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Weyl, Dirac, and Majorana neutrinos

A Weyl two-component spinor is a left ($L$)-handed1 particle state, $\psi_L$, which is necessarily associated by CPT with a right ($R$)-handed antiparticle state2 $\psi^c_R$. One refers to active (or ordinary) neutrinos as left-handed neutrinos which transform as $SU(2)$ doublets with a charged lepton partner. They therefore have normal weak interactions, as do their right-handed anti-lepton partners,

\left( \begin{array}{c} \nu_e \\ e^- \end{array} \ri...
...left( \begin{array}{c} e^+ \\ \nu^c_e \end{array} \right)_R$}. \end{displaymath} (1)

Sterile3 neutrinos are $SU(2)$-singlet neutrinos, which can be added to the standard model and are predicted in most extensions. They have no ordinary weak interactions except those induced by mixing with active neutrinos. It is usually convenient to define the $R$ state as the particle and the related $L$ anti-state as the antiparticle.
\begin{displaymath}N_R \stackrel{\rm CPT}{\longleftrightarrow} N^c_L. \end{displaymath} (2)

(Sterile neutrinos will sometimes also be denoted $\nu_s$.)

Mass terms describe transitions between right ($R$) and left ($L$)-handed states. A Dirac mass term, which conserves lepton number, involves transitions between two distinct Weyl neutrinos $\nu_L$ and $N_R$:

- L_{\rm Dirac} = m_D (\bar{\nu}_L N_R +
\bar{N}_R \nu_L)
= m_D \bar{\nu} \nu, \end{displaymath} (3)

where the Dirac field is defined as $\nu \equiv \nu_L + N_R$. Thus a Dirac neutrino has four components $ \nu_L, \; \nu_R^c, \; N_R, \;
N_L^c$, and the mass term allows a conserved lepton number $L = L_\nu
+ L_N$. This and other types of mass terms can easily be generalized to three or more families, in which case the masses become matrices. The charged current transitions then involve a leptonic mixing matrix (analogous to the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix), which can lead to neutrino oscillations between the light neutrinos.

For an ordinary Dirac neutrino the $\nu_L$ is active and the $N_R$ is sterile. The transition is $\Delta I = \frac{1}{2}$, where $I$ is the weak isospin. The mass requires $SU(2)$ breaking and is generated by a Yukawa coupling

\begin{displaymath}-L_{\rm Yukawa} = h_\nu (\bar{\nu}_e \bar{e})_L
\left( \begin{array}{c} \varphi^0 \\ \varphi^- \end{array} \right)
N_R + H.C. \end{displaymath} (4)

One has $m_D = h_\nu v/\sqrt{2}$, where the vacuum expectation value (VEV) of the Higgs doublet is $ v = \sqrt{2} \langle \varphi^o \rangle = ( \sqrt{2} G_F)^{-1/2} =
246$ GeV, and $h_\nu$ is the Yukawa coupling. A Dirac mass is just like the quark and charged lepton masses, but that leads to the question of why it is so small: one requires $h_{\nu_e} < 10^{-11}$ to have $m_{\nu_e} < 1$ eV.

A Majorana mass, which violates lepton number by two units $(\Delta L
= \pm 2)$, makes use of the right-handed antineutrino, $\nu^c_R$, rather than a separate Weyl neutrino. It is a transition from an antineutrino into a neutrino. Equivalently, it can be viewed as the creation or annihilation of two neutrinos, and if present it can therefore lead to neutrinoless double beta decay. The form of a Majorana mass term is

$\displaystyle - L_{\rm Majorana}$ $\textstyle =$ $\displaystyle \frac{1}{2} m_T (\bar{\nu}_L \nu_R^c +
\bar{\nu}^c_R \nu_L ) = \frac{1}{2} m_T \bar{\nu} \nu$  
  $\textstyle =$ $\displaystyle \frac{1}{2} m_T (\bar{\nu}_L C \bar{\nu}_L^T + H.C.),$ (5)

where $\nu = \nu_L +\nu^c_R$ is a self-conjugate two-component state satisfying $\nu = \nu^c = C \bar{\nu}^T$, where $C$ is the charge conjugation matrix. If $\nu_L$ is active then $\Delta I = 1$ and $m_T$ must be generated by either an elementary Higgs triplet or by an effective operator involving two Higgs doublets arranged to transform as a triplet.

One can also have a Majorana mass term

\begin{displaymath}- L_{\rm Majorana} = \frac{1}{2} m_N (\bar{N}^c_L N_R +
\bar{N}_R N^c_L ) \end{displaymath} (6)

for a sterile neutrino. This has $\Delta I = 0$ and thus can be generated by the VEV of a Higgs singlet4.

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Paul Langacker 2001-09-27