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Neutrino as a Majorana fermion Leptoquarks Examples 4-gluon
 interaction Contents

Neutrino as a Majorana fermion

It is well known that in the Standard Model only the left component of massless neutrino takes part in interactions. So one can describe neutrino by a Majorana field which has the same number of degrees of freedom as a left Dirac one. To realize such a particle in the framework of CompHEPone should add the following record to the table of particles:


Full name A A+ 2*spin mass width color aux
neutrino MN MN 1 0 0 1  



In terms of Dirac field $\Psi_\nu$ a neutrino appears in the Standard Model Lagrangian in the following way (Lagrangian of electroweak interactions.19)1:

$\displaystyle L_\nu$ $\textstyle =$ $\displaystyle \frac{i}{2}(\bar{\Psi_\nu} \gamma_\mu \partial^\mu \Psi_\nu -
(\p...
...ta_w} \cos{\Theta_w} } Z_\mu
\bar{\Psi}_\nu \gamma^\mu ( 1 - \gamma^5) \Psi_\nu$  
    $\displaystyle + \frac{e}{2\sqrt{2}\sin{\Theta_w} }
\left( W^-_\mu \bar{\Psi}_e ...
...^5)\Psi_\nu
+ W^+_\mu \bar{\Psi}_\nu \gamma^\mu (1 - \gamma^5)\Psi_e \right)\;,$ (13)

where $\Psi_e$ is the electron field. To rewrite it in terms of a Majorana fermion let us perform the substitution

\begin{displaymath}\Psi_\nu = \frac{1}{2}(1-\gamma^5) \psi_l + \frac{1}{2}(1+\gamma^5)
\psi_r\;,
\end{displaymath}

where $\psi_l$ and $\psi_r$ are Majorana fermions. Omitting Lagrangian for $\psi_r$ and applying the following identities for Majorana fermions

\begin{eqnarray*}
\frac{i}{4}(\bar{\psi} \gamma_\mu \gamma^5 \partial^\mu \psi -...
...psi})\gamma_\mu \gamma^5 \psi) = \bar{\psi} \gamma^\mu \psi &=&0
\end{eqnarray*}



which can be obtained by means of (11), we get

\begin{eqnarray*}
L_\nu &=& \frac{i}{4}(\bar{\psi}_l \gamma_\mu \partial^\mu \ps...
...{\psi}_l \gamma^\mu (1 - \gamma^5)\Psi_e \right)\;. \nonumber\\
\end{eqnarray*}



Here the first term is the free Lagrangian for a massless Majorana fermion. Other terms define the interaction. Using the definition (8) we can rewrite them in the CompHEP  notations:


A1 A2 A3 A4 Factor Lorentz part
MN MN Z   -EE/(2*SW*CW) G(m3)*G5
E1 MN W-   EE/(2*Sqrt2*SW) G(m3)*(1-G5)
MN e1 W+   EE/(2*Sqrt2*SW) G(m3)*(1-G5)



Let us emphasize that there are two identical neutrino fields in that term of the Lagrangian which describes the interaction of neutrinos with a $Z$-boson. It leads to the additional factor $2$ and to the symmetry property of the corresponding vertex. One of the typical mistakes in realization of such a vertex is an introduction of the G(m3)*(1-G5) term which breaks the symmetry property. Correct evaluation of the functional derivative (8) with the help of the identity (11) never produces such a term.



Footnote

where $Y=-1$, $\Psi_1=\Psi_\nu$, $\Psi_2=\Psi_e$, $g_2=e/\sin{\Theta_w}$, $g_1=e/\cos{\Theta_w}$

Leptoquarks Examples 4-gluon
 interaction Contents