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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.

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.

We report progress in our exploration of the finite-temperature phase structure of two-flavour lattice
QCD with twisted-mass Wilson fermions and a tree-level Symanzik-improved gauge action
for a temporal lattice size Nt = 8. Extending our investigations to a wider region of parameter
space we gain a global view of the rich phase structure. We identify the finite temperature transition/
crossover for a non-vanishing twisted-mass parameter in the neighbourhood of the zerotemperature
critical line at sufficiently high b . Our findings are consistent with Creutz’s conjecture
of a conical shape of the finite temperature transition surface. Comparing with NLO lattice
cPT we achieve an improved understanding of this shape.

We discuss the use of Wilson fermions with twisted mass for simulations of QCD thermodynamics.
As a prerequisite for a future analysis of the finite-temperature transition making use
of automatic O(a) improvement, we investigate the phase structure in the space spanned by the
hopping parameter k , the coupling b , and the twisted mass parameter m. We present results for
Nf = 2 degenerate quarks on a 163×8 lattice, for which we investigate the possibility of an Aoki
phase existing at strong coupling and vanishing m, as well as of a thermal phase transition at
moderate gauge couplings and non-vanishing m.

Pseudo-Critical Temperature and Thermal Equation of State from Nf = 2 Twisted Mass Lattice QCD
(2012)

We report about the current status of our ongoing study of the chiral limit of two-flavor QCD at finite temperature with twisted mass quarks. We estimate the pseudo-critical temperature Tc for three values of the pion mass in the range of mPS ~ 300 and 500 MeV and discuss different chiral scenarios. Furthermore, we present first preliminary results for the trace anomaly, pressure and energy density. We have studied several discretizations of Euclidean time up to Nt = 12 in order to assess the continuum limit of the trace anomaly. From its interpolation we evaluate the pressure and energy density employing the integral method. Here, we have focussed on two pion masses with mPS ~ 400 and 700 MeV.

We review our knowledge of the phase diagram of QCD as a function of temperature, chemical potential and quark masses. The presence of tricritical lines at imaginary chemical potential m = i p 3 T, with known scaling behaviour in their vicinity, puts constraints on this phase diagram, especially in the case of two light flavors. We show first results in our project to determine the finite-temperature behaviour in the Nf = 2 chiral limit.