Revisiting massless scalar two point functions on dS4

Scalar field theories in de Sitter space, as a special case of Quantum Field Theory in Curved spacetime, are currently popular research areas in physics. Scalar field theory is worth studying for several reasons among all the tensor field theories in de Sitter. For example, a well-established scalar field theory in de Sitter might pave the way for the construction of higher rank tensor fields in de Sitter, or massless scalar fields might explain the homogeneity of the Cosmic Wave Background with the inflation era of the universe. Hence, we mainly focused on massless scalar two-point functions because they exhibit different properties than massive ones; therefore, they require alternative approaches. From the representation theory perspective, in 4D de Sitter, among spin-0 particles m=0 belongs to Discrete Series, which requires special attention different from Principle and Complementary Series. While earlier works emphasize the lack of a unique vacuum and breaking of de Sitter symmetry as the main problems of massless two-point functions, these may be connected to an inadequate analysis of the boundary conditions. The recent ideas suggest more promising treatments for discrete series two-point functions \cite{takyon,shift-invariant}. The boundary conditions cannot completely constrain the form of the solution because of the lack of a preferred vacuum, while it is well defined for the massive case as the Bunch-Davies vacuum. Here we follow a Fourier Transformation procedure of two-point functions to address the boundary conditions for the massless Wightman Function and its singularities. Finally, we are left with one free parameter that denotes an overall constant contribution which can be set to zero. The form of singularities, on the other hand, coincides with the Minkowskian type singularities in the incident point limit. Additionally, we summarized two-contemporary studies and discussed their results.