The Q $$ mathcal{Q} $$ -cut representation of one-loop integrands and unitarity cut method

Recently, a new construction for complete loop integrands of massless field theories has been proposed, with on-shell tree-level amplitudes delicately incorporated into its algorithm. this new approach reinterprets integrands in a novel form, namely the $$ mathcal{q} $$ -cut representation. in this paper, by deriving one-loop integrands as examples, we elaborate in details the technique of this new representation, e.g., the summation over all possible $$ mathcal{q} $$ -cuts as well as helicity states for the non-scalar internal particle in the loop. moreover, we show that the integrand in the $$ mathcal{q} $$ -cut representation naturally reduces to the integrand in the traditional unitarity cut method for each given cut channel, providing a cross-check for the new approach.