In this paper, the one-loop CHY-integrands of bi-adjoint scalar theory has been reinvestigated. Differing from previous constructions, we have explicitly removed contributions from tadpole and massless bubbles when taking the forward limit of corresponding tree-level amplitudes. The way to remove those singular contributions is to exploit the idea of "picking poles", which is to multiply a special cross ratio factor with the role of isolating terms having a particular pole structure.

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An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [1, 2]. It has also been shown that the new method is a self-completed method in [3]. Analytic reduction coefficients can be easily produced by recursion relations in this method, where the Gram determinant appears in denominators. The singularity caused by Gram determinant is a well-known fact and it is important to address these divergences in a given frame. In this paper, we propose a systematical algorithm to deal with this problem in our method. The key idea is that now the master integral of the highest topology will be decomposed into combinations of master integrals of lower topologies. By demanding the cancellation of divergence for obtained general reduction coefficients, we solve decomposition coefficients as a Taylor series of the Gram determinant. Moreover, the same idea can be applied to other kinds of divergences.