See-saw mechanism and four light neutrino states
M. Czakon$^a$
J. Gluza$^{a,b}$ and M. Zra\l ek
hep-ph0107298.tex
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%\begin{titlepage}
\date{\today}
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%pagestyle{empty}
{\bf See-saw mechanism and four light neutrino states} \\
\vspace{2 cm}
{M.Czakon, J.Gluza and M.Zra\l ek}
\vskip 1.5cm
Department of Field Theory and Particle Physics \\
Institute of Physics, University of Silesia, Uniwersytecka 4\\
PL-40-007 Katowice, Poland

%e-mails:czakon,gluza,zralek@us.edu.pl
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{\bf Abstract}
\end{center}

A formal proof is given that in a see-saw type neutrino mass matrix with only two 
neutrino mass scales ($m_D \ll m_R$) and the maximal rank of $m_{R(D)}$
we can not get a fourth light sterile neutrino. 
\vspace{2 cm}

%\end{titlepage}


%\end{abstract}

%\maketitle

%\section{Introduction}

It has been known for long that extended gauge group models, such as
$SO(10)$ \cite{so10} or $SU(2)_L \times SU(2)_R \times
U(1)_{B-L}$ \cite{lr}, naturally develop a see-saw type neutrino mass matrix. 
Namely ($m_D$ is a $3\times n_R$ matrix and $m_R$ is a $n_R \times n_R$ matrix)
\begin{equation}
m_\nu = \left( \matrix{ 0 & m_D \cr 
m_D^T & m_R } \right),
\label{first}
\end{equation}
coupled with a large scale difference between $m_D$ and $m_R$, yields a mass
spectrum containing three light Majorana neutrinos, effectively described by
\begin{equation}
m_{light} \simeq m_D^T m_R^{-1} m_D.
\end{equation}
This leads to two different light $\Delta m^2$ mass scales, enabling one to understand
the solar and the atmospheric neutrino anomalies 

According to the current data (now at 99 \% c.l. \cite{99}) a fourth sterile 
neutrino is not necessary to explain the Superkamiokande data.
Suppose however that the LSND experiment \cite{lsnd} is confirmed and we wish also to explain 
its data in a language of neutrino oscillation phenomena. 
We then have to introduce a fourth light neutrino of sterile nature
(due to the invisible width
measurement at LEP). 
To avoid fine tunings of parameters and still have another
light neutrino, one requires the theory to have additional properties, like 
approximate horizontal symmetry \cite{smi},
exact parity symmetry \cite{par}, a discrete $Z_5$ symmetry \cite{z5}, global 
$S_3 \times Z_2$ symmetry \cite{gibb}, or even additional gauge
 ($SU(2)_S$) symmetry \cite{add}. For more examples see \cite{rev}.
Some phenomenological considerations have also appeared (see e.g. \cite{phen}).
Interestingly enough, a see-saw type mass matrix Eq.~(\ref{first}) 
can also lead to the fourth light neutrino.
This is realized by the so-called   singular see-saw mechanism \cite{sing,bi}.
The goal is achieved, by having an $m_R$ of rank $n_R-1$. However, this is not
enough, we still have to fine tune $m_R$ to the keV-MeV range. This last
unwanted problem can be circumvented by building a second stage of see-saw
structure. This still fits into the scheme Eq.~\ref{first}, but there are
in fact three scales, not two. 

Here we give a formal proof that with only two scales we can not get a fourth light
neutrino. The importance of this result lies in the fact that one may be tempted
to believe that the larger the mass matrix the more possibilities of choosing the masses
are available, and some symmetries may help getting light sterile neutrinos. 
The problem is defined in the following statement:
%It is not possible to impose any symmetries on  
% $m_D$ and/or $m_R$ matrices and to drag a heavy state connected with the $m_R$ matrix
%to the low mass spectrum.
%One might be tempted to inspect yet another idea. Namely, impose a symmetry
%connecting $m_D$ and $m_R$ in such a clever way, as to have the additional
%light neutrino from the right handed sector.

%In this short letter, we go a step further in analyzing the see-saw type models.
%We show that the only way of incorporating the LSND data into the scheme is
%precisely to have a small eigenvalue in the $m_R$ matrix, of which a special
%case is the singular see-saw mechanism. This obviously shows, that
%the confirmation of this experiment will be a disaster to the naturalness of
%the see-saw mechanism.

%The paper is organized as follows. In the next section, we analyze the case
%of a $5\times5$ matrix, which gives us a hint of the general situation. In section
%three we give the proof of our theorem. Conclusions end up the main text.
%An appendix is added for completeness.


%We are now in position to analyze the general case. 
%\underline{Statement}: \newline
Let $m_R$ be a matrix
of eigenvalues real positive and greater than some $M$, and let all of the moduli of elements 
of $m_D$ be much smaller than $M$, then the
spectrum of $m_\nu$ contains the full spectrum of $m_R$ with corrections of the order of
$m^2/M\equiv max(|(m_D)_{ij}|)^2/M$. That means that no
manipulation on $m_D$ and/or $m_R$ 
can move a mass from the heavy $m_R$ matrix into the light spectrum. 

The proof is a simple consequence of perturbation theory for finite
matrices \cite{reedsimon}. There, it is shown, that if we have a matrix of
the form 
\begin{equation}
M(\beta) = M^{(0)}+\beta M^{(1)},
\end{equation}
then, every non-degenerate eigenvalue of $M^{(0)}$ gives rise to a non-degenerate
eigenvalue of $M(\beta)$, being an analytic function of $\beta$ in some surrounding of zero.
Since we are interested in the heavy spectrum, the assumption of non-degeneracy
is quite general. In case of degenerate eigenvalues, we still can expand
the eigenvalues in series, which however will be analytic functions with branches.
We limit ourselves to the former case, but the reader should understand
that the theorem holds for the general case also.

We decompose $m_\nu$ as
\begin{eqnarray}
\left( \matrix{ 0 & m_D \cr
 m_D^T & m_R
} \right) &=& \left( \matrix{ 0 & 0 \cr
 0 & m_R
} \right)+ \left( \matrix{ 0 & m_D \cr
 m_D^T & 0
} \right) \nonumber \\ \nonumber \\
&=& M \left[ \left( \matrix{ 0 & 0 \cr
 0 & m_R/M
} \right)+ \beta \left( \matrix{ 0 & m_D/m \cr
 m_D^T/m & 0
} \right) \right]\nonumber \\ \nonumber \\
&\equiv & M(M^{(0)}+\beta M^{(1)}),
\end{eqnarray}
where $\beta =m/M$. The first matrix has all its elements greater than one, while
the second has all elements smaller than one, both are of course dimensionless.
The eigenvectors of $M^{(0)}$ are of the form (we chose without loss of generality 
a weak base for neutrinos in which $m_R$ is diagonal \cite{gl})
\begin{equation}
e_i = \left( \matrix{ 0 \cr
 \vdots \cr
 1 \cr
 \vdots \cr
 0 } \right).
\end{equation}
To find the first order correction, we expand the eigenvectors as
\begin{equation}
v_i = \sum_{j} \alpha_{ji} e_j.
\end{equation}
This gives us the following equation
\begin{equation}
(M^{(0)}+\beta M^{(1)}) v_i = \lambda_i v_i,
\end{equation}
which is solved into
\begin{equation}
\lambda_i = \lambda_i^{(0)}+\beta \frac{1}{\alpha_{ii}} \sum_j \alpha_{ji} e_i^T M^{(1)} e_j,
\end{equation}
where $\lambda^{(0)}$ are non-degenerate nonzero eigenvalues of $M^{(0)}$, which are
also eigenvalues of $m_R/M$. 
Obviously, only $\alpha_{ii}$ is ${\cal O}(1)$, and
$\alpha_{ji}$ for $j\neq i$ is ${\cal O} (\beta)$. 
The first order correction to 
$\lambda^{(0)}$ is therefore
\begin{equation}
\lambda_i^{(1)} = \lambda_i^{(0)}+\beta e_i^T M^{(1)} e_i.
\end{equation}
But this vanishes due to the nondiagonal form of $M^{(1)}$. Thus the first non-vanishing
correction to the large eigenvalues is of the order $M\beta^2 = m^2/M$, which
completes the proof. Remark, that the masses of the neutrinos are moduli of the
eigenvalues of $m_\nu$. The corrections to the moduli are however of the same order.

Using simple arguments based on perturbation series, we have shown
that a natural $m_R$ (no fine tunings and eigenvalues at the heavy 
scale) leads to three light neutrinos. Thus from the class of see-saw type models 
only a singular double
see-saw mechanism can accommodate the LSND data and an additional fourth light neutrino state.


\begin{center}
{\large \bf Acknowledgments}
\end{center}
This work was supported by the Polish Committee for Scientific Research under 
Grant No. 2P03B04919  and 2P03B05418. 

\begin{thebibliography}{99}
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Baryon Number in the Universe, eds. O.~Sawada and A.~Sugamoto 
(KEK,1979), p.95;
M.~Gell-Mann, P.~Ramond and R.~Slansky, in Supergravity, ed. by P.~Van
Neiuwenhuizen and D.~Freedman (North-Holland,Amsterdam,1979), p.315.
\bibitem{lr} R. Mohapatra and G. Senjanovic, Phys. Rev. Lett., {\bf 44} 
(1980) 912.
\bibitem{lsnd} C. Athanassopoulos et al., Phys. Rev. Lett. {\bf 77}(1996)3082;
Phys. Rev. {\bf C54} (1996) 268; 
Phys. Rev. {\bf C58} (1998) 2489; Phys. Rev. Lett. {\bf 81} (1998) 1774; 
S.J. Yellin, hep-ex/9902012.
\bibitem{99} T. Toshito, hep-ex/0105023.
\bibitem{smi} E. Chun, A.S. Joshipura, and A.Yu. Smirnov, Phys. Lett. {\bf B 357} 
(1995) 608.
\bibitem{par} R.~Foot and R.R.~Volkas,
%``Neutrino physics and the mirror world: How exact parity symmetry explains the solar neutrino deficit, the atmospheric neutrino anomaly and the LSND experiment,''
Phys.\ Rev.\ {\bf D52} (1995) 6595;
Z.G. Berezhiani and R.B. Mohapatra, Phys. Rev. {\bf D 52} (1995) 6607; 
R.N. Mohapatra, hep-ph/9903261.
\bibitem{z5} E.~Ma and P.~Roy,
%``Model of four light neutrinos prompted by all desiderata,''
Phys.\ Rev.\ {\bf D52} (1995) 4780;
E.~Ma, D.P.~Roy and U.~Sarkar,
%``A see-saw model for atmospheric and solar neutrino oscillations,''
Phys.\ Lett.\ {\bf B444} (1998) 391.
\bibitem{gibb}  S.C.~Gibbons, R.N.~Mohapatra, S.~Nandi and A.~Raychaudhuri,
%``A four-neutrino mixing scheme for observed neutrino data,''
Phys.\ Lett.\ {\bf B430} (1998) 296.
\bibitem{add} U. Sarkar, Phys. Rev. {\bf D59} 031301.
%\bibitem{55} S. Raby, hep-ph/9909279.
%\bibitem{mir} mir
\bibitem{rev} R.~N.~Mohapatra,
%``Theories of neutrino masses and mixings,''
hep-ph/9910365.
\bibitem{phen} 
%Phys. Rev. {\bf D 50} (1994) 3477; 
%J.T. Peltoniemi and J.W.F. Valle, Nucl. Phys. {\bf B 406} (1993) 409; 
%J.T. Peltoniemi, D. Tommasini and J.W.F. Valle, Phys. Lett. {\bf B 298} 
%(1993) 383; 
G.M. Fuller, J.R. Primack and Y.Z. Qian, Phys. Rev. {\bf D 52} (1995) 
1288; 
%D. Suematsu, Phys. Lett. {\bf B 392} (1997) 413; 
%A.Y. Smirnov and M. Tanimoto, Phys. Rev. {\bf D 55} (1997) 1665;
%;
% J.J. Gomez-Cadenas and G.M. Gonzalez-Garcia, Z. Phys. {\bf C 71} 
%(1996) 1996;
V. Barger, T.J. Weiler, and K. Whisnant, Phys. Lett. {\bf B 427} 
(1998) 97;  
S.M. Bilenky, C. Giunti, and W. Grimus, Eur. Phys. J. {\bf C 1} (1998) 247; Phys. Rev. {\bf D 57} (1998) 1920; 
S.M.~Bilenkii, C.~Giunti, W.~Grimus and T.~Schwetz,
%``Four-neutrino mixing and big-bang nucleosynthesis,''
Astropart.\ Phys.\ {\bf 11} (1999) 413;
N. Okada and O. Yasuda, Int. J. Mod. Phys. {\bf A 12} (1997) 3669;
J.M.~Gelb and S.P.~Rosen,
Phys.\ Rev.\ D {\bf 62} (2000) 013003,
%D.~Suematsu,
%``Neutrino oscillations in a 3nu(L) + 3nu(R) framework with five light  neutrinos,''
%hep-ph/9805305;
V.~Barger, S.~Pakvasa, T.J.~Weiler and K.~Whisnant,
%``Variations on four-neutrino oscillations,''
Phys.\ Rev.\ {\bf D58} (1998) 093016;
V.~Barger, Y.~Dai, K.~Whisnant and B.~Young,
%``Neutrino mixing, CP/T violation and textures in four-neutrino models,''
Phys.\ Rev.\ {\bf D59} (1999) 113010;
R.N. Mohapatra, S. Nussinov, Phys. Rev. {\bf D60} 013002;
%Q.Y.~Liu and A.Y.~Smirnov,
%``Neutrino mass spectrum with nu/mu --> nu/s oscillations of atmospheric  neutrinos,''
%Nucl.\ Phys.\ {\bf B524} (1998) 505;
S. Mohanty, D.P. Roy, U. Sarkar, Phys. Lett. {\bf B445} (1998) 185;
S.~M.~Bilenky, S.~Pascoli and S.~T.~Petcov,
%``Majorana neutrinos, neutrino mass spectrum, CP-violation and  neutrinoless double beta-decay. II: Mixing of four neutrinos,''
hep-ph/0104218.
%\bibitem{onl}  H. Minakata, Phys. Rev. {\bf D 52} (1995) 6630;
%G. Barenboim, F. Scheck, Phys. Lett. {\bf  B440} (1998) 332.
%\bibitem{mod}
%E. Ma and P. Roy, Phys. Rev. {\bf D 52} (1995) 4780; 
\bibitem{sing} E.J.~Chun, C.W.~Kim and U.W.~Lee,
%``Three neutrino Delta(m**2) scales and singular seesaw mechanism,''
Phys.\ Rev.\ {\bf D58} (1998) 093003;
\bibitem{bi}
Y.~Chikira, N.~Haba and Y.~Mimura,
%``The singular seesaw mechanism with hierarchical Dirac neutrino mass,''
Eur.\ Phys.\ J.\ C {\bf 16} (2000) 701.
\bibitem{reedsimon} M. Reed, B. Simon in ``Methods of modern mathematical physics'',
Academic Press, New York, London, 1972.
\bibitem{gl} J.~Gluza and M.~Zralek,
%``Higgs boson contributions to neutrino production in e- e+ collisions in a left-right symmetric model,''
Phys.\ Rev.\ D {\bf 51} (1995) 4695.
\end{thebibliography}

\end{document}
\section{Heuristic Example}

Let us focus for a short while on a simple example. Let $m_D$ be a
$3\times 2$ matrix, and $m_R$ a $2\times 2$. That is, we have 2 additional
sterile neutrino states. Following the remarks in the appendix, we can
write it as
\begin{equation}
\left( \begin{array}{ccccc} 0 & 0 & 0 & a & b \\ 0 & 0 & 0 & c & d \\
0 & 0 & 0 & e & f \\ a & c & e & M_1 & 0 \\ b & d & f & 0 & M_2
} \right) .
\end{equation}
Some of the $a,\dots,f$ may be complex, but it does not matter at present. 
We assume that $M_{1,2} \gg a,\dots,f$. Obviously one of the eigenvalues will
vanish, due to  linear dependence of the first three rows. We guess
that to have a fourth small mass, we need one of the last rows somehow 
approximately linearly dependent on the first two. We see however, that 
an exact linear dependence is impossible, because we have the same elements
$a,\dots,f$ in two different places. With a linear combination of the first
two rows, we can eliminate $M_1$ and $M_2$, but what remains is still 
linearly independent and leads to eigenvalues of the order of $M_{1,2}$.
Therefore, the strong correlation between the elements of the matrix 
leads us to conclude, that it may be impossible to have four light states
with a nonsingular $m_R$ matrix.

\section{Proof}




\title{See-saw mechanism and four light neutrino states}
\author{M. Czakon$^a$, J. Gluza$^{a,b}$ and M. Zra\l ek}

\address{$^a$ Department of Field Theory and Particle Physics, Institute 
of Physics, University of
Silesia, \\ Uniwersytecka 4, PL-40-007 Katowice, Poland \\
$^{b}$  DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany}

\date{\today}



\begin{abstract}\noindent

hep-ph0107298.extracted_bib
\begin{thebibliography}{99}
\bibitem{so10} T.~Yanagida, Proc. of the Workshop on the Unified Theory and 
Baryon Number in the Universe, eds. O.~Sawada and A.~Sugamoto 
(KEK,1979), p.95;
M.~Gell-Mann, P.~Ramond and R.~Slansky, in Supergravity, ed. by P.~Van
Neiuwenhuizen and D.~Freedman (North-Holland,Amsterdam,1979), p.315.
\bibitem{lr} R. Mohapatra and G. Senjanovic, Phys. Rev. Lett., {\bf 44} 
(1980) 912.
\bibitem{lsnd} C. Athanassopoulos et al., Phys. Rev. Lett. {\bf 77}(1996)3082;
Phys. Rev. {\bf C54} (1996) 268; 
Phys. Rev. {\bf C58} (1998) 2489; Phys. Rev. Lett. {\bf 81} (1998) 1774; 
S.J. Yellin, hep-ex/9902012.
\bibitem{99} T. Toshito, hep-ex/0105023.
\bibitem{smi} E. Chun, A.S. Joshipura, and A.Yu. Smirnov, Phys. Lett. {\bf B 357} 
(1995) 608.
\bibitem{par} R.~Foot and R.R.~Volkas,
%``Neutrino physics and the mirror world: How exact parity symmetry explains the solar neutrino deficit, the atmospheric neutrino anomaly and the LSND experiment,''
Phys.\ Rev.\ {\bf D52} (1995) 6595;
Z.G. Berezhiani and R.B. Mohapatra, Phys. Rev. {\bf D 52} (1995) 6607; 
R.N. Mohapatra, hep-ph/9903261.
\bibitem{z5} E.~Ma and P.~Roy,
%``Model of four light neutrinos prompted by all desiderata,''
Phys.\ Rev.\ {\bf D52} (1995) 4780;
E.~Ma, D.P.~Roy and U.~Sarkar,
%``A see-saw model for atmospheric and solar neutrino oscillations,''
Phys.\ Lett.\ {\bf B444} (1998) 391.
\bibitem{gibb}  S.C.~Gibbons, R.N.~Mohapatra, S.~Nandi and A.~Raychaudhuri,
%``A four-neutrino mixing scheme for observed neutrino data,''
Phys.\ Lett.\ {\bf B430} (1998) 296.
\bibitem{add} U. Sarkar, Phys. Rev. {\bf D59} 031301.
%\bibitem{55} S. Raby, hep-ph/9909279.
%\bibitem{mir} mir
\bibitem{rev} R.~N.~Mohapatra,
%``Theories of neutrino masses and mixings,''
hep-ph/9910365.
\bibitem{phen} 
%Phys. Rev. {\bf D 50} (1994) 3477; 
%J.T. Peltoniemi and J.W.F. Valle, Nucl. Phys. {\bf B 406} (1993) 409; 
%J.T. Peltoniemi, D. Tommasini and J.W.F. Valle, Phys. Lett. {\bf B 298} 
%(1993) 383; 
G.M. Fuller, J.R. Primack and Y.Z. Qian, Phys. Rev. {\bf D 52} (1995) 
1288; 
%D. Suematsu, Phys. Lett. {\bf B 392} (1997) 413; 
%A.Y. Smirnov and M. Tanimoto, Phys. Rev. {\bf D 55} (1997) 1665;
%;
% J.J. Gomez-Cadenas and G.M. Gonzalez-Garcia, Z. Phys. {\bf C 71} 
%(1996) 1996;
V. Barger, T.J. Weiler, and K. Whisnant, Phys. Lett. {\bf B 427} 
(1998) 97;  
S.M. Bilenky, C. Giunti, and W. Grimus, Eur. Phys. J. {\bf C 1} (1998) 247; Phys. Rev. {\bf D 57} (1998) 1920; 
S.M.~Bilenkii, C.~Giunti, W.~Grimus and T.~Schwetz,
%``Four-neutrino mixing and big-bang nucleosynthesis,''
Astropart.\ Phys.\ {\bf 11} (1999) 413;
N. Okada and O. Yasuda, Int. J. Mod. Phys. {\bf A 12} (1997) 3669;
J.M.~Gelb and S.P.~Rosen,
Phys.\ Rev.\ D {\bf 62} (2000) 013003,
%D.~Suematsu,
%``Neutrino oscillations in a 3nu(L) + 3nu(R) framework with five light  neutrinos,''
%hep-ph/9805305;
V.~Barger, S.~Pakvasa, T.J.~Weiler and K.~Whisnant,
%``Variations on four-neutrino oscillations,''
Phys.\ Rev.\ {\bf D58} (1998) 093016;
V.~Barger, Y.~Dai, K.~Whisnant and B.~Young,
%``Neutrino mixing, CP/T violation and textures in four-neutrino models,''
Phys.\ Rev.\ {\bf D59} (1999) 113010;
R.N. Mohapatra, S. Nussinov, Phys. Rev. {\bf D60} 013002;
%Q.Y.~Liu and A.Y.~Smirnov,
%``Neutrino mass spectrum with nu/mu --> nu/s oscillations of atmospheric  neutrinos,''
%Nucl.\ Phys.\ {\bf B524} (1998) 505;
S. Mohanty, D.P. Roy, U. Sarkar, Phys. Lett. {\bf B445} (1998) 185;
S.~M.~Bilenky, S.~Pascoli and S.~T.~Petcov,
%``Majorana neutrinos, neutrino mass spectrum, CP-violation and  neutrinoless double beta-decay. II: Mixing of four neutrinos,''
hep-ph/0104218.
%\bibitem{onl}  H. Minakata, Phys. Rev. {\bf D 52} (1995) 6630;
%G. Barenboim, F. Scheck, Phys. Lett. {\bf  B440} (1998) 332.
%\bibitem{mod}
%E. Ma and P. Roy, Phys. Rev. {\bf D 52} (1995) 4780; 
\bibitem{sing} E.J.~Chun, C.W.~Kim and U.W.~Lee,
%``Three neutrino Delta(m**2) scales and singular seesaw mechanism,''
Phys.\ Rev.\ {\bf D58} (1998) 093003;
\bibitem{bi}
Y.~Chikira, N.~Haba and Y.~Mimura,
%``The singular seesaw mechanism with hierarchical Dirac neutrino mass,''
Eur.\ Phys.\ J.\ C {\bf 16} (2000) 701.
\bibitem{reedsimon} M. Reed, B. Simon in ``Methods of modern mathematical physics'',
Academic Press, New York, London, 1972.
\bibitem{gl} J.~Gluza and M.~Zralek,
%``Higgs boson contributions to neutrino production in e- e+ collisions in a left-right symmetric model,''
Phys.\ Rev.\ D {\bf 51} (1995) 4695.
\end{thebibliography}