Cabibbo Angle and the Bootstrap Equation

A solution of the $c$-number bootstrap equations, $a_i=cd_{\mathrm{jk}}^{}{}_{}{}^i(a_j{a}_k+b_j{b}_k)$ and $b_i=2cd_{\mathrm{jk}}^{}{}_{}{}^i{a}_j{b}_k$, is obtained, where the amplitudes $a_i$ and $b_i$ represent the matrix elements of unitary octer-vector and axial-vector charges, respectively, $d_{\mathrm{jk}}^{}{}_{}{}^i$ is the completely symmetric invariant tensor of the third order, the unitary spin indices run from 1 to 8, and $c$ is an arbitrary constant. The solution depends on two parameters, which suggests that the bootstrap equations underdetermine the amplitudes. For a specific value of the parameters, one has a solution which satisfies the Cabibbo theory. It is also found in the present model that if the Cabibbo angle for the vector charges is different from the Cabibbo angle of the axial-vector charges, $\mathrm{CP}$ invariance must be violated. The matrix elements of the weak charges can be arbitrarily smaller than those of the electromagnetic charges.