Isoscalar quarks and leptons in an eight-dimensional space

Abstract

We present a model of isoscalar scalar quarks and leptons where family, quark-lepton, and color quantum numbers are generated by confining the particle with a harmonic-oscillator (HO) ``potential'' in an auxiliary (noncompact) four-dimensional Euclidean space R\ifmmode \tilde{}\else \~{}\fi{} $^{4}$. The field equation is (\ensuremath{\square}-\ensuremath{\square}\ifmmode \tilde{}\else \~{}\fi{}+${\ensuremath{\alpha}}^{4}$x\ifmmode \tilde{}\else \~{}\fi{} $^{2}$)\ensuremath{\psi}=0, where \ensuremath{\square} is the ordinary d'Alembertian, \ensuremath{\square}\ifmmode \tilde{}\else \~{}\fi{} is the Laplacian in R\ifmmode \tilde{}\else \~{}\fi{} $^{4}$, and ${\ensuremath{\alpha}}^{4}$ x\ifmmode \tilde{}\else \~{}\fi{} $^{2}$ is the HO potential. The R\ifmmode \tilde{}\else \~{}\fi{} $^{4}$ wave functions are basis functions for representations of SU(4)-symmetry operators.In an SU(3)\ifmmode\times\else\texttimes\fi{}U(1) decomposition, the SU(3) multiplets 1 (3,6, . . .) are identified as leptons (quarks, new kinds of Dirac particles) and the U(1) quantum number 0 (1,2,3, . . .) is identified with the first (second, third, fourth, . . .) family. We conjecture that weak isospin can also be generated if \ensuremath{\square}\ifmmode \tilde{}\else \~{}\fi{} is replaced by a suitable Dirac operator. (\ensuremath{\square} is simultaneously replaced by \ensuremath{\nabla}LSL to generatetrue Dirac particles.) A shortcoming of the model is that the Dirac-particle (mass${)}^{2}$ rises linearly with ascending SU(4) multiplets whereas it should rise exponentially. A solution for this problem might be dependent upon discovering a physical basis for the HO potential, if such exists. We couple a scalar ``electroweak (EW) gauge'' boson \ensuremath{\Phi} to the Dirac field \ensuremath{\psi} by generalizing the field equation to (\ensuremath{\square}-\ensuremath{\square}\ifmmode \tilde{}\else \~{}\fi{}+${\ensuremath{\alpha}}^{4}$ x\ifmmode \tilde{}\else \~{}\fi{} $^{2}$)\ensuremath{\psi}=-g\ensuremath{\Phi}\ensuremath{\psi}; we then derive the corresponding scattering equation and examine the first-order S-matrix element ${\mathrm{S}}_{f\mathrm{i}}^{(1)}$. If \ensuremath{\Vert}\ensuremath{\Phi}${\ensuremath{\Vert}}^{2}$ falls off much more slowly than does \ensuremath{\Vert}\ensuremath{\psi}${\ensuremath{\Vert}}^{2}$ when x\ifmmode \tilde{}\else \~{}\fi{} increases, then ${\mathrm{S}}_{f\mathrm{i}}^{(1)}$\ensuremath{\approxeq}${\ensuremath{\delta}}_{f\mathrm{i}}$ due to the orthogonality of the Dirac-particle wave functions; thus quarks and leptons, upon emitting or absorbing a scalar EW boson, may not change family number, transmute into one another, or, if quarks, change color. This is in approximate agreement with experiment. The R\ifmmode \tilde{}\else \~{}\fi{} $^{4}$ wave functions of the Dirac particles have rms ``radii'' x\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{\sim}${\ensuremath{\alpha}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\sim}several fm or more. This is not in conflict with upper size limits \ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}3}$ fm set by high-energy experiments since the latter can only measure size in ordinary space. To measure ``size'' in R\ifmmode \tilde{}\else \~{}\fi{} $^{4}$, one must probe the Dirac particle with a spectrum of R\ifmmode \tilde{}\else \~{}\fi{} $^{4}$ wave functions; however, there is only the one EW-boson R\ifmmode \tilde{}\else \~{}\fi{} $^{4}$ wave function in our model; hence, one can only ascertain the existence of the Dirac particle, not its structure in R\ifmmode \tilde{}\else \~{}\fi{} $^{4}$. Consequently, it is not necessary to compactify the extra dimensions to retain the apparent pointlike properties ofDirac particles. As remarked above, new families of Dirac particles are predicted, and also new kinds (neither leptons nor quarks). Perhaps the latter exist but have not been seen because they do not couple to EW bosons (i.e., the photon), just as leptons do not couple to gluons.