Subgroups Of SU Research Articles

Generators of the exceptional groupE8as bilinear quark and lepton fields

The quarks and leptons are assigned to the adjoint representation of the exceptional group ${\mathrm{E}}_{8}$ using decompositions under the subgroups SU(9) and [${\mathrm{SU}(3)]}^{4}$. Generators are constructed as linear combinations of bilinear quark and lepton fields. Closure of the algebra is used to determine the unknown coefficients of the linear combinations. It is noted that the Majorana spinors ${\ensuremath{\chi}}_{\ensuremath{\nu}}^{\ensuremath{\mu}}$ introduced to represent the adjoint representations of SU(9) and [${\mathrm{SU}(3)]}^{4}$ subgroups cannot be taken traceless. The trace ${\ensuremath{\chi}}_{\ensuremath{\nu}}^{\ensuremath{\mu}}$ should couple to the quark and lepton fields in order to close the algebra. The constraints on the bilinear fields which are of physical importance are introduced to obtain the right number of fermionic states in the adjoint representation. An attractive possibility of having an octet of strictly massless Majorana quarks and at least three massless Majorana leptons as a consequence of pure algebraic constraints is discussed. The exceptional subgroups ${\mathrm{E}}_{7}$ and ${\mathrm{E}}_{6}$ are identified and the explicit commutation relations are obtained. Using one assignment of ${\mathrm{E}}_{6}$ the role of color-singlet lepton-lepton and quark-antiquark currents is pointed out.

Nonscalar extension of shift operator techniques for SU (3) in an O(3) basis. I. Theory

A set of relations is set up which connect quadratic products of the shift operators Okl (k = 0,1,2), which are nonscalar with respect to the O(3) subgroup of SU (3). The usefulness of these relations is illustrated by the calculation of the eigenvalues of the scalar shift operator O0l for various irreducible representations ( p,q) of SU (3).

On reduction of the four-dimensional harmonic oscillator

This paper deals with reduction of the four-dimensional harmonic oscillator by use of a one-parameter subgroup U(1) of the symmetry group SU(4), U(1) being the symmetry subgroup generated by an ’’angular momentum.’’ The angular momentum determines in the energy surface S7 an ’’energy-momentum’’ manifold S3×S3 on which a subgroup SU(2)×SU(2) of SU(4) acts. The reduction process yields a manifold S3×S2 = S3×S3/U(1) on which SO(4) acts effectively.

On a ‘‘conformal’’ Kepler problem and its reduction

A ’’conformal’’ Kepler problem is defined in order to associate the Kepler problem with the harmonic oscillator. The four-dimensional conformal Kepler problem which shares an energy surface with the four-dimensional harmonic oscillator reduces to the ordinary three-dimensional Kepler problem. By use of the reduction the symmetry group SO(4) of the Kepler problem is brought out from a symmetry subgroup SU(2)×SU(2) of the conformal Kepler problem; the subgroup is the same as a subgroup of the symmetry group SU(4) of the harmonic oscillator.

Phase structure of non-Abelian lattice gauge theories

The phase structure of four-dimensional lattice gauge theories based on finite non-Abelian groups is studied by Monte Carlo computations. All models examined exhibit a two-phase structure with a first-order phase transition. In three systems where the gauge group is a discrete subgroup of SU(2) the critical temperature moves toward zero as the order of the group increases and the high-temperature phase has confining properties.

A unified theory of weak interactions

A gauge model for the weak interactions of the leptons (v e, e, μ, νμ) and the quarks (q p, qn,qλ,q p′) is presented in which deviations from universality, such as the Cabibbo suppression, are explicitly and spontaneously generated. The gauge group is, to begin with SU(4). There are three quartets of Higgs scalars with suitable vacuum expectation values, sufficient and necessary to give masses to all gauge bosons. It turns out that this gauge group is too ‘large’ and fails to account for many observed symmetries of weak interactions, especially electron-muon symmetry. This symmetry corresponds to a discrete transformationR which is an element of SU(4). To accommodate it, the gauge group is restricted to the subgroup of SU(4) which commutes withR. There are now 7 gauge bosons, 4 charged and 3 neutral. One pair of charged bosons is necessarily heavier than the other pair (denotedW ±) and two neutrals are necessarily heavier than the third (W 0). The electron and the muon become massive while the neutrinos and the quark fields remain massless.

Wigner coefficients for SU(6) ⊇ SU(3) ⊗ SU(2)

A technique for evaluating the Wigner coeficients for SU(6) with the physically interesting subgroups SU(3) ⊗SU(2) is outlined. Coefficients appropriate for coupling a single quark to a many quark wavefunction are tabulated for functions of nine or fewer quarks.

Mass splitting in SU(4)

A simple and rather general way to break SU(4) is to assume that the mass operator consists of a term which transforms as a scalar in the subgroup SU(3) c (which operates on u, d, and s quarks) and a term which transforms as a scalar in SU(3) s (which operates on u, d, and c). It is shown that this leads to a number of parallelogram laws between the masses. The generalized Okubo assumption falls within this framework and leads directly to Gell-Mann-Okubo relations between multiplets in the two subgroups. The specific SU(4) assumption of Okubo and others is a still further restriction. This group theoretic approach is compared with the results of the quark model.

The unitary representations of [formula omitted]L(4, R)

In the paper, nine series of unitary representations of the group G ̶ L(4, R ) are found. Moreover, there are obtained matrix elements of the generators of G ̶ L(4, R ) in an orthonormal basis. The elements of the basis are classified over the subgroup SU(2)⊗SU(2).

CP Violation and Off-Diagonal Neutral Currents

We discuss the CP violation based on the SU(2) ×U(1) ×SU_F(2) gauge model, where the new gauge bosons S_µ^i are introduced in association with the extra subgroup SU_F(2). The effective coupling constant G_S for the interaction mediated by the S_µ^i is found to be G_S/G_F > 5 ×10^-6. In this model the gauge bosons S_µ^i are coupled to off-diagonal neutral currents. The ratio R = [ Γ(K^± →π^± e µ) + Γ(K^± →π^± µe ] / Γ(K^- →π^0 µν_µ) is predicted to be R > 2 ×10^-9.

Classical solutions for SU(4) gauge fields: Interacting monopoles

The notion of local idempotents is introduced, and their relation to a class of solutions for $\mathrm{SU}(n)$ gauge fields is pointed out. This class includes the known monopole-type solutions for SU(2) and SU(3) gauge fields---coupled to scalars and spinors. Next, these ideas are used to study solutions for SU(4) gauge fields. The following classes of solutions are studied. Corresponding to two commuting SU(2) subgroups of SU(4) one has two monopole-type contributions from the space components, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{W}}(x)$, of the gauge field. They are directly coupled among themselves, the remaining SU(4) components providing a tensor-type interaction. They are also coupled to a scalar field $\ensuremath{\Phi}(x)$ and the time component ${W}_{0}(x)$. Two different possibilities for $\ensuremath{\Phi}(x)$ and ${W}_{0}(x)$ are considered in detail. An exact solution is given for a point monopole interacting with a particular system of finite mass. Simple variational calculations are used to obtain finite mass for the total system. Brief remarks are added concerning other possibilities; e.g., how pseudoparticles can be studied from our point of view.

The necessity for a Melosh-type transformation

The identification of null-plane U(6) W currents with U(6) W, consitituents is shown to exclude the existence of the 56(L z = 0) baryons. The subgroups SU(2) W, currents and U(3) × U(3) currents also lead to similar theoretical contradictions. These results, obtained by investigating the implications of these symmetries on the NNγ, ΔΔγ and ΔNγ vertices, lend additional support to the need of a Melosh-type transformation relating the constituent and current charges.

Composite magnetic monopoles for SU( N) spontaneously broken to U(1)

The point monopole solution found by Corrigan, Olive, Fairlie and Nuyts for SU(3) broken down to U(2) by an octet Higgs field is shown to survive when the symmetry is further broken to U(1), and therefore is a magnetic monopole in the traditional sense. The demonstration exploits a singular transformation from the Abelian gauge with a Dirac string, to a non-singular gauge in which the vector field has a manifest rotational symmetry. The procedure is generalized to SU( N) broken to U(1), always yielding the smallest strength monopole consistent with the Dirac quantization condition, and in some cases higher-strength monopoles as well. The manifest symmetry of the vector field corresponds to an angular momentum J 0 , different in general from the physical charge-pole angular momentum J . The latter generates a symmetry of the whole system of Higgs and vector fields. Even that symmetry fails for the interior of a finite-energy solution unless J and J 0 coincide, which happens only for the minimal monopole coupled to a charge in an SU(2) subgroup of SU( N). If SU( N) is not broken all the way to U(1), there can be solutions with exact J 0 symmetry whose possible significance is discussed. A theorem of Georgi and Glashow is used to show that SU(3) → U(1) could occur in a natural way if there were two Higgs fields.

On the structure of the irreducible representation basis for the exceptional group G2

Using the method of projection operators we have constructed the basis of the irreducible representation D ▪ of the exceptional Lie group G 2 corresponding to the reduction of this group to the subgroup SU 3. The basis is nonorthogonal but convenient for calculations. The matrices of the generators of the group G 2 in this basis have been found. The problem of additional quantum number ω required for the complete labelling of the basis vectors is considered. For this purpose we introduce the operator ω which is cubic with respect to the generators of the group G 2 and scalar with respect to the subgroup SU 3. The matrix of this operator has been calculated in the nonorthogonal basis. This matrix has a nondegenerate spectrum of eigenvalues ω which can be used as the missing quantum number.

Color-Symmetry Breaking and the Baryon Spectrum

In the colored-quark model, if the three vector gluons that correspond to an SU(2) subgroup of SU(3) are heavier than the other gluons, a quark-diquark structure for baryons results. Furthermore, the predicted baryon SU(6) representations are the 56 for even parity and the 70 for odd parity, in agreement with recent experimental indications.

Quark structure and octonions

The octonion (Cayley) algebra is studied in a split basis by means of a formalism that brings outs its quark structure. The groups SO(8), SO(7), and G2 are represented by octonions as well as by 8 × 8 matrices and the principle of triality is studied in this formalism. Reduction is made through the physically important subgroups SU(3) and SU(2) ⊗ SU(2) of G2, the automorphism group of octonions.

Current quarks, constituent quarks, and symmetries of resonance decays

The transformation between “current” quarks and “constituent” quarks recently suggested by Melosh is examined with respect to its predictions for pionic decays of resonances. It implies the use of SU (6) W for classifying particle states but not for describing decay processes. Instead, pion emission proceeds via ΔL z = 0, ± 1, where L is the internal (“quark”) orbital angular momentum. This decay symmetry is called SU (6) W ( ΔL Z = 0, ± 1) It is proven equivalent for any decay A → B + π (where A, B are arbitrary q q or qqq hadrons) to the 3P 0 quark-pair creation model for such decays, as formulated by Micu, Colglazier, Petersen and Rosner. The roles of final orbital angular momenta l and of SU (3) × SU (3) subgroups of SU (6) W are also discussed, and some new predictions are made for decays of meson resonances below 1700 MeV.

Unified Weak-Electromagnetic Gauge Schemes Based on the Three-Dimensional Unitary Group

SU(3)\ifmmode\times\else\texttimes\fi{}U(1) and chiral SU(3)\ifmmode\times\else\texttimes\fi{}SU(3)\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}U(1) models for the weak and electromagnetic interactions of leptons and hadrons are proposed. The usual quark model for hadrons as well as the usual Cabibbo rotation scheme is assumed. Symmetry-breaking effects due to inequalities in the masses of the gauge bosons are considered. By making a simple choice of the gauge boson masses based on the $U$-spin subgroup of SU(3), the undesired $|\ensuremath{\Delta}S|=2$ weak nonleptonic transitions can be completely eliminated and the undesired $|\ensuremath{\Delta}S|=1$ neutral current semileptonic transitions drastically suppressed. These models can reproduce the ordinary phenomenological weak interaction but are sufficiently flexible to give the possibility of interesting new effects for neutral leptonic and semileptonic processes, as well as to allow the possibility of an over-all arbitrariness in normalization for the effective nonleptonic interaction. Another consequence of the present scheme is the existence of very heavy leptons. A brief discussion of including the strong interactions in a grand unification is also presented.

Algebra of strengths generated by bilocal currents acting on “constituent” states

Within the context of a free field model, a unitary transformation is introduced which carries the entire [U(6) × U(6)] βW currents of the current system, with subgroup SU(6) W currents, into a newly defined [U(6) × U(6)] βW strong, with diagonal subgroup SU(6) W strong, which commutes with the boost in the z-direction. Analysis of strong current-hadron scattering amplitudes in the deep Regge region permits the extraction of the hadron couplings to vector and tensor Regge exchanges as the matrix elements of “strong” charges.

Decomposition of the Principal Series of Unitary Irreducible Representations of SU(2, 2) Restricted to the Subgroup SU(1,1)⊗SU(2)

An explicit form is given for the unitary irreducible representations of the principal nondegenerate series of the group SU(2, 2). These representations are then restricted to the subgroup SU(1,1)⊗SU(2) and are found to be equivalent to the regular representation of SU(1,1)⊗SU(2).