
\documentclass[12pt]{article}
\def\eq#1\en{\begin{equation} #1 \end{equation}}

\usepackage{graphicx,color}

\def\pp#1{\partial_{#1}}

\begin{document}

%%%%%%%%%%% CERN Titlepage %%%%%%%%%%%

\begin{titlepage}
\begin{flushright}
\begin{tabular}{l}
CERN-TH/2002-303\\
IUB-TH/021\\
%\\
December 2002
\end{tabular}
\end{flushright}

\vspace*{0.5truecm}

\begin{center}
\boldmath
{\Large \bf The photon-neutrino interaction in
non-commutative gauge field theory and astrophysical bounds}
\unboldmath

\vspace*{0.5cm}

\smallskip
\begin{center}
{\sc {P. Schupp$^2$, J. Trampeti\'{c}$^{3,4}$, J. Wess$^{1,5}$, and G. Raffelt$^1$}}\\

\vspace*{1cm}

{\sl $^1${Max-Planck-Institut f\"ur Physik, F\"ohringer Ring 6, \\
80805 Munich, Germany}}, \\ 
{\sl $^2${International University Bremen, 
School of Engeneering and Science, Campus Ring 1,
28759 Bremen, Germany}},\\
{\sl $^3${Theory Division, CERN, CH-1211 Geneva 23, Switzerland},\\
$^4${Theoretical Physics Division, Rudjer Bo\v skovi\' c Institute, 
Zagreb, Croatia}},\\
{\sl $^5${Theoretische Physik, Universit\"at M\"unchen, Theresienstr. 37, \\
80333 Munich, Germany}}
\end{center}
\vspace{0.5truecm}
%\Sl{D} \Sl{=}
\vspace{1.5truecm}

{\large\bf Abstract\\[10pt]} \parbox[t]{\textwidth}{In this letter we
propose a mechanism of left- and right-handed neutrino couplings to
photons, which arises naturally in non-commutative gauge field theory.
We estimate the predicted additional energy-loss in stars induced by
space-time non-commutativity.  The usual requirement that any new
energy-loss mechanism in globular cluster stars must not significantly
exceed the standard neutrino losses implies a scale of non-commutative
gauge theory above the scale of weak interactions.  }

\end{center}

\end{titlepage}

\thispagestyle{empty}
\vbox{}
\newpage

\setcounter{page}{1}

%%% end CERN title page %%%%%%%%%%%%%


\noindent
Neutrinos do not carry a $U(1)$
(electromagnetic) charge and hence do not directly couple to Abelian
gauge bosons (photons) -- at least not in a commutative setting. In the presence of space-time
non-commutativity, it is, however, possible to couple neutral particles to 
gauge bosons via a star commutator. The relevant covariant derivative is
\begin{equation}
\widehat D_\mu \widehat \psi = \partial_\mu \widehat \psi - i \kappa e \widehat A_\mu \star \widehat \psi
+ i \kappa e \widehat\psi \star \widehat A_\mu \; , \label{ncc}
\end{equation}
with star product $\star$ and a coupling constant $\kappa e$ that
corresponds to a multiple (or fraction) $\kappa$ of the positron
charge $e$.  The star product is associative but in general not
commutative -- otherwise the proposed coupling to the non-commutative
photon field $\widehat A_\mu$ would of course be zero.  In
(\ref{ncc}), one may think of the non-commutative neutrino field
$\widehat \psi$ as having left charge $+\kappa e$, right charge
$-\kappa e$ and total charge zero. From the perspective of non-Abelian
gauge theory, one could also say that the neutrino field is charged in
a non-commutative analogue of the adjoint representation with matrix
multiplication replaced by the star product $\star$.  >From a
geometric point of view the photons do not directly couple to the
``bare'' commutative neutrino fields, but rather modify the
non-commutative background.  The neutrinos propagate in that
background.

Kinematically, a decay of photons into neutrinos is of course only
allowed for off-shell photons. That is still true in a constant or
sufficiently slowly varying non-commutative background: Such a
background does not lead to a violation of four-momentum conservation,
although it may break other Lorentz symmetries.

Physically, such a coupling of neutral particles to gauge bosons is
possible because the non-commutative background is described by an
antisymmetric tensor $\theta^{\mu\nu}$ that plays the role of an
external field in the theory \cite{ws,sw}.  The star product in
(\ref{ncc}) is a (nonlocal) bilinear expression in the fields and
their derivatives that takes the form of a series in
$\theta^{\mu\nu}$.  To lowest order we obtain:
\[
\widehat D_\mu \widehat \psi = \partial_\mu \widehat \psi +  
\kappa e \theta^{\nu\rho} \, \partial_\nu\widehat A_\mu \, \partial_\rho \widehat \psi
\; 
.
\]
A similar expansion (Seiberg-Witten map) exists 
for the non-commutative fields $\widehat \psi$, $\widehat A_\mu$
in terms of $\theta^{\mu\nu}$, ordinary `commutative' fields $\psi$, $A_\mu$ and their derivatives.
The scale of non-commutativity $\Lambda_{\rm NC}$ is fixed by choosing dimensionless matrix elements
$c^{\mu\nu}=\Lambda^2_{\rm NC} \theta^{\mu\nu}$ of order one.
Gauge invariance requires that all $e$'s in the action are multiplied by $\kappa$. To the order
considered in this letter $\kappa$ can be absorbed in a rescaling of $\theta$, i.e.\ a rescaling of
the definition of $\Lambda_{\rm NC}$.

The coupling (\ref{ncc}) is part of an effective model of 
particle physics involving neutrinos and photons
on non-commutative space-time. It
describes the scattering of particles 
that enter from an asymptotically commutative
region into a non-commutative interaction region.
The model satisfies 
the following requirements \cite{sny}--\cite{behr}:
\begin{enumerate}
\item[(i)] Non-commutative effects are described perturbatively. The action is
written in terms of assymptotic commutative fields. 
\item[(ii)] The action is gauge-invariant under $U(1)$-gauge transformations.
\item[(iii)] It is possible to extend the model to a non-commutative electroweak
model based on the gauge group $U(1) \times SU(2)$. An appropriate non-commutative 
electroweak model with $\kappa =1$ 
can in fact be constructed with the same tools that were used for
the noncommutative standard model of \cite{calmet}.
\footnote{For a model where only the neutrino has dual left and right charges
$\kappa = 1$ is required by the gauge invariance of the action.}
\end{enumerate}

The action of such an effective model
differs from  the commutative theory essentially by the presence of star products
and Seiberg--Witten maps. The Seiberg--Witten maps~\cite{SW} are necessary to express
the non-commutative fields $\widehat \psi$, $\widehat A_\mu$ that appear in the action
and that transform under non-commutative gauge transformations, in terms
of their asymptotic commutative counterparts $\psi$ and $A_\mu$.
The coupling of matter fields to Abelian gauge bosons is a non-commutative
analogue of the usual minimal coupling scheme.

The action for a neutral fermion that couples to an Abelian gauge boson 
in a non-commutative background is:
\begin{equation}
S = \int d^4 x \left(\,\bar{\widehat \psi} \star 
i\gamma^\mu\widehat D_\mu \widehat\psi
-m \bar{\widehat \psi} \star \widehat\psi\right).
\label{1}
\end{equation}
Here $\widehat \psi = \psi +e \theta^{\nu\rho} A_\rho \pp\nu \psi + 
\mathcal{O}(\theta^2)$ and 
$\widehat A_\mu = A_\mu + \theta^{\rho\nu}A_{\nu}
\left[\partial_{\rho}A_{\mu}-\frac{1}{2}\partial_{\mu}A_{\rho}\right]+
 \mathcal{O}(\theta^2)$ 
is the Abelian NC gauge potential expanded by the Seiberg-Witten map. 

To first order in $\theta$, the gauge invariant action reads
\begin{eqnarray}
&&S = \int d^4 x \, \left\{ \bar \psi 
\left[i\gamma^\mu \pp\mu  - m\left(1+e\theta^{\mu\nu}A_{\mu\nu}\right)\right]\psi
\right. \label{2}\\
&&\left. 
+ ie \theta^{\mu\nu} \left[(\pp\mu \bar \psi) A_\nu \gamma^\rho (\pp\rho \psi)
-  (\pp\rho\bar \psi) A_\nu \gamma^\rho  (\pp\mu  \psi)
+  \bar \psi (\pp\mu A_\rho) \gamma^\rho  (\pp\nu \psi) \right]\right\}.
\nonumber
\end{eqnarray}
The above action represents a tree-level interaction of photons and neutrinos.
We could also call it ``the background field anomalous-contact'' interaction.


%\begin{center}
%{\it 2. The plasmon decay to neutrino--antineutrino pairs}
%\end{center}
The model is meant to provide an effective description of space-time
non-commutativity. We are working at tree level. 
There are many open questions when one tries to include loops, see
e.g.\ \cite{UV/IR}.

We shall now apply our model to the decay of plasmons into neutrino
anti-neutrino pairs induced by a hypothetical stellar non-commutative
space-time structure. The resulting neutrinos can escape from the star
and thereby lead to an energy-loss.  To obtain the ``transverse
plasmon'' decay rate in stars on the scale of non-commutativity, we
start with the action determining the $\gamma\nu\bar\nu$ interaction.
From Eq. (\ref{2}) we extract, for left or right and possibly massive
neutrinos, the following Feynman rule for the gauge invariant
${\gamma}(q)\to {\nu}(k'){\bar {\nu}}(k)$ vertex in momentum space:
\begin{equation}
{\Gamma}^{\mu}_{\rm L \choose \rm R}({\nu}{\bar {\nu}}{\gamma})
=ie\frac{1}{2}(1 \mp \gamma_5)
\left[(q\theta k)\gamma^{\mu}+({\not \!k}-2m_{\nu}){\widetilde q}^{\mu}-
{\not \!q}{\widetilde k}^{\mu}\right].
\label{5}
\end{equation}
In the case of massless neutrinos the vertex (\ref{5}) becomes totally symmetric:
\begin{eqnarray}
{\Gamma}^{\mu}_{\rm L \choose \rm R}({\nu}{\bar {\nu}}{\gamma})
=ie\frac{1}{2}(1 \mp \gamma_5){\theta}^{\mu\nu\tau}k_{\nu}q_{\tau},\;\;
\label{4}
{\theta}^{\mu\nu\tau}={\theta}^{\mu\nu}\gamma^{\tau}+{\theta}^{\nu\tau}\gamma^{\mu}+
{\theta}^{\tau\mu}\gamma^{\nu}.
\nonumber
\end{eqnarray}

In a stellar
plasma, the dispersion relation of photons is identical with that of 
a massive particle \cite{Jancovici}--\cite{Salati}
\begin{equation}
q^2 \equiv {\rm E}_\gamma^2-{\bf q}_\gamma^2=\omega_{\rm pl}^2
\end{equation}
with $\omega_{\rm pl}$ being the plasma frequency. 

>From the gauge-invariant amplitude 
${\cal M}_{\gamma {\nu} {\bar{\nu}}}$ in momentum space 
for plasmon (off-shell photon) decay to the
left and/or right massive neutrinos in the NCQED, we have: 
\begin{eqnarray}
\sum_{\rm pol.} |{\cal M}_{\gamma {\nu} {\bar{\nu}}}|^2 =
 4e^2\left[\left(q^2 -2m_{\nu}^2\right)
 \left(\frac{5}{2}m_{\nu}^2{\widetilde q}^2-(q\theta k)^2\right) 
 +m_{\nu}^2 q^2 ({\widetilde k}^2-{\widetilde k}{\widetilde q})\right],
\nonumber
\end{eqnarray}
from where phase-space integration then gives:
\begin{eqnarray}
&&\Gamma(\gamma_{\rm pl}\rightarrow {\bar{\nu}}_{\rm L\choose \rm R}\nu_{\rm L\choose \rm R})
= \frac{\alpha}{48}\frac{\omega^6_{\rm pl}}{{\rm E}_{\gamma}\Lambda^4_{\rm NC}}
\sqrt{1-4\frac{m_{\nu}^2}{\omega_{\rm pl}^2}}\\
&&\times \left[\left(1-28\frac{m^2_{\nu}}{\omega^2_{\rm pl}}+
72\frac{m^4_{\nu}}{\omega^4_{\rm pl}}\right)
\sum_{i=1}^{3}(c^{0i})^2 + 
2\frac{m^2_{\nu}}{\omega^2_{\rm pl}}
\left(1-4\frac{m^2_{\nu}}{\omega^2_{\rm pl}}\right)
\sum_{{i,j=1 \atop i<j}}^{3}(c^{ij})^2\right].
\nonumber
\end{eqnarray}
Here we have used the notation ${\widetilde q}^{\mu}={\theta^{\mu\nu}q_{\nu}}$.

In the above expression we
parametrize the $c_{0i}$'s by introducing the angles characterizing 
the background ${\theta}^{\mu\nu}$ field of the theory \cite{HK}:
\begin{eqnarray}
c_{01}=\cos\xi,\;\;c_{02}=\sin\xi \;\cos\zeta,\;\;c_{03}=\sin\xi \;\sin\zeta,
\nonumber
\end{eqnarray}
where $\xi$ is the angle between the ${\vec E}_{\theta}$ field and 
the direction of the incident beam,
i.e.\ the photon axes. The angle $\zeta$ defines the origin of the $\phi$ axis.  
The $c_{0i}$'s are not independent; in pulling out the overall scale 
$\Lambda_{\rm NC}$ we can always
impose the constraint ${\vec E}^2_{\theta}\equiv \sum_{i=1}^3 (c^{0i})^2=1$. 
Here we consider three physical cases:
$\xi=0,\;\pi/4,\;\pi/2$, which for $\zeta = \pi/2$ satisfy the imposed constraint.
This parametrization provides a good physical interpretation of the NC effects \cite{HK}.

In the rest frame of the medium, the decay rate of a ``transverse plasmon'',
of energy ${\rm E}_\gamma$ for the left--left and/or right--right 
massless neutrinos and for the constraint 
${\vec E}^2_{\theta}=1$, is given by
\begin{equation}
\Gamma_{\rm NC}
(\gamma_{\rm pl}\to \nu_{\rm L\choose \rm R}\bar\nu_{\rm L\choose \rm R})
=\frac{\alpha}{48}\,\frac{1}{\Lambda_{\rm NC}^4}\,
\frac{\omega_{\rm pl}^6}{{\rm E}_\gamma}\,.
\end{equation}

The Standard Model (SM) photon--neutrino interaction at tree level
does not exist. However, the effective photon--neutrino--neutrino
vertex $\Gamma^{\mu}_{\rm eff}(\gamma\nu\bar\nu)$ is generated through
1-loop diagrams, which are very well known in heavy-quark physics as
``penguin diagrams.'' Such effective interactions give a non-zero charge
radius as well as a contribution to the ``transverse plasmon''
decay rate \cite{Bernstein}--\cite{BPV};  for details 
and further references 
see
Ref.~\cite{as}. Note that the dipole moment operator $\sim
em_{\nu}G_{\rm F}{\bar\psi}\sigma_{\mu\nu}\psi A^{\mu\nu}$, also
generated by the ``neutrino-penguin diagram,'' gives very small
contributions because of the smallness of the neutrino mass, i.e.\
$m_{\nu}< 1$~eV \cite{nobel}.

The corresponding SM neutrino-penguin-loop result for the ``transverse plasmon'' decay rate is \cite{as}:
\begin{equation}
\Gamma_{\rm SM}\left(\gamma_{\rm pl}\to {\nu_{\rm L}}{\bar\nu}_{\rm L}\right)
=\frac{{\rm c}_{\rm v}^2 G_{\rm F}^2}{48\pi^2 \alpha}\;\frac{\omega_{\rm pl}^6}{{\rm E}_\gamma}.
\end{equation}
For $\nu_e$ we have ${\rm c}_{\rm v}=\frac{1}{2}+2\sin^2\Theta_{\rm W}$
while for $\nu_\mu$ and $\nu_\tau$ we have
${\rm c}_{\rm v}=-\frac{1}{2}+2\sin^2\Theta_{\rm W}$.  Comparing the
decay rates into all three neutrino families we thus need
to include a factor of~3 for the NC result, while 
${\rm c}_{\rm v}^2 = 0.79$ for the SM result \cite{rpp}. From the ratio of the rates 
\begin{equation}
\Re\equiv\frac{\sum_{\rm flavours}
\Gamma_{\rm NC}
\left(\gamma_{\rm pl}\to {\nu_{\rm L}}{\bar\nu}_{\rm L} + {\nu_{\rm R}}{\bar\nu}_{\rm R}\right)}
{\sum_{\rm flavours}\Gamma_{\rm SM}(\gamma_{\rm pl}\to {\nu_{\rm L}}{\bar\nu}_{\rm L})}
=\frac{6\pi^2\alpha^2}{{\rm c}_{\rm v}^2 G_{\rm F}^2\Lambda_{\rm NC}^4},
\end{equation}
we obtain:
\begin{equation}
\Lambda_{\rm NC}\;= \;\frac{80.8}{\Re^{1/4}} \;(\rm GeV).
\end{equation}
A standard argument involving globular cluster stars tells us that any
new energy-loss mechanism must not exceed the standard neutrino losses
by much; see section 3.1 in Ref. \cite{raffelt}.
Put another way, we should approximately
require $\Re<1$, translating into
\begin{equation}
\Lambda_{\rm NC}>\left(\frac{6\pi^2\alpha^2}{{\rm c}_{\rm v}^2 G_{\rm F}^2}\right)^{1/4}
\cong 81~{\rm GeV}\,.
\end{equation}
If sterile neutrinos ($\nu_{\rm R}$) do not exist
the scale of non-commutativity is approximately $\Lambda_{\rm NC}> 68~{\rm GeV}$.

%\begin{center}
%{\it 3. Discussion and conclusions}
%\end{center}

The advantage of our approach to the anomalous $\gamma\nu\bar\nu$
interaction, via non-commutative Abelian gauge field theory, lies in
the fact that, contrary to the SM approach, photons are also coupled
to the sterile neutrinos in the same, U(1)-gauge-invariant, way as the
left-handed ones.  The electromagnetic gauge invariance of the
$\gamma\nu\bar\nu$ amplitude comes automatically, since the starting
action is manifestly U(1)-gauge-invariant. The interaction (\ref{2})
produces extra contributions relative to the SM in the non-commutative
background.

The non-commutativity scale depends on the requirement $\Re<1$ and
from that aspect the constraint $\Lambda_{\rm NC}> 80~{\rm GeV}$,
obtained from the energy loss in the globular stellar clusters,
represents a lower bound on the scale of non-commutative gauge field
theories.  It also depends on the strength of the non-commutative
coupling which we have taken to be $\kappa=1$.

Compared to other bounds, see \cite{HK}, our limit on $\Lambda_{\rm
NC}$ is relatively low. However, it is based on a completely new
interaction channel and a completely different ``laboratory''
than other constraints and as such appears worth communicating.

%Finally, note that the bound $\Lambda_{\rm NC}> 80~{\rm GeV}$ applies
%to red giants and white dwarfs where the soft regime is relevant
%\cite{as}.  However, in supearnovae, etc., i.e.\ at higher energies,
%higher bounds on the scale of non-commutativity are expected
%\cite{radii}.

%We hope that, in the near future, more sophisticated methods to
%observe and more accurate techniques to measure the energy loss in the
%stellar clusters will produce more restricting limits to requirement
%$\Re<1$, something like $\Re<1/10$, and consequently firmer bound.  \\
%\vspace{0.1cm}

We would like to thank L.\ Stodolsky for stimulating discussions.
J.T.\ would like to thank L.\ Alvarez-Gaume and A.\ Armoni for many helpful discussions.

The work of J.T.\ is supported by the Ministry of Science and Technology 
of Croatia under Contract No. 0098002.
G.R.\ acknowledges partial support from the Deutsche
Forschungsgemeinschaft under grant No.\ SFB 375.


\begin{thebibliography}{99}

\bibitem{sny} 
H.~S.~Snyder,
%``Quantized Space-Time,''
Phys.\ Rev.\  {\bf 71}, 38 (1947).
%%CITATION = PHRVA,71,38;%%

\bibitem{Filk}
T.~Filk,
%``Divergencies In A Field Theory On Quantum Space,''
Phys.\ Lett.\ {\bf B376}, 53 (1996).
%%CITATION = PHLTA,B376,53;%%

\bibitem{Cerchiai}
           B.~L.~Cerchiai and J.~Wess,
           %``q-Deformed Minkowski Space based on a q-Lorentz Algebra,''
           Eur.\ Phys.\ J.\ C {\bf 5}, 553 (1998).
           %.
           %%CITATION = ;%%

\bibitem{con} A. Connes, Noncommutative Geometry (Academic Press, London, 1994);
              A. Connes and J. Lott, Nucl. Phys. {\bf 18}, 29 (1990) (Proc. Suppl.).

\bibitem{CDS} A. Connes, M.R. Douglas and A. Schwarz, JHEP {\bf 9802}, 003 (1998).

\bibitem{DH}  M.R. Douglas, and C. Hull, JHEP {\bf 9802}, 008 (1998);
              Y.-K.E. Cheung and M. Krogh, Nucl. Phys. {\bf B528}, 185 (1998);
              C.-S. Chu and P.-M. Ho, Nucl. Phys. {\bf B550}, 151 (1999) 
	      and {\bf B568}, 447 (2000).

\bibitem{rev}  M.R. Douglas and N.A. Nekrasov, 
               Rev. Mod. Phys. {\bf 73}, 977 (2001), ;
               R.J. Szabo, v3.


\bibitem{SW} N. Seiberg and E. Witten, JHEP {\bf 9909}, 032 (1999).

\bibitem{WESS} J.~Madore et al., %S.~Schraml, P.~Schupp and J.~Wess,
           %``Gauge theory on noncommutative spaces,''
           Eur. Phys. J. {\bf C16}, 161 (2000);
           %%CITATION = ;%
           B.~Jurco et al., %, S.~Schraml, P.~Schupp and J.~Wess,
           %``Enveloping algebra valued gauge transformations for non-Abelian gauge 
	   %groups on non-commutative spaces,''
           Eur. Phys. J. {\bf C17}, 521 (2000), ;
           %%CITATION = ;%%
           B.~Jurco,  P.~Schupp and J.~Wess,
           %``Nonabelian noncommutative gauge theory via noncommutative extra  dimensions,''
           Nucl. Phys. {\bf B604}, 148 (2001), ;
           %%CITATION = ;%%
           B.~Jurco et al., %, L.~Moller, S.~Schraml, P.~Schupp and J.~Wess,
           %``Construction of non-Abelian gauge theories on noncommutative spaces,''
           Eur. Phys. J. {\bf C21}, 383 (2001), ;
           %%CITATION = ;%%
           P.~Schupp,
           %``Non-Abelian gauge theory on noncommutative spaces,''
           PRHEP-hep2001/238 (2001), .
           %%CITATION = ;%%
           
\bibitem{Zumino}
           D.~Brace et al., %, B.~L.~Cerchiai, A.~F.~Pasqua, U.~Varadarajan and B.~Zumino,
           %``A cohomological approach to the non-Abelian Seiberg-Witten map,''
           JHEP {\bf 0106}, 047 (2001), ;
           %%CITATION = ;%%    
           A.~A.~Bichl et al., %J.~M.~Grimstrup, L.~Popp, M.~Schweda and R.~Wulkenhaar,
           %``Deformed QED via Seiberg-Witten map,''
           ; Eur. Phys. J. {\bf C24}, 165 (2002), .
           %%CITATION = ;%%     

\bibitem{calmet}  X. Calmet et al., 
           %``The standard model on non-commutative space-time,''
           Eur. Phys. J. {\bf C23}, 363 (2002), .
           %%CITATION = ;%%

\bibitem{behr}  W. Behr et al.,  v2; P. Aschieri et al., .		
			
\bibitem{ws} J. Wess: {\it Gauge theories on non-commutative spaces},
             Lecture given at the VIII Adriatic Meeting, Dubrovnik, 2001, to appear as  
             ``Lecture Notes in Physics'', Eds. J. Wess and J. Trampeti\' c (Springer, Heidelberg, 2002).

\bibitem{sw} P. Schupp: {\it Particle physics on non-commutative space-time}, ibid.
	     
\bibitem{Jancovici} B. Jancovici, Nuovo Cim. {\bf 25}, 428 (1962).

\bibitem{Altherr} T. Altherr and U. Kraemmer, Astroparticle Phys. {\bf 1}, 133 (1992);\\
                  E. Bratten and D. Segel, Phys. Rev. {\bf D48}, 1478 (1993).

\bibitem{Salati}  P. Salati,  Astroparticle Phys. {\bf 2}, 269 (1994).
                
\bibitem{HK}  I. Hinchliffe and N. Kersting, Phys. Rev. {\bf D64}, 116007 (2001); hep-ph/ 0205040;
              A. Anisimov et al., Phys. Rev. {\bf D65}, 085032, (2002), ;
              I. Mocioiu, M. Pospelov and R. Roiban, Phys. Lett. \ B {\bf 489}, 390 (2000), ;
              C.E. Carlson, C.D. Carone and R.F. Lebed,  
	      Phys. Lett. \ B {\bf 518}, 201 (2001), .
	      J.L. Hewett, F.J. Petriello and T.G. Rizzo, Phys. Rev. {\bf D64}, 075012 (2001),
               and Phys. Rev. {\bf D66}, 036001 (2002), . 

\bibitem{Bernstein} J. Bernstein and T.D. Lee, Phys. Rev. Lett. {\bf 11}, 512 (1963); \\
                    W.A. Bardeen, R. Gastmans
                    and B. Lautrup, Nucl. Phys. {\bf B46}, 319 (1972); 
		    S.Y. Lee, Phys. Rev. {\bf D6}, 1701 (1972); 
		    B.W. Lee and R.E. Shrock, Phys. Rev. {\bf D16}, 1444 (1977).
		    
\bibitem{dsm} J.M. Cornwall, Phys. Rev. {\bf D26}, 1453 (1982);
              J.M. Cornwall and J. Papavassiliou, Phys. Rev. {\bf D40}, 3474 (1989);
              J. Papavassiliou, Phys. Rev. {\bf D41}, 3179 (1990);
	      G. Degrassi, A. Sirlin and W. Marciano, Phys. Rev. {\bf D39}, 287 (1989);
              G. Degrassi and A. Sirlin, Phys. Rev. {\bf D46}, 287 (1992). 
	      
\bibitem{as}  T. Altherr and P. Salati, Nucl. Phys. {\bf B421}, 662 (1994).		      

\bibitem{BPV} J. Bernab\' eu, J. Papavassiliou and J. Vidal, ;
              L.G. Cabral-Rosetti, M. Moreno and A. Rosado, .	      

\bibitem{nobel} 
O.~Elgaroy et al., Phys. Rev. Lett. {\bf 89}, 061301 (2002). 
S.~Hannestad, .

%Super-Kamiokande Collaboration, Phys. Rev. Lett. {\bf
%                86}, 5651 (2001); Phys. Lett. {\bf B539}, 179 (2002).

\bibitem{UV/IR} S. Minwalla, M. Van Raamsdonk, N. Seiberg, ;
                A. Matusis, L. Susskind and N. Toumbas, JHEP {\bf 0012}, 002 (2000);
		C.P. Martin and D. Sanchez-Ruiz, Phys. Rev. Lett. {\bf 83}, 476 (1999); 
		H. Grosse, T. Krajewski and R. Wulkenhaar, ;
                A. Armoni, Nucl. Phys. {\bf B593}, 229 (2001);
                I. Chepelev and R. Roiban, JHEP {\bf 0005}, 037 (2000) and {\bf 0103}, 001 (2001);
                L. Griguolo and M. Pietroni, JHEP {\bf 0105}, 032 (2001); J.M. Grimstrup et al., ;
                F.R. Ruiz, Phys. Lett. \ B {\bf 502}, 274 (2001).   

\bibitem{rpp}  Review of Particle Properties, Eur. Phys. J. {\bf C15}, 1 (2000). 

\bibitem{raffelt} G. Raffelt, Annu. Rev. Nucl. Part. Sci. {\bf 49}, 163 (1999).

%\bibitem{radii} Work in progress.

\end{thebibliography}

\end{document}

















