Weak Decay of Nucleus: 3-Momentum Analysis

[Poirot]

Homework Statement

An example of a weak decay of a nucleus is (Z, A) → (Z +1, A) + τ^- + ν_τ-bar where (Z,A) represents a nucleus with Z protons and A−Z neutrons.

(d) Suppose the gauge boson in the above process is produced at rest. Assuming the neutrino is massless, what must the 3-momenta of the τ^- and neutrino be? [5 Marks]
(e) Now suppose that the neutrino is not massless. Express the mass of the gauge boson in terms of the lepton and neutrino masses and their 3-momenta. [3 Marks]
(f) Consider the case where the nucleus (Z,A) is at rest and the nucleus (Z +1,A) recoils with energy E along the X direction. The τ is observed to be moving along the Y direction (i.e. at right-angles to the nucleus’s recoil). What will the component of the 3-momenta of the neutrino be along the axis of the nucleus’s recoil? You should express your result in terms of E,Z,A,Mp and Mn. [7 Marks]

Homework Equations

E before= E after
3-momenta before= 3-momenta after
4-momentum before= 4 momentum after

The Attempt at a Solution

(d) Isolating the W_ bit of the process: W^-= τ^- + ν^τ-bar
so (M_W^-, 0) = (E_{τ_}, P_{τ_}) + (E_ντ, P_ντ)
And from 3-momentum conservation, the momenta of the neutrino and taon must be equal and opposite.
(My first issue is that I'm not sure that I can just isolate this part of the interaction, I just made an assumption from the Feynman diagram I drew). I also don't understand the significance of the neutrino being massless other than the |k_τ| = E_ν(bar)τ ?

(e) Applying 4-momentum conservation: M_W² = (E_τ + E_ν(bar)τ)² - (K _τ + K _ν(bar)τ)²
But I can't seem to work this through to gain an answer in terms of lepton neutrino masses and their 3-momentum.

(f) I'm not sure how to even approach this part, I can't figure out whether I need to write the 4 momenta's using A-Z etc. Any kind of hint or guidance would be great so I can have a good stab.

Thanks in advance for any help, it's greatly appreciated!

 

[mfb]

The problem statement is a collection of nonsense. Not your fault, but whoever wrote that problem should learn some particle physics.

(d) Isolating the W_ bit of the process: W^-= τ^- + ν^τ-bar
so (M_W^-, 0) = (E_{τ_}, P_{τ_}) + (E_ντ, P_ντ)

That would work if a nuclear decay could produce a real W boson. It cannot. Having (M_W,0) would violate energy/momentum conservation. Do (a) to (c) explicitely ask you to assume the nuclear decay could work like that? otherwise, I would use a lower energy.

And from 3-momentum conservation, the momenta of the neutrino and taon must be equal and opposite.

Right.

I also don't understand the significance of the neutrino being massless other than the |k_τ| = E_ν(bar)τ ?

That is exactly the significance. It allows to calculate the involved energies and momenta (up to the unknown direction).

(e) Applying 4-momentum conservation: M_W² = (E_τ + E_ν(bar)τ)² - (K _τ + K _ν(bar)τ)²
But I can't seem to work this through to gain an answer in terms of lepton neutrino masses and their 3-momentum.

You know how the energy depends on 3-momentum and mass.

(f) I'm not sure how to even approach this part, I can't figure out whether I need to write the 4 momenta's using A-Z etc. Any kind of hint or guidance would be great so I can have a good stab.

You'll need the 4-momenta of the nucleus before and afterwards, yes. And also the momenta of the other two particles in the final state.

 

[Poirot]

The problem statement is a collection of nonsense. Not your fault, but whoever wrote that problem should learn some particle physics.
That would work if a nuclear decay could produce a real W boson. It cannot. Having (M_W,0) would violate energy/momentum conservation. Do (a) to (c) explicitely ask you to assume the nuclear decay could work like that? otherwise, I would use a lower energy.Right.That is exactly the significance. It allows to calculate the involved energies and momenta (up to the unknown direction).
You know how the energy depends on 3-momentum and mass.You'll need the 4-momenta of the nucleus before and afterwards, yes. And also the momenta of the other two particles in the final state.

Thank you for you response, I will have a proper go at this in the morning but it's quite late now and I'm burnt out. But incase it helps, the rest of the question was:
(a) Write this process in terms of protons p and neutrons n. You can ignore any particles which don’t directly take part in the reaction. [1 Mark]
(b) Write this process in terms of quarks u and d. [2 Marks]
(c) Write this process in terms of the weak interaction’s gauge bosons (labelling the gauge boson involved). [2 Marks]

These were relatively simple and probably where some of my confusion lay due to their leading nature.

What did you mean by 'use a lower energy' for the W^-?

Thanks again, I'll crack on in the morning, I really appreciate your help! Sadly the whole module experience can be summed up by this disjointed question.

 

[mfb]

What did you mean by 'use a lower energy' for the W-?

There is no nuclear decay that would release 80 GeV. Decay energies are of the order of MeV, 4-5 orders of magnitude lower.

 

[Poirot]

There is no nuclear decay that would release 80 GeV. Decay energies are of the order of MeV, 4-5 orders of magnitude lower.

Does this mean I need to do anything with the calculation or is it just an unphysical example?
For (e): Using the 4-momentum conservation, I expanded out the brackets and as the momentum of the taon and neutrino are equal and opposite the angle between them must be 180 degrees, so the cosine from the dot product will be -1. This led to an answer of M²_W^- = M²_τ + M²_ντ - 2(√(M²_τ + P²_τ) √(M²_ντ + P²_ντ) + P_τP_ντ) Which has all the components asked for but it's not particularly pretty.

For (f): I wrote out the 4-momentum conservation: (ZMp + (A-Z)Mn, 0) = (E, Pp) + (Eτ, Pτ) + (Eν, Pν)
and I tried both 3 momentum conservation and energy conservation separately and 4 momentum conservation and using invariant mass^2. And with the fact that Pp⋅Pτ = 0 as perpendicular this clear a few things up but I couldn't get a nice answer. Am I doing something wrong?

 

[mfb]

Does this mean I need to do anything with the calculation or is it just an unphysical example?

That depends on the interpretation of the problem. You can ignore every physical reality, but is that the right approach for a physics question?

For (e), you can simplify it a bit more, but not much.

For (f), don't forget the binding energy. Using 4 momenta is easier than energy and momentum separately, but in the end you have three equations and three unknowns to solve for. You can neglect the neutrino mass again here.

 

[Poirot]

I don't understand what to do as this was a set exam question in past paper I'm working through.

For (e) I think I can see how to simplify it, can i pull out a factor of P² out of the square root as the 3-momenta are equal and opposite and then clean it up a little bit.

for (f) I'm not sure how to factor the binding energy into any equation. And ok I'll retry this with 4-momentum again neglecting the neutrino mass.

 

[mfb]

For (e) I think I can see how to simplify it, can i pull out a factor of P² out of the square root as the 3-momenta are equal and opposite and then clean it up a little bit.

That's what I was thinking about.

for (f) I'm not sure how to factor the binding energy into any equation. And ok I'll retry this with 4-momentum again neglecting the neutrino mass.

If there would be no binding energy, the process would just be the decay of a free neutron.