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It is well known that under the color-decomposition, one-loop amplitude ofgluons contains partial amplitudes of single and double trace structures, andparticularly all partial amplitudes of double trace structure can be expressedas a linear combination of partial amplitudes of single trace structure. Usingunitarity cut method, we prove that this result is the natural consequence oftree-level Kleiss-Kuijf relation. Generalizing the unitarity cut method totwo-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 expressedas a linear combination of partial amplitudes of leading-color single tracestructure. For partial amplitudes of subleading-color single trace structure,we have shown a very nontrivial Kleiss-Kuijf relation for six and seven-pointamplitudes, which is one new result of our paper and can not be obtained byU(1)-decoupling method. Mysteriously, when we consider the case of eightpoints, Kleiss-Kuijf relation must be modified for subleading-color singletrace partial amplitudes.

Relations of loop partial amplitudes in gauge theory by Unitarity cut method