Usual SU Research Articles

Hyperon magnetic moments and the1−−→0−++γdecays

Recent precision measurements of hyperon magnetic moments reveal that the moments of the $\ensuremath{\Lambda}$, ${\ensuremath{\Sigma}}^{+}$, and ${\ensuremath{\Xi}}^{0}$ are significantly smaller than the values predicted by the Coleman-Glashow formula or by the naive quark model. In the theoretical framework of SU(3) charge-current equal-time commutation relations and asymptotic SU(3) symmetry, the spin-nonflip sum rules imply that the usual SU(3) parametrization should apply not for the hyperon magnetic moments but for the product (anomalous magnetic moment)\ifmmode\times\else\texttimes\fi{}(hyperon mass). Thus, the gross features of the observed hyperon magnetic moments are explained in terms of these derived mass-scale factors. The consistency of the spin-nonflip and -flip sum rules also implies that the remaining discrepancies should be attributed to the SU(3) mixings between the hyperons and their higher-lying excited ${\mathrm{\textonehalf{}}}^{+}$ states. In the same theoretical framework, the status of the similar magnetic-moment interactions, ${1}^{\ensuremath{-}\ensuremath{-}}$\ensuremath{\rightarrow}${0}^{\ensuremath{-}+}$+$\ensuremath{\gamma}$, is also reviewed, by studying the consistency of the asymptotic SU(3) sum rules. It is concluded that the usual consideration of the related SU(3) mixing, especially the ones involving the $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$, is insufficient.

The Baker–Campbell–Haussdorff formula for the SU(2) supergroups

In the theory of supersymmetric SU(2) Yang–Mills fields described on the 8th dimensional superspace, the local gauge transformations constitute a group whose Lie algebra has its coefficients in the Weyl-spinorial-Grassmann algebra. The coefficients can be taken to be chiral elements, real or complex. This last case is the most general one and contains all others of interest for developing physical models. We present here a Baker–Campbell–Haussdorff formula for the complex SU(2) supergroup. The formula gives the finite form for each element of the supergroup in terms of the local fields entering into the infinitesimal (complex) superscalar generator. Thereafter we exhibit, as a particular case, a BCH-expression for the real SU(2) supergroup which completes previous results already reported for the chiral SU(2) supergroups. Moreover, it is shown that, despite the appearance of apparently nonperiodic polynomial terms, the global variety of the real SU(2) supergroup is contained in a variety having the same compact bosonic width as the usual SU(2) pure bosonic variety.

Dynamical symmetry breaking in the Lewis–Riesenfeld oscillator

Dynamical realization is given to the generator responsible for breaking the usual SU(3) invariance symmetry in the case of the Lewis–Riesenfeld time-dependent oscillator: H (t) = 1/2 𝒥3i[pipi+w2(t) qiqi]. This is achieved by finding a simple expression for the time-dependent Hamiltonian in the interaction picture: H=Ho+gH1. The breaking interaction is shown to transform as (6⊕6̄) of SU(3). Time-dependent dilations relate the broken Hamiltonian to the Lewis–Riesenfeld form. This scaling generates the algebra of the Weyl group and its role in oscillator noninvariance symmetries is considered.

Phenomenological SU(6) breaking of baryon wave functions and the chromodynamic spin-spin force

A contradiction is found between two successful models of SU(6) breaking. A quark-model mixing scheme (56,${0}^{+}$) + (70,${0}^{+}$) for the baryon octet has been devised to explain the ratio $\frac{{F}_{2}^{\mathrm{en}}(x)}{{F}_{2}^{\mathrm{ep}}(x)}$ in the valence-quark region and explains naturally other departures from the usual SU(6) predictions. On the other hand, the gluon-exchange model of SU(6) breaking accounts satisfactorily for the hadron spectrum splittings. The spin-spin contribution from this chromodynamic force is indeed shown to generate a (56,${0}^{+}$) + (70,${0}^{+}$) mixing of the octet. However, it yields a wrong sign for the mixing angle, thus pointing to a contradiction between spin-spin forces of one-gluon-exchange type and the deep-inelastic structure functions in the valence-quark region. Other spin-spin potentials, giving the right sign for the mixing angle, are shown to be also in difficulty, because of the hyperfine structure of excited levels. Finally, a careful discussion is made of the subtle $\ensuremath{\Sigma}$-$\ensuremath{\Lambda}$ effect in both approaches.

Tensorial mass operators for charmed baryons

Presents matrix elements of tensorial mass operators for baryons, between state vectors of the ground supermultiplet of the symmetric quark model. The approximate hadronic symmetry group is SU(8), obtained from the usual SU(6) by the inclusion of charm. These matrix elements lead directly to mass formulae in terms of which the mass spectrum of charmed baryons can be parametrized.

Generalized invariants for the time-dependent harmonic oscillator

A generalized class of invariants, I (t), for the three-dimensional, time-dependent harmonic oscillator is presented in both classical and quantum mechanics. For convenience a simple notation for types of harmonic oscillator is introduced. Two interpretations, one in terms of angular momentum and the other employing a canonical transformation, are offered for I (t). An invariant symmetric tensor, Imn(t), is constructed and shown to reduce to Fradkin’s invariant tensor for time-independent systems. The usual SU(3) (compact) or SU(2,1) (noncompact) is shown to be a noninvariance group for the time-dependent oscillator with S{U(2) ⊗U(1) } as the invariance subgroup. Extensions to anisotropic systems and the singular quadratic perturbation problem are discussed.

Restoring universality

In models of weak and electromagnetic interactions with six or more quarks, universality and, in some cases, the V-A structure of leptonic and semi-leptonic interactions have to be put in by hand. We show how both properties remerge naturally if replacing the usual SU(2) weak isospin group by a large SU(4) group, which also lends itself quite naturally to grand unifications (including strong interactions). The model needs eight quarks and two charged heavy leptons, and features, nesides universality and V-A, non-equality of neutral current scattering of ν's on leptons and on hadrons, and very reasonable relations for (bare) fermion masses.

Gauge model of weak and electromagnetic interactions with charm violating transitions

Extending the usual SU(2) ⊗ U(1) group of weak and electromagnetic interactions into a larger SU(2) ⊗ U(1) ⊗ U(1) group, it is shown that the extra freedom provided by the new generator allows a new neutral charm violating transition. Indeed apart from the two charged currents and the electromagnetic one there exist two neutral weak currents coupled to two distinct intermediate neutral bosons.

Phenomenological analysis of a fixed-sphere bag model

We perform a phenomenological analysis of baryon matrix elements in a fixed-sphere MIT bag model. The model consists of massive, noninteracting quarks which carry the usual SU(3) quantum numbers as well as an SU(3) of color. Two-quark matrix elements investigated include those of the $\ensuremath{\sigma}$ operator and axial-vector and vector currents and the magnetic moment, charge radius, and dipole moments of the electromagnetic current. Among our results are the masses 44 MeV for nonstrange quarks and 298 MeV for strange quarks. The model is successful in predicting static properties of the lowest-mass baryons, but has a problem in describing the radiative decay of the nucleon resonance, $N_{}^{*}{}_{13}{}^{}(1520)$. Four-quark matrix elements parity-violating nonleptonic decay amplitudes of the ${\frac{1}{2}}^{+}$ baryons, are investigated and are found to be poorly described by the model.

Symmetry principle and constraints on deep inelastic structure functions

It is shown that isospin invariance for hadron states and the usual SU(2) transformation properties of electromagnetic and weak currents (CVC hypothesis) impose strong restrictions on the weak structure functions in terms of the electromagnetic ones. Among others, the inequalities F v p+F v p F λ p +F λ n ⩽ 4, − ∞ ⩽ 1 2 (F v p −F v p )−3F λ p ⩽ 3 2 F λ n , are obtained. Furthermore, a quite restrictive scheme is obtained if one adds the following assumptions: 1. (i) Electromagnetic and weak currents are members of a SU(3) octet in the usual sense. 2. (ii) Absence of t-channel exotics. 3. (iii) Currents transform as nonets under U(3). We show that this scheme gives results which are very close to the SU(2) quark-parton light-cone models. In particular, we obtain 1 4 ⩽ F λ p F λ n ⩽4, F v p +F v p F λ p +F λ n ⩽ 18 5 , −6.11 F λ n ≈ (− 12+ 9 7 21 )F λ n ⩽ 1 2 (F v p −F v p )−3F v p ⩽ 3 2 F λ n The consequences of adding the SU(3) symmetry of hadron states are also investigated. All the statements which involve weak structure functions require as an additional assumption equality of vector and axial-vector contributions to the scattering amplitudes.

Analysis ofK−p→Λπ+π−in the Region of theΛ(1520)

The rate for the decay $\ensuremath{\Lambda}(1520)\ensuremath{-}\ensuremath{\Sigma}(1385)\ensuremath{\pi}$ has been measured in a study of the reaction sequence ${K}^{\ensuremath{-}}p\ensuremath{\rightarrow}\ensuremath{\Lambda}(1520)\ensuremath{\rightarrow}\ensuremath{\Sigma}(1385)\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\Lambda}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$. A sample of 9200 events of the type ${K}^{\ensuremath{-}}p\ensuremath{\rightarrow}\ensuremath{\Lambda}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ has been obtained in the Berkeley 25-in. hydrogen bubble chamber. The incident momenta range from 300 to 470 MeV/c. An energy-independent partial-wave analysis was performed using an isobar model. The model included a coherent mixture of six partial waves to describe all mass distributions, angular distributions, and the polarization of the $\ensuremath{\Lambda}$. The $\ensuremath{\Sigma}(1385)\ensuremath{\pi}$ decay mode was found to dominate the $\ensuremath{\Lambda}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay of the $\ensuremath{\Lambda}(1520)$, in disagreement with the results of a production reaction study of the same decay. The width for the decay was measured to be 1.40 \ifmmode\pm\else\textpm\fi{} 0.26 MeV. Mixing between the $\ensuremath{\Lambda}(1520)$ and the $\ensuremath{\Lambda}(1690)$ has been used in the past to describe the ${J}^{P}={\frac{3}{2}}^{\ensuremath{-}}$ baryons as SU(3) singlet and octet. Combining our rate for $\ensuremath{\Lambda}(1520)$ with a recently measured rate for $\ensuremath{\Lambda}(1690)\ensuremath{\rightarrow}\ensuremath{\Sigma}(1385)\ensuremath{\pi}$, we calculate the mixing angle $|\ensuremath{\theta}|=63\ifmmode^\circ\else\textdegree\fi{} \mathrm{to} 83\ifmmode^\circ\else\textdegree\fi{}$. This is in strong disagreement with the mixing angle derived from the two-body $D$-wave decays of the singlet and octet of -25 \ifmmode\pm\else\textpm\fi{} 6\ifmmode^\circ\else\textdegree\fi{}. Thus, some modification of the usual SU(3) description of these states needs to be made.

Conserved Tensor Currents, Isovector Photons, and the Algebraic Realization of Chiral Symmetry. I. The Algebra of Couplings

Weinberg's program for the algebraic realization of chiral symmetry is considered in an enlarged chiral model obtained by adjoining to the usual SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) algebra of currents, isovector and isoscalar tensor currents with equal-time commutation relations given by the quark model. The isovector tensor current is taken to be conserved, thereby permitting the introduction of "isovector photons" into the ordinary model. It is necessary to analyze asymptotically inelastic ($\ensuremath{\pi}+\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\gamma}+\ensuremath{\beta}$, $\ensuremath{\pi}+\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\gamma}+\ensuremath{\beta}$, $\ensuremath{\pi}+\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\gamma}+\ensuremath{\gamma}+\ensuremath{\beta}$) processes as well as elastic ($\ensuremath{\pi}+\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\beta}$, $\ensuremath{\gamma}+\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\gamma}+\ensuremath{\beta}$) processes to close the derived algebra of couplings, which is shown to be that of SU(4).

Some consequences of a modified quark model involving Schwinger terms

It is shown that the modified quark-model algebra of Mendes and Ne'eman reduces, in the SU(2) × SU(2) limit, to the usual SU(3) × SU(3) quark-model algebra.

Scaling properties in inelastic electron scattering with a fixed final multiplicity

The scaling property of inelastic electron scattering with a fixed final hadron multiplicity is analysed. The relation between the structure functions and the average multiplicity at infinite energy is discussed. A possibility exists that, if one excludes all soft mesons, the remaining hadron multiplicity at infinite energy may stay finite and not too high, in which case some insight may be gained as to why the observed structure functions appear to scale at a relatively low energy. By using the tree diagrams of the usual SU 2 phenomenological Lagrangian, explicit calculations of the structure functions W 1 and νW 2 are made for all two-body final channels, such as pπ 0, np +, etc.; it is found that, in this simple model, the scaling property is satisfied for these particular channels. Possible experimental consequences are discussed.

Hard-pion calculation without current algebra

Following the suggestions of earlier authors we explore the hypothesis that apart from fixing the scale of the weak axial-vector current, the main results of current algebra are largely implied in the concept of meson-pole dominance, which includes the famous PCAC principle. For definiteness we re-examine here the hard-pion analysis of the A 1 πϱ vertex by Schnitzer and Weinberg, not assuming anything about the commutator of two axial-vector currents but retaining all their other assumptions, specifically those about meson-pole dominance. We find, as expected, that these latter assumptions are powerful enough to determine the system up to the scale of the axial-vector current. As by-product of our analysis, we find a highly interesting corollary on Weinberg sum rules which allows us to exclude an otherwise permissible non-compact alternative to the usual SU(2) x SU(2) current algebra on mere covariance and positivity requirements.

Generalized chiral symmetry groups and the classification of hadrons

The existence of Lagrangian models partially invariant under groups that generalize chiral symmetry is shown. The implications of these models with regard to be classification of hadrons is discussed. The currents defined from the Lagrangian satisfy a current algebra which includes tensor currents and provides a generalization of the usual SU(3) ⊗ SU(3) current algebra with the extension of the PCAC condition to the more general currents.