(a)

gauge-group = Lie-group SU(3) 8 generators Gell-Mann-matrices λ_α, corresponding 8 massless spin-1 gauge bosons, 3 charges (colors) r g b, short range, pure quantum, asymptotically free (confinement),

energy scale $E_{col}=220\text{MeV}$ , with the zero-shift constant in the Callan-Symantzik relation to remove the singularity $c_{GE}=0.69$

confinement length scale $r_{col}=\frac{\hslash c}{E_{col}}=0.89\times {10}^{-15}\text{m}\approx r_{pr}$

cut-off energy $M_{col}=\frac{\hslash c}{r_{col}}=2.33\times {10}^{10}\text{eV}\approx 23\text{GeV}$

the pion π with mass 106MeV is the Yukawa field boson of the color interaction

Lagrangian $L_{col}=-\frac{1}{4}{G}^a{}_{\mu \nu }{G}^a{}^{\mu \nu }$

with the field tensor $G^a{}_{\mu \nu }={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$

covariant derivative $D_{\mu }\equiv {\partial }_{\mu }-ig{A}_{\mu }^a{\lambda }_a/2$ , where g is the coupling constant, A is the gluon gauge field, for eight different gluons $\alpha =\text{1},\cdots ,\text{8}$ , and whereλ_α is one of the eight Gell-Mann matrices, $\alpha =\text{1},\cdots ,\text{8}$ .

commutation relations with structure constants $\left[{\lambda }_a,{\lambda }_b\right]=2i{f}^{abc}{\lambda }_c$

(b)

gauge-group = Lie-group SU(4), 15 generators massless bosons spin = 1, short range, pure quantum

4 charges $r_L{}^{-}{r}_L{}^{+}{r}_R{}^{-}{r}_R{}^{+}$ for r-preon and $q_L{}^{-}{q}_L{}^{+}{q}_R{}^{-}{q}_R{}^{+}$ for q-preon resp. with charge + or − and helicity L or R, antiparticle $C\left(r_L{}^{+}\right)=r_R{}^{-}$ , $C\left(r_L{}^{-}\right)=r_R{}^{+}$

the weak-interaction is the Yukawa-limit of the hypercolor interaction

L-R-symmetry breaking $SU\left(4\right)=SU{\left(2\right)}_L\otimes SU{\left(1\right)}_R\otimes SU{\left(1\right)}_{em}$

covariant derivative $D_{\mu }\equiv {\partial }_{\mu }-ig{A}_{\mu }^a{\lambda }_a/2$

commutation relations with structure constants $\left[{\lambda }_a,{\lambda }_b\right]=2i{f}^{abc}{\lambda }_c$

field tensor $G^a{}_{\mu \nu }={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$

Lagrangian $L_{hc}=-\frac{1}{4}{G}^a{}_{\mu \nu }{G}^a{}^{\mu \nu }$

The coupling from the Callan-Symanzik equation is

$g_{hc}\left(m\right)=4\pi \sqrt{\frac{3}{76\sqrt{{\left(\log \left(\frac{m}{{\Lambda }_{hc}}\right)\right)}^2+c_{GE1}^2}}}$

critical energy ${\Lambda }_{hc}=2m\left(Z_0\right)=180\text{GeV}$ in analogy to the QCD, and zero-shift constant $c_{GE1}=\frac{1}{\log \left(\frac{m\left(t\right)}{m\left(d\right)}\right)}=0.095$

(c)

gauge-group = Lie-group SU(2) 3 generators Pauli matrices σ_j, corresponding 3 massive gauge bosons

Z, W⁺, W⁻, short range, pure quantum, are Yukawa bosons of hypercolor interaction

cut-off energy M_Z = 91.17 GeV,

length scale $r_{weak}={10}^{-17}\text{m}$

energy scale $E_{weak}\approx {10}^{-4}{E}_{em}=7\text{eV}$

Lagrangian, $L_1=-\frac{1}{4}{W}_{\mu \nu }^a{W}^{a\mu \nu }-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$ where

$W_{\mu \nu }^a={\partial }_{\mu }{W}_{\nu }^a-{\partial }_{\nu }{W}_{\mu }^a+g{f}^{abc}{W}_{\mu }^b{W}_{\nu }^c$ , $F_{\mu \nu }={\partial }_{\mu }{B}_{\nu }-{\partial }_{\nu }{B}_{\mu }$

where the physical gauge fields Z, W⁺, W⁻ are

$Z_{\mu }=\frac{g{W}_{\mu }^3+g\text{'}{B}_{\mu }}{\sqrt{g^2+g{\text{'}}^2}}=\cos {\theta }_W{W}_{\mu }^3+\sin {\theta }_W{B}_{\mu }$

$W_{\mu }^{\pm }=\frac{1}{\sqrt{2}}\left(W_{\mu }^1\pm i{W}_{\mu }^2\right)$

covariant derivatives left and right with Pauli matrices ${\sigma }_i$

$D_{\mu }R=\left({\partial }_{\mu }+ig\text{'}{B}_{\mu }\right)R$ , $D_{\mu }L=\left({\partial }_{\mu }+\frac{i}{2}g\text{'}{B}_{\mu }-\frac{i}{2}g{\sigma }_i{W}_{\mu }^i\right)L$