Quark Triplets Research Articles

Short distance analysis of weak interactions

We systematically formulate the short distance analysis of weak interactions, and show that in an asymptotically free theory one can calculate the dependence of any weak amplitude on the W boson and heavy quark masses. Our main purpose is to settle some conceptual questions, but we also derive some new quantitative results. For example, we show that in the SU(3) gauge theory with four quark triplets, the two W boson contribution to K L → μ μ will differ from the free quark theory estimate by a factor 25(1 − ( α(M W / α(m p′ ) 1 25 )( α(M W )/ α(m p′ )) 24 25 . We also show that the amplitudes for K + → π +e +e − will differ from the free quark theory estimate by a factor of 1.66( α(M W )/ α(m p′ )) 2 9 − 1.88( α(M W )/ α(m p′ ) −4 9 .

ηdecay into three pions and the on-mass-shell current algebra

We use the on-mass-shell current algebra based on the $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ $\ensuremath{\sigma}$ model for calculating the $G$-parity-violating decay $\ensuremath{\eta}\ensuremath{\rightarrow}3\ensuremath{\pi}$. The model we use consists of the triplet of quarks coupled to the SU(3) scalar and pseudoscalar fields, $\ensuremath{\sigma}$ and $\ensuremath{\varphi}$, and the chiral-symmetry-breaking operators ${u}_{8}$ and ${u}_{3}$ (tadpole terms). Owing to the terms ${\ensuremath{\varphi}}_{i}{\ensuremath{\varphi}}_{j}$ with $i \ensuremath{\ne} j = 0,3,\mathrm{and} 8$ in the Lagrangian, the axial-vector current sources and fields associated with the ${\ensuremath{\pi}}^{0}$, $\ensuremath{\eta}$, and ${\ensuremath{\eta}}^{\ensuremath{'}}$ particles are formed by mixtures of terms with opposite $G$ parity; the mixing coefficient $\ensuremath{\epsilon} = \frac{{u}_{3}}{(4{u}_{8})}$. Using these current operators, and having all particles on the mass shell, we evaluate the decay amplitude by the reduction formalism from the $〈3\ensuremath{\pi}|\ensuremath{\eta}〉$ matrix element, making use of pole dominance. The result is proportional to the coefficient $r = 4\ensuremath{\epsilon}$, which is determined by the tadpole parts of the masses and decay constants of the mesons through a set of relations which are derived from our Lagrangian. However, since these masses are not known satisfactorily, we first choose $r$ so as to obtain the observed $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay rate (which is proportional to ${r}^{2}$), and then we verify that such an $r$ value is consistent with the above set of relations. We find that $r = 2.12\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, and in turn the tadpole part of the ${K}^{0}\ensuremath{-}{K}^{+}$ mass difference ${{m}_{{k}^{0}}}^{2}\ensuremath{-}{{m}_{{k}^{+}}}^{2} = 0.0005$ ${\mathrm{GeV}}^{2}$. We also find the branching ratio and the ratio of the rates of the $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ decays in agreement with experiment. Comparing the decay amplitude obtained here with the one derived by the tree-graph method based on the $\mathrm{U}(3)\ensuremath{\bigotimes}\mathrm{U}(3)$ symmetry scheme, we realize the important role of the mixing of the ${\ensuremath{\varphi}}_{0}$, ${\ensuremath{\varphi}}_{8}$ and ${\ensuremath{\varphi}}_{3}$ field components, and the advantage of using the on-mass-shell current algebra for this process.

Seven-quark model for the spectrum of hadrons

We investigate a seven-quark model for the spectrum of hadrons. They consist of the usual triplet of quarks and an antitriplet and a singlet of heavy quarks. The $\ensuremath{\psi}$ particles are identified as the neutral members of the set of heavy-quark-antiquark bound states. One finds that $R=\frac{16}{3}$ in ${e}^{+}{e}^{\ensuremath{-}}$ scattering and $\ensuremath{\Gamma}(\ensuremath{\psi}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}):\ensuremath{\Gamma}({\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}):\ensuremath{\Gamma}({\ensuremath{\psi}}^{\ensuremath{'}\ensuremath{'}}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}):\ensuremath{\Gamma}({z}^{0}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}})=2:1:3:\frac{2}{3}$. The model is arranged to forbid strangeness-changing neutral weak currents by introducing a doublet of right-handed quarks. It is further shown that by enlarging the model to eight quarks one can live with only left-handed currents.

Quark theory with internal coordinates

The free-particle Dirac wave function $\ensuremath{\psi}(x)$ is generalized to $\ensuremath{\psi}(x){\ensuremath{\xi}}^{a}(z)$. Here, $z$ denotes a set of three complex coordinates, called internal coordinates, in an abstract complex three-dimensional space, called internal space, $a$ runs from 1 to 3, and ${\ensuremath{\xi}}^{a}(z)$ is assumed to contain a representation of the state of a quark triplet. The mass in the free-particle Dirac equation is replaced by a second-order operator $\ensuremath{-}{\ensuremath{\partial}}_{a}^{b}$ operating on ${\ensuremath{\xi}}^{a}(z)$. The so-modified Dirac equation is assumed to include a description of a free-quark triplet. Subsequently, symmetry-preserving interactions in space-time as well as in the internal space between two quark triplets are introduced. Two S${\mathrm{U}}_{3}$-symmetry-breaking interactions, one transforming like the eighth component of an S${\mathrm{U}}_{3}$ octet vector and the other like the S${\mathrm{U}}_{3}$ charge operator, are also introduced. A similarly generalized Bethe-Salpeter equation in the ladder approximation is obtained. This equation is treated in greater detail in the following paper in which the Gell-Mann-Okubo formula for pseudoscalar mesons is derived with the coefficients determined by given relations. Then, spherical coordinates and corresponding spherical harmonics in the internal space are introduced. Finally, the equation for a one-quark system is briefly treated.

Quark theory with internal coordinates

The free-particle Dirac wave functionψ(x) has been generalized toψ(x)Ξa(z). Here,z denotes a set of three complex coordinates, called internal coordinates, in an abstract complex three-dimensional space, called internal space;a runs from 1 to 3; and Ξa(z) is assumed to contain a representation of the state of a quark triplet. The mass in the free-particle Dirac equation is replaced by a second-order operator ∂ab operating on Ξa(z). The Dirac equation so modified is assumed to include a description of a free quark triplet. Subsequently, symmetry-preserving interactions between two quark triplets are introduced. TwoSU3 symmetry-breaking interactions, one transforming like the eighth component of anSU3 octet vector and the other like theSU3 charge operator, are also introduced. A similarly generalized Bethe-Salpeter equation in the ladder approximation was obtained. This equation has been treated in greater detail in an accompanying paper in which the Gell-Mann-Okubo formula for psuedoscalar mesons was derived with the coefficients determined by given relations. Then, spherical coordinates and corresponding spherical harmonics in the internal space are introduced. Finally, the equation for a one-quark system is briefly treated.

Leptonic decays of heavy pseudoscalar mesons

The authors identify the new particles psi (3100) and psi ' (3700) as the /b I/=0 members of a heavy-meson nonet and the wide bump at 4100 MeV, denoted by psi as the neutral /b I/=1 member of the corresponding octet. They propose that these hadronic states are built from an SU/sub 3/ triplet (p', n', lambda ') of heavy quarks which includes an isodoublet (p', n') and an isosinglet lambda ', just like ordinary hadrons are supposed to be made from the (p, n, lambda ) quarks. The electric charges of p' and n' are taken to be either /sup 2 ///sub 3/ and -/sup 1///sub 3/ or 1 and 0, respectively. Other quantum numbers are left unspecified in the present analysis.

The quark aspect of partons

It is shown in this article that the baryon nonet as well as the meson nonet can be represented as sets of three different triplets. The values of Q, Y, and 2T3 are the same as the corresponding values of the three integral quark triplets of Nambu. It is proposed to describe the Nambu quarks not in terms of the usualSU(3) ⊗SU(3)′, but rather in terms ofSU (3) ⊗SU (3) ′ ⊗SU″ (3) Like the baryons, such quarks will then be composites, made up of other quarks; the latter will also be made up of other quarks, and in this way we obtain a whole hierarchy of quarks. It is proposed to identify the totality of all of these states as partons.

Quark bags and local field theory. II. Confinement of Fermi and vector fields

We extend earlier work on obtaining the model of Chodos, Jaffe, Johnson, Thorn, and Weisskopf as a limit of a local field theory. As before, the discussion is on a classical level. We show how to confine Fermi and vector fields into the bag solutions found previously. In an appropriate limit we obtain exactly the Chodos et al. model for confinement of all fields except non-Abelian gauge fields. For the latter fields we argue that it is impossible from a local theory to obtain the same boundary conditions used in the Chodos et al. model. The boundary conditions we find do not have the virtue of completely eliminating color nonsinglet states from the theory. We propose an alternative scheme based on a single triplet of quarks and an Abelian gauge field. The baryons are bound states of three quarks and one additional scalar constituent.

Fourth triplet quarks for ψ

A phenomenological model is presented on the basis of experimental data on the ψ particles. A fourth triplet of quarks with no color are added to the Han-Nambu type quarks to form three 1 −states coupled to the one photon state.

Quark triplets in self-interaction and the light-cone structure of current commutators

A field theory of basic quark fields with current-current self-interaction is analyzed with respect to the leading singularity structure of current commutators on the light cone. Canonical leading singularities are found to be compatible with the four-fermion interaction whereas the structure of the bilocal operators differs essentially from the free field case.

Properties of hadron states containing a condensed phase of quark-antiquark excitations

We analyze how one-particle states can arise in a field theory of (three) quark triplets with current-current interactions which does not produce asymptotic quark states. A condensed phase of quark-antiquark pairs resembling the corresponding phase in a superconductor is responsible for the lack of stable states in the sectors with triality different from zero, provided that stable one-particle states indeed exist in the triality zero sectors, corresponding to stable configurations of valence quarks in the presence ofthe condensed phase. The precise dynamics of valence quarks is beyond the scope of the present model which is meant to illustrate a mechanism which prevents the basic quanta of the underlying fields to become asymptotically isolated and still eventually generates stable states in the hadronic sectors without inconsistencies.

Weak and electromagnetic interaction symmetry

An analogy between a triplet of quarks and a triplet of leptons is discussed in the framework of a model of weak and electromagnetic interactions possessing local SU(2) L × U(1) symmetry. After the symmetry is broken spontaneously by use of the Higgs-Kibble mechanism, two of the quark masses are non-zero. The relation of the quark masses to the chiral SU(3) × SU(3) model of strong interactions as well as the Cabibbo angle is discussed.

Leptonic and hadronic symmetries

Two models for a gauge theory of the weak and electromagnetic interactions of leptons and hadrons are considered. The first model is based on the group [SU(3)] 4 = SU(3) × SU(3) × SU(3) × SU(3), the fundamental representation of which is spanned by left and right-handed lepton and quark triplets. The lepton triplet involves the ν e and ν μ combined to a 4 component field, the e − and the μ +. A spontaneous symmetry breakdown is caused by the appearance of tadpoles. It is proved that the tadpoles responsible for the lepton and quark masses, by themselves, do not break the symmetry in a manner compatible with the observed universality properties of weak interactions. Many additional scalar fields are needed and they are classified into specific [SU(3)] 4 multiplets. The model can account for the suppression of | ΔS| = 1 neutral current induced semileptonic decays ( K → πν ν, K° → μ +μ − , etc…) A quartet model based on the symmetry U(4) × U(4) is discussed. The model is simpler, but has to confront the absence of “charmed” hadrons in the low ( m ⩽ 2−3 GeV) mass region.

SU(3) × SU(3) symmetry breaking in a simple model

A field-theoretical model, due to Lévy, is studied. It contains a triplet of quarks and a pseudoscalar and a scalar meson nonet. The original SU(3) × SU(3) symmetry is broken by terms linear in the scalar meson fields. A renormalization and regularization procedure is defined in order to remove the ultra-violet divergences. The possibility of a spontaneous breakdown of the symmetry is described and the Goldstone theorem is verified in the one-loop approximation. Moreover, the Ademollo-Gatto theorem is reproduced. The axial vector coupling constants differ in first order of the SU(3) symmetry breaking.

Mass Ratio of Quarks

The concept of "convective" currents proportional to the total momentum of a spin-\textonehalf{} particle-anti-particle fermion system, which was recently investigated by Barut, Cordero, and Ghirardi, is applied to a quark triplet system. The "convective" currents in the triplet system are introduced as an extra ${\mathrm{SU}}_{3}$ symmetry-breaking term. The field equation thus obtained involves certain constants; our particular choice of these constants leads to a relation for the quark masses that fixes the ratio $\frac{{m}_{1}}{{m}_{3}}$, where ${m}_{1}={m}_{2}$ is the mass of the light quarks, and ${m}_{3}$ is the mass of the heavy "strange" quark. The derived mass ratios provide a consistent interpretation of the mass spectra of pseudoscalar and vector mesons and the octet and decuplet baryons, if we assume that the mass of a composite elementary particle is directly proportional to the sum of the masses of its quark content.