Which particle decays more frequently in Pi+ decays: muon or electron?

[unscientific]

Homework Statement

Draw feynman diagrams for pi+ muon lepton decay and suggest which process is more likely.

Homework Equations

The Attempt at a Solution

[/B]
The feynman diagrams are:

The lepton decays proceed via the weak interaction W+ boson. This implies that e+ should be right-handed and neutrino is always left-handed. In the rest frame of W+, we get:

But this implies that total spin = 1, which is not true. Therefore, e+ must be left-handed?

 

[Orodruin]

Is the electron massless?

 

[unscientific]

Is the electron massless?

No, the electron is not massless.

 

[Orodruin]

No, the electron is not massless.

So, does the left-handed electron have definite helicity? (Or the right-handed positron?)

 

[unscientific]

So, does the left-handed electron have definite helicity? (Or the right-handed positron?)

Electron? I thought we are dealing with positrons here.I'm not sure if having mass implies that we can choose helicity as we want it. In this case, the only way this would work is if the positron having left-handed helicity (But the weak force couples more strongly to left-handed particles and right-handed anti-particles. Positron in this case is an anti-particle.)

 

[Orodruin]

Your problem relates to missing fundamental understanding on the difference between chirality and helicity. I could ask you about any particle for this.

So what is the difference between chirality and helicity?

 

[unscientific]

Your problem relates to missing fundamental understanding on the difference between chirality and helicity. I could ask you about any particle for this.

So what is the difference between chirality and helicity?

Helicity is the projection of the spin on the momentum. According to the restframe of the W+, the electron neutrino is left-handed and the positron is right handed. But as pointed above, total J = 0, so this must be untrue.

For a particle with mass, we can always go into a frame whereby the particle is moving left/right while the helicity remains unchanged. This implies helicity is not conserved in weak interactions as Z0 and W+ bosons have mass. For massless particles, there is no rest frame.

 

[Orodruin]

Helicity is the projection of the spin on the momentum. According to the restframe of the W+, the electron neutrino is left-handed and the positron is right handed. But as pointed above, total J = 0, so this must be untrue.

And herein lies your confusion of helicity and chirality. Right-hand chirality (which is what the W interacts with) is not equivalent to right-helicity for massive particles. The question is how they are related.

 

[unscientific]

And herein lies your confusion of helicity and chirality. Right-hand chirality (which is what the W interacts with) is not equivalent to right-helicity for massive particles. The question is how they are related.

Chirality is helicity in the limit where mass tends to zero. Chirality is an intrinsic property of the particle that is invariant. For massless particles, chirality and helicity are the same.

For a massive particle, we could always go into a frame where the particle is either moving left or right, so a massive particle can undertake any form of helicity. However, its chirality is invariant. Chirality isn't even mentioned in my lecturer's notes which is why it is so confusing..

 

[Buzz Bloom]

so a massive particle can undertake any form of helicity. However, its chirality is invariant.

Can someone please post a mathematical definition of the non-helicity chirality spin property of a massive particle, e.g., a neutrino?

 

[ConstantinW]

Helicity is the projection of the spin along the linear momentum of a subatomic particle. Chirality is the innate handedness of a particle. The weak interaction couples to negative chirality particle and positive chirality antiparticle states.

The Pi+ has a Jp state of 0- and, hence, its decay products also have to have a combined J of 0. Their spins have to anti-align. Now the neutrinos is produced in a negative chirality state and as it is massless it also has negative helicity (left handed). As we need the antilepton's and the neutrino's spin to anti-align, it also needs to be left handed (negative helicity). However, the weak force only couples to positive chirality antiparticles. Therefore, the electron has to be boosted in order to turn a positive chirality into a negative helicity. It the lepton was massless, this boost wouldn't be possible and the decay would be forbidden.

The muon is a lot heavier than the electron, so this boosting is a lot harder. Hence, Pi+ -> v_mu mu+ happens more often than Pi+ -> v_e e+.