Strong coupling expansion for Yang-Mills theory at finite temperature

Euclidean strong coupling expansion of the partition function is applied to lattice Yang-Mills theory at finite temperature, i.e. for lattices with a compactified temporal direction. The expansions have a finite radius of convergence and thus are valid only for $\beta<\beta_c$, where $\beta_c$ denotes the nearest singularity of the free energy on the real axis. The accessible temperature range is thus the confined regime up to the deconfinement transition. We have calculated the first few orders of these expansions of the free energy density as well as the screening masses for the gauge groups SU(2) and SU(3). The resulting free energy series can be summed up and corresponds to a glueball gas of the lowest mass glueballs up to the calculated order. Our result can be used to fix the lower integration constant for Monte Carlo calculations of the thermodynamic pressure via the integral method, and shows from first principles that in the confined phase this constant is indeed exponentially small. Similarly, our results also explain the weak temperature dependence of glueball screening masses below $T_c$, as observed in Monte Carlo simulations. Possibilities and difficulties in extracting $\beta_c$ from the series are discussed.Comment: 7 pages, 3 figures, 1 tables, corrected typo

Centre symmetric 3d effective actions for thermal SU(N) Yang-Mills from strong coupling series

We derive three-dimensional, Z(N)-symmetric effective actions in terms of Polyakov loops by means of strong coupling expansions, starting from thermal SU(N) Yang-Mills theory in four dimensions on the lattice. An earlier action in the literature, corresponding to the (spatial) strong coupling limit, is thus extended by several higher orders, as well as by additional interaction terms. We provide analytic mappings between the couplings of the effective theory and the parameters $N_\tau,\beta$ of the original thermal lattice theory, which can be systematically improved. We then investigate the deconfinement transition for the cases SU(2) and SU(3) by means of Monte Carlo simulations of the effective theory. Our effective models correctly reproduce second order 3d Ising and first order phase transitions, respectively. Furthermore, we calculate the critical couplings $\beta_c(N_\tau)$ and find agreement with results from simulations of the 4d theory at the few percent level for $N_\tau=4-16$.Comment: 27 pages, 21 figures; final version published in JHEP; attached the corresponding Erratum (ref. JHEP 1107:014,2011, DOI 10.1007/JHEP07(2011)014) for ease of consultatio

The pressure of strong coupling lattice QCD with heavy quarks, the hadron resonance gas model and the large N limit

In this paper we calculate the pressure of pure lattice Yang-Mills theories and lattice QCD with heavy quarks by means of strong coupling expansions. Dynamical fermions are introduced with a hopping parameter expansion, which also allows for the incorporation of finite quark chemical potential. We show that in leading orders the results are in full agreement with expectations from the hadron resonance gas model, thus validating it with a first principles calculation. For pure Yang-Mills theories we obtain the corresponding ideal glueball gas, in QCD with heavy quarks our result equals that of an ideal gas of mesons and baryons. Another finding is that the Yang-Mills pressure in the large N limit is of order $\sim N^0$ to the calculated orders, when the inverse 't Hooft coupling is used as expansion parameter. This property is expected in the confined phase, where our calculations take place.Comment: 12 pages, 4 figure

The deconfinement transition of finite density QCD with heavy quarks from strong coupling series

Starting from Wilson's action, we calculate strong coupling series for the Polyakov loop susceptibility in lattice gauge theories for various small N_\tau in the thermodynamic limit. Analysing the series with Pad\'e approximants, we estimate critical couplings and exponents for the deconfinement phase transition. For SU(2) pure gauge theory our results agree with those from Monte-Carlo simulations within errors, which for the coarser N_\tau=1,2 lattices are at the percent level. For QCD we include dynamical fermions via a hopping parameter expansion. On a N_\tau=1 lattice with N_f=1,2,3, we locate the second order critical point where the deconfinement transition turns into a crossover. We furthermore determine the behaviour of the critical parameters with finite chemical potential and find the first order region to shrink with growing \mu. Our series moreover correctly reflects the known Z(N) transition at imaginary chemical potential.Comment: 18 pages, 7 figures, typos corrected, version published in JHE

Onset Transition to Cold Nuclear Matter from Lattice QCD with Heavy Quarks

Lattice QCD at finite density suffers from a severe sign problem, which has so far prohibited simulations of the cold and dense regime. Here we study the onset of nuclear matter employing a three-dimensional effective theory derived by combined strong coupling and hopping expansions, which is valid for heavy but dynamical quarks and has a mild sign problem only. Its numerical evaluations agree between a standard Metropolis and complex Langevin algorithm, where the latter is free of the sign problem. Our continuum extrapolated data clearly show a first order phase transition building up at $\mu_B \approx m_B$ as the temperature approaches zero. An excellent description of the data is achieved by an analytic solution in the strong coupling limit.Comment: Four pages, three figures; uses REVTeX-4. Version accepted by PRL. Title changed upon request by the Editor

Onset Transition to Cold Nuclear Matter from Lattice QCD with Heavy Quarks

Lattice QCD at finite density suffers from a severe sign problem, which has so far prohibited simulations of the cold and dense regime. Here we study the onset of nuclear matter employing a three-dimensional effective theory derived by combined strong coupling and hopping expansions, which is valid for heavy but dynamical quarks and has a mild sign problem only. Its numerical evaluations agree between a standard Metropolis and complex Langevin algorithm, where the latter is free of the sign problem. Our continuum extrapolated data approach a first order phase transition at µB ≈ mB as the temperature approaches zero. An excellent description of the data is achieved by an analytic solution in the strong coupling limit. PACS numbers: 05.70. Fh,11.15Ha,12.38.Gc Keywords: QCD phase diagram, lattice gauge theory, sign problem QCD at zero temperature is expected to exhibit the so-called silver blaze property: when a chemical potential for baryon number µ B is switched on in the grand canonical partition function, initially all observables should be completely independent of µ B . This changes abruptly once the chemical potential exceeds a critical value µ Bc , for which the baryon number jumps from zero to a finite value and a transition to a condensed state of nuclear matter takes place. The reason for this behavior is the mass gap in the fermionic spectrum, where the nucleon mass m B represents the lowest baryonic energy that can be populated once µ B ≈ m B . While this picture is easy to see in terms of energy levels of nucleons in a Hamiltonian language, it is elusive in the fundamental formulation of QCD thermodynamics in terms of a path integral. There, chemical potential enters through the Dirac operators of the quark fields, and hence all Dirac eigenvalues are shifted for any value of µ B . The silver blaze property thus requires some exact cancellations for µ B &lt; m B . An analytic derivation of the silver blaze property from the path integral exists only for the related case of finite isospin chemical potential where Bose-Einstein condensation of pions sets in at µ I = m π /2. A numerical demonstration of the effect by means of lattice QCD has so far been impossible due to the so-called sign problem. For finite baryon chemical potential the action becomes complex, prohibiting its use in a Boltzmann factor for Monte Carlo approaches with importance sampling. Several approximate methods have been devised to circumvent this problem. These are valid in the regime µ &lt; ∼ T , where they give consistent results (for a recent review see In this work we show that cold and dense lattice QCD is accessible within a 3d effective theory of Polyakov loops, which has been derived from the full lattice theory with Wilson fermions by means of strong coupling and hopping parameter expansions The lattice QCD partition function with Wilson gauge action S g [U ] and f = 1, . . . , N f quark flavours with Wilson fermion matrix Q(κ f , µ f ) can be written a

Glueballs, gluon condensate, and pure glue QCD below T_c

A quasiparticle description of pure glue QCD thermodynamics at T<T_c is proposed and compared to recent lattice data. Given that a gas of glueballs with constant mass cannot quantitatively reproduce the early stages of the deconfinement phase transition, the problem is to identify a relevant mechanism leading to the observed sudden increase of the pressure, trace anomaly, etc. It is shown that the strong decrease of the gluon condensate near T_c combined with the increasing thermal width of the lightest glueballs might be the trigger of the phase transition.Comment: 5 pages, 5 figures; analysis refined in v2, explanations added; v3 to appear in EPJ