The non-equivalence of Weyl and Majorana neutrinos with standard-model gauge interactions
Max-Planck-Institut f\"ur Physik (Werner-Heisenberg-Institut)
F\"ohringer Ring 6
D-80805 M\"unchen
Germany
\noindent It is shown that a Majorana neutrino with identical phenomenology as a standard-model Weyl neutrino obeys a Lagrangian different by a factor $\sqrt{2}$ in the weak-interaction term from the one that follows from the standard model. Assuming that the standard model {does} hold in good approximation for the weak interactions of a Majorana neutrino e.g. a charged-current production cross section a factor 2 larger than the observed one is predicted. From this it is concluded that the neutrino is not of Majorana type. This conclusion does {not} forbid the possible existence of a Weyl neutrino with a Majorana mass term and ensuing lepton-number violating phenomena (like e.g. neutrino-less double beta decay). The paper is not in contradiction with any published literature, because it analyses the formal proof of equivalence between Weyl and Majorana neutrino for the first time under the assumption that the standard model is {quantitatively} correct for the weak interactions of the neutrino.
hep-ph0108052.tex
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\begin{document}


\title{The non-equivalence of Weyl and Majorana 
neutrinos with standard-model gauge interactions}
\author{
        R.Plaga\thanks
        {plaga@mppmu.mpg.de, see http://hegra1.mppmu.mpg.de/majorana for
additional material on this manuscript}
        \\Max-Planck-Institut f\"ur Physik (Werner-Heisenberg-Institut)
        \\ F\"ohringer Ring 6
        \\D-80805 M\"unchen, Germany
        }
\maketitle
%\centerline{format is PRL}
\begin{abstract}
\noindent
It is shown that a Majorana neutrino with identical
phenomenology as a standard-model
Weyl neutrino obeys a Lagrangian
different by a factor $\sqrt{2}$ in the weak-interaction term 
from the one that follows from the standard
model. Assuming that the standard model {\it does} hold
in good approximation for the weak interactions of
a Majorana neutrino e.g. a charged-current production 
cross section a factor 2 larger
than the observed one is predicted.
From this it is concluded that the neutrino is not of Majorana type.
This conclusion does {\it not} forbid the possible existence
of a Weyl neutrino with a Majorana mass term
and ensuing lepton-number violating phenomena (like
e.g. neutrino-less double beta decay).
The paper is not in contradiction with any published
literature, because it analyses 
the formal proof of equivalence between Weyl and Majorana
neutrino for the first time under the assumption that
the standard model is {\it quantitatively} 
correct for the weak interactions of the neutrino. 
\end{abstract}
\section{Introduction}
\subsection{A remark about the meaning of the present paper's
conclusion}
In section \ref{proof} of this paper I show that the observed neutrinos -
in the massless limit -
are definitely of Weyl and not Majorana type. 
This conclusion only holds under the
following two, widely used, {\it assumptions}:
\\
A.the standard model 
{\it quantitatively} describes the weak 
interactions of the neutrino to good approximation
\\
B. standard quantum field theory applies 
to good approximation for the description
of the neutrino field
\\
As discussed in section \ref{weylmass}
my conclusion holds also
if Weyl neutrinos have finite Majorana masses.
This is why the present manuscript's
claim is not as far reaching as it might first sound: all
of the lepton-number violating phenomena connected with 
massive Majorana neutrinos,
such as neutrino-less double beta decay and all 
Majorana mass related small-mass
generating mechanisms proposed for them (such as the see-saw
mechanism) remain viable for a {\it Weyl neutrino with a Majorana mass
term}. On the other hand
- because the term ``Majorana neutrino'' is precisely mathematically
defined (section \ref{definition}) 
- the manuscript's conclusion is nontrivial
(it solves a longstanding question of neutrino
physics) and potentially important in the development of new theories.

\subsection{Outline of the paper}
The review of the precise definition of the Majorana-neutrino
field in section \ref{definition} makes clear
that {\it mathematically} Weyl and Majorana fields are different.
Langrangians
predicted by the standard model for two mathematically different
fermion fields - even if they have the same degrees of freedom -
must not necessarily lead to the same phenomenology\footnote{
An example where this is obvious is e.g. a Weyl neutrino and
a massless electron.}.
Below I derive the Lagrangians giving rise to the same
phenomenology (briefly outlined in the
next paragraph ``1.'') and the Lagrangians derived
from the standard model (briefly outlined in
following paragraph ``2.'') and find
they are {\it not} identical.
\\
1. In section \ref{uniequi}
- as a review of work from several authors in the late 1950s -
it is shown that a unitary similarity 
transformation (the ``Pauli I'' transformation)
between Weyl and Majorana neutrino fields exists.
One concludes that Weyl neutrinos with a standard-model (SM) 
Lagrangian L$_{\rm Weyl}^{\rm SM}$ 
can be unitarily transformed to a Majorana neutrino
with a Lagrangian L$_{\rm Maj}^{\rm Weyl-equivalent}$ 
which is uniquely
specified by the condition that any similarity transformation
leaves the form of all field equations unmodified.   
A Majorana neutrino {\it obeying this Lagrangian} is then
phenomenologically completely equivalent to a Weyl neutrino.
\\
2. Since the late
1970s the interaction of the left-handed Weyl neutrino $\nu_L$
is uniquely specified by the standard model {\it without
any reference to the observed 
neutrino's properties}, 
(analogous to the top quark, whose weak-interaction properties were all
specified before its actual discovery).
Because a Majorana-neutrino field 
can be decomposed to Weyl-neutrino components
via the ``Pauli I'' transformation, one
can derive the standard-model Lagrangian of a Majorana neutrino
L$_{\rm Maj}^{SM}$. One finds that L$_{\rm Maj}^{\rm SM}$ 
$\neq$ L$_{\rm Maj}^{\rm Weyl-equivalent}$
In other words:
the ``Pauli I'' transformation - that brings a Weyl to a
phenomenologically equivalent Majorana neutrino - is not
SU(2) invariant. 
\\
If the observed neutrino were 
of Majorana type 
it {\it had} to be phenomenologically equivalent to a Weyl
neutrino with standard-model interactions 
to not contradict experimental facts (e.g.
about neutrino cross sections).
Under assumption A. in the introduction we can then conclude that the
observed neutrino must be of Weyl type.
The end of section \ref{uniequi} discusses why this conclusion
is not in contradiction with the pubished literature on the
``Majorana Dirac confusion theorem''.
\\
Section \ref{weylmass} of the paper is devoted to showing in
detail how a Weyl neutrino with a Majorana mass term
can violate lepton number without ever being its own
anti particle and section \ref{summary} concludes.

\section{The definition of a Majorana-neutrino field}
\label{definition}
For definitness I first discuss the quantitative definition
of a ``Majorana neutrino field''. In this I follow the standard
literature.
Let $\nu$ be a 4-component Dirac neurino field\footnote{
I do not discuss the possible
case of Dirac neutrino masses in this paper.}
Then, using ``east-coast'' notation (imaginary time)
the 2-component Weyl field $\nu_L$ is defined\footnote{
I do not explicitely include the condition of fermion number
conservation in this definition, as is sometimes
done in the literature.} as\cite{weyl}:
\begin{equation}
\nu_L = {(1+\gamma_5) \over 2} \nu_L = P_L \nu_L
\label{weyl}
\end{equation}
The 
2-component Majorana field $\nu_M$ is defined via the following
relations by which the neutrino is its own antiparticle
\cite{majorana}:
\begin{equation}
\nu_M = C \nu{_M}^{\dagger T} \gamma_4 \equiv
 \nu_M^c; \: \nu_M={1\over\sqrt{2}}(\nu + \nu^c)
\label{maj}
\end{equation}
Here C is the charge conjugation matrix, $\dagger$ the
hermitian conjugate and T a transpose acting
only on the spinor,
$^c$ symbolizes charge conjugation and
a conventional ``creation phase factor''\cite{kayb}
was set to 1
\footnote{ 
This defining constraint for Majorana fields (``Majorana neutrinos
are eigenstates of C'')
is universally accepted 
in the literature, also
after the discovery of parity violation (e.g. eq.(19)
in Pauli's paper\cite{pauli57} written after the
discovery of parity violation, or e.g. eq.(10) in 
Ref.\cite{nieves}). Even if one
prefers Kayser's alternative characterization of
Majorana neutrino as a state that
is turned into itself with reversed helicity under CPT 
\cite{kayb}, eq.(\ref{maj}) has
to hold.
E.g. in a textbook by Mohapatra and Pal
\cite{moha} the CPT properties of Majorana
neutrinos are derived in section 4.4.3 {\it using}
their eq.(4.16) which is identical to eq.(\ref{maj}).
Berestetski\u{i} et al.\cite{landau} argue that
there is no problem with condition (\ref{maj}) 
even in the presence of weak interactions
because it is invariant not only to
CPT but also with respect to each
of these transformations seperately.}.
The definition in eq.(\ref{maj})
after the semicolon defines the field normalization and
it can be easily shown to
be the one that fulfills the usual
field-anticommutation axioms of quantum-field theory:
\begin{equation}
\left[ \nu_M(\vec{x},t),\eta(\vec{x}^{\prime},t) \right]_+=
i\delta(\vec{x}-\vec{x}^{\prime})
\end{equation}
where $\eta$ is the field which is canonical conjugate to
$\nu_M$.
This field normalisation 
({\it with} the factor 1/$\sqrt{2}$) is generally used
in the literature\cite{kayb,mannheim,fukugita,lee,roman2} 
\footnote{Only Bilenky and Petcov\cite{bilenky} leave the 
factor 1/$\sqrt{2}$ out, for reasons that are not clear
(they quote Ref.\cite{mannheim} that {\it does} use it
as standard reference for the definition of a Majorana
fermion.)}.
\\
Clearly the conditions eq.(\ref{weyl}) and eq.(\ref{maj}) are
mathematically mutually exclusive; a Weyl particle can never be
its own antiparticle.

\section{The unitary equivalence of Weyl and Majorana neutrino
fields, also called ``Dirac-Majorana confusion theorem''}
\label{uniequi}
For the present argument it is
sufficient to consider 
only charged currents in the low energy limit.
The standard-model (SM) Lagrangian for 
massless Weyl neutrino fields is then:
\begin{equation}
L_{\rm Weyl}^{\rm SM}=
\bar{\nu}_L \gamma_{\mu} {\partial \over {\partial x_\mu}} \nu_L
+ i g/\sqrt{2} \left[W_{\mu}^- \bar{e}_L \gamma_{\mu} P_L \nu_L + H.C. \right]
\label{dirl1}
\end{equation}
here $g={e \over{sin(\theta_w)}}$.
Let us now answer the following question: What Lagrangian
``$L_{\rm Maj}^{\rm Weyl-equivalent}$'' 
must hold Majorana neutrino, so that it shows a phenomenology
identical to the one of the Weyl neutrino with Lagrangian (\ref{dirl1})?  
\\
Pauli\cite{pauli57} specified 
the following ``Pauli I'' transformation ``U$_1$''
which transforms a neutrino field $\nu$
into $\nu^{\prime}$:   
\begin{equation}
\nu^{\prime}=U_1 \nu U_1^{-1}={1\over \sqrt{2}}(\nu - \gamma_5 \nu^c)
\label{t1}
\end{equation}
This transformation
\footnote{The explicit form
of U$_1$ can be found in 
Refs.\cite{ryan,fukugita}} 
can be easily shown to be {\it unitary} but does not
conserve a SU(2) invariance of a Lagrangian.
Similarity transformations leave 
the form of operator
equations (i.e. in particular
the field equations and anticommutation
relations) unmodified and
the expectation values of field operators 
do not change under a unitary transformation
of field operator together with 
the field states\cite{roman2,pauli57}.
Therefore the phenomenology remains unchanged 
if one replaces $\nu$ by $\nu^{\prime}$ everywhere.
\\
For the special case
$\nu=\nu_L$ eq.(\ref{t1}) reads (h=helicity)\footnote{The Pauli I 
transformation of $\nu_L$ is 
$U_1 \nu_L U_1^{-1} = U_1 P_L \nu U_1^{-1} \neq  P_L (U_1 \nu U_1^{-1})  $}:
\\
\begin{equation}
\nu^{\prime}={1\over \sqrt{2}} (\nu_L - \gamma_5 (\nu_L)^c)=
{1\over \sqrt{2}} (\nu_L + (\nu_L)^c) = \nu_M(h=-1)
\label{maj2}
\end{equation}
From the invariance of the field equations,
the Majorana Lagrangian is obtained by replacing
$\nu_L$ with $\nu_M$ in equation (\ref{dirl1})
\begin{eqnarray}
L_{\rm Maj}^{\rm Weyl-equivalent}=
\bar{\nu}_M(h=+1) 
\gamma_{\mu} {\partial \over {\partial x_\mu}} \nu_M(h=-1)
\nonumber
\\
+ i g/\sqrt{2} 
\left[ W_{\mu}^- \bar{e}_L \gamma_{\mu} P_L \nu_M(h=-1) + H.C.\right]
\label{dirm1}
\end{eqnarray}
The ``Pauli I'' transformation 
eq.(\ref{t1}) 
does not include the electron field and is therefore
not equivalent to a mere representation change
of the theory. It is thus not at all clear if
this Lagrangian still obeys the standard model (see next section).
%or equivalently (insertion in (\ref{dirl2})):
%\begin{equation}
%L_{Maj}=
%P_L \bar{\nu}_M(h=+1) 
%\gamma_{\mu} {\partial \over {\partial x_\mu}} P_L \nu_M(h=-1)
%\\
%+ i g \left[W_{\mu}^- P_L \bar{e}_L \gamma_{\mu} P_L \nu_M(h=-1) + H.C.\right]
%\label{dirm2}
%\end{equation}
In the late 1950s (i.e. long before
the formulation of the standard model)
- with no reason whatsoever to
exclude the validity of L$_{\rm Maj}^{\rm Weyl-equivalent}$ for
neutrinos - various authors
\cite{ryan,serpe,radicati,lueders58,kahana57,
mclennan,case} could only conclude that
massless $\nu_L$ and $\nu_M$
(helicity=$-$1 states) (and 
analogously $\bar{\nu}_L$ and $\bar{\nu}_M$
(helicity=+1 states)) are phenomenologically 
completely equivalent 
(this conclusion was later also called ``Dirac - Majorana confusion
theorem''\cite{kayb}).
The  ``Dirac - Majorana confusion
theorem'' was never discussed in the literature under the
assumption of {\it quantitative} validity of the standard model.
The difference between Majorana and Weyl neutrino is of a {\it purely}
quantitative character (a factor $\sqrt{2}$) all qualitative properties
are the same (e.g. in the massless case both Majorana and Weyl neutrinos
conserve lepton number).
Kayser\cite{kayb} and  Zra\l ek\cite{zralek} state the confusion
theorem's validity
under the assumption that the weak interaction is left handed
(``qualitative validity'' of the standard model),
a correct statement which is not in contradiction with the present paper.


\section{Proof that  L$_{\rm Maj}^{\rm Weyl-equivalent}$ $\neq$
L$_{\rm Maj}^{\rm SM}$}
\label{proof}
Let us calculate 
the Lagrangian L$_{\rm Maj}^{SM}$ 
for massless Majorana neutrinos
that is predicted by the Standard model.
Applying P$_L$ onto eq.(\ref{maj2}) one gets:
\begin{equation}
\nu_L = \sqrt{2} P_L \nu_M(h=-1)
\label{rep1}
\end{equation}
One can also show that \cite{li,haber}:
\begin{equation}
\bar{\nu}_L \gamma_{\mu} {\partial \over {\partial x_\mu}}
\nu_L = \bar{\nu}_M(h=+1) 
\gamma_{\mu} {\partial \over {\partial x_\mu}} \nu_M(h=-1)
\label{rep2}
\end{equation}
Replacing the kinetic term in Lagrangian (\ref{dirl1}) using
eq.(\ref{rep2}) and $\nu_L$ in the interaction term using
eq.(\ref{rep1}) one gets:
\begin{equation}
L_{\rm Maj}^{\rm SM}= \bar{\nu}_M(h=+1) 
\gamma_{\mu} {\partial \over {\partial x_\mu}} \nu_M(h=-1)
+ i g
\left[ W_{\mu}^- \bar{e}_L \gamma_{\mu} P_L \nu_M(h=-1) + H.C.\right]
\label{dirm2}
\end{equation}
%In eq.(\ref{dirm2}) apply the projection operators 
%to the Majorana fields as defined in eq.(\ref{maj2});
%\begin{equation}
%L_{Maj} =
%{1 \over \sqrt{2}} \bar{\nu}_L \gamma_{\mu} {\partial \over {\partial x_\mu}}
%{1 \over \sqrt{2}} \nu_L
%+ i g \left[W_{\mu} \bar{e}_L \gamma_{\mu}
% {1 \over \sqrt{2}} \nu_L + H.C.\right]
%\label{dirm3}
%\end{equation}
%Multiplication by 2 (overall factors are irrelevant
%in a Lagrangian) gives the final result:
%\begin{equation}
%L_{Maj} =
% \bar{\nu}_L \gamma_{\mu} {\partial \over {\partial x_\mu}}
% \nu_L
%+ i g \sqrt{2} \left[ W_{\mu} \bar{e}_L \gamma_{\mu} \nu_L + H.C.\right]
%\label{dirm4}
%\end{equation}
The charged-current coupling constant 
in eq.(\ref{dirm2}) is seen to be a factor $\sqrt{2}$ larger than
in eq.(\ref{dirm1}) the two Lagrangians are thus different.
\\
The numerical value $g = {e \over{sin(\theta_w)}}$
is determined in the standard-model
gauge theory by considering only neutral-current
(for $sin(\theta_w)$)
and electromagnetic (for $e$)
reactions of the electron, i.e. without reference to 
neutrino properties.
One numerically different coupling constant 
in the two otherwise identical Lagrangians eq.(\ref{dirm1}) and 
eq.(\ref{dirm2})
is a difference which persists
to the phenomenological level (i.e.
the application of Feynman rules). In other words:
if the neutrino is a Majorana particle and its gauge
interactions are the one of the standard model, charged-current
reactions of the neutrino would have a factor 2 larger
cross section than observed.
If we assume the strict validity of the 
standard model {\it gauge sector}
a priori (see assumption A
in the introduction)
the observed neutrino, if massless must be a Weyl
neutrino, i.e. definitely not its own antiparticle.
This conclusion rests {\bf only} on the 
quantitative consideration of the
charged current ``source'' term 
$i g  W_{\mu}^-  \bar{e}_L \gamma_{\mu} \nu_L$;
as long as only kinetic, mass and the form
of the interaction term are considered
({\it as is done in all 
equivalence proofs in the literature!}) 
Majorana and Weyl fields are seen to be completely equivalent.
%\\
%Some authors (e.g.\cite{li,haber}) define the
%Majorana field with a field normalization different from
%the one of the other fermions -
%e.g. they omit the factor (${1 \over \sqrt{2}}$)
%in the second expression of eq.(\ref{maj}).
%This procedure amounts
%to a change in the basic postulates
%of field theory but is mathematically identical
%to requiring a charged-current coupling constant $g$
%increased by a factor $\sqrt{2}$
%together with the usual field normalization.
%Even with this normalisation the qualitative conclusions
%of this paper remain unchanged.
   
\section{Weyl neutrinos with Majorana masses}
\label{weylmass}
It could be that Lagrangian
(\ref{dirl1}) for a Weyl  neutrino contains a small Majorana
mass term. In this section I first review the formal
reason why this leads to a lepton-number violating theory
and then analyze how this is possible for a particle
that is never its own antiparticle.
\\
Lepton conservation is induced, according
Noether's theorem, by the invariance
of (\ref{dirl1}) under the following continuous 
transformation group.
The charged lepton field e and 
neutrino field $\nu$ are simultaneously 
transformed via\cite{roman2}:
\begin{eqnarray}
\nu^{\prime}= e^{i \alpha} \nu
\nonumber
\\
\bar{\nu}^{\prime} =\bar{\nu} e^{- i \alpha} 
\nonumber
\\
e^{\prime} = e^{i \alpha} e
\label{tou}
\end{eqnarray}
Considering
$\alpha$ infinitesimal for the 
infinitesimal field transformation
$\delta \Psi$, Noether's theorem yields 
lepton conservation.
The standard model Lagrangian
with the addition of a non standard-model
Majorana mass term $m_{Maj}$:   
\begin{equation}
L_{\rm Weyl}^{\rm SM} =
\bar{\nu}_L \gamma_{\mu} {\partial \over {\partial x_\mu}} \nu_L
+ i g/\sqrt{2} \left[W_{\mu}^- \bar{e}_L \gamma_{\mu} \nu_L + H.C. \right]
+ \left[ m_{Maj} {\bar{\nu}_L} (\nu_L)^c + H.C. \right]
\label{dirl1m}
\end{equation}
The treatment of section \ref{uniequi} continues to hold.
This means:
\\
1. Lagrangian eq.(\ref{dirl1m}) is phenomenologically
equivalent to the Lagrangian
\begin{eqnarray}
L_{\rm Maj}^{\rm Weyl-equivalent}=
\bar{\nu}_M
\gamma_{\mu} {\partial \over {\partial x_\mu}} \nu_M
+ i g/\sqrt{2} \left[ W_{\mu}^- \bar{e}_L \gamma_{\mu} \nu_M(h=-1) 
+ H.C.\right]
\nonumber
\\
+ \left[ m_{Maj} \nu_M  {\bar{\nu}_M} + H.C. \right]
\label{dirma1}
\end{eqnarray}
2. assuming the validity of the standard-model
gauge sector the neutrino is definitely
not a Majorana field.
\\
m$_{Maj}$ violates
the invariance of eq.(\ref{dirl1m}) under the transformation
group eq.(\ref{tou}) because the Majorana mass term acquires
a phase of e$^{2 i \alpha}$ under 
transformation (\ref{tou}).
\\
What is the mechnanism with which a Weyl field, which is
never its own charge conjugate, violates lepton number? 
Consider the state of a Weyl field with a Majorana 
mass in an inertial
frame at which it is a rest:
\begin{equation}
\gamma_4  i {\partial \over {\partial t}} \nu_L(\rm{rest})
= m_{Maj}  (\nu_L)^c(\rm{rest})
\end{equation}
A solution to this equation in the Weyl representation is:
$\nu_L(\rm{rest})$ = $\left( \begin{array}{c} 0 \\ \phi_0+i \sigma_2 
\phi_0^*\end{array} \right)$ with
$\phi_0= a e^{imt}$. This can be rewritten
as:
\begin{equation}
\nu_L(rest) = {1\over \sqrt{2}}
(\nu_D(\rm{rest})+\nu_D(\rm{rest})^c)_L = \nu_ L + (\nu^c)_L
\end{equation}
where $\nu_D(\rm{rest})$ = $\left( \begin{array}{c} \phi_0 \\ \phi_0 
\end{array} \right)$ describes a Dirac particle at rest.
This result means: in the rest frame
the Weyl spinor consists of the 
components ``left-handed neutrino $\nu_L$''
(helicity = $-$1) and ``left-handed antineutrino $(\nu^c)_L$''
(helicity=+1) which are {\it not}
their respective charge conjugates. 
A Lorentz boost along the z-axis can be shown to transform
$\nu_L(\rm{rest})$ (with helicity=0) into a state
which is predominantly $\nu_L$ (helicity=$-$1)
or $(\nu^c)_L$ (helicity=+1). I.e. depending on the inertial
frame, a massive Weyl particle is predominantly particle
or antiparticle. In no frame it is its own antiparticle, however. 
\\
With a Majorana mass
term a Weyl field can thus violate lepton conservation,
{\it without being its own antiparticle}.

\section{Summary}
\label{summary} 
Within the present framework of field 
theory, a theory
 ``A'': ``the neutrino is a Majorana particle and
its weak-interaction is characterised
by Lagrangian eq.(\ref{dirm1}) (which is different from the
one expected in standard model)'' and theory ``B'':
``the neutrino is a Weyl particle
and the standard-model gauge sector 
is strictly valid'' are phenomenologically completely
equivalent. Therefore - without a powerful theory like
the standard model that 
quantitatively predicts the form of the neutrino
weak interaction {\it without any recourse to measured neutrino properties} -
to go from theory ``A'' to ``B'' is merely a change of designations.
This is how the equivalence between Weyl and Majorana neutrinos
became conventional wisdom.
However, 25 years of impressive experimental
confirmations of the standard model
convinced most particle physicists
that the gauge sector of future theories
is quantitatively described by this theory to 
good approximation. Under this - now very plausible -
assumption, theory ``B'' is realized in nature,
i.e. Majorana's idea of hermitian fermion fields
describing neutral fermions is not realized in nature
for the neutrino.
%The main physics content of this note is:
%{\it A theory in which the neutrino is a Majorana field
%with standard field normalization
%(i.e. (\ref{maj}) fulfilled)
%and the standard-model gauge sector 
%is quantitatively valid, is not realized in nature.}
%The manuscript finds that if, and only if, the stated assumptions
%are fulfilled
%the neutrino observed in nature happens to be no Majorana fermion.
This is a nontrivial constraint on all future theories.
I do not claim that a Majorana theory is inconsistent
in any sense: I only say that experimental results happen
to prefer a Weyl neutrino, without offering any theoretical
reason why this should be so. 
The present paper does not contradict any
publication in a refereed journal, because none 
analysed the {\it formal proof} of Weyl - Majorana equivalence
under the assumption of {\it quantitative} validity of the standard model.
\\
{\bf Acknowledgements}
I sincerely thank H.Haber, B.Kayser, W.Marciano and S.Pezzoni for
extensive discussions and explanations.
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\end{document}


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