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A ##SU(2)## doublets, Majorana Fermions and Higgs

  1. Jan 1, 2017 #1
    Say ##L## and ##L^{c}## are a pair of ##SU(2)## doublets (electroweak-charge fermions) and ##N_{1}## and ##N_{1}^{c}## are a pair of neutral Majorana fermions.

    Say that these fermions couple to the Higgs via Yukawa coupling and have vector masses ##M_0## and ##M_1## respectively:

    $$M_{0}LL^{c} + M_{1}N_{1}N_{1}^{c} + YHLN_{1}^{c} + Y^{c}H^{\dagger}L^{c}N_{1}$$


    What is the difference between ##L## and ##L^{c}##?

    What does the superscript ##c## signify?
     
  2. jcsd
  3. Jan 1, 2017 #2

    Orodruin

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    Charge conjugation. Your ##LL^c## term breaks SU(2) gauge invariance.
     
  4. Jan 1, 2017 #3
    How does the ##LL^{c}## term break ##SU(2)## gauge invariance?
     
  5. Jan 1, 2017 #4

    Orodruin

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    I really meant to say SU(2)xU(1). You are missing a number of bars on your fermion fields. The term ##\bar L L^c## is not a hypercharge singlet because ##\bar L## and ##L^c## have the same hypercharge.
     
  6. Jan 1, 2017 #5
    But, ##L## is the complex conjugate of ##L^{c}##. So, isn't ##LL^{c}## a scalar?

    Why then do we need to have ##\bar{L}L^{c}##?
     
  7. Jan 1, 2017 #6

    Orodruin

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    Without the bar your expression is not Lorentz invariant.
     
  8. Jan 1, 2017 #7
    The Lagrangian is taken from equation (1.1) in page 2 of the article in the link https://arxiv.org/abs/1609.06320.

    In the article, there is no bar on ##L##. What am I getting wrong here?
     
  9. Jan 1, 2017 #8

    Orodruin

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    They are being sloppy. Any specialist reading that is going to understand what they mean.
     
  10. Jan 1, 2017 #9
    Okay, in the Dirac Lagrangian, it is possible to have the mass term ##m\bar{\psi}\psi##.

    So, why can't we have the term ##M_{0}\bar{L}L## and not ##M_{0}\bar{L}L^{c}## here?
     
  11. Jan 1, 2017 #10

    Orodruin

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    The first one because you cannot have a mass term involving two left-handed fields. The second because it violates gauge invariance.
     
  12. Jan 2, 2017 #11
    So, let me get this right:

    the correct term is ##M_{0}LL^{c}## and not ##M_{0}\bar{L}L^{c}##?
     
  13. Jan 3, 2017 #12
    Also, why is ##M_0## called the vector mass and not simply the mass?
     
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