  We consider the complete set of observables for collider searches for
indirect effects of new heavy physics. They consist of $SU(3)_{\rm C}\times
U(1)_{\rm EM}$ invariant interaction terms/operators that parameterize
deviations from the Standard Model. We show that, under very general
assumptions, the leading deviations from the Standard Model are given by a
finite number of `primary' operators, with the remaining operators given by
`Mandelstam descendants' whose effects are suppressed by powers of Mandelstam
variables divided by the mass scale $M$ of the heavy physics. We explicitly
determine all 3 and 4-point primary operators relevant for Higgs signals at
colliders by using the correspondence between on-shell amplitudes and
independent operators. We give a detailed discussion of the methods used to
obtain this result, including a new analytical method for determining the
independent operators. The results are checked using the Hilbert series that
counts independent operators. We also give a rough sketch of the phenomenology,
including unitarity bounds on the interaction strengths and rough estimates of
their importance for Higgs decays at the HL-LHC. These results motivate further
exploration of Higgs decays to $Z\bar{f}f$, $W\bar{f}f'$, $\gamma \bar{f}f$,
and $Z\gamma\gamma$.
