Understanding the Unitarity of CKM Matrix and Its Conservation of Probability

[plasmon]

This is a known fact that CKM matrix, a matrix that is used to connect the weak interaction eigenstates to mass eigenstates is unitary. I have studied that this is due to the conservation of probability. i.e. an up type quark will decay into exactly three type of d quarks, nothing more.

 

[genneth]

I'm not sure what exactly you're trying to say --- it's all correct and true, but why did you decide to post that as a message?

 

[plasmon]

I just want to express the fact that an up quark can only decay to three down types of quarks only and thus the unitarity of ckm matrix mathematically.

 

[malawi_glenn]

What kind of respons do you want plasmon?

 

[Atakor]

Hello, I don't understand the aim of the post too..
but this has raised a question in my mind..
(thanks Plasmon !)

in the leptonic sector, the neutrino also have a CKM-analogue matrix right?
but we don't know yet if it is unitary or not..
if it is just related to probabilites.. where's the problem ?

 

[Vanadium 50]

There is a lepton analogue called the MNS matrix, for Maki, Nakagawa and Sakata.

The CKM (and MNS) matrices are unitary by construction. Essentially they are a change in basis - a rotation, if you will - so must be unitary. A separate question is experimental: is the 3x3 matrix that we can measure unitary? If the answer is "no", that's proof for new physics: e.g. a 4th family.

For quarks, the answer is that we cannot tell. Partly this is because the effect of new physics can be made arbitrarily small, but there's a little more to it than that. For example, given only the upper 2x2 CKM matrix, one cannot infer the existence of a 3rd family. Additionally, the numbers don't work out nicely: Vtb is so close to 1 that one needs a measurement of single top production good to better than 0.1% to see the effect of other quarks.

 

[plasmon]

Dear All

I guess we can express it as

W-|u>=Vud |d>+Vus |s>+Vub |b>
W-|c>=Vcd |d>+Vcs |s>+Vcb |b>
W-|t>=Vtd |d>+Vts |s>+Vtb |b>

and |d> , |s>, |b> are orthogonal states (not necessary due to GIM mechanism)

<u|W-(adj) W-|u>=1, <c,t|W-(adj) W-|u>=0

=> V as a unitary matrix

Am i right in making above assumptions.

 

[Atakor]

Thanks Vanadium for your enriching contribution.
another question, I may be off topic but it's in my mind, so I hope this won't bother you too much :)

Is the eventual non-unitarity of the MNS matrix related to the CP violation ?

 

[Vanadium 50]

Is the eventual non-unitarity of the MNS matrix related to the CP violation ?

There is no "eventual non-unitarity of the MNS matrix". If there are only three families of neutrinos and no new physics, you will get a unitary matrix.

 

[Atakor]

Vanadium 50,

the eventuality is as you stated it : "If there are only three families of neutrinos and no new physics"

..

 

[masudr]

Atakor

The word "if" does not imply an eventuality.

 

[Atakor]

Atakor

The word "if" does not imply an eventuality.

Really ?

There is no "eventual non-unitarity of the MNS matrix". If there are only three families of neutrinos and no new physics, you will get a unitary matrix.

==
if we say 'there are only 3 families' THEN MNS is unitary,
otherwise (if you don't like the word 'eventuality') MNS doesn't have to be Unitary.

quiet clear for me...