Constraining momentum space CFT correlators with consistent position space OPE limit and the collider bound

Consistency with position space OPE limit requires momentum space CFT correlators to have only total energy singularity. We show that this requirement gives a simple proof of the known result that the parity-odd structure cannot exist for three-point correlators of exactly conserved currents with spins $s_i,s_j,s_k$, when triangle inequality $s_i\le s_j+s_k$ is violated. We also show that even for parity even correlation functions the properties are different inside and outside the triangle inequality. It was previously shown that if we allow for weakly broken higher spin symmetry, parity-odd correlators can exist even when triangle inequality is violated. In this paper we establish a relation between non-conservation Ward-Takahashi (WT) identities for weakly broken currents using known WT identities for exactly conserved currents. This allows us to calculate the parity violating results outside the triangle inequality using parity-even free bosonic and free fermionic results. In general, there is one parity-odd structure and two parity-even structures for three-point functions. It can be shown that the coefficient of one of the parity-even and odd parts can be combined into a complex parameter $c$ when correlators are expressed in spinor-helicity variables. When this complex parameter takes real value $c=\pm 1$ it corresponds to either the free boson or free fermion theory. When $c$ is a pure phase, it corresponds to Chern-Simons matter theories. Furthermore, re-expressing known results for conformal collider bounds we see that $\lvert c\rvert\le 1$ for generic 3d CFTs and $\lvert c\rvert\le f(\Delta_{gap})$ for holographic CFTs.