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.
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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.
The authors declare no competing financial interests.
Nature thanks M. Paola and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Extended data figures and tables
Extended Data Figure 1 The probability distribution of the eigenvalues corresponding to the would-be zero modes.
The plot shows the results of dynamical lattice QCD calculations with nf = 2 + 1 + 1 flavours of staggered quarks at a temperature of T = 240 MeV. The result is obtained using the chirality method described in the text. The different colours refer to different lattice spacings; see key at right. Nt is the number of lattice points in the temporal direction. Note that it is inversely proportional to the lattice spacing: a = 1/(TNt).
Extended Data Figure 2 Expectation value of the weight factors in different topological sectors, 〈w〉Q, as a function of the lattice spacing squared.
The plot shows lattice results with nf = 3 + 1 flavours of staggered quarks at a temperature of T = 300 MeV. For clarity we plot the results as a function of 10/Nt2. Different colours correspond to different topological charge (Q) sectors. Error bars, s.e.m.
Extended Data Figure 3 The lattice spacing dependence of the topological susceptibility χ obtained from three different methods described in the main text, namely standard, ratio and reweighting.
For the reweighting method, a continuum extrapolation is also shown. The plot shows lattice results with nf = 2 + 1 + 1 flavours of staggered quarks at a temperature of T = 150 MeV. At this relatively low temperature the standard (‘brute force’) method still cannot provide three lattice spacings, which extrapolate to the proper continuum limit, though they correspond to very fine lattices with Nt = 12, 16 and 20. Error bars are s.e.m. and smaller than the symbols.
Extended Data Figure 4 Lattice spacing dependence of the topological susceptibility obtained from four different methods described in the text, namely, standard, ratio, reweighting and integral.
The plot shows lattice results with nf = 2 + 1 + 1 flavours of staggered quarks at a temperature of T = 300 MeV. For the ratio method, a misleading continuum extrapolation using Nt = 8, 10 and 12 is shown with a dashed line. For the reweighting and integral methods, continuum extrapolations are shown with bands. Error bars, s.e.m.
Extended Data Figure 5 Histograms of the topological charge (Q) for different lattice spacings in simulations under the constraint Q > 0.5.
The relative fraction of configurations in each bin is plotted. The centre of the peaks, denoted by z, is also given. The plot shows pure gauge theory simulations at T ≈ 6Tc.
Extended Data Figure 6 Histograms of the topological charge from fixed sector simulations for Q = 0–8.
The relative fraction of configurations in each bin is plotted. The sector boundaries are defined using a z factor, as described in the text. Note that the relative weights between the histograms are not included in the plot. These can be determined from the fixed sector integral technique. The plot shows pure gauge theory simulations at a temperature of T = 5Tc. Colour coding: red, 8 × 163 lattice with fixed topological charges Q = 0–8; green, 8 × 323 lattice with fixed Q = 1; blue, 8 × 643 lattice with fixed Q = 8.
The plot shows bQ as defined in equation (2) from pure gauge theory simulations on 8 × 163 lattices at temperature T = 5Tc. The different points correspond to independent simulation streams and different topological sectors. A good fit can be obtained assuming ergodicity and that the action difference scales linearly with the topological charge, see equation (5). The horizontal lines correspond to this fit at the given Q values and the reduced χ2 is shown at top right. Error bars are s.e.m. and smaller than the symbols.
Extended Data Figure 8 Lattice-spacing and finite-volume dependence of the decay exponent of the topological susceptibility, b.
The decay exponent b as a function of the lattice spacing squared shows the continuum extrapolation (upper panel) whereas b as a function of the linear extent of the lattice represents the infinite volume extrapolation (lower panel). The plots shows pure gauge theory simulations at temperature T = 6Tc. The lines are obtained from a joint fit, which takes into account both finite spacing and finite size effects. For the exponent we obtain b = 7.1(3) in the continuum and infinite volume limit at this particular temperature. This is in good agreement with our previous estimate from the direct method5, where we obtained b = 7.1(4). Error bars, s.e.m.
The results shown are from an earlier direct simulation5 and from the novel fixed sector integral technique. Upper panel, the temperature dependence of decay exponent b; lower panel, temperature dependence of the topological susceptibility itself. Filled red and pink circles show the lattice results for the decay exponent on 8 × 323 and 8 × 643 lattices, respectively. The open circles show the topological susceptibility from the fixed sector integral technique. The green band refers to the result of ref. 5, the key shows the abbreviated arXiv preprint number. The black arrow indicates the Stefan–Boltzmann limit. We also show the result from the DIGA with blue bands, see, for example, ref. 33. To convert the result into units of Tc we used from ref. 15. The width of the DIGA prediction reflects the change of the renormalization scale from 1/2 πT to 2 πT. For the exponent b we see a good agreement for temperatures above ~4Tc, for smaller temperatures the lattice tends to give smaller values than the DIGA. In the case of the susceptibility, the DIGA underestimates the lattice result by about an order of magnitude, this has already been observed in ref. 5. The ratio at T = 2.4Tc is K = 11.1(2.6), where the error is dominated by the lattice calculation. Error bars, s.e.m.
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Borsanyi, S., Fodor, Z., Guenther, J. et al. Calculation of the axion mass based on high-temperature lattice quantum chromodynamics. Nature 539, 69–71 (2016) doi:10.1038/nature20115
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