  We construct a crossing symmetric basis for conformal four-point functions in
momentum space by requiring consistent factorization. Just as scattering
amplitudes factorize when the intermediate particle is on-shell, non-analytic
parts of conformal correlators enjoy a similar factorization in momentum space.
Based on this property, Polyakov, in his pioneering 1974 work, introduced a
basis for conformal correlators which manifestly satisfies the crossing
symmetry. He then initiated the bootstrap program by requiring its consistency
with the operator product expansion. This approach is complementary to the
ordinary bootstrap program, which is based on the conformal block and requires
the crossing symmetry as a consistency condition of the theory. Even though
Polyakov's original bootstrap approach has been revisited recently, the
crossing symmetric basis has not been constructed explicitly in momentum space.
In this paper we complete the construction of the crossing symmetric basis for
scalar four-point functions with an intermediate operator with a general spin,
by using new analytic expressions for three-point functions involving one
tensor. Our new basis manifests the analytic properties of conformal
correlators. Also the connected and disconnected correlators are manifestly
separated, so that it will be useful for the study of large $N$ CFTs in
particular.
