D. Toublan

Kaon condensation and Goldstone's theorem

Abstract We consider QCD at a nonzero chemical potential for strangeness. At a critical value of the chemical potential equal to the kaon mass, kaon condensation occurs through a continuous phase transition. We show that in the limit of exact isospin symmetry a Goldstone boson with the dispersion relation E ∼ p 2 appears in the kaon condensed phase. At the same time, the number of the Goldstone bosons is less than the number of broken generators. Both phenomena are familiar in nonrelativistic systems. We interpret our results in terms of a Goldstone boson counting rule found previously by Nielsen and Chadha. We also formulate a criterion sufficient for the equality between the number of Goldstone bosons and the number of broken generators.

Thermodynamics of chiral symmetry at low densities

Abstract The phase diagram of two-color QCD as a function of temperature and baryon chemical potential is considered. Using a low-energy chiral Lagrangian based on the symmetries of the microscopic theory, we determine, at the one-loop level, the temperature dependence of the critical chemical potential for diquark condensation and the temperature dependence of the diquark condensate and baryon density. The prediction for the temperature dependence of the critical chemical potential is consistent with the one obtained for a dilute Bose gas. The associated phase transition is shown to be of second order for low temperatures and first order at higher temperatures. The tricritical point at which the second order phase transition ends is determined. The results are carried over to QCD with quarks in the adjoint representation and to ordinary QCD at a non-zero chemical potential for isospin.

The QCD phase diagram at nonzero temperature, baryon and isospin chemical potentials in random matrix theory

We introduce a random matrix model with the symmetries of QCD at finite temperature and chemical potentials for baryon number and isospin. We analyze the phase diagram of this model in the chemical potential plane for different temperatures and quark masses. We find a rich phase structure with five different phases separated by both first and second order lines. The phases are characterized by the pion condensate and the chiral condensate for each of the flavors. In agreement with lattice simulations, we find that in the phase with zero pion condensate the critical temperature depends in the same way on the baryon number chemical potential and on the isospin chemical potential. At nonzero quark mass, we find, remarkably, that the critical end point at nonzero temperature and baryon chemical potential is split in two by an arbitrarily small isospin chemical potential. As a consequence, there are two crossovers that separate the hadronic phase from the quark-gluon plasma phase at high temperature. Detailed analytical results are obtained at zero temperature and in the chiral limit.

QCD with two colors at finite baryon density at next-to-leading order

We study QCD with two colors and quarks in the fundamental representation at finite baryon density in the limit of light quark masses. In this limit the free energy of this theory reduces to the free energy of a chiral Lagrangian which is based on the symmetries of the microscopic theory. In earlier work this Lagrangian was analyzed at the mean field level and a phase transition to a phase of condensed diquarks was found at a chemical potential of half the diquark mass (which is equal to the pion mass). In this article we analyze this theory at next-to-leading order in chiral perturbation theory. We show that the theory is renormalizable and calculate the next-to-leading order free energy in both phases of the theory. By deriving a Landau-Ginzburg theory for the order parameter we show that the finite one-loop contribution and the next-to-leading order terms in the chiral Lagrangian do not qualitatively change the phase transition. In particular, the critical chemical potential is equal to half the next-to-leading order pion mass, and the phase transition is second order. PACS: 11.30.Rd, 12.39.Fe, 12.38.Lg, 71.30.+hAbstract We study QCD with two colors and quarks in the fundamental representation at finite baryon density in the limit of light-quark masses. In this limit the free energy of this theory reduces to the free energy of a chiral Lagrangian which is based on the symmetries of the microscopic theory. In earlier work this Lagrangian was analyzed at the mean-field level and a phase transition to a phase of condensed diquarks was found at a chemical potential of half the diquark mass (which is equal to the pion mass). In this article we analyze this theory at next-to-leading order in chiral perturbation theory. We show that the theory is renormalizable and calculate the next-to-leading order free energy in both phases of the theory. By deriving a Landau–Ginzburg theory for the order parameter we show that the finite one-loop contribution and the next-to-leading order terms in the chiral Lagrangian do not qualitatively change the phase transition. In particular, the critical chemical potential is equal to half the next-to-leading order pion mass, and the phase transition is of second order.

The phase diagram of four flavor SU(2) lattice gauge theory at nonzero chemical potential and temperature

Abstract SU(2) lattice gauge theory with four flavors of quarks is simulated at nonzero chemical potential μ and temperature T and the results are compared to the predictions of effective Lagrangians. Simulations on 164 lattices indicate that at zero T the theory experiences a second order phase transition to a diquark condensate state. Several methods of analysis, including equation of state fits suggested by Chiral Perturbation Theory, suggest that mean-field scaling describes this critical point. Nonzero T and μ are studied on 123×6 lattices. For low T, increasing μ takes the system through a line of second order phase transitions to a diquark condensed phase. Increasing T at high μ, the system passes through a line of first order transitions from the diquark phase to the quark–gluon plasma phase. Metastability is found near the first order line. Presumably, there is a tricritical point along this line of transitions. We estimate its position to be consistent with theoretical predictions.

Diquark condensation at nonzero chemical potential and temperature

Abstract SU (2) lattice gauge theory with four flavors of quarks is studied at nonzero chemical potential μ and temperature T by computer simulation and Effective Lagrangian techniques. Simulations are done on 8 4 , 8 3 ×4 and 12 3 ×6 lattices and the diquark condensate, chiral order parameter, Wilson line, fermion energy and number densities are measured. Simulations at a fixed, nonzero quark mass provide evidence for a tricritical point in the μ – T plane associated with diquark condensation. For low T , increasing μ takes the system through a line of second order phase transitions to a diquark condensed phase. Increasing T at high μ , the system passes through a line of first order transitions from the diquark phase to the quark–gluon plasma phase. Using Effective Lagrangians we estimate the position of the tricritical point and ascribe its existence to trilinear couplings that increase with μ and T .

Diquark and pion condensation in random matrix models for two color QCD

We introduce a random matrix model with the symmetries of QCD with two colors at nonzero isospin and baryon chemical potentials and temperature. We analyze its phase diagram and find phases with condensation of pion and diquark states in addition to the phases with spontaneously broken chiral symmetries. In the limit of small chemical potentials and quark masses, we reproduce the mean field results obtained from chiral Lagrangians. As in the case of QCD with three colors, the presence of two chemical potentials breaks the flavor symmetry and leads to phases that are characterized by different behaviors of the chiral condensates for each flavor. In particular, the phase diagram we obtain is similar to QCD with three colors and three flavors of quarks of equal masses at zero baryon chemical potential and nonzero isospin and strange chemical potentials. A tricritical point of the superfluid transitions found in lattice calculations and from an analysis in terms of chiral Lagrangians does not appear in the random matrix model. Remarkably, at fixed isospin chemical potential, for the regions outside of the superfluid phases, the phase diagrams in the temperature--baryon chemical potential plane for two colors and three colors are qualitatively the same.

Statistical properties of the spectrum of the QCD Dirac operator at low energy

Abstract We analyze the statistical properties of the spectrum of the QCD Dirac operator at low energy in a finite box of volume L 4 by means of partially quenched Chiral Perturbation Theory, a low-energy effective field theory based on the symmetries of QCD. We derive the two-point spectral correlation function from the discontinuity of the chiral susceptibility. For eigenvalues much smaller than m c = F 2 / ΣL 2 , where F is the pion decay constant and Σ is the absolute value of the quark condensate, our result for the two-point correlation function coincides with the result previously obtained from chiral Random Matrix Theory (chRMT). The departure from the chRMT result above that scale is due to the contribution of the nonzero momentum modes. In terms of the variance of the number of eigenvalues in an interval containing n eigenvalues on average, it amounts to a crossover from a log n -behavior to a n 2 log n -behavior.

The PseudoGoldstone spectrum of two color QCD at finite density

We examine the spectrum of two-color lattice QCD with one staggered quark field (four flavors) at a finite chemical potential

A large Nc perspective on the QCD phase diagram

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