STM-D-0969Paper2006Published and peer-reviewed

SU(2) and SU(3) Yang–Mills thermodynamics and some implications

Ralf Hofmann

Abstract and summary · read the original at the source

In one page

Ralf Hofmann’s short paper is a guided tour of a programme he had spent years building: how to do the thermodynamics of a strongly interacting gauge theory exactly, without expanding in a small coupling that is not actually small. The move is to coarse-grain over the theory’s topological lumps — calorons — until a macroscopic, inert field emerges that carries the ground state. Do that for SU(2) and SU(3), the mathematics behind the weak force and the strong force, and each turns out to have three phases with a different vacuum in each: a hot deconfining one, a narrow magnetic one in which monopoles condense, and a cold confining one whose excitations are spin-one-half particles made of knotted vortex loops. The vacuum here is not a blank stage; it has a computable energy density and a negative pressure. Hofmann then states the implication he is chasing — that the photon is the massless mode of one such theory, its scale set by the temperature of the microwave background — and lists what would be seen if that is right.

Why it matters hereChapter 2 argues that the vacuum is a structured medium rather than an empty background, and this is one of the few places where the structure is derived rather than assumed: a ground state with a temperature, a pressure and an energy density that fall out of Yang–Mills theory itself. Chapter 13 keeps the frameworks that try to hold the vacuum, the particles and the dark sector in one hand, and this paper is that attempt in its most compact form, with observations attached to it.

What it claims

01Yang–Mills thermodynamics can be done analytically and nonperturbatively by subjecting the theory’s topological field configurations — calorons and anticalorons — to an optimised spatial coarse-graining. What emerges is a macroscopic, inert scalar field whose dynamics defines the ground state, and the fundamental gauge symmetry is broken in successive stages as the temperature falls: SU(2) and SU(3) each have a deconfining, a preconfining and a confining phase.Section 1, Introduction; Section 2, Deconfining phase
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02The ground state is not a free parameter. The potential of the coarse-grained field follows from its first-order BPS equation rather than the other way round, so the usual freedom to shift a potential by a constant is absent; and the mass of any fluctuation of that field is far larger than both the resolution scale and the temperature, so quantum and statistical fluctuations of it are absent altogether. In the deconfining and preconfining phases the ground-state pressure is negative and the vacuum energy density it generates rises linearly with temperature.Section 2, the derivation around Equation 3; Section 1, Introduction, third paragraph
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03The effective gauge coupling is nearly constant through most of the deconfining phase and diverges logarithmically at its lower edge: the coupling is 8.89 at a dimensionless temperature of 13.87 for SU(2), and 7.26 at 9.475 for SU(3). At that edge the screened magnetic monopoles become massless and condense, which is what drives the transition. The preconfining phase that follows is narrow — its window runs from 8.478 down to 7.075 for SU(2) and from 7.376 down to 6.467 for SU(3) — and it ends when the coupling of the dual theory itself diverges.Section 2, the paragraph quoting the critical couplings; Section 3, Preconfining phase, closing paragraph
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04Below the second transition the excitations change statistics. Single centre-vortex loops become abundant and condense into a new ground state; a loop that twists onto itself carries a monopole at the intersection and becomes a massive spin-one-half fermion, with mass equal to the number of self-intersections times the confining Yang–Mills scale. The ground-state pressure of that phase is exactly zero well below the scale, and the entropy density vanishes at the transition — at that point the theory generates nothing but a cosmological constant.Section 4, Confining phase; Section 5, Thermodynamical quantities
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05The implication Hofmann pursues is that the photon is the massless mode of an SU(2) theory whose deconfining boundary sits at the 2.7 kelvin temperature of the microwave background, rather than of the standard hypercharge U(1). On that postulate the present ground-state energy density of the theory comes out below 0.4 per cent of the measured dark-energy density, so an additional mechanism is required, and the candidate he names is a Planck-scale axion of mass around ten to the minus thirty-six electronvolts, caught on the slope of its potential by cosmological friction.Section 6, first implication
What to watch
06What to watch, because the paper says exactly what would be seen: a spectral gap at low frequency modifying black-body spectra at temperatures not far above that of the microwave background, which would also explain why the cold dilute clouds between the spiral arms of the outer galaxy are atomic rather than molecular hydrogen and why that state is stable; a photon that stays massless for at most a further two billion years before acquiring a Meissner mass, after which the ground state of the universe is superconducting; and, if the spectral gap is confirmed, six extra relativistic degrees of freedom at the megaelectronvolt freeze-out that put the standard account of light-element synthesis under strain unless the Fermi coupling there is larger than assumed.Section 6, second and third implications, with Figure 5
What to watch

Read it · abstract

Abstract

We sketch the development of effective theories for SU(2) and SU(3) Yang–Mills thermodynamics. The most important results are quoted and some implications for particle physics and cosmology are discussed.

Ralf Hofmann, Institut für Theoretische Physik, Universität Heidelberg, SU(2) and SU(3) Yang–Mills thermodynamics and some implications, Modern Physics Letters A 21, pages 999 to 1016 (2006); received 30 March 2006, revised 24 November 2006. The preprint is arXiv:hep-th/0603241.

(Abstract only — no other text of the paper is reproduced here; see the rights note above. On this site, the long account this paper condenses is at /library/stm-fb901bd417, the later treatment of the thermal ground state and its nonthermal probes at /library/stm-9a9190b65f, the self-intersecting centre-vortex loop that becomes a charged lepton at /library/stm-e1f7b72444, the charged-lepton spectra that follow at /library/stm-dcaf382795, the axial anomaly applied to galaxies and the dark universe at /library/stm-b06591e288, and the modified MIT bag whose vacuum structure sets the scene at /library/stm-6fa6fa9f58.)

The way in

https://doi.org/10.1142/s0217732306020457PUBLICATION. Modern Physics Letters A volume 21, pages 999 to 1016 (2006), copyright World Scientific; the same volume carries an erratum at page 3049 and a further note at page 3053. Crossref records no licence for the article. The batch record printed the title in capitals as the journal sets it; it is given here in normal case. WHAT WAS READ. The text read for this page is the author’s arXiv posting hep-th/0603241, version 3 of 27 November 2006, which arXiv labels under the legacy licence assumed for submissions of 1991 to 2003 — a distribution licence to arXiv, not a Creative Commons licence. The paper is therefore abstract-only here: the abstract below is the published one, it is the only text of the paper reproduced anywhere on this page, and every claim is located to a numbered section, equation or figure of that arXiv version. AUTHOR. Ralf Hofmann, then at the Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16; the paper was received 30 March 2006 and revised 24 November 2006. This is the short companion to his long 2005 account, and the two are read together. NUMBERS. Where the paper writes an inequality or an exponent, this page states it in words, because the page format requires it. RELATED PAGES: see the cross-links at the foot of this page.

How to cite it

Ralf Hofmann (2006) SU(2) and SU(3) Yang–Mills thermodynamics and some implications. doi:10.1142/s0217732306020457

Where it sits in the curriculum

What the vacuum is