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

[spaghetti3451]

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?

 

[Orodruin]

Charge conjugation. Your $LL^c$ term breaks SU(2) gauge invariance.

 

[spaghetti3451]

How does the $LL^{c}$ term break $SU(2)$ gauge invariance?

 

[Orodruin]

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.

 

[spaghetti3451]

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.

But, $L$ is the complex conjugate of $L^{c}$. So, isn't $LL^{c}$ a scalar?

Why then do we need to have $\bar{L}L^{c}$?

 

[Orodruin]

Without the bar your expression is not Lorentz invariant.

 

[spaghetti3451]

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?

 

[Orodruin]

They are being sloppy. Any specialist reading that is going to understand what they mean.

 

[spaghetti3451]

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?

 

[Orodruin]

The first one because you cannot have a mass term involving two left-handed fields. The second because it violates gauge invariance.

 

[spaghetti3451]

So, let me get this right:

the correct term is $M_{0}LL^{c}$ and not $M_{0}\bar{L}L^{c}$?

 

[spaghetti3451]

Also, why is $M_0$ called the vector mass and not simply the mass?