Weak decays- why do they occur?

[WilliamD]

I've heard the weak force described as essentially "a force between particles with flavor" just as the EM is "a force between particles with charge". Mathematically, I understand the concept of the CKM matrix being used to evaluate the weak force eigenstates of quarks, but I'm confused as to how the flavor of one of these quarks can be changed simply through an interaction with other quarks. I feel like this would be analogous to two electrons interacting through the EM force and then one of them just losing its electric charge.

The best explanation I've been able to come up for myself is that one quark is eliciting the release of a W or Z boson from another quark. I'm sure this is incorrect.

This is probably an utterly stupid question, but can anyone explain this to me?

 

[BenTheMan]

I think the thing to remember is that the weak gauge bosons are charged under the electromagnetism, too. The two forces are very different, so comparing the two with a simple analogy is dangerous. Also, it really doesn't make sense to separate the two forces. And, when you learn the structure of the standard model more deeply, you'll find out that you CAN'T separate the two forces, like you can separate out QCD.

So, some John Baez ASCII figures:

|e-
^
|
|~~~~~~~photon
|
^
|e-

This is how the electromagnetic force acts on an electron---a happy electron floats along, and absorbs a photon. This is all you need to know for electromagnetism :)

The weak force is a bit more complex, because the W boson is charged:

|d
^
|
|~~~~~~~W-
|
^
|u

So a up quark is floating along, and it absorbs a W- boson, and turns into a down quark.

Finally, the neutral current in the SM conserves flavor:

|u
^
|
|~~~~~~~~ Z
|
^
|u

If you know a little quantum mechanics, and a little about SU(2), then I can tell you that the flavor conservation is an artifact of \sigma_3 being diagonal.

 

[BenTheMan]

Looking back on my reply, I don't know if I answered your question properly. Please let me know!

 

[arivero]

It seems the OP is also worried about the CKM matrix. It is right to be worried about it; the |u eigenstate that couples to a W- is a charge eigenstate, while the |u' eigenstate living in the propagator is a mass eigenstate. Thus, what happens, if I understand rightly, is that 1) the initial u' becomes a linear combination of u, c, t; 2) each of them have a probability to transform in their corresponding d, s, b; and 3) each of them become projected into some state d',s',b'

The sequence of projections is abstracted in the CKM matrix; but it is more important that it constitutes the GIM mechanism.

 

[arivero]

An even deeper point is that a single flavoured quark, as seen in Dirac equation, can not anhiquilate against its antiquark.

 

[arivero]

An even deeper point is that a single flavoured quark, as seen in Dirac equation, can not anhiquilate against its antiquark.

Hmm I should elaborate a bit on this. Consider for instance a quark s; the 4-component dirac equation is the combination of two particles, s and anti-s, as a mass eigenstate. Thus in order to collide and emit an electroweak particle, they must be in a charge eigenstate. It is not a real issue, because all the generations have the same coupling. But it is kind of amazing that the decay rate of Upsilon is, in proportion to its mass, slower than the decay rate of J/Psi.

 

[BenTheMan]

It is not a real issue, because all the generations have the same coupling.

Did I read this properly?

 

[arivero]

Did I read this properly?

Hmm, perhaps I am wrong. I was telling only that all the three d,s,b have the same electroweak charges, thus the same electroweak couplings.


But now that you stress it, I am not so sure about if the total cross section for, say, electroweak annihilation of b and anti-b quarks, has some dependence on the CKM matrix. Should we revisit some calculation here in this thread?

 

[BenTheMan]

They have the same electroweak couplings, but you didn't specify this originally.

And the electroweak cross sections should all depend on phase space factors and group theory factors, probably, not the CKM matrix.

 

[WilliamD]

I didn't think the CKM has anything to do with electroweak annihilation cross sections, but rather gives the likelihood of flavor change from one quark to another which can be used to calculate weak force eigenstates of quarks, but not the cross section of electroweak annihilation.

Basically, I didn't think the CKM has anything to do with annihilation.

?

 

[BenTheMan]

Quarks change flavor by emitting W+ or W- bosons. Whenever an up quark emits a W- boson, it has the opportunity to change to a down quark, a strange quark, or a bottom quark. The following LaTeX was ripped off of wikipedia:

\begin{bmatrix} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{bmatrix} \begin{bmatrix} \left| d \right \rangle \\ \left| s \right \rangle \\ \left| b \right \rangle \end{bmatrix} = \begin{bmatrix} \left| d' \right \rangle \\ \left| s' \right \rangle \\ \left| b' \right \rangle \end{bmatrix}

V_{ij} = \begin{bmatrix} 0.97383 & 0.2272 & 0.00396 \\ 0.2271 & 0.97296 & 0.04221 \\ 0.00814 & 0.04161 & 0.999100 \end{bmatrix}.

Look at the top row. (Square the numbers to get %'s.) When a u quark emits a W- boson, it changes to a d quark 94.6% of the time, an s quark 5.2% of the time, and a b quark 0.2% of the time.

==================

Again, I reread your post and think I didn't quite answer your question.

The CKM matrix doesn't have anything to do with annihilation---you are correct. The Z boson and the photon do with that, for example.

 

[samalkhaiat]

I feel like this would be analogous to two electrons interacting through the EM force and then one of them just losing its electric charge.

Think of (W^{+},W^{-},Z^{0}) in terms of the generators (Pauli matrices) of the group SU(2), where the u-quark is an upper spinor

|u \rangle \equiv \left( \begin{array}{c} 1 \\ 0 \\ \end{array} \right)

and d-quark is the lower spinor

|d \rangle \equiv \left( \begin{array}{c} 0 \\ 1 \\ \end{array} \right)

Now if you write

<br /> W^{+} \equiv \frac{1}{2} ( \sigma_{1} + i \sigma_{2} ) = \left( \begin{array}{rr} 0 &amp; 1 \\ 0 &amp; 0 \\ \end{array} \right)<br />

<br /> W^{-} \equiv \frac{1}{2} ( \sigma_{1} - i \sigma_{2} ) = \left( \begin{array}{rr} 0 &amp; 0 \\ 1 &amp; 0 \\ \end{array} \right)<br />

and

Z^{0} = \sigma_{3}

you can now represent the processes

W^{+} + d \rightarrow u \ \mbox{by} \ \ W^{+}|d \rangle = |u \rangle

W^{-} + u \rightarrow d \ \mbox{by} \ \ W^{-}|u \rangle = |d \rangle

and

Z^{0} + (u , d ) \rightarrow ( u , d ) \ \mbox{by} \ \ Z^{0}|u (d) \rangle = |u (-d) \rangle

So, it is all about the difference between U(1) (electrodynamics) and SU(2) (weak-dynamics).

sam

 

[BenTheMan]

The subtlety here is that sam is working in an interaction basis, with a non-diagonal mass matrix. If you move the CKM matrix out of the W vertex, you have to stick it into the quark mass matrix.

 

[WilliamD]

Sam, that clears a lot of things up, thanks a lot.

Ben, that only applies if you are are worried about the weak force eigenstates of these quarks, rather than just the interaction itself (regardless of the cross sections of these interactions), right?

 

[BenTheMan]

The two conventions are the same---you just change what you mean by "up quark".