pith. sign in

arxiv: 2510.23033 · v2 · submitted 2025-10-27 · ✦ hep-ph · astro-ph.CO· hep-th

Exploring the Landscape of Spontaneous CP Violation in Supersymmetric Theories

Fangchao Liu , Shota Nakagawa , Yuichiro Nakai , Yaoduo Wang This is my paper

Pith reviewed 2026-05-18 04:48 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COhep-th
keywords spontaneous CP violationsupersymmetrystrong CP problemCKM matrixR-symmetryspurion formalismsoft SUSY breaking
0
0 comments X

The pith

Supersymmetry enables spontaneous CP violation that solves the strong CP problem by protecting scales and suppressing operators.

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

This paper shows how supersymmetry can realize spontaneous CP violation to address the strong CP problem. CP symmetry is exact in the Lagrangian but broken spontaneously in the vacuum to generate the CKM phase without a large strong CP phase. Supersymmetry protects the scale of this violation from radiative corrections and suppresses higher dimensional operators that could induce a strong CP phase. The authors study two cases: the exact supersymmetric limit with an extended spurion formalism and R-symmetry constraints, and a model with breaking at an intermediate scale using soft terms and nonperturbative effects.

Core claim

Supersymmetry provides a natural framework to accommodate spontaneous CP violation by protecting the scale of SCPV from radiative corrections and suppressing problematic higher-dimensional operators generating a strong CP phase. In the exact SUSY limit, the spurion formalism is extended to identify the necessary condition for stabilizing CP-violating phases, and radial vacuum expectation values are stabilized through R-symmetry constraints on the superpotential. In a second construction, CP is spontaneously broken at an intermediate scale along pseudo-flat directions stabilized by soft SUSY breaking and non-perturbative effects of a gauge theory, predicting light scalars in the SCPV sector

What carries the argument

The spurion formalism extended to supersymmetric theories together with R-symmetry constraints on the superpotential to stabilize CP-violating phases and radial VEVs.

If this is right

  • Generation of the observed CKM phase without introducing a nonzero strong CP phase.
  • Protection of the SCPV scale from radiative corrections.
  • Suppression of higher-dimensional operators that generate a strong CP phase.
  • Prediction of light scalars whose masses are set by the SUSY breaking scale.

Where Pith is reading between the lines

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

  • Future experiments could search for these light scalars as a signature of the mechanism.
  • This approach may be combined with other supersymmetric solutions to hierarchy or flavor problems.
  • Similar symmetry-based stabilizations could apply to other spontaneous symmetry breaking scenarios in particle physics.

Load-bearing premise

Stabilizing the CP-violating phases requires extending the spurion formalism and using R-symmetry constraints on the superpotential to fix the radial vacuum expectation values.

What would settle it

A collider search failing to detect light scalars with masses around the SUSY breaking scale would challenge the intermediate scale model.

read the original abstract

The strong CP problem remains one of the most important unresolved issues in the Standard Model. Spontaneous CP violation (SCPV) is a promising approach to the problem by assuming that CP is an exact symmetry of the Lagrangian but broken spontaneously at the vacuum, which enables the generation of the observed Cabibbo-Kobayashi-Maskawa (CKM) phase without reintroducing a nonzero strong CP phase. Supersymmetry (SUSY) provides a natural framework to accommodate such a mechanism, as SUSY can not only protect the scale of SCPV from radiative corrections but also suppress problematic higher-dimensional operators generating a strong CP phase. In the present study, we explore the realization of SCPV in two distinct SUSY scenarios. First, we investigate SCPV in the exact SUSY limit by extending the spurion formalism developed in non-supersymmetric theories to identify the necessary condition for stabilizing CP-violating phases, and by analyzing the stabilization of radial vacuum expectation values through R-symmetry constraints on the superpotential. Second, we construct a model in which CP is spontaneously broken at an intermediate scale along pseudo-flat directions, stabilized by soft SUSY breaking and non-perturbative effects of a gauge theory. The latter setup predicts light scalars in the SCPV sector whose masses are determined by the SUSY breaking scale.

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 manuscript explores spontaneous CP violation (SCPV) in supersymmetric theories to address the strong CP problem. It examines two scenarios: (1) SCPV in the exact SUSY limit, achieved by extending the non-supersymmetric spurion formalism to identify conditions for stabilizing CP-violating phases and imposing R-symmetry constraints on the superpotential to stabilize radial vacuum expectation values; (2) a model in which CP is spontaneously broken at an intermediate scale along pseudo-flat directions, with stabilization provided by soft SUSY-breaking terms and non-perturbative effects from a gauge theory. The second scenario predicts light scalars in the SCPV sector whose masses are set by the SUSY-breaking scale.

Significance. If the constructions hold, the work offers a SUSY framework for SCPV that protects the SCPV scale from radiative corrections and suppresses higher-dimensional operators capable of generating a strong CP phase. The explicit prediction of light scalars tied to the SUSY-breaking scale provides a falsifiable signature that could be tested at future colliders or in precision flavor experiments. The extension of the spurion approach and the use of R-symmetry constraints represent a systematic exploration of the SCPV landscape in SUSY.

major comments (2)
  1. [Exact SUSY limit] Exact SUSY limit section: The extension of the spurion formalism is presented as identifying the necessary conditions for stabilizing CP-violating phases, yet no explicit derivation of the effective potential or minimization conditions is given. This is load-bearing for the central claim, as the holomorphy of the superpotential restricts phase-dependent operators to holomorphic combinations; without the full scalar potential (including Kähler contributions) it remains unclear whether a stable CP-odd minimum is actually obtained.
  2. [Intermediate-scale model] Intermediate-scale model section: The stabilization of pseudo-flat directions by soft SUSY breaking and non-perturbative gauge effects is asserted to produce light scalars whose masses are determined by the SUSY-breaking scale, but no explicit computation of the scalar mass matrix or the resulting spectrum is provided. This undermines the quantitative prediction and the claim that the setup is radiatively stable.
minor comments (2)
  1. Notation for the spurion fields and R-charges should be defined more explicitly at first use to improve readability.
  2. The abstract and introduction would benefit from a brief statement of the key assumptions (e.g., exact R-symmetry in the first scenario) to orient the reader.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each major comment below and indicate the revisions we plan to make.

read point-by-point responses

  1. Referee: [Exact SUSY limit] Exact SUSY limit section: The extension of the spurion formalism is presented as identifying the necessary conditions for stabilizing CP-violating phases, yet no explicit derivation of the effective potential or minimization conditions is given. This is load-bearing for the central claim, as the holomorphy of the superpotential restricts phase-dependent operators to holomorphic combinations; without the full scalar potential (including Kähler contributions) it remains unclear whether a stable CP-odd minimum is actually obtained.

    Authors: We agree that an explicit derivation of the effective potential, including the contributions from the Kähler potential, would clarify how the CP-violating phases are stabilized and confirm the existence of a stable minimum. The manuscript relies on the holomorphy and R-symmetry constraints to argue for the conditions, but we will add a detailed minimization of the scalar potential in the revised version to address this point directly. revision: yes

  2. Referee: [Intermediate-scale model] Intermediate-scale model section: The stabilization of pseudo-flat directions by soft SUSY breaking and non-perturbative gauge effects is asserted to produce light scalars whose masses are determined by the SUSY-breaking scale, but no explicit computation of the scalar mass matrix or the resulting spectrum is provided. This undermines the quantitative prediction and the claim that the setup is radiatively stable.

    Authors: We acknowledge that providing an explicit computation of the scalar mass matrix would strengthen the quantitative aspects of the prediction for the light scalars. While the stabilization by soft terms and non-perturbative effects is outlined, we will include the mass matrix calculation and spectrum analysis in the revised manuscript to better support the claims of radiative stability and the mass scale being set by SUSY breaking. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends independent prior formalisms

full rationale

The paper extends the spurion formalism from non-supersymmetric theories to SUSY to identify necessary conditions for stabilizing CP-violating phases and analyzes radial VEV stabilization via R-symmetry constraints on the superpotential. These steps rely on established SUSY properties (holomorphy, R-symmetry) and prior non-SUSY work rather than defining results in terms of themselves or renaming fitted inputs as predictions. No load-bearing self-citations, ansatz smuggling, or uniqueness theorems imported from the same authors are indicated in the abstract or described chain. The central claims remain independent of the target results and are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard domain assumptions about SUSY and CP symmetry plus an extension of the spurion formalism; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption CP is an exact symmetry of the Lagrangian
    Basis for spontaneous CP violation stated in the abstract.
  • domain assumption SUSY protects the scale of SCPV from radiative corrections and suppresses higher-dimensional operators
    Claimed as the natural framework in the abstract.

pith-pipeline@v0.9.0 · 5775 in / 1423 out tokens · 59015 ms · 2026-05-18T04:48:32.213131+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

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

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Nelson-Barr Models with Vector-Like Quark Doublets

    hep-ph 2026-04 unverdicted novelty 7.0

    Nelson-Barr models with vector-like quark doublets suppress hadronic CP violation to three loops via an accidental symmetry in the renormalizable theory, yielding phenomenologically viable solutions distinct from sing...

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · cited by 1 Pith paper · 31 internal anchors

  1. [1]

    An Improved Experimental Limit on the Electric Dipole Moment of the Neutron

    C.A. Baker et al.,An Improved experimental limit on the electric dipole moment of the neutron,Phys. Rev. Lett.97(2006) 131801 [hep-ex/0602020]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  2. [2]

    A Revised Experimental Upper Limit on the Electric Dipole Moment of the Neutron

    J.M. Pendlebury et al.,Revised experimental upper limit on the electric dipole moment of the neutron,Phys. Rev. D92(2015) 092003 [1509.04411]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  3. [3]

    Measurement of the Permanent Electric Dipole Moment of the Neutron,

    C. Abel et al.,Measurement of the Permanent Electric Dipole Moment of the Neutron,Phys. Rev. Lett.124(2020) 081803 [2001.11966]

    work page arXiv 2020

  4. [4]

    Nelson,Naturally Weak CP Violation,Phys

    A.E. Nelson,Naturally Weak CP Violation,Phys. Lett. B136(1984) 387

    work page 1984

  5. [5]

    Barr,Solving the Strong CP Problem Without the Peccei-Quinn Symmetry,Phys

    S.M. Barr,Solving the Strong CP Problem Without the Peccei-Quinn Symmetry,Phys. Rev. Lett.53(1984) 329

    work page 1984

  6. [6]

    Barr,A Natural Class of Nonpeccei-quinn Models,Phys

    S.M. Barr,A Natural Class of Nonpeccei-quinn Models,Phys. Rev. D30(1984) 1805

    work page 1984

  7. [7]

    Bento, G.C

    L. Bento, G.C. Branco and P.A. Parada,A Minimal model with natural suppression of strong CP violation,Phys. Lett. B267(1991) 95

    work page 1991

  8. [8]

    Challenges for the Nelson-Barr Mechanism

    M. Dine and P. Draper,Challenges for the Nelson-Barr Mechanism,JHEP08(2015) 132 [1506.05433]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  9. [9]

    Barr and G

    S.M. Barr and G. Segre,Spontaneous CP violation and supersymmetry,Phys. Rev. D48 (1993) 302

    work page 1993

  10. [10]

    Supersymmetry and the Nelson-Barr Mechanism

    M. Dine, R.G. Leigh and A. Kagan,Supersymmetry and the Nelson-Barr mechanism,Phys. Rev. D48(1993) 2214 [hep-ph/9303296]. – 24 –

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  11. [11]

    Evans, C

    J. Evans, C. Han, T.T. Yanagida and N. Yokozaki,Complete solution to the strongCP problem: Supersymmetric extension of the Nelson-Barr model,Phys. Rev. D103(2021) L111701 [2002.04204]

    work page arXiv 2021

  12. [12]

    Fujikura, Y

    K. Fujikura, Y. Nakai, R. Sato and M. Yamada,Baryon asymmetric Universe from spontaneous CP violation,JHEP04(2022) 105 [2202.08278]

    work page arXiv 2022

  13. [13]

    Feruglio, M

    F. Feruglio, M. Parriciatu, A. Strumia and A. Titov,Solving the strong CP problem without axions,JHEP08(2024) 214 [2406.01689]

    work page arXiv 2024

  14. [14]

    Feruglio and R

    F. Feruglio and R. Ziegler,CPon Dark Matter,JHEP03(2025) 102 [2411.08101]

    work page arXiv 2025

  15. [15]

    Feruglio, A

    F. Feruglio, A. Marrone, A. Strumia and A. Titov,Solving the strong CP problem in string-inspired theories with modular invariance,JHEP08(2025) 076 [2505.20395]

    work page arXiv 2025

  16. [16]

    Spontaneous CP violation and the strong CP problem

    L. Vecchi,Spontaneous CP violation and the strong CP problem,JHEP04(2017) 149 [1412.3805]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  17. [17]

    Valenti and L

    A. Valenti and L. Vecchi,Super-soft CP violation,JHEP07(2021) 152 [2106.09108]

    work page arXiv 2021

  18. [18]

    Girmohanta, S.J

    S. Girmohanta, S.J. Lee, Y. Nakai and M. Suzuki,A natural model of spontaneous CP violation,JHEP12(2022) 024 [2203.09002]

    work page arXiv 2022

  19. [19]

    Asadi, S

    P. Asadi, S. Homiller, Q. Lu and M. Reece,Chiral Nelson-Barr models: Quality and cosmology,Phys. Rev. D107(2023) 115012 [2212.03882]

    work page arXiv 2023

  20. [20]

    Bai and G.N

    Y. Bai and G.N. Wojcik,Variant Nelson-Barr mechanism with minimal flavor violation, JHEP04(2023) 063 [2212.07459]

    work page arXiv 2023

  21. [21]

    Murai and K

    K. Murai and K. Nakayama,Revisiting the minimal Nelson-Barr model,JHEP11(2024) 098 [2407.16202]

    work page arXiv 2024

  22. [22]

    Murai and K

    K. Murai and K. Nakayama,Domain walls in Nelson-Barr axion model,JHEP09(2025) 099 [2412.19456]

    work page arXiv 2025

  23. [23]

    Ferro-Hernandez, S

    R. Ferro-Hernandez, S. Morisi and E. Peinado,Axionless strong CP problem solution: The spontaneous CP-violation case,Phys. Rev. D111(2025) 073009 [2407.18161]

    work page arXiv 2025

  24. [24]

    Jiang and N

    Y. Jiang and N. Yokozaki,Spontaneous CP violation, sterile neutrino dark matter and leptogenesis,Phys. Lett. B862(2025) 139331 [2408.13990]

    work page arXiv 2025

  25. [25]

    Solving the strong CP problem with supersymmetry

    G. Hiller and M. Schmaltz,Solving the Strong CP Problem with Supersymmetry,Phys. Lett. B514(2001) 263 [hep-ph/0105254]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  26. [26]

    Strong-weak CP hierarchy from non-renormalization theorems

    G. Hiller and M. Schmaltz,Strong Weak CP Hierarchy from Nonrenormalization Theorems, Phys. Rev. D65(2002) 096009 [hep-ph/0201251]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  27. [27]

    Nakagawa, Y

    S. Nakagawa, Y. Nakai and Y. Wang,Spontaneous CP violation in supersymmetric QCD, JHEP09(2024) 105 [2406.01260]

    work page arXiv 2024

  28. [28]

    Group-theoretic Condition for Spontaneous CP Violation

    H.E. Haber and Z. Surujon,A Group-theoretic Condition for Spontaneous CP Violation, Phys. Rev. D86(2012) 075007 [1201.1730]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  29. [29]

    Varieties of vacua in classical supersymmetric gauge theories

    M.A. Luty and W. Taylor,Varieties of vacua in classical supersymmetric gauge theories, Phys. Rev. D53(1996) 3399 [hep-th/9506098]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  30. [30]

    M. Dine, L. Randall and S.D. Thomas,Baryogenesis from flat directions of the supersymmetric standard model,Nucl. Phys. B458(1996) 291 [hep-ph/9507453]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  31. [31]

    Flat directions in the scalar potential of the supersymmetric standard model

    T. Gherghetta, C.F. Kolda and S.P. Martin,Flat directions in the scalar potential of the supersymmetric standard model,Nucl. Phys. B468(1996) 37 [hep-ph/9510370]. – 25 –

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  32. [32]

    Kim,A Common Scale for the Invisible Axion, Local SUSY GUTs and Saxino Decay, Phys

    J.E. Kim,A Common Scale for the Invisible Axion, Local SUSY GUTs and Saxino Decay, Phys. Lett. B136(1984) 378

    work page 1984

  33. [33]

    Rajagopal, M.S

    K. Rajagopal, M.S. Turner and F. Wilczek,Cosmological implications of axinos,Nucl. Phys. B358(1991) 447

    work page 1991

  34. [34]

    Cosmological Axion Problem in Chaotic Inflationary Universe

    S. Kasuya, M. Kawasaki and T. Yanagida,Cosmological axion problem in chaotic inflationary universe,Phys. Lett. B409(1997) 94 [hep-ph/9608405]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  35. [35]

    Axinos as Dark Matter

    L. Covi, H.-B. Kim, J.E. Kim and L. Roszkowski,Axinos as dark matter,JHEP05(2001) 033 [hep-ph/0101009]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  36. [36]

    Cosmological implications of supersymmetric axion models

    M. Kawasaki, K. Nakayama and M. Senami,Cosmological implications of supersymmetric axion models,JCAP03(2008) 009 [0711.3083]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  37. [37]

    Inflation from a Supersymmetric Axion Model

    M. Kawasaki, N. Kitajima and K. Nakayama,Inflation from a Supersymmetric Axion Model, Phys. Rev. D82(2010) 123531 [1008.5013]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  38. [38]

    Cosmological aspects of inflation in a supersymmetric axion model

    M. Kawasaki, N. Kitajima and K. Nakayama,Cosmological Aspects of Inflation in a Supersymmetric Axion Model,Phys. Rev. D83(2011) 123521 [1104.1262]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  39. [39]

    K.J. Bae, K. Choi and S.H. Im,Effective Interactions of Axion Supermultiplet and Thermal Production of Axino Dark Matter,JHEP08(2011) 065 [1106.2452]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  40. [40]

    Cosmology of the DFSZ axino

    K.J. Bae, E.J. Chun and S.H. Im,Cosmology of the DFSZ axino,JCAP03(2012) 013 [1111.5962]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  41. [41]

    Peccei-Quinn extended gauge-mediation model with vector-like matter

    K. Nakayama and N. Yokozaki,Peccei-Quinn extended gauge-mediation model with vector-like matter,JHEP11(2012) 158 [1204.5420]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  42. [42]

    Thermal Effects on Saxion in Supersymmetric Model with Peccei-Quinn Symmetry

    T. Moroi and M. Takimoto,Thermal Effects on Saxion in Supersymmetric Model with Peccei-Quinn Symmetry,Phys. Lett. B718(2012) 105 [1207.4858]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  43. [43]

    Axions : Theory and Cosmological Role

    M. Kawasaki and K. Nakayama,Axions: Theory and Cosmological Role,Ann. Rev. Nucl. Part. Sci.63(2013) 69 [1301.1123]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  44. [44]

    Explosive Axion Production from Saxion

    Y. Ema and K. Nakayama,Explosive Axion Production from Saxion,Phys. Lett. B776 (2018) 174 [1710.02461]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  45. [45]

    R.T. Co, L.J. Hall and K. Harigaya,QCD Axion Dark Matter with a Small Decay Constant, Phys. Rev. Lett.120(2018) 211602 [1711.10486]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  46. [46]

    Y. Ema, D. Hagihara, K. Hamaguchi, T. Moroi and K. Nakayama,Supersymmetric Flaxion, JHEP04(2018) 094 [1802.07739]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  47. [47]

    Y. Ema, R. Jinno, K. Nakayama and J. van de Vis,Preheating from target space curvature and unitarity violation: Analysis in field space,Phys. Rev. D103(2021) 103536 [2102.12501]

    work page arXiv 2021

  48. [48]

    Nakai and M

    Y. Nakai and M. Suzuki,Axion Quality from Superconformal Dynamics,Phys. Lett. B816 (2021) 136239 [2102.01329]

    work page arXiv 2021

  49. [49]

    Co and M

    R.T. Co and M. Yamada,Axion cogenesis without isocurvature perturbations,Phys. Rev. D 110(2024) 055009 [2312.17730]

    work page arXiv 2024

  50. [50]

    The Nelson-Barr Relaxion

    O. Davidi, R.S. Gupta, G. Perez, D. Redigolo and A. Shalit,Nelson-Barr relaxion,Phys. Rev. D99(2019) 035014 [1711.00858]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  51. [51]

    M. Dine, G. Perez, W. Ratzinger and I. Savoray,Nelson-Barr ultralight dark matter,Phys. Rev. D111(2025) 015049 [2405.06744]. – 26 –

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  52. [52]

    Theories with Gauge-Mediated Supersymmetry Breaking

    G.F. Giudice and R. Rattazzi,Theories with gauge mediated supersymmetry breaking,Phys. Rept.322(1999) 419 [hep-ph/9801271]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  53. [53]

    Supersymmetry Breaking and Gauge Mediation

    R. Kitano, H. Ooguri and Y. Ookouchi,Supersymmetry Breaking and Gauge Mediation,Ann. Rev. Nucl. Part. Sci.60(2010) 491 [1001.4535]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  54. [54]

    The Cosmological Axino Problem

    C. Cheung, G. Elor and L.J. Hall,The Cosmological Axino Problem,Phys. Rev. D85(2012) 015008 [1104.0692]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  55. [55]

    McNamara and M

    J. McNamara and M. Reece,Reflections on Parity Breaking,2212.00039

    work page arXiv

  56. [56]

    IS THERE A DOMAIN WALL PROBLEM?

    G.R. Dvali and G. Senjanovic,Is there a domain wall problem?,Phys. Rev. Lett.74(1995) 5178 [hep-ph/9501387]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  57. [57]

    Nonrestoration of spontaneously broken P, CP and PQ at high temperature

    G.R. Dvali, A. Melfo and G. Senjanovic,Nonrestoration of spontaneously broken P and CP at high temperature,Phys. Rev. D54(1996) 7857 [hep-ph/9601376]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  58. [58]

    Nakagawa, Y

    S. Nakagawa, Y. Nakai, Y.-C. Qiu, L. Wang and Y. Wang,High Reheating Temperature without Axion Domain Walls,2509.24812. – 27 –

    work page arXiv