Thread: CKM matrix parameters View Single Post

 Quote by LAHLH If you look at (#4)^2+(#5)^2, you get an equation constraining just the magnitudes, a1,a2,a3,a4,a5,a6; similarly (#6)^2+(#7)^2 constraints just a1,a2,a3,a7,a8,a9, and (#8)^2+(#9)^2 constraints just a4,a5,a6,a7,a8,a9. How do I get the three equations for just the phases $b_i$ out of these that you mention?
How do you get rid of the $b_i$ when you do (4)^2 + (5)^2 ?

for (4) and (4)^2 I get:
$\alpha_1 \alpha_4 cos(\beta_1 - \beta_4) + \alpha_2 \alpha_5 cos(\beta_2 - \beta_5) = -\alpha_3 \alpha_6 cos(\beta_3 - \beta_6)$

$(\alpha_1 \alpha_4 )^2 cos^2(\beta_1 - \beta_4) + (\alpha_2 \alpha_5)^2 cos^2(\beta_2 - \beta_5) + 2(\alpha_1 \alpha_4 \alpha_2 \alpha_5)cos(\beta_1 - \beta_4) cos(\beta_2 - \beta_5) = (\alpha_3 \alpha_6)^2 cos^2(\beta_3 - \beta_6)$

for (5) and (5)^2 I get:
$\alpha_1 \alpha_4 sin(\beta_1 - \beta_4) + \alpha_2 \alpha_5 sin(\beta_2 - \beta_5) = -\alpha_3 \alpha_6 sin(\beta_3 - \beta_6)$

$(\alpha_1 \alpha_4 )^2 sin^2(\beta_1 - \beta_4) + (\alpha_2 \alpha_5)^2 sin^2(\beta_2 - \beta_5) + 2(\alpha_1 \alpha_4 \alpha_2 \alpha_5)sin(\beta_1 - \beta_4) sin(\beta_2 - \beta_5) = (\alpha_3 \alpha_6)^2 sin^2(\beta_3 - \beta_6)$

so for (4)^2 + (5)^2 I get:
$(\alpha_1 \alpha_4 )^2 + (\alpha_2 \alpha_5)^2 + 2\alpha_1 \alpha_4 \alpha_2 \alpha_5 (cos(\beta_1 - \beta_4) cos(\beta_2 - \beta_5) + sin(\beta_1 - \beta_4) sin(\beta_2 - \beta_5) )$

$= (\alpha_1 \alpha_4 )^2 + (\alpha_2 \alpha_5)^2 + 2\alpha_1 \alpha_4 \alpha_2 \alpha_5 (cos(\beta_1 - \beta_4 - \beta_2 + \beta_5)) = (\alpha_3 \alpha_6)^2$

What am I doing wrong here?