THE PRESSURE OF A GLUON FIRE, AFTER TWENTY YEARS

Quarks are bound together by gluons, the carriers of the strong force. Heat matter enough and this binding is released: gluons roam free in a plasma. To study that state with fewer complications, physicists often drop the quarks and keep only the gluons — a theory called pure Yang-Mills. It keeps the key features of the real strong force, the authors note, while being simple enough to be simulated very precisely on computers.

One basic question about this plasma: what pressure does it exert at a given temperature?

An answer built step by step

At very high temperature, the force between gluons weakens, measured by a small number g, the coupling. The pressure can then be written as a series: a main term, then corrections proportional to g², g³, g⁴, and so on, each smaller than the last. The main term is the pressure of free gluons, called the Stefan-Boltzmann limit.

The authors retrace the history. The g⁴ term with a logarithm was found in 1982; the full g⁴ term only in 1994, because the three-loop calculations were so hard; the g⁵ term soon after. At the next order, g⁶, things get harder still. Part of it comes from slow magnetic fluctuations in the plasma that cannot be computed by this kind of expansion at all. A trick that splits the plasma into separate scales — fast, slower, slowest — handled that part, and computer simulations of a simplified three-dimensional version supplied its value.

That left one piece: the contribution of the fastest scale, which requires adding up every vacuum diagram with four loops. “Long considered intractable,” the team writes, “this problem has remained open for over two decades.”

65 diagrams, a million hours

Keijo Kajantie, Pablo Navarrete, Risto Paatelainen, Leon Sandbote, York Schröder, Kaapo Seppänen and Aleksi Vuorinen — from the Universities of Helsinki and Turku, the University of Bío-Bío in Chile and the University of Bern — took it on.

There are 65 four-loop diagrams of gluons (and auxiliary “ghost” particles).
Working out their algebra produced some 100,000 integrals, which reduce to 21 standard forms. Ten cannot be split into simpler pieces, and nine of them had never been computed.
To evaluate them, the team used a method called thermal loop-tree duality, combined with new ways of subtracting the infinities that appear at very high and very low momenta. What remained was finite and could be integrated numerically by random sampling: about one million CPU-hours on supercomputers.

As a check, the dependence on an arbitrary choice in the calculation (the gauge parameter) cancelled out, as it must. The infinities from the different scales also cancelled exactly, as the theory requires.

The number that was missing

The key new coefficient comes out at 6236 ± 40. An earlier estimate, inferred indirectly from simulations, gave 6000 ± 600 — consistent, but fifteen times less precise. In the authors’ usual notation, the full g⁶ coefficient of the pressure for SU(3) — the version with three “colours”, the one simulated on the lattice — is −3591.2, with an uncertainty of ±16.9 from this calculation and ±250 from the magnetic part computed on the lattice.

Theory meets simulation

The payoff shows when the series is compared with four-dimensional lattice simulations, which have computed this pressure up to a thousand times the temperature at which gluons break free (about 260 MeV). Lower orders behave badly: the g³ and g⁴ results overshoot the free-gluon limit, and g⁵ drops well below the simulations. With the g⁶ term included, using a standard way of reorganising the series, the theory falls in quantitative agreement with the lattice data down to a few times that critical temperature.

According to the authors, this is only the second time the thermodynamics of a hot quantum field theory has been worked out to this order, and the first in a physical gauge theory. While finishing, they learned that an independent group at the University of Granada was attempting the same calculation; the two teams checked that their main results agree.

Next on their list: the g⁷ term, and the full strong force with quarks and nonzero quark densities. That would reach regimes beyond today’s lattice simulations, relevant, the authors write, for mergers of neutron stars.