On general BCJ relation at one-loop level in Yang-Mills theory

The tree-level BCJ relations reveal a duality between color and kinematic structure, this can be used to reduce the number of independent color-ordered amplitudes. At one-loop level in Yang-Mills theory, we investigate a similar BCJ relation in this paper. The four-point one-loop example in N - 4 SYM gives hints about the relations between integrands. The five-point example suggests that the general formula can be proven by the unitary-cut method. We will then prove a 'general' BCJ relation for one-loop integrands by D-dimension unitary cut, which can be regarded as a non-trivial generalization of the (fundamental)BCJ relation given by Boels and Isermann in [17, 18].