How can we measure chirality in experiments?

Question

Chirality is a concept quite different from helicity. These two concepts only happen to have the same numerical value for massless particles. I understand that we can measure helicity, but how can we determine chirality for massive particles?

Some thoughts:

• Helicity is the projection of spin in the direction of momentum $h= \frac{ \hat p \hat s}{p}$. For massive particles helicity is not Lorentz invariant (we can boost in a frame where the particle moves in the other direction), but conserved.
• Chirality is a concept from representation theory that tells us how a given object transforms under Lorentz transformations. Chirality is Lorentz invariant, but not conserved. Something is left-chiral if it is described by an object transforming according to the $(\frac{1}{2}, 0)$ representation of the Lorentz group (a left-chiral spinor). Analogous right-chiral is defined as transforming according to the $(0,\frac{1}{2})$ representation of the Lorentz group. We need to describe particles/fields by Dirac spinors, because left-chiral particles tranform into right-chiral particles and vice versa as time moves on. The chirality operator for Dirac spinors is $\gamma_5$.

Therefore these two concepts are really different at first sight. The only time I know that chirality was "measured" was when parity violation was discovered.

Parity is violated, because only left-chiral fields interact via the weak force. This was discovered by the famous Wu experiment, which discovered that neutrinos are always left-handed. At this time neutrinos were considered to be massless and therefore it was concluded that only left-chiral neutrinos interact weakly. (Recall that for massless particles we have left-chiral=left-handed). In how far is this line of thought still valid, although neutrinos have mass?

Answer 1

In theory, it is possible to measure chirality using the exact method that you mentioned. If you have an electron and you are unsure its chirality, you can send it through a dense bucket of $W$ bosons and see if it interacts with them. If it did, then you know that you had a left handed electron.

There is, however, an important issue with this idea (and no its not the issue of figuring out how to produce a bucket of $W$ bosons). The problem is that the electron can change chirality and back again so even if you measured its chirality one moment, it doesn't mean that it had a same chirality a little time earlier.

One way to avoid this problem, is to accelerate the electrons. In this case the time it takes an electron to flip its chirality is $\sim 1/E $. This is easily seen using ordinary QM: \begin{equation} P(e_L \rightarrow e_R) = \left| \left\langle e _L \right| \exp \left( - i H t \right) \left| e _R \right\rangle \right| ^2 = \sin^2(- E t ) \end{equation} where $ H = \gamma M = \gamma \left( \begin{array}{cc} 0 & m _e \\ m _e & 0 \end{array} \right) $, $\left| e _L \right\rangle = ( 1 , 0 ) ^T , \left| e _R \right\rangle = ( 0, 1 ) ^T$, and $ E = \gamma m _e $.

If $1/E\gg $ the time it takes the electrons to pass through the apparatus then you can confidently say that the electrons you started with are the ones you measured and so you can measure its chirality.

Notice that this idea relies on going to the high energy limit, where chirality=helicity, so it may not be exactly what you were looking for. But while the two quantum numbers are seemingly completely different, our ability to measure chirality is closely linked to helicity.

While the above idea was quite hypothetical, you could in principle produce high energy chiral eigenstate electrons and collide them with positrons. Then you could measure the products of the collisions and infer if the initial states were indeed left or right chiral.

Lastly, the property of neutrino masses does not change whether or not (hypothetical) right handed neutrinos can interact weakly. Due to the necessary quantum numbers of the right handed neutrinos they would have to be standard model singlets. For this reason, they cannot interact with the weak force.