Quark-number Chemical Potential Research Articles

Functional renormalization group study of thermodynamic geometry around the phase transition of quantum chromodynamics

We investigate the thermodynamic geometry of the quark-meson model at finite temperature, T, and quark number chemical potential, μ. We extend previous works by the inclusion of fluctuations exploiting the functional renormalization group approach. We use recent developments to recast the flow equation into the form of an advection-diffusion equation. We adopt the local potential approximation for the effective average action. We focus on the thermodynamic curvature, R, in the (μ,T) plane, in proximity of the chiral crossover, up to the critical point of the phase diagram. We find that the inclusion of fluctuations results in a smoother behavior of R near the chiral crossover. Moreover, for small μ, R remains negative, signaling the fact that bosonic fluctuations reduce the capability of the system to completely overcome the fermionic statistical repulsion of the quarks. We investigate in more detail the small μ region by analyzing a system in which we artificially lower the pion mass, thus approaching the chiral limit in which the crossover is actually a second order phase transition. On the other hand, as μ is increased and the critical point is approached, we find that R is enhanced and a sign change occurs, in agreement with mean field studies. Hence, we completely support the picture that R is sensitive to a crossover and a phase transition, and provides information about the effective behavior of the system at the phase transition. Published by the American Physical Society 2024

Nonanalyticity, sign problem, and Polyakov line in a Z3 -symmetric heavy quark model at low temperature: Phenomenological model analyses

The nonanalyticity and the sign problem in the Z3-symmetric heavy quark model at low temperature are studied phenomenologically. For the free heavy quarks, the nonanalyticity is analyzed in the relation to the zeros of the grand canonical partition function. The Z3-symmetric effective Polyakov-line model (EPLM) in strong coupling limit is also considered as an phenomenological model of Z3-symmetric QCD with large quark mass at low temperature. We examine how the Z3-symmetric EPLM approaches to the original one in the zero-temperature limit. The effects of the Z3-symmetry affect the structure of zeros of the microscopic probability density function at the nonanalytic point. The average value of the Polyakov line can detect the structure, while the other thermodynamic quantities are not sensible to the structure in the zero-temperature limit. The effect of the imaginary quark chemical potential is also discussed. The imaginary part of the quark number density is very sensitive to the symmetry structure at the nonanalytical point. For a particular value of the imaginary quark number chemical potential, large quark number may be induced in the vicinity of the nonanalytical point.

Complex Langevin Simulations of QCD at Finite Density – Progress Report

We simulate lattice QCD at finite quark-number chemical potential to study nuclear matter, using the complex Langevin equation (CLE). The CLE is used because the fermion determinant is complex so that standard methods relying on importance sampling fail. Adaptive methods and gauge-cooling are used to prevent runaway solutions. Even then, the CLE is not guaranteed to give correct results. We are therefore performing extensive testing to determine under what, if any, conditions we can achieve reliable results. Our earlier simulations at β = 6/g2 = 5.6, m = 0.025 on a 124 lattice reproduced the expected phase structure but failed in the details. Our current simulations at β = 5.7 on a 164 lattice fail in similar ways while showing some improvement. We are therefore moving to even weaker couplings to see if the CLE might produce the correct results in the continuum (weak-coupling) limit, or, if it still fails, whether it might reproduce the results of the phase-quenched theory. We also discuss action (and other dynamics) modifications which might improve the performance of the CLE.

Understanding QCD at high density from aZ3-symmetric QCD-like theory

We investigate QCD at large mu/T by using Z_3-symmetric SU(3) gauge theory, where mu is the quark-number chemical potential and T is temperature. We impose the flavor-dependent twist boundary condition on quarks in QCD. This QCD-like theory has the twist angle theta as a parameter, and agrees with QCD when theta=0 and becomes symmetric when theta=2\pi/3. For both QCD and the Z_3-symmetric SU(3) gauge theory, the phase diagram is drawn in mu--T plane with the Polyakov-loop extended Nambu--Jona-Lasinio model. In the Z_3-symmetric SU(3) gauge theory, the Polyakov loop varphi is zero in the confined phase appearing at T \lsim 200 MeV. The perfectly confined phase never coexists with the color superconducting (CSC) phase, since finite diquark condensate in the CSC phase breaks Z_3 symmetry and then makes varphi finite. When mu \gsim 300 MeV, the CSC phase is more stable than the perfectly confined phase at T \lsim 100 MeV. Meanwhile, the chiral symmetry can be broken in the perfectly confined phase, since the chiral condensate is Z_3 invariant. Consequently, the perfectly confined phase is divided into the perfectly confined phase without chiral symmetry restoration in a region of mu \lsim 300 MeV and T \lsim 200 MeV and the perfectly confined phase with chiral symmetry restoration in a region of \mu \gsim 300 MeV and 100 \lsim T \lsim 200 MeV. The basic phase structure of Z_3-symmetric QCD-like theory remains in QCD. We show that in the perfectly confined phase the sign problem becomes less serious because of \varphi=0, using the heavy quark theory. We discuss a lattice QCD framework to evaluate observables at \theta=0 from those at \theta=2\pi/3.

Novel Lifshitz point for chiral transition in the magnetic field

Based on the generalized Ginzburg–Landau theory, chiral phase transition is discussed in the presence of magnetic field. Considering the chiral density wave we show that chiral anomaly gives rise to an inhomogeneous chiral phase for nonzero quark-number chemical potential. Novel Lifshitz point appears on the vanishing chemical potential line, which may be directly explored by the lattice QCD simulation.

CHARGED PION CONDENSATION PHENOMENON OF DENSE BARYONIC MATTER INDUCED BY FINITE VOLUME: THE NJL2 MODEL CONSIDERATION

The properties of two-flavored massless Nambu–Jona-Lasinio (NJL) model in (1+1)-dimensional R1 × S1 space–time with compactified space coordinate are investigated in the presence of isospin and quark number chemical potentials μI, μ. The consideration is performed in the large Nc limit, where Nc is the number of colored quarks. It is shown that at L = ∞ (L is the length of the circumference S1) the charged pion condensation (PC) phase with zero quark number density is realized at arbitrary nonzero μI and for rather small values of μ. However, at arbitrary finite values of L the phase portrait of the model contains the charged PC phase with nonzero quark number density (in the case of periodic boundary conditions for quark fields). Hence, finite sizes of the system can serve as a factor promoting the appearance of the charged PC phase in quark matter with nonzero baryon densities. In contrast, the phase with chiral symmetry breaking may exist only at rather large values of L.

Inhomogeneous charged pion condensation phenomenon in theNJL2model with quark number and isospin chemical potentials

The properties of two-flavored massive Nambu--Jona-Lasinio model in (1+1)-dimensional spacetime are investigated in the presence of isospin and quark number chemical potentials. The consideration is performed in the large-$N_c$ limit, where $N_c$ is the number of colored quarks. It is shown in the framework of this model that charged pion condensation phenomenon of dense quark/hadron isotopically asymmetric matter is rather a spatially inhomogeneous than a homogeneous one.

LargeNcequivalence and baryons

In the large ${N}_{c}$ limit, gauge theories with different gauge groups and matter content sometimes turn out to be ``large ${N}_{c}$ equivalent,'' in the sense of having a set of coincident correlation functions. Large ${N}_{c}$ equivalence has mainly been explored in the glueball and meson sectors. However, a recent proposal to dodge the fermion sign problem of QCD with a quark number chemical potential using large ${N}_{c}$ equivalence motivates investigating the applicability of large ${N}_{c}$ equivalence to correlation functions involving baryon operators. Here we present evidence that large ${N}_{c}$ equivalence extends to the baryon sector, under the same type of symmetry realization assumptions as in the meson sector, by adapting the classic Witten analysis of large ${N}_{c}$ baryons.

Theta vacuum and entanglement interaction in the three-flavor Polyakov-loop extended Nambu-Jona-Lasinio model

We investigate theta-vacuum effects on the QCD phase diagram for the realistic $2+1$ flavor system, using the three-flavor Polyakov-extended Nambu-Jona-Lasinio (PNJL) model and the entanglement PNJL model as an extension of the PNJL model. The theta-vacuum effects make the chiral transition sharper. For large theta-vacuum angle the chiral transition becomes first order even if the quark number chemical potential is zero, when the entanglement coupling between the chiral condensate and the Polyakov loop is taken into account. We finally propose a way of circumventing the sign problem on lattice QCD with finite theta.

Quark-mass dependence of the three-flavor QCD phase diagram at zero and imaginary chemical potential: Model prediction

We draw the three-flavor phase diagram as a function of light- and strange-quark masses for both zero and imaginary quark-number chemical potential, using the Polyakov-loop extended Nambu-Jona-Lasinio model with an effective four-quark vertex depending on the Polyakov loop. The model prediction is qualitatively consistent with 2+1 flavor lattice QCD prediction at zero chemical potential and with degenerate three-flavor lattice QCD prediction at imaginary chemical potential.

Violations of parity and charge conjugation in theθvacuum with imaginary chemical potential

Charge conjugation (C) and Parity (P) are exact symmetries at theta =pi and Theta = mu/(iT)=pi, where theta is the parameter of the so-called theta-vacuum, mu is the imaginary quark-number chemical potential and T is the temperature. Spontaneous breakings of these discrete symmetries are investigated by the the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. At zero T, P symmetry is spontaneously broken while C symmetry is conserved. As T increases, P symmetry is restored just after C symmetry is spontaneously broken, so that either P or C symmetry or both the symmetries are spontaneously broken for any T. The chiral-symmetry restoration and the deconfinement transition at theta =Theta =0 are remnants of the P restoration and the C breaking at theta =Theta =pi, respectively.

Entanglement between deconfinement transition and chiral symmetry restoration

We extend the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model by introducing an effective four-quark vertex depending on Polyakov loop. The effective vertex generates entanglement interactions between Polyakov loop and chiral condensate. The new model is consistent with lattice QCD data at imaginary quark-number chemical potential and real and imaginary isospin chemical potentials, particularly on strong correlation between the chiral and deconfinement transitions and also on the quark-mass dependence of the order of the Roberge-Weiss endpoint predicted by lattice QCD very lately. We investigate an influence of the entanglement interactions on a location of the tricritical point at real isospin chemical potential and a location of the critical endpoint at real quark-number chemical potential.

PION PROPERTIES AT FINITE DENSITY

In this talk, we report our recent work on the pion weak decay constant (Fπ) and pion mass (mπ) using the nonlocal chiral quark model with the finite quark-number chemical potential (μ) taken into account. Considering the breakdown of Lorentz invariance at finite density, the time and space components are computed separately, and the corresponding results turn out to be: [Formula: see text] and [Formula: see text] at μc ≈ 320 MeV , respectively. Using the in-medium Gell-Mann-Oakes-Renner (GOR) relation, we show that the pion mass increases by about 15% at μc.

Pion weak decay constant at finite density from the instanton vacuum

We investigate the pion weak decay constant (Fπ) and pion mass (mπ) at finite density within the framework of the nonlocal chiral quark model from the instanton vacuum with the finite quark-number chemical potential (μ) taken into account. We mainly focus on the Nambu–Goldstone phase below the critical value of the chemical potential μc≈320 MeV, which is determined consistently within the present framework. The breakdown of Lorentz invariance at finite density being considered, the time (Fπt) and space (Fπs) components are computed separately, and the corresponding results turn out to be: Fπt=82.96 MeV and Fπs=80.29 MeV at μc, respectively. Using the in-medium Gell-Mann–Oakes–Renner (GOR) relation, we show that the pion mass increases by about 15% at μc.

Lattice QCD at finite temperature and density in the phase-quenched approximation

QCD at a finite quark-number chemical potential $\ensuremath{\mu}$ has a complex fermion determinant, which precludes its study by standard lattice QCD simulations. We therefore simulate lattice QCD at finite $\ensuremath{\mu}$ in the phase-quenched approximation, replacing the fermion determinant with its magnitude. (The phase-quenched approximation can be considered as simulating at finite isospin chemical potential $2\ensuremath{\mu}$ for ${N}_{f}/2$ $u$-type and ${N}_{f}/2$ $d$-type quark flavors.) These simulations are used to study the finite-temperature transition for small $\ensuremath{\mu}$, where there is some evidence that the position (and possibly the nature) of this transition is unchanged by this approximation. We look for the expected critical endpoint for 3-flavor QCD. Here, it has been argued that the critical point at zero $\ensuremath{\mu}$ would become the critical endpoint at small $\ensuremath{\mu}$, for quark masses just above the critical mass. Our simulations indicate that this does not happen, and there is no such critical endpoint for small $\ensuremath{\mu}$. We discuss how we might adapt techniques used for imaginary $\ensuremath{\mu}$ to improve the signal/noise ratio and strengthen our conclusions, using results from relatively low statistics studies.

Susceptibilities and speed of sound from the Polyakov-Nambu-Jona-Lasinio model

We present the Taylor expansion coefficients of the pressure in quark number chemical potential ${\ensuremath{\mu}}_{0}={\ensuremath{\mu}}_{B}/3={\ensuremath{\mu}}_{u}={\ensuremath{\mu}}_{d}$, for the strongly interacting matter as described by the Polyakov-Nambu-Jona-Lasinio (PNJL) model for two light degenerate flavors of quarks $u$ and $d$. The expansion has been done up to eighth order in ${\ensuremath{\mu}}_{0}$, and the results are consistent with recent estimates from Lattice. We have further obtained the specific heat ${C}_{V}$, squared speed of sound ${v}_{s}^{2}$, and the conformal measure $\mathcal{C}$.

Stressed pairing in conventional color superconductors is unavoidable

At sufficiently high densities, cold dense three-flavor quark matter is in the color-flavor locked (CFL) phase, in which all nine quarks pair in a particularly symmetric fashion. Once the heaviness of the strange quark (mass ${m}_{s}$) and the requirements of electric and color neutrality are taken into account, the CFL pattern of color superconductivity requires the pairing of quarks that would, in the absence of pairing, have Fermi momenta that differ by of order ${m}_{s}^{2}/\ensuremath{\mu}$, with $\ensuremath{\mu}$ the quark number chemical potential. This means that at sufficiently small $\ensuremath{\mu}$, the ``stress'' on the pairing is large enough that the system can lower its energy by breaking pairs, resulting in some unconventional color superconductor which includes gapless excitations, spatial inhomogeneity, counter-propagating currents, or all three. In this paper we ask whether there is some less symmetric but still conventional pattern of pairing that can evade the stress. In other words, is there a pattern of pairing in which, once electric and color neutrality are imposed by suitable chemical potentials, pairing only occurs among those quarks whose Fermi momenta would be equal in the absence of pairing? We use graph-theoretical methods to classify 511 patterns of conventional color-superconducting pairing, and show that none of them meet this requirement. All feel a stress, and all can be expected to become unstable to gapless modes at a density comparable to that at which the CFL phase becomes unstable.

Finite temperature transition for 2-flavor lattice QCD at finite isospin density

We simulate 2-flavour lattice QCD at finite isospin chemical potential mu_I, for temperatures close to the finite temperature transition from hadronic matter to a quark-gluon plasma. The mu_I dependence of the transition coupling is observed and used to estimate the decrease in the transition temperature with increasing mu_I. These simulations are performed on an 8^3 times 4 lattice at 3 different quark masses. Our estimate of the magnitude of the fluctuations of the phase of the fermion determinant at small quark-number chemical potential mu, suggest that the position of the small mu and small mu_I transitions should be the same for mu_I=2mu, and we argue that the nature of these transitions should be the same. For all mu_I < m_pi the smoothness of these transitions and the values of the Binder cumulant B_4, indicate that these transitions are mere crossovers, and show no sign of the expected critical endpoint. For mu_I > m_pi and a small isospin (I_3) breaking term lambda, we do find evidence of a critical endpoint which would indicate that, for lambda=0, there is a tricritical point on the phase boundary where the pion condensate evaporates, where this phase transition changes from second to first order.

Quenched lattice QCD at finite isospin density and related theories

We study quenched QCD at finite chemical potential, $\mu_I$, for the third component of isospin and quenched two-colour QCD at finite chemical potential, $\mu$, for quark number. In contrast to the quenched approximation to QCD at finite quark-number chemical potential, the quenched approximations to these theories behave similarly to the full theories. The reason is that these theories have real positive fermion determinants. In both of these theories there is some critical chemical potential above which the charge coupled to the chemical potential is spontaneously broken. In each case, the transition appears to be second order. We study the scaling properties near the critical point using scaling functions suggested by effective (chiral) Lagrangians and find evidence for scaling with mean-field critical exponents in each case. The subtleties associated with observing the critical scaling of these theories are discussed.