\newsection{Introduction}
Undoubtedly, the usual particle physics \sm\ is among the most successful physical theories
and so far it has passed all the precision tests and is capable of
explaining all the present data, or those phenomena and concepts which can be accommodated
within its mathematical structure, such as quarks and neutrino mass and mixing.
The only unobserved, or perhaps theoretically  less elegant, part is the Higgs sector.

Although being experimentally so successful, perhaps its only weak point is the large
number of theoretically undetermined parameters. Mainly motivated by this point, there has
been a lot of work devoted  to formulating theories beyond \sm, through which one can find
some relations between the parameters of the \sm\ and in this way reduce the number of free
parameters. Among these very different attempts  one can mention the grand unified theories
(GUT's) and the minimal supersymmetric \sm\ (MSSM). 


In this work we construct a model beyond the \sm\ from a completely different perspective, i.e. the
\sm\ on a \nc space-time, the \nc\sm\ (NCSM).
Noncommutative space-time can be presented by the so-called
Moyal plane, with the coordinates and their conjugate
momentum operators, $\hat{x}_\mu,\ \hat{p}_\nu$, satisfying
\ba
[\hat{x}_\mu,\hat{x}_\nu]=i\theta_{\mu\nu}\ &,& \ \ \ \theta_{\mu\nu}=-\theta_{\nu\mu}\ ,\cr 
[\hat{x}_\mu,\hat{p}_\nu]=i\hbar\eta_{\mu\nu}\ &,& \ \ \
[\hat{p}_\mu,\hat{p}_\nu]=0\ .
\ea
In the above, $\theta_{\mu\nu}$, the \ncy\ parameter (usually taken as a constant tensor), 
is of dimension of $(length)^2$. As it
is seen, the Lorentz symmetry is lost, but, we expect to find the manifest Lorentz
symmetry at low energies, $E^2\theta\ll 1$ (at least if we ignore the quantum
corrections), where $\theta$ is the dimensionful scale of the $\theta_{\mu\nu}$ tensor. 
Then, one should define field theory on the \nc space-times, \nc field
theory. To pass to \nc field theories, it is enough to replace the usual product of the
fields in the (\com) action, by the Moyal \sp\footnote{We note that this recipe cannot be
used for gauge theories other than \Un.}
\begin{eqnarray}\label{star}
(f\star g)(x)&=&{\rm e}^{{i\over 2}\theta_{\mu\nu}
\partial_{x_{\mu}}\partial_{y_{\nu}}}f(x)g(y)\Big|_{x=y}\cr
&=&f(x)g(x)+{i\over2}\theta_{\mu\nu}
\partial_{{\mu}}f\partial_{{\nu}}g+{\cal{O}}({\theta^2})\ .
\end{eqnarray} 
Introducing this \sp\ into the actions has some non-trivial consequences both at
the classical (tree) and quantum (loop) levels. 

At classical level, among these consequences, we would like to mention the restrictions it
imposes on the gauge theories: only the \nc $U(n)$ gauge theories have a simple \nc
extension and we cannot even have \nc $SU(n)$ gauge theories. Furthermore,
the \rep s for the \un\ algebra are restricted to those of \nbyn\ hermitian matrices
\cite{nogo}. Also, \ncy\
imposes severe restrictions on the fermions and their charges \cite{{nogo},{Haya}}.
We shall discuss these points in more detail in the next section. The other interesting
classical consequence of \ncy\ is the inherent $C$ and $CP$ violation in the \nc field
theories \cite{CPT}.

As for the quantum level, we can mention the loop calculations and \renormy\ discussions. 
During the past two years there has been a large number of articles on that subject  
 (see, e.g.,
\cite{Filk}-\cite{Loriano})
\footnote{For a string theory survey on \nc issues, see \cite{SW}.}. From all
these results here we mention only two:
\newline
{\it i)} In general, the unitarity of \nc field theories is
related to having a space-like \ncy, i.e. $\theta_{\mu\nu}\theta^{\mu\rho}$ as a matrix should
be positive definite \cite{Unitary};
\newline
{\it ii)}  An intrinsic and general feature of the \nc field theories is the so-called
IR/UV mixing \cite{Sei1}: although we can usually remove the UV divergences in the \nc
version of the usual \com\ \renorme\ theories by adding proper counter-terms (and hence the
theory is UV
\renorme), upon sending the UV cut-off to infinity we remain with some new IR divergences.
There have been three proposals to resolve this IR divergence problem
\cite{{Sei1},{Sei2}};\cite{Hard};\cite{Wien}, among which are  
the \nc hard resummassion \cite{Hard}, and/or introducing a new way of regularization
\cite{Wien}; we believe that, one way or another, this problem can be
removed.

In particular we would like to point out that the \nc gauge theories
\cite{{Carmello},{Loriano}}, the
\nc version of real $\phi^4$ theory \cite{{Sei1},{Amicu},{Roiban}} as well as the complex
$\phi^4$ theory \cite{Aref} and the \nc version of QED (NCQED) \cite{{Haya},{Ihab}} have
been shown to be one-loop \renorme.

There have also been many attempts to study the phenomenological consequences of \nc
field theories. However, most of them are aimed at accommodating  the extra \nc
contributions within the error bars of the present data
\cite{Zdecay}-\cite{Banks}. A rigorous
and robust mathematical framework which is not suffering from the charge quantization
problem \cite{{nogo},{Haya}} and the extra \Uo\ factors (in the \Un\ gauge theory
compared to $SU(n)$) \cite{{Armoni},{Loriano}} is not yet constructed. This is exactly what
we would like to do in this paper. We will show how, by fixing the gauge group of the \nc\sm\
(NCSM) to \Uth$\times$\Ut$\times$\Uo\ and reducing the two extra \Uo\ factors through the
appropriate Higgs particles, the number
of possible particles in each family (which are six: left-handed leptons, right-handed
charged lepton, left-handed quarks, right-handed up quarks, right-handed down quarks and
Higgs) is fixed, as well as their hyper-charges (and hence the electric charge). We would like to
emphasize that the existence of the Higgs particle, in our model, is an unavoidable outcome. 
As a consequence, 
two extra massive gauge bosons and two extra massive scalar particles will appear.

{\it Convention:}

In order to make a distinction between these two types of scalar fields which we have:
the one(s) which we use for the {\it reduction} of the extra \Uo\ symmetries and the usual \sm\
Higgs, which is used for breaking the electro-weak symmetry, we call the former one as
"{\it Higgsac}" and
keep the "Higgs" for the usual Higgs doublet
\footnote{The suffix "ac" stems either 
from the word "acommutative" (i.e. not commutative) or from the diminutive suffix in Persian,
similar to "ino" in Italian, and hence "Higgsac" is equivalent to small Higgs. We use this
terminology to distinguish these scalars from the usual Higgs and also the Higgsinos of MSSM.}. 

The paper is organized as follows. In section 2, we review the problems
and restrictions for constructing a \nc version of \sm\ and discuss a mechanism
to resolve these problems. 
In section 3, in order to show how our procedure works in practice, we work out the details
of the reduction of  the extra \Uo\ factor(s) and show how this also resolves the
charge quantization problem, for the particular case of the \nc version of QCD+QED
which can be denoted  by $NC(SU_c(3)\times U(1))$ gauge theory. 
In section 4, which in a sense is the main part of the paper,  
we construct  the NCSM \footnote{ The \nc geometry has been
previously used to build a theory beyond \sm\ (see, e.g., \cite{connes}). However, we would like to
remark that in our model we use a completely different approach.}. 
We start with the \Uth$\times$\Ut$\times$\Uo\
gauge theory and reduce the two extra \Uo\ factors by introducing two extra Higgsac
particles in proper \rep s. Then, we proceed with introducing matter fields and discuss
in detail how the hyper-charges are fixed to those of the usual \sm. 
%Therefore, recalling the arguments of Refs. \cite{Anomaly}, our NCSM is anomaly free; 
%we will discuss this in more detail in the last section. 
In section 5, we work out the
details of the electro-weak symmetry breaking. In this way we define the photon, $Z$ and $W^\pm$
fields. Then, in the fermionic part, we discuss the interaction terms for the fermions and compare
them with the usual \sm\ as well as the corresponding Yukawa couplings and mass terms. 
In section 6, among several new features of NCSM,
we mention the neutrino dipole moment and the \nc correction to the weak-mixing angle,
$\theta_W$, or more precisely to the $\rho$ parameter and ${m^2_W\over m^2_Z}$ ratio. 
In this way we impose some upper bounds on the masses of two extra massive gauge
boson as well as on the \ncy\ parameter. Finally in section 7, we discuss some of the open
questions. 
 

  
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\newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}
\newcommand{\bea}{\begin{array}}
\newcommand{\eea}{\ene{array}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def \parth {{\partial\over\partial\theta_{\mu\nu}}}
\def \parmu {\partial_{\mu}}
\def \parnu {\partial_{\nu}}
\def \ola  {\overleftarrow}
\def \ora  {\overrightarrow}
\def \zz  {{\mathbb Z}}
\def \cc  {{\mathbb C}}
\def \rr  {{\mathbb R}}
\def \hh {{\mathbb H}}
\def \oo {{\cal O}}
\def \aa {{\cal A}}
%%%%%%%%%%%%%%
\def\rep{representation}
\def\trans{transformation}
\def\renorme{renormalizable}
\def\renormy{renormalizability}
\def\hyper{hyper-charge}
\def\lh{left-handed}
\def\rh{right-handed}
\def\nc{noncommutative }
\def\ncy{noncommutativity}
\def\com{commutative}
\def\sm{Standard Model}
\def\Uo{$U_{\star}(1)$}
\def\Ut{$U_{\star}(2)$}
\def\Uth{$U_{\star}(3)$}
\def\Uott{$U_{\star}(3)\times U_{\star}(2)\times U_{\star}(1)$}
\def\Un{$U_{\star}(n)$}
\def\uo{$u_{\star}(1)$}
\def\ut{$u_{\star}(2)$}
\def\uth{$u_{\star}(3)$}
\def\un{$u_{\star}(n)$}
\def\sp{$\star$-product}
\def \da {\dagger}
\def\um{$u_{\star}(m)$}
\def\Um{$U_{\star}(m)$}
\def\nbyn {$n\times n$}
\def \l {\lambda}
\def\af{anti-fundamental}
%\def\rh{right-handed}
%\def\lh{left-handed}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%






\begin{document}
 
\thispagestyle{empty} \setcounter{page}0
\renewcommand{\baselinestretch}{0.1}
\begin{flushright}
hep-th/0107055\\[-0.4cm]
HIP-2001-25/TH\\[-0.4cm] 
IPM/P-2001/024\\[-0.4cm] 
\today
\end{flushright}
%\vspace{1cm}
\renewcommand{\baselinestretch}{1.2}
\begin{center}
{\Large \bf{Noncommutative Standard Model:\\
Model Building}}
\vskip 1cm
{\bf M. Chaichian$^{\dagger}$,
P. Pre\v{s}najder$^{\dagger,a}$, M. M. Sheikh-Jabbari$^{\dagger,b,c}$
\ \ and \ \ {{A. Tureanu}}$^{\dagger}$}
\vskip 0.2cm
{\it $^{\dagger}$High Energy Physics Division, Department of
Physics,
University of Helsinki\\
\ \ {and}\\
\ \ Helsinki Institute of Physics,
P.O. Box 64, FIN-00014 Helsinki, Finland\\
$^a$Department of Theoretical Physics, Comenius University, Mlynsk\'{a} dolina, SK-84248 Bratislava,
Slovakia \\
$^{b}$ The Abdus Salam ICTP,
Strada Costiera 11,Trieste, Italy\\
$^c$Institute for Studies in Theoretical Physics and Mathematics\\[-0.4cm] 
P.O. Box 19395-5531, Tehran, Iran}
\end{center}
\vskip 1cm

\begin{abstract}
A \nc version of the usual electro-weak theory is constructed. We
discuss how to overcome the two major problems: 1) although we can have  
\nc $U(n)$ (which we denote by $U_\star(n)$) gauge theory we cannot have
\nc $SU(n)$
and 2) the charges in \nc QED are quantized to just $0, \pm
1$.   We show how the problem with charge quantization, as
well as with the gauge group, can be resolved by taking $U_{\star}(3)\times
U_{\star}(2)\times U_{\star}(1)$ gauge group and reducing the extra
$U(1)$ factors in an appropriate way.
Then we proceed with building the \nc version of the standard model by specifying the 
proper representations for the entire particle content of the theory, 
the gauge bosons, the fermions and Higgs. We also present the full action 
for the \nc \sm\ (NCSM). In addition, among several  
peculiar features of our model, we address the {\it inherent} CP
violation and new neutrino interactions.   
\end{abstract}
\newpage

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\input{problems}
\input{QECD}
\input{main}
\input{Higgs}
\input{specific}
\input{Outlook}
\input{ref}
      
\end{document} 







\newsection{The major problems in constructing NCSM and \\ [- 0.6cm] the
proposal to resolve them}

In this section we recapitulate the problems one encounters in building a \nc version of the \sm\
and present the way out of them. 
These problems and restrictions which we classify in three sets, are all imposed by 
the mathematical (group theoretical) structure of \nc gauge theories.
However, first let us review some related information about the usual \sm. The usual \sm\ in
the gauge bosons sector contains 8 (massless) gluons, 1 (massless) photon and 3
(massive) weak gauge bosons. We have integrated the information about the matter fields and
their charges in the following table. 

\begin{table}[ht]\label{tab:sc}
\vspace{0.3cm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Particles& Electric charge & $SU(2)$ weak charge& Hyper-charge& Colour charge \\
\hline
LH electron& $-1$ & $-{1\over 2}$ & $-1$ & none \\
LH neutrino& $0$  & $+{1\over 2}$ & $-1$ & none\\
RH electron& $-1$ & $0$         & $-2$   & none\\
\hline
LH up quark& $+{2\over 3}$ & $+{1\over 2}$ & $+{1\over 3}$ & has \\
LH down quark&$-{1\over 3}$& $-{1\over 2}$ & $+{1\over 3}$ & has \\
RH up quark& $+{2\over 3}$ & $0$           & $+{4\over 3}$ & has \\
RH down quark&$-{1\over 3}$& $0$           & $-{2\over 3}$ & has \\
\hline
Higgs      & $0$ & $-{1\over 2}$ & $+1$ & none\\
\hline
\end{tabular}
\end{center}
\caption{LH=left handed, RH=right handed}
\end{table}  

Now we are ready to discuss the three major problems.

\subsection{Problems}

{\it i) Charge quantization problem}:

As it was shown in \cite{Haya}, the charges for the matter fields coupled to the \Uo\ theory must be 
quantized to just $0,\pm 1$, depending on the \rep\ of particles. This is due to the fact
that in a sense the \Uo\ theory is a non-Abelian theory (for a more detailed discussion we refer
to \cite{{Haya},{nogo}}). Now, we face the first and the most challenging obstacle:
As we explicitly see from the table, 
not all the electric or hyper-charges of the particles fulfill this
condition. So, not only we are not able to construct NCQED, but going
to the electro-weak level (and considering the \hyper s) makes the problem worse and we
face a larger variety of non-quantized \hyper s.
\newline
{\it ii) The extra gauge fields}:

According to \nc group theoretical arguments (e.g. see \cite{nogo}), the \Uo\ sub-group of \Un\
is not a normal sub-group and therefore mathematically it is not possible to define a \nc
$SU(n)$ algebra (or group). However, even if we ignore this mathematical fact and drop
the corresponding \Uo\ gauge field in the \Un\ gauge theory action, the remaining
theory is not \renorme\ \cite{{Armoni},{Loriano}}. Consequently, as a direct generalization
of the $SU_c(3)\times SU_L(2)\times U(1)$ gauge theory, one cannot avoid two extra
$U(1)$ factors, i.e., two extra gauge fields appearing in NCSM.
\newline
{\it iii) The no-go theorem}:

In \cite{nogo}, based on group theoretical arguments, we have proved a no-go theorem stating
that:
\newline
a) the local \un\ {\it algebra} only admits the irreducible \nbyn\
matrix-representation. Hence the gauge fields are in \nbyn\ matrix
form, while the matter fields {\it can only be} in fundamental, adjoint or singlet states;
\newline
b) for any gauge group consisting of several simple-group factors, the matter fields can
transform nontrivially under {\it at most two} \nc group factors. In other words, the matter
fields cannot carry more than two \nc gauge group charges. 
%\newline

The a) restriction is actually what we have already had in the usual \sm, i.e. all the gauge
bosons as well as the matter fields are sitting in the \rep s which are also allowed in the
\nc case. However, as for the b) criterion, it is clear from the table that the
particles coupled to gluons, the quarks, carry {\it three} different charges,
i.e. \hyper, weak $SU(2)$ charge and colour charge.
%\newline

Before explaining our procedure to resolve the above mentioned problems,
however, we would like to make a comment on the no-go theorem.
The arguments of \cite{nogo}, and in particular part b), are based on the invariance of the action
under the {\it finite} gauge transformations.
In other words, to define the gauge transformation for the matter fields we have considered 
the {\it group} factors, while in principle it is also possible to define these gauge
transformations only with the {\it algebra} (i.e. the infinitesimal gauge transformations),
in which case one can relax the condition b) \footnote{We would like to thank L. Bonora for a
discussion on this point.}. For the usual Lie-groups and algebras 
where the group elements are obtained through the simple exponentiation of the algebra
elements, of course the infinitesimal and finite gauge transformations are resulting in the
same physics (at least for Yang-Mills theories).
However, this is not always the case, a famous example being the Chern-Simons theories in
which, although the theory is invariant under infinitesimal gauge transformations, the
invariance under finite gauge transformations is not immediate. As a result, to have a
well-defined quantum Chern-Simons theory, the level should be quantized, which  in
turn is  an implication of finite gauge transformations.
For the \nc groups when the gauge group involves more than one simple \Un\ factor, the
relation between the algebra and the corresponding group is not given by a simple
star-exponentiation \cite{nogo}. We believe that it is the invariance under
the {\it finite} gauge transformations which is indeed fundamental, and of course this also
covers the infinitesimal gauge invariance. 

\subsection{The way out}

To show the way out of the above mentioned problems we recall two facts:
\newline
{\it I)} In the usual physical models,  there is always a $U(1)$ factor together with the
$SU(n)$ factors, i.e. $SU_c(3)\times U_q(1)$ for QCD+QED and $SU_c(3)\times
SU_L(2)\times U_Y(1)$  for the \sm.
\newline       
{\it II)} If we define the photon (or the hyper-photon) through a linear combination of two
(or three) \Uo\ fields, although the charge for each \Uo\ factor is quantized restrictively to 0 and $\pm1$, 
there is the
chance to find more variety of charges (but still quantized). Furthermore, this shows a way out
of the implications of part b) of our no-go theorem.

Hopefully there is a standard and well-known procedure to implement the above two
facts: the Higgs symmetry breaking scenario.
Hence our recipe is to 
start with \Uth$\times$\Uo\ (or \Uott) and {\it reduce} two
(or three) \Uo\ factors to {\it one} \Uo\ factor,  through one (or two) proper 
Higgsac field(s).
We would like to emphasize that in order to reduce a symmetry through the Higgs mechanism it
is necessary that the Higgs is in a non-singlet \rep\ of that symmetry. Therefore, in our
case, the Higgsac field(s) should be charged {\it only} under the \Uo\ sub-group of \Uth\
(and \Ut) as well as under the individual \Uo. As discussed in \cite{nogo}, indeed the \Un\
group enjoys the property of having the needed \Uo\ as a sub-group.

In sections 3 and 4, we will explicitly and in details show how the above   
observation works and how tightly it fits into the existing matter content of the \sm.


     
\vskip 1cm

{\bf Acknowledgments:}

M.C. and A.T. are indebted to C. Montonen for enlightening remarks. M.M. Sh-J. would like to thank 
Y. Farzan, A. Smirnov and J. Ellis for fruitful discussions. 
The financial support of the Academy of Finland under the Project No. 163394
is greatly acknowledged. The work of P.P. was partially supported by VEGA project 1/7069/20.

   
\vskip 1cm






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\end{thebibliography}
\newsection{Some specific features of NCSM}

In the previous section,  we worked out in detail the 
fermions-gauge bosons interaction terms in the NCSM. 
In general, one can classify the new ingredients of the NCSM in two sets:

First are those coming from the group theoretical structure of the model and do not depend on
the
\ncy\ parameter explicitly. This set is mainly a consequence of having two extra massive gauge
bosons, $G^0_{\mu}$
and $W^0_{\mu}$. Although we did not present it, almost all the fermions  interact with the new
massive gauge bosons, $G^0_{\mu}$ and $W^0_{\mu}$. Such interaction terms effectively will give
rise to Fermi's four-fermion interaction, where its coupling (up to some numeric factors) is
$G_F({m_Z\over m_{W^0}})^2$. Another important effect of these new massive gauge bosons is the
correction to the physical $Z$-particle, and in particular to its mass. We will discuss this in details
in the subsection 6.2 and in this way we impose some lower bounds on the masses of these new
massive gauge bosons. 

The second class of new features in the NCSM are the interaction terms coming from the \sp\ and
(at least at the classical level) in the \com\ limit, i.e. $\theta\to 0$, they vanish explicitly.
In other words, all the particles, besides the usual charge, up to the first order in
$\theta_{\mu\nu}$, also carry dipole charge which is proportional to the \ncy\
parameter \cite{{Lamb},{Ren}}. From these new interaction terms here we discuss that of 
neutrino-photon coupling and from there we obtain a lower bound on the \ncy\ scale. 




\subsection{Neutrino dipole moment}

As we have explicitly shown in the previous section, and in particular in (\ref{n-gamma}), neutrino in the NCSM
undergoes a new type of interaction: the neutrino-photon vertex. Unlike all the other photon-fermion
interactions in the NCSM, this vertex is a {\it chiral} one, i.e. the only existing neutrino, the
left-handed $\nu$, appears in this interaction term. More precisely, in the \nc case, we do
not need necessarily a right-handed neutrino to have a coupling to the electro-magnetic field and therefore the
neutrino, without being massive, can carry dipole charges.

On the other hand, there are very strong (astro-physical) bounds on the neutrino-photon interactions and 
especially the neutrino dipole moment \cite{Data}. In fact, this bounds can be translated to a lower bound 
on the \ncy\ scale, $\Lambda_{NC}$, defined as:
\be
\theta_{\mu\nu}={1\over \Lambda^2_{NC}} \epsilon_{\mu\nu}\ ,
\ee
where $\epsilon_{\mu\nu}$ is  a dimensionless anti-symmetric parameter, whose elements
are of the order of one. 

It is well-known that neutrino has a considerable effect in the stellar cooling process. However,
according to
the \sm, they only participate in the weak interactions through massive $W^\pm$ and $Z$. In this way any
direct photon-neutrino interaction such as what we have here, can speed up the cooling process, which in
turn will change the whole star evolution. To avoid drastic changes in this respect (which have
not been observed) the strength of neutrino-photon interaction should be smaller compared to that of $Z$. 
To materialize the above argument, let us expand (\ref{n-gamma}) up to the first order in
$\theta_{\mu\nu}$:
\be
{\cal L}_{\nu-\gamma}=-ie\bar{\nu}\gamma^{\mu}[\nu,A_{\mu}]_{\star}=-e\ 
\bar{\nu}\gamma^{\mu}\left(\theta_{\alpha\beta}\partial_{\alpha}A_\mu \partial_{\beta}\nu\right)
+{\cal O}(\theta^2)\ .
\ee
As we see, in the above interaction the derivative of neutrino appears (as well as that of the
photon field $A_{\mu}$). Then, one can read off the effective neutrino \nc dipole moment:
\be
d_{\nu}= e{1\over \Lambda^2_{NC}}\ E_{\nu}\ , 
\ee
where $E_{\nu}$ is the energy of the neutrino. For the case at hand, the solar neutrino
problem, 
$E_\nu \simeq 10 MeV$ and the corresponding bound on the magnitude of dipole moment is
\cite{neutrino}
\be
d_\nu \lesssim 0.1\times 10^{-10}\ \mu_{B}\ ,
\ee
where $\mu_B={e\hbar\over 2m_e c}$ is the Bohr magneton 
\footnote{In fact this bound is coming from the consideration of Red Giant cooling
process. 
There are some weaker and also stronger bounds on the neutrino dipole moment coming from
some other sources. Since in our model we do not have right-handed neutrinos we cannot
use the stronger bound of $10^{-13}\mu_B$.}. 
Therefore, one can readily
obtain the lower bound on the \ncy\ scale
\be
\Lambda_{NC}\gtrsim  10^3 \ GeV\ .
\ee
Of course, this bound is based on a rough estimate and a more detailed calculation and survey can
improve this bound. Also we note that this bound is of the same order as the previous bounds coming from
the Lamb-shift \cite{Lamb} and the Lorentz-violation considerations \cite{{Harvey}}.
  


\subsection{Corrections to the weak-mixing angle}

As we have discussed previously in section five, the physical $Z$-particle, which is an
eigen-state 
of the mass matrix after the electro-weak symmetry breaking, besides the $W^3_{\mu}$, the
 hyper-photon $Y_{\mu}$ now receives a contribution from the other two new massive gauge
bosons, $G^0_{\mu}$ and $W^0_{\mu}$, while the photon field is only made out of $W^3_{\mu}$
and $Y_{\mu}$, in such a way that at the end $Z_{\mu}$ and the photon field $A_{\mu}$ are 
ortho-normal states.
However, as we have explicitly shown, these contributions are suppressed by the 
$({m_Z\over m_{W^0}})^2$ ratio, eq. (\ref{Zmass}). On the other hand, the $W^\pm$ gauge
bosons remain
the same as in the usual \sm, $W^\pm_{\mu}={1\over \sqrt 2}(W^1_{\mu}\pm iW^2_{\mu})$.
Therefore, the ${m_Z\over m_{W}}$ ratio now receives a correction, as indicated in
(\ref{massratio}).
We remind that the weak-mixing angle $\theta^0_W$, is still defined through the ratio of
hyper-photon coupling and the weak coupling: ${g'\over g_2}=\tan\theta^0_W$.

In the usual \sm, although the parameter
$$
\rho=({m_Z\over m_{W}})^2\cos^2\theta^0_W\ 
$$
at classical (tree) level is equal to one, it receives quantum (loop) corrections, see e.g.
\cite{Altarel1}. In fact, one of the precision tests of the \sm\ is to evaluate these corrections to
$\rho$ and compare them to the corresponding experimental data \cite{{Altarel1},{Altarel2}}.
Here we use the conventions and notations of \cite{Altarel1} to parameterize these corrections:
\ba
\left. ({m_Z\over m_{W}})^2=({m_Z\over m_{W}})^2\right|_{B}\ (1+1.43\epsilon_1-1.00
\epsilon_2-0.86
\epsilon_3)\ ,
\ea
where the $\epsilon_i$ show the "large" asymptotic contributions, up to the leading linearized
approximation and
$$
\left. ({m_Z\over m_{W}})^2\right|_{B}=0.768905\ 
$$
is the $Z$ and $W$ mass ratio in the Born approximation.
With the latest data used in \cite{Altarel2}, the {\it predicted} values of $\epsilon$
variables in the usual \sm, which do depend on Higgs and top quark masses,
are given in table 2. 
\begin{table}[ht]\label{tab:epsilon}
\vspace{0.3cm}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$\epsilon_i\ \times 10^{+3}$ & $m_t=174.3-5.1$ & $m_t=174.3$ & $m_t=174.3+5.1$ \\
\hline
$\epsilon_1\ $ & $5.1$   & $5.6$  & $6.0$ \\
$\epsilon_2\ $ & $-7.2$  & $-7.4$ & $-7.6$ \\
$\epsilon_3\ $ & $5.4$   & $5.4$  & $5.3$   \\
\hline
\end{tabular}
\end{center}
\caption{\sm\ predictions for $\epsilon$ variables, at $m_H=113 GeV$}
\end{table}
However, the observed values of $\epsilon_i$'s obtained from all combined hadronic, leptonic
and Higgs measurements are:
\ba\label{exper}
\epsilon_1 &=& (5.4\pm 1.0) \times 10^{-3}\  ,\cr   
\epsilon_2 &=& (-9.7\pm 1.2) \times 10^{-3}\ ,\cr   
\epsilon_3 &=& (5.4\pm 0.9) \times 10^{-3}\ .
\ea
Comparing the  \sm\ model results and the observed values (\ref{exper}), the \nc corrections
should be smaller than the difference between these two values. More explicitly, 
\be
\cos^2\theta^0_W\Big[(\frac{m_Z}{m_{G^0}})^2\sin^2\delta_{23}
+(\frac{m_Z}{m_{W^0}})^2\cos^2\delta_{23}\cos^2\delta_{11'}\Big]\lesssim (2.014\pm
3.404) \times
10^{-3}\ .
\ee  
On the other hand, 
\ba
\tan\delta_{23}=\left.{2\over 3}\sqrt{{\alpha_{QED}\over \alpha_s}{1\over
\sin^2\theta^0_W}}\ \ \right|_{m_Z} 
= 0.354\ ,
\ea
where in the above we have used the data given in \cite{Data} \footnote{Using the relations 
defining $\delta_{11'}$ we find that: $\ \sin^2\delta_{11'}={\tan^2\theta^0_W\over
\cos^2\delta_{23}}=0.3383$.}.
Now, if we assume that $m_{G^0}\simeq m_{W^0}$, we can find a lower bound on $m_{G^0}$:
\be
m_{G^0} \gtrsim  2.5\times 10 \ m_Z\ .
\ee
 

