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

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