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\begin{document}

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\preprint{LMU-TPW 02/01 (v2)}
\preprint{IRB-TH-2/02 (v2)}

%Title of paper
\title{The $Z\,\rightarrow \,\gamma  \gamma,\;gg$ 
Decays in the Noncommutative Standard Model}

\author{W.~Behr}
%\email[]{Your e-mail address}
%\homepage[]{Your web page}
%\thanks{}
%\altaffiliation{}
\affiliation{Theoretische Physik, Universit\"at M\"unchen, 
Theresienstr. 37, 80333 M\"unchen, Germany}
\email[]{wess , schupp @theorie.physik.uni-muenchen.de}
\author{N.G.~Deshpande}
\affiliation{Institute of Theoretical Science, University of Oregon, 
Eugene, OR94703, USA}
\email[]{desh@oregon.uoregon.edu}
\author{G.~Duplan\v{c}i\'{c}}
\affiliation{Theoretical Physics Division, 
Rudjer Bo\v{s}kovi\'{c} Institute, P.O.Box 180,
10002 Zagreb, Croatia}
\email[]{josip, gorand@thphys.irb.hr}
\author{P.~Schupp}
\affiliation{Theoretische Physik, Universit\"at M\"unchen, 
Theresienstr. 37, 80333 M\"unchen, Germany}
\author{J.~Trampeti\' c}
\thanks{Address after September 1, 2002: Theory Division, CERN, CH-1211, Geneve 23, Schwitzerland}
\affiliation{Theoretical Physics Division, 
Rudjer Bo\v{s}kovi\'{c} Institute, P.O.Box 180,
10002 Zagreb, Croatia}
\author{J.~Wess}
\affiliation{Theoretische Physik, Universit\"at M\"unchen, 
Theresienstr. 37, 80333 M\"unchen, Germany}
\affiliation{Max-Planck-Institut f\"ur Physik, F\"ohringer Ring 6, 
80805 M\"unchen, Germany}

\date{\today}

\begin{abstract}
On noncommutative spacetime, the Standard Model (SM) allows
new, usually SM forbidden, triple gauge boson interactions.
In this letter we propose the SM strictly forbidden $Z\rightarrow \gamma\gamma$ and $Z\rightarrow gg$
decay modes coming from the gauge sector of the Noncommutative Standard Model (NCSM) as a 
place where noncommutativity could be experimentally discovered.
\end{abstract}

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\pacs{12.60.Cn, 13.38.Dg, 02.40.Gh}
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%\section{Introduction}
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%\section{\label{}}

The idea that coordinates may not commute 
can be traced back to Heisenberg.
A simple way to introduce a noncommutative structure into
spacetime is to promote the usual spacetime 
coordinates $x$ to noncommutative (NC) coordinates 
$\hat x$ with~\cite{sny}
\begin{equation}
\left[{\hat x}^{\mu},{\hat x}^{\nu} \right]=i\theta^{\mu\nu}, \quad
\left[\theta^{\mu\nu},{\hat x}^{\rho} \right]=0,                \label{CR}
\end{equation}
were $\theta^{\mu\nu}$ is a constant, real, antisymmetric matrix.
The noncommutativity scale $\Lambda_{NC}$ is fixed
by choosing dimensionless matrix elements 
$c^{\mu\nu}=\Lambda_{NC}^2\,
\theta^{\mu\nu}$ of order one.
The original motivation to study such a scenario
was the hope that the introduction
of a fundamental scale
could deal with the infinities of
quantum field theory in a natural way.
The simple commutation relation (\ref{CR})
with constant $\theta^{\mu\nu}$ 
fails to provide a complete regularization~\cite{Filk},
but more complicated noncommutative structures
can indeed introduce spacetime lattice structures into the
theory that are compatible with a deformation of \emph{continuous} 
spacetime symmetries (see, e.g., \cite{Cerchiai}). This is in contrast
to the situation in ordinary lattice
field theory, where only discrete translation symmetries survive.
Aside from these technical merits, the possibility of 
a noncommutative structure 
of spacetime is of interest in its own right
and its experimental discovery would be a result of fundamental importance. 

Noncommutative gauge theory has become a focus of interest
in string theory and M-theory with the work given in Ref.\cite{CDS}. 
Noncommutativity of spacetime
is very natural in string theory and can be
understood as an effect of the interplay of closed and open strings.
The commutation relation (\ref{CR}) 
enters in string theory through
the Moyal-Weyl star product
\begin{equation}
f \star g = \sum_{n=0}^\infty \frac{\theta^{\mu_1 \nu_1} 
\cdots \theta^{\mu_n \nu_n}}{(-2i)^nn!}  
\partial_{\mu_1}\ldots\partial_{\mu_n} f
\cdot\partial_{\nu_1}\ldots\partial_{\nu_n} g.
\end{equation} 
For coordinates: $x^\mu \star x^\nu - x^\nu \star x^\mu = i \theta^{\mu\nu}$.
The tensor $\theta^{\mu\nu}$ is determined by a NS  $B^{\mu\nu}$-field and
the open string metric $G^{\mu\nu}$~\cite{DH}, which both
depend on a given closed string background. 
The effective physics on D-branes is most naturally captured by
noncommutative $U(N)$ gauge theory, but it can also be described by
ordinary gauge theory. Both descriptions
are related by the Seiberg-Witten (SW) map~\cite{SW}, which 
expresses noncommutative gauge fields in terms of fields
with ordinary ``commutative'' gauge transformation properties.

Experimental signatures of noncommutativity have been discussed from the point of 
view of collider physics~\cite{AY,HPR,HK,MPR} as 
well as low-energy non-accelerator experiments~\cite{MPR,ABDG,CCL}. 
Two widely disparate sets of bounds on $\Lambda_{NC}$
can be found in the literature: bounds of order $10^{11}$ $GeV$~\cite{ABDG} 
or higher~\cite{MPR}, and 
bounds of a few TeV's from colliders~\cite{AY,HPR,HK}.
All these limits rest on one or more of the following assumptions which 
may have to be modified: 
(1) $\theta$ is constant across distances that are very large 
compared with the NC scale; 
(2) unrealistic gauge groups; 
(3) noncommutativity down to low energy scales.

Noncommutative gauge field theory (NCGFT) as it appears 
in string theory is strictly speaking limited to
the case of $U(N)$ gauge groups. 
A method of constructing models on noncommutative spacetime
with more realistic gauge groups and particle content
has been developed in a series of papers by the 
Munich group~\cite{WESS} and~\cite{Zumino}
culminating in the construction of the Noncommutative Standard Model
(NCSM)~\cite{WESS1}. 
This construction for a given NC space rests on few basic
concepts: covariant coordinates, 
locality, gauge equivalence, and
consistency conditions. It is independent
of string theory but uses some of its ideas, in particular
the Seiberg-Witten map, in an essential way.
The problems that are solved in this approach include, in addition to the
introduction of more general gauge groups, 
the charge quantization problem of NC Abelian gauge theories
and the construction of covariant Yukawa couplings.
 
There are two essential points in which NC gauge  theories differ 
from standard gauge theories. 
The first point is the breakdown of Lorentz invariance with
respect to a fixed non-zero $\theta^{\mu\nu}$ background 
(which obviously fixes preferred directions) and the other
is the appearance of new interactions 
(three-photon coupling, for example) and the modification of standard ones. 
Both properties have a common origin and appear in a number of phenomena. 

In this letter we consider strictly SM forbidden decays
coming from the gauge sector of the NCSM,
which could be probed in high energy collider experiments.
This sector is particularly 
interesting from the theoretical point of view. 
It is also a place where the 
approaches of different groups show the greatest differences.
Our main results are summarized in equations (\ref{eqn0}) to (\ref{eqn3}).

The action of NC gauge theory
resembles that of ordinary Yang-Mills theory, but with
star products in addition to ordinary matrix multiplication.
The general form of the gauge-invariant action 
for gauge fields is \cite{WESS1}
\begin{equation}
S_{gauge} =-\frac{1}{2}\int d^4x {\bf Tr}\frac{1}{{\bf G}^2}
{\widehat F}_{\mu\nu} \star {\widehat F}^{\mu\nu}.
\end{equation}
Here ${\bf Tr}$ is a trace 
and ${\bf G}$ is an operator that encodes the
coupling constants of the theory. Both will be discussed in detail below.
The NC field strength is 
\begin{equation}
{\widehat F}_{\mu\nu} = \partial_{\mu} {\widehat V}_{\nu} - \partial_{\nu} {\widehat V}_{\mu}
- i[{\widehat V}_{\mu}\stackrel{\star}{,}{\widehat V}_{\nu}]
\end{equation}
and ${\widehat V}_{\mu}$ is the NC analog of the gauge vector potential. 
The Seiberg-Witten maps are used to express the noncommutative fields and 
parameters as functions of 
ordinary fields and parameters and their derivatives. 
This automatically ensures a restriction to the correct degrees of freedom.
For the NC vector potential the SW map yields
\begin{equation}
{\widehat V}_{\xi}=V_{\xi}+\frac{1}{4}{\theta}^{\mu\nu}
\{V_{\nu},(\partial_{\mu}V_{\xi}+F_{\mu\xi})\}+{\cal O}\left(\theta^2 \right),
\end{equation}
where $F_{\mu\nu}\equiv \partial_{\mu}V_{\nu} - \partial_{\nu}V_{\mu} - i[V_{\mu},V_{\nu}]$
is the ordinary field strength and
$V_{\mu}$ is the whole gauge potential for the 
gauge group $G_{SM}\equiv SU(3)_C \times SU(2)_L \times U(1)_Y$
\begin{equation}
V_{\mu}=g'{\cal A}_{\mu}(x)Y + g\sum^3_{a=1}B_{\mu,a}(x)T^a_L + g_s\sum^8_{b=1}G_{\mu,b}(x)T^b_S.
\end{equation}
It is important to realize that the choice of the representation
in the definition of the trace ${\bf Tr}$
has a strong influence on the theory in the noncommutative case.
The reason for this is, that owing to the Seiberg-Witten map, terms
of higher than quadratic order in the Lie algebra generators will 
appear in the trace.
The choice of the trace corresponds to a choice of the representation
of the gauge group. 
The adjoint representation would certainly be a
natural choice, at least for the non-Abelian gauge fields.
This choice and some other simple choices do not lead to 
new triple gauge boson interactions and, in particular, show no
triple-photon vertices~\cite{WESS1,ASCH}.

The action that we present here
should be understood as an effective theory. 
From this point of view, all representations
of gauge fields that appear in the SM have to be considered in
the definition of the trace. We call the resulting model
the non-minimal NCSM. 
Consequently, according to \cite{WESS1}, we choose a trace over all particles 
with different quantum numbers in the model that have
covariant derivatives acting on them.
In the SM, these are, for each generation, five multiplets
of fermions and one Higgs multiplet. 
The operator ${\bf G}$, which determines the coupling constants of the theory,
must commute with all generators 
$(Y,T^a_L,T^b_S)$ of the gauge group,
so that it does not spoil the trace property of ${\bf Tr}$. 
This implies that ${\bf G}$ 
takes on constant values $g_1,\ldots,g_6$
on the six multiplets (Table 1 in Ref.~\cite{WESS1}).
The operator ${\bf G}$ is in general a function of $Y$ 
and the casimirs of $SU(2)$ and $SU(3)$.
However, because of the special assignment of hypercharges in the
SM it is possible to express ${\bf G}$ solely in terms of $Y$.

The action up to linear order in $\theta$ 
allows new triple gauge boson interactions that are forbidden in the SM
and has the following form
\begin{eqnarray}
\lefteqn{S_{gauge}=-\frac{1}{4}\int \hspace{-1mm}d^4x\, f_{\mu \nu} f^{\mu \nu}}
 \label{action2} \\
& &\hspace{-5mm}{}
-\frac{1}{2}\int \hspace{-1mm}d^4x\, {\rm Tr}\left( F_{\mu \nu} F^{\mu \nu}\right)
-\frac{1}{2}\int\hspace{-1mm} d^4x\, {\rm Tr}\left( G_{\mu \nu} G^{\mu \nu}\right)
\nonumber \\
& &\hspace{-5mm}{}
+g_s \,\theta^{\rho\tau}\hspace{-2mm}
\int\hspace{-1mm} d^4x\, {\rm Tr}
\left(\frac{1}{4} G_{\rho \tau} G_{\mu \nu} - G_{\mu \rho} G_{\nu \tau}\right)G^{\mu \nu}\nonumber \\
& &\hspace{-5mm}{}+{g'}^3\kappa_1{\theta^{\rho\tau}}\hspace{-2mm}\int \hspace{-1mm}d^4x\,
\left(\frac{1}{4}f_{\rho\tau}f_{\mu\nu}-f_{\mu\rho}f_{\nu\tau}\right)f^{\mu\nu}
 \nonumber \\
& &\hspace{-5mm}{}+g'g^2\kappa_2 \, \theta^{\rho\tau}\hspace{-2mm}\int
\hspace{-1mm} d^4x \sum_{a=1}^{3}
\left[(\frac{1}{4}f_{\rho\tau}F^a_{\mu\nu}-
f_{\mu\rho}F^a_{\nu\tau})F^{\mu\nu,a}\!+c.p.\right]
 \nonumber \\
& &\hspace{-5mm}{}+g'g^2_s\kappa_3\, \theta^{\rho\tau}\hspace{-2mm}\int
\hspace{-1mm} d^4x \sum_{b=1}^{8}
\left[(\frac{1}{4}f_{\rho\tau}G^b_{\mu\nu}-
f_{\mu\rho}G^b_{\nu\tau})G^{\mu\nu,b}\!+c.p.\right], \nonumber 
\end{eqnarray}
where $c.p.$ means cyclic permutations in $f$.
Here $f_{\mu\nu}$, $F^a_{\mu\nu}$, and $G^b_{\mu\nu}$ are the physical field strengths corresponding 
to the groups $U(1)_Y$, $SU(2)_L$, and $SU(3)_C$, respectively. 
The constants $\kappa_1$, $\kappa_2$, and $\kappa_3$ are functions of $1/g_i^2,\; (i=1,...6)$ 
and have the following form:
\begin{eqnarray}
\kappa_1 &=& -\frac{1}{g^2_1}-\frac{1}{4g^2_2}+\frac{8}{9g^2_3}-\frac{1}{9g^2_4}+\frac{1}{36g^2_5}
+\frac{1}{4g^2_6},
\nonumber \\
\kappa_2 &=& -\frac{1}{4g^2_2}+\frac{1}{4g^2_5}+\frac{1}{4g^2_6},
\nonumber \\
\kappa_3 &=& +\frac{1}{3g^2_3}-\frac{1}{6g^2_4}+\frac{1}{6g^2_5}.
\end{eqnarray}
In order to match the SM action at zeroth order in $\theta$, three consistency conditions
have been imposed in (\ref{action2}):
\begin{eqnarray}
\frac{1}{{g'}^2} &=& \frac{2}{g^2_1}+\frac{1}{g^2_2}+\frac{8}{3g^2_3}+\frac{2}{3g^2_4}+\frac{1}{3g^2_5}
+\frac{1}{g^2_6},
\nonumber \\
\frac{1}{g^2}&=& \frac{1}{g^2_2}+\frac{3}{g^2_5}+\frac{1}{g^2_6},\nonumber \\
\frac{1}{g_s^2}&=& \frac{1}{g^2_3}+\frac{1}{g^2_4}+\frac{2}{g^2_5}.
\end{eqnarray}
These three conditions together with the requirement that 
$1/g_i^2 > 0$, define a three-dimensional simplex in
the six-dimensional moduli space spanned by $1/g_1^2,...,1/g_6^2$. 
Since the last three couplings in (\ref{action2}) are 
not uniquely fixed by the NCSM, they need to be determined through the 
various types of physical processes, like decays and collisions, unpolarized and polarized, ect.

From the action (\ref{action2}) we extract the 
neutral triple-gauge boson terms which are not present in the SM Lagrangian. 
In terms of physical fields ($A,Z,G$) they are
\begin{eqnarray}
{\cal L}_{\gamma\gamma\gamma}&=&\frac{e}{4} \sin2{\theta_W}\;{\rm K}_{\gamma\gamma\gamma}
{\theta^{\rho\tau}}A^{\mu\nu}\left(A_{\mu\nu}A_{\rho\tau}-4A_{\mu\rho}A_{\nu\tau}\right),\nonumber\\
{\rm K}_{\gamma\gamma\gamma}&=&\frac{1}{2}\; gg'(\kappa_1 + 3 \kappa_2);  \label{L1}\\
& & \nonumber \\
{\cal L}_{Z\gamma\gamma}&=&\frac{e}{4} \sin2{\theta_W}\,{\rm K}_{Z\gamma \gamma}\,
{\theta^{\rho\tau}}
\left[2Z^{\mu\nu}\left(2A_{\mu\rho}A_{\nu\tau}-A_{\mu\nu}A_{\rho\tau}\right)\right.\nonumber\\
& & +\left. 8 Z_{\mu\rho}A^{\mu\nu}A_{\nu\tau} - Z_{\rho\tau}A_{\mu\nu}A^{\mu\nu}\right], \nonumber \\
{\rm K}_{Z\gamma\gamma}&=&\frac{1}{2}\; \left[{g'}^2\kappa_1 + \left({g'}^2-2g^2\right)\kappa_2\right]; \label{L2}\\
& &\nonumber \\
{\cal L}_{ZZ\gamma}&=&{\cal L}_{Z\gamma\gamma}(A\leftrightarrow Z),\nonumber \\
{\rm K}_{ZZ\gamma}&=&\frac{-1}{2gg'}\; \left[{g'}^4\kappa_1 + g^2\left(g^2-2{g'}^2\right)\kappa_2\right]; \label{L3}\\
& &\nonumber \\
{\cal L}_{ZZZ}&=&{\cal L}_{\gamma\gamma\gamma}(A\to Z),\nonumber\\
{\rm K}_{ZZZ}&=&\frac{-1}{2g^2}\; \left[{g'}^4\kappa_1 + 3g^4\kappa_2\right]; \label{L4}\\
& &\nonumber \\
{\cal L}_{Zgg}&=&{\cal L}_{Z\gamma\gamma}(A\to G^b), \nonumber \\
{\rm K}_{Zgg}&=&\frac{g^2_s}{2} \left[1+(\frac{{g'}}{g})^2\right]\kappa_3; \label{L5}\\
& &\nonumber \\
{\cal L}_{\gamma gg}&=&{\cal L}_{Zgg}(Z\rightarrow A), \nonumber \\
{\rm K}_{\gamma gg}&=&\frac{-g^2_s}{2}\;
\left[\frac{g}{g'}+\frac{g'}{g}\right]\kappa_3, \label{L6}
\end{eqnarray} 
where $A_{\mu\nu} \equiv \partial_{\mu}A_{\nu} -
\partial_{\nu}A_{\mu}$, ect.

Fig.(\ref{fig1}) shows the three-dimensional simplex that bounds 
allowed values for the dimensionless coupling constants
${\rm K}_{\gamma\gamma\gamma}$, ${\rm K}_{Z\gamma\gamma}$ 
and ${\rm K}_{Zgg}$. For any choosen point within simplex 
in Fig.(\ref{fig1}) the remaining three coupling constants (\ref{L3},\ref{L4},\ref{L6}), i.e.
${\rm K}_{Z Z \gamma}$, ${\rm K}_{Z Z Z}$ 
and ${\rm K}_{\gamma g g}$ respectively, are uniquely fixed by the NCSM.
This is true for any combination of three coupling constants from equations (\ref{L1}) to (\ref{L6}).
\begin{figure}
 \resizebox{0.45\textwidth}{!}{%
  \includegraphics{simplex.eps}}
 \caption{The three-dimensional simplex that bounds possible values
 for the coupling constants ${\rm K}_{\gamma\gamma\gamma}$, 
 ${\rm K}_{Z\gamma\gamma}$ 
 and ${\rm K}_{Zgg}$ at the  $M_Z$ scale. The vertices of the simplex are:\\
 ($-0.184$,\,$-0.333$,\,$0.054$), ($-0.027$,\,$-0.340$,\,$-0.108$),\\
 ($0.129$,\,$-0.254$,\,$0.217$),
 ($-0.576$,\,$0.010$,\,$-0.108$),\\ ($-0.497$,\,$-0.133$,\,$0.054$), and 
 ($-0.419$,\,$0.095$,\,$0.217$).}
 \label{fig1}
\end{figure}

Experimental evidence for noncommutativity coming from the gauge sector, should be searched for 
in processes which involve the  above vertices. 
The simplest and most natural choice are the 
$Z\rightarrow \gamma\gamma, \;gg$ decays, allowed for real (on-shell) particles.
All other simple processes, such as
$\gamma \rightarrow \gamma \gamma, \;gg$, and $Z\rightarrow Z\gamma, \;ZZ$,
are on-shell forbidden by kinematics. 
The $Z\rightarrow \gamma\gamma, \: gg$ decays are strictly
forbidden in the SM by Lorentz invariance and spin statistics, therefore
they both could serve as a clear signal for the existence of 
spacetime noncommutativity.
  
The $Z\rightarrow \gamma\gamma$ process has a tiny SM 
background from the rare $Z\rightarrow \pi^0\gamma,\;\eta\gamma$ decays. 
At high energies, the two photons from the $\pi^0$ or $\eta$ decay
are too close to be separated and they are seen in the electromagnetic calorimeter as 
a single high-energy photon \cite{EXP}. The SM
branching ratios for these rare decays are of order $10^{-11}$
to $10^{-10}$ \cite{ALT}. This is much smaller than the experimental upper bounds
which are of order $10^{-5}$ for the all three branching ratios 
($Z\rightarrow \gamma\gamma,\; \pi^0\gamma,\; \eta\gamma$) \cite{rpp}. 
The experimental upper bound, obtained from the $e^+e^-\rightarrow \gamma\gamma$ annihilation, 
for $\Gamma_{Z \rightarrow \gamma\gamma}$ is $< 1.3\times 10^{-4} GeV$ \cite{rpp}.

The $Z\rightarrow gg$ decay mode should be observed in $Z\rightarrow 2\;{\rm jets}$ processes.
However, it could be smothered by the strong 
$Z\rightarrow q{\bar q}$ background, i.e. by hadronization, which also
contains NC contributions. Since
the hadronic width of the $Z$ is in good agreement with the QCD corrected SM, 
the $Z\rightarrow gg$ can at most be a few percent.
Taking into account the discrepancy between the experimentally 
observed hadronic width for the $Z$-boson 
and the theoretical estimate based on the radiatively corrected SM, 
we estimate the upper bound for any new hadronic
mode, like $\Gamma_{Z \rightarrow gg}$, to be $\sim 10^{-3}\; GeV$ \cite{rpp}.

The decay of the $Z$-boson into two photons 
was previously considered in \cite{MPR}, but 
the model used differs from that which we present here. 
It suffers from the charge quantization problem,
the ${\cal L}_{Z\gamma \gamma}$ is not gauge-invariant and, it is not 
clear how it deals with the problem of choosing gauge groups other than $U(N)$.

We now derive the partial widths for the $Z(p) \rightarrow \gamma (k)\,\gamma (k')$ decay.
Care has to be taken when one tries to compute matrix elements in NCGFT. In our model, the 
\emph{in} and \emph{out} states can be taken to be ordinary \emph{commutative} particles.
Quantization is straightforward to the order in $\theta$ that we have considered;
Feynman rules can be obtained either via the Hamiltonian formulation or directly from the
Lagrangian; a rather convenient property of the action, relevant to computations, is
its symmetry under ordinary gauge transformations in addition to noncommutative ones  \cite{foot1}.
From the Lagrangian ${\cal L}_{Z\gamma \gamma}$, it is easy to write 
the gauge-invariant amplitude ${\cal M}_{Z\rightarrow \gamma\gamma}$ in momentum space, which gives: 
\begin{eqnarray}
\sum_\mathrm{spins}\,|{\cal M}_{Z\rightarrow \gamma \gamma}|^2 
= -{\theta}^2 + \frac{8}{M^2_Z}(p{\theta}^2 p)
- \frac{16}{M^4_Z}(k{\theta}k')^2 \, .
\label{eqn0}
\end{eqnarray}
From above equation and in the $Z$-boson rest frame, the partial width
of the $Z \rightarrow \gamma\gamma$ decay is
\begin{equation}
\Gamma_{Z\rightarrow \gamma\gamma} 
= \frac{\alpha}{12} M^5_Z \sin^2 2\theta_W {\rm K}^2_{Z\gamma \gamma} 
\left[\frac{7}{3}({\vec {\theta}}_T)^2+({\vec {\theta}}_S)^2\right],
\label{eqn1}
\end{equation}
where ${\vec {\theta}}_T=\{{\theta^{01}},{\theta^{02}},{\theta^{03}}\}$ 
and ${\vec {\theta}}_S=\{{\theta^{23}},{\theta^{13}},{\theta^{12}}\}$, are
responsible for time-space and space-space noncommutativity, respectively. 
This result differs essentially from that given in \cite{MPR}
where the $\Gamma_{Z\rightarrow \gamma\gamma}$
partial width depends only on time-space noncommutativity.

For the $Z$-boson at rest and polarized in the 
direction of the $3$-axis, we find that the \emph{polarized} partial width is
\begin{eqnarray}
& &\Gamma_{Z^3 \rightarrow \gamma\gamma }\;=\;
\frac{\alpha}{4} \;M^5_Z \;\sin^2 2\theta_W\;{\rm K}^2_{Z\gamma \gamma} 
\nonumber \\
& &\times \left[\frac{2}{5}
\left(({\theta}^{01})^2+({\theta}^{02})^2\right)
+\frac{23}{15}({\theta}^{03})^2+({\theta}^{12})^2\right]. \label{eqn2}
\end{eqnarray}
In the absence of time-space noncommutativity 
%(which by itself may have conceptual problems owing to causality breaking, etc.),
a sophisticated, sensibly arranged  polarization
experiment could in principal determine the vector of ${\vec {\theta}}_S$. 
A NC structure of spacetime may depend on the matter that is present. 
In our case it is conceivable that the direction of ${\vec {\theta}}_{T,S}$
may be influenced by the polarization of the $Z$ particle.
In this case, our result for the \emph{polarized} partial width is particularly relevant.

Due to the same Lorentz structure of
the Lagrangians ${\cal L}_{Z\gamma\gamma}$ and ${\cal L}_{Zgg}$ we find
\begin{eqnarray}
\frac{\Gamma_{Z\rightarrow gg}}{\Gamma_{Z\rightarrow \gamma\gamma}}\;=\;
\frac{\Gamma_{Z^3\rightarrow gg}}{\Gamma_{Z^3\rightarrow \gamma\gamma}}\;=\;
8\frac{{\rm K}^2_{Zgg}}{{\rm K}^2_{Z\gamma \gamma}}. \label{eqn3}
\end{eqnarray}
The factor of eight in the above ratios is due to color.

In order to estimate the NC parameter from upper bounds 
$\Gamma^{exp}_{Z \rightarrow \gamma\gamma} < 1.3 \times 10^{-4} GeV$ and
$\Gamma^{exp}_{Z \rightarrow gg} < 1 \times 10^{-3}\; GeV$ \cite{rpp}
it is necessary to determine the range of couplings ${\rm K}_{Z\gamma\gamma}$ and ${\rm K}_{Zgg}$.
\begin{figure}
 \resizebox{0.45\textwidth}{!}{%
  \includegraphics{fig1.ps}}
 \caption{The allowed region for ${\rm K}_{Z\gamma\gamma}$ and ${\rm K}_{Zgg}$ 
at the $M_Z$ scale, projected from the simplex given in Fig 1. The vertices of the polygon are:
$(-0.254,\, 0.217)$, 
$(-0.333,\, 0.054)$, 
$(-0.340,\, -0.108)$, 
$(0.010,\, -0.108)$ and 
$(0.095, \,0.217)$.}
 \label{fig2a}
\end{figure}
The allowed region for coupling constants ${\rm K}_{Z\gamma\gamma}$ and ${\rm K}_{Zgg}$ 
is given in Fig.(\ref{fig2a}).
Since ${\rm K}_{Z\gamma\gamma}$ and ${\rm K}_{Zgg}$ could be zero
simultaneously it is not possible to extract an upper bound on $\theta$ 
only from the above experimental upper bounds alone.

To succeed in estimating $\theta$, we 
should consider an extra interaction from the NCSM gauge sector, in
particular triple photon vertices.
\begin{figure}
 \resizebox{0.45\textwidth}{!}{%
  \includegraphics{fig2.ps}}
 \caption{The allowed region for ${\rm K}_{Z\gamma\gamma}$ 
 and ${\rm K}_{\gamma\gamma\gamma}$  at the $M_Z$ scale, projected from
 the simplex given in Fig 1. The vertices of the polygon are:
 $(-0.333,\, -0.184)$, $(-0.340,\, -0.027)$, $(-0.254,\, 0.129)$, $(0.095,\, -0.419)$, 
 $(0.0095, \,-0.576)$, and $(-0.133,\, -0.497)$.}
 \label{fig2b}
\end{figure}
From the simplex in Fig.(\ref{fig1}) we find that the triplet of
coupling constants 
${\rm K}_{\gamma\gamma\gamma}$, ${\rm K}_{Z\gamma\gamma}$ and ${\rm K}_{Zgg}$,
as well as the pair of couplings ${\rm K}_{\gamma\gamma\gamma}$ and ${\rm K}_{Z\gamma\gamma}$
{\it cannot vanish simultaneously} (see e.g. Fig.(\ref{fig2b})) 
and that is possible to estimate $\theta$ 
from the NCSM gauge sector through a combination of various types of
processes containing the $\gamma\gamma\gamma$ and $Z\gamma\gamma$ vertices.
These are processes of the type $2 \rightarrow 2$, such as 
$e^+e^-\rightarrow \gamma\gamma$, 
$e\gamma \rightarrow e\gamma$, and $\gamma\gamma \rightarrow e^+e^-$ 
in leading order.
The analysis has to be carried out in the same way as in Ref.\cite{HPR}.
Theoreticaly consistent modifications of relevent vertices are, however, necessary. 
\begin{figure}
 \resizebox{0.45\textwidth}{!}{%
  \includegraphics{fig3.ps}}
 \caption{The allowed region for ${\rm K}_{\gamma\gamma\gamma}$
 and ${\rm K}_{Zgg}$ at the $M_Z$ scale, projected from the simplex
 given in Fig 1. The vertices of the polygon are: $(-0.108,\, -0.576)$, $(-0.108,\, -0.027)$, 
 $(0.217,\, 0.129)$, $(0.217,\, -0.419)$, and $(0.054,\, -0.497)$.}
 \label{fig2c}
\end{figure}
Finally, we present the allowed region for pair of couplings ${\rm K}_{\gamma\gamma\gamma}$ and
${\rm K}_{Zgg}$ in Fig.(\ref{fig2c}). 
Note, that
Figs.(\ref{fig2a}) to (\ref{fig2c}) represent projections of pairs of coupling constants
from the three dimensional simplex spanned by the constants 
${\rm K}_{\gamma\gamma\gamma}$, ${\rm K}_{Z\gamma\gamma}$ and ${\rm K}_{Zgg}$.

The structure of our main results (\ref{eqn0}) to (\ref{eqn3}) remains the same
for  $SU(5)$ and $SU(3)_C \times SU(3)_L \times SU(3)_R$ GUT's that
embed the NCSM that is based on the SW map \cite{Desh,ASCH}; only the coupling
constants change. In the particular case of $SO(10)$ GUT
there is no triple gauge boson coupling \cite{ASCH}.
This is due to the same Lorentz structure of 
the gauge boson couplings  ${Z\gamma\gamma}$ and ${Zgg}$ in our NCSM and 
in the above GUT's, understood underlying theories for the NCSM.
In the GUT framework the triple gauge couplings could be uniquely fixed.
However, the GUT couplings have to be evolved down
to the TeV scale. This requires additional theoretical work, 
and it is a subject for another study.

In this letter we propose two SM strictly forbidden decay modes, namely,
$Z \rightarrow \gamma\gamma, gg$, as a possible signature of the NCSM. 
An experimental discovery of $Z \rightarrow \gamma\gamma, gg$
decays would certainly indicate a violation of the presently accepted SM and definitive apperance
of new physics. To determine whether such SM breaking is ultimatly comming from
spacetime noncommutativity or some other source would require a tremendous amount
of additional theoretical and experimental work, and is beyond the scope of the present letter.

In conclusion, the gauge sector of the nonminimal NCSM is an excellent place to discover
spacetime noncommutativity experimentally, but not the best place to find bounds that exclude it.
We hope that the importance of a possible
discovery of noncommutativity of spacetime will convince
experimentalists to look for SM forbidden decays in the gauge sector.
A good reason for this is that the sensitivity to 
the noncommutative parameter $\theta^{\mu\nu}$ could be in a range of the next
generation of linear colliders with a c.m.e. around a few TeV's. 
%\subsection{}
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\begin{acknowledgments}
We would like to thank P.\ Aschieri, B.\ Jur\v co and H.~\v Stefan\v ci\' c for helpful discussions.
One of us (NGD) 
would like to thank the University of Hawaii Theory Group for hospitality.
This work was supported by the Ministry of Science and 
Technology of the Republic of Croatia under Contract No. 0098002, and by the 
US Department of Energy, Grant No.\ 
DE-FG06-85ER 40224.
\end{acknowledgments}

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