- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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
  • Letter
  • Open Access

Topological leptogenesis

Juven Wang*,†

PDF

Phys. Rev. Research 8, L022024 – Published 5 May, 2026

DOI: https://doi.org/10.1103/r763-lq1b

Abstract

In the standard lore, the baryon asymmetry of the present universe is attributed to the leptogenesis from the sterile right-handed neutrino with heavy Majorana fermion mass decaying into the Standard Model's leptons in the very early universe—called the Majorana leptogenesis, while the electroweak sphaleron causes baryogenesis at a later time. In this work, we propose a mechanism, named topological leptogenesis, to explain the lepton asymmetry. Topological leptogenesis replaces some of the sterile neutrinos by introducing a gapped topological order sector (whose low energy exhibits topological quantum field theory with long-range entanglement) that can cancel the baryon minus lepton (B±L) mixed gauge-gravitational anomaly of the standard model . But unlike the Majorana mass gap that causes the continuous (B−L) and lepton number L symmetry breaking down to the fermion parity Z2F, the gapped topological order preserves a discrete (B±L) symmetry, with an extra gauge group symmetry extension over the discrete (B±L). Then, the beyond-the-standard-model dark matter consists of topological quantum matter, such that the gapped nonparticle excitations of extended line and surface defects with fractionalization and anyon charges can decay into the Standard Model particles. In addition, gravitational leptogenesis can be regarded as an intermediate step (between Majorana particle leptogenesis and topological nonparticle leptogenesis) to mediate such decaying processes from the highly entangled gapped topological order excitations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (71)

  1. A. D. Sakharov, Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe, Pis’ma Zh. Eksp. Teor. Fiz. 5, 32 (1967).
  2. M. Fukugita and T. Yanagida, Baryogenesis without grand unification, Phys. Lett. B 174, 45 (1986).
  3. S. Davidson, E. Nardi, and Y. Nir, Leptogenesis, Phys. Rep. 466, 105 (2008).
  4. F. R. Klinkhamer and N. S. Manton, A saddle-point solution in the weinberg-salam theory, Phys. Rev. D 30, 2212 (1984).
  5. V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, On the anomalous electroweak baryon number nonconservation in the early universe, Phys. Lett. B 155, 36 (1985).
  6. S. L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177, 2426 (1969).
  7. J. S. Bell and R. Jackiw, A PCAC puzzle: π0→γγ in the σ model, Nuovo Cimento A 60, 47 (1969).
  8. G. 't Hooft, Symmetry breaking through Bell-Jackiw anomalies, Phys. Rev. Lett. 37, 8 (1976).
  9. A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59, 85 (1975).
  10. S. H.-S. Alexander, M. E. Peskin, and M. M. Sheikh-Jabbari, Leptogenesis from gravity waves in models of inflation, Phys. Rev. Lett. 96, 081301 (2006).
  11. T. Eguchi and P. G. O. Freund, Quantum gravity and world topology, Phys. Rev. Lett. 37, 1251 (1976).
  12. L. Alvarez-Gaume and E. Witten, Gravitational anomalies, Nucl. Phys. B 234, 269 (1984).
  13. P. Putrov and J. Wang, Categorical symmetry of the standard model from gravitational anomaly, Phys. Rev. D 110, 125028 (2024).
  14. J. Wang, Anomaly and cobordism constraints beyond the standard model: Topological force, arXiv:2006.16996 [hep-th].
  15. J. Wang, Anomaly and cobordism constraints beyond grand unification: Energy hierarchy, arXiv:2008.06499 [hep-th].
  16. J. Wang, Ultra Unification: Unified model beyond grand unification, Phys. Rev. D 103, 105024 (2021).
  17. M. Cheng, J. Wang, and X. Yang, Boundary topological orders of (4+1)d fermionic Z2NF symmetry protected topological states, Phys. Rev. B 113, 125105 (2026).
  18. S.-S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. Math. 99, 48 (1974).
  19. A. S. Schwarz, The partition function of degenerate quadratic functional and ray-singer invariants, Lett. Math. Phys. 2, 247 (1978).
  20. E. Witten, Quantum field theory and the jones polynomial, Commun. Math. Phys. 121, 351 (1989).
  21. M. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math. 68, 175 (1989).
  22. X.-G. Wen, Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017).
  23. P. Putrov, J. Wang, and S.-T. Yau, Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions, Ann. Phys. 384, 254 (2017).
  24. J. Wang, K. Ohmori, P. Putrov, Y. Zheng, Z. Wan, M. Guo, H. Lin, P. Gao, and S.-T. Yau, Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions, Prog. Theor. Exp. Phys. 2018, 053A01 (2018).
  25. F. Wilczek, Quantum mechanics of fractional spin particles, Phys. Rev. Lett. 49, 957 (1982).
  26. Fractional statistics and anyon superconductivity, edited by F. Wilczek (World Scientific, 1990).
  27. C. G. Callan, Jr. and J. A. Harvey, Anomalies and fermion zero modes on strings and domain walls, Nucl. Phys. B 250, 427 (1985).
  28. J. Wang, Z. Wan, and Y.-Z. You, Cobordism and deformation class of the standard model, Phys. Rev. D 106, L041701 (2022).
  29. J. Wang, Z. Wan, and Y.-Z. You, Proton stability: From the standard model to beyond grand unification, Phys. Rev. D 106, 025016 (2022).
  30. D. Tong, Line operators in the standard model, J. High Energy Phys. 07 (2017) 104.
  31. c1 is obtained by integrating over c1≡∫12πTrF locally with nontrivial transition functions between patches. c2 is obtained by integrating over c2≡∫−18π2Tr(F∧F)+12(12πTrF)2 locally with nontrivial transition functions between patches. It is more common to consider the su(2) sphaleron because the instanton number on a compact space like four-sphere S4 can still be nonzero. But the u(1) instanton number on S4 is trivial; only that on those such as S2×S2 with nontrivial lower homotopy can be nonzero.
  32. See Refs. [65, 66] for the gravitational leptogenesis realization in Einstein gravity versus modified gravity. See Ref. [67] for discussions on Dirac or Majorana neutrino mass scenarios for such leptogenesis.
  33. G. Dvali and L. Funcke, Small neutrino masses from gravitational θ-term, Phys. Rev. D 93, 113002 (2016).
  34. G. Dvali, L. Funcke, and T. Vachaspati, Time- and space-varying neutrino mass matrix from soft topological defects, Phys. Rev. Lett. 130, 091601 (2023).
  35. J. McNamara and C. Vafa, Cobordism classes and the swampland, arXiv:1909.10355 [hep-th].
  36. ALEPH, DELPHI, L3, and OPAL Collaborations, Precision electroweak measurements on the Z resonance, Phys. Rep. 427, 257 (2006).
  37. J. Erler and P. Langacker, Constraints on extended neutral gauge structures, Phys. Rev. Lett. 84, 212 (2000).
  38. P. Langacker, The physics of heavy z' gauge bosons, Rev. Mod. Phys. 81, 1199 (2009).
  39. Let us examine the order of magnitude of gravitational contribution of d★jL=(−Nf+nνR)p1(TM)24=−(−3+nνR)124×8π2122ε̃μναβRabμνRbaαβd4x.In the Friedmann-Lemaître-Robertson-Walker (FLRW) metric that describes a homogeneous isotropic expanding universe, the Pontryagin class p1 vanishes. So, there is no classical gravitational anomaly source in FLRW. The nonzero p1 requires CP-odd elliptically polarized chiral gravitational waves [10], inflationary tensor backgrounds [68, 69, 70], etc. The Hubble expansion rate is H=ȧa for a dimensionless scale factor a in a radiation-dominated universe. The Friedmann equation simplifies to H2=8πG3ρ with the energy density of a relativistic thermal plasma ρ=π230g*T4, with g* effective number of relativistic degrees of freedom and some, so H≃8π390g*T2MPlanck with MPlanck∼1019 GeV and g*∼109. For Type I seesaw leptogenesis, T∼MMajorana≳1010 GeV due to Davidson-Ibarra bound [71], so H≃15×T2MPlanck≳10×1010×21019∼100 GeV. Let us assume an appropriate classical gravitational background. Thus, the gravitational anomaly can contribute to the lepton number violation of order dnLdt∝RR̃∝H4∼108GeV4. Consider the lepton number density nL over entropy density s∼T3, for Type I Majorana seesaw leptogenesis [2] nLs∼dnLdt×Δts∼H4×H−1T3∼H3T3∼1061030∼10−24. Reference [10] under a classical chiral gravitational wave background obtains nLs∼10−22. However, this scale is much smaller (about 10−12to10−14 magnitude smaller) in comparison of the experimental lepton asymmetry nL or baryon asymmetry nB relative to the entropy density s, which is experimentally measured as nLs; also,YB≡nBs∼10−10. Let us further evaluate the suppression of the three-loop effect of Eq.  (12) involving two fermionic triangles and one box involving graviton propagators. Each graviton vertex introduces a factor 1MPlanck, so four graviton vertices contribute 1MPlanck4; taking into account two graviton propagator (1k2)2∼T−4 and fermion loop integrals (∫d4k)2∼T8, we get a suppression of T4MPlanck4 for Eq.  (12) diagram. This is indeed a rather small contribution compared with the usual order of Type I leptogenesis [2].
  40. K. Dick, M. Lindner, M. Ratz, and D. Wright, Leptogenesis with Dirac neutrinos, Phys. Rev. Lett. 84, 4039 (2000).
  41. H. Murayama and A. Pierce, Realistic Dirac leptogenesis, Phys. Rev. Lett. 89, 271601 (2002).
  42. F. Wilczek and A. Zee, Operator analysis of nucleon decay, Phys. Rev. Lett. 43, 1571 (1979).
  43. F. Wilczek and A. Zee, Conservation or Violation of B – L in Proton Decay, Phys. Lett. B 88, 311 (1979).
  44. Y. Tachikawa and K. Yonekura, Why are fractional charges of orientifolds compatible with Dirac quantization? SciPost Phys. 7, 058 (2018).
  45. I. Garcia-Etxebarria and M. Montero, Dai-Freed anomalies in particle physics, J. High Energy Phys. 08 (2019) 003.
  46. C.-T. Hsieh, Discrete gauge anomalies revisited, arXiv:1808.02881 [hep-th].
  47. M. Guo, K. Ohmori, P. Putrov, Z. Wan, and J. Wang, Fermionic finite-group gauge theories and interacting symmetric/crystalline orders via cobordisms, Commun. Math. Phys. 376, 1073 (2020).
  48. Z. Wan and J. Wang, Beyond standard models and grand unifications: Anomalies, topological terms, and dynamical constraints via cobordisms, J. High Energy Phys. 07 (2020) 062.
  49. J. Wang, X.-G. Wen, and E. Witten, Symmetric gapped interfaces of SPT and SET states: Systematic constructions, Phys. Rev. X 8, 031048 (2018).
  50. C. Arf, Untersuchungen über quadratische Formen in Körpern der Charakteristik 2, I, J. Reine Angew. Math. 1941, 148 (1941).
  51. E. H. Brown, Generalizations of the Kervaire invariant, Ann. Math. 95 368 (1972).
  52. A. Kapustin, R. Thorngren, A. Turzillo, and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, J. High Energy Phys. 12 (2015) 052.
  53. L. M. Krauss and F. Wilczek, Discrete gauge symmetry in continuum theories, Phys. Rev. Lett. 62, 1221 (1989).
  54. J. Davighi, B. Gripaios, and N. Lohitsiri, Global anomalies in the Standard Model(s) and Beyond, J. High Energy Phys. 07 (2020) 232.
  55. S. Koren, A note on proton stability in the standard model, Universe 8, 308 (2022).
  56. Z. Wan, J. Wang, and Y.-Z. You, Topological responses of the standard model gauge group, arXiv:2412.21196 [hep-th].
  57. J. Wang, Topological quantum dark matter via global anomaly cancellation, arXiv:2502.21319 [hep-th].
  58. Z. Wan and J. Wang, Anomalous (3+1)d fermionic topological quantum field theories via symmetry extension, arXiv:2512.25038 [hep-th].
  59. Z. Wan, J. Wang, and S.-T. Yau, Three Families and Pontryagin Class Mod 3: Topological order via symmetry extension, 2025, https://lims.ac.uk/paper/solve-three-generations/, https://lims.ac.uk/documents/solve-three-generations.pdf
  60. C. Cordova and K. Ohmori, Anomaly constraints on gapped phases with discrete chiral symmetry, Phys. Rev. D 102, 025011 (2020).
  61. A. Debray, W. Ye, and M. Yu, How to build anomalous (3+1)d topological quantum field theories, arXiv:2510.24834 [math-ph].
  62. A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in three dimensions and group cohomology, Phys. Rev. Lett. 112, 231602 (2014).
  63. A. Debray, M. Yu, and W. Ye, Symmetric gapped states and symmetry-enforced gaplessness in 3-dimension, arXiv:2602.12335 [hep-th].
  64. S. Koren, Cosmological lithium solution from discrete gauged B − L, Phys. Rev. Lett. 131, 091003 (2023).
  65. A. Maleknejad, Chiral gravity waves and leptogenesis in inflationary models with non-abelian gauge fields, Phys. Rev. D 90, 023542 (2014).
  66. A. Maleknejad, Gravitational leptogenesis in axion inflation with SU(2) gauge field, J. Cosmol. Astropart. Phys. 2016, 027 (2016).
  67. P. Adshead, A. J. Long, and E. I. Sfakianakis, Gravitational leptogenesis, reheating, and models of neutrino mass, Phys. Rev. D 97, 043511 (2018).
  68. D. H. Lyth and Y. Rodriguez, The Inflationary prediction for primordial non-Gaussianity, Phys. Rev. Lett. 95, 121302 (2005).
  69. D. H. Lyth, Generating the curvature perturbation at the end of inflation, J. Cosmol. Astropart. Phys. 2005, 006 (2005).
  70. R. R. Caldwell and C. Devulder, Axion gauge field inflation and gravitational leptogenesis: A lower bound on B modes from the matter-antimatter asymmetry of the universe, Phys. Rev. D 97, 023532 (2018).
  71. S. Davidson and A. Ibarra, A Lower bound on the right-handed neutrino mass from leptogenesis, Phys. Lett. B 535, 25 (2002).

Outline

Information

Phys. Rev. Research 8, L022024– Published 5 May, 2026
  • Received 21 January 2025
  • Accepted 13 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation