Dimension 5 operator for neutron dipole moment

Question

In the HW for an online effective field theory course that I'm working through the professor says that "With the full standard model gauge group it is not possible to write down a dimension five operator (which contributes to the neutron dipole moment) without breaking gauge symmetry". But what is wrong with  the operator,

$$i \bar{u} _R \sigma^{\mu\nu} u_R B _{\mu\nu} $$

for example (here I define $B_{\mu\nu} \equiv \partial _\mu B_\nu -  \partial _\nu B_\mu$, where $B_\mu$ is the hypercharge vector boson)? I think it should give an electric contribution since it forms the photon and $Z$ boson. 

Answer 1

It does not respect Lorentz symmetry. Specifically, you should have \(\bar{u}_R\sigma^{\mu\nu}u_L\) or \(\bar{u}_L\sigma^{\mu\nu}u_R\) to have Lorentz invariance, but then you lost \(SU(2)_L\) invariance.

Comments

Ah, yes of course. Thanks so much. I thought it would be something silly.