Kobayashi Maskawa Matrix Error ?

[Anne-Sylvie]

Hello everyone,

I'm studying some lectures on electroweak interactions and I was reading the paper from Kobayashi and Maskawa, particulary the matrix :

But these matrix have to be unitary... And my professor said that it was not the case. When I compute the product of this matrix and its conjugate transpose, I actually don't find the identity matrix but I don't rely on my computation...

Is there REALLY an error in the paper ?
(I computed for many values of θ1, θ2 and θ3 numerically the product and I never have the identity.)

 

[Bill_K]

If you look at the Wikipedia page on the CKM Matrix, they show that the "standard" parametrization can be factored into three Euler-angle rotations. This makes it obvious that the result is unitary. And although they don't do it, the "original" KM parametrization can be factored in a similar manner.

(If only I knew how to write a matrix!) But anyway, it's a rotation by θ₃ in the 23 plane, followed by a rotation by θ₁ in the 12 plane, followed by a rotation by θ₂ again in the 23 plane. The matrix is correct as you give it.

 

[Avodyne]

And the matrix as you give it is unitary.

 

[Anne-Sylvie]

Thanks very much for your answer.
In fact, the matrix that I give is unitary as you said.
But this is not EXACTLY the matrix as in the paper of Kobayashi and Maskawa ;
The V_3,3 term have been corrected. In the paper they write a sin(θ3) term in place of a cos(θ3) term.

Thank you !