  We propose the Z -> gamma gamma decay, a process strictly forbidden in the
standard model, as a signal suitable for the search of noncommutativity of
coordinates at very short distances. We compute the Z -> gamma gamma partial
widthin the framework of the recently proposed renormalizable gauge sector of
the noncommutative standard model. The one-loop renormalizability is obtained
for the model containing the usual six representations of matter fields of the
first generation. Even more, the noncommutative part is finite or free of
divergences, showing that perhaps new interaction symmetry exists in the
noncommutative gauge sector of the model. Discovery of such symmetry would be
of tremendous importance in further search for the violation of the Lorentz
invariance at very high energies. Experimental possibilities of Z -> gamma
gamma decay are analyzed and a firm bound to the scale of the noncommutativity
parameter is set around 1 TeV.
