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 of second order.     

Fluctuations as probe of the QCD phase transition and freeze-out in heavy ion collisions at LHC and RHIC

We discuss the relevance of higher order cumulants of net baryon number fluctuations for the analysis of freeze-out and critical conditions in heavy ion collisions at LHC and RHIC. Using properties of O(4) scaling functions, we discuss the generic structure of these higher cumulants at vanishing baryon chemical potential and apply chiral model calculations to explore their properties at non-zero baryon chemical potential. We show that the ratios of the sixth to second and eighth to second order cumulants of the net baryon number fluctuations change rapidly in the transition region of the QCD phase diagram. Already at vanishing baryon chemical potential they deviate considerably from the predictions of the hadron resonance gas model which reproduce the second and fourth order cumulants of the net proton number fluctuations at RHIC. We point out that the sixth order cumulants of baryon number and electric charge fluctuations remain negative at the chiral transition temperature. Thus, they offer the possibility to probe the proximity of the chemical freeze-out to the crossover line.     

The thermodynamic properties of weakly interacting quark-gluon plasma via the one-gluon exchange interaction

The thermodynamic properties of the quark-gluon plasma (QGP), as well as its phase diagram, are calculated as a function of baryon density (chemical potential) and temperature. The QGP is assumed to be composed of the light quarks only, i.e., the up and down quarks, which interact weakly, and the gluons which are treated as they are free. The interaction between quarks is considered in the framework of the one gluon exchange model which is obtained from the Fermi liquid picture. The bag model is used, with fixed bag pressure (B)for the nonperturbative part, and the quantum chromodynamics (QCD) coupling is assumed to be constant, i.e., with no dependence on the temperature or the baryon density. The effect of weakly interacting quarks on the QGP phase diagram are shown and discussed. It is demonstrated that the one-gluon exchange interaction for the massless quarks has considerable effect on the QGP phase diagram and it causes the system to reach to the confined phase at the smaller baryon densities and temperatures. The pressure of excluded volume hadron gas model is also used to find the transition phase diagram. Our results depend on the values of bag pressure and the QCD coupling constant. The latter does not have a dramatic effect on our calculations. Finally, we compare our results with the thermodynamic properties of strange quark matter and the lattice QCD prediction for the QGP transition critical temperature.     

Second to tenth order susceptibilities of conserved charges within a modified Nambu-Jona-Lasinio model

We discuss the sign and energy dependence of second to tenth order susceptibilities of the baryon number,charge number, and strangeness for the analysis of critical conditions in heavy ion collisions in the LHC and RHIC by applying a modified Nambu-Jona-Lasinio model. This model is fitted to the quark condensate of the lattice QCD result at finite temperature and zero baryon chemical potential. The presence of a critical point made these susceptibilities deviate considerably from a Hadron-Resonance-Gas model that shows no criticality. The sign, magnitude, and energy dependence of these higher order fluctuations hint towards the existence and location of a critical point that could be discovered in future heavy ion collision experiments.     

No-go theorem for critical phenomena in large-N(c) QCD

We derive some rigorous results on the chiral phase transition in QCD and QCD-like theories with a large number of colors, N(c), based on the QCD inequalities and the large-N(c) orbifold equivalence. We show that critical phenomena and associated soft modes are forbidden in flavor-symmetric QCD at finite temperature T and finite but not so large quark chemical potential μ for any nonzero quark mass. In particular, the critical point in QCD at a finite baryon chemical potential μ(B)=N(c)μ is ruled out, if the coordinate (T, μ) is outside the pion condensed phase in the corresponding phase diagram of QCD at a finite isospin chemical potential μ(I)=2μ.     

On the chiral phase transition in hadronic matter

A qualitative analysis of the chiral phase transition in QCD with two massless quarks and nonzero baryon density is performed. It is assumed that, at zero baryonic density, ρ=0, the temperature phase transition is of the second order. Due to a specific power dependence of baryon masses on the chiral condensate, the phase transition becomes of the first order at the temperature T=T_ph(ρ) for ρ>0. At temperatures T_cont(ρ)> _T>_Tph(ρ), there is a mixed phase consisting of the quark phase (stable) and the hadron phase (unstable). At the temperature T=T_cont(ρ), the system experiences a continuous transition to the pure chirally symmetric phase.     

Phase transitions in quark matter and behaviour of physical observables in the vicinity of the critical end point

We study the chiral phase transition at finite T and μ _B within the framework of the SU(3) Nambu-Jona-Lasinio (NJL) model. The QCD critical end point (CEP) and the critical line at finite temperature and baryonic chemical potential are investigated: the study of physical quantities, such as the baryon number susceptibility near the CEP, will provide complementary information concerning the order of the phase transition. We also analyze the information provided by the study of the critical exponents around the CEP.     

QCD phase diagram according to the center group

We study an effective theory for QCD at finite temperature and density which contains the leading center symmetric and center symmetry breaking terms. The effective theory is studied in a flux representation where the complex phase problem is absent and the model becomes accessible to Monte Carlo techniques also at finite chemical potential. We simulate the system by using a generalized Prokof'ev-Svistunov worm algorithm and compare the results to a low temperature expansion. The phase diagram is determined as a function of temperature, chemical potential, and quark mass. The shape and quark mass dependence of the phase boundaries are as expected for QCD. The transition into the deconfined phase is smooth throughout, without any discontinuities or critical points.     

利用格点量子色动力学研究手征平滑过渡温度和手征相变温度

回顾了最近关于手征平滑过渡温度和手征相变温度的研究结果。首先给出了在零重子化学势能下的手征平滑过渡温度为156.5(1.5) MeV,其次,给出了在非零重子化学势能下手征相转变曲线的二阶及四阶曲率分别为0.012(4)和0.000(4)。接着讨论了在格点QCD中第一次得到的量子色动力学的手征相变温度。在热力学极限、连续极限及手征极限下,我们得到手征相变温度为132$^{+3}_{-6}$ MeV。     

SU(3) Polyakov Linear-Sigma Model: Magnetic Properties of QCD Matter in Thermal and Dense Medium

The linear-sigma model, in which information about confining gluons is included through the Polyakov-loop potential (PLSM), is considered in order to perform a systematic study for various magnetic properties of QCD matter under extreme conditions of high temperatures and densities and finite magnetic field strengths. The introduction of magnetic field to the PLSM Lagrangian requires suitable utilization of Landau quantization, modification in the dispersion relations, and momentum-space dimension-reduction. We observed that increasing the magnetic field leads to filling-up lower Landau levels first and decreasing the number of occupied levels. We conclude that the population of Landau levels is most sensitive to the magnetic field and to the quark charges. The influences of finite magnetic field on the temperature dependence of chiral and deconfinement order-parameter(s) are studied. We present estimations for the magnetization, the magnetic susceptibility, the permeability, and the catalytic properties of QCD matter as functions of temperature. The dependences of the resulting freeze-out parameters, temperatures, and baryon chemical potentials on the corresponding magnetic field strengths have been analyzed, as well. These calculations are compared with recent lattice QCD simulations, whenever available. We conclude that the QCD matter seems to have paramagnetic property at temperatures greater than the critical one. There is an evidence for weak diamagnetic property at low temperatures. Last but not least, we observe that the magnetic catalysis is inverse, namely, the critical temperatures decrease with increasing the magnetic field.     

The QCD phase boundary from quark–gluon dynamics

We study one-flavor QCD at finite temperature and chemical potential using the functional renormalization group. We discuss the chiral phase transition in QCD and its order with its underlying mechanism in terms of quarks and gluons and analyze the dependence of the phase-transition temperature on small quark chemical potentials. Our result for the curvature of the phase boundary at small quark chemical potential relies on only a single input parameter, the value of the strong coupling at the Z mass scale.     

Lattice results on QCD at high temperature and non-zero baryon number density

Studies of QCD thermodynamics on the lattice now can be performed with an almost realistic quark mass spectrum and on quite large lattices. This will soon allow a controlled extrapolation to the continuum limit. We present recent results on the QCD equation of state, discuss deconfining and chiral symmetry restoring aspects of the QCD transition at vanishing chemical potential and show results on baryon number, electric charge and strangeness fluctuations. We briefly discuss the generic structure of Taylor expansion coefficients in the vicinity of the chiral phase transition and comment on the determination of the anticipated chiral critical point within the framework of Taylor expansions of the QCD partition function.     

QCD at finite temperature and density within the fRG approach: an overview

In this paper, we present an overview on recent progress in studies of QCD at finite temperature and densities within the functional renormalization group (fRG) approach. The fRG is a nonperturbative continuum field approach, in which quantum, thermal and density fluctuations are integrated successively with the evolution of the renormalization group (RG) scale. The fRG results for the QCD phase structure and the location of the critical end point (CEP), the QCD equation of state (EoS), the magnetic EoS, baryon number fluctuations confronted with recent experimental measurements, various critical exponents, spectral functions in the critical region, the dynamical critical exponent, etc, are presented. Recent estimates of the location of the CEP from first-principle QCD calculations within fRG and Dyson–Schwinger equations, which pass through lattice benchmark tests at small baryon chemical potentials, converge in a rather small region at baryon chemical potentials of about 600 MeV. A region of inhomogeneous instability indicated by a negative wave function renormalization is found with μ_B ≳ 420 MeV. It is found that the non-monotonic dependence of the kurtosis of the net-proton number distributions on the beam collision energy observed in experiments could arise from the increasingly sharp crossover in the regime of low collision energy.     

Pion decay constant at finite temperature

The values of the pion decay constant, and of the bilinear fermionic condensate, for temperatures from zero up to the critical temperature for chiral phase transition, are calculated using our composite operator approach to finite temperature QCD. For small temperatures our results agree with those found with other approaches. Near the critical temperature we recover the universal behaviour typical of second order phase transition. We have assumed that effects responsible for deconfinement do not strongly perturb our chiral approach for two and possibly three flavours.     

Center-symmetric effective theory for two-color quark matter

We revisit the center-symmetric dimensionally reduced effective theory for two-color Yang-Mills theory at high temperature. This effective theory includes an order parameter for deconfinement and thus allows to broaden the range of validity of the conventional three-dimensional effective theory (EQCD) towards the confining phase transition. We extend the previous results by including the effects of massive quarks with nonzero baryon chemical potential. The parameter space of the theory is constrained by leading-order matching to the Polyakov loop effective potential of two-color QCD. Once all the parameters are fixed, the effective theory can provide model-independent predictions for the physics above the deconfinement transition, thus bridging the gap between large-scale numerical simulations and semi-analytical calculations within phenomenological models.     

Calculation of Partition Function of QCD at Finite Chemical Potential

In equilibrium statistical field theory, the partition function has fundamental importance. In this paper we propose a direct and general method for calculating the partition function and equation of state of QCD at finite chemical potential. It is found that the partition function is totally determined by the dressed quark propagator at finite chemical potential up to a multiplicative constant. From this a criterion for the phase transition between the Nambu and the Wigner phases is obtained. This general method is applied to two specific cases： the free quark theory and QCD with a model dressed quark propagator having confinement features. In the first case, the standard Fermi distribution at T = 0 is reproduced. In the second case, we apply the conclusion in previous works to obtain the dressed quark propagator at finite chemical potential and find the unphysical result that the baryon number density vanishes for all values of chemical potential. The reason for this result is discussed.     

Phase diagrams in nonlocal Polyakov-Nambu-Jona-Lasinio models constrained by lattice QCD results

Based on lattice QCD-adjusted SU(2)f nonlocal Polyakov-Nambu-Jona-Lasinio (PNJL) models, we investigate how the location of the critical endpoint in the QCD phase diagram depends on the strenght of the vector meson coupling, as well as the Polyakov-loop (PL) potential and the form factors of the covariant model. The latter are constrained by lattice QCD data for the quark propagator. The strength of the vector coupling is adjusted such as to reproduce the slope of the pseudocritical temperature for the chiral phase transition at low chemical potential extracted recently from lattice QCD simulations. Our study supports the existence of a critical endpoint in the QCD phase diagram albeit the constraint for the vector coupling shifts its location to lower temperatures and higher baryochemical potentials than in the case without it.     

Lowering Critical Temperature in the Extended Quark Sigma Model at Finite Temperature

A phase transition, a critical temperature, and meson masses are studied in the extended quark sigma model, in which the effective mesonic potential is extended to include eighth-order mesonic interactions. The second derivative of the effective mesonic potential is applied to calculate the effective sigma and pion masses as functions of temperature. We find that the critical temperature assumes a lower value in comparison with that of original quark sigma model. A comparison with recent calculations of lattice QCD is introduced. The behavior of the phase transition remains unchanged when the higher-order mesonic interactions are included. We find that the spontaneous symmetry-breaking condition is necessary to satisfy the Goldstone theorem at low temperatures.     

Hadron-quark phase coexistence in a hybrid MIT-Bag model

The phase transition of hadronic to quark matter and the boundaries of the hadron-quark coexistence phase are studied within the two Equation of State (EoS) models. The relativistic effective mean-field approach with constant and density-dependent meson-nucleon couplings is used to describe hadronic matter, and the MIT-Bag model is adopted to describe quark matter. The boundaries of the mixed phase for different Bag constants are obtained solving the Gibbs equations. We notice that the dependence on the Bag parameter of the critical temperatures (at zero chemical potential) can be well reproduced by a fermion ultrarelativistic quark gas model, without contribution from the hadron part. At variance, the critical chemical potentials (at zero temperature) are very sensitive to the EoS of the hadron sector. Hence, the contribution of the hadronic interaction is much more relevant for the determination of the transition to the quark-gluon plasma at finite baryon density and low T . Moreover, in the low-temperature and finite chemical potential region no solutions of the Gibbs conditions are existing for small Bag-constant values, B < (135 MeV)⁴ . Isospin effects in asymmetric matter appear important in the high chemical-potential regions at lower temperatures, of interest for the inner-core properties of neutron stars and for heavy-ion collisions at intermediate energies.