Quark mixing and energy conservation

[Jezza]

We've recently been looking at the hadronic decays of the W boson. In this one example, we looked at possible decays for the W boson being produced near its resonance peak, meaning the centre of mass energy is sufficient to produce u,d,c,s & b quarks. However, because we're below the mass of the top quark, and the coupling of the b to the u and c quarks is tiny, we neglected it in our calculations. Nonetheless, there is still a small chance of getting a b in the decay owing to the small but non-zero coupling of the b to the d and the s.

What confuses me is that all the discussions I've seen so far make no reference to energy conservation. Suppose we observe the decays of (virtual) W bosons at a centre of mass energy above the mass of the c but below the mass of the b. (I am assuming we can observe such processes). Energy conservation implies we should see no b production, and yet the small coupling of the b to the d and the s implies we should have a small chance of seeing a b produced. i.e. the coupling doesn't seem to necessarily satisfy energy conservation.

So my question: What happens to the coupling of the b to the d and s quarks at centre of mass energies below the mass of the b quark?

 

[Vanadium 50]

The coupling stays the same. You just don't get any events in that channel, just as you don't get (real) W decays to t+b.

 

[Jezza]

Hmm... I've been viewing this 'coupling' as the probability amplitude of a quantum superposition. That is the W decays (for example) into an u and an anti-d', where d' is a superposition of the d, s and b with mostly d, some s and a small amount of b. For us to not get any events with a b, that means the amount of b in d' must be zero, implying the coupling of the bottom to the down is now zero. Is this then the wrong way to think about it?

 

[Vanadium 50]

Yes, it's the wrong way. The probability of interaction is the matrix element (including the coupling) squared multiplied by phase space. The probability is zero if the matrix element is zero, but it's also zero if the phase space is zero.

 

[mfb]

Suppose we observe the decays of (virtual) W bosons at a centre of mass energy above the mass of the c but below the mass of the b. (I am assuming we can observe such processes).

Decays of b-hadrons have this case if there is no lighter b-hadron to decay to.