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Lattice Yang-Mills theories at finite temperature can be mapped onto effective 3d spin systems, thus facilitating their numerical investigation. Using strong-coupling expansions we derive effective actions for Polyakov loops in the SU(2) and SU(3) cases and investigate the effect of higher order corrections. Once a formulation is obtained which allows for Monte Carlo analysis, the nature of the phase transition in both classes of models is investigated numerically, and the results are then used to predict – with an accuracy within a few percent – the deconfinement point in the original 4d Yang-Mills pure gauge theories, for a series of values of Nt at once.

In quantum field theories at finite temperature spectral functions describe how particle systems behave in the presence of a thermal medium. Although data from lattice simulations can in principle be used to determine spectral function characteristics, existing methods rely on the extraction of these quantities from temporal correlators, which requires one to circumvent an illposed inverse problem. In these proceedings we report on a recent approach that instead utilises the non-perturbative constraints imposed by field locality to extract spectral function information directly from spatial correlators. In particular, we focus on the application of this approach to lattice QCD data of the spatial pseudo-scalar meson correlator in the temperature range 220−960 MeV, and outline why this data supports the conclusion that there exists a distinct pion state above the chiral pseudo-critical temperature Tpc.

Perturbation theory for non-abelian gauge theories at finite temperature is plagued by infrared
divergences which are caused by magnetic soft modes ~ g2T, corresponding to gluon fields of
a 3d Yang-Mills theory. While the divergences can be regulated by a dynamically generated
magnetic mass on that scale, the gauge coupling drops out of the effective expansion parameter
requiring summation of all loop orders for the calculation of observables. Some gauge invariant
possibilities to implement such infrared-safe resummations are reviewed. We use a scheme based
on the non-linear sigma model to estimate some of the contributions ~ g6 of the soft magnetic
modes to the QCD pressure through two loops. The NLO contribution amounts to ~ 10% of the
LO, suggestive of a reasonable convergence of the series.

The so-called sign problem of lattice QCD prohibits Monte Carlo simulations at finite baryon
density by means of importance sampling. Over the last few years, methods have been developed
which are able to circumvent this problem as long as the quark chemical potential is m=T <~1.
After a brief review of these methods, their application to a first principles determination of the
QCD phase diagram for small baryon densities is summarised. The location and curvature of the
pseudo-critical line of the quark hardon transition is under control and extrapolations to physical
quark masses and the continuum are feasible in the near future. No definite conclusions can as
yet be drawn regarding the existence of a critical end point, which turns out to be extremely quark
mass and cut-off sensitive. Investigations with different methods on coarse lattices show the lightmass
chiral phase transition to weaken when a chemical potential is switched on. If persisting on
finer lattices, this would imply that there is no chiral critical point or phase transition for physical
QCD. Any critical structure would then be related to physics other than chiral symmetry breaking.

The chiral critical surface is a surface of second order phase transitions bounding the region of
first order chiral phase transitions for small quark masses in the fmu;d;ms;mg parameter space.
The potential critical endpoint of the QCD (T;m)-phase diagram is widely expected to be part of
this surface. Since for m = 0 with physical quark masses QCD is known to exhibit an analytic
crossover, this expectation requires the region of chiral transitions to expand with m for a chiral
critical endpoint to exist. Instead, on coarse Nt = 4 lattices, we find the area of chiral transitions
to shrink with m, which excludes a chiral critical point for QCD at moderate chemical potentials
mB < 500 MeV. First results on finer Nt = 6 lattices indicate a curvature of the critical surface
consistent with zero and unchanged conclusions. We also comment on the interplay of phase
diagrams between the Nf = 2 and Nf = 2+1 theories and its consequences for physical QCD.

Phase transitions in a non-perturbative regime can be studied by ab initio Lattice Field Theory methods. The status and future research directions for LFT investigations of Quantum Chromo-Dynamics under extreme conditions are reviewed, including properties of hadrons and of the hypothesized QCD axion as inferred from QCD topology in different phases. We discuss phase transitions in strong interactions in an extended parameter space, and the possibility of model building for Dark Matter and Electro-Weak Symmetry Breaking. Methodological challenges are addressed as well, including new developments in Artificial Intelligence geared towards the identification of different phases and transitions.

The QCD equation of state is not often discussed in cosmology. However, the relic density of
weakly interacting massive particles (WIMPs) depends on the entropy and the expansion rate of
the Universe when they freeze out, at a temperature in the range 400 MeV – 40GeV, where QCD
corrections are still important. We use recent analytic and lattice calculations of the QCD pressure
to produce a new equation of state suitable for use in relic density calculations. As an example,
we show that relic densities calculated by the dark matter package DarkSUSY receive corrections
of several per cent, within the observational accuracy of the Planck CMB mission, due for launch
in 2007.

A lot of effort in lattice simulations over the last years has been devoted to studies of the QCD deconfinement transition. Most state-of-the-art simulations use rooted staggered fermions, while Wilson fermions are affected by large systematic uncertainties, such as coarse lattices or heavy sea quarks. Here we report on an ongoing study of the transition, using two degenerate flavours of nonperturbatively O(a) improved Wilson fermions. We start with Nt = 12 and 16 lattices and pion masses of 600 to 450 MeV, aiming at chiral and continuum limits with light quarks.

Recently, an approximate SU(4) chiral spin-flavour symmetry was observed in multiplet patterns of QCD meson correlation functions, in a temperature range above the chiral crossover. This symmetry is larger than the chiral symmetry of massless QCD, and can only arise effectively when colour-electric quark-gluon interactions dynamically dominate the quantum effective action. At temperatures about three times the crossover temperature, these patterns disappear again, indicating the screening of colour-electric interactions, and the expected chiral symmetry is recovered. In this contribution we collect independent evidence for such an intermediate temperature range, based on screening masses and the pion spectral function. Both kinds of observables behave non-perturbatively in this window, with resonance-like peaks for the pion and its first excitation disappearing gradually with temperature. Using symmetry arguments and the known behaviour of screening masses at small densities, we discuss how this chiral spin symmetric band continues into the QCD phase diagram.

For a long time, strong coupling expansions have not been applied systematically in lattice QCD thermodynamics, in view of the success of numerical Monte Carlo studies. The persistent sign problem at finite baryo-chemical potential, however, has motivated investigations using these methods, either by themselves or combined with numerical evaluations, as a route to finite density physics. This article reviews the strategies, by which a number of qualitative insights have been attained, notably the emergence of the hadron resonance gas or the identification of the onset transition to baryon matter in specific regions of the QCD parameter space. For the simpler case of Yang–Mills theory, the deconfinement transition can be determined quantitatively even in the scaling region, showing possible prospects for continuum physics.