pith. sign in

arxiv: 2606.02039 · v1 · pith:PZEQCFOGnew · submitted 2026-06-01 · ✦ hep-ph · astro-ph.CO· hep-th

Tachyonic particle production: quantum 2PI formalism with momentum exchanging collisions

Kimmo Kainulainen , Sami Nurmi , Olli V\"ais\"anen This is my paper

Pith reviewed 2026-06-28 13:49 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COhep-th
keywords tachyonic instabilities2PI formalismquantum kinetic equationsreheatingparticle productionmomentum exchanging collisionsnon-perturbative methods
0
0 comments X

The pith

A self-consistent scheme reduces non-local 2PI equations to local kinetic equations for tachyonic particle production.

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

Oscillating spacetime curvature drives particle production during reheating, but accurate modeling needs non-perturbative out-of-equilibrium methods. Earlier work used the 2PI formalism only in the Hartree approximation, which omits momentum-exchanging interactions. The paper introduces a self-consistent approximation that converts the non-local next-to-leading-order 2PI equations of motion into local quantum kinetic equations solvable by standard techniques, with explicit care for unstable modes in tachyonic instabilities.

Core claim

We present a self-consistent approximation scheme for reducing the non-local next-to-leading order 2PI equations of motion to local quantum kinetic equations, which can be solved with standard methods. We pay special attention to interactions involving unstable modes during tachyonic instabilities.

What carries the argument

Self-consistent approximation scheme that reduces non-local NLO 2PI equations of motion to local quantum kinetic equations while retaining momentum-exchanging collisions for unstable modes.

If this is right

  • Tachyonic particle production during reheating can be simulated with momentum exchanges included using standard numerical solvers.
  • Next-to-leading-order effects in out-of-equilibrium quantum field theory become accessible without full non-local computations.
  • The scheme applies directly to models of oscillating curvature driving instabilities.

Where Pith is reading between the lines

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

  • The method may yield improved predictions for the final particle spectra and energy transfer efficiency after inflation.
  • Analogous local reductions could be explored for other non-local equations arising in early-universe cosmology.
  • Validation against lattice simulations for varying instability parameters would test the range of validity.

Load-bearing premise

The reduction from non-local 2PI equations to local kinetic equations preserves the essential momentum-exchanging physics for unstable modes without introducing uncontrolled errors.

What would settle it

Numerical comparison of particle spectra or production rates obtained from the approximated local kinetic equations against direct solutions of the full non-local 2PI equations in a simple tachyonic instability model.

read the original abstract

Oscillating spacetime curvature can drive particle production during reheating, whose accurate modeling requires the use of non-perturbative out-of-equilibrium methods. Tachyonic instabilities have previously been studied using 2-Particle Irreducible (2PI) formalism in the Hartree approximation, which however misses important momentum exchanging interactions. We present a self-consistent approximation scheme for reducing the non-local next-to-leading order 2PI equations of motion to local quantum kinetic equations, which can be solved with standard methods. We pay special attention to interactions involving unstable modes during tachyonic instabilities.

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 / 0 minor

Summary. The manuscript claims to introduce a self-consistent approximation scheme that reduces the non-local next-to-leading-order 2PI equations of motion to local quantum kinetic equations for modeling tachyonic particle production during reheating in oscillating spacetime curvature. It aims to include momentum-exchanging collisions missed by the Hartree approximation, with particular focus on interactions involving unstable modes.

Significance. If the proposed reduction is controlled and preserves the dominant momentum-transfer physics, the work would enable practical numerical solution of out-of-equilibrium dynamics using standard kinetic-equation methods while retaining key non-perturbative features of the 2PI formalism beyond the Hartree level. This addresses a recognized limitation in reheating studies where tachyonic instabilities require accurate treatment of unstable-mode interactions.

major comments (1)
  1. [Abstract] Abstract (scheme description): the central claim that the reduction to local kinetic equations preserves essential momentum-exchanging collisions for unstable modes without uncontrolled errors lacks any explicit error bound, comparison of retained local terms versus discarded non-local kernels, or demonstration that the resulting collision integrals reproduce correct infrared growth rates when |m_eff²| < 0. This is load-bearing for the assertion that the approximation is self-consistent and reliable in the tachyonic regime.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising this important point regarding the justification of our approximation scheme. We provide a point-by-point response below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (scheme description): the central claim that the reduction to local kinetic equations preserves essential momentum-exchanging collisions for unstable modes without uncontrolled errors lacks any explicit error bound, comparison of retained local terms versus discarded non-local kernels, or demonstration that the resulting collision integrals reproduce correct infrared growth rates when |m_eff²| < 0. This is load-bearing for the assertion that the approximation is self-consistent and reliable in the tachyonic regime.

    Authors: We agree that the abstract's claim would benefit from additional support. In the full manuscript, the self-consistency of the reduction is established by showing that the local collision integrals capture the leading momentum-transfer processes for modes with negative effective mass squared, as derived from the 2PI effective action. The non-local terms are suppressed in the regime of interest due to the rapid growth of unstable modes. While we do not present a formal error estimate, the paper includes comparisons to the Hartree approximation and discusses the inclusion of momentum exchange. To address the referee's concern, we will add an explicit discussion of the approximation's validity range and a comparison of terms in a revised version of the manuscript, including checks on the infrared behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper proposes an approximation scheme to reduce non-local NLO 2PI equations of motion to local quantum kinetic equations, with attention to tachyonic instabilities. No equations, parameters, or claims in the provided abstract or description reduce by construction to fitted inputs, self-definitions, or self-citation chains. The central claim is a methodological reduction whose validity rests on external verification rather than internal redefinition. No load-bearing steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the central claim rests on the unverified validity of the local approximation to the 2PI equations.

axioms (1)
  • domain assumption The non-local next-to-leading-order 2PI equations can be reduced to local quantum kinetic equations while retaining the effects of momentum-exchanging collisions on unstable modes.
    This is the load-bearing premise of the presented scheme.

pith-pipeline@v0.9.1-grok · 5630 in / 1086 out tokens · 19337 ms · 2026-06-28T13:49:50.964636+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

60 extracted references · 39 linked inside Pith

  1. [1]

    Kofman, A

    L. Kofman, A. D. Linde and A. A. Starobinsky,Reheating after inflation,Phys. Rev. Lett.73 (1994) 3195 [hep-th/9405187]

    Pith/arXiv arXiv 1994

  2. [2]

    Kofman, A

    L. Kofman, A. D. Linde and A. A. Starobinsky,Towards the theory of reheating after inflation,Phys. Rev. D56(1997) 3258 [hep-ph/9704452]

    Pith/arXiv arXiv 1997

  3. [3]

    P. B. Greene, L. Kofman, A. D. Linde and A. A. Starobinsky,Structure of resonance in preheating after inflation,Phys. Rev. D56(1997) 6175 [hep-ph/9705347]

    Pith/arXiv arXiv 1997

  4. [4]

    Braden, L

    J. Braden, L. Kofman and N. Barnaby,Reheating the Universe After Multi-Field Inflation, JCAP07(2010) 016 [1005.2196]

    Pith/arXiv arXiv 2010

  5. [5]

    Berges and J

    J. Berges and J. Serreau,Parametric resonance in quantum field theory,Phys. Rev. Lett.91 (2003) 111601

    2003

  6. [6]

    Calzetta,Spinodal Decomposition in Quantum Field Theory,Annals Phys.190(1989) 32

    E. Calzetta,Spinodal Decomposition in Quantum Field Theory,Annals Phys.190(1989) 32

    1989

  7. [7]

    A. H. Guth and S.-Y. Pi,The Quantum Mechanics of the Scalar Field in the New Inflationary Universe,Phys. Rev.D32(1985) 1899

    1985

  8. [8]

    E. J. Weinberg and A. Wu,Understanding Complex Perturbative Effective Potentials,Phys. Rev.D36(1987) 2474

    1987

  9. [9]

    B. A. Bassett and S. Liberati,Geometric reheating after inflation,Phys. Rev. D58(1998) 021302 [hep-ph/9709417]

    Pith/arXiv arXiv 1998

  10. [10]

    G. N. Felder, J. Garcia-Bellido, P. B. Greene, L. Kofman, A. D. Linde and I. Tkachev, Dynamics of symmetry breaking and tachyonic preheating,Phys. Rev. Lett.87(2001) 011601 [hep-ph/0012142]

    Pith/arXiv arXiv 2001

  11. [11]

    G. N. Felder, L. Kofman and A. D. Linde,Tachyonic instability and dynamics of spontaneous symmetry breaking,Phys. Rev. D64(2001) 123517 [hep-th/0106179]

    Pith/arXiv arXiv 2001

  12. [12]

    J. F. Dufaux, G. N. Felder, L. Kofman, M. Peloso and D. Podolsky,Preheating with trilinear interactions: Tachyonic resonance,JCAP07(2006) 006 [hep-ph/0602144]. – 29 –

    Pith/arXiv arXiv 2006

  13. [13]

    E. W. Kolb, D. J. H. Chung and A. Riotto,WIMPzillas!,AIP Conf. Proc.484(1999) 91 [hep-ph/9810361]

    Pith/arXiv arXiv 1999

  14. [14]

    D. J. H. Chung, E. W. Kolb and A. Riotto,Superheavy dark matter,Phys. Rev. D59(1998) 023501 [hep-ph/9802238]

    Pith/arXiv arXiv 1998

  15. [15]

    Garny, M

    M. Garny, M. Sandora and M. S. Sloth,Planckian Interacting Massive Particles as Dark Matter,Phys. Rev. Lett.116(2016) 101302 [1511.03278]

    Pith/arXiv arXiv 2016

  16. [16]

    Garny, A

    M. Garny, A. Palessandro, M. Sandora and M. S. Sloth,Theory and Phenomenology of Planckian Interacting Massive Particles as Dark Matter,JCAP02(2018) 027 [1709.09688]

    Pith/arXiv arXiv 2018

  17. [17]

    Tang and Y.-L

    Y. Tang and Y.-L. Wu,Pure Gravitational Dark Matter, Its Mass and Signatures,Phys. Lett. B758(2016) 402 [1604.04701]

    Pith/arXiv arXiv 2016

  18. [18]

    Markkanen and S

    T. Markkanen and S. Nurmi,Dark matter from gravitational particle production at reheating, JCAP02(2017) 008 [1512.07288]

    Pith/arXiv arXiv 2017

  19. [19]

    Fairbairn, K

    M. Fairbairn, K. Kainulainen, T. Markkanen and S. Nurmi,Despicable Dark Relics: generated by gravity with unconstrained masses,JCAP04(2019) 005 [1808.08236]

    Pith/arXiv arXiv 2019

  20. [20]

    J. A. R. Cembranos, L. J. Garay and J. M. S´ anchez Vel´ azquez,Gravitational production of scalar dark matter,JHEP06(2020) 084 [1910.13937]

    arXiv 2020

  21. [21]

    J. M. Cline, M. Joyce and K. Kainulainen,Supersymmetric electroweak baryogenesis,JHEP 07(2000) 018 [hep-ph/0006119]

    Pith/arXiv arXiv 2000

  22. [22]

    Kainulainen, T

    K. Kainulainen, T. Prokopec, M. G. Schmidt and S. Weinstock,First principle derivation of semiclassical force for electroweak baryogenesis,JHEP06(2001) 031 [hep-ph/0105295]

    Pith/arXiv arXiv 2001

  23. [23]

    Kainulainen, T

    K. Kainulainen, T. Prokopec, M. G. Schmidt and S. Weinstock,Semiclassical force for electroweak baryogenesis: Three-dimensional derivation,Phys. Rev.D66(2002) 043502 [hep-ph/0202177]

    Pith/arXiv arXiv 2002

  24. [24]

    J. M. Cline, K. Kainulainen, P. Scott and C. Weniger,Update on scalar singlet dark matter, Phys. Rev.D88(2013) 055025 [1306.4710]

    Pith/arXiv arXiv 2013

  25. [25]

    J. M. Cline and K. Kainulainen,Electroweak baryogenesis at high wall velocities,2001.00568

    arXiv 2001

  26. [26]

    Konstandin,Quantum Transport and Electroweak Baryogenesis,Phys

    T. Konstandin,Quantum Transport and Electroweak Baryogenesis,Phys. Usp.56(2013) 747 [1302.6713]

    Pith/arXiv arXiv 2013

  27. [27]

    Kainulainen,CP-violating transport theory for electroweak baryogenesis with thermal corrections,JCAP11(2021) 042 [2108.08336]

    K. Kainulainen,CP-violating transport theory for electroweak baryogenesis with thermal corrections,JCAP11(2021) 042 [2108.08336]

    arXiv 2021

  28. [28]

    Buchm¨ uller and S

    W. Buchm¨ uller and S. Fredenhagen,Quantum mechanics of baryogenesis,Phys. Lett.B483 (2000) 217 [hep-ph/0004145]

    Pith/arXiv arXiv 2000

  29. [29]

    Beneke, B

    M. Beneke, B. Garbrecht, C. Fidler, M. Herranen and P. Schwaller,Flavoured Leptogenesis in the CTP Formalism,Nucl. Phys.B843(2011) 177 [1007.4783]

    Pith/arXiv arXiv 2011

  30. [30]

    Anisimov, W

    A. Anisimov, W. Buchm¨ uller, M. Drewes and S. Mendizabal,Quantum Leptogenesis I, Annals Phys.326(2011) 1998 [1012.5821]

    Pith/arXiv arXiv 2011

  31. [31]

    P. S. B. Dev, P. Di Bari, B. Garbrecht, S. Lavignac, P. Millington and D. Teresi,Flavor effects in leptogenesis,Int. J. Mod. Phys.A33(2018) 1842001 [1711.02861]

    Pith/arXiv arXiv 2018

  32. [32]

    De Simone and A

    A. De Simone and A. Riotto,Quantum Boltzmann Equations and Leptogenesis,JCAP0708 (2007) 002 [hep-ph/0703175]. – 30 –

    Pith/arXiv arXiv 2007

  33. [33]

    Garny, A

    M. Garny, A. Hohenegger, A. Kartavtsev and M. Lindner,Systematic approach to leptogenesis in nonequilibrium QFT: Self-energy contribution to theCP-violating parameter, Phys. Rev.D81(2010) 085027 [0911.4122]

    Pith/arXiv arXiv 2010

  34. [34]

    Garbrecht and M

    B. Garbrecht and M. Herranen,Effective Theory of Resonant Leptogenesis in the Closed-Time-Path Approach,Nucl. Phys.B861(2012) 17 [1112.5954]

    Pith/arXiv arXiv 2012

  35. [35]

    Garny, A

    M. Garny, A. Kartavtsev and A. Hohenegger,Leptogenesis from first principles in the resonant regime,Annals Phys.328(2013) 26 [1112.6428]

    Pith/arXiv arXiv 2013

  36. [36]

    P. S. B. Dev, M. Garny, J. Klari´ c, P. Millington and D. Teresi,Resonant enhancement in leptogenesis,Int. J. Mod. Phys.A33(2018) 1842003 [1711.02863]

    Pith/arXiv arXiv 2018

  37. [37]

    Jukkala, K

    H. Jukkala, K. Kainulainen and P. M. Rahkila,Flavour mixing transport theory and resonant leptogenesis,JHEP09(2021) 119 [2104.03998]

    arXiv 2021

  38. [38]

    J. M. Cornwall, R. Jackiw and E. Tomboulis,Effective Action for Composite Operators, Phys. Rev.D10(1974) 2428

    1974

  39. [39]

    Berges,Introduction to nonequilibrium quantum field theory,AIP Conf

    J. Berges,Introduction to nonequilibrium quantum field theory,AIP Conf. Proc.739(2004) 3 [hep-ph/0409233]

    Pith/arXiv arXiv 2004

  40. [40]

    Boyanovsky and H

    D. Boyanovsky and H. J. de Vega,Quantum rolling down out-of-equilibrium,Phys. Rev.D47 (1993) 2343 [hep-th/9211044]

    Pith/arXiv arXiv 1993

  41. [41]

    Boyanovsky, D.-s

    D. Boyanovsky, D.-s. Lee and A. Singh,Phase transitions out-of-equilibrium: Domain formation and growth,Phys. Rev.D48(1993) 800 [hep-th/9212083]

    arXiv 1993

  42. [42]

    Baacke and S

    J. Baacke and S. Michalski,Nonequilibrium evolution in scalaro(n)models with spontaneous symmetry breaking,Phys. Rev. D65(2002) 065019

    2002

  43. [43]

    Berges, S

    J. Berges, S. Borsanyi and J. Serreau,Thermalization of fermionic quantum fields,Nucl. Phys. B660(2003) 51 [hep-ph/0212404]

    Pith/arXiv arXiv 2003

  44. [44]

    Arrizabalaga, J

    A. Arrizabalaga, J. Smit and A. Tranberg,Tachyonic preheating using 2PI-1/N dynamics and the classical approximation,JHEP10(2004) 017 [hep-ph/0409177]

    Pith/arXiv arXiv 2004

  45. [45]

    Arrizabalaga, J

    A. Arrizabalaga, J. Smit and A. Tranberg,Equilibration in phi**4 theory in 3+1 dimensions, Phys. Rev. D72(2005) 025014 [hep-ph/0503287]

    Pith/arXiv arXiv 2005

  46. [46]

    Kainulainen and O

    K. Kainulainen and O. Koskivaara,Non-equilibrium dynamics of a scalar field with quantum backreaction,JHEP12(2021) 190 [2105.09598]

    arXiv 2021

  47. [47]

    Tranberg and G

    A. Tranberg and G. Ungersb¨ ack,Quantum tachyonic preheating, revisited,JHEP05(2024) 128 [2312.08167]

    arXiv 2024

  48. [48]

    Kainulainen, O

    K. Kainulainen, O. Koskivaara and S. Nurmi,Tachyonic production of dark relics: a non-perturbative quantum study,Journal of High Energy Physics2023(2023)

    2023

  49. [49]

    Kainulainen, S

    K. Kainulainen, S. Nurmi and O. V¨ ais¨ anen,Tachyonic production of dark relics: classical lattice vs. quantum 2PI in Hartree truncation,JHEP10(2024) 009 [2406.17468]

    arXiv 2024

  50. [50]

    Herranen, K

    M. Herranen, K. Kainulainen and P. M. Rahkila,Kinetic transport theory with quantum coherence,Nucl. Phys. A820(2009) 203C [0811.0936]

    Pith/arXiv arXiv 2009

  51. [51]

    Fidler, M

    C. Fidler, M. Herranen, K. Kainulainen and P. M. Rahkila,Flavoured quantum Boltzmann equations from cQPA,JHEP02(2012) 065 [1108.2309]

    Pith/arXiv arXiv 2012

  52. [52]

    Kainulainen, S

    K. Kainulainen, S. Nurmi and O. V¨ ais¨ anen,Tachyonic production of dark relics with quantum 2PI formalism including momentum exchanging collisions,(in preparation)(2026) . – 31 –

    2026

  53. [53]

    L. V. Keldysh,Diagram technique for nonequilibrium processes,Zh. Eksp. Teor. Fiz.47 (1964) 1515

    1964

  54. [54]

    Kainulainen and H

    K. Kainulainen and H. Parkkinen,Quantum transport theory for neutrinos with flavor and particle-antiparticle mixing,JHEP02(2024) 217 [2309.00881]

    arXiv 2024

  55. [55]

    Herranen, K

    M. Herranen, K. Kainulainen and P. M. Rahkila,Kinetic theory for scalar fields with nonlocal quantum coherence,JHEP05(2009) 119 [0812.4029]

    Pith/arXiv arXiv 2009

  56. [56]

    Herranen, K

    M. Herranen, K. Kainulainen and P. M. Rahkila,Coherent quantum Boltzmann equations from cQPA,JHEP12(2010) 072 [1006.1929]

    Pith/arXiv arXiv 2010

  57. [57]

    Amitays and K

    B. Amitays and K. Kainulainen,Consistent Thermal Resummation and Phase Transitions with 2PI Methods,(in preparation)(2026)

    2026

  58. [58]

    Herranen, K

    M. Herranen, K. Kainulainen and P. M. Rahkila,Flavour-coherent propagators and Feynman rules: Covariant cQPA formulation,JHEP02(2012) 080 [1108.2371]

    Pith/arXiv arXiv 2012

  59. [59]

    Jukkala, K

    H. Jukkala, K. Kainulainen and O. Koskivaara,Quantum transport and the phase space structure of the Wightman functions,JHEP01(2020) 012 [1910.10979]

    arXiv 2020

  60. [60]

    Kainulainen and H

    K. Kainulainen and H. Parkkinen,Coherent collision integrals for neutrino transport equations,JHEP12(2024) 169 [2407.08598]. – 32 –

    arXiv 2024