I want to deduce the Feynman rules for neutrino magnetic moment, and there are Dirac and Majorana terms for this

$$\mathcal{L}_D=\mu_{ij}\bar{\nu}_{iL}\sigma_{\alpha\beta}\ \nu_{jR}F^{\alpha\beta}$$

$$\mathcal{L}_M=\mu_{ij}\bar{\nu}_i^c\sigma_{\alpha\beta}\ \nu_j F^{\alpha\beta}$$

I can get the Feynman rules for Dirac case using field theory techniques but I don't see how knowing that the Majorana field has the condition $\nu^c=\nu$ can change the results?!

any good refs on this?