pith. sign in

arxiv: 2507.13011 · v2 · submitted 2025-07-17 · ✦ hep-th

Physics-informed operator flows and observables

Friederike Ihssen , Jan M. Pawlowski This is my paper

Pith reviewed 2026-05-19 04:49 UTC · model grok-4.3

classification ✦ hep-th
keywords physics-informed renormalisation groupoperator flowscorrelation functionsvertex expansionzero-dimensional phi-four theoryquantum field theoryrenormalisation group flows
0
0 comments X

The pith

Physics-informed renormalisation group flows extended to general operators capture every correlation function of the quantum field theory.

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

The paper develops physics-informed renormalisation group flows for arbitrary operators rather than restricting them to the fundamental field. It argues that these operator flows furnish all correlation functions of the theory without additional external input. The authors present the construction as the completion of an earlier PIRG framework, so that its computational simplifications become available for general applications. They illustrate the approach with an exact analytic treatment of zero-dimensional phi-four theory, where a vertex expansion generates the full set of one- through ten-point functions.

Core claim

Operator PIRGs provide a comprehensive access to all correlation functions of the quantum field theory under investigation. The operator PIRGs can be seen as a completion of the PIRG-approach, whose qualitative computational simplification and structural insights are now fully accessible for general applications. The potential of this setup is assessed within a simple analytic example of the zero-dimensional phi-four theory for which the generating functions of the fundamental field are computed within a vertex expansion, using the one- to ten-point functions.

What carries the argument

Operator PIRGs: renormalisation-group flow equations written directly for general composite operators, which generate the complete tower of correlation functions through a vertex expansion.

If this is right

  • Every correlation function of the theory becomes directly accessible from the operator PIRG equations.
  • The computational and structural advantages of the original PIRG method extend to the full set of observables.
  • No further truncations beyond the vertex expansion are required to recover the complete information.
  • The zero-dimensional example demonstrates explicit reconstruction of generating functions up to order ten.

Where Pith is reading between the lines

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

  • The same operator-flow construction could be applied to higher-dimensional toy models to check whether the completeness persists beyond zero dimensions.
  • Direct comparison with lattice or functional-renormalisation-group results for the same theory would provide an independent numerical test of the extracted correlators.
  • The method may allow extraction of composite-operator expectation values that are otherwise obtained only through additional Ward identities or external constraints.

Load-bearing premise

That flows defined for general operators automatically encode every correlation function once the vertex expansion is performed, without hidden truncations or missing external data.

What would settle it

A mismatch between the ten-point functions obtained from the operator PIRG vertex expansion and the independently known exact generating function in the zero-dimensional phi-four model.

Figures

Figures reproduced from arXiv: 2507.13011 by Friederike Ihssen, Jan M. Pawlowski.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic depiction of the flows for the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Schematic depiction of operator flows of the derived pairs [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: RG-time dependence of the two-point function on [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Different definitions for the map of the field [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Field dependence of composite operator flows of the connected part of the two-point function [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: RG-time dependence of the operator flows for the [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Reconstruction of the generating functionals of correlation functions of the fundamental field, the 1PI effective [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Relative error ∆Γ [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

We discuss physics-informed renormalisation group flows (PIRGs) for general operators. We show that operator PIRGs provide a comprehensive access to all correlation functions of the quantum field theory under investigation. The operator PIRGs can be seen as a completion of the PIRG-approach, whose qualitative computational simplification and structural insights are now fully accessible for general applications. The potential of this setup is assessed within a simple analytic example of the zero-dimensional $\phi^4$-theory for which the generating functions of the fundamental field are computed within a vertex expansion, using the one- to ten-point functions.

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 / 1 minor

Summary. The manuscript introduces physics-informed renormalisation group flows (PIRGs) for general operators. It claims that operator PIRGs provide comprehensive access to all correlation functions of the quantum field theory under investigation, completing the PIRG framework. This is assessed via a vertex expansion in zero-dimensional φ⁴ theory, computing the generating functions of the fundamental field using the one- to ten-point functions.

Significance. If the general construction holds, the work would extend PIRGs to operators and thereby make their qualitative simplifications and structural insights available for broader QFT applications. The explicit analytic computation of one- to ten-point functions in the zero-dimensional example is a concrete strength that demonstrates the vertex expansion in a controlled setting.

major comments (1)
  1. Abstract: the central claim that operator PIRGs yield complete information on all correlation functions without further truncations or external data is supported only by the zero-dimensional analytic example. In this setting the generating functional reduces to an ordinary integral with no momentum integrals or loop structure, so the hierarchy closes trivially; no explicit verification is supplied that the same operator-flow construction remains closed once spatial dependence or non-trivial propagators are restored.
minor comments (1)
  1. The abstract refers to 'the vertex expansion, using the one- to ten-point functions' but does not state the truncation order or supply error estimates for the generating function.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading of our manuscript and the constructive feedback. We address the major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: Abstract: the central claim that operator PIRGs yield complete information on all correlation functions without further truncations or external data is supported only by the zero-dimensional analytic example. In this setting the generating functional reduces to an ordinary integral with no momentum integrals or loop structure, so the hierarchy closes trivially; no explicit verification is supplied that the same operator-flow construction remains closed once spatial dependence or non-trivial propagators are restored.

    Authors: We agree that the explicit analytic verification of the closure for all correlation functions (via the vertex expansion up to ten-point functions) is performed in the zero-dimensional φ⁴ theory, which indeed reduces to an ordinary integral without momentum dependence. This controlled setting was chosen to allow a fully analytic demonstration of how operator PIRGs determine the complete hierarchy of n-point functions from the flow equations without external data or additional truncations beyond the vertex expansion itself. The general operator-flow construction and the associated flow equations for arbitrary operators are derived in a dimension-independent manner in the main text, relying on the same physics-informed structure as the original PIRG approach for the effective action. By construction, the operator flows close the system for correlation functions in the same way the standard PIRG closes for the effective potential and vertices; the zero-dimensional case serves as a non-trivial test of this closure at high orders. That said, we acknowledge that an explicit numerical or analytic check with momentum-dependent propagators in d>0 would provide further confirmation. We will revise the abstract to clarify that the comprehensive access is demonstrated through the general construction and validated analytically in the zero-dimensional example, and we will add a short paragraph in the conclusions discussing the extension to higher-dimensional theories with non-trivial propagators. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit zero-dimensional verification

full rationale

The manuscript extends the existing PIRG framework to general operators and asserts comprehensive access to all correlation functions. This is supported by a structural argument plus an explicit analytic vertex expansion (one- to ten-point functions) in zero-dimensional φ⁴ theory. No equation or claim reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the zero-dimensional computation functions as an independent, parameter-free check within its domain. Self-citations to prior PIRG work are not required to close the central completeness statement.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard renormalisation-group framework plus the domain assumption that operator flows can be defined without loss of information; the zero-dimensional example uses a finite-order vertex truncation as a practical choice.

free parameters (1)
  • Vertex expansion order
    Computation limited to one- through ten-point functions constitutes a truncation whose accuracy is not quantified in the abstract.
axioms (1)
  • domain assumption Physics-informed renormalisation group flows can be consistently defined for arbitrary operators
    This is the central extension invoked to reach all correlation functions.

pith-pipeline@v0.9.0 · 5614 in / 1230 out tokens · 56591 ms · 2026-05-19T04:49:13.011428+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We discuss physics-informed renormalisation group flows (PIRGs) for general operators... operator PIRGs provide a comprehensive access to all correlation functions... vertex expansion, using the one- to ten-point functions.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages · 21 internal anchors

  1. [1]

    First we remark that the relation (25) translates into a similar one for the effec- tive action Γϕ[ϕ, JO]

    Generalised operator flow equation We proceed by deriving the flow equation for a general operator ˆO within this setup. First we remark that the relation (25) translates into a similar one for the effec- tive action Γϕ[ϕ, JO]. This effective action is obtained as the Legendre transform of ln Zϕ with respect to Jϕ while keeping JO as a variable, Γϕ[ϕ, JO]...

    work page

  2. [2]

    Operator PIRGs With (30) we are now fully prepared for extending the PIRG-approach to flows of general operators, such as cor- relation functions of the fundamental field. Specifically we shall use the generality of the flowing field ˙ϕ, that is that of general choices of the composite operators ˆϕ, to unlock the full potential of the generalised operator...

    work page

  3. [3]

    This choice was already alluded to at the end of Section III B 1

    Simplified operator PIRGs As a first application of the operator PIRG we consider the choice FO[ϕ, JO] = 0 , (40) in the generalised operator flow (30) with FO defined in (30b). This choice was already alluded to at the end of Section III B 1. With the choice (40) the generalised operator flow reduces to a variant of the operator flow (17a) with the effec...

    work page

  4. [4]

    The restrictions mirror those on ˙ϕ, which were put forward in [1, 4]

    Local and global existence In the following we discuss constraints on the first- order contribution ˙ϕ(1) in (45). The restrictions mirror those on ˙ϕ, which were put forward in [1, 4]. We have recalled them below (8) in Section II A. The additional constraints are labelled accordingly, (i) Local existence of ˙ϕ(1) k [ϕ] in (45): One has to show that ther...

    work page

  5. [5]

    Rij ϕ Gil δϕO nj δϕl # . (48) With (30) and (48), the integrability condition (47) reads ∂t⟨ ˆOn⟩ + ∂t ϕO nm δΓϕ δϕm − ∂t

    Integrability constraint While the global requirement(ii) is far more restrictive than the local one, (i), the latter is still very helpful. A minimal consistency constraint that follows from (i) is the integrability of the transformation, ∂t , δ δJO Γϕ[ϕ, JO] = 0 . (47) The use of (47) and its violation is twofold. To begin with, approximations will lead...

    work page

  6. [6]

    This may be seen as the analogue of a classical target action

    Non-triviality of trivialising flowing fields ˙ϕ(1) We close this Section with an analysis of operator flows with trivialising flowing fields ˙ϕ(1): they are defined by a vanishing right hand side of the generalised operator flow (17), leading to ∂tOϕ = 0: the target operator is the one at k = Λ. This may be seen as the analogue of a classical target acti...

    work page

  7. [7]

    (A1) Then, the generalised flow equation provides a linear ODE for ˙ϕ(ϕ), see (60c)

    Cumulants-preserving computation of ⟨ ˆφ2⟩ For the classical target action (60b) the flow of the effective action reduces to a simple shift, ∂tΓT (ϕ) = ∂tCk . (A1) Then, the generalised flow equation provides a linear ODE for ˙ϕ(ϕ), see (60c). This leaves us with two con- stant degrees of freedom that need to be fixed: the flow of the constant Ck and the ...

    work page

  8. [8]

    As discussed in Sec- tion IV B we may use different splits of the operator O2, such as (69) or (73)

    PIRG operator flows with total derivative flows We compute the flow of the two-point function from the total derivative flow (43). As discussed in Sec- tion IV B we may use different splits of the operator O2, such as (69) or (73). Here we use O2 = O2,cϕ + ϕ2 , (A6) in (73), leading to the representation (74) of the total derivative flow. Its solution req...

    work page

  9. [9]

    (A8) Then we have to resort to the generalised operator flow (30) as (A8) induces a JO-dependence for ˙ϕ(ϕ, JO)

    PIRG operator flows with JO-independent composite operator ˆϕ In case one shies away from yet a further implicit defi- nition for the operator ˆϕ, which was used for the deriva- tion of the total derivative in Section III B 3, we may use a JO-independent composite operator ˆϕ with ∂ ˆϕ ∂JO = 0 . (A8) Then we have to resort to the generalised operator flow...

    work page

  10. [10]

    Ihssen and J

    F. Ihssen and J. M. Pawlowski, (2024), arXiv:2409.13679 [hep-th]

    work page arXiv 2024

  11. [11]

    J. M. Pawlowski, Annals Phys. 322, 2831 (2007), arXiv:hep-th/0512261

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  12. [12]

    Exact evolution equation for the effective potential

    C. Wetterich, Phys. Lett. B 301, 90 (1993), arXiv:1710.05815 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  13. [13]

    Ihssen and J

    F. Ihssen and J. M. Pawlowski, (2025), arXiv:2503.22638 [hep-th]

    work page arXiv 2025

  14. [14]

    Bonanno, F

    A. Bonanno, F. Ihssen, and J. M. Pawlowski, (2025), arXiv:2504.03437 [hep-th]

    work page arXiv 2025

  15. [15]

    Baldazzi, K

    A. Baldazzi, K. Falls, and R. Ferrero, Annals Phys. 440, 168822 (2022), arXiv:2112.02118 [hep-th]

    work page arXiv 2022

  16. [16]

    Wetterich, Nucl

    C. Wetterich, Nucl. Phys. B 1008, 116707 (2024), arXiv:2402.04679 [hep-th]

    work page arXiv 2024

  17. [17]

    Wetterich, Z

    C. Wetterich, Z. Phys. C 72, 139 (1996), arXiv:hep- ph/9604227

    work page arXiv 1996

  18. [18]

    Renormalization Flow of Bound States

    H. Gies and C. Wetterich, Phys. Rev. D 65, 065001 (2002), arXiv:hep-th/0107221

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  19. [19]

    Universality of spontaneous chiral symmetry breaking in gauge theories

    H. Gies and C. Wetterich, Phys. Rev. D 69, 025001 (2004), arXiv:hep-th/0209183

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  20. [20]

    Exact flow equation for composite operators

    S. Floerchinger and C. Wetterich, Phys. Lett. B 680, 371 (2009), arXiv:0905.0915 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  21. [21]

    W.-j. Fu, J. M. Pawlowski, and F. Rennecke, Phys. Rev. D 101, 054032 (2020), arXiv:1909.02991 [hep-ph]

    work page arXiv 2020

  22. [22]

    Ihssen, J

    F. Ihssen, J. M. Pawlowski, F. R. Sattler, and N. Wink, (2024), arXiv:2408.08413 [hep-ph]

    work page arXiv 2024

  23. [23]

    Lamprecht, Diploma thesis Heidelberg University (2007)

    F. Lamprecht, Diploma thesis Heidelberg University (2007). 21

    work page 2007

  24. [24]

    Application of the functional renormalization group to Bose gases: from linear to hydrodynamic fluctuations

    F. Isaule, M. C. Birse, and N. R. Walet, Phys. Rev. B 98, 144502 (2018), arXiv:1806.10373 [cond-mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  25. [25]

    Isaule, M

    F. Isaule, M. C. Birse, and N. R. Walet, Annals Phys. 412, 168006 (2020), arXiv:1902.07135 [cond-mat.quant- gas]

    work page arXiv 2020

  26. [26]

    Ihssen and J

    F. Ihssen and J. M. Pawlowski, SciPost Phys. 15, 074 (2023), arXiv:2207.10057 [hep-th]

    work page arXiv 2023

  27. [27]

    Salmhofer, Annalen Phys

    M. Salmhofer, Annalen Phys. 16, 171 (2007), arXiv:cond- mat/0607289

    work page arXiv 2007

  28. [28]

    F. J. Wegner, Journal of Physics C: Solid State Physics 7, 2098 (1974)

    work page 2098

  29. [29]

    Baldazzi, R

    A. Baldazzi, R. B. A. Zinati, and K. Falls, SciPost Phys. 13, 085 (2022), arXiv:2105.11482 [hep-th]

    work page arXiv 2022

  30. [30]

    Baldazzi and K

    A. Baldazzi and K. Falls, Universe 7, 294 (2021), arXiv:2107.00671 [hep-th]

    work page arXiv 2021

  31. [31]

    Knorr, (2022), arXiv:2204.08564 [hep-th]

    B. Knorr, (2022), arXiv:2204.08564 [hep-th]

    work page arXiv 2022

  32. [32]

    Baldazzi, K

    A. Baldazzi, K. Falls, Y. Kluth, and B. Knorr, (2023), arXiv:2312.03831 [hep-th]

    work page arXiv 2023

  33. [33]

    Ohta and M

    N. Ohta and M. Yamada, (2025), arXiv:2506.03601 [hep- th]

    work page arXiv 2025

  34. [34]

    A note on scaling arguments in the effective average action formalism

    C. Pagani, Phys. Rev. D 94, 045001 (2016), arXiv:1603.07250 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  35. [35]

    T. K. Herbst, J. Luecker, and J. M. Pawlowski, (2015), arXiv:1510.03830 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  36. [36]

    Geometric operators in the asymptotic safety scenario for quantum gravity

    M. Becker and C. Pagani, Phys. Rev. D 99, 066002 (2019), arXiv:1810.11816 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  37. [37]

    Becker and C

    M. Becker and C. Pagani, Universe 5, 75 (2019)

    work page 2019

  38. [38]

    Houthoff, A

    W. Houthoff, A. Kurov, and F. Saueressig, JHEP 04, 099, arXiv:2002.00256 [hep-th]

    work page arXiv 2002

  39. [39]

    X.-Z. Luo, D. Luo, and R. G. Melko, (2024), arXiv:2403.03199 [quant-ph]

    work page arXiv 2024

  40. [40]

    J. M. Pawlowski, M. M. Scherer, R. Schmidt, and S. J. Wetzel, Annals Phys. 384, 165 (2017), arXiv:1512.03598 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  41. [41]

    On the construction of renormalized gauge theories using renormalization group techniques

    C. Becchi, (1996), arXiv:hep-th/9607188

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  42. [42]

    Realization of symmetry in the ERG approach to quantum field theory

    Y. Igarashi, K. Itoh, and H. Sonoda, Prog. Theor. Phys. Suppl. 181, 1 (2010), arXiv:0909.0327 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  43. [43]

    Ihssen and J

    F. Ihssen and J. M. Pawlowski, (2023), arXiv:2305.00816 [hep-th]

    work page arXiv 2023

  44. [44]

    Ihssen, D

    F. Ihssen, D. Krojer, and J. M. Pawlowski, in preparation (2025)

    work page 2025

  45. [45]

    Falls, (2025), arXiv:2504.17851 [hep-th]

    K. Falls, (2025), arXiv:2504.17851 [hep-th]

    work page arXiv 2025

  46. [46]

    W.-j. Fu, C. Huang, J. M. Pawlowski, Y.-y. Tan, and L.-j. Zhou, (2025), arXiv:2502.14388 [hep-ph]

    work page arXiv 2025

  47. [47]

    D. F. Litim and J. M. Pawlowski, Phys. Rev. D 66, 025030 (2002), arXiv:hep-th/0202188

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  48. [48]

    D. F. Litim and J. M. Pawlowski, Phys. Rev. D 65, 081701 (2002), arXiv:hep-th/0111191

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  49. [49]

    D. F. Litim and J. M. Pawlowski, Phys. Lett. B 546, 279 (2002), arXiv:hep-th/0208216

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  50. [50]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, Phys. Rept. 910, 1 (2021), arXiv:2006.04853 [cond-mat.stat-mech]

    work page arXiv 2021

  51. [51]

    Reuter and C

    M. Reuter and C. Wetterich, Nucl. Phys. B 417, 181 (1994)

    work page 1994

  52. [52]

    Reuter, Phys

    M. Reuter, Phys. Rev. D 57, 971 (1998), arXiv:hep- th/9605030

    work page arXiv 1998

  53. [53]

    Renormalization Group Flow and Equation of State of Quarks and Mesons

    B.-J. Schaefer and H.-J. Pirner, Nucl. Phys. A 660, 439 (1999), arXiv:nucl-th/9903003

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  54. [54]

    Zappala, Phys

    D. Zappala, Phys. Rev. D 66, 105020 (2002), arXiv:hep- th/0202167

    work page arXiv 2002

  55. [55]

    Susceptibilities near the QCD (tri)critical point

    B.-J. Schaefer and J. Wambach, Phys. Rev. D75, 085015 (2007), arXiv:hep-ph/0603256

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  56. [56]

    J. A. Dietz and T. R. Morris, JHEP 04, 118, arXiv:1502.07396 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  57. [57]

    Background independence in a background dependent renormalization group

    P. Labus, T. R. Morris, and Z. H. Slade, Phys. Rev. D 94, 024007 (2016), arXiv:1603.04772 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  58. [58]

    Gauge-invariant fields and flow equations for Yang-Mills theories

    C. Wetterich, Nucl. Phys. B 934, 265 (2018), arXiv:1710.02494 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  59. [59]

    Renormalized Functional Renormalization Group

    S. Lippoldt, Phys. Lett. B 782, 275 (2018), arXiv:1804.04409 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  60. [60]

    Bonanno, S

    A. Bonanno, S. Lippoldt, R. Percacci, and G. P. Vacca, Eur. Phys. J. C 80, 249 (2020), arXiv:1912.08135 [hep- th]

    work page arXiv 2020

  61. [61]

    Falls, Eur

    K. Falls, Eur. Phys. J. C 81, 121 (2021), arXiv:2004.11409 [hep-th]

    work page arXiv 2021

  62. [62]

    Mandric and T

    V.-M. Mandric and T. R. Morris, Phys. Rev. D 107, 065012 (2023), arXiv:2210.00492 [hep-th]

    work page arXiv 2023

  63. [63]

    Wetterich, Phys

    C. Wetterich, Phys. Lett. B 864, 139435 (2025), arXiv:2403.17523 [hep-th]

    work page arXiv 2025

  64. [64]

    Pagani and H

    C. Pagani and H. Sonoda, Phys. Rev. D 111, 105006 (2025), arXiv:2408.03625 [hep-th]

    work page arXiv 2025

  65. [65]

    Grossi and N

    E. Grossi and N. Wink, SciPost Phys. Core 6, 071 (2023), arXiv:1903.09503 [hep-th]

    work page arXiv 2023

  66. [66]

    Grossi, F

    E. Grossi, F. J. Ihssen, J. M. Pawlowski, and N. Wink, Phys. Rev. D104, 016028 (2021), arXiv:2102.01602 [hep- ph]

    work page arXiv 2021

  67. [67]

    Koenigstein, M

    A. Koenigstein, M. J. Steil, N. Wink, E. Grossi, and J. Braun, Phys. Rev. D 106, 065013 (2022), arXiv:2108.10085 [cond-mat.stat-mech]

    work page arXiv 2022

  68. [68]

    M. J. Steil and A. Koenigstein, Phys. Rev. D106, 065014 (2022), arXiv:2108.04037 [cond-mat.stat-mech]

    work page arXiv 2022

  69. [69]

    Koenigstein, M

    A. Koenigstein, M. J. Steil, N. Wink, E. Grossi, J. Braun, M. Buballa, and D. H. Rischke, Phys. Rev. D106, 065012 (2022), arXiv:2108.02504 [cond-mat.stat-mech]

    work page arXiv 2022

  70. [70]

    Ihssen, J

    F. Ihssen, J. M. Pawlowski, F. R. Sattler, and N. Wink, Comput. Phys. Commun. 300, 109182 (2024), arXiv:2207.12266 [hep-th]

    work page arXiv 2024

  71. [71]

    Ihssen, F

    F. Ihssen, F. R. Sattler, and N. Wink, Phys. Rev. D 107, 114009 (2023), arXiv:2302.04736 [hep-th]

    work page arXiv 2023

  72. [72]

    F. R. Sattler and J. M. Pawlowski, (2024), arXiv:2412.13043 [hep-ph]

    work page arXiv 2024

  73. [73]

    Zorbach, A

    N. Zorbach, A. Koenigstein, and J. Braun, (2024), arXiv:2412.16053 [cond-mat.stat-mech]

    work page arXiv 2024

  74. [74]

    W. R. Inc., Mathematica, Version 14.2, champaign, IL, 2024

    work page 2024