Bootstrap Equation and the Cabibbo Angle

It is conjectured that the weak and electromagnetic properties of hadrons are determined by self-consistency requirements. Possible solutions of the $c$-number bootstrap equations $a_i=c{d_{\mathrm{jk}}}^i(a_j{a}_k+b_j{b}_k)$ and $b_i=2c{d_{\mathrm{jk}}}^i{b}_k$ are studied, where the amplitudes $a_i$ and $b_i$ represent the matrix elements of unitary octet 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. When all expect the amplitudes $a_3$ and $a_8$ are zero, a solution is $a_3=\sqrt{3}{a}_8$, which holds for the isoscalar ($a_8$) and isovector ($a_3$) components of electromagnetic interaction. When slight deviations from $a_3=\sqrt{3}{a}_8$ are considered, the physical Cabibbo angle can be artifically obtained. More general solutions are considered in which the relative sizes of the amplitudes are obtained in closed form but the Cabibbo angle is $\frac{1}{4}\pi$. It appears that one must consider slight deviations from $a_3=\sqrt{3}{a}_8$ or include higher-order terms in the amplitudes to obtain the physical Cabibbo angle.