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The renormalisation-group flow of scalar-fermion theories is a gradient flow through four loops — once the beta shift is subtracted from the beta function.

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2026-08-04 06:51 UTC pith:B57CC2KG

load-bearing objection Careful and honest paper, but its four-loop gradiency claim is conditional on an uncomputed beta shift; the lower-order results and the eF matching are genuinely solid. the 3 major comments →

arxiv 2511.01971 v2 pith:B57CC2KG submitted 2025-11-03 hep-th cond-mat.stat-mechhep-ph

Gradient RG Flow in Scalar-Fermion QFTs

William H. Pannell , William Patrick Ronayne , Andreas Stergiou This is my paper
classification hep-th cond-mat.stat-mechhep-ph MSC 81T1781T40 PACS 11.10.Hi
keywords renormalisation groupgradient flowbeta shiftscalar-fermion theoriesepsilon expansionconformal field theory fixed pointsfour-loop beta functionsscheme independence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks whether the renormalisation-group flow of theories with scalars and fermions can be written as the gradient of a single function on coupling space, which would guarantee a quantity that decreases along every flow. It shows that the answer is yes through four loops in d=4 and d=4−ϵ, provided one uses the beta shift: the flow vector is B = β − Sg, not the standard beta function β. Satisfying the gradient equations turns out to be equivalent to 1016 scheme-independent constraints on the beta-function coefficients, and every constraint that can be checked with known results is satisfied. The paper also finds that among fixed points of these theories, those with nonzero beta shift — genuine CFTs with nonvanishing beta functions, whose couplings rotate along the flow — are common and become the majority as the number of fields grows. If right, this means the gradient structure of the RG survives the addition of fermions, and that a large class of conformal field theories would be missed by searches that set β=0.

Core claim

The paper claims that for the most general scalar-fermion theory in d=4 and d=4−ϵ, the RG flow generated by B = β − (Sg) is a Riemannian gradient through 3/4 loop order. This holds if and only if a set of 955 (d=4) plus 61 (d=4−ϵ) scheme-independent constraints on the beta-function and beta-shift coefficients are satisfied; all constraints that can be verified with existing four-loop data are satisfied. The standard beta function β alone fails once the shift can contribute, starting at 2/3 loop order. At fixed points of B with β ≠ 0, the couplings trace limit cycles under flavour rotations, and these are genuine CFTs. The paper exhibits two analytic examples and shows numerically that such f

What carries the argument

The central object is the beta shift S, a contribution to the trace of the energy-momentum tensor from operator mixing with non-conserved spin-one currents. It corrects the beta function to B = β − (Sg). The gradient flow equation ∂A/∂g = G B, with A the would-be monotone and G a Riemannian metric on coupling space, is expanded order by order using tensor structures built from quartic and Yukawa couplings. All vacuum graphs through O(λ^5, …, y^10) are enumerated, and the resulting linear system is reduced to the constraint equations. A primitive-diagram family of constraints fixes the fermion-metric normalization g0 = 24, which also makes A match the sphere free-energy quantity at low orders

Load-bearing premise

The whole gradient-flow conclusion rests on the as-yet-uncomputed four-loop coefficients of the beta shift satisfying the 64 constraints listed in Appendix B and the two ε-dependent constraints; if even one fails, the B function is not gradient at 3/4 loops.

What would settle it

Compute the four-loop beta-shift coefficients S_ij and P_ab in a general scalar-fermion theory and check the 64 equalities in Appendix B and the two in equation (3.38). A single violation would show B = β − Sg is not a Riemannian gradient at 3/4 loops; full agreement would confirm the paper's central claim. The list includes exact predictions such as S4_2 = 7/24 + π^4/120.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Fixed points of the RG are zeros of B = β − Sg, not of β; at such points the couplings undergo flavour rotations, and the physically relevant anomalous-dimension matrix is γ + S.
  • Gradient flow at 3/4 loops is equivalent to 1016 scheme-independent constraints; the constraints on β alone are violated in the minimal-subtraction scheme, and the beta shift repairs them.
  • In the ε expansion, fixed points with nonzero beta shift already exist for two scalars and one Weyl fermion, and their proportion grows sharply with field content (to roughly two-thirds for N_s=4, N_f=2).
  • The A-function constructed as the gradient potential can be matched at low order to the sphere free energy with a single field-independent coefficient choice, extending the eF-conjecture toward a full gradient-flow statement.

Where Pith is reading between the lines

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

  • Inference: If the four-loop beta shift is ever computed, the 64 constraints in Appendix B plus the two ε-dependent constraints provide an unambiguous binary test — one violated equality would show the gradient picture breaks at this order, while full agreement would extend it to the deepest order claimed.
  • Inference: Systematic searches for infrared fixed points in the ε expansion that solve β=0 will miss most of the conformal theories once N_s ≥ 3 and N_f ≥ 2; such searches should instead solve B=0 at order ε^3.
  • Inference: The constraint-counting method should transfer to general gauge-Yukawa theories, where a beta shift also exists; verifying the analogous constraints there would broaden the gradient property beyond the scalar-fermion class.
  • Inference: The matching of A to the sphere free energy with a single coefficient suggests eF may admit a gradient-flow extension; a concrete test would be computing both quantities at a third, less symmetric fixed point.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the gradient property of the renormalisation-group vector field in general scalar-fermion theories, with the beta shift B = β − (S g) playing a central role. In d = 4 and d = 4 − ε, the authors enumerate all relevant diagrams through three/four loops, solve the gradient flow equation ∂A = G B order by order, and derive a large set of algebraic constraints on the beta-function coefficients and on the beta-shift coefficients. The lower-order constraints are checked against known MS results and against the known three-loop beta shift; the four-loop beta shift is not computed. The paper also matches the constructed A-function to the sphere free energy eF for the Gross–Neveu–Yukawa and Nambu–Jona-Lasinio–Yukawa CFTs, fixing one coefficient a3_4 = 8. Finally, a numerical search over one-loop fixed points for small numbers of scalars and fermions shows that a substantial and increasing fraction have non-zero beta shift.

Significance. If the four-loop beta-shift constraints are eventually verified, this would be a significant extension of the gradient-flow programme to scalar-fermion theories, giving further evidence that the proper RG vector field is B and that the beta shift is essential for Riemannian gradient flow, and providing an explicit construction of A up to three/four loops. The paper is transparent about the conditional nature of the four-loop check and about the distinction between one-loop roots and B=0 fixed points. Strengths of the presentation include the explicit enumeration of diagrams, the scheme-invariance analysis, the detailed list of constraints in Appendix B, the matching of A to eF in two models, and the ancillary Mathematica file that allows independent inspection. The numerical observation that beta-shift fixed points become increasingly common is intriguing, but it is not yet a statement about solutions of B=0 to the order needed.

major comments (3)
  1. [Sec. 3.1/3.2, Appendix B] The central claim that the B-function is gradient through three/four loops is conditional on the four-loop beta shift satisfying the 64 constraints (B.3)–(B.66) and the two ε-dependent constraints (3.38). The manuscript states at §3.1 and §3.2 that 'a true check would require the calculation of these coefficients at four loops', and the fixed coefficient list in (3.32) is explicitly contingent on that calculation. Since S4 and P4 are never computed, the result is a set of necessary and sufficient constraints on an uncalculated object, not a verification of gradiency. The abstract and introduction should be reworded so that the four-loop conclusion is presented as a consistency condition with strong lower-loop support, or the four-loop shift must be computed and checked.
  2. [Sec. 4, Table 3] The numerical census counts roots of the one-loop beta functions and labels any such root with non-zero S as a beta-shift fixed point. But a CFT fixed point requires B=0 to the appropriate order in ε; at order ε^3, where S first contributes, a one-loop root need not extend to a solution of B=0. The paper verifies B=0 explicitly only for the two analytic examples in Section 2, not for the hundreds of numerically found candidates. Therefore the conclusion that 'the space of CFTs is dominated by those with non-zero beta shift' is not established by this census. The candidates should be re-checked against the B=0 equations, or the claim should be restricted to one-loop beta-function roots with non-zero shift.
  3. [Sec. 3, diagram enumeration] The treatment of odd-trace diagrams is an additional assumption: the text says they are included but their coefficients are 'set to zero as necessary'. If such coefficients are not independently computed, then the statement that the analysis covers general scalar-fermion beta functions is not fully supported. The assumption should be stated as a restriction on the class of theories (e.g., those with a four-dimensional Weyl uplift), or the odd-trace coefficients should be computed and included in the constraints.
minor comments (5)
  1. [Sec. 3.4] The match of A to eF in the Gross–Neveu–Yukawa and NJL–Yukawa models is a significant check, but it uses one fitted coefficient a3_4 = 8. The paper correctly notes that this is a single coefficient independent of N; the text could state more explicitly that this is a consistency check of the eF-extension conjecture, not a proof.
  2. [Sec. 2.1] In the two analytic examples, the text says that the beta-function equations 'cannot be continued beyond two loops' and that B removes the obstruction. It would be helpful to display the non-zero constant term in the beta function of y2 at order ε^3, so the reader can see the cancellation explicitly without reconstructing it from the formulas.
  3. [Sec. 3.3] The scheme-invariance argument is clear, but the notation in equations (3.39)–(3.41), especially the sum over sub-tensor structures, is compressed. A one-sentence explanation of how T^n_m split under the action on β would improve readability.
  4. [Appendix B] The list of constraints (B.3)–(B.66) would be more usable if the diagrammatic definitions of the individual S4_k and P4_k coefficients were available in the same document, rather than only via the ancillary file. At minimum, a table matching the coefficient labels to the drawing numbers would help.
  5. [General] There are minor typographical issues, e.g., 'N_S' vs 'Ns' in Section 2 and the label '0/1loop' in Table 1. These do not affect the results.

Circularity Check

0 steps flagged

No significant circularity; the four-loop gradiency claim is unverified but not circular.

full rationale

The central derivation is self-contained: gradient-flow constraints are obtained by inserting the diagrammatic expansions (3.1) into (1.5), eliminating A and G coefficients order by order, and comparing with beta-function/beta-shift coefficients quoted from independent computations. The beta shift is not defined by the constraints; it is that of (2.2) from [10,14,36], and the three-loop values are checked against the constraints rather than fitted. The eF matching fixes one A coefficient, a3_4 = 8, from the Gross–Neveu–Yukawa result and then verifies the same value at the Nambu–Jona-Lasinio–Yukawa fixed point, so the second comparison is an independent check, not a fit recycled as a prediction. The four-loop beta-shift constraints (B.3)–(B.66) and (3.38) are not verified; the paper explicitly says 'a true check would require the calculation of these coefficients at four loops' (§3.2). That is an unverified empirical/computational gap, not circularity: no constraint is defined to be satisfied by construction, and no fitted parameter is renamed as a prediction. Self-citations such as [5,6,10] are used for context and for the previously established beta-shift formalism; they do not supply the load-bearing equation that is then called a derivation in this paper. The numerical census (§4) counts one-loop roots with non-zero shift and is labeled as such; it does not feed back into the gradient-flow constraints. Consequently no circular step can be exhibited.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

Selections are dominated by input data from external four-loop computations and unverified predictions for the four-loop beta shift. The central gradient claim does not require new dynamics, but does require the four-loop beta shift to satisfy constraints that are currently beyond direct calculation. The A/eF matching introduces one fitted coefficient but this does not feed back into the beta constraints.

free parameters (1)
  • a3_4 = 8
    Coefficient in the A-function fixed by matching A to the eF free energy at the Gross-Neveu-Yukawa fixed point (§3.4). It is then checked against the Nambu-Jona-Lasinio-Yukawa fixed point, so it is one fitted parameter with two constraints.
axioms (5)
  • domain assumption The four-loop beta functions of [35], with odd-trace coefficients handled as described, are correct inputs.
    All 1016 constraint equations are built from these coefficients; any error or ad hoc zeroing of odd-trace coefficients changes the derived S/P constraints.
  • ad hoc to paper Odd-trace diagrams in the beta function can be included with coefficients set to zero as necessary to maintain a 4d uplift.
    The paper does not specify exactly which coefficients are zeroed or prove consistency; this affects the four-loop beta function and hence the constraints (§3, before Eq. (3.3)).
  • ad hoc to paper The four-loop beta shift satisfies the 64 constraints in Appendix B and the two epsilon-dependent constraints (3.38).
    The central four-loop gradiency claim is conditional on these unverified constraints; the authors state a true check would require calculation of the shift at four loops.
  • domain assumption The eF values from [26,27] are valid benchmarks for matching A.
    Used to fix a3_4 = 8; if the eF conjecture failed, the matching would be coincidence. This does not affect the beta-function constraints.
  • domain assumption Perturbative MS scheme and epsilon expansion are valid; unitarity and O(Ns) x U(Nf) symmetry with conserved J^mu_A hold.
    Framework of the entire calculation, including the existence of the beta shift and the meaning of B = β − (Sg).

pith-pipeline@v1.3.0-alltime-deepseek · 26422 in / 16375 out tokens · 150617 ms · 2026-08-04T06:51:21.182741+00:00 · methodology

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read the original abstract

The gradient property of the renormalisation group (RG) is examined to four-loop order in scalar-fermion systems in $d=4$ and $d=4-\varepsilon$ dimensions. The crucial role played by the beta shift, which is a modification of the standard dim-reg beta function, is elucidated, and specific conditions that it needs to satisfy for the RG flow to be gradient are derived. Over a thousand gradient-flow conditions are found, all of which are scheme-independent and satisfied whenever the full set of results needed to check them is available. It is shown, in the framework of the $\varepsilon=4-d$ expansion, that the space of conformal field theories (CFTs) is dominated by those with non-zero beta shift as the number of fields grows. Physical properties of CFTs obtained as solutions where the beta functions are not zero in the $\varepsilon$ expansion are discussed.

Figures

Figures reproduced from arXiv: 2511.01971 by Andreas Stergiou, William H. Pannell, William Patrick Ronayne.

Figure 1
Figure 1. Figure 1: Results of numerical search for Ns = 2, Nf = 1. We find 13 fixed points, with 1 out of 13, or ∼ 8%, having non-zero beta shift. −1 −0.5 0 0.5 1 −0.2 −0.1 0 0.1 0.2 0.3 R T ′ Scalar-fermion fixed points for Ns = 2 and Nf = 2 [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Results of numerical search for Ns = 2, Nf = 2. We find 31 fixed points, with 4 out of the 31, or ∼ 13%, having non-zero beta shift. 26 [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Results of numerical search for Ns = 2, NfD = 1. We find 11 fixed points, with 3 out of the 11, or ∼ 27% having non-zero beta shift. −1 −0.5 0 0.5 1 −0.2 0 0.2 0.4 R T ′ Scalar-fermion fixed points for Ns = 3 and Nf = 1 [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Results of numerical search for Ns = 3, Nf = 1. We find 44 fixed points, with 16 out of the 44, or ∼ 36% having non-zero beta shift. 27 [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Results of numerical search for Ns = 3, Nf = 2. We find 403 fixed points, with 276 out of the 403, or ∼ 68% having non-zero beta shift. −1 −0.5 0 0.5 1 −0.2 0 0.2 0.4 0.6 R T ′ Scalar-fermion fixed points for Ns = 4 and Nf = 1 [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results of numerical search for Ns = 4, Nf = 1. We find 75 fixed points, with 38 out of the 75, or ∼ 51% having non-zero beta shift. 28 [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Results of numerical search for Ns = 4, Nf = 2. We find 294 fixed points, with 190 out of the 294, or ∼ 65% having non-zero beta shift. −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.5 0 0.5 R T ′ Scalar-fermion fixed points for Ns = 2 and NfD = 2 [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Results of numerical search for Ns = 2, NfD = 2. We find 32 fixed points, with 4 out of the 32, or ∼ 13% having non-zero beta shift. 5. Conclusion The results of this work provide further evidence for the gradient structure of the renormalisation group. The non-trivial nature of the gradient conditions we get, which are satisfied whenever 29 [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

46 extracted references · 30 linked inside Pith · cited by 3 Pith papers

  1. [1]

    Critical exponents in 3.99 dimensions

    K. G. Wilson & M. E. Fisher,“Critical exponents in 3.99 dimensions”, Phys. Rev. Lett.28, 240 (1972)

    1972

  2. [2]

    Gradient Flow and the Renormalization Group

    D. J. Wallace & R. K. P. Zia,“Gradient Flow and the Renormalization Group”, Phys. Lett. A 48, 325 (1974)

    1974

  3. [3]

    Gradient Properties of the Renormalization Group Equations in Multicomponent Systems

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

    1975

  4. [4]

    Scheme invariants in ϕ4 theory in four dimensions

    I. Jack & C. Poole,“Scheme invariants in ϕ4 theory in four dimensions”, Phys. Rev. D98, 065011 (2018),arXiv:1806.08598 [hep-th]

    Pith/arXiv arXiv 2018

  5. [5]

    Gradient properties of perturbative multiscalar RG flows to six loops

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

    Pith/arXiv arXiv 2024

  6. [6]

    Gradient flows and the curvature of theory space

    W. H. Pannell & A. Stergiou,“Gradient flows and the curvature of theory space”, JHEP2509, 117 (2025),arXiv:2502.06940 [hep-th]

    Pith/arXiv arXiv 2025

  7. [7]

    The a-function in six dimensions

    J. A. Gracey, I. Jack & C. Poole,“The a-function in six dimensions”, JHEP1601, 174 (2016), arXiv:1507.02174 [hep-th]

    Pith/arXiv arXiv 2016

  8. [8]

    Gradient properties of ϕ3 in d = 6 −ε

    L. Benfatto & O. Zanusso,“Gradient properties of ϕ3 in d = 6 −ε ”, arXiv:2507.20761 [hep-th]

    Pith/arXiv arXiv

  9. [9]

    Analogs for thec Theorem for Four-dimensional Renormalizable Field Theories

    I. Jack & H. Osborn,“Analogs for thec Theorem for Four-dimensional Renormalizable Field Theories”, Nucl. Phys. B343, 647 (1990)

    1990

  10. [10]

    Limit Cycles and Conformal Invariance

    J.-F. Fortin, B. Grinstein & A. Stergiou,“Limit Cycles and Conformal Invariance”, JHEP 1301, 184 (2013),arXiv:1208.3674 [hep-th]

    Pith/arXiv arXiv 2013

  11. [11]

    Scale without Conformal Invariance: An Example

    J.-F. Fortin, B. Grinstein & A. Stergiou,“Scale without Conformal Invariance: An Example”, Phys. Lett. B704, 74 (2011),arXiv:1106.2540 [hep-th]

    Pith/arXiv arXiv 2011

  12. [12]

    Scale without Conformal Invariance: Theoretical Foundations

    J.-F. Fortin, B. Grinstein & A. Stergiou,“Scale without Conformal Invariance: Theoretical Foundations”, JHEP1207, 025 (2012),arXiv:1107.3840 [hep-th]. 36

    Pith/arXiv arXiv 2012

  13. [13]

    Constraints on RG Flow for Four Dimensional Quantum Field Theories

    I. Jack & H. Osborn,“Constraints on RG Flow for Four Dimensional Quantum Field Theories”, Nucl. Phys. B883, 425 (2014),arXiv:1312.0428 [hep-th]

    Pith/arXiv arXiv 2014

  14. [14]

    General gauge-Yukawa-quarticβ-functions at 4-3-2-loop order

    J. Davies, F. Herren & A. E. Thomsen,“General gauge-Yukawa-quarticβ-functions at 4-3-2-loop order”, JHEP2201, 051 (2022),arXiv:2110.05496 [hep-ph]

    Pith/arXiv arXiv 2022

  15. [15]

    Limit Cycles in Four Dimensions

    J.-F. Fortin, B. Grinstein & A. Stergiou,“Limit Cycles in Four Dimensions”, JHEP1212, 112 (2012),arXiv:1206.2921 [hep-th]

    Pith/arXiv arXiv 2012

  16. [16]

    Scale and Conformal Invariance in Quantum Field Theory

    J. Polchinski,“Scale and Conformal Invariance in Quantum Field Theory”, Nucl. Phys. B303, 226 (1988)

    1988

  17. [17]

    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, 486 (1991)

    1991

  18. [18]

    Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory

    A. B. Zamolodchikov,“Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory”, JETP Lett.43, 730 (1986)

    1986

  19. [19]

    Seeing a c-theorem with holography

    R. C. Myers & A. Sinha,“Seeing a c-theorem with holography”, Phys. Rev. D82, 046006 (2010),arXiv:1006.1263 [hep-th]

    Pith/arXiv arXiv 2010

  20. [20]

    Towards the F-Theorem: N=2 Field Theories on the Three-Sphere

    D. L. Jafferis, I. R. Klebanov, S. S. Pufu & B. R. Safdi,“Towards the F-Theorem: N=2 Field Theories on the Three-Sphere”, JHEP1106, 102 (2011),arXiv:1103.1181 [hep-th]

    Pith/arXiv arXiv 2011

  21. [21]

    F-Theorem without Supersymmetry

    I. R. Klebanov, S. S. Pufu & B. R. Safdi,“F-Theorem without Supersymmetry”, JHEP1110, 038 (2011),arXiv:1105.4598 [hep-th]

    Pith/arXiv arXiv 2011

  22. [22]

    On the RG running of the entanglement entropy of a circle

    H. Casini & M. Huerta,“On the RG running of the entanglement entropy of a circle”, Phys. Rev. D85, 125016 (2012),arXiv:1202.5650 [hep-th]

    Pith/arXiv arXiv 2012

  23. [23]

    On Renormalization Group Flows in Four Dimensions

    Z. Komargodski & A. Schwimmer,“On Renormalization Group Flows in Four Dimensions”, JHEP1112, 099 (2011),arXiv:1107.3987 [hep-th]

    Pith/arXiv arXiv 2011

  24. [24]

    Averaged null energy and the renormalization group

    T. Hartman & G. Mathys,“Averaged null energy and the renormalization group”, JHEP2312, 139 (2023),arXiv:2309.14409 [hep-th]

    Pith/arXiv arXiv 2023

  25. [25]

    Scale without Conformal Invariance at Three Loops

    J.-F. Fortin, B. Grinstein & A. Stergiou,“Scale without Conformal Invariance at Three Loops”, JHEP1208, 085 (2012),arXiv:1202.4757 [hep-th]

    Pith/arXiv arXiv 2012

  26. [26]

    Generalized F-Theorem and the ϵ Expansion

    L. Fei, S. Giombi, I. R. Klebanov & G. Tarnopolsky,“Generalized F-Theorem and the ϵ Expansion”, JHEP1512, 155 (2015),arXiv:1507.01960 [hep-th]

    Pith/arXiv arXiv 2015

  27. [27]

    Yukawa CFTs and Emergent Supersym- metry

    L. Fei, S. Giombi, I. R. Klebanov & G. Tarnopolsky,“Yukawa CFTs and Emergent Supersym- metry”, PTEP2016, 12C105 (2016),arXiv:1607.05316 [hep-th]

    Pith/arXiv arXiv 2016

  28. [28]

    Two Loop Background Field Calculations for Arbitrary Background Fields

    I. Jack & H. Osborn,“Two Loop Background Field Calculations for Arbitrary Background Fields”, Nucl. Phys. B207, 474 (1982)

    1982

  29. [29]

    Background Field Calculations in Curved Space-time. 1. General Formalism and Application to Scalar Fields

    I. Jack & H. Osborn,“Background Field Calculations in Curved Space-time. 1. General Formalism and Application to Scalar Fields”, Nucl. Phys. B234, 331 (1984). 37

    1984

  30. [30]

    General Background Field Calculations With Fermion Fields

    I. Jack & H. Osborn,“General Background Field Calculations With Fermion Fields”, Nucl. Phys. B249, 472 (1985)

    1985

  31. [31]

    Two Loop Renormalization Group Equations in a General Quantum Field Theory. 1. Wave Function Renormalization

    M. E. Machacek & M. T. Vaughn,“Two Loop Renormalization Group Equations in a General Quantum Field Theory. 1. Wave Function Renormalization”, Nucl. Phys. B222, 83 (1983)

    1983

  32. [32]

    Two Loop Renormalization Group Equations in a General Quantum Field Theory. 2. Yukawa Couplings

    M. E. Machacek & M. T. Vaughn,“Two Loop Renormalization Group Equations in a General Quantum Field Theory. 2. Yukawa Couplings”, Nucl. Phys. B236, 221 (1984)

    1984

  33. [33]

    Two Loop Renormalization Group Equations in a General Quantum Field Theory. 3. Scalar Quartic Couplings

    M. E. Machacek & M. T. Vaughn,“Two Loop Renormalization Group Equations in a General Quantum Field Theory. 3. Scalar Quartic Couplings”, Nucl. Phys. B249, 70 (1985)

    1985

  34. [34]

    General quartic β-function at three loops

    T. Steudtner & A. E. Thomsen,“General quartic β-function at three loops”, JHEP2410, 163 (2024),arXiv:2408.05267 [hep-ph]

    Pith/arXiv arXiv 2024

  35. [35]

    Four Loop Renormalisation Group Equations in General Gross-Neveu-Yukawa Theories

    T. Steudtner,“Four Loop Renormalisation Group Equations in General Gross-Neveu-Yukawa Theories”,arXiv:2507.18710 [hep-th]

    arXiv

  36. [36]

    Explorations in scalar fermion theories: β-functions, supersymmetry and fixed points

    I. Jack, H. Osborn & T. Steudtner,“Explorations in scalar fermion theories: β-functions, supersymmetry and fixed points”, JHEP2402, 038 (2024),arXiv:2301.10903 [hep-th]

    Pith/arXiv arXiv 2024

  37. [37]

    Introducing RGBeta: a Mathematica package for the evaluation of renormal- ization groupβ-functions

    A. E. Thomsen,“Introducing RGBeta: a Mathematica package for the evaluation of renormal- ization groupβ-functions”, Eur. Phys. J. C81, 408 (2021),arXiv:2101.08265 [hep-ph]

    Pith/arXiv arXiv 2021

  38. [38]

    Scalar-fermion fixed points in theε expansion

    W. H. Pannell & A. Stergiou,“Scalar-fermion fixed points in theε expansion”, JHEP2308, 128 (2023),arXiv:2305.14417 [hep-th]

    Pith/arXiv arXiv 2023

  39. [39]

    Seeking fixed points in multiple coupling scalar theories in theε expansion

    H. Osborn & A. Stergiou,“Seeking fixed points in multiple coupling scalar theories in theε expansion”, JHEP1805, 051 (2018),arXiv:1707.06165 [hep-th]

    Pith/arXiv arXiv 2018

  40. [40]

    On ambiguities and divergences in perturbative renormalization group functions

    F. Herren & A. E. Thomsen,“On ambiguities and divergences in perturbative renormalization group functions”, JHEP2106, 116 (2021),arXiv:2104.07037 [hep-th]

    Pith/arXiv arXiv 2021

  41. [41]

    Four-loop critical exponents for the Gross-Neveu-Yukawa models

    N. Zerf, L. N. Mihaila, P. Marquard, I. F. Herbut & M. M. Scherer,“Four-loop critical exponents for the Gross-Neveu-Yukawa models”, Phys. Rev. D96, 096010 (2017), arXiv:1709.05057 [hep-th]

    Pith/arXiv arXiv 2017

  42. [42]

    xTras: A field-theory inspired xAct package for Mathematica

    T. Nutma,“xTras: A field-theory inspired xAct package for Mathematica”, Com- puter Physics Communications185, 1719 (2014)

    2014

  43. [43]

    IGraph/M: graph theory and network analysis for Mathematica

    S. Horvát, J. Podkalicki, G. Csárdi, T. Nepusz, V. Traag, F. Zanini & D. Noom,“IGraph/M: graph theory and network analysis for Mathematica”, Journal of Open Source Software8, 4899 (2023)

    2023

  44. [44]

    Quantum periods: A Census of ϕ4-transcendentals

    O. Schnetz,“Quantum periods: A Census of ϕ4-transcendentals”, Commun. Num. Theor. Phys. 4, 1 (2010),arXiv:0801.2856 [hep-th]

    Pith/arXiv arXiv 2010

  45. [45]

    The Galois coaction onϕ4 periods

    E. Panzer & O. Schnetz,“The Galois coaction onϕ4 periods”, Commun. Num. Theor. Phys. 11, 657 (2017),arXiv:1603.04289 [hep-th]. 38

    Pith/arXiv arXiv 2017

  46. [46]

    Interpolating between a and F

    S. Giombi & I. R. Klebanov,“Interpolating between a and F”, JHEP1503, 117 (2015), arXiv:1409.1937 [hep-th]. 39

    Pith/arXiv arXiv 2015