The following term appeared in a Lagrangian I found

$$\bar{\psi}\sigma^{\mu \nu}\psi F_{\mu \nu}.$$

I have never seen this term before, or even the $\bar{\psi}\sigma^{\mu \nu}\psi$ tensor. I was wondering what the physical interpretation of these terms are. It appears to be a coupling between the fields and the particles but it is different from previous Lagrangians terms of this form that I have seen previously.

  • $\begingroup$ Although I could guess it, it's always good to define your notation - what is $\sigma^{\mu\nu}$? $\endgroup$ – ACuriousMind Nov 16 '16 at 16:10
  • $\begingroup$ It's the commutator of the gamma matrices $[\gamma^{\mu}\gamma^{\nu}]$ $\endgroup$ – T-Ray Nov 16 '16 at 19:33

Well it is still a vertex with three legs: two for the fermion field and one for the photon field. However in this case it is a derivative coupling, because it connect with the derivative of the photon field. For this reason it has the "wrong" dimension (at least for the normal case of 4D space-time and the usually mass dimensions of these fields). Therefore, it would need a dimension parameter as coefficient to correct the dimension. Probably a term one would only find in an effective field theory.

Other than this it would also have different symmetry properties that would distinguish it from the usual gauge interaction.


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