QCD with Chemical Potential and Matrix Models

FIGURE 7. Drawings of preserved tadpole specimen of Boophis rhodoscelis (ZSM 0630/2008) from National Park of Ranomafana (developmental stage 33). (a) dorsal view; (b) lateral view; (c) oral disk.

In this paper we carry out a low-temperature scan of the phase diagram of dense two-color QCD with $N_f=2$ quarks. The study is conducted using lattice simulation with rooted staggered quarks. At small chemical potential we observe the hadronic phase, where the theory is in a confining state, chiral symmetry is broken, the baryon density is zero and there is no diquark condensate. At the critical point $\mu = m_{\pi}/2$ we observe the expected second order transition to Bose-Einstein condensation of scalar diquarks. In this phase the system is still in confinement in conjunction with non-zero baryon density, but the chiral symmetry is restored in the chiral limit. We have also found that in the first two phases the system is well described by chiral perturbation theory. For larger values of the chemical potential the system turns into another phase, where the relevant degrees of freedom are fermions residing inside the Fermi sphere, and the diquark condensation takes place on the Fermi surface. In this phase the system is still in confinement, chiral symmetry is restored and the system is very similar to the quarkyonic state predicted by SU($N_c$) theory at large $N_c$.

We revisit two-color, two-flavor chiral perturbation theory at finite isospin and baryon density. We investigate the phase diagram obtained varying the isospin and the baryon chemical potentials, focusing on the phase transition occurring when the two chemical potentials are equal and exceed the pion mass (which is degenerate with the diquark mass). In this case, there is a change in the order parameter of the theory that does not lend itself to the standard picture of first order transitions. We explore this phase transition both within a Ginzburg-Landau framework valid in a limited parameter space and then by inspecting the full chiral Lagrangian in all the accessible parameter space. Across the phase transition between the two broken phases the order parameter becomes an SU(2) doublet, with the ground state fixing the expectation value of the sum of the magnitude squared of the pion and the diquark fields. Furthermore, we find that the Lagrangian at equal chemical potentials is invariant under global SU(2) transformations and construct the effective Lagrangian of the three Goldstone degrees of freedom by integrating out the radial fluctuations.

We study the equation of state at finite temperature and density in two-flavor QCD with the renormalization group improved gluon action and the clover-improved Wilson quark action on a ${16}^{3}\ifmmode\times\else\texttimes\fi{}4$ lattice. Along the lines of constant physics at ${m}_{\mathrm{PS}}/{m}_{\mathrm{V}}=0.65$ and 0.80, we compute the second and forth derivatives of the grand canonical partition function with respect to the quark chemical potential ${\ensuremath{\mu}}_{q}=({\ensuremath{\mu}}_{u}+{\ensuremath{\mu}}_{d})/2$ and the isospin chemical potential ${\ensuremath{\mu}}_{I}=({\ensuremath{\mu}}_{u}\ensuremath{-}{\ensuremath{\mu}}_{d})/2$ at vanishing chemical potentials, and study the behaviors of thermodynamic quantities at finite ${\ensuremath{\mu}}_{q}$ using these derivatives for the case ${\ensuremath{\mu}}_{I}=0$. In particular, we study density fluctuations at nonezero temperature and density by calculating the quark number and isospin susceptibilities and their derivatives with respect to ${\ensuremath{\mu}}_{q}$. To suppress statistical fluctuations, we also examine new techniques applicable at low densities. We find a large enhancement in the fluctuation of the quark number when the density increased near the pseudocritical temperature, suggesting a critical point at finite ${\ensuremath{\mu}}_{q}$ terminating the first order transition line between hadronic and quark-gluon-plasma phases. This result agrees with the previous results using staggered-type quark actions qualitatively. Furthermore, we study heavy-quark free energies and Debye screening masses at finite density by measuring the first and second derivatives of these quantities for various color channels of heavy quark-quark and quark-antiquark pairs. The results suggest that, to the leading order of ${\ensuremath{\mu}}_{q}$, the interaction between two quarks becomes stronger at finite densities, while that between quark and antiquark becomes weaker.

Quantum Chromodynamics (QCD) at finite temperature and density is currently a subject of great interest, due to both advances in experimental heavy-ion collisions and theoretical improvement of our understanding in recent years. Simulations of the theory on finite space-time lattices provide important theoretical advances. But in particular for investigating QCD phase transitions, finite quark masses and finite volumes in the simulations need to be taken into account. The investigation of finite-volume effects in QCD has a long tradition in the framework of chiral perturbation theory and random matrix theory and can provide useful tools for the analysis. With regard to a description of the chiral phase transition in QCD, it is very important to correctly take into account the effects of long-range fluctuations such as pions as the Goldstone bosons of spontaneous chiral symmetry breaking. It is natural to employ Renormalization Group methods for this purpose. Together with Hans-Jurgen Pirner, we have initiated work in this direction, and there has been an increasing interest in the investigation of such finite-volume effects. I will give an overview over recent developments.

We investigate the phase structure of (2+Nf)-flavor QCD, where two light flavors and Nf massive flavors exist, to discuss the feasibility of the electroweak baryogenesis in realistic technicolor scenario and to understand properties of finite density QCD. Because an appearance of a first order phase transition at finite temperature is a necessary condition for the baryogenesis, it is important to study the nature of finite temperature phase transition. Applying the reweighting method, the probability distribution function of the plaquette is calculated in the many-flavor QCD. Through the shape of the distribution function, we determine the critical mass of heavy flavors terminating the first order region, and find it to become larger with Nf. We moreover study the critical line at finite density and the first order region is found to become wider as increasing the chemical potential. We discuss how the properties of real (2+1)-flavor QCD at finite temperature and density can be extracted from simulations of many-flavor QCD.

We study the phase structure of QCD at finite temperature and density by\nnumerical simulations on a lattice. The most important point for the numerical\nstudy at finite density is treatment of the sign problem. We propose a method\nto avoid the sign problem, which is based on a cumulant expansion of the\ncomplex phase in the density of state method combined with the reweighting\nmethod. Using the method, we study the critical point terminating a first order\nphase transition line in lattice QCD at high temperature and density.\n

Electron self-energy in (2 + 1) topologically massive QED at finite temperature and density

We study the behavior of the pion dispersion relation in a pion medium at finite density and temperature. We introduce a pion chemical potential to describe the finite pion number density and argue that such description is valid during the hadronic phase of a relativistic heavy-ion collision between chemical and thermal freeze-out. We make use of an effective Lagrangian that explicitly respects chiral symmetry through the enforcement of the chiral Ward identities. The pion dispersion relation is computed through the computation of the pion self-energy in a non-perturbative fashion by giving an approximate solution to the Schwinger-Dyson equation for this self-energy. The dispersion relation is described in terms of a density and temperature dependent mass and an index of refraction which is also temperature, density as well as momentum dependent. The index of refraction is larger than unity for all values of the momentum for finite $\mu$ and $T$. We conclude by exploring some of the possible consequences for the propagation of pions through the boundary between the medium and vacuum.

We consider the thermodynamics of three-flavor QCD in the pion-condensed phase at nonzero isospin chemical potential (μI) and vanishing temperature using chiral perturbation theory in the isospin limit. The transition from the vacuum phase to a superfluid phase with a Bose-Einstein condensate of charged pions is shown to be second order and takes place at μI=mπ. We calculate the pressure, isospin density, and energy density to next-to-leading order in the low-energy expansion. Our results are compared with recent high-precision lattice simulations as well as previously obtained results in two-flavor chiral perturbation theory. The agreement between the lattice results and the predictions from three-flavor chiral perturbation theory is very good for μI<200 MeV. For larger values of μI, the agreement between lattice data and the two-flavor predictions is surprisingly good and better than with the three-flavor predictions. Finally, in the limit ms≫mu=md, we show that the three-flavor observables reduce to the two-flavor observables with renormalized parameters. The disagreement between the results for two-flavor and three-flavor χPT can largely be explained by the differences in the measured low-energy constants.

We extend our previous analysis of gauge and Dirac fields in the presence of a chemical potential. We consider an alternative thermal operator which relates in a simple way the Feynman graphs in QED at finite temperature and charge density to those at zero temperature but nonzero chemical potential. Several interesting features of such a factorization are discussed in the context of the thermal photon and fermion self-energies.

EOS of nuclear matter at finite temperature and density in the 1/ N expansion of the relativistic σ-ω model

We investigate pion superfluidity and its effect on meson properties and equation of state at finite temperature and isospin and baryon densities in the frame of a standard flavor SU(2) NJL model. In mean field approximation to quarks and random phase approximation to mesons, the critical isospin chemical potential for pion superfluidity is exactly the pion mass in the vacuum, and corresponding to the isospin symmetry spontaneous breaking, there is in the pion superfluidity phase a Goldstone mode which is the linear combination of the normal sigma and charged pion modes. We calculate numerically the gap equations for the chiral and pion condensates, the phase diagrams, the meson spectra, and the equation of state, and compare them with that obtained in other effective models. The competitions between pion superfluidity and color superconductivity at finite baryon density and between pion and kaon superfluidity at finite strangeness density in flavor SU(3) NJL model are briefly discussed.

We study the phase diagram of QCD at finite isospin density using two flavors of staggered quarks. We investigate the low temperature region of the phase diagram where we find a pion condensation phase at high chemical potential. We started a basic analysis of the spectrum at finite isospin density. In particular, we measured pion, rho and nucleon masses inside and outside of the pion condensation phase. In agreement with previous studies in two-color QCD at finite baryon density we find that the Polyakov loop does not depend on the density in the staggered formulation.

Color–flavor unlocking and phase diagram with self-consistently determined strange-quark masses