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We use on-shell methods to calculate tree-level effective field theory (EFT)amplitudes, with no reference to the EFT operators. Lorentz symmetry, unitarityand Bose statistics determine the allowed kinematical structures. As aby-product, the number of independent EFT operators simply follows from the setof polynomials in the Mandelstam invariants, subject to kinematicalconstraints. We demonstrate this approach by calculating several amplitudeswith a massive, SM-singlet, scalar ($h$) or vector ($Z^\prime$) particlecoupled to gluons. Specifically, we calculate $hggg$, $hhgg$ and $Z^\prime ggg$amplitudes, which are relevant for the LHC production and three-gluon decays ofthe massive particle. We then use the results to derive the massless-$Z^\prime$amplitudes, and show how the massive amplitudes decompose into themassless-vector plus scalar amplitudes. Amplitudes with the gluons replaced byphotons are straightforwardly obtained from the above.

Effective Field Theory Amplitudes the On-Shell Way: Scalar and Vector Couplings to Gluons