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\begin{document}





\preprint{BUTP-2003/02}
\preprint{CERN-TH/2003-012}
\preprint{IUB-TH/031}

\title{Non-commutative ``$\star$--charge radius'' and ``$\star$--dipole moment''
of the neutrino}
\author{P. Minkowski}
\email{Peter.Minkowski@cern.ch}
\affiliation{Theory Division, CERN, CH-1211 Geneva 23, Switzerland}
\affiliation{Institute for Theoretical Physics, University of Bern,
CH-3012 Bern, Switzerland}
\author{P. Schupp}
\email{P.Schupp@iu-bremen.de}
\affiliation{International University Bremen, 
School of Engineering and Science, Campus Ring 1,
28759 Bremen, Germany}
\author{J. Trampeti\'{c}}
\email{Josip.Trampetic@cern.ch}
\affiliation{Theory Division, CERN, CH-1211 Geneva 23, Switzerland}
\affiliation{Theoretical Physics Division, Rudjer Bo\v skovi\' c Institute, 
Zagreb, Croatia}

\date{\today}




\begin{abstract}
In this letter we propose to obtain a lower bound for the mass scale of space-time non-commutativity,
considering the photon--neutrino interaction. 
We evaluate a non-commutative ``$\star$--dipole moment'' and ``$\star$--charge
radius''.
{}From the ``$\star$--dipole moment'' 
we obtain the bound $\Lambda_{\rm NC}\;\geq M^*=2\,\pi\,(\sqrt 2
G_F)^{-1/2}{\simeq}\;1.5$ TeV by comparing it with the standard model contribution
assuming it to be correct. 
This is a generic estimate, whereas details depend on the neutrino mass matrix.
Next we find that the ``$\star$--charge radius'' depends on the non-commutativity between space and time as 
$\langle r^2_{\nu}\rangle^2_{\rm NC}=3\sum_{i=1}^3 ({\theta}^{0i})^2$, and is 
$\stackrel{<}{\sim} 0.3 \times 10^{-33}\; {\rm cm}^2$ at a scale of non-commutativity of $\Lambda_{\rm NC}
\geq 1.5\;\rm TeV$.
This number is smaller than the bounds obtained from astrophysics. 
\end{abstract}

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In this letter we compare the hypothetical neutrino photon interaction induced by space time 
non-commutativity~\cite{rstw}, with the electomagnetic properties of neutrinos: charge radius
and dipole moments. The latter probe distances equivalent to $10^{-31}$ cm as seen through electromagnetic
interactions at long range.

The action of the effective model that we would like to study,
differs from the commutative theory by the presence of $\star$-products
and Seiberg--Witten (SW) maps~\cite{SW,WESS,Zumino,calmet}. In the presence of space-time
non-commutativity,  neutral particles can couple to 
gauge bosons via a ${\star}$-commu\-ta\-tor 
\begin{equation}
D^{\mbox{\scriptsize NC}}_\mu \widehat \psi = \partial_\mu \widehat \psi - i e \widehat A_\mu \star \widehat \psi
+ i e \widehat\psi \star \widehat A_\mu \;,
\label{ncc}
\end{equation}
where a hat denotes non-commutative fields that are  expanded in terms of regular fields
via SW maps.
The \mbox{$\star$-}products originate from antisymmetric tensor fields
that can conceivably be traced back to an extension of gravity.
In the language of quantized gravity non-commutative effects belong to target space,
a quantum deformation of classical base space.
The \mbox{$\star$-}products  induce on target space an algebraic structure of position 
operators that define non-commutative space-time.
Observable effects are in any case not necessarily fixed to the Planck scale.


The effective model of neutrino and photon interactions
in non-commutative space-time \cite{rstw}
provides a description of  the interaction of particles 
that enter from an asymptotically commutative
region into a non-commutative interaction region \cite{WESS}. 
The expansion in the form proposed 
in \cite{SW,WESS,Zumino,calmet} is understood as a perturbative description of
non-commutativity of the target space variables. 
This approach currently neither tries to 
describe the dynamics of the non-commutative structures
nor does it include non-perturbative effects.
The action, written in terms of commutative fields, 
is gauge-invariant under $U(1)_{\rm em}$-gauge transformations.
The requirements satisfied by our model are summarized in \cite{ws}.
For related work on non-commutative gauge theory and phenomenology,
see \cite{HK}.

Expanding the $\star$-product in (\ref{ncc}) to first order in the antisymmetric
(Poisson) tensor $\theta^{\mu\nu}$, we find the following covariant derivative
on neutral spinor fields
\begin{equation}
D^{\mbox{\scriptsize NC}}_\mu \widehat \psi = \partial_\mu \widehat \psi + 
e \theta^{\nu\rho} \, \partial_\nu\widehat A_\mu \, \partial_\rho \widehat \psi
\; .
\label{ncc1}	
\end{equation}
We shall treat $\theta^{\mu\nu}$ as a constant background field of strength 
$|\theta^{\mu\nu}|=1/\Lambda^2_{NC}$ that models the non-commutative structure
of space-time in the neighbourhood of the interaction region~
\footnote{We assume $\theta$ to be constant in first approximation,
but only in the interaction region.
A NC background ``ether'' that is truely constant on macroscopic distances and time scales
is essentially ruled out \cite{HK}. This is compatible with our expectation that
NC effects are relevant only at very short distances and high energies.}.
As $\theta^{\mu\nu}$ is not invariant under Lorentz transformations, the
neutrino field can pick up angular momentum in the interaction.

The gauge-invariant action for a neutral fermion that couples to an Abelian gauge boson 
via (\ref{ncc1}) is 
\begin{eqnarray}
S &=& \int d^4 x \, \left\{ \bar \psi 
\left[i\gamma^\mu \pp\mu  - m\left(1+e\theta^{\mu\nu}F_{\mu\nu}\right)\right]\psi
\right.
\nonumber\\
&-& \left.ie \left[{\bar{\psi}}\Big(\frac{1}{2}\theta^{\mu\nu}{\gamma^{\rho}} +
\theta^{\nu\rho} \gamma^{\mu}\Big) F_{\mu\nu}\pp\rho\psi \right]\right\}
\label{ncc2}
\end{eqnarray}
up to first order in $\theta$ \cite{rstw,tram}.
In the derivative part of the action (\ref{ncc2}),
the neutrino couples to the gradient of the field strengths 
(modulo partial integration.)
This is an interaction of the Stern-Gerlach type \cite{sg}.


The standard model (SM) photon-neutrino effective vertex is
generated through 1-loop diagrams, the so-called ``neutrino-penguin'' diagrams. 

The hermitian flavour antysymmetric magnetic and electric dipole moment operators 
in Majorana 4-component notation are
\begin{eqnarray}
{D \choose D_5}^{\alpha\beta}_{jk} = 
e\,\psi^{\top}_{\rm j}\left[\rm C \;\sigma^{\alpha\beta}{i \mbox{\bf 1} \choose \gamma_5}\right]\,\psi_{\rm k}\;.
\label{2a}
\end{eqnarray}
The flavour indices j, k denote the the $\nu$--flavours in weak interaction space set by the
charged lepton flavours $e^{\mp}, \, \mu^{\mp},\, \tau^{\mp}$. They receive contributions
proportional to the neutrino masses, of the chiral and flavour specific form
\begin{eqnarray}
d_{\rm jk}=m^{(\nu)}_{\rm jk}\Delta_{\rm jk},\;\; 
\Delta_{\rm jk}=L\left(m^{\ell^{-}}_{\rm j}\right)-L\left(m^{\ell^{-}}_{\rm k}\right)
\nonumber
\\
m^{(\nu)}_{\rm jk}=m^{(\nu)}_{\rm kj}, \,\,\, d_{\rm jk}=-d_{\rm kj}, \;\rm and \;\rm h.c.,
\label{2b}
\end{eqnarray}
where the L's necessarilly contain dependence on the masses of charged leptons.  
In Eq. (\ref{2b}) $m^{(\nu)}_{\rm jk}$ denotes the complex 3$\times$3 symmetric neutrino mass matrix, 
reduced from the six flavour 
case to the three light mass eigenfields (eigenstates), in chiral flavour space
The electric transition dipole moment is proportional to the imaginary part of the mass matrix $m^{(\nu)}$,
while the magnetic one is given by the real part. \cite{mink}.
The dipole moments $d_{\rm jk}$ are complex antysymmetric in lepton flavour space. In the neutrino mass extended SM
the leading contribution is found to be
\begin{eqnarray}
\Delta_{\rm jk}\cong \frac{eG_F}{2\sqrt2\,\pi^2} 
{\rm log} \left(\frac{m^{\ell^{-}}_{\rm k}}{m^{\ell^{-}}_{\rm j}}\right)=
\frac{e}{M^{*2}}{\rm log} \left(\frac{m^{\ell^{-}}_{\rm k}}{m^{\ell^{-}}_{\rm j}}\right),
\label{2c}
\end{eqnarray}
by implementing the Majorana condition. Here $M^*=2\pi\,v=1.55$ TeV, where $v=(\sqrt 2\, G_F)^{-1/2}=246$ GeV 
represents the vacuum expetation value of the scalar Higgs field. 

Note, the flavour cancellation mechanism operating for the light Majorana neutrino dipole moments. 
It is generated through the smallness of light neutrino masses controled by their heavy counterparts,
combined with the just logarithmic charged mass asymmetry in Eq.\ (\ref{2c}). 
For Dirac neutrinos we refer to \cite{Voloshin:ty,VVoloshin:ty}.

Independently the same mechanism allows us to calculate the rate of
Higgs decay into two light neutrinos. 
In the case where heavy--light mixing
is neglected the decay rate of the 1--doublet Higgs scalar to all three light neutrino flavours,
denoted by $\zeta =1,2,3$, is determined and proportional to
the square of their masses. The relevant ratio of rates is 
\begin{eqnarray}
\frac{\Gamma\left(H\rightarrow (\nu\bar\nu)_{\zeta}\right)}{\Gamma\left(H\rightarrow \tau^+\tau^-\right)}\;
\sim \; 3\times 10^{-23}\,[{|m_{\zeta}|}/{10^{-2}\,\rm eV}]^2.
\label{2ch}
\end{eqnarray}

The neutrino dipole moments (\ref{2b}) violate lepton number by $\pm2$ and for 
complex neutrino mass matrix violate independently CP.

The electromagnetic dipole moments universally reduce to the three light 
mass eigenfields (eigenstates) of neutrinos and antineutrinos,
even though the mediating interactions procede through light and heavy weak 
interaction eigenfields (eigenstates) involving
minimally six flavours.
This extends the analogous situation pertaining to neutrino and antineutrino 
oscillations, valid at energies, in production and 
detection, far below the masses of heavy neutrino flavors \cite{mink}. 
The CKM--like parametrization is generated by diagonalizing the light--flavour 
mass matrix in (\ref{2b}). 
The corresponding analytic structure is quite definite,
yet often globally referred to as see--saw mechanism \cite{gell}.
Specific mass patterns for light neutrino flavors arising from approximate discrete
symmetries are discussed in \cite{fri}.

The dipole moments receive very small contributions because of the smallness of the neutrino mass, 
$|m_{\nu}|\simeq 10^{-2}$ eV \cite{nobel}.
The order of magnitude of both, electric and magnetic transition moments is given by:
\begin{eqnarray}
{\cal O}\left(d^{\rm mag}_{\rm el}(\nu)\right) = \left(\frac{|m_{\nu}|}{10^{-2}\,\rm eV}\right)\frac{e\,{\cal C}}{2m_e}\;,
\label{6a}
\end{eqnarray}
where the dimensionless quantity 
\begin{eqnarray}
{\cal C} = \frac{2m_e\,G_F}{2\sqrt2\,\pi^2}{10^{-2}\,\rm eV} \cong 0.43\times 10^{-20}.
\label{6b}
\end{eqnarray}
Transforming the electric dipole moment to [$\rm e\,cm$] units Eq.(\ref{6b}) gives 
\begin{eqnarray}
{\cal O}\left(d_{\rm el}(\nu)\right) \cong 0.82 \times 10^{-31}
 \;\left(\frac{|m_{\nu}|}{10^{-2}\,\rm eV}\right)\;[\rm e\,cm]\,,
\label{6c}
\end{eqnarray}
while for the magnetic dipole moment we have
\begin{eqnarray}
\mbox{}\hspace{-1.5em} {\cal O}\left(d^{\rm mag}(\nu)\right) 
&\cong& 2.09 \times 10^{-31} \left(\frac{|m_{\nu}|}{10^{-2}\,\rm eV}\right)[\rm MeV/T] 
\nonumber \\
&\cong& 0.38 \times 10^{-9}\left(\frac{|m_{\nu}|}{10^{-2}\,\rm eV}\right) [\rm rad/(sT)]\,.
\label{6d}
\end{eqnarray}

The order of magnitude of electric and magnetic dipole moments, (\ref{6c},\ref{6d}), suggests a bridge to Planck
distances i.e. to gravitational interactions.
Note that the light neutrino flavours are protected by the smallness of their mass over all scales ranging from the 
Bohr radius to $\sim 10^{-17}$ cm. On the line of the non-commutative effects this protection mechanisms is not
valid in general. This protection mechanism is restricted only to non-gravity interactions. Hovever, owing to the gravity
coupling to energy this mechanism automatically applies, while for the extensions of gravity
(for instance non-commutativity) this is not garanteed.

It has become an unwarranted tacit implication in many discussions of how the SM is to be extended to include
non-vanishing neutrino masses, that together with the introduction of mass 
the number of chiral neutrino flavours doubles.
The words used are most confusing in our mind and say: massive neutrinos are assumed to be Dirac spin 1/2 fermions.
While such a doubling of basic $\nu$-states cannot readily be ruled out, we assume here that it is
not natural and thus follow the opposed logic, where the statement is: neutrinos are Majorana spin 1/2 fermions.

We can extract an upper limit on the $\star$-gradient interaction by replacing  
$\pp\mu \psi$ by $m_\nu \psi$.
The strenght of the interaction (\ref{ncc2}) becomes $m_{\nu}|e\,\theta \,F|$ to be compared with dipole
interaction $|F\, d(\nu)|$ from Eq. (\ref{6c}). Thus we derive the bound on $\Lambda_{NC}$
\begin{eqnarray}
\Lambda_{NC} \geq \sqrt{\frac{em_{\nu}}{d(\nu)}}\simeq 1.55 \,{\rm TeV} = M^*.
\end{eqnarray}
We would like to point out that this bound on the non-commutativity scale involves
only  basic properties of neutrinos and photons.\\

Next we determine the radius of the photon--neutrino interaction, 
evaluating the quantity that we call 
the neutrino $\star$-charge radius $\langle r^2_{\nu}\rangle_{\rm NC}$.  
Since non-commutativity can be a source of ``transvers plasmon'' decay 
into neutrino--antineutrino pairs \cite{rstw}, this is to be compared with 
the same process induced by the bona fide neutrino charge radius defined by the axial
electromagnetic form factor:
\begin{eqnarray}
\langle r^2\rangle =6\left[{\frac {\partial F_{\nu}(q^2)}{\partial q^2}}\right]_{q^2=0}; \
\left[F_{\nu}(q^2)\right]_{q^2 \to 0} \longrightarrow
\frac{q^2}{6}\langle r^2\rangle.
\label{1}
\end{eqnarray}

The electromagnetic interaction 
in the limit of massless Majorana neutrinos \cite{Jancovici}--\cite{hnr}
corresponds to:
\begin{eqnarray}
\langle r^2\rangle \cong \frac{G_F}{2\sqrt2\,\pi^2} \;{\rm log} \left(\frac{m_W}{m_{\ell}}\right)
=\frac{1}{M^{*2}} \;{\rm log} \left(\frac{m_W}{m_{\ell}}\right)
\label{2}
\end{eqnarray}
It contributes to the ``transverse plasmon'' decay rate \cite{Bernstein}--\cite{BPV}. 


To estimate the $\star$-charge radius $\langle r^2_{\nu}\rangle_{\rm NC}$ we first 
evaluate the partial width:
\begin{eqnarray}
\Gamma_{\rm SM}(\gamma \rightarrow {\bar\nu}_{\rm L}\nu_{\rm L})=
\frac{\alpha}{4}\frac{q^2}{E_{\gamma}}F^2_{\nu}(q^2),
\label{7}
\end{eqnarray}
which gives the pure charge radius induced rate. 
The next step is to compare (\ref{7}) with the non-commutative rate
$\Gamma_{\rm NC}(\gamma \rightarrow {\bar\nu}_{\rm L}\nu_{\rm L})$.


{}From Eq. (\ref{ncc2}) we extract, for the left--chiral massless neutrinos, 
the following gauge invariant amplitude for  
the ${\gamma}(q)\to {\nu}(k'){\bar {\nu}}(k)$ vertex in momentum space \cite{rstw,tram}:
\begin{eqnarray}
{\cal M}_{\gamma\nu\bar\nu} & = & ie\bar\psi_L({\theta}^{\mu\nu\tau}k_{\nu}q_{\tau})\psi_L\,\epsilon_{\mu}(q),
\nonumber \\
{\theta}^{\mu\nu\tau} & = & {\theta}^{\mu\nu}\gamma^{\tau}+{\theta}^{\nu\tau}\gamma^{\mu}+
{\theta}^{\tau\mu}\gamma^{\nu}. \label{6}
\end{eqnarray}
The amplitude (\ref{6})
for off-shell photon decay to  massless Majorana neutrinos, 
leads in the c.m. system to the following rate \cite{rstw} \footnote{Note that 
photons can also coupled to sterile neutrinos 
in the same, electromagnetically gauge-invariant way as the left-handed neutrions
in non-commutative Abelian gauge field theory. 
With sterile neutrinos, the
contribution from the
interaction (\ref{ncc2}) to the SM 
in a non-commutative background   doubles
 \cite{rstw,tram}.}: 
\begin{eqnarray}
&&\Gamma_{\rm NC}(\gamma\rightarrow {\bar{\nu}}_{\rm L} \nu_{\rm L})
= \frac{\alpha}{48}\frac{q^6}{{\rm E}_{\gamma}\Lambda^4_{\rm NC}}\sum_{i=1}^{3}(c^{0i})^2 ,
\label{8}
\end{eqnarray}
where the non-commutativity scale $\Lambda_{\rm NC}$ enters choosing dimensionless matrix elements
$c^{\mu\nu}=\Lambda^2_{\rm NC} \theta^{\mu\nu}$ of order one.
The $c^{0i}$'s are not independent. In pulling out the overall scale 
$\Lambda_{\rm NC}$, we can always
impose the constraint $ \sum_{i=1}^3 (c^{0i})^2 \equiv {\vec E}^2_{\theta}=1$ \cite{HK}. 

The partial width (\ref{8}) is next compared with the rate (\ref{7}) induced by 
the loops in the SM \cite{Bernstein}--\cite{BPV}.
The ratio $\Re$ of the decay rates (\ref{8}) and (\ref{7}), for massless neutrinos is given by
\begin{equation}
\Re=\frac{1}{12}\frac{q^4}{\Lambda_{\rm NC}^4}\frac{{\vec E}^2_{\theta}}{F^2_{\nu}(q^2)}
=\frac{3}{\Lambda_{\rm NC}^4}\frac{{\vec E}^2_{\theta}}{\langle r^2_{\nu}\rangle^2}.
\end{equation}

Assuming that any
new mechanism contributing to the charge radius should not exceed the SM value
we can at most have $\Re = 1 $ at the scale of non-commutativity, translating into
\begin{equation}
\langle r^2_{\nu}\rangle =  \frac{\sqrt3}{\Lambda_{\rm NC}^2} \stackrel{<}{\sim} 
\frac{\sqrt3}{(M^*)^2},
\label{9}
\end{equation}
where the constraint ${\vec E}^2_{\theta}=1$ is imposed. 
We have found the $\star$-charge radius, which in a sense 
represents the radius or range of non-commutativity, to be a simple function of the noncommutativity scale.

Note that there are other polarization phenomena induced by the non-commutativity
tensor $\theta^{\mu\nu}$, which would involve correlations between spin and momenta. 
These, however, were integrated out in our estimate. The motivation to do so lies in the fact that 
at this stage our model for the photon--neutrino interaction represents only the tree-level effective 
non-commutative gauge field theory in which the  question of renormalization procedure
is not addressed \cite{UV/IR}. 

{}From Eq. (\ref{9}) we have   
$\langle r^2_{\nu}\rangle_{\rm NC} \,\stackrel{<}{\sim}
0.3 \times 10^{-33}\; {\rm cm}^2$, for a scale of non-commutativity 
${ \Lambda_{\rm NC} \stackrel{>}{\sim} 1.5}$ TeV.
This bound is stronger than the charge radius bounds obtained in Refs. \cite{Salati}--\cite{hnr}.

It is important to emphasize that, in the Dirac case by the degeneracy of $\nu_{\rm L}$ and $\nu_{\rm R}$ 
the relevant amplitudes are coherently amplified yielding more stringent limits.
Then the right-handed neutrinos can still be produced through
$e^+e^- \to \nu_{\rm R}\bar\nu_{\rm R}$ mechanism \cite{hnr},
and therefore constraints from nucleosynthesis as well as from the Supernova 1987A apply, 
yielding $|\langle r^2_{\nu}\rangle| \stackrel{<}{\sim} 7\times 10^{-33}$ cm$^2$ \cite{gm1} and
$|\langle r^2_{\nu}\rangle| \stackrel{<}{\sim} 2\times 10^{-33}$ cm$^2$ \cite{gm}, respectively.

We have compared the SM neutrino charge radius and electromagnetic transition dipole moments
with their analogs arising from a theory of non-commutative space-time.
We hope that this comparison sheds light on the magnitude of associated phenomena so induced.
Our results relate to a set of physics problems involving mass and electromagnetic
properties of neutrinos. 
It appears open to us wether any cosmological implications for dark forms of
energy/matter might arrise, because we can not exclude that some dark form object is in fact 
consequence of neutrino condensation, no matter how small the interaction is that
induces it.

\begin{acknowledgements}

We would like to thank  G. Raffelt for suggesting the study of the neutrino charge radius
to find limits on NC and J. Wess for many
helpful discussions leading to the construction of the model. 
The work of P.M. is supported by the Swiss National Science Foundation.
The work of J.T. is supported by the Ministry of Science and Technology 
of Croatia under Contract No. 0098002.

\end{acknowledgements}

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