Chirality is not conserved in propagation (e.g., electron oscillates between left and right chiral). But is chirality conserved in interactions? What is a good reference paper or book for this notion of chiral conservation (or not) in interactions?

As an example, in QED Bhabha scattering say a left-chiral electron and a right-chiral positron interact, both being annihilated and thereby creating a virtual photon which after propagation an electron and positron are then created. Is the newly created electron required to be left-chiral (as opposed to right-chiral or a superposition of chiralities)? Likewise for the positron and its right-chirality?

As a side note, I'm not concerned about how we know the initial electron and positron are purely left and right chiral, respectively. As a theoretical question, I'm presuming it.


1 Answer 1


Any good QFT book would spend lots of space on vector-like (~nonchiral) interactions such as QED & QCD, and then contrast them to chiral ones for the weak interactions. I gather you are cool with the WP sample calculation.

The basic vertices are equal mixtures of both chiralities, $$ eA_\mu \overline \psi \gamma^\mu \psi = e A_\mu \overline \psi _R \gamma^\mu \psi_L +eA_\mu \overline \psi_L \gamma^\mu \psi_R \\ = e A_\mu \overline {\psi _L} ~\gamma^\mu \psi_L +eA_\mu \overline {\psi_R} ~\gamma^\mu \psi_R , $$ so in the t-channel (exchange) an electron or positron emitting or absorbing a virtual photon preserves its chirality.

In the s-channel (annihilation & creation), as you say, the annihilants have to have opposite chiralities, and the products as well. The two types of diagrams add and interfere. The virtual photons are off-mass-shell, so like massive.

The completeness relations $ \sum_{s=1,2}{u^{(s)}_p \bar{u}^{(s)}_p} = {p\!\!/} + m $ involve the fermion masses, so contaminate/violate chiralities, in principle, proportionally to m, which you monitor doing your traces.

(If you used helicity amps, instead, you'd note that in the center of mass of the annihilants, the spin-1 rotation matrices would favor emission of the creation products in the direction of the original helicities, in a telltale $1+\cos\theta$ distribution. So at right angle emission, you've lost memory of the original helicities, and basically also chiralities, since helicity and chirality are mismatched by m/p.)


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