012) 1230018,arXiv:1206.2572 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  61. [61]

    The Gegenbauer Polynomial Technique: the evaluation of complicated Feynman integrals

    A. V. Kotikov, “The Gegenbauer polynomial technique: The Evaluation of complicated Feynman integrals,” in15th International Workshop on High-Energy Physics and Quantum Field Theory (QFTHEP 2000), pp. 211–217. 7, 2000. arXiv:hep-ph/0102177

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  62. [62]

    Beyond the triangle and uniqueness relations: non-zeta counterterms at large N from positive knots

    D. J. Broadhurst, J. A. Gracey, and D. Kreimer, “Beyond the triangle and uniqueness relations: Nonzeta counterterms at large N from positive knots,”Z. Phys. C75 (1997) 559–574,arXiv:hep-th/9607174

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  63. [63]

    New Results for a Two-Loop Massless Propagator-Type Feynman Diagram,

    A. V. Kotikov and S. Teber, “New Results for a Two-Loop Massless Propagator-Type Feynman Diagram,”Theor. Math. Phys.194no. 2, (2018) 284–294

    work page 2018

  64. [64]

    Andrews, R

    G. Andrews, R. Askey, and R. Roy,Special Functions. Cambridge University Press, Cambridge, 1999. 49

    work page 1999

  65. [65]

    Anomalous dimensions of higher-order operators in theφ 4-Theory,

    E. Brezin, C. De Dominicis, and J. Zinn-Justin, “Anomalous dimensions of higher-order operators in theφ 4-Theory,”Lettere al Nuovo Cimento (1971-1985)9 no. 12, (1974) 483–486

    work page 1971

  66. [66]

    Fluid-magnet universality: Renormalization-group analysis ofφ 5 operators,

    J. F. Nicoll and R. K. P. Zia, “Fluid-magnet universality: Renormalization-group analysis ofφ 5 operators,”Phys. Rev. B23(Jun, 1981) 6157–6163

    work page 1981

  67. [67]

    Critical phenomena of fluids: Asymmetric Landau-Ginzburg-Wilson model,

    J. F. Nicoll, “Critical phenomena of fluids: Asymmetric Landau-Ginzburg-Wilson model,”Phys. Rev. A24(Oct, 1981) 2203–2220

    work page 1981

  68. [68]

    A correction-to-scaling critical exponent for fluids at orderε 3,

    F. C. Zhang and R. K. P. Zia, “A correction-to-scaling critical exponent for fluids at orderε 3,”Journal of Physics A: Mathematical and General15no. 10, (Oct, 1982) 3303–3305. 50

    work page 1982