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arxiv: 2605.22436 · v1 · pith:HFD6562Knew · submitted 2026-05-21 · 🧮 math-ph · hep-th· math.MP

A perturbative approach to the Wetterich equation for Bosonic and Fermionic interacting fields

Beatrice Costeri This is my paper

Pith reviewed 2026-05-22 02:03 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MP
keywords Wetterich equationrenormalization group flowlocal potential approximationperturbative algebraic quantum field theoryscalar fieldsDirac fieldsNash-Moser theoremcurved spacetime
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The pith

Perturbative algebraic methods give explicit beta functions and prove local solutions for the Wetterich RG equation in scalar and Dirac models on curved backgrounds.

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

The paper examines the Lorentzian Wetterich renormalization group flow equation for interacting quantum fields on curved spacetimes using perturbative algebraic quantum field theory. It derives the flow equations in the local potential approximation for two interacting scalar fields and for self-interacting Dirac fields, then computes the associated beta functions. Local existence and uniqueness of solutions to these equations is shown by adapting the Nash-Moser theorem. The scalar model also features an asymmetric interaction that formally corresponds to the Martin-Siggia-Rose stochastic formalism. These steps establish a consistent perturbative framework for renormalization group studies directly in Lorentzian signature.

Core claim

Within perturbative algebraic quantum field theory the Wetterich equation for mutually interacting scalars on globally hyperbolic spacetimes and for self-interacting Dirac fields on spin manifolds reduces, under the local potential approximation, to explicit evolution equations whose beta functions are calculated; the same equations are shown to possess unique local solutions via an adaptation of the Nash-Moser theorem.

What carries the argument

The local potential approximation of the perturbative Wetterich equation combined with the Nash-Moser theorem for proving local well-posedness of the resulting flow.

If this is right

  • Beta functions for the interaction couplings are obtained for both the bosonic and fermionic models.
  • The asymmetric scalar potential indicates a direct formal link to stochastic field-theoretic descriptions.
  • Local-in-scale existence and uniqueness hold for the RG flow equations of both classes of fields.
  • The construction applies on general globally hyperbolic backgrounds and on spin manifolds respectively.

Where Pith is reading between the lines

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

  • This framework could be used to numerically integrate the flows and locate fixed points for fields in specific curved geometries such as cosmological spacetimes.
  • The connection to stochastic dynamics opens the possibility of transferring methods between renormalization group analysis and non-equilibrium statistical mechanics.
  • Analogous local well-posedness results may extend to other field theories or to higher-order truncations of the effective potential.

Load-bearing premise

The Nash-Moser theorem strategy previously applied to a related renormalization group equation can be adapted without major modification to the flow equations of the scalar and Dirac models studied here.

What would settle it

A direct computation of the beta functions in Minkowski space that disagrees with standard perturbative results, or an explicit construction of initial data for which the flow equation immediately develops a singularity.

read the original abstract

We study the Lorentzian Wetterich Renormalization Group (RG) flow equation for interacting quantum fields on curved backgrounds within the framework of perturbative Algebraic Quantum Field Theory (pAQFT). Specifically, we consider two classes of models: two mutually interacting scalar fields on globally hyperbolic spacetimes without boundary and, under the further assumption that the underlying background is spin, self-interacting Dirac fields. In both cases, we derive the corresponding RG flow equations within a Local Potential Approximation and compute the beta functions for the relevant couplings. For the scalar model, we also discuss an asymmetric interaction potential which is formally reminiscent of the Martin-Siggia-Rose description of a stochastic dynamics, thereby indicating a possible connection between Lorentzian algebraic RG methods [DDP+24] and stochastic field-theoretic models, [Duc25]. In addition, we address the local well-posedness of the resulting flow equations. Adapting the strategy detailed in [DP23] and based on the the Nash-Moser theorem, we prove local existence and uniqueness of solutions for both the scalar and the Dirac models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a perturbative treatment of the Lorentzian Wetterich equation in pAQFT for two mutually interacting scalar fields and for self-interacting Dirac fields on globally hyperbolic backgrounds. Within the Local Potential Approximation it derives the corresponding RG flow equations, computes the beta functions for the relevant couplings, discusses an asymmetric scalar potential that formally resembles the Martin-Siggia-Rose stochastic dynamics, and proves local existence and uniqueness of solutions to the flow equations by adapting the Nash-Moser-based strategy of reference [DP23].

Significance. If the beta-function derivations and the well-posedness argument are correct, the work supplies explicit RG flows and a rigorous local existence result for Wetterich-type equations in curved spacetime, thereby extending the algebraic RG framework of [DDP+24] and [Duc25]. The explicit LPA computations and the attempt to adapt a functional-analytic existence theorem constitute the main technical contributions.

major comments (1)
  1. [well-posedness section / existence proof] The local well-posedness claim (abstract and the section containing the existence proof) rests on an adaptation of the Nash-Moser theorem from [DP23]. The manuscript must supply explicit verification that the Wetterich operator arising from the LPA truncation on a general globally hyperbolic background satisfies the required tame estimates and that the loss-of-derivatives remains compatible with the theorem’s hypotheses; the curved metric, the pAQFT regularization, and the specific form of the LPA potential each alter the functional-analytic setting relative to the models treated in [DP23]. Without these estimates the existence/uniqueness statement is not yet justified.
minor comments (2)
  1. [Dirac model derivation] Clarify the precise definition of the LPA truncation used for the Dirac model (e.g., which spinor bilinears are retained) and state whether the resulting beta functions are computed at a fixed background or include back-reaction terms.
  2. [scalar model with asymmetric potential] The discussion of the asymmetric scalar potential and its link to stochastic dynamics would benefit from an explicit comparison of the resulting beta functions with those of the symmetric case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need to strengthen the presentation of the well-posedness result. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [well-posedness section / existence proof] The local well-posedness claim (abstract and the section containing the existence proof) rests on an adaptation of the Nash-Moser theorem from [DP23]. The manuscript must supply explicit verification that the Wetterich operator arising from the LPA truncation on a general globally hyperbolic background satisfies the required tame estimates and that the loss-of-derivatives remains compatible with the theorem’s hypotheses; the curved metric, the pAQFT regularization, and the specific form of the LPA potential each alter the functional-analytic setting relative to the models treated in [DP23]. Without these estimates the existence/uniqueness statement is not yet justified.

    Authors: We agree that the current exposition would benefit from more explicit verification of the tame estimates in the adapted setting. In the revised manuscript we will expand the well-posedness section to provide a step-by-step check that the Wetterich operator obtained from the LPA truncation satisfies the required tame estimates on a general globally hyperbolic background. We will verify that the loss-of-derivatives remains compatible with the hypotheses of the Nash-Moser theorem used in [DP23], taking into account the curved metric through the pAQFT regularization and confirming that the concrete form of the LPA potential does not introduce further obstructions. This will be done by adapting the estimates of [DP23] to the present models while keeping the functional-analytic framework intact. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivations of LPA flow equations and beta functions are independent; well-posedness adapts external strategy

full rationale

The paper derives the Lorentzian Wetterich RG flow equations within Local Potential Approximation for two interacting scalar fields and for self-interacting Dirac fields on globally hyperbolic backgrounds, computes the beta functions for relevant couplings, and adapts the Nash-Moser theorem strategy from the cited reference [DP23] to establish local existence and uniqueness. These steps are presented as explicit calculations in the pAQFT setting rather than reductions to previously fitted quantities or self-defined inputs. Citations to [DDP+24] and [Duc25] provide context for possible connections to stochastic models but are not load-bearing for the central claims. The adaptation claim does not constitute a tautological reuse or self-citation chain that forces the result by construction, so the derivation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper rests on standard domain assumptions of perturbative algebraic quantum field theory and Lorentzian geometry; no new free parameters or invented entities are introduced in the abstract.

axioms (3)
  • domain assumption Spacetimes are globally hyperbolic without boundary for the scalar model
    Explicitly stated as the geometric setting for the two mutually interacting scalar fields.
  • domain assumption The background admits a spin structure for the Dirac model
    Required for the self-interacting Dirac fields under the further assumption given in the abstract.
  • domain assumption The Local Potential Approximation suffices to extract the beta functions
    Used to derive the RG flow equations for the relevant couplings.

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Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    Araki, Mathematical theory of Quantum Fields, International series of monographs on physics , Oxford University Press, (1999), 236p

    H. Araki, Mathematical theory of Quantum Fields, International series of monographs on physics , Oxford University Press, (1999), 236p

    work page 1999

  2. [2]

    Bastiani, ``Applications différentiables et variétés différentiables de dimension infinie'' , J

    A. Bastiani, ``Applications différentiables et variétés différentiables de dimension infinie'' , J. d'Analyse mathématique 13(1), 1-114 (1964)

    work page 1964

  3. [3]

    Non-perturbative renormalization flow in quantum field theory and statistical physics

    J. Berges, N. Tetradis and C. Wetterich, ``Non-perturbative renormalization flow in quantum field theory and statistical physics'' , Phys. Rep. 363(4--6) (2002), 223--386, 10.1016/S0370-1573(01)00098-9

    work page doi:10.1016/s0370-1573(01)00098-9 2002

  4. [4]

    Bonicelli, B

    A. Bonicelli, B. Costeri, C. Dappiaggi and P. Rinaldi, ``A microlocal investigation of stochastic partial differential equations for spinors with an application to the Thirring model'' , Math. Phys. Anal. Geom. 27 (2024), 16, 10.1007/s11040-024-09488-7

    work page doi:10.1007/s11040-024-09488-7 2024

  5. [5]

    Bonicelli, C

    A. Bonicelli, C. Dappiaggi and N. Drago, ``An Algebraic Correspondence Between Stochastic Differential Equations and the Martin–Siggia–Rose Formalism'' , Annales Henri Poincaré, Springer International Publishing, (2025)

    work page 2025

  6. [6]

    A. G. Borges and I. L. Shapiro, ``Derivative expansion in a two-scalar field theory'' , Phys. Rev. D 111(12) (2025), 125019, 10.1103/rynr-2znv

    work page doi:10.1103/rynr-2znv 2025

  7. [7]

    Brunetti, C

    R. Brunetti, C. Dappiaggi, K. Fredenhagen and J. Yngvason, Advances in algebraic quantum field theory, Springer (2015), 455 p

    work page 2015

  8. [8]

    Brunetti, M

    R. Brunetti, M. D\"utsch and K. Fredenhagen, ``Perturbative Algebraic Quantum Field Theory and the Renormalization Groups'' , Adv. Theor. Math. Phys., (2009)

    work page 2009

  9. [9]

    Brunetti, K

    R. Brunetti, K. Fredenhagen and M. Kohler, ``The microlocal spectrum condition and Wick polynomials of the free fields on curved spacetimes'' , Commun. in Math. Phys., 237 (2003), 31--68

    work page 2003

  10. [10]

    Carosso, ``Stochastic renormalization group and gradient flow'' , Journal of High Energy Physics, 1 (2020), 172

    A. Carosso, ``Stochastic renormalization group and gradient flow'' , Journal of High Energy Physics, 1 (2020), 172

    work page 2020

  11. [11]

    Catellier and K

    R. Catellier and K. Chouk, ``Paracontrolled distributions and the 3-dimensional stochastic quantization equation'' , Ann. Probab. 46(5) (2018), 2621--2679, 10.1214/17-AOP1235

    work page doi:10.1214/17-aop1235 2018

  12. [12]

    Costeri, C

    B. Costeri, C. Dappiaggi and M. Goi, ``Conservation law and trace anomaly for the stress-energy tensor of a self-interacting scalar field'' , Ann. Henri Poincar\'e 27 (2026), 1843--1884, 10.1007/s00023-025-01580-0

    work page doi:10.1007/s00023-025-01580-0 2026

  13. [13]

    Costeri, C

    B. Costeri, C. Dappiaggi, B. A. Ju\'arez-Aubry and R. D. Singh, ``The Hadamard parametrix on half-Minkowski with Robin boundary conditions: Fundamental solutions and Hadamard states'' , Ann. Henri Poincar\'e (2026), 10.1007/s00023-026-01702-2

    work page doi:10.1007/s00023-026-01702-2 2026

  14. [14]

    D'Angelo and N

    E. D'Angelo and N. Pinamonti, ``Local solutions of the RG flow equations from the Nash-Moser theorem'' , Communications in Mathematical Physics, 405(9), 200 (2024)

    work page 2024

  15. [15]

    D'Angelo, N

    E. D'Angelo, N. Drago, N. Pinamonti and K. Rejzner, ``An algebraic QFT approach to the Wetterich equation on Lorentzian manifolds'' , Ann. Henri Poincar\'e 25 (2024), 2295--2352, 10.1007/s00023-023-01348-4

    work page doi:10.1007/s00023-023-01348-4 2024

  16. [16]

    Dappiaggi, N

    C. Dappiaggi, N. Drago and P. Rinaldi, ``The algebra of Wick polynomials of a scalar field on a Riemannian manifold'' , Reviews in Mathematical Physics 32 (2019) no. 8, 2050023

    work page 2019

  17. [17]

    Dappiaggi, T.-P

    C. Dappiaggi, T.-P. Hack and N. Pinamonti, The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor, Reviews in Mathematical Physics 21 (2009), 1241--1312

    work page 2009

  18. [18]

    Dappiaggi, F

    C. Dappiaggi, F. Nava and L. Sinibaldi, ``On the interplay between boundary conditions and the Lorentzian Wetterich equation'' , Rev. Math. Phys. 37(2) (2025), 2450031, 10.1142/S0129055X24500314

    work page doi:10.1142/s0129055x24500314 2025

  19. [19]

    Drago, T.-P

    N. Drago, T.-P. Hack and N. Pinamonti, The generalised principle of perturbative agreement and the thermal mass, Annales Henri Poincaré 18, no. 3 (2017), 807--868

    work page 2017

  20. [20]

    Duch, ``Lecture notes on flow equation approach to singular stochastic PDEs" , arXiv preprint (2025), [arXiv:2511.07120 [math.PR]]

    P. Duch, ``Lecture notes on flow equation approach to singular stochastic PDEs" , arXiv preprint (2025), [arXiv:2511.07120 [math.PR]]

    work page arXiv 2025

  21. [21]

    P. Duch, M. Gubinelli and P. Rinaldi, ``Parabolic stochastic quantisation of the fractional ^4_3 model in the full subcritical regime'' , arXiv:2303.18112 [math.PR] (2023)

    work page arXiv 2023

  22. [22]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier and N. Wschebor, ``The nonperturbative functional renormalization group and its applications'' , Phys. Rep. 910 (2021), 1--114, 10.1016/j.physrep.2021.01.001

    work page doi:10.1016/j.physrep.2021.01.001 2021

  23. [23]

    Freidel, J

    L. Freidel, J. Padua-Arg \"u elles, S. Schander and M. Schiffer, ``Non-Perturbative S -matrix Renormalization'' , [arXiv:2509.00156 [hep-th]]

    work page arXiv

  24. [24]

    T. H. Gr \"o nwall, ``Note on the derivatives with respect to a parameter of the solutions of a system of differential equations'' , Annals of Mathematics, Second Series, 20, no. 4 (1919), 292--296

    work page 1919

  25. [25]

    [GK22] G

    M. Gubinelli, P. Imkeller and N. Perkowski, ``Paracontrolled distributions and singular PDEs'' , Forum Math. Pi 3 (2015), e6, 10.1017/fmp.2015.2

    work page doi:10.1017/fmp.2015.2 2015

  26. [26]

    Gubinelli, H

    M. Gubinelli, H. Koch and T. Oh, ``Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity'' , J. Eur. Math. Soc. 26(3) (2024), 817--874, 10.4171/JEMS/1294

    work page doi:10.4171/jems/1294 2024

  27. [27]

    Haag, Local quantum physics: Fields, Particles, Algebras, Theoretical and Mathematical Physics, Springer-Verlag, (1996), 360p

    R. Haag, Local quantum physics: Fields, Particles, Algebras, Theoretical and Mathematical Physics, Springer-Verlag, (1996), 360p

    work page 1996

  28. [28]

    Hamilton ``The inverse function theorem of Nash and Moser'', Bull

    R. Hamilton ``The inverse function theorem of Nash and Moser'', Bull. Amer. Math. Soc. (N.S.) 7(1): 65-222 (1982)

    work page 1982

  29. [29]

    S. W. Hawking and G. F. R. Ellis, ``The large scale structure of space-time'' , Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge (1973)

    work page 1973

  30. [30]

    H\"ormander, The Analysis of Linear Partial Differential Operators I, Springer (1990), 440p

    L. H\"ormander, The Analysis of Linear Partial Differential Operators I, Springer (1990), 440p

    work page 1990

  31. [31]

    S. R. Ju\'arez Wysozka, P. Kielanowski, E. Uribe Longoria and L. Vazquez Mercado, ``Two interacting scalar fields: practical renormalization'' , arXiv:2104.09681 [hep-th] (2021), 10.48550/arXiv.2104.09681

    work page doi:10.48550/arxiv.2104.09681 2021

  32. [32]

    H. B. Lawson Jr. and M.-L. Michelsohn, Spin Geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989

    work page 1989

  33. [33]

    P. C. Martin, E. D. Siggia and H. A. Rose, ``Statistical Dynamics of Classical Systems" , Phys. Rev. A 8 (1973), 423

    work page 1973

  34. [34]

    Michal, ``Differential calculus in linear topological spaces" , Proc

    A.D. Michal, ``Differential calculus in linear topological spaces" , Proc. Natl. Acad. Sci. USA 24(8), 340-342 (1938)

    work page 1938

  35. [35]

    Michor, ``A convenient setting for differential geometry and global analysis I

    P.W. Michor, ``A convenient setting for differential geometry and global analysis I. II." , Cahiers Topol. Geo. Diff 25, 63-109 (1984)

    work page 1984

  36. [36]

    Milnor, Remarks on the infinite dimensional Lie groups, (1984)

    J. Milnor, Remarks on the infinite dimensional Lie groups, (1984)

    work page 1984

  37. [37]

    Moretti, ``Direct -function approach and renormalization of one-loop stress tensors in curved spacetimes'' , Phys

    V. Moretti, ``Direct -function approach and renormalization of one-loop stress tensors in curved spacetimes'' , Phys. Rev. D 56(12) (1997), 7797--7819, 10.1103/PhysRevD.56.7797

    work page doi:10.1103/physrevd.56.7797 1997

  38. [38]

    Moretti, ``Comments on the stress-energy tensor operator in curved spacetime'' , Commun

    V. Moretti, ``Comments on the stress-energy tensor operator in curved spacetime'' , Commun. Math. Phys. 232 (2003), 189--221, 10.1007/s00220-002-0702-7

    work page doi:10.1007/s00220-002-0702-7 2003

  39. [39]

    T. R. Morris, ``The exact renormalization group and approximate solutions'' , Int. J. Mod. Phys. A 9(14) (1994), 2411--2450, 10.1142/S0217751X94000972

    work page doi:10.1142/s0217751x94000972 1994

  40. [40]

    Moser, ``A rapidly convergent iteration method and non-linear partial differential equations

    J. Moser, ``A rapidly convergent iteration method and non-linear partial differential equations. I'' , Ann. Scuola Norm. Sup. Pisa 20(2) (1966), 265--315

    work page 1966

  41. [41]

    Moser, ``A rapidly convergent iteration method and non-linear differential equations

    J. Moser, ``A rapidly convergent iteration method and non-linear differential equations. II'' , Ann. Scuola Norm. Sup. Pisa 20(3) (1966), 499--535

    work page 1966

  42. [42]

    The imbedding problem for Riemannian manifolds.Annals of Mathematics, 2nd Ser., 63(1):20–63, 1956.doi:10.2307/1969989

    J. Nash, ``The imbedding problem for Riemannian manifolds'' , Ann. of Math. 63(1) (1956), 20--63, 10.2307/1969989

    work page doi:10.2307/1969989 1956

  43. [43]

    Neeb, Infinite-dimensional Lie groups, Monastir Summer school, TU Darmstadt Preprint, 2433, (2006)

    K.H. Neeb, Infinite-dimensional Lie groups, Monastir Summer school, TU Darmstadt Preprint, 2433, (2006)

    work page 2006

  44. [44]

    Polchinski,Renormalization and Effective Lagrangians, Nucl

    J. Polchinski, ``Renormalization and effective lagrangians'' , Nucl. Phys. B 231(2) (1984), 269--295, 10.1016/0550-3213(84)90287-6

    work page doi:10.1016/0550-3213(84)90287-6 1984

  45. [45]

    Rejzner, Perturbative Algebraic Quantum Field Theory, Springer , (2016), p

    K. Rejzner, Perturbative Algebraic Quantum Field Theory, Springer , (2016), p. 180

    work page 2016

  46. [46]

    Rejzner, ``Fermionic fields in the functional approach to classical field theory'' , Reviews in Mathematical Physics, 23(09) (2011), 1009--1033

    K. Rejzner, ``Fermionic fields in the functional approach to classical field theory'' , Reviews in Mathematical Physics, 23(09) (2011), 1009--1033

    work page 2011

  47. [47]

    Trèves, Topological vector spaces, distributions and kernels, Academic Press, INC

    F. Trèves, Topological vector spaces, distributions and kernels, Academic Press, INC. (1967), p. 563

    work page 1967

  48. [48]

    Wetterich, ``Average action and the renormalization group equations'' , Nucl

    C. Wetterich, ``Average action and the renormalization group equations'' , Nucl. Phys. B 352(3) (1991), 529--584, 10.1016/0550-3213(91)90099-J

    work page doi:10.1016/0550-3213(91)90099-j 1991

  49. [49]

    Wetterich, ``Exact evolution equation for the effective potential'' , Phys

    C. Wetterich, ``Exact evolution equation for the effective potential'' , Phys. Lett. B 301(1) (1993), 90--94, 10.1016/0370-2693(93)90726-X

    work page doi:10.1016/0370-2693(93)90726-x 1993

  50. [50]

    K. G. Wilson and J. Kogut, ``The renormalization group and the expansion'' , Phys. Rep. 12(2) (1974), 75--199, 10.1016/0370-1573(74)90023-4

    work page doi:10.1016/0370-1573(74)90023-4 1974

  51. [51]

    Zanella and E

    J. Zanella and E. Calzetta, ``Renormalization group and nonequilibrium action in stochastic field theory'' , Phys. Rev. E 66(3) (2002), 10.1103/PhysRevE.66.036134

    work page doi:10.1103/physreve.66.036134 2002