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\begin{document}
\Title{Parameters of the Neutrino sector in tau decays}
\bigskip\bigskip
%+\addtocontents{toc}{{\it D. Reggiano}}
%+\label{ReggianoStart}
\begin{raggedright}
{\it Darius Jur\v{c}iukonis, Thomas Gajdosik, Andrius Juodagalvis, Tomas Sabonis \index{Jur\v{c}iukonis D.}\\
\bigskip
Vilnius University\\
Universiteto St 3, LT-01513 Vilnius,\\ LITHUANIA}
\bigskip\bigskip
\end{raggedright}
\section{Introduction}
The smallness of the neutrino masses can be well understood within the see-saw \index{see-saw}
mechanism (type I). After spontaneous symmetry breaking of the
Standard Model gauge group one obtains a $(n_L+n_R)\times(n_L+n_R)$
Majorana mass matrix $M_{\nu}$ for neutrinos. The mixing between the
$n_R$ "right-handed" singlet fermions and the neutral parts of the
$n_L$ lepton doublets gives masses for the neutrinos which are of the
size expected from neutrino oscillations.
The diagonalization of the mass matrix gives rise to a split spectrum
consisting of heavy and light states of neutrinos given by $U^T
M_{\nu}U=\mathrm{diag}(m^{\mathrm{light}}_{n_L},m^{\mathrm{heavy}}_{n_R})$. For the case
$n_R=1$ we diagonalize $M_{\nu}$ with a rotation
matrix determined by two angles, two masses, and Majorana phases. For the case $n_R=2$ we diagonalize the mass matrix
with a unitary matrix determined by complex parameters, four
masses, and Majorana phases. In both cases we take $n_L=3$.
We calculate the one-loop radiative corrections to the mass parameters
which produce mass terms for the neutral leptons. In both cases we
numerically analyse light neutrino masses as functions of the heavy
neutrino masses.
\section{Discussion}
At the tree level the mass terms for the neutrinos can be written in a compact form
as a mass term with a $(n_L+n_R) \times (n_L+n_R)$ symmetric mass matrix $M_{\nu} =
\left( \renewcommand{\arraystretch}{0.2} \begin{array}{cc} \scriptstyle 0 & \scriptstyle M_D^T \\ \scriptstyle M_D & \scriptstyle \hat{M}_R \end{array} \right)$,
where $M_D$ is $n_L \times n_R$ Dirac neutrino mass matrix and $\hat{M}_R$ is a diagonal matrix. $M_{\nu}$ can be diagonalized as $U^T M_{\nu}\, U = \hat m
= \mathrm{diag} \left( m_1, m_2, \ldots, m_{n_L+n_R} \right)$,
where the $m_i$ are real and non-negative.
In order to implement the seesaw mechanism~\cite{SV80}
we assume that the elements of $M_D$ are of order $m_D$
and those of $M_R$ are of order $m_R$,
with $m_D \ll m_R$.
Then, the neutrino masses $m_i$ with $i=1, 2, \ldots, n_L$
are of order $m_D^2/m_R$, while those with $i = n_L+1, \ldots, n_L+n_R$ are of order $m_R$.
One-loop corrections to the mass matrix,
i.e.\ the self energies, are determined by the neutrino interactions
with the $Z$ boson, the neutral Goldstone boson $G^0$, and the Higgs
boson $H^0$~\cite{Grimus:2002nk}. Each diagram contains a divergent
piece but when summing up the three contributions the result turns out
to be finite~\cite{Sierra}.
First we consider the minimal extension of the standard model adding
only one right-handed field $\nu_R$ to three left-handed fields
contained in $\nu_L$. We use the parametrization of $M_D=m_D \vec{a}^T$ with $|\vec{a}|=1$. Working at tree level, we can construct the diagonalization matrix $U$ from two diagonal matrices of phases and three rotation matrices $U={\hat U}_{\phi}(\phi_i)U_{12}(\alpha_1)U_{23}(\alpha_2)U_{34}(\beta){\hat U}_i$, where the angle $\beta$ is determined by the masses of light and heavy neutrinos. The values of $\phi_i$ and $\alpha_i$ can be chosen to cover variations in $M_D$. The radiative corrections give mass to the second lightest neutrino. The third lightest neutrino remains massless.
If we add two singlet fields $\nu_R$ to three left-handed fields
$\nu_L$, the radiative corrections give masses to all three light
neutrinos. Now we parametrize $M_D=\left(\renewcommand{\arraystretch}{0.2} \begin{array}{c} \scriptstyle m_{D_2} \vec{a}^T \\ \scriptstyle m_{D_1} \vec{b}^T \end{array} \right)$ with two vectors, which $|\vec{a}|=1$ and $|\vec{b}|=1$. The diagonalization matrix for tree level $U=U_{12}(\alpha_1,\alpha_2)U_{\mathrm{egv}}(\beta_i){\hat U}_{\phi}(\phi_i)$ is composed
of a rotation matrix, an eigenmatrix of $U_{12}^T M_{\nu}M^{\dagger}_{\nu}U_{12}^\ast$ and a diagonal phase matrix, respectively.
The full results with discussions are presented in~\cite{jurc}. For the case $n_R=1$ we can match the differences of the calculated light neutrino masses to $\Delta m^2_{\odot}$ and $\Delta m^2_\mathrm{atm}$ only for a mass of the heavy singlet of TeV scale.
In the case $n_R=2$ we obtain three non vanishing masses of light neutrinos for normal hierarchy. The numerical analysis shows that the values of light neutrino masses (especially of the lightest mass) depend on the choice of the heavy neutrino masses. The radiative corrections generate the lightest neutrino mass and have a big impact on the second lightest neutrino mass.
In future we plan to apply our parametrization to study the $\tau$
polarization coming from the decay of a $W$ boson in the data of the
CMS experiment at LHC and thus determine restrictions to the
parameters of the neutrino sector.
\bigskip
\noindent This work was supported by European Union Structural Funds project "Postdoctoral Fellowship Implementation in Lithuania".
\begin{thebibliography}{99}
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\bibitem{SV80} J.~Schechter and J.~W.~F.~Valle, Phys. Rev. D 22 (1980) 2227.
\bibitem{Grimus:2002nk}
W.~Grimus and L.~Lavoura, Phys.Lett. B 546 (2002) 86.
\bibitem{Sierra}
D.~Aristizabal Sierra and C.~E.~Yaguna, JHEP 1108 (2011) 013.
\bibitem{jurc}
D.~Jur\v{c}iukonis, T.~Gajdosik, A.~Juodagalvis, T.~Sabonis, { \it in preparation}.
\end{thebibliography}
\end{document}