Letter | Published:

Calculation of the axion mass based on high-temperature lattice quantum chromodynamics

Nature volume 539, pages 6971 (03 November 2016) | Download Citation


Unlike the electroweak sector of the standard model of particle physics, quantum chromodynamics (QCD) is surprisingly symmetric under time reversal. As there is no obvious reason for QCD being so symmetric, this phenomenon poses a theoretical problem, often referred to as the strong CP problem. The most attractive solution for this1 requires the existence of a new particle, the axion2,3—a promising dark-matter candidate. Here we determine the axion mass using lattice QCD, assuming that these particles are the dominant component of dark matter. The key quantities of the calculation are the equation of state of the Universe and the temperature dependence of the topological susceptibility of QCD, a quantity that is notoriously difficult to calculate4,5,6,7,8, especially in the most relevant high-temperature region (up to several gigaelectronvolts). But by splitting the vacuum into different sectors and re-defining the fermionic determinants, its controlled calculation becomes feasible. Thus, our twofold prediction helps most cosmological calculations9 to describe the evolution of the early Universe by using the equation of state, and may be decisive for guiding experiments looking for dark-matter axions. In the next couple of years, it should be possible to confirm or rule out post-inflation axions experimentally, depending on whether the axion mass is found to be as predicted here. Alternatively, in a pre-inflation scenario, our calculation determines the universal axionic angle that corresponds to the initial condition of our Universe.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.


  1. 1.

    & CP conservation in the presence of instantons. Phys. Rev. Lett. 38, 1440–1443 (1977)

  2. 2.

    A new light boson? Phys. Rev. Lett. 40, 223–226 (1978)

  3. 3.

    Problem of strong P and T invariance in the presence of instantons. Phys. Rev. Lett. 40, 279–282 (1978)

  4. 4.

    , & Lattice QCD input for axion cosmology. Phys. Rev. D 92, 034507 (2015)

  5. 5.

    et al. Axion cosmology, lattice QCD and the dilute instanton gas. Phys. Lett. B 752, 175–181 (2016)

  6. 6.

    , , , & Topological susceptibility from Nf = 2 + 1 + 1 lattice QCD at nonzero temperature. J. Phys. Conf. Ser. 668, 012123 (2016)

  7. 7.

    et al. Axion phenomenology and θ-dependence from Nf = 2 + 1 lattice QCD. J. High Energy Phys. 2016, 155 (2016)

  8. 8.

    , & The topological susceptibility in finite temperature QCD and axion cosmology. Phys. Lett. B (2016)

  9. 9.

    & Axion cosmology revisited. Phys. Rev. D 82, 123508 (2010)

  10. 10.

    Confinement of quarks. Phys. Rev. D 10, 2445–2459 (1974)

  11. 11.

    , , , & The order of the quantum chromodynamics transition predicted by the standard model of particle physics. Nature 443, 675–678 (2006)

  12. 12.

    & Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11, 395–408 (1975)

  13. 13.

    & Analytic smearing of SU(3) link variables in lattice QCD. Phys. Rev. D 69, 054501 (2004)

  14. 14.

    Exactly massless quarks on the lattice. Phys. Lett. B 417, 141–144 (1998)

  15. 15.

    ., , , & Precision SU(3) lattice thermodynamics for a large temperature range. J. High Energy Phys. 2012, 56 (2012)

  16. 16.

    et al. High-precision scale setting in lattice QCD. J. High Energy Phys. 2012, 10 (2012)

  17. 17.

    et al. Ab initio determination of light hadron masses. Science 322, 1224–1227 (2008)

  18. 18.

    et al. Ab initio calculation of the neutron-proton mass difference. Science 347, 1452–1455 (2015)

  19. 19.

    , , & NNLO hard-thermal-loop thermodynamics for QCD. Phys. Lett. B 696, 468–472 (2011)

  20. 20.

    et al. Review of particle physics. Chin. Phys. C 38, 090001 (2014)

  21. 21.

    Properties and uses of the Wilson flow in lattice QCD. J. High Energy Phys. 2010, 71 (2010); Erratum. J. High Energy Phys. 2014, 92 (2014)

  22. 22.

    & Standard Model thermodynamics across the electroweak crossover. J. Cosmol. Astropart. Phys. 2015, 035 (2015)

  23. 23.

    et al. The QCD transition temperature: results with physical masses in the continuum limit II. J. High Energy Phys. 2009, 088 (2009)

  24. 24.

    et al. Is there still any Tc mystery in lattice QCD? Results with physical masses in the continuum limit III. J. High Energy Phys. 2010, 073 (2010)

  25. 25.

    , & Staggered eigenvalue mimicry. Phys. Rev. D 70, 094502 (2004)

  26. 26.

    Theoretical issues with staggered fermion simulations. In Proceedings of Science Vol. LAT2005, 021, (2006)

  27. 27.

    & Lattice QCD without topology barriers. J. High Energy Phys. 2011, 036 (2011)

  28. 28.

    , & Metadynamics surfing on topology barriers: the CP(N-1) case. J. High Energy Phys. 2016, 89 (2015)

  29. 29.

    Lattice QCD on non-orientable manifolds. Preprint at (2015)

  30. 30.

    , , , & Topological susceptibility at high temperature on the lattice. J. High Energy Phys. 2016, 021 (2016)

  31. 31.

    , & Improving the chiral properties of lattice fermions. Phys. Rev. D 67, 054501 (2003)

  32. 32.

    , & Topological structure in the SU(2) vacuum. Nucl. Phys. B 505, 417–441 (1997)

  33. 33.

    & Confronting instanton perturbation theory with QCD lattice results. Phys. Lett. B 459, 249–258 (1999)

Download references


We thank M. Dierigl, M. Giordano, S. Krieg, D. Nogradi and B. Toth for discussions. This project was funded by the DFG (grant SFB/TR55) and by OTKA (grant OTKA-K-113034). T.G.K. is supported by the Hungarian Academy of Sciences under ‘Lendulet’ grant no. LP 2011-011. The work of J.R. is supported by the Ramon y Cajal Fellowship 2012-10597 and by FPA2015-65745-P (MINECO/FEDER). The computations were performed on JUQUEEN at Forschungszentrum Jülich, on SuperMUC at Leibniz Supercomputing Centre in München, on Hazel Hen at the High Performance Computing Center in Stuttgart, on QPACE in Wuppertal and on GPU clusters in Wuppertal, Budapest and Debrecen.

Author information


  1. Department of Physics, University of Wuppertal, D-42119 Wuppertal, Germany

    • S. Borsanyi
    • , Z. Fodor
    • , J. Guenther
    • , K.-H. Kampert
    • , A. Pasztor
    •  & K. K. Szabo
  2. Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52428 Jülich, Germany

    • Z. Fodor
    • , T. Kawanai
    • , S. W. Mages
    •  & K. K. Szabo
  3. Institute for Theoretical Physics, Eötvös University, H-1117 Budapest, Hungary

    • Z. Fodor
    • , S. D. Katz
    •  & F. Pittler
  4. MTA-ELTE Lendület Lattice Gauge Theory Research Group, H-1117 Budapest, Hungary

    • S. D. Katz
    •  & F. Pittler
  5. Institute for Nuclear Research of the Hungarian Academy of Sciences, H-4026 Debrecen, Hungary

    • T. G. Kovacs
  6. University of Zaragoza, E-50009 Zaragoza, Spain

    • J. Redondo
  7. Max Planck Institut für Physik, D-80803 Munich, Germany

    • J. Redondo
  8. Deutsches Elektronen-Synchrotron DESY, D-22607 Hamburg, Germany

    • A. Ringwald


  1. Search for S. Borsanyi in:

  2. Search for Z. Fodor in:

  3. Search for J. Guenther in:

  4. Search for K.-H. Kampert in:

  5. Search for S. D. Katz in:

  6. Search for T. Kawanai in:

  7. Search for T. G. Kovacs in:

  8. Search for S. W. Mages in:

  9. Search for A. Pasztor in:

  10. Search for F. Pittler in:

  11. Search for J. Redondo in:

  12. Search for A. Ringwald in:

  13. Search for K. K. Szabo in:


S.B. and S.W.M. developed the fixed sector integral, T.G.K. and K.K.S. the eigenvalue reweighting, and F.P. the odd flavour overlap techniques. S.B., J.G., S.D.K, T.G.K., T.K., S.W.M., A.P., F.P. and K.K.S. wrote the necessary codes, carried out the runs and determined the EoS and χ(T). A.P., J.R. and A.R. calculated the DIGA prediction. K.-H.K., J.R. and A.R. worked out the experimental set-up. Z.F. wrote the main paper and coordinated the project.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Z. Fodor.

Reviewer Information

Nature thanks M. Paola and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data

Supplementary information

PDF files

  1. 1.

    Supplementary Information

    This file contains Supplementary Text and Data, Supplementary Figures 1-28, Supplementary Tables 1-9 and additional references.

About this article

Publication history






Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.