Question about the Majorana mass term

[phypar]

The Majorana mass term is expressed from a single Weyl spinor. But I am a little confused by the expression. For example, see Eq. (2) in http://arxiv.org/pdf/hep-ph/0410370v2.pdf

$\mathcal{L}=\frac{1}{2}m(\chi^T\epsilon \chi+h.c.)$

Here $\chi$ is the Weyl spinor and $\epsilon = i\sigma^2$ is the antisymmetric tensor.

But when I do a simple calculation:
$\chi^T\epsilon \chi =(\chi^T\epsilon \chi)^T = \chi^T(-\epsilon) \chi$

Here I used $\epsilon^T = -\epsilon$

therefore $\chi^T\epsilon \chi =0$

So can anyone tell me what is wrong here? What is missing here? Thanks a lot

 

[andrien]

you will get an extra minus sign while taking the hemitian conjugate.It is special with σ₂.try with a two component spinor of (a b) form explicitly.

 

[Bill_K]

I don't see any σ₂ in the expression he's given.

It's just χ^Tεχ = χ₁χ₂ - χ₂χ₁ ≡ 0.

 

[andrien]

actually that does not vanish.

 

[phypar]

Thanks for all the replies. I found the mistake in my calculation. $\chi_1$ and $\chi_2$ are Grassman variables, thus satisfy the anti-commutation relation, which means $\chi_1\chi_2-\chi_2\chi_1 \neq 0$

 

[Bill_K]

In that case, I'm interested in your solution to Exercise 1.4, which says it is zero.

 

[fzero]

In that case, I'm interested in your solution to Exercise 1.4, which says it is zero.

Because of the Grassmann statistics, $2\chi_1^a\chi_2^b=\epsilon^{\alpha\beta}\chi^a_\alpha\chi_\beta^b$ is symmetric in $a,b$.

 

[andrien]

A simple calculation yields x₁²,x₂² type term,which are zero.

 

[fzero]

A simple calculation yields x₁²,x₂² type term,which are zero.

That term cannot appear in the expression in exercise 1.4. What appears in the expression is an $\epsilon_{ab}$ for the internal indices: there's no 22-component!