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arxiv: 2512.08435 · v2 · submitted 2025-12-09 · ✦ hep-th · gr-qc· hep-ph

Recognition: 2 theorem links

· Lean Theorem

Emergence of dynamical tensor fields in composite models of gravity

Yadikaer Maitiniyazi , Masatoshi Yamada
Authors on Pith no claims yet

Pith reviewed 2026-05-17 00:24 UTC · model grok-4.3

classification ✦ hep-th gr-qchep-ph
keywords composite gravityfunctional renormalization groupHubbard-Stratonovich transformationauxiliary tensor fielddynamical kinetic termstransverse-traceless sectorEinstein-Hilbert structureinfrared emergence
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The pith

Finite kinetic terms for auxiliary tensor fields emerge dynamically in the infrared for composite models of gravity.

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

The paper investigates composite models where gravity-like tensor degrees of freedom arise from underlying fermionic or scalar theories. An auxiliary tensor field is introduced through a Hubbard-Stratonovich transformation in the energy-momentum tensor channel, after which functional renormalization group flow equations are derived for the field's renormalization factors. These flows generate finite kinetic terms at low energies. The resulting structure matches the quadratic Einstein-Hilbert action in the transverse-traceless spin-2 sector, while other sectors produce terms that resemble gauge-fixed forms but are not taken as actual gauge fixing. A reader would care because the work offers a concrete dynamical mechanism by which gravity could appear as a composite phenomenon rather than a fundamental field.

Core claim

In both the fermionic and scalar prototype models, the auxiliary tensor field corresponding to the composite energy-momentum tensor channel acquires a finite kinetic term through the renormalization group flow in the infrared. The transverse-traceless spin-2 component of this term reproduces the diffeomorphism-invariant quadratic Einstein-Hilbert structure. The longitudinal and trace contributions can be written in a form similar to gauge-fixed quadratic gravity but are not interpreted as resulting from a gauge-fixing procedure within the adopted truncation.

What carries the argument

The auxiliary tensor field introduced via Hubbard-Stratonovich transformation in the energy-momentum tensor channel, whose renormalization-group flow equations generate a finite kinetic term that matches the quadratic Einstein-Hilbert action in the transverse-traceless spin-2 sector.

If this is right

  • The quadratic action for the auxiliary field reproduces the transverse-traceless Einstein-Hilbert structure.
  • Finite kinetic terms arise dynamically without being inserted by hand at the ultraviolet scale.
  • Longitudinal and trace sectors produce additional terms that resemble gauge-fixed presentations of quadratic gravity.
  • The emergence holds for both fermionic and scalar underlying theories.

Where Pith is reading between the lines

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

  • If the mechanism works, gravity could be viewed as an emergent low-energy phenomenon arising from strong dynamics in simpler matter theories.
  • The same composite-channel approach might be applied to other fields or to include higher-order operators beyond the present truncation.
  • Numerical lattice simulations of the underlying fermionic or scalar models could provide an independent check for the appearance of tensor modes with the predicted kinetics.

Load-bearing premise

The chosen truncation of the effective average action and the specific Hubbard-Stratonovich channel for the composite tensor are assumed to capture the dominant dynamics without large contributions from omitted operators.

What would settle it

A direct computation of the renormalization-group flow for the auxiliary-field renormalization factor that yields a vanishing or infinite kinetic-term coefficient in the infrared would falsify the claim of dynamical generation.

Figures

Figures reproduced from arXiv: 2512.08435 by Masatoshi Yamada, Yadikaer Maitiniyazi.

Figure 1
Figure 1. Figure 1: Single graviton exchange process in the fermionic (left) and scalar [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

We investigate composite models of gravity and explore how dynamical tensor fields can emerge within the functional renormalization group framework. We consider two prototype models: a fermionic theory and a scalar theory. In both cases, an auxiliary tensor field is introduced via a Hubbard-Stratonovich transformation, corresponding to the composite channel associated with the energy-momentum tensor. We derive the flow equations for the field renormalization factors of the auxiliary tensor field and demonstrate that finite kinetic terms are dynamically generated in the infrared regime. Agreement with the diffeomorphism-invariant quadratic Einstein-Hilbert structure can be established in the transverse-traceless spin-2 sector, while the remaining contributions reside in the longitudinal and trace sectors. Although these terms can be cast into a form reminiscent of gauge-fixed presentations of the quadratic Einstein-Hilbert action, we do not interpret them as originating from a genuine gauge-fixing procedure within the present truncation.

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 paper explores composite models of gravity using the functional renormalization group. For fermionic and scalar prototype theories, an auxiliary tensor field is introduced via Hubbard-Stratonovich transformation in the energy-momentum tensor channel. Flow equations are derived for the field renormalization factors of this auxiliary field, demonstrating dynamical generation of finite kinetic terms in the infrared. The transverse-traceless spin-2 sector matches the quadratic Einstein-Hilbert structure, while longitudinal and trace contributions remain and can be cast in a gauge-fixed-like form without being interpreted as actual gauge fixing.

Significance. If the central claim survives extensions beyond the current truncation, the work would indicate a mechanism for emergent tensor degrees of freedom and gravitational kinetics from matter composites, providing a novel route to diffeomorphism-invariant structures in the IR without a fundamental metric. The explicit matching in the TT spin-2 sector strengthens the result, though the non-invariant sectors highlight the need for further interpretation.

major comments (2)
  1. [Derivation of flow equations and truncation] The truncation of the effective average action to running Z factors for the auxiliary tensor (introduced via HS in the EMT channel) is load-bearing for the claim of IR kinetic-term generation. The flow equations for these Z factors are derived from the Wetterich equation, but without explicit inclusion or analysis of mixing with scalar potentials, higher-derivative operators, or additional tensor structures, it remains unclear whether back-reaction alters the reported approach to finite non-zero IR values. This directly impacts the robustness of the dynamical emergence result.
  2. [IR regime and spin-2 sector analysis] The agreement with the diffeomorphism-invariant quadratic Einstein-Hilbert structure is stated only for the transverse-traceless spin-2 sector. The paper notes that remaining contributions reside in longitudinal and trace sectors and can be recast in a form reminiscent of gauge-fixed quadratic EH, but does not demonstrate that these do not feed back into the Z flows or modify the IR fixed-point behavior under the chosen regulator and projection.
minor comments (2)
  1. [Methods] Clarify the precise form of the regulator and the spin-2 projection used in the flow equations, as these choices can influence the reported IR generation of kinetics.
  2. [Discussion of sectors] The abstract and text distinguish the longitudinal/trace terms from genuine gauge fixing, but a brief comparison table or explicit decomposition would improve readability of how these terms relate to standard EH gauge-fixed presentations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major points below and have revised the text to better articulate the scope of our truncation and the implications of the sector projections.

read point-by-point responses

  1. Referee: [Derivation of flow equations and truncation] The truncation of the effective average action to running Z factors for the auxiliary tensor (introduced via HS in the EMT channel) is load-bearing for the claim of IR kinetic-term generation. The flow equations for these Z factors are derived from the Wetterich equation, but without explicit inclusion or analysis of mixing with scalar potentials, higher-derivative operators, or additional tensor structures, it remains unclear whether back-reaction alters the reported approach to finite non-zero IR values. This directly impacts the robustness of the dynamical emergence result.

    Authors: We agree that restricting the effective average action to the running renormalization factors Z for the auxiliary tensor constitutes a significant truncation. The flow equations are obtained by substituting the ansatz into the Wetterich equation and projecting onto the two-point structures of the auxiliary field. Within this approximation the matter loops generate finite non-vanishing IR values for the kinetic terms. We acknowledge that mixing with scalar potentials or higher-derivative operators is omitted and could quantitatively affect the fixed-point values. In the revised manuscript we have added an explicit discussion of this limitation in Section 3 and the conclusions, together with an outline of how a larger truncation could be constructed in future work. The qualitative emergence result therefore holds strictly inside the present truncation. revision: partial

  2. Referee: [IR regime and spin-2 sector analysis] The agreement with the diffeomorphism-invariant quadratic Einstein-Hilbert structure is stated only for the transverse-traceless spin-2 sector. The paper notes that remaining contributions reside in longitudinal and trace sectors and can be recast in a form reminiscent of gauge-fixed quadratic EH, but does not demonstrate that these do not feed back into the Z flows or modify the IR fixed-point behavior under the chosen regulator and projection.

    Authors: The flow equations are projected onto the transverse-traceless spin-2 sector by means of the standard tensor projectors and a regulator that respects the relevant symmetry properties. This projection renders the longitudinal and trace modes orthogonal to the TT kinetic term, so they do not enter the beta functions for the TT renormalization factors in the current truncation. The remaining sectors are shown to admit a parametrization analogous to a gauge-fixed quadratic Einstein-Hilbert action, but we do not interpret them as arising from an actual gauge-fixing procedure. We have inserted a clarifying paragraph explaining that any feedback from these sectors would require an enlarged truncation that includes their full dynamics; such an extension lies beyond the present scope. The reported IR behavior in the TT sector is therefore stable under the approximations employed. revision: partial

Circularity Check

0 steps flagged

No circularity: FRG flows for auxiliary-field Z factors are computed from standard Wetterich equation in truncation

full rationale

The paper introduces an auxiliary tensor via Hubbard-Stratonovich transformation in the EMT channel, truncates the effective average action to include running field renormalization factors Z for this field, and derives the beta functions for those Z's from the functional renormalization group equation. The claim that finite kinetic terms are generated in the IR follows from integrating the resulting flow equations; this is a dynamical output of the truncated system rather than a redefinition of inputs or a fitted parameter renamed as prediction. No load-bearing self-citation, uniqueness theorem, or ansatz smuggling is present in the derivation chain, and the transverse-traceless sector agreement with quadratic Einstein-Hilbert is an emergent feature of the flow, not imposed by construction. The truncation assumption is stated explicitly but does not create circularity in the reported steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on the standard FRG framework and the assumption that the Hubbard-Stratonovich auxiliary field faithfully represents the composite energy-momentum tensor channel. No new particles or forces are postulated beyond the auxiliary field itself.

axioms (2)
  • standard math The Wetterich equation governs the scale dependence of the effective average action in the chosen truncation.
    Invoked to derive the flow equations for the field renormalization factors.
  • domain assumption The composite channel corresponds to the energy-momentum tensor of the underlying fermionic or scalar theory.
    Used to motivate the Hubbard-Stratonovich transformation for the auxiliary tensor field.
invented entities (1)
  • Auxiliary tensor field no independent evidence
    purpose: To represent the composite operator associated with the energy-momentum tensor.
    Introduced via Hubbard-Stratonovich transformation to facilitate the study of dynamical generation.

pith-pipeline@v0.9.0 · 5449 in / 1396 out tokens · 33765 ms · 2026-05-17T00:24:19.911189+00:00 · methodology

discussion (0)

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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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We introduce an auxiliary tensor field corresponding to a composite field of the energy-momentum tensor by means of the Hubbard-Stratonovich transformation. We derive the flow equations for the field renormalization factors of the auxiliary tensor field and show the tensor field becomes dynamical in the infrared regime.

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

    Relation between the paper passage and the cited Recognition theorem.

    ∂tZHi = k² Ai Nf κψ² / (4π)² with Ai = −7/48, 1/4, −5/24, 7/96; resulting ratios ZH1(0)/ZH0(0) = −12/7 etc. interpreted as gauge-fixed quadratic Einstein-Hilbert

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

45 extracted references · 45 canonical work pages · 16 internal anchors

  1. [1]

    Nambu and G

    Y . Nambu and G. Jona-Lasinio,Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. 1.,Phys. Rev.122(1961) 345

    work page 1961

  2. [2]

    Nambu and G

    Y . Nambu and G. Jona-Lasinio,Dynamical model of elementary particles based on an analogy with superconductivity. II.,Phys. Rev.124(1961) 246

    work page 1961

  3. [3]

    Terazawa, Y

    H. Terazawa, Y . Chikashige, K. Akama and T. Matsuki, Simple Relation Between the Fine Structure and Gravitational Constants,Phys. Rev. D15(1977) 1181

    work page 1977

  4. [4]

    Terazawa, Y

    H. Terazawa, Y . Chikashige, K. Akama and T. Matsuki, Model of the Nambu-Jona-Lasinio Type for Gravity,J. Phys. Soc. Jap.43(1977) 5

    work page 1977

  5. [5]

    Akama, Y

    K. Akama, Y . Chikashige, T. Matsuki and H. Terazawa, Gravity and Electromagnetism as Collective Phenomena: A Derivation of Einstein’s General Relativity,Prog. Theor. Phys.60(1978) 868

    work page 1978

  6. [6]

    Akama, Y

    K. Akama, Y . Chikashige and T. Matsuki,Unified Model of the Nambu-Jona-Lasinio Type for the Gravitational and Electromagnetic Forces,Prog. Theor. Phys.59(1978) 653

    work page 1978

  7. [7]

    Akama,An Attempt at Pregeometry: Gravity With Composite Metric,Prog

    K. Akama,An Attempt at Pregeometry: Gravity With Composite Metric,Prog. Theor. Phys.60(1978) 1900

    work page 1978

  8. [8]

    Terazawa and K

    H. Terazawa and K. Akama,Subquark Pregeometry With Spontaneously Broken Conformal Invariance,Phys. Lett. B97(1980) 81

    work page 1980

  9. [9]

    Akama,PREGEOMETRY INCLUDING FUNDAMENTAL GAUGE BOSONS,Phys

    K. Akama,PREGEOMETRY INCLUDING FUNDAMENTAL GAUGE BOSONS,Phys. Rev. D24 (1981) 3073

    work page 1981

  10. [10]

    Akama,Quantum Mechanical Equivalence of Two Lagrangian Formalisms in ’Scalar Pregeometry’,Prog

    K. Akama,Quantum Mechanical Equivalence of Two Lagrangian Formalisms in ’Scalar Pregeometry’,Prog. Theor. Phys.61(1979) 687. 5

    work page 1979

  11. [11]

    Akama and H

    K. Akama and H. Terazawa,Pregeometric Origin of the Big Bang,Gen. Rel. Grav.15(1983) 201

    work page 1983

  12. [12]

    Akama and I

    K. Akama and I. Oda,Topological pregauge pregeometry, Phys. Lett. B259(1991) 431

    work page 1991

  13. [13]

    Terazawa,Various actions for pregeometry,Prog

    H. Terazawa,Various actions for pregeometry,Prog. Theor. Phys.86(1991) 337

    work page 1991

  14. [14]

    Akama and I

    K. Akama and I. Oda,Chern-Simons pregeometry,Prog. Theor. Phys.89(1993) 215

    work page 1993

  15. [15]

    Akama and I

    K. Akama and I. Oda,BRST quantization of pregeometry and topological pregeometry,Nucl. Phys. B397(1993) 727

    work page 1993

  16. [16]

    Akama and I

    K. Akama and I. Oda,BRST quantization of pregeometry and topological pregeometry,Adv. Appl. Clifford Algebras 4(1994) 259

    work page 1994

  17. [17]

    Gravity from Spinors

    C. Wetterich,Gravity from spinors,Phys. Rev. D70(2004) 105004 [hep-th/0307145]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  18. [18]

    Spinor Gravity

    A. Hebecker and C. Wetterich,Spinor gravity,Phys. Lett. B574(2003) 269 [hep-th/0307109]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  19. [19]

    On the origin of the difference between time and space

    C. Wetterich,Spontaneous symmetry breaking origin for the difference between time and space,Phys. Rev. Lett.94 (2005) 011602 [hep-th/0405223]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  20. [20]

    Wetterich,Pregeometry and euclidean quantum gravity, Nucl

    C. Wetterich,Pregeometry and euclidean quantum gravity, Nucl. Phys. B971(2021) 115526 [2101.07849]

    work page arXiv 2021

  21. [21]

    Wetterich,Cosmology from pregeometry,Phys

    C. Wetterich,Cosmology from pregeometry,Phys. Rev. D 104(2021) 104040 [2104.14013]

    work page arXiv 2021

  22. [22]

    Wetterich,Pregeometry and spontaneous time-space asymmetry,JHEP06(2022) 069 [2101.11519]

    C. Wetterich,Pregeometry and spontaneous time-space asymmetry,JHEP06(2022) 069 [2101.11519]

    work page arXiv 2022

  23. [23]

    Floreanini and R

    R. Floreanini and R. Percacci,Topological pregeometry, Mod. Phys. Lett. A5(1990) 2247

    work page 1990

  24. [24]

    Matsuzaki, S

    S. Matsuzaki, S. Miyawaki, K.-y. Oda and M. Yamada, Dynamically emergent gravity from hidden local Lorentz symmetry,Phys. Lett. B813(2021) 135975 [2003.07126]

    work page arXiv 2021

  25. [25]

    G. E. V olovik,Gravity from Symmetry Breaking Phase Transition,J. Low Temp. Phys.207(2022) 127 [2111.07817]

    work page arXiv 2022

  26. [26]

    Maitiniyazi, S

    Y . Maitiniyazi, S. Matsuzaki, K.-y. Oda and M. Yamada, Irreversible vierbein postulate: Emergence of spacetime from quantum phase transition,Phys. Rev. D109(2024) 106018 [2309.16230]

    work page arXiv 2024

  27. [27]

    Maitiniyazi, S

    Y . Maitiniyazi, S. Matsuzaki, K.-y. Oda and M. Yamada, Spacetime and Planck mass generation from scale-invariant degenerate gravity,Phys. Rev. D111 (2025) 046002 [2411.17238]

    work page arXiv 2025

  28. [28]

    Oda,Emergence of General Relativity from Cosmological Constant Via Ghost Condensation, 2509.23648

    I. Oda,Emergence of General Relativity from Cosmological Constant Via Ghost Condensation, 2509.23648

    work page arXiv

  29. [29]

    Huang, M

    L. Huang, M. Kawaguchi, Y . Maitiniyazi, S. Matsuzaki, A. Tomiya and M. Yamada,Functional renormalization group study of a four-fermion model with CP violation, Phys. Rev. D111(2025) 074015 [2411.07027]

    work page arXiv 2025

  30. [30]

    Exact evolution equation for the effective potential

    C. Wetterich,Exact evolution equation for the effective potential,Phys. Lett.B301(1993) 90 [1710.05815]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  31. [31]

    Flow Equations for N Point Functions and Bound States

    U. Ellwanger,FLow equations for N point functions and bound states,Z. Phys. C62(1994) 503 [hep-ph/9308260]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  32. [32]

    T. R. Morris,The Exact renormalization group and approximate solutions,Int. J. Mod. Phys. A9(1994) 2411 [hep-ph/9308265]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  33. [33]

    D. F. Litim,Optimized renormalization group flows,Phys. Rev. D64(2001) 105007 [hep-th/0103195]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  34. [34]

    Renormalization Flow of Bound States

    H. Gies and C. Wetterich,Renormalization flow of bound states,Phys. Rev. D65(2002) 065001 [hep-th/0107221]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  35. [35]

    Universality of spontaneous chiral symmetry breaking in gauge theories

    H. Gies and C. Wetterich,Universality of spontaneous chiral symmetry breaking in gauge theories,Phys. Rev. D 69(2004) 025001 [hep-th/0209183]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  36. [36]

    J. M. Pawlowski,Aspects of the functional renormalisation group,Annals Phys.322(2007) 2831 [hep-th/0512261]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  37. [37]

    Exact flow equation for composite operators

    S. Floerchinger and C. Wetterich,Exact flow equation for composite operators,Phys. Lett. B680(2009) 371 [0905.0915]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  38. [38]

    From Quarks and Gluons to Hadrons: Chiral Symmetry Breaking in Dynamical QCD

    J. Braun, L. Fister, J. M. Pawlowski and F. Rennecke, From Quarks and Gluons to Hadrons: Chiral Symmetry Breaking in Dynamical QCD,Phys. Rev. D94(2016) 034016 [1412.1045]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  39. [39]

    Chiral symmetry breaking in continuum QCD

    M. Mitter, J. M. Pawlowski and N. Strodthoff,Chiral symmetry breaking in continuum QCD,Phys. Rev. D91 (2015) 054035 [1411.7978]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  40. [40]

    A. K. Cyrol, M. Mitter, J. M. Pawlowski and N. Strodthoff, Nonperturbative finite-temperature Yang-Mills theory, Phys. Rev. D97(2018) 054015 [1708.03482]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  41. [41]

    A. K. Cyrol, M. Mitter, J. M. Pawlowski and N. Strodthoff, Nonperturbative quark, gluon, and meson correlators of unquenched QCD,Phys. Rev. D97(2018) 054006 [1706.06326]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  42. [42]

    Bound state properties from the Functional Renormalisation Group

    R. Alkofer, A. Maas, W. A. Mian, M. Mitter, J. París-López, J. M. Pawlowski et al.,Bound state properties from the functional renormalization group, Phys. Rev. D99(2019) 054029 [1810.07955]. 6

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  43. [43]

    T. Denz, M. Mitter, J. M. Pawlowski, C. Wetterich and M. Yamada,Partial bosonization for the two-dimensional Hubbard model,Phys. Rev. B101(2020) 155115 [1910.08300]

    work page arXiv 2020

  44. [44]

    W.-j. Fu, J. M. Pawlowski and F. Rennecke,QCD phase structure at finite temperature and density,Phys. Rev. D 101(2020) 054032 [1909.02991]

    work page arXiv 2020

  45. [45]

    Goertz, Á

    F. Goertz, Á. Pastor-Gutiérrez and J. M. Pawlowski, Gauge-fermion cartography: From confinement and chiral symmetry breaking to conformality,Phys. Rev. D112 (2025) 034029 [2412.12254]. 7

    work page arXiv 2025