# Neutrinos and anti-neutrinos in the Standard Model

In standard model neutrinos and the left handed electron forms SU(2) doublet.

1. What about the anti-neutrinos in the standard model? Do they also form some doublet?

2. If neutrinos have tiny masses will it not imply indirectly and conclusively that right-handed neutrinos must exist in nature?

EDIT : Neutrinos will have Majorana mass term if they are Majorana fermion. Is that right? Now, if neutrinos are Majorana fermions, will they have definite handedness? For example, does $\nu_M=\begin{pmatrix}\nu_L\\ i\sigma^2\nu_L^*\end{pmatrix}$ have definite handedness? Therefore, doesn't it imply that if neutrinos are massive then a right-handed component of it $\begin{pmatrix} 0\\ i\sigma^2\nu_L^*\end{pmatrix}$ must exist? Although we are not using $\nu_R$ to construct this column, does it imply $\nu_M$ do not have a right handed component? It is the column $\nu_M$ which we should call a neutrino. Then it has both the components. However, one can say that a purely right-handed neutrino need not exist if the neutrino is a Majorana fermion. Therefore, it seems that if neutrinos are massive a right handed component of it must exist (be it a Dirac particle as well as a Majorana particle). Correct me if I am wrong.

The antineutrinos do indeed form a doublet. The particle-antiparticle conjugation operator is usually denoted by $\hat{C}$ and is defined through: \begin{equation} \hat{ C}: \psi \rightarrow \psi ^c = C \bar{\psi} ^T \end{equation} where $C \equiv i \gamma _2 \gamma _0$. So given a neutrino you can always get its complex conjugate with this operator: \begin{equation} \nu _L ^{\,\,c } = i \gamma _2 \gamma _0 ( \overline{\nu _L} ) ^T \end{equation} Its easy to check this that this antineutrino is actually right handed, by applying a left projector onto it.
With regards to your second question, no having neutrino masses does not imply that there exist right handed neutrinos. This is because neutrinos could have Majorana masses ($\frac{m}{2} \nu _L \nu _L +h.c.$) as well as Dirac masses $m( \overline{\nu_L} \nu_R + h.c.)$. Majorana masses could arise if for example there exists a heavy Higgs which is a triplet under $SU(2)_L$ (which can be rise to what's known as a type 2 See-saw mechanism).
• Neutrinos will have Majorana mass term if they are Majorana fermion. Is that right? Now, if neutrinos are Majorana fermions, will they have definite handedness? For example, does $\nu_M=\begin{pmatrix}\nu_L\\ i\sigma^2\nu_L^*\end{pmatrix}$ have definite handedness? Therefore, doesn't it imply that if neutrinos are massive then a right-handed component of it must exist? Although we are not using $\nu_R$ to construct this column, does it imply $\nu_R$ do not have a right handed component? The column is what we should call a neutrino. Then it has both the components. Correct me if I am wrong.
• I'm not sure about your Majorana mass term, shouldn't it read $\bar\nu_L \nu_R^c$? Which for active, ordinary left-handeded neutrinos, could only originate from a triplet? For a sterile, we can write such a term straight away. Mar 14 '14 at 19:38
• It is Majorana since $\nu = \nu_L + \nu_R^c = \nu^c$ Mar 14 '14 at 19:40