Quark mixing and energy conservation

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Discussion Overview

The discussion revolves around the hadronic decays of the W boson, particularly focusing on the implications of energy conservation in relation to quark mixing and the production of b quarks in specific energy regimes. Participants explore the theoretical aspects of coupling between quarks and how this relates to observable decay processes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that while the coupling of the b quark to d and s quarks is small, it still allows for a non-zero chance of b production in W boson decays, raising questions about energy conservation in these processes.
  • Another participant asserts that the coupling remains unchanged, indicating that no events will occur in that channel when energy is insufficient, similar to the absence of real W decays to t+b.
  • A different perspective suggests that the coupling should be viewed as a probability amplitude in a quantum superposition, questioning whether the absence of b production implies a zero coupling in certain conditions.
  • One participant corrects the previous notion, explaining that the probability of interaction depends on the matrix element and phase space, indicating that zero probability can arise from either factor.
  • A later post introduces the idea that decays of b-hadrons may exhibit similar characteristics when no lighter b-hadron is available for decay, further complicating the discussion on energy thresholds.

Areas of Agreement / Disagreement

Participants express differing views on the implications of coupling and energy conservation, with no consensus reached on how to interpret the relationship between quark mixing and observable decay processes in the context of energy thresholds.

Contextual Notes

The discussion highlights the complexity of coupling dynamics and energy conservation in particle decays, with participants acknowledging the limitations of their assumptions regarding quark interactions and decay channels.

Jezza
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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?
 
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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.
 
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?
 
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.
 
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Jezza said:
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.
 

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