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I am struggling to understand the meaning of a term in a Lagrangian of a paper I am reading. I think it is a notation issue. The term is:

$$\lambda_e \bar{\nu}_L^c l_L \Sigma^{\dagger}_e$$

where $\Sigma$ is an extra scalar added to the SM and in this case, it should be a singlet under $SU(2)$ [from the text of the paper]. I do not understand what $\bar{\nu}_L^c$ is then. Cause to be $SU(2)$ invariant, I would need 2 doublets with opposite transformation under the symmetry. So is $\bar{\nu}_L^c$ a doublet? I am quite sure $l_L$ is the lepton doublet $(\nu_L,e_L)$.

And if $\Sigma$ is a $SU(2)$ singlet, why do we have the dagger and not just ${}^*$?

https://arxiv.org/abs/1305.6587, eq (2.6)

Urb
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TheoPhy
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    Indeed, the dagger is a sloppy affectation: it means star, complex conjugate. The doublets saturated then are $l_L$ and $\bar\nu^c_L$, a right chiral field which is also an SU(2) doublet. The term is then Lorentz invariant. Read up on conjugate spinors and Majorana masses. – Cosmas Zachos Dec 14 '20 at 18:49
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    I do not understand how the $\bar{\nu}_L^c$ doublet is composed. I see that it can be a right-handed neutrino but how come is it a doublet? Why would I use this notation for a doublet. I would write something like $\bar{l}^c_L$ beeing ($\bar{\nu}_L^c$, $\bar{e}_L^c$) – TheoPhy Dec 14 '20 at 23:03
  • You have a good point. I would assume it is the conjugate rep of the very same multiplet, group theoretically, and they use a different name to remind you of this up-down transposition. This way, a singlet positive scalar balances charge versus each component in the dot product. They appear very sloppy. – Cosmas Zachos Dec 14 '20 at 23:47

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