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Superconducting quark matter in SU(2) colour group
Abstract
The effects of superconductivity of the quark matter in SU(2) colour group are studied. We analyze the QCD-inspired model of interaction of the quarks with the four quark contact interaction which represents relativistic extension of the BCS model. We construct, using the Gor'kov's propagator approach to description of the superconductivity, explicit expressions for the quark propagator, for the anomalous Green's functions, discuss their transformation properties, transformation properties of the order parameters, and find dispersion laws for quasiparticle modes in the one flavour quark matter. There exist two types of solutions of the superconductivity equations corresponding to condensation of the quark Cooper pairs in the triplet spin state. We show that the gap function depends on directions of the quasiparticle momentum and in the massless limit the energy gap vanishes for some directions of the momentum. We estimate the order parameters and critical temperaturs of the phase transition from the normal to the superconducting state, and give expressions for the fermion number density and spin density in the two states of the quark matter. We discuss also analogy between the superconducting quark matter and the nuclear matter in the case of SU(2) colour group. Two quarks in colourless state in SU(2) colour group carry fermion number and should therefore be identified with baryons, so the Cooper pairs in the quark matter can be considered as a kind of baryons present in the quark matter. The superconducting quark matter can be considered as a mixture of quarks and baryons. Such a type of the baryonic matter is shown to obey saturation condition — the well known property of nuclear matter in the real world. The possibility of extension of the model considered to the SU(3) colour group is discussed.
Supplementary resource (1)
... The techniques are inspired by Le Verrier's paper [3]. A closed analytical form for the recursion solutions [4,5,6,7] can be obtained by highlighting specific aspects of matrix algebra based on Cayley-Hamilton's theorem [8]. Trace identities are also known to exist for adjugate matrices [4,5,7] and Pfaffians [7]. ...
... A closed analytical form for the recursion solutions [4,5,6,7] can be obtained by highlighting specific aspects of matrix algebra based on Cayley-Hamilton's theorem [8]. Trace identities are also known to exist for adjugate matrices [4,5,7] and Pfaffians [7]. Other noteworthy developments include the analytical relationship between modules of eigenvector components and diagonal minor determinants and eigenvalues [9] (for a recent review, see [10]), and particular results for mass matrices in the theory of Majorana neutrinos [11]. ...
... Trace identities for a 2 × 2 matrix A give [5,7] adj ...
Eigenvectors associated with non-degenerate eigenvalues are shown to correspond to columns of the adjugate of the characteristic matrix. Degenerate eigenvalues are associated with eigenvectors that correspond to reduced complement tensors of minor determinants of the characteristic matrix. These observations are corroborated by a description of the non-degenerate two-level system and the Dirac equation, which exhibits twofold spin degeneracy of energy eigenvalues. Trace identities for the reduced order-one complement tensor and the diagonal sum of minor determinants are also presented.
... where the sum runs over sets of non-negative integers (k1,k2,…,kn) satisfying the equation Reference [1] gives a combinatorial proof of the equation. Here we give an alternative Proof. ...
... A combinatorial proof of Lemma 2 is given in Ref. [1]. Here we offer an alternative Proof. ...
... Levi-Civita symbol of the dimension n, 1 ≤ il , jl ≤ n for l = 1,2,…,n.Definition.A Pfaffian of a 2n×2n skew-symmetric matrix A is a number Levi-Civita symbol of the dimension 2n, 1 ≤ il , jl ≤ 2n for l = 1,2,…,n.The results of Ref.[1] are stated as Lemmas 1 and 2: Lemma 1. Let C be a n×n matrix, ...
We derive an expression for the product of the Pfaffians of skew-symmetric matrices A and B as a finite sum of the traces of powers of AB and an expression for the inverse matrix A$^{-1}$, or equivalently B$^{-1}$ as a finite-order polynomial of AB with the coefficients depending on the traces of powers of AB.
... This may be computed explicitly from χ as follows. Supposing χ has degree n, then we have the following beastly formula relating the determinant of the character of an element to the characters of the element's powers (obtained straightforwardly from the formula for an arbitrary square matrix given in [10]): ...
We show how the theory of characters can be used to analyze an anomaly corresponding to chiral fermions carrying an arbitrary representation of a gauge group that is finite, but otherwise arbitrary. By way of example, we do this for some groups of relevance for the study of quark and lepton masses and mixings.
... Instead, it seems a good idea to study one of the key parts of (14), the matrix T −1 , to develop a vision about the context. For the case in which only five anchor nodes are used, we use the Cayley-Hamilton decomposition to compute the inverse of the 4 × 4 matrix T [37] (31) ...
In this paper we consider the problem of improving unknown node localization by using differential received signal strength (DRSS). Many existing localization approaches, especially those using the least squares methods, either ignore nonlinear constraint among model parameters or utilize them inefficiently. In this paper, we develop four DRSS-based localization methods by utilizing different combinations of covariance and weight matrices. Each method constructs a two-stage procedure. During the first stage, an initial coarse position estimate is obtained. The second stage results the refined localization by accounting for nonlinear dependency among estimator variables. The proper choice among these proposed methods may be offered, depending on a particular signal to noise range. We implement these methods and compare them with some of the state-of-art methods in this particular problem domain and verify their performances by simulations.
Several types of dynamics at stationarity can be described in terms of a Markov jump process among a finite number N of representative sites. Before dealing with the dynamical aspects, one basic problem consists in expressing the a priori steady-state occupation probabilities of the sites. In particular, one wishes to go beyond the mere black-box computational tools and find expressions in which the jump rate constants appear explicitly, therefore allowing for a potential design/control of the network. For strongly connected networks admitting a unique stationary state with all sites populated, here we express the occupation probabilities in terms of a formula that involves powers of the transition rate matrix up to order N − 1. We also provide an expression of the derivatives with respect to the jump rate constants, possibly useful in sensitivity analysis frameworks. Although we refer to dynamics in (bio)chemical networks at thermal equilibrium or under nonequilibrium steady-state conditions, the results are valid for any Markov jump process under the same assumptions.
The main purpose and motivation of this article is to create a linear transformation on the polynomial ring of rational numbers. A matrix representation of this linear transformation based on standard fundamentals will be given. For some special cases of this matrix, matrix equations including inverse matrices, the Bell polynomials will be given. With the help of these equations, new formulas containing different polynomials, especially the Bernoulli polynomials, will be given. Finally, by applying the Laplace transform to the generating function for the Bernoulli polynomials, we derive some novel formulas involving the Hurwitz zeta function and infinite series.
Techniques are developed for constructing amplitudes of neutrino-related processes in terms of the neutrino mass matrix, with no reference to the neutrino mixing matrix. The amplitudes of neutrino oscillations in vacuum and medium, quasi-elastic neutrino scattering, β decays and double-β decays are considered. The proposed approach makes extensive use of Frobenius covariants within the framework of Sylvester’s theorem on matrix functions. The in-medium dispersion laws are found in quadratures for three flavors ofMajorana neutrinos as an application of the developed formalism. The in-medium dispersion laws for Dirac neutrinos can be determined in the general case by searching for the roots of a polynomial of degree 6. In the rest frame of baryonic matter, the minimum energy of both Majorana and Dirac neutrinos is achieved at a neutrino momentum equal to half the mean-field potential. In such cases, Dirac neutrinos occupy a hollow Fermi sphere at zero temperature and low chemical potentials. Fitting experimental data in terms of the neutrino mass matrix can provide better statistical accuracy in determining the neutrino mass matrix compared to methods using the neutrino mixing matrix at intermediate stages.
We consider pseudo-Anosov mapping classes on a closed orientable surface of genus $g$ that fix a rank $k$ subgroup of the first homology of the surface. We first show that there exists a uniform constant $C>0$ so that the minimal asymptotic translation length on the curve complex among such pseudo-Anosovs is bounded below by $C \over g(2g-k+1)$. This interpolates between results of Gadre-Tsai and of the first author and Shin, who treated the cases of the entire mapping class group ($k = 0$) and the Torelli subgroup ($k = 2g$), respectively. We also discuss possible strategy to obtain an upper bound. Finally, we construct a pseudo-Anosov on a genus $g$ surface whose maximal invariant subspace is of rank $2g-1$ and the asymptotic translation length is of $\asymp 1/g$ for all $g$. Such pseudo-Anosovs are further shown to be unable to normally generate the whole mapping class groups. As Lanier-Margalit proved that pseudo-Anosovs with small translation lengths on the Teichm\"uller spaces normally generate mapping class groups, our observation provides a restriction on how small the asymptotic translation lengths on curve complexes should be if the similar phenomenon holds for curve complexes.
We show that flat bands can be categorized into two distinct classes, that is, singular and nonsingular flat bands, by exploiting the singular behavior of their Bloch wave functions in momentum space. In the case of a singular flat band, its Bloch wave function possesses immovable discontinuities generated by the band-crossing with other bands, and thus the vector bundle associated with the flat band cannot be defined. This singularity precludes the compact localized states from forming a complete set spanning the flat band. Once the degeneracy at the band crossing point is lifted, the singular flat band becomes dispersive and can acquire a finite Chern number in general, suggesting a new route for obtaining a nearly flat Chern band. On the other hand, the Bloch wave function of a nonsingular flat band has no singularity, and thus forms a vector bundle. A nonsingular flat band can be completely isolated from other bands while preserving the perfect flatness. All one-dimensional flat bands belong to the nonsingular class. We show that a singular flat band displays a novel bulk-boundary correspondence such that the presence of the robust boundary mode is guaranteed by the singularity of the Bloch wave function. Moreover, we develop a general scheme to construct a flat band model Hamiltonian in which one can freely design its singular or nonsingular nature. Finally, we propose a general formula for the compact localized state spanning the flat band, which can be easily implemented in numerics and offer a basis set useful in analyzing correlation effects in flat bands.
The rapid development of cloud computing promotes a wide deployment of data and computation outsourcing to cloud service providers by resource-limited entities. Based on a pay-per-use model, a client without enough computational power can easily outsource large-scale computational tasks to a cloud. Nonetheless, the issue of security and privacy becomes a major concern when the customer’s sensitive or confidential data is not processed in a fully trusted cloud environment. Recently, a number of publications have been proposed to investigate and design specific secure outsourcing schemes for different computational tasks. The aim of this survey is to systemize and present the cutting-edge technologies in this area. It starts by presenting security threats and requirements, followed with other factors that should be considered when constructing secure computation outsourcing schemes. In an organized way, we then dwell on the existing secure outsourcing solutions to different computational tasks such as matrix computations, mathematical optimization, and so on, treating data confidentiality as well as computation integrity. Finally, we provide a discussion of the literature and a list of open challenges in the area.
Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
Assuming that ultra-dense matter behaves as a fluid of quarks rather than hadrons, the authors investigate the possible superfluid order parameters which may arise.
The physical properties of the astronomical compact objects are explored
on a theoretical and observational basis in a textbook designed for a
one-semester beginning-graduate-level astrophysics course. Overlapping
topics from solid-state, nuclear, and particle physics, relativity
theory, and hydrodynamics are considered. Subjects discussed include
star deaths and the formation of the compact objects, the cold equation
of state below and above neutron dip, white dwarfs and their cooling,
perturbations of fluid configurations, rotation and magnetic fields,
pulsars, compact X-ray sources, accretion onto compact objects,
gravitational radiation, supermassive stars, stellar collapse, and
supernova explosions.
Ideas and techniques known in quantum electrodynamics have been applied to the Bardeen-Cooper-Schrieffer theory of superconductivity. In an approximation which corresponds to a generalization of the Hartree-Fock fields, one can write down an integral equation defining the self-energy of an electron in an electron gas with phonon and Coulomb interaction. The form of the equation implies the existence of a particular solution which does not follow from perturbation theory, and which leads to the energy gap equation and the quasi-particle picture analogous to Bogoliubov's. The gauge invariance, to the first order in the external electromagnetic field, can be maintained in the quasi-particle picture by taking into account a certain class of corrections to the chargecurrent operator due to the phonon and Coulomb interaction. In fact, generalized forms of the Ward identity are obtained between certain vertex parts and the self-energy. The Meissner effect calculation is thus rendered strictly gauge invariant, but essentially keeping the BCS result unaltered for transverse fields. It is shown also that the integral equation for vertex parts allows homogeneous solutions which describe collective excitations of quasi-particle pairs, and the nature and effects of such collective states are discussed.
We note the following: The quark model implies that superdense matter (found in neutron-star cores, exploding black holes, and the early big-bang universe) consists of quarks rather than of hadrons. Bjorken scaling implies that the quarks interact weakly. An asymptotically free gauge theory allows realistic calculations taking full account of strong interactions.
It is shown that at some temperature quark releasing occurs due to the string condensation. The properties of the arising phase are investigated.
DOI:https://doi.org/10.1103/PhysRev.104.1189