What is the relationship between scattering and helicity in neutrino collisions?

[goronx]

Hi, I have a problem involving helicity.

Homework Statement

In a scattering
$$e^{+}+e^{-} \rightarrow \nu_{\mu}+\overline{\nu}_{\mu}$$
I have to determine for which values of the helicity of initial particles the cross section is not 0.

Homework Equations

The Attempt at a Solution

On a first moment I thought to use the conservation of angular momentum, but the boson involved is either a photon or a Z: they have unitary spin, so that I have to find an other way.

I may say the the neutrino is a massless particle described by a left-handed Weyl field, but I don't know if it is still valid in the case of a weak neutral current.

 

[sgd37]

I don't think the standard model has adjusted for small mass neutrinos and still considers them massless

 

[goronx]

This is an extract from my book:

If we neglect the small masses indicated by the oscillations experiments, the neutrinos in the Standard Model are described by massless left-handed Weyl fields. The neutrino has h =−1/2 and the antineutrino has h =+1/2.

Anyway, if you know an other way to solve the problem you can tell me.

 

[sgd37]

well the helicity of the photon is +/-1 so i would presume to conserve that

 

[goronx]

Ok, let's say that both neutrino and anti-neutrino have fixed helicity (respectively -1 and +1), what's the next step?
I thought that, because the have fixed and opposite helicity, they have also the same spin, but I don't know what to do now.

Is it important if the propagator is a photon (spin 1) or an Higgs boson (spin 0)?

 

[sgd37]

well it doesn't matter for a massive neutrino field and does matter for a massless neutrino field since that can only have one possible helicity combination

 

[goronx]

In this case they are considered as massless particles, so their helicity is fixed.
As next step, I thought that the starting point is to think them in the center of mass system, then I can represent neutrino and antineutrino with spins into the same direction because of what we said about helicity.

But now I have a Z or an Higgs' boson and they have different spins: how can I deduce something on inital particles' spins?
Maybe I have to say how they are related with neutrino's and anti-neutrino's spins?

 

[sgd37]

sorry what next step and why are you considering the bosons, just conserve helicity

 

[goronx]

The problem is that I don't think that exists something like the conservation of helicity.

 

[sgd37]

right so pair annihilation producing two photons has nothing to do with helicity being conserved

 

[goronx]

Why are you talking about two photons?

Sorry, but may I ask you to be more explicit on this argument? Really, I'm trying to solve this "simple" problem since 1 week, if I had any idea regarding how to proceed I would have done by miself. Thank you.

1- I can't find nothing about conservation of helicity and, if such a thing exists, I could say that the initial particles have different helicity but I don't know which one is + or -.
2-I would like to use angular momentum conservation, but it does not seem to be the right situation.

 

[vela]

On a first moment I thought to use the conservation of angular momentum, but the boson involved is either a photon or a Z: they have unitary spin, so that I have to find an other way.

Since neutrinos are involved, it's a weak interaction, so the process isn't mediated by a photon. It's either the W or the Z.

 

[goronx]

Ok, I did not think about it, thanks.

 

[sgd37]

Well just the Z in order to conserve charge. Just go to the COM mass frame and conserve angular momentum. And how can you not have helicity conservation if you have to have momentum conservation and angular momentum conservation. I don't like this question for it's many unknowns but if they are massless then that reaction can't happen since essentially you have a helicity +/-1 boson spitting out a system with 0 helicity. There exist other possibilities but those have loops in them. But if we ignore the middle bit then we can just simply say that you have to have zero helicity on the left as well

 

[vela]

You can draw a Feynman diagram of the process with a W.

 

[goronx]

Well just the Z in order to conserve charge. Just go to the COM mass frame and conserve angular momentum. And how can you not have helicity conservation if you have to have momentum conservation and angular momentum conservation.

Maybe I'm wrong (and it is my mistake), but I thought that having a boson with unitary spin (such as Z) is a problem for the conservation of angular momentum.

Considering the two neutrinos with opposed helicity (particle and antiparticle), they must have spin directed into the same direction, conventionally the z axis.
But, in a Feynman diagram, there's the Z boson and the conservation of angular momentum is valid on the z axis.

In this case I have 3 possible states
$$\left|1,1\right\rangle=\left|+;+\right\rangle$$
$$\left|1,-1\right\rangle=\left|-;-\right\rangle$$
$$\left|1,0\right\rangle=\frac{1}{\sqrt{2}}(\left|+;-\right\rangle+\left|-;+\right\rangle)$$
and the last one has spins directed in opposite directions so, if I'm not making a mistake, it's not possible to use the conservation of angular momentum when the propagator boson has unitary spin.

I don't like this question for it's many unknowns but if they are massless then that reaction can't happen since essentially you have a helicity +/-1 boson spitting out a system with 0 helicity. There exist other possibilities but those have loops in them. But if we ignore the middle bit then we can just simply say that you have to have zero helicity on the left as well

People told me that it is possible without considering loops.

 

[vela]

In this case I have 3 possible states
$$\left|1,1\right\rangle=\left|+;+\right\rangle$$
$$\left|1,-1\right\rangle=\left|-;-\right\rangle$$
$$\left|1,0\right\rangle=\frac{1}{\sqrt{2}}(\left|+;-\right\rangle+\left|-;+\right\rangle)$$
and the last one has spins directed in opposite directions so, if I'm not making a mistake, it's not possible to use the conservation of angular momentum when the propagator boson has unitary spin.

How did you jump from a Z can be in a |1 0> state to the Z can't mediate the interaction? The Z could be in one of the other states, right?

 

[sgd37]

I don't understand what your problem is. You can't create an integral spin system from just one half spin particle (without susy) but you have two here and that has j = 1 the same as the boson. I think you may be confusing j and m eigenvalues or something else but a photon with j=1 can still have an m = 0 state. What exactly do you mean by

"and the last one has spins directed in opposite directions so, if I'm not making a mistake, it's not possible to use the conservation of angular momentum when the propagator boson has unitary spin"

 

[goronx]

The only certain thing is that I'm confused. :tongue:

Let's fix this first doubt.
I have this:
$$\left|\uparrow,\downarrow\right\rangle=\left|0;0\right\rangle+\left|1;0\right\rangle$$
Does it mean that if I have two particles with opposite spins I can be or in a singulet state or in a triplet one?
On the contrary, if I have two parallel spins I'm certainly in a triplet state, right?

 

[sgd37]

well there should be some normalization coefficients but yes that is true, you have a 50 percent chance of being in the j=0 or in the m= 0 of the j=1 . You also have

$$\left| \downarrow \uparrow \right \rangle = \frac {1}{\sqrt{2}} ( \left|1;0\right\rangle - \left|0;0\right\rangle )$$

 

[vela]

The only certain thing is that I'm confused. :tongue:

Let's fix this first doubt.
I have this:
$$\left|\uparrow,\downarrow\right\rangle=\left|0;0\right\rangle+\left|1;0\right\rangle$$
Does it mean that if I have two particles with opposite spins I can be or in a singulet state or in a triplet one?
On the contrary, if I have two parallel spins I'm certainly in a triplet state, right?

Right. If the spins are aligned, you know the total z-component is +1 or -1, so the total angular momentum has to be s=1, i.e. you're in a triplet state. If the spins are opposite, then that's a linear combination of the triplet |1 0> and singlet |0 0> states.

 

[goronx]

Ok, thank you both.

So, neutrino and anti-neutrino have parallel spins, which means that I'm in a triplet state.
The Z has spin=1 so I can use the conservation of angular momentum, there's no problem with it, right?

This implies that also initial particles must have parallel spins and, because they are again a particle and an antiparticle, they have opposed helicity for sure.
But how to decide which one has helcity +1 or -1, positron or electron?
The exercise does not tell anything about being or not in the massless limit, there must be an additional consideration to do.