Right handed neutrino identity

[lalo_u]

I am reading Mohapatra's book: "Massive Neutrinos in Physics and Astrophysics". At the beginning of chapter 7, it is sought expressions where the right neutrino was considered in the Electroweak Standard Model.
Everything was fine until I found the expression [itex]\overline{N^c_{l'L}}\nu^c_{lR}=\overline{\nu_{lL}}N_{l'R}[/itex].
Where [itex]N_{l'R}[/itex] is the right handed field associated with right handed neutrinos and the subsctipts [itex]l,l'[/itex] indicate the lepton flavors.

Well, I'm trying to prove this, but I get stuck on the following expression:

[itex]\overline{N^c_{l'L}}\nu^c_{lR} =\overline{\left(N_{l'L}\right)^c}\left(\nu_{lR}\right)^c
= \overline{N_{l'R}}\nu_{lL}¿?\overline{\nu_{lL}}N_{l'R}[/itex]
And I'm assuming that they are Majorana neutrinos.

To complete the test I should justify why the conjugate for the last expression can be taken and remain unchanged, someone could help?

 

[andrien]

It seems what you have wriiten is correct.Those fermionic fields have anticommuting properties.Try to do an explicit calculation with a two component spinor.

 

[lalo_u]

OK Andrien, but if i only take an anticommuting propetry, the fields change places (wih some minus sign). The question is why i can take the conjugate for the last expression and remains tha same...

 

[andrien]

just think about if N_lR is the right handed neutrino(v_lR),then majorana mass term will look like v_R^-v_l or equivalently you can write because majorana mass term will have only left handed and express the right handed part by a charge conjugation.So it will look like v_l^-cv_l and then you can write it something like x^Tεx(because of realness of majorana spinor,x denotes the two component spinors),so if you take the conjugate of the expression.It is same.you can extend this argument or if you want to verify then you can take a two component spinor like (x₁ x₂) and notice the anticommuting property of grassmann variable along with the relation of dirac conjugate to hermitian conjugate to obtain the result.

 

[lalo_u]

I got it, Andrien. I had shown only for the full Majorana fermion, and not for any of their helicities that was what I wanted. I considered its components, as you suggested, and I did.

Thank you.