quantum field theory - Why does the weak force distinguish left and right handedness?

I'm wondering why the weak interaction only affects left-handed particles (and right-handed antiparticles).

Before someone says "because thats just the way nature is" :-), let me explain what I find needs an explanation:

In the limit of massless fermions, chirality (handedness) becomes helicity $(\vec S \cdot \hat p)$. Now, helicity is a property of the state of motion of an object in space. It is pretty unobvious to me how the internal symmetry $SU(2) \times U(1)$ would "know" about it, and be able to distinguish the two different helicity states of motion.

On a more technical level, IIRC, left and right handed spinors are distinguished by their transformation properties under certain space-time transformations, and are defined independent of any internal symmetry. If we want to get the observed V-A / parity violating behavior, we have to plug in a factor of $(1 - \gamma^5)$ explicitly into the Lagrangian.

Is there any reason this has to be like this? Why is there no force coupling only to right handed particles? Why is there no $(1 + \gamma^5)$ term? Maybe it exists at a more fundamental level, but this symmetry is broken?

Answer

I think you are sort of reversing the logic of chirality and helicity in the massless limit. Chirality defines which representation of the lorentz group your Weyl spinors transform in. It doesn't 'become' helicity, helicity 'becomes' chirality in the massless limit. That is, chirality is what it is, and it defines a representation of a group and that can't change. This other thing we have defined called helicity just happens to be the same thing in a particular limit.

Now once you take the massless limit the Weyl fermions are traveling at the speed of light you can no longer boost to a frame that switches the helicity. I think its best to think of a fermion mass term as an interaction in this case and remember that the massive term of a Dirac fermion is a bunch of left and right- handed Weyl guys bumping up into one another along the way. Conversely if you want to talk about a full massive Dirac fermion that travels less than c and you can boost to change the helicity, but that full Dirac fermion isn't the thing carrying weak charge, only a 'piece' of it is.

See this blog post on helicity and chirality.

As far as the left-right symmetry being broken people have certainly built models along these lines but I don't think they have worked out.

Does this answer your question?