Fierz Transformation Research Articles

Covariant density functional theory with localized exchange terms

A new density-dependent point-coupling covariant density functional PCF-PK1 is proposed, where the exchange terms of the four-fermion terms are local and are taken into account with the Fierz transformation. The coupling constants of the PCF-PK1 functional are determined by empirical saturation properties and ab initio equation of state and proton-neutron Dirac mass splittings for nuclear matter as well as the ground-state properties of selected spherical nuclei. The success of the PCF-PK1 is illustrated with properties of the infinite nuclear matter and finite nuclei including the ground-state properties and the Gamow-Teller resonances. In particular, the PCFPK1 eliminates the spurious shell closures at Z = 58 and Z = 92, which exist commonly in many covariant density functionals without exchange terms. Moreover, the Gamow-Teller resonances are nicely reproduced without any adjustable parameters, and this demonstrates that a self-consistent description for the Gamow-Teller resonances can be achieved with the localized exchange terms in the PCF-PK1. ?

Interacting ud and uds quark matter at finite densities and quark stars

The stability and equation of state of quark matter are studied within both two-flavor and ($2+1$)-flavor Nambu-Jona-Lasinio (NJL) models including the vector interactions. With a free parameter $\ensuremath{\alpha}$, the Lagrangian is constructed by two parts, the original NJL Lagrangian and the Fierz transformation of it, as $\mathcal{L}=(1\ensuremath{-}\ensuremath{\alpha}){\mathcal{L}}_{\mathrm{NJL}}+\ensuremath{\alpha}{\mathcal{L}}_{\text{Fierz}}$. We find that there is a possibility for both $ud$ nonstrange and $uds$ strange matter being absolute stable, depending on the interplay of the confinement with quark vector interaction and the exchange interaction channels. The calculated quark star properties can reconcile with the recently measured masses and radii of PSR $\mathrm{J}0030+0451$ and PSR $\mathrm{J}0740+6620$, as well as the tidal deformability of GW170817. Furthermore, the more strongly interacting quark matter in the nonstrange stars allows a stiffer equation of state and consequently a higher maximum mass ($\ensuremath{\sim}2.7\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$) than the strange ones ($\ensuremath{\sim}2.1\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$). The sound velocities in strange and nonstrange quark star matter are briefly discussed compared to those of neutron star matter.

Spectral fluctuations in the Sachdev-Ye-Kitaev model

We present a detailed quantitative analysis of spectral correlations in the Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Matrix Theory (RMT) behavior are due to a small number of long-wavelength fluctuations (of the order of the number of Majorana fermions N) from one realization of the ensemble to the next one. These modes can be parameterized effectively in terms of Q-Hermite orthogonal polynomials, the main contribution being due to scale fluctuations for which we give a simple analytical estimate. Our numerical results for N = 32 show that only the lowest eight polynomials are needed to eliminate the nonuniversal part of the spectral fluctuations. The covariance matrix of the coefficients of this expansion can be obtained analytically from low-order double-trace moments. We evaluate the covariance matrix of the first six moments and find that it agrees with the numerics. We also analyze the spectral correlations in terms of a nonlinear σ-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlation functions to two-loop order. The wide correlator is given by the sum of the universal RMT result and corrections whose lowest-order term corresponds to scale fluctuations. However, the loop expansion of the σ-model results in an ill-behaved expansion of the resolvent, and it gives universal RMT fluctuations not only for q = 4 or higher even q-body interactions, but also for the q = 2 SYK model albeit with a much smaller Thouless energy while the correct result in this case should have been Poisson statistics. In our numerical studies we analyze the number variance and spectral form factor for N = 32 and q = 4. We show that the quadratic deviation of the number variance for large energies appears as a peak for small times in the spectral form factor. After eliminating the long-wavelength fluctuations, we find quantitative agreement with RMT up to an exponentially large number of level spacings for the number variance or exponentially short times in the case of the spectral form factor.

Strange quark stars within proper time regularized ( 2+1 )-flavor NJL model

In this work we use the equation of state (EOS) of (2+1)-flavor Nambu-Jona-Lasinio (NJL) model to study the structure of the strange quark star. With a new free parameter $\alpha$, the Lagrangian is constructed by two parts, the original NJL Lagrangian and the Fierz transformation of it, as $\mathcal{L}=(1-\alpha)\mathcal{L}_{NJL}+\alpha\mathcal{L}_{Fierz}$. To determine the range of $\alpha$, we compare the binding energies in the 2-flavor and (2+1)-flavor cases. We also consider the constraints of chemical equilibrium and electric charge neutrality in the strange quark star and choose six representative EOSs with different $\alpha$ and $B$ (bag constant) to study their influence on the structure of the strange quark star. As a result, we find that a larger $\alpha$ and a smaller $B$ corresponds to a heavier star with a stiffer EOS. Furthermore, the heaviest strange quark star is in agreement with not only the recent mass observation of PSR J0740+6620 and the X-ray observations on radius measurements, but also the constraint on tidal deformability of GW170817.

Hidden role of antiunitary operators in Fierz transformation

We show that whenever the symmetry group of a field theory commutes with one or more antiunitary operators $T$, which do not have to but may represent the reversal of physical time, the number of linearly independent contact two-body (quartic) terms is determined by the number of tensors that are even, or by the number of tensors that are odd, under such $T$. The choice depends on the sign of $T^2$ and on the statistics of the fields. The theorem enables one to circumvent the usual computation of the Fierz matrix in determining the independent interaction terms. Some physical examples of current interest in many-body physics are discussed.

Thermodynamics of a generalized graphene-motivated (2+1) D Gross–Neveu model beyond the mean field within the Beth–Uhlenbeck approach

Abstract We investigate the thermodynamics at finite density of a generalized $(2 + 1)$D Gross–Neveu model of $N$ fermion species with various types of four-fermion interactions. The motivation for considering such a generalized schematic model arises from taking the Fierz transformation of an effective Coulomb current–current interaction and certain symmetry-breaking interaction terms, as considered for graphene-type models in Ref. [29]. We then apply path-integral bosonization techniques, based on the large-$N$ limit, to derive the thermodynamic potential. This includes the leading-order mean-field (saddle point) contribution as well as the next-order contribution of Gaussian fluctuations of exciton fields. The main focus of the paper is then the investigation of the thermodynamic properties of the resulting fermion–exciton plasma. In particular, we derive an extended Beth–Uhlenbeck form of the thermodynamic potential, and discuss the Levinson theorem and the decomposition of the phase of the exciton correlation into resonant and scattering parts.

Novel self-consistent mean field approximation and its application in strong interaction phase transitions * * Supported in part by the National Natural Science Foundation of China (11690030, 11475085, 11535005) and National Major state Basic Research and Development of China (2016YFE0129300)

We propose a novel self-consistent mean field approximation method by means of a Fierz transformation, taking the Nambu-Jona-Lasinio model as an example. This new self-consistent mean field approximation introduces a new free parameter α to be determined experimentally. When α assumes the value of 0.5, the approximation reduces to the mean field calculation commonly used in the past. Subsequently, we study the influence of the undetermined parameter α on the phase diagram of the two-flavor strong interaction matter. The value of α plays a crucial role in the strong interaction phase diagram, as it not only changes the position of the phase transition point of strong interaction matter, but also affects the order of the phase transition. For example, when α is greater than the critical value , then the strong interaction matter phase diagram no longer has a critical end point. In addition, in the case of zero temperature and finite density, we found that when α > 1.044, the pseudo-critical chemical potential corresponds to ~4–5 times the saturation density of the nuclear matter, which agrees with the expected results from the picture of the hadrons degree of freedom. The resulting equations of state of strong interaction matter at low temperatures and high densities will have an important impact on studies concerning the mass radius relationship of neutron stars and the merging process of binary neutron stars.

Effects of Fierz transformation on gap equation and CEP at finite chemical potential and finite temperature in Hartree–Fock approximation

Three representations of Nambu–Jona-Lasinio interaction Lagrangians are constructed by Fierz transformation, and gap equation and critical end point in these three representations at finite temperature and finite chemical potential are studied. We find that the values of the quark mass and the critical end point are different in these three representations because of the consideration of Hartree–Fock approximation in our calculation.

SPIN POLARIZATION IN HIGH DENSITY QUARK MATTER

We investigate the occurrence of a ferromagnetic phase transition in high density hadronic matter (e.g., in the interior of a neutron star). This could be induced by a four-fermion interaction analogous to the one which is responsible for chiral symmetry breaking in the Nambu–Jona-Lasinio model, to which it is related through a Fierz transformation. Flavor SU(2) and flavor SU(3) quark matter are considered. A second-order phase transition is predicted at densities about 5 times the normal nuclear matter density. It is also found that in flavor SU(3) quark matter, a first-order transition from the so-called 2 flavor super-conducting phase to the ferromagnetic phase arises. The color-flavor-locked phase may be completely hidden by the FP.

Localized form of Fock terms in nuclear covariant density functional theory

In most of the successful versions of covariant density functional theory in nuclei, the Fock terms are not included explicitly, which leads to local functionals and forms the basis of their widespread applicability at present. However, it has serious consequences for the description of Gamow-Teller resonances (GTR) and spin-dipole resonances (SDR) which can only be cured by adding further phenomenological parameters. Relativistic Hartree-Fock models do not suffer from these problems. They can successfully describe the GTR and SDR as well as the isovector part of the Dirac effective mass without any additional parameters. However, they are non-local and require considerable numerical efforts. By the zero-range reduction and the Fierz transformation, a new method is proposed to take into account the Fock terms in local functionals, which retains the simplicity of conventional models and provides proper descriptions of the spin-isospin channels and the Dirac masses.

Random matrix model for chiral and color-flavor locking condensates

We study the phase diagram of a chiral random matrix model with three quark flavors at finite temperature and chemical potential, taking the chiral and diquark condensates as independent order parameters. Fixing the ratio of the coupling strengths in the quark-antiquark and quark-quark channels applying the Fierz transformation, we find that the color-flavor locked (CFL) phase is realized at large chemical potential, while the ordinary chirally broken phase appears in the region with small chemical potential. We investigate responses of the phases by changing small quark masses in the cases with three equal-mass flavors and with $2+1$ flavors. In the case with three equal-mass flavors, we find that the finite masses make the CFL phase transition line move to the higher-density region. In the case with $2+1$ flavors, we find the two-flavor color-superconducting phase at the medium-density region as a result of the finite asymmetry between the flavors, as well as the CFL phase at the higher-density region.

Bounds in 4D conformal field theories with global symmetry

We explore the constraining power of OPE associativity in 4D conformal field theory with a continuous global symmetry group. We give a general analysis of crossing symmetry constraints in the 4-point function ⟨ϕϕϕ†ϕ†⟩, where ϕ is a primary scalar operator in a given representation R. These constraints take the form of ‘vectorial sum rules’ for conformal blocks of operators whose representations appear in R⊗R and . The coefficients in these sum rules are related to the Fierz transformation matrices for the invariant tensors. We show that the number of equations is always equal to the number of symmetry channels to be constrained. We also analyze in detail two cases—the fundamental of SO(N) and the fundamental of SU(N). We derive the vectorial sum rules explicitly, and use them to study the dimension of the lowest singlet scalar in the ϕ × ϕ† OPE. We prove the existence of an upper bound on the dimension of this scalar. The bound depends on the conformal dimension of ϕ and approaches 2 in the limit For several small groups, we compute the behavior of the bound at . We discuss implications of our bound for the conformal technicolor scenario of electroweak symmetry breaking.

Hybrid Neutron Stars Based on a Modified PNJL Model

We discuss a three-flavor Nambu--Jona-Lasinio (NJL) type quantum field theoretical approach to the quark matter equation of state (EoS) with scalar diquark condensate, isoscalar vector mean field and Kobayashi-Maskawa-'t Hooft (KMT) determinant interaction. While often the diquark and vector meson couplings are considered as free parameters, we will fix them here to their values according to the Fierz transformation of a one-gluon exchange interaction. In order to estimate the effect of a possible change in the vacuum pressure of the gluon sector at finite baryon density we exploit a recent modification of the Polyakov-loop NJL (mPNJL) model which introduces a parametric density dependence of the Polyakov-loop potential also at T=0, thus being relevant for compact star physics. We use a Dirac-Brueckner-Hartree-Fock (DBHF) EoS for the hadronic matter phase and discuss results for mass-radius relationships following from a solution of the TOV equations for such a hybrid EoS in the context of observational constraints from selected objects.

Calculation of the three-body decay amplitude with high-order Fierz transformation

We present a scheme to make use of the permutation operators of the SU(4)-group to extend the Fierz identity from the usual two-bilinears case to the three-bilinears one. Using this technique, we investigate the non-resonance contributions to the charmonium decays into three pseudoscalar mesons, e.g. and so on. The result shows that the branching fractions for the direct three-body decays of ψ(2S) and J/ψ are less than 10−6. This indicates that the direct three-body decays of J/ψ and ψ(2S) are highly suppressed.

Dynamical chiral symmetry breaking, color superconductivity, and Bose–Einstein condensation in an [formula omitted]-invariant supersymmetric Nambu–Jona-Lasinio model at finite temperature and density

We investigate the phenomena of the dynamical chiral symmetry breaking (DCSB), color superconductivity (CSC), and Bose–Einstein condensation (BEC) in a supersymmetric (SUSY) vector-like SU ( N c ) gauge model at finite temperature and density. Both the N = 1 four-dimensional and N = 2 three-dimensional cases are considered. We employ the N = 1 four-dimensional generalized SUSY Nambu–Jona-Lasinio model ( N = 1 generalized SNJL 4 ) with a chemical potential as the model Lagrangian. The N = 2 three-dimensional theory is obtained by a simple dimensional reduction scheme of the four-dimensional counterpart. In order to realize the DCSB and BCS-type CSC in this model, we introduce a SUSY soft mass term. After adopting the method of SUSY auxiliary fields with the Fierz transformation in color and flavor spaces, we discuss several possible breaking schemes of the global symmetries of the model. The integrations of the auxiliary fields of composites in the effective potential are performed by using the steepest descent approximation. Under the finite-temperature Matsubara formalism, the gap equations are derived and solved. The roles of both the boson and fermion sectors in the BEC, DCSB and CSC are examined by the quasiparticle excitation spectra and the gap equations. The physical properties of the DCSB and the CSC are studied in detail. Examination on the physical property of the BEC is beyond scope of this paper. It is found that the BEC, DCSB and CSC can coexist under a condition of model parameters. Several important observations are obtained in the method of the construction of the SUSY BCS-type theory, starting from a SUSY field theoretical framework.

The S-matrix of the Faddeev-Reshetikhin model, diagonalizability andPTsymmetry

We study the question of diagonalizability of the Hamiltonian for the Faddeev-Reshetikhin (FR) model in the two particle sector. Although the two particle S-matrix element for the FR model, which may be relevant for the quantization of strings on $AdS_{5}\times S^{5}$, has been calculated recently using field theoretic methods, we find that the Hamiltonian for the system in this sector is not diagonalizable. We trace the difficulty to the fact that the interaction term in the Hamiltonian violating Lorentz invariance leads to discontinuity conditions (matching conditions) that cannot be satisfied. We determine the most general quartic interaction Hamiltonian that can be diagonalized. This includes the bosonic Thirring model as well as the bosonic chiral Gross-Neveu model which we find share the same S-matrix. We explain this by showing, through a Fierz transformation, that these two models are in fact equivalent. In addition, we find a general quartic interaction Hamiltonian, violating Lorentz invariance, that can be diagonalized with the same two particle S-matrix element as calculated by Klose and Zarembo for the FR model. This family of generalized interaction Hamiltonians is not Hermitian, but is $PT$ symmetric. We show that the wave functions for this system are also $PT$ symmetric. Thus, the theory is in a $PT$ unbroken phase which guarantees the reality of the energy spectrum as well as the unitarity of the S-matrix.

Light hadron spectra in the constituent quark model with the Kobayashi–Kondo–Maskawa-’t Hooft effective UA (1) symmetry breaking interaction

We make a critical comparison of several versions of instanton-induced interactions present in the literature, all based on ITEP group’s extension to three colours and flavours of ’t Hooft’s effective lagrangian, with the predictions of the phenomenological Kobayashi–Kondo–Maskawa (KKM) chiral quark lagrangian. We analyze the effects of all versions of the effective U A (1) symmetry breaking interactions on light hadron spectra in the non-relativistic constituent quark model. We show that the KKMT force, when used as a residual hyperfine interaction reproduces the correct ordering of pseudoscalar and vector mesons even without explicitly taking chiral symmetry into account. Moreover, the nucleon spectra are also correctly reproduced, only the Roper resonance remains too high, albeit lower than usual, at 1660 MeV. The latter’s lower than expected mass is not due to a small excitation energy, as in the Glozman–Riska (GR) model, but to a combination of colour, flavour, and spatial wave function properties that enhance the relevant matrix elements. The KKMT interaction explicitly depends on flavour and spin of the quarks, but unlike the GR flavour–spin one it has a firm footing in QCD. In the process we provide several technical advances, in particular we show the first explicit derivation of the three-body Fierz transformation and apply it to the KKM interaction. We also discuss the ambiguities associated with the colour degree of freedom.

COLOR SUPERCONDUCTIVITY AT MODERATE BARYON DENSITY

This article focuses on the two-flavor color superconducting phase at moderate baryon density. In order to simultaneously investigate the chiral phase transition and the color superconducting phase transition, the Nambu–Gorkov formalism is extended to treat the quark-antiquark and diquark condensates on an equal footing. The competition between the chiral condensate and the diquark condensate is analyzed. The cold dense charge neutral two-flavor quark system is investigated in detail. Under the local charge neutrality condition, the ground state of two-flavor quark matter is sensitive to the coupling strength in the diquark channel. When the diquark coupling strength is around the value obtained from the Fierz transformation or from fitting the vacuum bayron mass, the ground state of charge neutral two-flavor quark matter is in a thermal stable gapless 2SC (g2SC) phase. The unusual properties at zero as well as nonzero temperatures and the chromomagnetic properties of the g2SC phase are reviewed. Under the global charge neutrality condition, assuming the surface tension is negligible, the mixed phase composed of the regular 2SC phase and normal quark matter is more favorable than the g2SC phase. A hybrid nonstrange neutron star is constructed.

Generalized Fierz identities

Low energy weak interaction calculations with fermions frequently involve a superposition of quartic products of Dirac spinors, in which the order of the spinors is not the same in all the contributing terms. A common trick that is used to bring them to a uniform ordering is the Fierz transformation. We show that the standard Fierz transformation quoted in textbooks is one element of a class of transformations that exist between fermion amplitudes that are written with different, arbitrary reorderings, of the spinors. The explicit form of all such transformation matrices is given, and in addition to the usual Lorentz scalar quartic products, we consider the pseudoscalar ones. Moreover, we include the cases in which some, or all, of the spinors in the original amplitude appear in the transformed amplitudes in their Lorentz-invariant complex conjugate form. Such manipulations and formulas are useful in practical calculations, as we show with some examples.

Description of nuclear systems within the relativistic Hartree–Fock method with zero-range self-interactions of the scalar field

An exact method is suggested to treat the nonlinear self-interactions (NLSI) in the relativistic Hartree–Fock (RHF) approach for nuclear systems. We consider here the NLSI constructed from the relativistic scalar nucleon densities including products of six and eight fermion fields. This type of NLSI corresponds to the zero-range limit of the standard cubic and quartic self-interactions of the scalar field. The method to treat the NLSI uses the Fierz transformation, which enables one to express the exchange (Fock) components in terms of the direct (Hartree) ones. The method is applied to nuclear matter and finite nuclei. It is shown that, in the RHF formalism, the NLSI, which are explicitly isovector-independent, generate scalar, vector and tensor nucleon self-energies with a strong isovector dependence. This strong isovector structure of the self-energies is due to the exchange terms of the RHF method. Calculations are carried out with a parametrization containing five free parameters. The model allows a description of both types of systems compatible with experimental data.