  It is well known that under the color-decomposition, one-loop amplitude of
gluons contains partial amplitudes of single and double trace structures, and
particularly all partial amplitudes of double trace structure can be expressed
as a linear combination of partial amplitudes of single trace structure. Using
unitarity cut method, we prove that this result is the natural consequence of
tree-level Kleiss-Kuijf relation. Generalizing the unitarity cut method to
two-loop (triple cut in this case), we show that, unlike the one-loop case,
partial amplitudes of double and triple trace structures can not be expressed
as a linear combination of partial amplitudes of leading-color single trace
structure. For partial amplitudes of subleading-color single trace structure,
we have shown a very nontrivial Kleiss-Kuijf relation for six and seven-point
amplitudes, which is one new result of our paper and can not be obtained by
U(1)-decoupling method. Mysteriously, when we consider the case of eight
points, Kleiss-Kuijf relation must be modified for subleading-color single
trace partial amplitudes.
