Dynamic density and spin responses of a superfluid Fermi gas in the BCS–BEC crossover: Path integral formulation and pair fluctuation theory

We present a standard field theoretical derivation of the dynamic density and spin linear response functions of a dilute superfluid Fermi gas in the BCS–BEC crossover in both three and two dimensions. The derivation of the response functions is based on the elegant functional path integral approach which allows us to calculate the density–density and spin–spin correlation functions by introducing the external sources for the density and the spin density. Since the generating functional cannot be evaluated exactly, we consider two gapless approximations which ensure a gapless collective mode (Goldstone mode) in the superfluid state: the BCS–Leggett mean-field theory and the Gaussian-pair-fluctuation (GPF) theory. In the mean-field theory, our results of the response functions agree with the known results from the random phase approximation. We further consider the pair fluctuation effects and establish a theoretical framework for the dynamic responses within the GPF theory. We show that the GPF response theory naturally recovers three kinds of famous diagrammatic contributions: the Self-Energy contribution, the Aslamazov–Lakin contribution, and the Maki–Thompson contribution. We also show that unlike the equilibrium state, in evaluating the response functions, the linear (first-order) terms in the external sources as well as the induced order parameter perturbations should be treated carefully. In the superfluid state, there is an additional order parameter contribution which ensures that in the static and long wavelength limit, the density response function recovers the result of the compressibility (compressibility sum rule). We expect that the f$f$-sum rule is manifested by the full number equation which includes the contribution from the Gaussian pair fluctuations. The dynamic density and spin response functions in the normal phase (above the superfluid critical temperature) are also derived within the Nozières–Schmitt–Rink (NSR) theory.     

Chiral condensate and chemical freeze-out

We consider a chemical freeze-out mechanism which is based on a strong medium dependence of the rates for inelastic flavor-equilibrating collisions based on the delocalization of hadronic wave functions and growing hadronic radii when approaching the chiral restoration. We investigate the role of mesonic (pion) and baryonic (nucleon) fluctuations for melting the chiral condensate in the phase diagram in the (T, μ)-plane. We apply the PNJL model beyond mean-field and present an effective generalization of the chiral perturbation theory result which accounts for the medium dependence of the pion decay constant while preserving the GMOR relation. We demonstrate within a schematic resonance gas model consisting of a variable number of pionic and nucleonic degrees of freedom that within the above model a quantitative explanation of the hadonic freeze-out curve and its phenomenological conditions can be given.     

Chiral anomaly and nucleon properties in the Nambu-Jona-Lasinio model with vector mesons

We consider the extended SU(3) Nambu and Jona-Lasinio model with explicit vector couplings in the presence of external fields. We study the chiral anomaly in this model and its implications on the properties of the nucleon described as a chiral soliton of three valence quarks bounded in mesonic background fields. For the model to reproduce the QCD anomaly it is necessary to subtract suitable local and polynomial counterterms in the external and dynamical vector and axial-vector fields. We compute the counterterms explicitly in a vector-gauge-invariant regularization, and obtain modifications to the total effective action and vector and axial currents. We study the numerical influence of those counterterms in the two-flavour version of model with dynamical σ, π, , A and ω mesons. We find that, for time-independent hedgehog configurations, the numerical effects in the nucleon mass, the isoscalar nucleon radius and the axial coupling constant are negligibly small.     

High precision mean-field results for lattice gauge theories: with special attention paid to the gauge dependence of mean-field perturbation theory to finite order

We do mean-field perturbation theory for U(1) lattice gauge theory in the axial gauge, and evaluate corrections from fluctuations up to fourth order for the free energy and plaquette energy. Comparing with similar results previously obtained in the Feynman gauge we find, to those orders studied, a gauge dependence of the size of the first correction term neglected with one exception. This gauge dependence decreases rapidly as the order of the approximation is increased. To any finite order, results in axial gauge are better approximations than results in the Feynman gauge. We speculate why. Assuming it to be generally true, we evaluate the first correction beyond the one-loop mean-field approximation to the free energy of SU(2) gauge theory with Wilson action in the axial gauge. This correction brings the mean-field result very close to Monte Carlo results for β > 1.6. It also makes the mean-field result identical, within a narrow margin, to ressumed strong coupling results in the interval 1.6 < β < 2.4, thus showing the absence of a phase transition.For both groups studied, we find that the asymptotic series of mean-field perturbation theory give much better approximations than do ordinary weak coupling series.     

QCD susceptibilities and nuclear-matter saturation in a relativistic chiral theory

We investigate the evolutions with density of the QCD scalar susceptibility and of the sigma mass in a chiral relativistic theory of nuclear matter, in the mean-field approximation. In order to reach saturation we need to introduce the scalar response of the nucleons. The consequences are a quite mild density dependence of the sigma mass and the progressive decoupling of the quark density fluctuations from the nucleonic ones at large densities.     

Density fluctuations in the presence of spinodal instabilities

Density fluctuations resulting from spinodal decomposition in a nonequilibrium first-order chiral phase transition are explored. We show that such instabilities generate divergent fluctuations of conserved charges along the isothermal spinodal lines appearing in the coexistence region. Thus, divergent density fluctuations could be a signal not only for the critical end point but also for the first-order phase transition expected in strongly interacting matter. We also compute the mean-field critical exponent at the spinodal lines. Our analysis is performed in the mean-field approximation to the Nambu-Jona-Lasinio model formulated at finite temperature and density. However, our main conclusions are expected to be generic and model independent.     

Aspects of the phase diagram in (P)NJL-like models

We discuss three applications of NJL- and PNJL-like models to assess aspects of the QCD phase diagram: First, we study the effect of mesonic correlations on the pressure below and above the finite temperature phase transition within a non-local PNJL model beyond the mean-field approximation. Second, we reconstruct the phase boundary of an NJL model from a Taylor expansion of the chiral susceptibility about μ=0 and compare the result with the exact phase boundary. Finally, we demonstrate the realization of the “non-standard scenario” for the critical surface in a three-flavor PNJL model with a μ-dependent determinant interaction.     

Calculations of multiquark functions in effective models of strong interaction

In this paper we present our results of the investigation of multiquark equations in the Nambu-Jona-Lasinio model with chiral symmetry of SU(2) group in the mean-field expansion. To formulate the mean-field expansion we have used an iteration scheme of solution of the Schwinger-Dyson equations with the fermion bilocal source. We have considered the equations for Green functions of the Nambu-Jona-Lasinio model up to third step for this iteration scheme. To calculate the high-order corrections to the mean-field approximation, we propose the method of the Legendre transformation with respect to the bilocal source, which allows effectively to take into account the symmetry constraints related with the chiral Ward identity. We discuss also the problem of calculating the multiquark functions in the mean-field expansion for Nambu-Jona-Lasinio-type models with other types of the multifermion sources.     

Conserving approximations for response functions of the Fermi gas in a random potential

One- and two-electron Green functions are simultaneously needed to determine the responsefunctions of the electron gas in a random potential. Reliable approximations must retainconsistency between the two types of Green functions expressed via Ward identities so thattheir output is compliant with macroscopic symmetries and conservation laws. Such aconsistency is not directly guaranteed when summing nonlocal corrections to the local(dynamical) mean field. We analyze the reasons for this failure and show how the full Wardidentity can generically be implemented in the diagrammatic approach to the vertexfunctions without breaking the analytic properties of the self-energy. We use thelow-energy asymptotics of the conserving two-particle vertex determining the singular partof response and correlation functions to derive an exact representation of the diffusionconstant in terms of Green functions of the perturbation theory. We then calculateexplicitly the leading vertex corrections to the mean-field diffusion constant due tomaximally-crossed diagrams.     

Critical properties of the half-filled Hubbard model in three dimensions

By means of the dynamical vertex approximation (DΓA) we include spatial correlations on all length scales beyond the dynamical mean-field theory (DMFT) for the half-filled Hubbard model in three dimensions. The most relevant changes due to nonlocal fluctuations are (i) a deviation from the mean-field critical behavior with the same critical exponents as for the three dimensional Heisenberg (anti)ferromagnet and (ii) a sizable reduction of the Néel temperature (T(N)) by ~30% for the onset of antiferromagnetic order. Finally, we give a quantitative estimate of the deviation of the spectra between DΓA and DMFT in different regions of the phase diagram.     

The magnetic form factors of 3He and 3H with mesonic-exchange corrections: Effects of the D-state and the role of short-range correlations

We show that the D-state of the trinucleon and the short-range behavior of the S- and D-state wave functions are extremely important in determining the magnetic form factors of the three-body system. We present a complete calculation of the form factors taking into account the conventional impulse approximation for the nuclear current and contributions from mesonic exchange currents. The D-state wave function has been determined by perturbation theory.     

Propagation of p-polarized electromagnetic waves obliquely incident on stratified random media: Random phase approximation

We consider the propagation of p-polarized electromagnetic waves obliquely incident on stratified random dielectric media. Using the invariant imbedding method generalized to random media and applying the random phase approximation, we derive a simple analytical expression of the localization length and calculate the disorder-averaged reflectance and transmittance and the fluctuations of the localization length and the reflectance as functions of the incident angle. We also calculate the disorder-averaged intensity profile of the magnetic field inside the random medium. We find that within the random phase approximation, the p wave can be delocalized and transmitted completely at a certain critical incident angle, which is bigger than the Brewster angle in the uniform case.     

Meson-exchange corrections to the nuclear weak axial charge density in the hard pion model and 0+ 3 0− transitions in A = 16 nuclei

Starting with the hard-pion model based on a minimal chiral invariant phenomenological lagrangian, the two-body part of the time component of the weak axial-vector current is constructed in the tree approximation. Pion, rho- and A₁-meson exchanges are considered. The mesonic exchange operator obtained is applied to describe the purely weak axial 0⁺?0^?, ΔT = 1 transition in the nuclear A = 16 system. In order to treat nuclear structure correlation effects, explicit use of shell-model wave functions with configuration mixing is made. We confirm the large enhancement of the nuclear weak axial charge density with respect to impulse approximation.     

Quark dynamics and strong meson decays

The strong decays of meson resonances are treated in a dynamical quark model. The model is formulated in the framework of general field theory and therefore all calculations are fully relativistic covariant. Spectrum and wave functions are derived from a Bethe-Salpeter equation describing the binding of heavy quarks by a smooth, very strong interaction. The mesonic vertices are calculated with help of these BS amplitudes in triangle graph approximation, thereby guaranteeing a symmetric treatment of all mesons involved in the process. A particular spin dependence of the interaction has the consequence that the super-strong quark binding forces lead to mesonic forces of moderate strength only (saturation of quark forces!).The applications refer to the decays of vector mesons, tensor mesons, scalar mesons into pseudoscalars and vectors and are extended to resonances in the R region. In particular, we include the decays into two pions of the radial excited ?′(1600). The helicity structure of the decays of the axial vector meson is discussed.     

Effects of mesonic correlations in the QCD phase transition

The finite-temperature phase transition of strongly interacting matter is studied within a nonlocal chiral quark model of the NJL type coupled to a Polyakov loop. In contrast to previous investigations which were restricted to the mean-field approximation, mesonic correlations are included by evaluating the quark-antiquark ring sum. For physical pion masses, we find that the pions dominate the pressure below the phase transition, whereas above T _c the pressure is well described by the mean-field-approximation result. For large pion masses, as realized in lattice simulations, the meson effects are suppressed. The text was submitted by the authors in English.     

Nucleon properties in a few mode bag — an effective approximation to QCD

The Hamiltonian of Quantum Chromodynamics is approximated by restricting the spatial dependencies of the quark-antiquark and gluon fields to a small number of modes (few-mode approximation). As a working hypothesis, we assume that the ground states of the dynamical sectors of the theory are automatically gauge invariant so that Gauss's law need not be strictly satisfied by a trial state. We derive and apply a method for explicit calculation of expectation values for wavefunctions in colour space depending on the twoSU (3) invariants acting as dynamical variables. An interacting ground state is constructed by a dynamical dressing transformation of the bare fermionic vacuum which may be regarded as a few-mode approximation to a quark-gluon (or mesonic) condensate. Based on this vacuum a nucleon state is formulated quite similar to the solition bag construction. Some of its effective properties are calculated by a variational calculation and comparison is made with experimental data as well as with other theoretical models of bag-type.     

Phase transition and the nucleon as a soliton in the Nambu-Jona-Lasinio model at finite temperature and density

We study some bulk thermodynamical characteristics, meson properties and the nucleon as a baryon-number-one soliton in hot quark matter in the NJL model as well as in hot nucleon matter in a hybrid NJL model in which the Dirac sea of quarks is combined with a Fermi sea of nucleons. In both cases, working in the mean-field approximation, we find a chiral phase transition from the Goldstone to the Wigner phase. At finite density the chiral order parameter and the constituent quark mass have a non-monotonic temperature dependence — at finite temperatures not close to the critical one they are less affected than in cold matter. Whereas quark matter is rather soft against thermal fluctuations and the corresponding chiral phase transition is smooth, nucleon matter is much stiffer and the chiral phase transition is very sharp. The thermodynamical variables show large discontinuities which is an indication for a first-order phase transition. We solve the B = 1 solitonic sector of the NJL model in the presence of external hot quark and nucleon media. In the hot medium at intermediate temperature the soliton is more bound and less swelled than in the case of cold matter. At some critical temperature, which for nucleon matter coincides with the critical temperature for the chiral phase transition, we find no more a localized solution. According to this model scenario one should expect a sharp phase transition from nucleon to quark matter.