How Does the Dirac Conjugation Operator Affect Majorana Neutrino Mass Terms?

[rkrsnan]

In books I find that the Majorana mass term for the neutrinos is given by $$m_L \nu_L^T C^\dagger \nu$$ where C is Dirac Conjugation operator. How does C look like if I write $$\nu_L$$ as in terms of its two components $$\left(\begin{array} (\nu_{L1} \\ \nu_{L2} \end{array} \right)$$?

Is $$C= (i\sigma^2) =\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right)$$?

Thanks for your help!

 

[Physics Monkey]

Yes, that looks ok to me. You can read about this in the Dirac chapter of Peskin, there is even a problem about it as I recall.

 

[rkrsnan]

Thanks, the expression is correct. I was confused earlier because when I expand it I get $$\nu_1 \nu_2 - \nu_2 \nu_1$$ which I thought is zero. Then it didn't occur to me that the fields are fermionic and they anticommute.