Isoscalar Factors Research Articles

Spectral content of isoscalar nucleon form factors

The nucleon strange vector and isoscalar electromagnetic form factors are studied using a spectral decomposition. The $K\overline{K}$ contribution to the electric and magnetic radii as well as the magnetic moment is evaluated to all orders in the strong interaction using an analytic continuation of experimental $\mathrm{KN}$ scattering amplitudes and bounds from unitarity. The relationship between nonresonant and resonant $K\overline{K}$ contributions to the form factors is demonstrated, and values for the vector and tensor $\ensuremath{\varphi}N\overline{N}$ couplings are derived. The $K\overline{K}$ spectral functions are used to evaluate the credibility of model calculations for the strange quark vector current form factors.

KK¯continuum and isoscalar nucleon form factors

We analyze the isoscalar vector current form factors of the nucleon using dispersion relations. In addition to the usual vector meson poles, we account for the $K\overline{K}$-continuum contribution by drawing upon a recent analytic continuation of $\mathrm{KN}$ scattering amplitudes. For the Pauli form factor all strength in the \ensuremath{\varphi} region is already given by the continuum contribution, whereas for the Dirac form factor additional strength in the \ensuremath{\varphi} region is required. The pertinent implications for the leading strangeness moments are demonstrated as well. We derive a reasonable range for the leading moments which is free of assumptions about the asymptotic behavior of the form factors. We also determine the $\ensuremath{\varphi}\mathrm{NN}$ coupling constants from the form factor fits and directly from the $K\overline{K}\ensuremath{\rightarrow}N\overline{N}$ partial waves and compare the resulting values.

Isoscalar Factors of the Permutation Group

We propose a phase convention for the Clebsch-Gordan coefficients of Sn and present tables of isoscalar factors for S3, S4, S5, and S6. The isoscalar factors have been calculated from a recursion relation due to Hamermesh and by using symmetry properties of the Clebsch-Gordan coefficients. Applications to the study of multiquark systems are discussed.

Sensitivity of tensor analyzing power in the processd+p→d+Xto the longitudinal isoscalar form factor of the Roper resonance electroexcitation

The tensor analyzing power of the process $d + p \to d + X$, for forward deuteron scattering in the momentum interval 3.7 to 9 GeV/c, is studied in the framework of $\omega$ exchange in an algebraic collective model for the electroexcitation of nucleon resonances. We point out a special sensitivity of the tensor analyzing power to the isoscalar longitudinal form factor of the Roper resonance excitation. The main argument is that the $S_{11}(1535)$, $D_{13}(1520)$ and $S_{11}(1650)$ resonances have only isovector longitudinal form factors. It is the longitudinal form factor of the Roper excitation, which plays an important role in the $t-$dependence of the tensor analyzing power. We discuss possible evidence of swelling of hadrons with increasing excitation energy.

A recursive method for the construction of irreducible representations of the orthogonal group O(n)

An algorithm for the construction of O(n) Gel’fand–Zetlin states in an invariant vector space is developed. The states are calculated recursively using a new type of coefficient of fractional parentage. These coefficients are the eigenvectors of the Gel’fand invariants. It is shown that the calculation of the Gel’fand invariants' matrix elements can be reduced to the evaluation of a single generator at each step. This algorithm provides a new approach to the calculation of the Clebsh–Gordon coefficients and isoscalar factors for the orthogonal group and can be applied to construct a basis function with well-defined orthogonal symmetry for physical systems where separation between collective motions and intrinsic motions, associated with the group O(n), is of interest.

The induced representations of Brauer algebras and the Clebsch-Gordan coefficients of

Induced representations of from with are discussed. The induction coefficients (IDCs) or the outer-product reduction coefficients of with up to a normalization factor are derived by using the linear equation method. Weyl tableaux for the corresponding Gel'fand basis of SO(n) are defined. The assimilation method for obtaining Clebsch-Gordan coefficients of SO(n) in the Gel'fand basis for no modification rule involved couplings from IDCs of Brauer algebras is proposed. Some isoscalar factors of for the resulting irrep with are tabulated.

From canonical tensor operators of SU(3) and (3) to bi-orthogonal coupling coefficients: explicit expansion

Expansion of the matrix elements of SU(3) and canonical tensor operators in terms of the bi-orthogonal coupling coefficients and overlaps of the Draayer-Akiyama construction are considered. Special bi-orthogonal extremal isoscalar factors (with subscripts as multiplicity labels and proportional to q-Racah (q-6j) coefficients or the generalized Wilson-Rahman rational bi-orthogonal functions in terms of balanced or hypergeometric series) are used as a natural basis for extremal matrix elements of the highest weight component of the canonical tensor operators of SU(3) in the generating function approach of Biedenharn, Lohe and Louck. The expansion that is obtained (triple sum), together with previously derived asymmetric seed isofactors and elementary overlaps, gives the explicit overlap coefficients and can be used to derive SU(3) and canonical tensor operators as well as new explicit normalized seed isofactors with Regge-type symmetry, specified for the minimal null space case.

Nucleon vector strangeness form factors: multi-pion continuum and the OZI rule

We estimate the 3 \pi continuum contribution to the nucleon strange quark vector current form factors, including the effect of a 3 \pi <----> \rho \pi resonance. We find the magnitude of this OZI-violating contribution to be comparable to that of typical OZI-allowed contributions. We also study the isoscalar electromagnetic form factors, and find that the presence of a \rho \pi resonance in the multi-pion continuum may generate an appreciable contribution.

Dispersion analysis of the strange vector form factors of the nucleon

We analyze the nucleon matrix element of the strange quark vector current in a nucleon-model-independent dispersive approach with input from the current world data set for the isoscalar electromagnetic form factors. The update of Jaffe's minimal three-pole Ansatz for the spectral functions yields a 40% larger (Sachs) strangeness radius ${(r}_{s}^{2}{)}_{\mathrm{Sachs}}=0.20 {\mathrm{fm}}^{2}$ and a by 20% reduced magnitude of the strangeness magnetic moment ${\ensuremath{\mu}}_{s}=\ensuremath{-}0.26$. In the pole approximation these values are shown to be upper bounds. After extending the Ansatz in order to implement the asymptotic QCD momentum dependence (which the three-pole form factors cannot reproduce), we find the magnitude of the three-pole results reduced by up to a factor of 2.5. The signs of the leading moments originate primarily from the large $\ensuremath{\varphi}$-meson couplings and are generic in the pole approximation.

Complementary group resolution of the SU(n) outer multiplicity problem. I. The Littlewood rules and a complementary U(2n−2) group structure

A complementary group to SU(n) is found that realizes all features of the Littlewood rules for Kronecker products of SU(n) representations. This is accomplished by considering a state of SU(n) to be a special Gel’fand state of the complementary group U(2n−2) with labels of the latter used to distinguish multiple occurrences of irreducible representations of SU(n) (irreps) in the SU(n)×SU(n)↓SU(n) decomposition that is obtained from the Littlewood rules. Furthermore, this realization also helps us to determine SU(n)⊃SU(n−1)×U(1) Reduced Wigner Coefficients (RWCs, frequently called Isoscalar Factors) and Clebsch–Gordan Coefficients [CGCs, or full (nonreduced) Wigner Coefficients] of SU(n), using algebraic or numeric methods, in either the canonical or a noncanonical basis. New explicit formulas for the SU(3) and SU(4) multiplicities are obtained by using this technique.

Towards the canonical tensor operators of u q(3). I. The maximal null space case

Generalizing the SU(3) canonical tensor operator concept (Biedenharn and Louck) to the quantum algebra uq(3), the Wigner–Clebsch–Gordan coefficients of uq(3) with repeating irreducible representations are considered. Extremal projectors of the quantum algebra uq(3) in terms of the ordered generator polynomials are used for evaluation of the bilinear combinations of the uq(3) canonical isoscalar factors. Explicit expressions of the uq(3) isofactors, corresponding to the maximal null space case of the uq(3) unit canonical tensor operators, and their normalization factors (denominator functions) are presented. The transposition and conjugation phase factors for the SU(3) and uq(3) canonical isofactors are correlated with phases and zeros of boundary isofactors. Invariance of the canonical isofactors (or absence of such invariance) under interchange of the tensor operator and the initial or final state parameters is correlated with the existence and invariance (or numerical degeneracy) of the usual splitting (distinctive) conditions. Some oversights of previous publications are disclosed.

Program for generating tables of SU(3) ⊃ SU(2) ⊗ U(1) coupling coefficients

A C-Language program which tabulates the isoscalar factors and Clebsch-Gordan coefficients for products of representations in SU(3) ⊃ SU(2) ⊗ U(1) is presented. These are efficiently computed using recursion relations, and the results are presented in exact precision as square roots of rational numbers. Output is in LaTeX format.

Scaling behavior in soliton models

In the framework of chiral soliton models we study the behavior of static nucleon properties under rescaling of the parameters describing the effective meson theory. In particular we investigate the question of whether the Brown-Rho scaling laws are general features of such models. When going beyond the simple Skyrme model we find that restrictive constraints need to be imposed on the mesonic parameters in order to maintain these scaling laws. Furthermore, in the case when vector mesons are included in the model it turns out that the isoscalar form factor no longer scales according to these laws. Finally we note that, in addition to the exact scaling laws of the model, one may construct approximate local scaling laws, which depend of the particular choice of Lagrangian parameters.

SU3 isoscalar factors

A summary of the properties of the Wigner–Clebsch–Gordan coefficients and iso- scalar factors for the group SU3 in the SU2⊗U1 decomposition is presented. The outer degeneracy problem is discussed in detail with a proof of a conjecture (Braunschweig’s) which has been the basis of previous work on the SU3 coupling coefficients. Recursion relations obeyed by the SU3 isoscalar factors are produced, along with an algorithm which allows numerical determination of the factors from the recursion relations. The algorithm produces isoscalar factors which share all the symmetry properties under permutation of states and conjugation which are familiar from the SU2 case. The full set of symmetry properties for the SU3 Wigner–Clebsch–Gordan coefficients and isoscalar factors are displayed.

FACTORIZATION OF THE MATRIX ELEMENTS OF THREE-ELECTRON OPERATORS USED IN CONFIGURATION-INTERACTION STUDIES OF THE ATOMICfSHELL

Single-electron excitations of the atomicfshell are usually taken into account in analyses of lanthanide and actinide spectra by including six three-electron scalar operatorstiin the Hamiltonian. Their matrix elements have been factorized for allfNby using the Wigner–Eckart theorem applied to Racah's groupsSO(7) andG2. The two component parts, namely a reduced matrix element inG2and an isoscalar factor, are tabulated. This provides a compact representation of the values of the matrix elements and also enables unusual selection rules and proportionalities to be exposed.

Tables of SU(3) Isoscalar Factors

The Clebsch-Gordan coefficients of SU(3) are useful in calculations involving baryons and mesons, as well as in calculations involving arbitrary numbers of quarks. For the latter case, one needs the coupling constants between states of nonintegral hypercharges. The existing published tables are insufficient for many such applications, and therefore this collection has been compiled. This report supplies the isoscalar factors required to reconstruct the Clebsch-Gordan coefficients for a large set of products of representations.

Relations between isoscalar charge form factors of two- and three-nucleon systems.

Theoretical predictions for the deuteron and the isoscalar trinucleon charge form factors are compared. Correlations between them are found. Linear relations hold for the position of the diffraction minimum and for the position and height of the secondary maximum. The linear relation for the diffraction minimum does not pass through the experimental data.

Young invariant decomposition of spin-interacting operators

The Young invariant decomposition of totally symmetric spin-interacting operators is examined from the viewpoint of outer-product isoscalar factors of the symmetric group. The coefficients required for the decomposition of oneand two-body operators are constructed and given explicitly. The formalism enables a separate treatment of the spatial and spin parts of the matrix elements of the spin-interacting operators.

Coulomb energy averaged over the nlN‐atomic states with a definite spin

AbstractA purely group‐theoretical approach (for which the symmetric group plays a central role), based upon the use of properties of fractional‐parentage coefficients and isoscalar factors, is developed for the derivation of the Coulomb energy averaged over the states, with a definite spin, arising from an atomic configuration nlN. © 1995 John Wiley &amp; Sons, Inc.

Two-loop calculations with vertex corrections in the Walecka model.

Two-loop corrections with scalar and vector form factors are calculated for nuclear matter in the Walecka model. The on-shell form factors are derived from vertex corrections within the framework of the model and are highly damped at large spacelike momenta. The two-loop corrections are evaluated first by using the one-loop parameters and mean fields and then by refitting the total energy/baryon to empirical nuclear matter saturation properties. The modified two-loop corrections are significantly smaller than those computed with bare vertices. Contributions from the anomalous isoscalar form factor of the nucleon are included for the first time. The effects of the implicit density dependence of the form factors, which arise from the shift in the baryon mass, are also considered. Finally, necessary extensions of these calculations are discussed.