Primary Observables for Top Quark Collider Signals

At the HL-LHC and future high energy colliders, a sample of a billion top quarks will be produced, allowing precision searches for new physics in top quark decay and production. To aid in this endeavor, we characterize the independent three and four point on-shell amplitudes involving top quarks, under the assumption of $SU(3)_c\times U(1)_{em}$ invariance. The four point amplitudes factorize into primary and descendent amplitudes, where descendants are primaries multiplied by Mandelstam variables. By enumerating the allowed amplitudes, we can check for amplitude redundancies to find the number of independent terms and convert those into a Lagrangian which parameterizes these amplitudes. These results are then cross checked by utilizing the Hilbert series to count the number of independent Lagrangian operators. Interestingly, we find situations where the Hilbert series has cancellations which, if na\"{i}vely interpreted, would lead to the incorrect conclusion that that there are no primary operators at a given mass dimension. We characterize the four fermion ($ffff$) and two fermion, two gauge boson ($ffVV$) operators respectively up to dimension 12 and 13. Finally, by combining unitarity bounds on the coupling strengths and simple estimates of the branching ratio sensitivities, we highlight interesting amplitudes for top quark decay that should be studied more closely at the HL-LHC. Of those highlighted, there are both new charge current and flavor changing neutral current decays that occur at dimension 8 and 10 in SMEFT.