It is shown that U(3) L × U(3) R eight-quark interactions stabilize the asymmetric ground state of the well-known model with four-quark Nambu-Jona-Lasinio and six-quark 't Hooft interactions. The result remains when the reduced SU(3) flavour symmetry is explicitly broken by the general current quark mass term withm u =m d =m s .

The necessity of adding higher order multiquark interactions to the three flavor NJL model with U A (1) breaking, in order to stabilize its vacuum, is discussed.

It is shown that the spontaneous magnetization occurs due to the anomalous magnetic moments of quarks in the high-density quark matter under the tensor-type four-point interaction. The spin polarized condensate for each flavor of quark appears at high baryon density, which leads to the spontaneous magnetization due to the anomalous magnetic moments of quarks. The implications to the strong magnetic field in the compact stars is discussed.

The effect of strong magnetic fields on the equation of state (EoS) for compact stars described with density dependent relativistic hadronic models is studied. A comparison with other mean-field relativistic models is done. It is shown that the largest differences between models occur for low densities and that the magnetic field affects the crust properties of star, namely its extension. PACS numbers: 26.60.-c 26.60.Kp 97.60.Jd 24.10.Jv I. INTRODUCTIONThe study of very asymmetric nuclear matter is presently an important issue due to the radioactive beams which will be operating in the near future and which will allow the investigation of a region of the nuclear matter phase space unaccessible till recently. Asymmetric nuclear matter is of particular interest for the description of stellar matter of compact stars.Compact star properties depend a lot on the model used to describe the hadronic equation of state (EoS). In particular relativistic nuclear mean-field models [1,2] are very popular to describe stellar matter because causality will always be satisfied. The imposition of constraints, both coming from measured star properties or from relativistic heavy ion collisions in the laboratory, is essential to test the different models [3].Magnetars are neutron stars which may have surface magnetic fields larger that 10 15 G [4-6] and which were discovered at the x-ray and γ-ray energies (for a review refer [7]). They are identified with the anomalous x-ray pulsars (AXP) and soft γ-ray repeaters. Taking as reference the critical

In order to get an irreducible representation of u(M) algebra, we develop a canonical theory of mixed state and discuss the dual property of mutually commutable u(M) algebras which have the Casimir operators in common. With the aid of this dual property, we can obtain an irreducible representation of u(M) algebra which is useful for investigations of nuclear dynamics, where generators of u(M) algebra given by one-body operators of fermion are expressed in terms of bosons. As an example, we derive the Holstein-Primakoff representation of u(M) generators. We also derive the irreducible representation of su(3) algebra. * ) Results on the irreducible representation given here is purely mathematical and independent of the atomic nuclear model. However, the present treatment of u(M) algebra will make clear the physical image or the physical ingredient of the irreducible representation.

It is shown that eight-quark interactions, which are needed to stabilize the ground state of the combined three flavor Nambu -- Jona-Lasinio and 't Hooft Lagrangians, play also an important role in determining the critical temperature at which transitions occur from the dynamically broken chiral phase to the symmetric phase.Comment: 4 pages, 2 figure

Examples are given of non-Hermitian Hamiltonian operators which have a real spectrum. Some of the investigated operators are expressed in terms of the generators of the Weyl-Heisenberg algebra. It is argued that the existence of an involutive operatorĴ which renders the HamiltonianĴ-Hermitian leads to the unambiguous definition of an associated positive definite norm allowing for the standard probabilistic interpretation of quantum mechanics. Non-Hermitian extensions of the PoeschlTeller Hamiltonian are also considered. Hermitian counterparts obtained by similarity transformations are constructed.

Catalysis of dynamical symmetry breaking by a constant magnetic field in (3 + 1) dimensions is considered. We use the three flavour NambuJona-Lasinio type model with 't Hooft and eight-quark interaction terms. It is shown that the multi-quark interactions introduce new additional features to this phenomenon: (a) the local minimum of the effective potential catalyzed by the constant magnetic field is smoothed out with increasing strength of the field at the characteristic scale H ∼ 10 19 G, (b) the multi-quark forces generate independently another local minimum associated with a larger dynamical fermion mass. This state may exist even for multi-quark interactions with a subcritical set of couplings, and is globally stable with respect to a further increase of the magnetic field. © 2007 Elsevier B.V. All rights reserved. PACS: 11.30.Rd; 11.30.Qc It has been shown in a series of papers [1][2][3] that in (2 + 1) and (3 + 1) dimensions a constant magnetic field H = 0 catalyzes the dynamical symmetry breaking leading to a fermion mass even at the weakest attractive four-fermion interaction between particles, and the symmetry is not restored at any arbitrarily large H . Soon thereafter it became also clear [4][5][6] that the zero-energy surface of the lowest Landau level (LLL) plays a crucial role in the dynamics of such fermion pairing. It has been found that the dynamics of the fermion pairing in the homogeneous magnetic field is essentially (1 + 1)-dimensional, and a deep analogy of this phenomenon with the dynamics of electron pairing in BCS [7] has been stressed. The generated fermion mass, M dyn , turned out to be much smaller than the Landau gap ∼ √ |eH |. The existence of a zero-energy surface in the spectrum of a Dirac particle is ensured for any homogeneous magnetic field with a fixed direction by a quantum mechanical supersymme-* Corresponding author.E-mail address: osipov@nu.jinr.ru (A.A. Osipov).try of the corresponding second-order Dirac Hamiltonian [8]. This aspect of the phenomenon appears to be a quite exceptional situation and indicates that the dynamical generation of mass is not so universal as one would expect by extrapolating the results obtained for homogeneous or unidirectional [9] magnetic field profiles. For instance, it has been demonstrated by Ragazzon [10] that the Nambu-Jona-Lasinio (NJL) model [11] minimally coupled to a background magnetic field with variable direction does not possess a massive phase until the coupling constant exceeds some critical value. Obviously, in this case one faces the conventional scenario of dynamical chiral symmetry breaking, where the magnetic field does not play an essential role. Conversely, having in mind that homogeneous magnetic fields can act as strong catalysts of chiral symmetry breaking, one might ask what is the effect caused by the strong interaction, when higher order multi-fermion interactions are present. These extensions of the NJL model are well known [12][13][14], for instance, the four-quark U(3) L × U(3) R chiral symmetric Lagr...