Finite density QCD with a canonical approach

We present a canonical method where the properties of QCD are directly obtained as a function of the baryon density rho, rather than the chemical potential mu. We apply this method to the determination of the phase diagram of four-flavor QCD. For a pion mass m_pi \sim 350 MeV, the first-order transition between the hadronic and the plasma phase gives rise to a co-existence region in the T-rho plane, which we study in detail, including the associated interface tension. We obtain accurate results for systems containing up to 30 baryons and quark chemical potentials mu up to 2 T. Our T-mu phase diagram agrees with the literature when mu/T \lesssim 1. At larger chemical potential, we observe a ``bending down'' of the phase boundary. We compare the free energy in the confined and deconfined phase with predictions from a hadron resonance gas and from a free massless quark gas respectively.Comment: 6 pages, 9 figures, proceedings of "Workshop on Computational Hadron Physics", Cyprus, Sept. 200

QCD at zero baryon density

While the grand canonical partition function Z_{GC}(mu) with chemical potential mu explicitly breaks the Z_3 symmetry with the Dirac determinant, the canonical partition function at fixed baryon number Z_C(B) is manifestly Z_3-symmetric. We compare Z_{GC}(mu=0) and Z_C(B=0) formally and by numerical simulations, in particular with respect to properties of the deconfinement transition. Differences between the two ensembles, for physical observables characterising the phase transition, vanish with increasing lattice size. We show numerically that the free energy density is the same for both ensembles in the thermodynamic limit.Comment: Lattice2003(nonzero), 3 pages, 5 figure

QCD at Zero Baryon Density and the Polyakov Loop Paradox

We compare the grand canonical partition function at fixed chemical potential mu with the canonical partition function at fixed baryon number B, formally and by numerical simulations at mu=0 and B=0 with four flavours of staggered quarks. We verify that the free energy densities are equal in the thermodynamic limit, and show that they can be well described by the hadron resonance gas at T T_c. Small differences between the two ensembles, for thermodynamic observables characterising the deconfinement phase transition, vanish with increasing lattice size. These differences are solely caused by contributions of non-zero baryon density sectors, which are exponentially suppressed with increasing volume. The Polyakov loop shows a different behaviour: for all temperatures and volumes, its expectation value is exactly zero in the canonical formulation, whereas it is always non-zero in the commonly used grand-canonical formulation. We clarify this paradoxical difference, and show that the non-vanishing Polyakov loop expectation value is due to contributions of non-zero triality states, which are not physical, because they give zero contribution to the partition function.Comment: 21 pages, 7 figure

The θ\theta-term, CPN−1^{N-1} Model and the Inversion Approach in the Imaginary θ\theta Method

The weak coupling region of CP$^{N-1}$ lattice field theory with the $\theta$-term is investigated. Both the usual real theta method and the imaginary theta method are studied. The latter was first proposed by Bhanot and David. Azcoiti et al. proposed an inversion approach based on the imaginary theta method. The role of the inversion approach is investigated in this paper. A wide range of values of $h=-{\rm Im} \theta$ is studied, where $\theta$ denotes the magnitude of the topological term. Step-like behavior in the $x$-$h$ relation (where $x=Q/V$, $Q$ is the topological charge, and $V$ is the two dimensional volume) is found in the weak coupling region. The physical meaning of the position of the step-like behavior is discussed. The inversion approach is applied to weak coupling regions.Comment: PTPTEX, 17 pages with 13 figures. Some sentences were correcte

QCD at small baryon number

We consider the difficulties of finite density QCD from the canonical formalism. We present results for small baryon numbers, where the sign problem can be controlled, in particular by supplementing the mu=0 sampling with imaginary mu ensembles. We initiate the thermodynamic study of few-nucleon systems, starting with the measurement of the free energy of a few baryons in the confined and deconfined phase. We present a simple model for the phase transition, whose results are in good agreement with the literature, but extend to lower temperatures.Comment: Lattice2004(nonzero), 3 pages, 3 figures. 1 reference adde

The 3-state Potts model as a heavy quark finite density laboratory

The 3-D Z(3) Potts model is a model for finite temperature QCD with heavy quarks. The chemical potential in QCD becomes an external magnetic field in the Potts model. Following Alford et al.\cite{Alford_et_al}, we revisit this mapping, and determine the phase diagram for an arbitrary chemical potential, real or imaginary. Analytic continuation of the phase transition line between real and imaginary chemical potential can be tested with precision. Our results show that the chemical potential weakens the heavy-quark deconfinement transition in QCD.Comment: 6 pages and 7 figures. talk presented at Lattice 2005 (non-zero temperature and density

Testing Dimensional Reduction in SU(2) Gauge Theory

At high temperature, every $(d+1)$-dimensional theory can be reformulated as an effective theory in $d$ dimensions. We test the numerical accuracy of this Dimensional Reduction for (3+1)-dimensional SU(2) by comparing perturbatively determined effective couplings with lattice results as the temperature is progressively lowered. We observe an increasing disagreement between numerical and perturbative values from $T=4 T_c$ downwards, which may however be due to somewhat different implementations of dimensional reduction in the two cases.Comment: Lattice2001(hightemp), AMS-LaTeX v1.2, 3 pages with 3 figure

Critical point in finite density lattice QCD by canonical approach

We propose a method to find the QCD critical point at finite density calculating the canonical partition function ${\cal Z}_{\rm C} (T,N)$ by Monte-Carlo simulations of lattice QCD, and analyze data obtained by a simulation with two-flavor p4-improved staggered quarks with pion mass $m_{\pi} \approx 770 {\rm MeV}$. It is found that the shape of an effective potential changes gradually as the temperature decreases and a first order phase transition appears in the low temperature and high density region. This result strongly suggests the existence of the critical point in the $(T, \mu_q)$ phase diagram.Comment: 4 pages, 2 figures, To appear in the conference proceedings for Quark Matter 2009, March 30 - April 4, Knoxville, Tennesse

Glueball masses in 4d U(1) lattice gauge theory using the multi-level algorithm

We take a new look at plaquette-plaquette correlators in 4d compact U(1) lattice gauge theory which are separated in time, both in the confined and the deconfined phases. From the behaviour of these correlators we extract glueball masses in the scalar as well as the axial-vector channels. Also in the deconfined phase, the non-zero momentum axial-vector correlator gives us information about the photon which appears as a massless particle in the spectrum. Using the Luescher - Weisz multi-level algorithm, we are able to go to large time separations which were not possible previously.Comment: 18 pages, 7 figures and 3 tables. Minor notational change

Breakdown of staggered fermions at nonzero chemical potential

The staggered fermion determinant is complex when the quark chemical potential mu is nonzero. Its fourth root, used in simulations with dynamical fermions, will have phase ambiguities that become acute when Re mu is sufficiently large. We show how to resolve these ambiguities, but our prescription only works very close to the continuum limit. We argue that this regime is far from current capabilities. Other procedures require being even closer to the continuum limit, or fail altogether, because of unphysical discontinuities in the measure. At zero temperature the breakdown is expected when Re mu is greater than approximately half the pion mass. Estimates of the location of the breakdown at nonzero temperature are less certain.Comment: 6 pages RevTeX, 2 figures. Returning to v5 after erroneous replacement. Apologie