- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Quantal theory of restoration of strong CP symmetry

Nemanja Kaloper*

  • *Contact author: kaloper@physics.ucdavis.edu
PDF

Phys. Rev. D 113, 075020 – Published 16 April, 2026

DOI: https://doi.org/10.1103/s167-h4dh

Abstract

We propose a mechanism for relaxing a gauge theory CP violating phase in discrete steps to very small values. The idea is that the CP violating phase includes the magnetic dual of a 4-form flux that can discharge by the nucleation of membranes. Inside the bubbles surrounded by the membranes, the total CP violating phase is reduced. When the bubbles are produced rapidly during radiation domination in the early Universe, near the chiral symmetry breaking scale, they will collide and percolate, melting away into gauge theory radiation and dramatically relaxing CP violation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (83)

  1. S. Weinberg, Opening Address-Neutrinos ’78, Contribution to: Neutrino 78, 6th International Conference on Neutrino Physics, (Lafayette, IN, USA, 1978).
  2. A. Aurilia, H. Nicolai, and P. K. Townsend, Hidden constants: The theta parameter of QCD and the cosmological constant of N=8 supergravity, Nucl. Phys. B176, 509 (1980).
  3. A. Aurilia, D. Christodoulou, and F. Legovini, A classical interpretation of the bag model for hadrons, Phys. Lett. 73B, 429 (1978).
  4. M. J. Duff and P. van Nieuwenhuizen, Quantum inequivalence of different field representations, Phys. Lett. B 94B, 179 (1980).
  5. S. R. Coleman, More about the massive Schwinger model, Ann. Phys. (N.Y.) 101, 239 (1976).
  6. C. Rosenzweig, J. Schechter, and C. G. Trahern, Is the effective Lagrangian for QCD a sigma model?, Phys. Rev. D 21, 3388 (1980).
  7. P. Di Vecchia and G. Veneziano, Chiral dynamics in the large n limit, Nucl. Phys. B171, 253 (1980).
  8. P. Nath and R. L. Arnowitt, The U(1) problem: Current algebra and the theta vacuum, Phys. Rev. D 23, 473 (1981).
  9. E. Witten, Large N chiral dynamics, Ann. Phys. (N.Y.) 128, 363 (1980).
  10. N. Ohta, Vacuum structure and chiral charge quantization in the large N limit, Prog. Theor. Phys. 66, 1408 (1981); 67, 993(E) (1982).
  11. M. Lüscher, The secret long range force in quantum field theories with instantons, Phys. Lett. 78B, 465 (1978).
  12. A. D’Adda, M. Luscher, and P. Di Vecchia, A 1/n expandable series of nonlinear sigma models with instantons, Nucl. Phys. B146, 63 (1978).
  13. E. Witten, Instantons, the quark model, and the 1/n expansion, Nucl. Phys. B149, 285 (1979).
  14. G. Dvali and A. Vilenkin, Cosmic attractors and gauge hierarchy, Phys. Rev. D 70, 063501 (2004).
  15. G. Dvali, Large hierarchies from attractor vacua, Phys. Rev. D 74, 025018 (2006).
  16. G. Dvali, Three-form gauging of axion symmetries and gravity, arXiv:hep-th/0507215.
  17. G. Dvali, A vacuum accumulation solution to the strong CP problem, Phys. Rev. D 74, 025019 (2006).
  18. G. Dvali and G. R. Farrar, Strong CP problem with 10**32 standard model copies, Phys. Rev. Lett. 101, 011801 (2008).
  19. A. Aurilia, Y. Takahashi, and P. K. Townsend, The U(1) problem and the Higgs mechanism in two-dimensions and four-dimensions, Phys. Lett. 95B, 265 (1980).
  20. R. Jackiw and C. Rebbi, Vacuum periodicity in a Yang-Mills quantum theory, Phys. Rev. Lett. 37, 172 (1976).
  21. C. G. Callan, Jr., R. F. Dashen, and D. J. Gross, The structure of the gauge theory vacuum, Phys. Lett. 63B, 334 (1976).
  22. M. A. Shifman, Domain walls and decay rate of the excited vacua in the large N Yang-Mills theory, Phys. Rev. D 59, 021501 (1999).
  23. M. M. Forbes and A. R. Zhitnitsky, Domain walls in QCD, J. High Energy Phys. 10 (2001) 013.
  24. G. R. Dvali and Z. Kakushadze, Large N domain walls as D-branes for N=1 QCD string, Nucl. Phys. B537, 297 (1999).
  25. S. Dubovsky, A. Lawrence, and M. M. Roberts, Axion monodromy in a model of holographic gluodynamics, J. High Energy Phys. 02 (2012) 053.
  26. N. Kaloper, An alternative to axion, Phys. Rev. D 113, L011701 (2026).
  27. A. Kosowsky, M. S. Turner, and R. Watkins, Gravitational waves from first order cosmological phase transitions, Phys. Rev. Lett. 69, 2026 (1992).
  28. M. Kamionkowski, A. Kosowsky, and M. S. Turner, Gravitational radiation from first order phase transitions, Phys. Rev. D 49, 2837 (1994).
  29. G. Veneziano, U(1) Without Instantons, Nucl. Phys. B159, 213 (1979).
  30. J. B. Kogut and L. Susskind, How to solve the eta →3 pi problem by seizing the vacuum, Phys. Rev. D 11, 3594 (1975).
  31. C. Vafa and E. Witten, Parity conservation in QCD, Phys. Rev. Lett. 53, 535 (1984).
  32. G. Gabadadze, Modeling the glueball spectrum by a closed bosonic membrane, Phys. Rev. D 58, 094015 (1998).
  33. G. Gabadadze and M. Shifman, QCD vacuum and axions: What’s happening?, Int. J. Mod. Phys. A 17, 3689 (2002).
  34. A. M. Polyakov and A. A. Belavin, Metastable states of two-dimensional isotropic ferromagnets, JETP Lett. 22, 245 (1975).
  35. A. D’Adda, P. Di Vecchia, and M. Lüscher, Confinement and chiral symmetry breaking in CPn−1 models with quarks, Nucl. Phys. B152, 125 (1979).
  36. N. Kaloper and A. Lawrence, London equation for monodromy inflation, Phys. Rev. D 95, 063526 (2017).
  37. A. Aurilia, The problem of confinement: From two-dimensions to Four-dimensions, Phys. Lett. 81B, 203 (1979).
  38. N. Kaloper, Hidden variables of gravity and geometry and the cosmological constant problem, Phys. Rev. D 106, 065009 (2022).
  39. N. Kaloper, Pancosmic relativity and nature’s hierarchies, Phys. Rev. D 106, 044023 (2022).
  40. R. D. Peccei and H. R. Quinn, CP conservation in the presence of instantons, Phys. Rev. Lett. 38, 1440 (1977).
  41. S. Weinberg, A new light boson?, Phys. Rev. Lett. 40, 223 (1978).
  42. F. Wilczek, Problem of strong P and T invariance in the presence of instantons, Phys. Rev. Lett. 40, 279 (1978).
  43. J. E. Kim, Weak interaction singlet and strong CP invariance, Phys. Rev. Lett. 43, 103 (1979).
  44. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Can confinement ensure natural CP invariance of strong interactions?, Nucl. Phys. B166, 493 (1980).
  45. A. R. Zhitnitsky, On possible suppression of the axion hadron interactions. (In Russian), Sov. J. Nucl. Phys. 31, 260 (1980).
  46. M. Dine, W. Fischler, and M. Srednicki, A simple solution to the strong CP problem with a harmless axion, Phys. Lett. 104B, 199 (1981).
  47. N. Kaloper and L. Sorbo, Where in the string landscape is quintessence, Phys. Rev. D 79, 043528 (2009).
  48. N. Kaloper and L. Sorbo, A natural framework for chaotic inflation, Phys. Rev. Lett. 102, 121301 (2009).
  49. N. Kaloper, A. Lawrence, and L. Sorbo, An ignoble approach to large field inflation, J. Cosmol. Astropart. Phys. 03 (2011) 023.
  50. B. Julia and G. Toulouse, The many defect problem: Gauge like variables for ordered media containing defects, J. Phys. Lett. 40, 396 (1979).
  51. F. Quevedo and C. A. Trugenberger, Phases of antisymmetric tensor field theories, Nucl. Phys. B501, 143 (1997).
  52. P. Gnadig, P. Hasenfratz, J. Kuti, and A. S. Szalay, The quark bag model with surface tension, Phys. Lett. 64B, 62 (1976).
  53. J. D. Brown and C. Teitelboim, Dynamical neutralization of the cosmological constant, Phys. Lett. B 195, 177 (1987).
  54. J. D. Brown and C. Teitelboim, Neutralization of the cosmological constant by membrane creation, Nucl. Phys. B297, 787 (1988).
  55. J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
  56. S. R. Coleman, The fate of the false vacuum. 1. Semiclassical theory, Phys. Rev. D 15, 2929 (1977); 16, 1248(E) (1977).
  57. C. G. Callan, Jr. and S. R. Coleman, The fate of the false vacuum. 2. First quantum corrections, Phys. Rev. D 16, 1762 (1977).
  58. J. Garriga, Nucleation rates in flat and curved space, Phys. Rev. D 49, 6327 (1994).
  59. A. H. Guth and E. J. Weinberg, Could the universe have recovered from a slow first order phase transition?, Nucl. Phys. B212, 321 (1983).
  60. M. S. Turner, E. J. Weinberg, and L. M. Widrow, Bubble nucleation in first order inflation and other cosmological phase transitions, Phys. Rev. D 46, 2384 (1992).
  61. K. Freese and D. Spolyar, Chain inflation: ’Bubble bubble toil and trouble’, J. Cosmol. Astropart. Phys. 07 (2005) 007.
  62. Y. B. Zeldovich, I. Y. Kobzarev, and L. B. Okun, Cosmological consequences of the spontaneous breakdown of discrete symmetry, Zh. Eksp. Teor. Fiz. 67, 3 (1974).
  63. J. E. Kim, Light pseudoscalars, particle physics and cosmology, Phys. Rep. 150, 1 (1987).
  64. V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Dynamics of bubbles in general relativity, Phys. Rev. D 36, 2919 (1987).
  65. T. Banks, M. Dine, and N. Seiberg, Irrational axions as a solution of the strong CP problem in an eternal universe, Phys. Lett. B 273, 105 (1991).
  66. V. Anastassopoulos, S. Andrianov, R. Baartman, M. Bai, S. Baessler, J. Benante, M. Berz, M. Blaskiewicz, T. Bowcock, K. Brown et al., A storage ring experiment to detect a proton electric dipole moment, Rev. Sci. Instrum. 87, 115116 (2016).
  67. A. S. Zhevlakov and V. E. Lyubovitskij, Deuteron EDM induced by CP violating couplings of pseudoscalar mesons, Phys. Rev. D 101, 115041 (2020).
  68. D. Kharzeev, R. D. Pisarski, and M. H. G. Tytgat, Possibility of spontaneous parity violation in hot QCD, Phys. Rev. Lett. 81, 512 (1998).
  69. I. E. Halperin and A. Zhitnitsky, Axion potential, topological defects and CP odd bubbles in QCD, Phys. Lett. B 440, 77 (1998).
  70. D. Kharzeev, R. D. Pisarski, and M. H. G. Tytgat, Parity odd bubbles in hot QCD, arXiv:hep-ph/9808366.
  71. K. Buckley, T. Fugleberg, and A. Zhitnitsky, Can theta vacua be created in heavy ion collisions?, Phys. Rev. Lett. 84, 4814 (2000).
  72. D. Kharzeev and A. Zhitnitsky, Charge separation induced by P-odd bubbles in QCD matter, Nucl. Phys. A797, 67 (2007).
  73. B. Holdom, Two U(1)’s and Epsilon Charge Shifts, Phys. Lett. 166B, 196 (1986).
  74. J. L. Feng, J. March-Russell, S. Sethi, and F. Wilczek, Saltatory relaxation of the cosmological constant, Nucl. Phys. B602, 307 (2001).
  75. J. Garriga and A. Megevand, Coincident brane nucleation and the neutralization of lambda, Phys. Rev. D 69, 083510 (2004).
  76. A. R. Brown, Schwinger pair production at nonzero temperatures or in compact directions, Phys. Rev. D 98, 036008 (2018).
  77. A. R. Brown and A. Dahlen, Small steps and giant leaps in the landscape, Phys. Rev. D 82, 083519 (2010).
  78. A. R. Brown and A. Dahlen, Giant leaps and minimal branes in multi-dimensional flux landscapes, Phys. Rev. D 84, 023513 (2011).
  79. R. Easther, J. T. Giblin, Jr, L. Hui, and E. A. Lim, A new mechanism for bubble nucleation: Classical transitions, Phys. Rev. D 80, 123519 (2009).
  80. M. Kleban, K. Krishnaiyengar, and M. Porrati, Flux discharge cascades in various dimensions, J. High Energy Phys. 11 (2011) 096.
  81. T. C. Bachlechner, K. Eckerle, O. Janssen, and M. Kleban, Multiple-axion framework, Phys. Rev. D 98, 061301 (2018).
  82. M. Demirtas, N. Gendler, C. Long, L. McAllister, and J. Moritz, PQ axiverse, J. High Energy Phys. 06 (2023) 092.
  83. N. Kaloper, Axion flux monodromy discharges relax the cosmological constant, J. Cosmol. Astropart. Phys. 11 (2023) 032.

Outline

Information

Phys. Rev. D 113, 075020– Published 16 April, 2026
  • Received 31 December 2025
  • Accepted 30 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation