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The order of the quantum chromodynamics transition predicted by the standard model of particle physics

Nature volume 443, pages 675–678 (2006)Cite this article

Abstract

Quantum chromodynamics (QCD) is the theory of the strong interaction, explaining (for example) the binding of three almost massless quarks into a much heavier proton or neutron—and thus most of the mass of the visible Universe. The standard model of particle physics predicts a QCD-related transition that is relevant for the evolution of the early Universe. At low temperatures, the dominant degrees of freedom are colourless bound states of hadrons (such as protons and pions). However, QCD is asymptotically free, meaning that at high energies or temperatures the interaction gets weaker and weaker1,2, causing hadrons to break up. This behaviour underlies the predicted cosmological transition between the low-temperature hadronic phase and a high-temperature quark–gluon plasma phase (for simplicity, we use the word ‘phase’ to characterize regions with different dominant degrees of freedom). Despite enormous theoretical effort, the nature of this finite-temperature QCD transition (that is, first-order, second-order or analytic crossover) remains ambiguous. Here we determine the nature of the QCD transition using computationally demanding lattice calculations for physical quark masses. Susceptibilities are extrapolated to vanishing lattice spacing for three physical volumes, the smallest and largest of which differ by a factor of five. This ensures that a true transition should result in a dramatic increase of the susceptibilities. No such behaviour is observed: our finite-size scaling analysis shows that the finite-temperature QCD transition in the hot early Universe was not a real phase transition, but an analytic crossover (involving a rapid change, as opposed to a jump, as the temperature varied). As such, it will be difficult to find experimental evidence of this transition from astronomical observations.

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Figure 1: Susceptibilities for the light quarks for N t = 4 and for N t = 6 as a function of 6/ g 2 , where g is the gauge coupling.
Figure 2: Normalized susceptibilities T4/(m2Δ χ ) for the light quarks for given aspect ratios ( r ) as functions of the lattice spacing.
Figure 3: Continuum extrapolated susceptibilities T4/(m2Δ χ ) as a function of 1/( Tc3V).
Figure 4: The line of constant physics.

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Acknowledgements

We thank F. Csikor, A. Dougall, K.-H. Kampert, M. Nagy, Z. Rácz and D. J. Schwarz for discussions. This research was partially supported by a DFG German Science grant, OTKA Hungarian Science grants and an EU research grant. The computations were carried out on PC clusters at the University of Budapest and Wuppertal with next-neighbour communication architecture29 and on the BlueGene/L machine in Jülich. A modified version of the publicly available MILC code (http://physics.indiana.edu/~sg/milc.html) was used.

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Authors and Affiliations

  1. Department of Physics, University of Wuppertal, D-42097, Wuppertal, Germany

    Y. Aoki, Z. Fodor, S. D. Katz & K. K. Szabó

  2. Institute for Theoretical Physics, Eötvös University, H-1117, Budapest, Hungary

    G. Endrődi, Z. Fodor & S. D. Katz

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  1. Y. Aoki
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Correspondence to S. D. Katz.

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Aoki, Y., Endrődi, G., Fodor, Z. et al. The order of the quantum chromodynamics transition predicted by the standard model of particle physics. Nature 443, 675–678 (2006). https://doi.org/10.1038/nature05120

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