The Role of M\"obius Constants and Scattering Functions in CHY Scalar Amplitudes

The integrations leading to the Cachazo-He-Yuan (CHY) double-color $n$-point massless scalar amplitude are carried out one integral at a time. M"obius invariance dictates the final amplitude to be independent of the three M"obius constants ${\sigma }_r,{\sigma }_s,{\sigma }_t$, but their choice affects integrations and the intermediate results. The effect of the M"obius constants, the two colors, and the scattering functions on each integration is investigated. A systematic way to carry out the $n-3$ integrations is explained, each exposing one of the $n-3$ propagators of the Feynman diagrams. Two detailed examples are shown to illustrate the procedure, one a five-point amplitude, and the other a nine-point amplitude.