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

1. Jan 1, 2017

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?

2. Jan 1, 2017

Orodruin

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

3. Jan 1, 2017

spaghetti3451

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

4. Jan 1, 2017

Orodruin

Staff Emeritus
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.

5. Jan 1, 2017

spaghetti3451

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

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

6. Jan 1, 2017

Orodruin

Staff Emeritus
Without the bar your expression is not Lorentz invariant.

7. Jan 1, 2017

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?

8. Jan 1, 2017

Orodruin

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

9. Jan 1, 2017

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?

10. Jan 1, 2017

Orodruin

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

11. Jan 2, 2017

spaghetti3451

So, let me get this right:

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

12. Jan 3, 2017

spaghetti3451

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