Are preons dyons? Naturalness of three generations

In the present paper a model of preons and their composites is constructed in the framework of the superstring-inspired flipped $E_6\times {\tilde{E}}_6$ gauge symmetry group which reveals a generalized dual symmetry. Here $E_6$ and ${\tilde{E}}_6$ are nondual and dual sectors of the theory with hyperelectric $g$ and hypermagnetic $\tilde{g}$ charges, respectively. We follow the old idea by J. Pati presuming that preons are dyons, which in our model are confined by hyper-magnetic strings—composite $\mathbf{N}=1$ supersymmetric non-Abelian flux tubes created by the condensation of spreons near the Planck scale. Considering preons belonging to the 27-plet of $E_6$, we have investigated the breakdown of $E_6$ (and ${\tilde{E}}_6$) near the Planck scale into the $SU(6)\times U(1)$ (and $\widetilde{SU(6)}\times \widetilde{U(1)}$) gauge group and shown that the six types of strings having fluxes ${\Phi }_n=n{\Phi }_0$ $(n=\pm 1,\pm 2,\pm 3)$ produce three (and only three) generations of composite quark-leptons and bosons. The model predicts the existence of the Family replicated gauge group $[E_6{]}^3$ near the Planck scale. The runnings of the corresponding fine structure constants are investigated, and the critical coupling constants at the two phase transition points near the Planck scale are calculated. The model explains the hierarchies of masses in the standard model naturally. The compactification in a space-time with five dimensions and its influence on form-factors of composite objects are also briefly discussed.

Figure 1

This figure presents the running of the inverse gauge coupling constants ${\alpha }_i^{-1}(x)$ for $i=1$, 2, 3, $X$, $Z$, $X1$, $Z1$, 5, 10 of the chain $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y\to SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Z\times U(1{)}_X\to SU(5)\times U(1{)}_X\to SU(5)\times U(1{)}_{Z1}\times U(1{)}_{X1}\to SO(10)\times U(1{)}_{X1}\to E_6$, corresponding to the breakdown of the flipped $SU(5)$ to the supersymmetric (MSSM) $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Z\times U(1{)}_X$ gauge symmetry group with Higgs bosons belonging to the $5_h+{\overline{5}}_h$, ${10}_H+{\overline{10}}_H$, 24-dimensional adjoint $A$ and higher representations of the flipped $SU(5)$. The final unification group is flipped $E_6$ at the supersuperGUT scale $M_{\mathrm{SSG}}\approx {10}^{18}\mathrm{GeV}$ with ${\alpha }^{-1}(M_{\mathrm{SSG}})\approx 22$ for $M_{\mathrm{SUSY}}=10\mathrm{TeV}$ and seesaw scale $M_{\mathrm{SS}}={10}^{11}\mathrm{GeV}$.

Figure 2

The figure provides a qualitative description of the running of ${\alpha }^{-1}(x)$ near the Planck scale predicted by the present preonic model. The region $AD$ corresponds to the Family replicated gauge group of symmetry $[E_6{]}^3$, which arises at the scale $M_{\mathrm{FR}}$ given by point $D$. The point $A$ at the scale of energy $\mu =M_{\mathrm{crit}}$ indicates that hyper-electric preonic strings exist for $\mu \ge M_{\mathrm{crit}}$. The point $B$ corresponds to the scale $\mu ={\tilde{M}}_{\mathrm{crit}}$ and indicates that hyper-magnetic preonic strings exist for $\mu \le {\tilde{M}}_{\mathrm{crit}}$. The curve $AB$ corresponds to the region of energies where spreons are condensed near the Planck scale, giving both hyper-electric and hyper-magnetic, preonic strings. For $\mu \ge {\tilde{M}}_{\mathrm{crit}}$ we have the running of ${\alpha }^{-1}(x)$ for monopolic “quark-leptons.” The point $C$ corresponds to the Planck scale and gives ${\alpha }^{-1}(M_{\mathrm{Pl}})=3$.

Figure 3

Preons are bound by hyper-magnetic strings: (a), (b) correspond to the string configurations of composite particles belonging to the 27-plet of the flipped $E_6$ gauge symmetry group; (c) represents a closed string describing a graviton.

Figure 4

Vector gauge bosons belonging to the 78 representation of the flipped $E_6$ and Higgs scalars—singlets of $E_6$—are composite objects created (a) by fermionic preons $P$, $P^{\mathbf{c}}$ and (b) by scalar preons $P_s$, $v$. Both of them are confined by hyper-magnetic strings.