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Дата изменения: Wed Aug 9 20:40:47 2000
Дата индексирования: Mon Oct 1 22:51:01 2012
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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 (.19)¹:

$$L_\nu = \frac{i}{2}(\bar{\Psi_\nu} \gamma_\mu \partial^\mu \Psi_\nu - (\p... ...ta_w} \cos{\Theta_w} } Z_\mu \bar{\Psi}_\nu \gamma^\mu ( 1 - \gamma^5) \Psi_\nu$$

$$+ \frac{e}{2\sqrt{2}\sin{\Theta_w} } \left( W^-_\mu \bar{\Psi}_e ... ...^5)\Psi_\nu + W^+_\mu \bar{\Psi}_\nu \gamma^\mu (1 - \gamma^5)\Psi_e \right)\;,$$ (13)

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

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

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

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