pith. sign in

arxiv: 2606.25154 · v1 · pith:J74347SHnew · submitted 2026-06-23 · ✦ hep-th · gr-qc

Diffeomorphic Scalar Duality

Claudia de Rham , Andrew J. Tolley This is my paper

Pith reviewed 2026-06-25 21:51 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords scalar effective field theorydualityfield-dependent diffeomorphismS-matrix equivalenceEinstein-Cartan formalismtorsionsecond-order equations of motion
0
0 comments X

The pith

Every local scalar effective field theory admits a duality to an infinite class of theories with distinct Lagrangians but identical S-matrix.

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

The paper establishes that any local scalar effective field theory can be mapped to many others through a field-dependent diffeomorphism. This map produces a different Lagrangian yet leaves all scattering amplitudes unchanged, so the two theories are physically equivalent. The transformation is not a local field redefinition and therefore evades the usual equivalence criteria. When the theories are coupled to gravity the duality survives only if the Einstein-Cartan formulation is used, in which the vielbein transforms while the spin connection stays fixed.

Core claim

Every local scalar effective field theory admits a new kind of duality to an infinite class of local scalar field theories with distinct Lagrangians. The duality map takes the form of a field-dependent diffeomorphism, and cannot be obtained via purely local field redefinitions, nevertheless the dual theory has an identical S-matrix. The subset of interactions that maintain second-order equations of motion is non-trivially mapped into themselves under this transformation. When coupled to gravity in the Einstein-Cartan formalism the duality maps torsion-free configurations to configurations with non-zero torsion.

What carries the argument

The field-dependent diffeomorphism that implements the duality between scalar Lagrangians while preserving the S-matrix.

If this is right

  • Interactions that produce second-order equations of motion are mapped among themselves.
  • Generic scalar theories can be coupled to gravity while preserving the duality when the Einstein-Cartan formalism is adopted.
  • Torsion-free gravitational configurations are mapped to dual configurations that carry non-zero torsion.
  • A family of first-order gravitational theories is closed under the duality.
  • In the weak-gravity limit the duality closes on scalar theories that are kinetically mixed with the graviton.

Where Pith is reading between the lines

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

  • The existence of this duality enlarges the space of apparently distinct scalar effective theories that are actually equivalent at the level of observables.
  • The forced appearance of torsion suggests that torsionful geometries may be required to realize all dual descriptions when gravity is dynamical.
  • One could search for an explicit closed-form expression for the duality map in simple polynomial theories and check whether it generates new integrable models.

Load-bearing premise

The duality can be extended to gravity only by transforming the vielbein while holding the spin connection fixed, generically producing torsion in the dual theory.

What would settle it

Explicitly construct the dual Lagrangian for a concrete scalar interaction such as a cubic term, then compute its tree-level four-point scattering amplitude and verify whether it agrees with the original amplitude.

read the original abstract

We show that every local scalar effective field theory admits a new kind of duality to an infinite class of local scalar field theories with distinct Lagrangians. The duality map takes the form of a field-dependent diffeomorphism, and cannot be obtained via purely local field redefinitions, nevertheless the dual theory has an identical $S$-matrix. The subset of interactions that maintain second-order equations of motion is non-trivially mapped into themselves under this transformation. We show how to couple generic scalar field theories to gravity in a way that preserves the duality. Crucially, this requires working in the Einstein-Cartan formalism, with the vielbein and spin connection treated as independent variables. When coupling to massless gravity, the duality is interpreted as a local field redefinition in which the vielbein transforms while the spin connection is held fixed; consequently, a torsion-free configuration is generically mapped to a dual configuration with non-zero torsion. We specify the general family of first-order gravitational theories that map into themselves under the duality. In the weak gravitational field limit, these reduce to scalar theories kinetically mixed with the graviton, which themselves form a family closed under the duality.

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

0 major / 4 minor

Summary. The manuscript claims that every local scalar effective field theory admits a duality, realized by a field-dependent diffeomorphism, to an infinite family of distinct local scalar theories that share an identical S-matrix. The map cannot be reproduced by ordinary local field redefinitions. The subclass of interactions yielding second-order equations of motion is closed under the duality. The construction is extended to gravity by coupling in the Einstein-Cartan formalism, with the vielbein transforming while the spin connection remains fixed; this maps torsion-free configurations to ones with torsion. A general family of first-order gravitational theories is identified that is closed under the duality, reducing in the weak-field limit to kinetically mixed scalar-graviton theories that are likewise closed.

Significance. If the central construction holds, the result supplies a new, non-local-redefinition duality for scalar EFTs that preserves the S-matrix by design and is closed on the second-order-EOM sector. The explicit Einstein-Cartan coupling rule and the identification of self-dual gravitational families constitute concrete, falsifiable extensions. These features would be of interest to the EFT and gravitational effective-field-theory communities.

minor comments (4)
  1. The abstract states that the duality 'cannot be obtained via purely local field redefinitions'; the manuscript should supply an explicit, low-order example (e.g., a cubic or quartic interaction) demonstrating that the generated map lies outside the equivalence class of local redefinitions.
  2. Section 3 (or the relevant section deriving the S-matrix equality) should include a direct verification that the on-shell scattering amplitudes agree at least through one-loop order for a non-trivial interaction, rather than relying solely on the formal diffeomorphism argument.
  3. The statement that the second-order-EOM sector is 'non-trivially mapped into itself' would benefit from an explicit counter-example showing that a generic higher-derivative term is sent outside the sector, to make the closure claim sharper.
  4. Notation for the field-dependent diffeomorphism (presumably introduced around Eq. (X)) should be made uniform between the pure-scalar and gravity-coupled sections to avoid reader confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs an explicit duality map realized as a field-dependent diffeomorphism that sends any local scalar EFT to a distinct local scalar theory while preserving the S-matrix by the properties of the map. The subset of second-order interactions is shown to be closed under the transformation, and the gravity coupling is defined directly in Einstein-Cartan variables with the vielbein transforming and the spin connection fixed. No step in the provided abstract or description reduces a claimed result to a fitted parameter, a self-citation chain, or an ansatz smuggled from prior work; the central statements are direct consequences of the stated transformation rules rather than re-labelings or statistical fits. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5726 in / 1058 out tokens · 28732 ms · 2026-06-25T21:51:20.651537+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

76 extracted references · 54 linked inside Pith

  1. [1]

    R. G. Leigh,Dirac-Born-Infeld Action from Dirichlet Sigma Model,Mod. Phys. Lett. A4 (1989) 2767

    1989

  2. [2]

    D. T. Son,Low-energy quantum effective action for relativistic superfluids,hep-ph/0204199

    Pith/arXiv arXiv

  3. [3]

    Endlich, A

    S. Endlich, A. Nicolis, R. Rattazzi and J. Wang,The Quantum mechanics of perfect fluids, JHEP04(2011) 102, [1011.6396]

    Pith/arXiv arXiv 2011

  4. [4]

    Dubovsky, L

    S. Dubovsky, L. Hui, A. Nicolis and D. T. Son,Effective field theory for hydrodynamics: thermodynamics, and the derivative expansion,Phys. Rev. D85(2012) 085029, [1107.0731]

    Pith/arXiv arXiv 2012

  5. [5]

    Hull,Duality Symmetries,2504.09672

    C. Hull,Duality Symmetries,2504.09672

    arXiv

  6. [6]

    Hull,A New Action for Superstring Field Theory,2508.03902

    C. Hull,A New Action for Superstring Field Theory,2508.03902

    arXiv

  7. [7]

    de Rham, M

    C. de Rham, M. Fasiello and A. J. Tolley,Galileon Duality,Phys. Lett. B733(2014) 46–51, [1308.2702]

    Pith/arXiv arXiv 2014

  8. [9]

    T. L. Curtright and D. B. Fairlie,A galileon primer,arXiv:1212.6972

    arXiv

  9. [10]

    Chagoya and G

    J. Chagoya and G. Tasinato,A geometrical approach to degenerate scalar-tensor theories,JHEP 02(2017) 113, [arXiv:1610.07980]

    Pith/arXiv arXiv 2017

  10. [11]

    Baratella, P

    P. Baratella, P. Creminelli, M. Serone and G. Trevisan,Inequivalence of Coset Constructions for Spacetime Symmetries: Coupling with Gravity,Phys. Rev. D93(2016) 045029, [1510.01969]. – 53 –

    Pith/arXiv arXiv 2016

  11. [12]

    Fasiello and A

    M. Fasiello and A. J. Tolley,Cosmological stability bound in massive gravity and bigravity, JCAP1312(2013) 002, [arXiv:1308.1647]

    Pith/arXiv arXiv 2013

  12. [13]

    Kamefuchi, L

    S. Kamefuchi, L. O’Raifeartaigh and A. Salam,Change of variables and equivalence theorems in quantum field theories,Nucl. Phys.28(1969) 529

    1969

  13. [14]

    J. S. R. Chisholm,Change of variables in quantum field theories,Nucl. Phys.26(1961) 469

    1961

  14. [15]

    Salam and J

    A. Salam and J. A. Strathdee,Equivalent formulations of massive vector field theories,Phys. Rev. D2(1970) 2869

    1970

  15. [16]

    H. J. Borchers, ¨Uber die mannigfaltigkeit der interpolierenden felder zu einer kausalen s-matrix, Il Nuovo Cimento15(1960) 784–794

    1960

  16. [17]

    Nicolis, R

    A. Nicolis, R. Rattazzi and E. Trincherini,The galileon as a local modification of gravity,Phys. Rev. D79(2009) 064036

    2009

  17. [18]

    G. R. Dvali, G. Gabadadze and M. Porrati,4-D gravity on a brane in 5-D Minkowski space, Phys. Lett. B485(2000) 208–214, [hep-th/0005016]

    Pith/arXiv arXiv 2000

  18. [19]

    G. R. Dvali, G. Gabadadze and M. Porrati,Metastable gravitons and infinite volume extra dimensions,Phys. Lett. B484(2000) 112–118, [hep-th/0002190]

    Pith/arXiv arXiv 2000

  19. [20]

    M. A. Luty, M. Porrati and R. Rattazzi,Strong interactions and stability in the DGP model, JHEP09(2003) 029, [hep-th/0303116]

    Pith/arXiv arXiv 2003

  20. [21]

    de Rham and G

    C. de Rham and G. Gabadadze,Generalization of the fierz-pauli action,Phys. Rev. D82 (2010) 044020

    2010

  21. [22]

    de Rham, G

    C. de Rham, G. Dvali, S. Hofmann, J. Khoury, O. Pujolas, M. Redi et al.,Cascading gravity: Extending the Dvali-Gabadadze-Porrati model to higher dimension,Phys. Rev. Lett.100(2008) 251603, [0711.2072]

    Pith/arXiv arXiv 2008

  22. [23]

    de Rham and G

    C. de Rham and G. Gabadadze,Selftuned Massive Spin-2,Phys. Lett.B693(2010) 334–338, [1006.4367]

    Pith/arXiv arXiv 2010

  23. [24]

    de Rham, G

    C. de Rham, G. Gabadadze, D. Pirtskhalava, A. J. Tolley and I. Yavin,Nonlinear Dynamics of 3D Massive Gravity,JHEP06(2011) 028, [1103.1351]

    Pith/arXiv arXiv 2011

  24. [25]

    de Rham, G

    C. de Rham, G. Gabadadze and A. J. Tolley,Resummation of massive gravity,Phys. Rev. Lett. 106(2011) 231101

    2011

  25. [26]

    de Rham, G

    C. de Rham, G. Gabadadze, L. Heisenberg and D. Pirtskhalava,Cosmic Acceleration and the Helicity-0 Graviton,Phys. Rev. D83(2011) 103516, [1010.1780]

    Pith/arXiv arXiv 2011

  26. [27]

    de Rham, G

    C. de Rham, G. Gabadadze and A. J. Tolley,Helicity Decomposition of Ghost-free Massive Gravity,JHEP11(2011) 093, [1108.4521]

    Pith/arXiv arXiv 2011

  27. [28]

    N. A. Ondo and A. J. Tolley,Complete Decoupling Limit of Ghost-free Massive Gravity,JHEP 11(2013) 059, [1307.4769]

    Pith/arXiv arXiv 2013

  28. [29]

    de Rham and A

    C. de Rham and A. J. Tolley,The Encyclopedia of Cosmology: Volume 1: Modified Gravity,

  29. [30]

    de Rham and A

    C. de Rham and A. J. Tolley,DBI and the Galileon reunited,JCAP05(2010) 015, [1003.5917]

    Pith/arXiv arXiv 2010

  30. [31]

    Hinterbichler, M

    K. Hinterbichler, M. Trodden and D. Wesley,Multi-field galileons and higher co-dimension branes,Phys. Rev. D82(2010) 124018, [1008.1305]. – 54 –

    Pith/arXiv arXiv 2010

  31. [32]

    Trodden and K

    M. Trodden and K. Hinterbichler,Generalizing Galileons,Class. Quant. Grav.28(2011) 204003, [1104.2088]

    Pith/arXiv arXiv 2011

  32. [33]

    Garoffolo, K

    A. Garoffolo, K. Hinterbichler and M. Trodden,Multi-Galileons in curved space,JHEP09 (2025) 115, [2505.08865]

    arXiv 2025

  33. [34]

    G. Goon, K. Hinterbichler, A. Joyce and M. Trodden,Galileons as Wess-Zumino Terms,JHEP 06(2012) 004, [1203.3191]

    Pith/arXiv arXiv 2012

  34. [35]

    Burrage, C

    C. Burrage, C. de Rham and L. Heisenberg,de Sitter Galileon,JCAP05(2011) 025, [1104.0155]

    Pith/arXiv arXiv 2011

  35. [36]

    Deffayet, G

    C. Deffayet, G. Esposito-Farese and A. Vikman,Covariant Galileon,Phys. Rev.D79(2009) 084003, [0901.1314]

    Pith/arXiv arXiv 2009

  36. [37]

    Gabadadze, K

    G. Gabadadze, K. Hinterbichler, J. Khoury, D. Pirtskhalava and M. Trodden,A Covariant Master Theory for Novel Galilean Invariant Models and Massive Gravity,Phys. Rev. D86 (2012) 124004, [1208.5773]

    Pith/arXiv arXiv 2012

  37. [38]

    Andrews, G

    M. Andrews, G. Goon, K. Hinterbichler, J. Stokes and M. Trodden,Massive Gravity Coupled to Galileons is Ghost-Free,Phys. Rev. Lett.111(2013) 061107, [1303.1177]

    Pith/arXiv arXiv 2013

  38. [39]

    G. Goon, A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, K. Hinterbichler, S. Mukohyama and M. Trodden,Galileons Coupled to Massive Gravity: General Analysis and Cosmological Solutions,JCAP08(2014) 008, [1402.5424]

    Pith/arXiv arXiv 2014

  39. [40]

    Nicolis and R

    A. Nicolis and R. Rattazzi,Classical and quantum consistency of the DGP model,JHEP06 (2004) 059, [hep-th/0404159]

    Pith/arXiv arXiv 2004

  40. [41]

    de Rham, G

    C. de Rham, G. Gabadadze, L. Heisenberg and D. Pirtskhalava,Nonrenormalization and naturalness in a class of scalar-tensor theories,Phys. Rev.D87(2013) 085017, [1212.4128]

    Pith/arXiv arXiv 2013

  41. [42]

    de Rham, L

    C. de Rham, L. Heisenberg and R. H. Ribeiro,Quantum Corrections in Massive Gravity,Phys. Rev. D88(2013) 084058, [1307.7169]

    Pith/arXiv arXiv 2013

  42. [43]

    G. Goon, K. Hinterbichler, A. Joyce and M. Trodden,Aspects of Galileon Non-Renormalization,JHEP11(2016) 100, [1606.02295]

    Pith/arXiv arXiv 2016

  43. [44]

    A. I. Vainshtein,To the problem of nonvanishing gravitation mass,Phys. Lett. B39(1972) 393–394

    1972

  44. [45]

    Babichev and C

    E. Babichev and C. Deffayet,An introduction to the Vainshtein mechanism,Class. Quant. Grav.30(2013) 184001, [1304.7240]

    Pith/arXiv arXiv 2013

  45. [46]

    de Rham,Massive gravity,arXiv:1401.4173

    C. de Rham,Massive gravity,arXiv:1401.4173

    Pith/arXiv arXiv

  46. [47]

    Hinterbichler and A

    K. Hinterbichler and A. Joyce,Hidden symmetry of the Galileon,Phys. Rev. D92(2015) 023503, [1501.07600]

    Pith/arXiv arXiv 2015

  47. [48]

    Cachazo, S

    F. Cachazo, S. He and E. Y. Yuan,Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM,JHEP07(2015) 149, [1412.3479]

    Pith/arXiv arXiv 2015

  48. [49]

    Cheung, K

    C. Cheung, K. Kampf, J. Novotny and J. Trnka,Effective Field Theories from Soft Limits of Scattering Amplitudes,Phys. Rev. Lett.114(2015) 221602, [1412.4095]

    Pith/arXiv arXiv 2015

  49. [50]

    J. J. M. Carrasco and L. Rodina,UV considerations on scattering amplitudes in a web of theories,Phys. Rev. D100(2019) 125007, [1908.08033]. – 55 –

    arXiv 2019

  50. [51]

    Burrage, C

    C. Burrage, C. de Rham, D. Seery and A. J. Tolley,Galileon inflation,JCAP1101(2011) 014, [1009.2497]

    Pith/arXiv arXiv 2011

  51. [52]

    A. J. Tolley, Z.-Y. Wang and S.-Y. Zhou,New positivity bounds from full crossing symmetry, JHEP05(2021) 255, [2011.02400]

    arXiv 2021

  52. [53]

    Creminelli, M

    P. Creminelli, M. Serone, G. Trevisan and E. Trincherini,Inequivalence of Coset Constructions for Spacetime Symmetries,JHEP02(2015) 037, [1403.3095]

    Pith/arXiv arXiv 2015

  53. [54]

    Kampf and J

    K. Kampf and J. Novotny,Unification of Galileon Dualities,JHEP10(2014) 006, [1403.6813]

    Pith/arXiv arXiv 2014

  54. [55]

    Epstein,On the borchers class of a free field,Il Nuovo Cimento27(1963) 886–893

    H. Epstein,On the borchers class of a free field,Il Nuovo Cimento27(1963) 886–893

    1963

  55. [56]

    A. R. Solomon and M. Trodden,Higher-derivative operators and effective field theory for general scalar-tensor theories,JCAP02(2018) 031, [1709.09695]

    Pith/arXiv arXiv 2018

  56. [57]

    Deffayet, X

    C. Deffayet, X. Gao, D. A. Steer and G. Zahariade,From k-essence to generalised Galileons, Phys. Rev.D84(2011) 064039, [1103.3260]

    Pith/arXiv arXiv 2011

  57. [58]

    V. N. Gribov,Quantization of Nonabelian Gauge Theories,Nucl. Phys. B139(1978) 1

    1978

  58. [59]

    Dudal, S

    D. Dudal, S. P. Sorella, N. Vandersickel and H. Verschelde,Gribov no-pole condition, Zwanziger horizon function, Kugo-Ojima confinement criterion, boundary conditions, BRST breaking and all that,Phys. Rev. D79(2009) 121701, [0904.0641]

    Pith/arXiv arXiv 2009

  59. [60]

    de Rham, L

    C. de Rham, L. Keltner and A. J. Tolley,Generalized galileon duality,Phys. Rev. D90(2014) 024050, [1403.3690]

    Pith/arXiv arXiv 2014

  60. [61]

    G. L. Goon, K. Hinterbichler and M. Trodden,Symmetries for galileons and dbi scalars on curved space,JCAP07(2011) 017, [arXiv:1103.5745]

    Pith/arXiv arXiv 2011

  61. [62]

    de Rham and A

    C. de Rham and A. J. Tolley,Vielbein to the rescue? Breaking the symmetric vielbein condition in massive gravity and multigravity,Phys. Rev. D92(2015) 024024, [1505.01450]

    Pith/arXiv arXiv 2015

  62. [63]

    Lovelock,The Einstein tensor and its generalizations,J

    D. Lovelock,The Einstein tensor and its generalizations,J. Math. Phys.12(1971) 498–501

    1971

  63. [64]

    Lovelock,The four-dimensionality of space and the einstein tensor,J

    D. Lovelock,The four-dimensionality of space and the einstein tensor,J. Math. Phys.13(1972) 874–876

    1972

  64. [65]

    Deffayet, S

    C. Deffayet, S. Deser and G. Esposito-Farese,Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress-tensors,Phys. Rev.D80(2009) 064015, [0906.1967]

    Pith/arXiv arXiv 2009

  65. [66]

    G. W. Horndeski,Second-order scalar-tensor field equations in a four-dimensional space,Int. J. Theor. Phys.10(1974) 363–384

    1974

  66. [67]

    Gubitosi, F

    G. Gubitosi, F. Piazza and F. Vernizzi,The Effective Field Theory of Dark Energy,JCAP02 (2013) 032, [1210.0201]

    Pith/arXiv arXiv 2013

  67. [68]

    Zumalacarregui and J

    M. Zumalacarregui and J. Garcia-Bellido,Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian,Phys. Rev.D89 (2014) 064046, [1308.4685]

    Pith/arXiv arXiv 2014

  68. [69]

    Deffayet and D

    C. Deffayet and D. A. Steer,A formal introduction to Horndeski and Galileon theories and their generalizations,Class. Quant. Grav.30(2013) 214006, [1307.2450]

    Pith/arXiv arXiv 2013

  69. [70]

    Armendariz-Picon, T

    C. Armendariz-Picon, T. Damour and V. F. Mukhanov,k - inflation,Phys. Lett. B458(1999) 209–218, [hep-th/9904075]. – 56 –

    Pith/arXiv arXiv 1999

  70. [71]

    Armendariz-Picon, V

    C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt,Essentials of k essence,Phys. Rev. D63(2001) 103510, [astro-ph/0006373]

    Pith/arXiv arXiv 2001

  71. [72]

    Babichev, C

    E. Babichev, C. Deffayet and R. Ziour,k-Mouflage gravity,Int. J. Mod. Phys. D18(2009) 2147–2154, [0905.2943]

    Pith/arXiv arXiv 2009

  72. [73]

    Keltner and A

    L. Keltner and A. J. Tolley,UV properties of Galileons: Spectral Densities,1502.05706

    Pith/arXiv arXiv

  73. [74]

    Zwanziger,Local and Renormalizable Action From the Gribov Horizon,Nucl

    D. Zwanziger,Local and Renormalizable Action From the Gribov Horizon,Nucl. Phys. B323 (1989) 513–544

    1989

  74. [75]

    Vandersickel and D

    N. Vandersickel and D. Zwanziger,The Gribov problem and QCD dynamics,Phys. Rept.520 (2012) 175–251, [1202.1491]

    Pith/arXiv arXiv 2012

  75. [76]

    F. G. Scholtz and S. V. Shabanov,Gribov versus BRST,Annals Phys.263(1998) 119–132, [hep-th/9701051]

    Pith/arXiv arXiv 1998

  76. [77]

    Rogers,Gauge fixing and BFV quantization,Class

    A. Rogers,Gauge fixing and BFV quantization,Class. Quant. Grav.17(2000) 389–397, [hep-th/9902133]. – 57 –

    Pith/arXiv arXiv 2000