One-loop CHY Integrand For Scalar Field And Solutions Of Scattering Equations

This thesis contains two parts of contents.The first part is about the relations between the solutions of CHY scattering equation and the kinematics of particles.We also study the contribution of singular solutions and the regular solutions.In the second part,we construct the one-loop CHY-integrands of bi-adjoint scalar theory by multiply the cross ratio factor.In the first part we review the previous research of the relations of the solutions and kinematics,and improve it to more general case.Moreover,by exploiting this result,we analyse the contribution to the measure part and amplitude of singular and regular solutions.In the second part,the one-loop CHY-integrands of bi-adjoint scalar theory has been rein-vestigated.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.