  We study "on-shell constructibility" of tree amplitudes from recursion
relations in general 4-dimensional local field theories with any type of
particles, both massless and massive. Our analysis applies to renormalizable as
well as non-renormalizable interactions, with or without supersymmetry. We
focus on recursion relations that arise from complex deformations of all
external momenta. Under certain conditions, these "all-line shift recursion
relations" imply the MHV vertex expansion. We derive a simple sufficient
criterion for the validity of the all-line shift recursion relations. It
depends only on the mass dimensions of the coupling constants and on the sum of
helicities of the external particles. Our proof is strikingly simple since it
just relies on dimensional analysis and little-group transformation properties.
In particular, the results demonstrate that all tree amplitudes with n>4
external states are constructible in any power-counting renormalizable theory.
Aspects of all-line shift constructibility are illustrated in numerous
examples, ranging from pure scalar theory and the massless Wess-Zumino model to
theories with higher-derivative interactions, gluon-Higgs fusion, and Z-boson
scattering. We propose a sharp physical interpretation of our constructibility
criterion: the all-line shift fails precisely for those classes of n-point
amplitudes that can receive local contributions from independent
gauge-invariant n-field operators.
