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The Role of the Hidden Color Channel in Some Interesting Dibaryon Candidates

Nowadays, exploring dibaryon candidates has attracted much attention, both theoretically and experimentally. It is important to find a reasonable model to predict the possible dibaryon candidates. The chiral SU(3) quark model is just one of the most successful models, with which we can reasonably explain the experimental binding energies of baryon’s ground state and the properties of deuteron, NN and YN scattering processes. By utilizing the same set of model parameters, we predicted the nonstrange d* dibaryon with a binding energy of 84MeV, which is consistent with a recent experiment in which we also found that the hidden color (CC) channel plays an important role in forming this bound state. Due to the theoretical investigation of the CC channel being scarce for dibaryons, we explore other possible and interesting dibaryon candidates in the present work. According to the symmetry properties, we chose six interesting candidates, including strangeness 0,−1,−5,−6 systems. All the hidden color wave functions were built, and the spin-flavor-color matrix elements were systematically evaluated. Then, we applied these obtained matrix elements to further dynamically solve the corresponding resonating group method’s equation in a coupled-channel calculation. The results show that the coupling to the CC channel plays an significant role in forming each spin S = 3 state, where tensor coupling is also included and has an obvious effect in forming each S = 0 state. The present work is significant in helping us to acquire deeper understanding of the effects of the hidden color channel and QCD phenomenology.

The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case

We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of {W}_{0,{mathfrak{q}}^{pm 2}} contains diagonal fields (hr,1, hr,1) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in {mathfrak{q}}^{pm 2} . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.

Unitary representations of Poincaré group P ( 1 , n ) in S O ( 1 , n ) -basis

This paper concerns the problem of reduction of unitary irreducible representations of the Poincaré group $\mathrm{P}(1,n)$ with respect to representations of its subgroup $\mathrm{SO}(1,n)$. Based on a generalization of the Wigner-Eckart theorem, we obtain matrix elements of the shift operators in the $\mathrm{SO}(1,n)$-basis. <br><br> Робота присвячена проблемі редукції унітарних незвідних представлень групи Пуанкаре $P(1, n)$ відносно представлень її підгрупи $SO(1, n)$. На основі узагальнення теореми Вігнера-Еккарта отримано матричні елементи операторів зсуву в $SO(1, n)$-базисі.

On matrix elements of the -representation in a space of functions on -matrices

ABSTRACTWe obtain matrix elements and kernel of representation operators for one realization of-representation in an appropriate space of functions defined on matrix cone. Indeed, we show that the matrix elements can be expressed in terms of product of two Gaussian hypergeometric functions and the kernel of integral operator corresponding to representation linear operator can be expressed in terms of Bessel functions of the first kind or modified Bessel functions of the second kind.

Measurement of the matrix elements for the decays η′→ηπ+π− and η′→ηπ0π0

Based on a sample of $1.31\times10^9$ $J/\psi$ events collected with the BESIII detector, the matrix elements for the decays $\eta^{\prime}\to\eta\pi^{+}\pi^{-}$ and $\eta^{\prime}\to\eta \pi^{0}\pi^{0}$ are determined using 351,016 $\eta^{\prime}\to(\eta\to\gamma\gamma)\pi^+\pi^-$ and 56,249 $\eta^{\prime}\to(\eta\to\gamma\gamma)\pi^0\pi^0$ events with background levels less than 1\%. Two commonly used representations are used to describe the Dalitz plot density. We find that an assumption of a linear amplitude does not describe the data well. A small deviation of the obtained matrix elements between $\eta^{\prime}\to\eta\pi^{+}\pi^{-}$ and $\eta^{\prime}\to\eta \pi^{0}\pi^{0}$ is probably caused by the mass difference between charged and neutral pions or radiative corrections. No cusp structure in $\eta^{\prime}\to\eta \pi^{0}\pi^{0}$ is observed.

Schematic interactions with many degeneracies

In previous works, we examined the spectra for systems of two protons and two neutrons, in a single [Formula: see text] shell calculation, by obtaining matrix elements from experiment. More recently, we considered the schematic interactions in the same model space. We continue in this vein here. The present work and the former can be regarded as two bookends on a bookshelf.

Polarimetric Measurement of Jones Matrix of a Twisted Nematic Liquid Crystal Spatial Light Modulator

A polarimetric experimental method was developed to determine the Jones matrix elements of transparent optical materials without sign ambiguity. A set of polarization dependent transmittance data of the samples was measured with polarizer - sample - analyzer system and another set of data was measured with polarizer - sample - quarter-wave plate - analyzer. Two data sets were compared and mathematically analyzed to obtain the correct signs of the elements of the matrix. The Jones matrix elements of a quarter-wave plate were determined to check the validity of the method. The experimentally obtained matrix elements of the quarter-wave plate were consistent with the theoretical expectations. The same method was applied to obtain the Jones matrix elements of a twisted nematic liquid crystal panel.

Application of diophantine integration to hartree-fock and configuration interaction calculations

The methods of open and closed Diophantine integration have been applied to obtain matrix elements relevant to unrestricted Hartree-Fock (uhf) and configuration interaction (ci) calculations for atoms and molecules. In the present formulation matrix elements of the effective Hamiltonian are evaluated directly without the intermediate step of generating molecular integrals. The method appears to be superior to conventional techniques for large polyatomic systems; it can handle arbitrary analytic or numerical basis sets, and is easily adapted to include operators such as the spin–orbit interaction in the Hamiltonian. Preliminary results for both uhf and ci methods are quite encouraging, showing that reasonably accurate energies and wave functions can be obtained with moderate computational effort. Results of calculations on the Sc2F and MnF6 complexes, as well as several small atoms and molecules, are briefly discussed. Importance sampling and the use of distribution functions is shown to accelerate convergence of the integrals.

Representations of the Poincaré Lie algebra from unitary representations of SO(1,4)

An algebraic isomorphism of an extension of the Lie field of so(1,4) and an extension of the Poincaré Lie field was given in Havlicek and Moylan [J. Math Phys. 34, 5320 (1993)]. Here we use the isomorphism described in that paper in order to obtain matrix elements of the Poincaré Lie algebra generators in a normalizable basis for an arbitrary unitary spherical principal series representation of SO0(1,4).

Isospin character of low-lying states in 56Fe.

Low-lying states in {sup 56}Fe, up to an excitation energy of about 4 MeV, have been investigated by means of inelastic proton and deuteron scattering experiments at {ital E}{sub {ital p}}=65 and 400 MeV and at {ital E}{sub {ital d}}=56 MeV, respectively. Measured cross sections and analyzing powers have been compared with coupled-channels calculations using collective form factors; calculations in both the Schr{umlt o}dinger and Dirac formalisms have been carried out for the proton data. For each probe, the matrix elements have been deduced for transitions from the ground state and from the 2{sub 1}{sup +} state to six quadrupole (2{sup +}) states to one octupole (3{sub 1}{sup {minus}}) and two hexadecapole (4{sub 1}{sup +} and 4{sub 2}{sup +}) states. The obtained matrix elements and the previous values from {gamma} decay or electron inelastic scattering have been used to evaluate the isospin character of the transitions. To discuss the quadrupole mixed-symmetry states in {sup 56}Fe, the deduced neutron ({ital M}{sub {ital n}}) and proton ({ital M}{sub {ital p}}) components of the matrix elements, or equivalently the isoscalar ({ital M}{sub {ital s}}) and isovector ({ital M}{sub {ital v}}) parts, have been compared with theoretical calculations based on the neutron-proton interacting bosonmore » model and on the shell model evaluated in a full {ital f}-{ital p} configuration space. {copyright} {ital 1996 The American Physical Society.}« less

A hybrid method for the evaluation of the matrix elements of the Coulomb potential

An analysis of the method of obtaining matrix elements of the Coulomb potential by numerical integration of the ‘exact’ potential, a strategy proposed by Termath and Handy, reveals that this method will be inaccurate in large systems because of the long-range nature of the Coulomb forces. The results can be improved substantially if one integrates the long-range part of the potential analytically. This has been achieved — at virtually no extra cost — by introducing a simple model density whose electrostatic potential has the correct long-range behaviour and whose Coulomb matrix elements can easily be evaluated analytically. The model density can be designed to reduce errors which arise in the fully numerical method due to the electrostatic potential in the cores of heavy atoms.

Beyond the single universe

Wormholes are going to play a rather crucial role in any theory of quantum gravity. In this paper a method is discussed for describing the statistics associated with a single wormhole in terms of probabilities and correlation functions, and their generating functionals, relative both to an observer living in Euclidean time and an observer in the asympotic region. We obtain matrix elements between vacuum states, filtered by the quantum state of the wormhole, of the same form as those which were recently discussed by Hawking and others for Abelian matter fields. For non-Abelian gauge fields, the matrix elements turn out to be given in terms of the field intensities rather than the fields themselves. In generalizing the procedure for obtaining wormhole-filtered generating functionals, we had to introduce the notion of cosmological spin which turned out to be responsible for providing the quantum state of a single universe with an observable character. It has been also seen that baby universes are in highly non-classical states describable as localized squeezed states of space-time.

A new boson realization for the isotropic three-dimensional oscillator

Two new closely related boson realizations of the Heisenberg–Weyl algebra are derived that are directly equivalent to the introduction of spherical coordinates for the isotropic three-dimensional harmonic oscillator. This provides another way to obtain matrix elements of all physical operators connecting angular-momentum eigenstates, which are represented here by simple products of boson operators acting on the vacuum. Two of these bosons carry the angular-momentum quantum numbers while the third is a novel ‘‘radial boson.’’ In this approach, one begins by finding the nonunitary Dyson and then the unitarized Holstein–Primakoff realizations for the spectrum-generating SU(1,1)×SO(3) algebra of the oscillator. The corresponding mappings of the Heisenberg–Weyl generators are then obtained from their tensor properties under SU(1,1)×SO(3). The Dyson map of the Heisenberg–Weyl generators is not uniquely determined in this way, but the Holstein–Primakoff map is nevertheless unique. The possibility exists for generalization to isotropic oscillators of higher dimension, which are of interest in connection with nuclear collective motion.

First and second derivative matrix elements for linear and out‐of‐plane bending motion

AbstractSimplified formulas for first and second derivatives of the internal coordinates with respect to Cartesian coordinates are reported for linear and out‐of‐plane bending motion. They are expressed in a local coordinate system and then rotated to a space‐fixed Cartesian coordinate system. For linear motion the important points are: (1) the invariance of the energy with respect to translational and rotational coordinate transformations and (2) the presentation of derivatives of the energy in terms of (α − π)2 rather than (α − π), where α is the bond angle, to avoid indeterminant forms. By factoring the second derivatives of the energy with respect to (α − π) and sinα, analysis in this local coordinate system avoids division by sinα as α → π. The formulas describe linear systems without the need to project the motion onto two perpendicular planes. When the angle is exactly π two degenerate directions arise, and the coordinate axes perpendicular to the axis of the linear sequence of atoms may be chosen arbitrarily. Further refinements and clarifications of methods reported previously to obtain matrix elements for linear and out‐of‐plane motion are presented.

Meaning of the BRS Lagrangian theory.

A simplified treatment of the Becchi-Rouet-Stora (BRS) Lagrangian theory is presented. With this treatment we show that the BRS Lagrangian theory in general, and the Feynman-gauge field theory in particular, are effective theories, not the physical theory, and the Feynman gauge is not, strictly speaking, a gauge. The relationship between the quantum states in the BRS Lagrangian theory and those in the physical theory is explicitly given. We also show that one may obtain matrix elements of gauge-invariant operators in the physical theory by calculating corresponding ones in the BRS Lagrangian theory. The formulas which equate such matrix elements are called correspondence formulas. The correspondence formula for the $S$ matrix enables us to equate the scattering amplitudes in the physical theory with those in the BRS Lagrangian theory, thus a proof of the unitary of the Feynman-gauge (as well as other covariant gauges) Feynman rules is rendered unnecessary. This treatment can be applied to various gauge field theories and the examples of the pure Yang-Mills theory and a gauge field theory with a Higgs field is explicitly worked out.

Algebraic solution of a general quadrupole Hamiltonian in the interacting boson model.

A method based on an eigenmode condition and projection techniques is presented for solution of a general quadrupole Hamiltonian in the interacting boson model. It is shown that the intrinsic states obtained from the eigenmode condition provide a zeroth order solution to the diagonalization of the corresponding quadrupole Hamiltonian. The method is in effect a 1/N expansion, where N is the boson number, ideally suited for the deformed nuclei for which Napprox. =12--16. Because of certain cancellations with the normalization, zeroth order solutions of the intrinsic states are sufficient to obtain matrix elements to order O(1/N/sup 2/).

Matrix elements for the Morse potential using ladder operators

AbstractThe algebra of the two‐dimensional harmonic oscillator is exploited to obtain matrix elements between eigenstates of the Morse potential. This follows after mapping the latter into the radial equation of the former problem by means of a change of variable and the use of the angular variable as a dummy variable.

From determinants to spin eigenfunctions—a simple algorithm

AbstractCalculation of blocks of matrix elements between determinants associated with two fixed orbital configurations is very easy. A simple method to obtain the whole blocks and then transform them to the spinadapted basis is described. The method is suitable for many‐particle operators, the number of operations to obtain matrix elements being independent of the number of orbitals or electrons. Some applications of the proposed algorithm, and possible extensions to eigenfunctions of Ľ2 and other operators, are discussed.

Ladder operators for orbital angular momentum

Three pairs of abstract operators are presented which serve as ladder operators for the orbital angular momentum quantum numbers l and m. These operators are used to prove the restriction of l to integral values and also to obtain matrix elements for orbital angular momentum state vectors. The calculations are based entirely on an application of the abstract (Dirac) operator method to orbital angular momentum.

A generalized internal axis method for high barrier tunneling problems, as applied to the water dimer

When more than one large-amplitude vibrational motion is present in a molecule, it is often not possible to define a global internal-axis-method (IAM) coordinate system and set of basis functions. In the present work, a method is presented for extending the IAM treatment to tunneling problems in such cases, using as an illustration a model for the water dimer with three large-amplitude vibrational coordinates. The method involves the construction of two different sets of local IAM-like coordinate systems. The first of these contains n coordinate systems, one for the small neighborhood surrounding each of the n equilibrium frameworks. The second contains on the order of n 2 2 coordinate systems, one for each feasible tunneling path between each pair of frameworks. Basis functions written in the second set of local IAM-like coordinates are used to determine the complex phase factors associated in this method with tunneling matrix elements of the phenomenological rotational Hamiltonian in the high barrier limit. These phase factors govern the way in which the various real tunneling frequencies in the molecule constructively and/or destructively interfere in the Hamiltonian matrix elements and final energy expressions. Various mathematical approximations are involved in using the local IAM-like basis sets to obtain matrix elements; the full extent of the adverse effects of these approximations will not be known until an attempt to fit experimental data is carried out.