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Question about the Majorana mass term

  1. Mar 11, 2013 #1
    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

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

    Here [itex]\chi[/itex] is the Weyl spinor and [itex]\epsilon = i\sigma^2 [/itex] is the antisymmetric tensor.

    But when I do a simple calculation:
    [itex]\chi^T\epsilon \chi =(\chi^T\epsilon \chi)^T = \chi^T(-\epsilon) \chi [/itex]

    Here I used [itex]\epsilon^T = -\epsilon[/itex]

    therefore [itex]\chi^T\epsilon \chi =0 [/itex]

    So can anyone tell me what is wrong here? What is missing here? Thanks a lot
     
    Last edited: Mar 11, 2013
  2. jcsd
  3. Mar 11, 2013 #2
    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.
     
  4. Mar 11, 2013 #3

    Bill_K

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

    It's just χTεχ = χ1χ2 - χ2χ1 ≡ 0.
     
  5. Mar 11, 2013 #4
    actually that does not vanish.
     
  6. Mar 12, 2013 #5
    Thanks for all the replies. I found the mistake in my calculation. [itex]\chi_1[/itex] and [itex]\chi_2[/itex] are Grassman variables, thus satisfy the anti-commutation relation, which means [itex]\chi_1\chi_2-\chi_2\chi_1 \neq 0[/itex]
     
  7. Mar 12, 2013 #6

    Bill_K

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    In that case, I'm interested in your solution to Exercise 1.4, which says it is zero. :confused:
     
  8. Mar 13, 2013 #7

    fzero

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    Because of the Grassmann statistics, [itex]2\chi_1^a\chi_2^b=\epsilon^{\alpha\beta}\chi^a_\alpha\chi_\beta^b[/itex] is symmetric in [itex]a,b[/itex].
     
  9. Mar 15, 2013 #8
    A simple calculation yields x12,x22 type term,which are zero.
     
  10. Mar 15, 2013 #9

    fzero

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